government of karnataka j{uvktbs.kar.nic.in/new/website textbooks/class10/10th...i government of...

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Government of Karnataka

MATHEMATICS

Karnataka Text Book Society (R.)100 Feet Ring Road,

Banashankari 3rd Stage, Bengaluru - 85

2015

10

Tenth Standard

J{UV

_amR>r _mÜ`_Marathi Medium

B¶Îmm Xhmdr

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Preface

The Textbook Society, Karnataka has been engaged in producing new textbooksaccording to the new syllabi which in turn are designed on NCF – 2005 since June 2010.Textbooks are prepared in 12 languages; seven of them serve as the media of instruction.From standard 1 to 4 there is the EVS, mathematics and 5th to 10th there are three coresubjects namely mathematics, science and social science.

NCF – 2005 has a number of special features and they are:· connecting knowledge to life activities· learning to shift from rote methods· enriching the curriculum beyond textbooks· learning experiences for the construction of knowledge· making examinations flexible and integrating them with classroom experiences· caring concerns within the democratic policy of the country· make education relevant to the present and future needs.· softening the subject boundaries- integrated knowledge and the joy of learning.· the child is the constructor of knowledge

The new books are produced based on three fundamental approaches namely.Constructive approach, Spiral Approach and Integrated approachThe learner is encouraged to think, engage in activities, master skills and competencies.The materials presented in these books are integrated with values. The new books arenot examination oriented in their nature. On the other hand they help the learner in thetotal development of his/her personality, thus help him/her become a healthy member ofa healthy society and a productive citizen of this great country, India.Mathematics is essential in the study of various subjects and in real life. NCF 2005proposes moving away from complete calculations, construction of a framework of concepts,relate mathematics to real life experiences and cooperative learning.Many students have a maths phobia and in order to help them overcome this phobia,jokes, puzzles, riddles, stories and games have been included in textbooks. Each conceptis introduced through an activity or an interesting story at the primary level. Thecontributions of great Indian mathematicians are mentioned at appropriate places.Textbooks for students X have a special significance. As any other new textbook they helplearners’ master skills and competencies and at the same time there is going to be apublic examination based on them.

The Textbook Society expresses grateful thanks to the chairpersons, writers,scrutinisers, artists, staff of DIETs and CTEs and the members of the Editorial Board and

printers in helping the Text Book Society in producing these textbooks.

Nagendra KumarManaging Director

Karnataka Textbook Society®Bengaluru, Karnataka

Prof G.S. MudambadithayaCoordinator

Curriculum Revision and Textbook PreparationKarnataka Textbook Society®

Bengaluru, Karnataka

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Chairperson’s note

This tenth standard mathematics textbook is prepared for the revised syllabusof Karnataka State based on the recommendations made by NCF 2005. It isdesigned to support the two major suggestions made by NCF 2005.

The higher aim of mathematics education should be to develop the innerresources in children. That is, development of mental abilities such as logi-cal and abstract thinking, reasoning, analysing, problem solving etc.

The mode of transaction should be based on contructivism, which facilitatethe learners to construct their own knowledge.

Conceptual understanding of basic ideas and problem solving are the twomain components of mathematics learning. Hence, this textbook is prepared insuch a way to facilitate mathematics learning by doing mathematics and wellbalanced text and exercises.

The salient features of this text book are

Each unit begins with real life situations or activities that engage studentsin learning tasks.

Each concept in each unit is provided with suitable exploring activities andillustrations.

The text is presented in simple language and logical manner with suitablehands on and minds on activities.

The units where theorems are included, both practical activity for stating thetheorem and logical proof for proving the theorem are discussed.

Each theorem is followed by numerical problems and riders. Equalweightage is given to problems and riders on theorems, its converse andcorollories.

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Questions are raised wherever possible to promote thinking and doanalysis.

Additional information is provided in boxes with the heading “know this”.This is to motivate students and create interest in the subject.

Questions and statements with headings “Try” and “Discuss” are givenfor promoting inquiry and co-operative learning strategies.

Each unit ends with a flow chart where all the ideas learnt in the unit areprovided through concept mapping. Students with the guidance of teacherscan generate better concept maps.

This text book is made to be both learner friendly and teacher friendly. We,the Tenth standard mathematics text book committee members hope that thetext book will pave a suitable path for making mathematics learning joyful andmeaningful.

We would be greatful for constructive suggestions and comments fromexperts, teachers, students and parents for further improvement of this book.

Dr. G. VijayakumariChairperson

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Text Book Committee

Chairperson :Dr. Vijaya Kumari G. - Associate Professor, Vijaya Teachers College (CTE),4th block, Jayanagar, Bengaluru - 11.Members :1) Dr. Y.B. Venkatesh - Lecturer, Govt. P.U. College, Bidadi, Ramanagara District.2) Sri V. Ravikumar - Senior Lecturer, DIET, Ilakal, Bagalkot District.3) Sri Narasappa - Head Master, Sri Saraswati Vidyaniketana, Dommasandra,

Bengaluru.4) Sri T.K. Prasanna Murthy - Asst. teacher, Vijaya High School, Jayanagar,

Bengaluru.5) Smt. M. N. Deepa Rao - Asst. Teacher, Sri. N.K.S.E.H.S, D.V.V. Gujarati Shala,

Majestic Circle, Bengaluru.6) Sri Tharakesh (Artist) Drawing Teacher, Government High School,

Bannitalapura, Gundlupet Taluk, Charamarajanagar Dt.

Scrutinizers :

1) Dr. B.J. Venkatachala, Prof. H.B.C.S.E (T.I.F.R) Mathematics Dept. IISC,Bengaluru.

2) Sri K. Krishna Iyengar, Retired B.E.O., Bengaluru.

Editorial Committee Members:1) Dr. R. Raveendra - N.C.E.R.T Director (Retd.), Banashankari III stage,

Bengaluru.2) Dr. B.S. Upadhyaya - Professor, Extension & Education Division, R.I.E., Mysuru.3) Dr. V.S. Prasad - Professor, Mathematics Department, R.I.E., Mysuru.4) Dr. Sharath Sure - Asst. Professor, Azim Premji University, P.E.S. School of

Engineering Campus, Electronics City, Bengaluru.Chief Advisor :1) Sri Nagendra Kumar, Managing Director, KTBS, Bengaluru.2) Sri Panduranga (Incharge), Deputy Director, KTBS, Bengaluru.

Translator Committee :1. Sri. A. S. Desai (Chairman), (H.M.) B.K. Model High School, Camp, Belagavi.

2. Sri. J. U. Ghadi, (H. M.) New girls High School, Goaves, Shahapur, Belagavi.

3. Sri. A. V. Patil, (A.M.) Marathi Vidyalaya, Vijaypura.

4. Sri. P. J. Patil (A.M.) The Volkart Academy Gokak Falls, Gokak.

Chief Co-ordinator :Prof. G.S. Mudambaditaya, Co-ordinator, Curriculum revision and textbookpreparation, KTBS, Bengaluru.

Programme Co-ordinator :Smt. Shashikala R.N. - Assistant Director, KTBS, Bengaluru.

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1. gQ> 1 - 15

2. H«$‘ 16 - 55

3. dmñVd g§»¶m 56 - 70

4. H«$‘doe Am{U g§doe 71- 94

5. g§^mì¶Vm 95 - 120

6. g§»¶memó 121 - 145

7. H$aUr 146 - 161

8. ~hþnXr 162 - 190

9. dJ©g‘rH$aUo 191 - 224

10 dVw©io 225 - 260

11. g‘ê$n {ÌH$moU 261 - 300

12. nm¶W°Jmoag à‘o¶ 301 - 313

13. {ÌH$moU{‘Vr 314 - 348

14 gh{ZXo}eH$ y{‘Vr 349 - 368

15 ‘mnZo 369 - 407

n¶m©¶r àûZ 408- 417

J{UVr Z‘wZo (nwadUr) 418-421

A.Z§ àH$aUo n¥ð> H«$‘m§H$

AZwH«$‘{UH$m

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1*gQ>m§Mm g§¶moJ Am{U N>oXZ ¶mdarb

H«$‘{Zanoj Am{U gmhM¶© JwUY‘©.* gQ>m§Mm g§¶moJ Am{U N>oXZ ¶mdarb H«$‘{Zanoj Am{U gmhM¶© JwUY‘mªMm nS>Vmim..* gQ>mdarb {dVaU JwUY‘©.* S>r>-‘m°J©ÝgMm {Z¶‘* n(A) + n(B) = n(A B) + n(A B).

Am°JñQ>g {S> ‘m°J©Z (1806-1871)¶m§Mm OÝ‘ ^maVm‘Ü¶o Vm{‘iZmSy> ‘Yrb‘XÿamB© ¶oWo Pmbm hmoVm. Vo 7 ‘{hÝ¶m§MoAgVmZm Ë¶m§Mo Hw$Qw>§~ B§½b§S>‘Ü¶o ñWbm§VarVPmbo. Ë¶m§Zr Amnbo {ejU B§½b§S>‘Yrb{Ì{ZVr {dÚmb¶ Ho$på~«O ‘Ü¶o KoVbo.{S>‘m°J©ÝgMm {Z¶‘ gQ>m§Mm g§¶moJ, N>oXZAm{U H$moQ>r ¶m ‘yb yV {H«$¶m§er g§~§YrVAmho.

¶m KQ>H$m‘wio Vwåhmbm ¶m§Mo gwb^rH$aU hmoVo,* g§¶moJ Am{U N>oXZ gQ>mdarb H«$‘{Zanoj Am{U gmhM¶© JwUY‘mªMo ñnï>rH$aU* gQ>m§Mm g§¶moJ Am{U N>oXZ ¶mdarb H«$‘{Zanoj Am{U gmhM¶© JwUY‘© nS>VmiUo.* gQ>m§À`m {dVaU JwUY‘m©Mo ñnï>rH$aU Am{U nS>VmiUo.* S>r-‘m°J©ÝgMm {Z¶‘ ñnï> H$aUo.* S>r-‘m°J©ÝgMm {Z¶‘ nS>VmiUo.* gQ>mVrb g§~§Y ñnï> H$aUo.. n(A) + n(B) = n(A B) + n(A B)* gQ>m§Mm g§~§Y dmnéZ CXmhaUo gmoS>{dUo.* doZ AmH¥$Vr H$mT>Uo.

God created infinity, and man, unable tounderstand infinity, created finite sets.

-Anonymous

Sets gQ

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Sets 3

A B = B A. Am{U A B = B A

XmoZ gQ>m §Mm g§¶moJAm{U N> ooXZ gQ> hm H«$‘{Zanoj AgVmo.

{Q>n : EH$mM doir XmoZ gQ> KodyZ, H$moUË¶mhr g§»¶oMo g§¶moJ qH$dm N>oXZ gQ> aMbo OmD$ eH$VmV. g§¶moJ

Am{U N>oXZ gQ> ho Zoh‘r H«$‘{Zanoj AgVmV.

(iii) XmoZ g§¶moJ gQ>m§Mm gmhM¶© JwUY‘© :(Associatine property of union of two sets)

A, B Am{U C ¶m VrZ gQ>m§Mm {dMma H$am.

A= {a, b, c, d }, B = {b, c, e, f }, C = {c, e, f, g }

àW‘ AmnU A Am{U B gQ>m§Mm g§¶moJ gQ> H$mTy>.

A B = {a, b, c, d } {b, c, e, f } = {a, b, c, d, e, f }

AmVm, AmnU ¶m ’$b{ZînÎmr gQ>mMm C gQ>mdarb g§¶moJ gQ> H$mTy>.

(A B) C = {a, b, c, d, e, f } {c, e, f, g }

(A B) C = {a, b, c, d, e, f, g } .....(i)

¶oWo, AmnU àW‘ (A B) H$mTy> Am{U Z§Va (A B) gQ>mMm C gQ>mdarb g§¶moJ gQ> H$mTy>.

’$b {ZînÎmr H$moUVr? àW‘ AmnU B Am{U C g§¶moJ gQ> H$mTy> Am{U Z§Va Ë¶m§Mm A darb g§¶moJ gQ>

H$mTy>.

B C H$mTm

B C = {b, c, e, f } {c, e, f, g} = {b, c, e, f, g }

AmVm AmnU A gQ>mMm (B C) darb g§¶moJ gQ> H$mTy>.

A (B C) = {a, b, c, d } {b, c, e, f, g }

A (B C) = {a, b, c, d, e, f, g} ......(ii)

(i) Am{U (ii), ¶m§Mr VwbZm H$éZ, Amåhmbm {Xgob H$s Vo g‘mZ AmhoV.

Amåhr Agm {ZîH$f© H$mTy> H$s,

(A B) C = A (B C)

gQ>m§Mm g§¶moJ hm gmhM¶© AgVmo.

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Sets 5

(i) Am{U (ii) ¶m§Mr VwbZm H$am. Amåhmbm {Xgob H$s XmoÝhr g‘mZ AmhoV åhUyZ AmnU {ZîH$f© H$mTy> H$s

(A B) C = A (B C)

gQ>m§Mm N> oXZ hm gmhM¶© AgVmo.doZ AmH¥$Vr H$mTy>Z N>oXZ gQ>m§Mm gmhM¶© JwUY‘© VnmgyZ nmhÿ.

nm¶ar 1 : A, B Am{U C. ho gQ> AmhoV. nm¶ar 2 : A B nm¶ a r 3 : (A B) C

2

4

6

8

A

1

2

4

3

B

4

6

5

7

C A B68

13

24

C4 56 7

138

65 7

24

A B

C

.......(i)

nm¶ar 4 : B C nm¶ar 5 : A (B C)B C1

23

5

64

12

3

5

64

A2 64 8

138

65 7

24

A B

C.............(ii)

g_r (i) Am{U (ii) _Yrb aoIm§H$sV ^mJmMr VwbZm H$ê$Z AmnU Agm {ZnH$f© H$mT>y H$s (A B) C = A (B C)

(V) gQ>m§Mm {dVaU JwUY‘© :(Distributive property of Sets )

Amåhr Aä¶mgbobo Amho H$s, H«$‘{Zanoj JwUY‘© Am{U gmhM¶© JwUY‘© g§¶moJ Am{U N>oXZ gQ>mdarb

{H«$¶mgmR>r bmJy nS>VmV.

ñ‘aU H$am H$s, darb àË¶oH$ JwUY‘© ñnï> H$aVmZm gQ>m§À¶m {H«$¶on¡H$s ’$º$ EH$M, EH$ Va g§¶moJ qH$dm

N>oXZ {H«$`m dmnabr Amho.

gQ>mdarb XmoÝhr {H«$¶m N>oXZ Am{U g§¶moJ ¶m VrZ gQ>mgmR>r dmnaUo e³¶ Amho H$m? Oa Vgo Agob Va¶m KQ>Zo‘Ü¶o H$moUVm JwUY‘© AmT>iVmo. ¶m{df¶r Aä¶mg H$é.

¶m {df¶r AmnU Aä¶mg H$ê$.

g§¶moJ gQ>mMm N> oXZ gQ>mdarb {dVaU JwUY‘©:

Imbrb VrZ gQ>m§Mm {dMma H$am.

K= {3, 5, 7, 9}, L = {5, 8, 9}, M = {1, 2, 3, 9}

K (L M) H$mT>m.

L Am{U M. Mm N>oXZ gQ> H$mT>m.

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Sets 7

K (L M) = {x, y, z, t} {r, s, t, y, z} = {t, y, z}.....(i)

AmVm AmnU K d L; Am{U K d M ¶m§Mm N>oXZ gQ> H$mTy>, Am{U Z§Va ¶m XmoÝhtMm g§¶moJ gQ> H$mTy>.

K L = {x, y, z, t} {y, z} = {y, z}K M = {x, y, z, t} {r, s, t} = { t }

AmVm, K L Am{U K M ¶m gQ>m§Mm g§¶moJ gQ> H$mTy>.

(K L) (K M) = {y, z} {t}(K L) (K M) = {t, y, z}..................... .(ii)

(i) Am{U (ii) déZ AmnU nmhÿ eH$Vmo H$s, XmoÝhr gQ> gmaIoM AmhoV.

¶mdéZ AmnU {ZîH$f© H$mTy> eH$Vmo H$s, K (L M) = (K L) (K M)

gQ>mdarb {H«$¶mgmR>r {dVaU JwUY‘© bmJy nS>Vmo.

gQ>m§Mm N>oXZ gQ> hm g§¶moJ gQ>mdarb {dVaU JwUY‘© Amho.

AmVm AmnU doZ AmH¥$Vr H$mTy>Z N>oXZ gQ>m§Mm g§¶moJ gQ>mdarb {dVaU JwUY‘© VnmgyZ nmhÿ.

i.e. K (L M) = (K L) (K M)

S>mdr ~mOy = K (L M) COdr ~mOy = (K L) (K M)

K L

K L

M

K

K L

L

(( )) MKM

M

K L

K

L

L

K

MM

M

K

K L M

L

( )

(i) Am{U (ii)‘Yrb aoIm§H$sV ^mJmMr VwbZm H$éZ, AmnU {ZîH$f© H$mTy> eH$Vmo H$s,K (L M) = (K L) (K M)

gQ>m§Mm N>oXZ gQ> hm g§`moJ gQ>mda {dVaU JwUY_© Amho.

S>r-‘m°J©ÝgMm {Z¶‘ (De Morgan's Law)

AmVm n¶ªV AmnU gQ>mdarb {H«$¶m§da H«$‘{Zanoj, gmhM¶© Am{U {dVaU JwUY‘© bmJy nS>Vmo ho Aä¶mgbo Amho.

ñ‘aU H$am H$s, {Xboë¶m gQ>mMm H$moQ>rgQ> gwÜXm H$gm H$mT>bm OmVmo ho AmnU Aä¶mgbo Amho.

{Xboë¶m XmoZ gQ>m§Mm H$moQ>r gQ>mer H$moUVmhr g§~§Y Amho H$m? ¶mMm Aä¶mg H$é.

Imbrb H$moï>H$ nmhm. àË¶oH$ CXmhaUm‘Ü¶o XmoZ gQ> {Xbobo AmhoV. Ë¶m§Mm H$moQ>rgQ> H$mT>m Am{U àË¶oH$

ñV§ m‘Ü¶o Z‘yX Ho$boë¶m {H«$¶m H$am. n{hbo EH$ Vw‘À¶mgmR>r Ho$bobo Amho. ¶m‘m{hVrdéZ Vwåhr H$moUVm {ZîH$f©H$mTy> eH$Vm H$m?

............. .(ii)................ .(i)

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Sets 9

(i) S>r>-‘m°J©ÝgMo {Z¶‘ (i) : (A B) I = AI B I S>r-‘m°J©ÝgMo {Z¶‘ (ii) (A B) I = AI BI

A B A B

BA

A B

A B

A BA B A B

BA

A B

A B

A B

A B

I S>r-‘m°J©ÝgMo {Z¶‘

* (A B)I = AI BI

* (A B)I = AI BI

ñdmÜ¶m¶ 1.1

1. g§¶moJ Am{U N>oXZ gQ>m§Mm H«$‘{Zanoj JwUY‘© Vnmgm.

A = {l, m, n, o, p, q} B = {m, n, o, r, s, t}

2. P = {a, b, c, d, e}, Q = {a, e, i, o, u} Am{U R = {a, c, e, g}, {Xbobo AmhoV Va g§¶moJ Am{U N>oXZgQ>m§Mm gmhM¶© JwUY‘© Vnmgm. 3. Oa A = {–3, –1, 0, 4, 6, 8, 10}, B = {–1, –2, 3, 4, 5, 6}, Am{U C = {–6, –4, –2, 2, 4, 6}

Va A (B C) = (A B) (A C) ho XmIdm.4. Oa U = {4, 8, 12, 16, 20, 24, 28}, A = {8, 16, 24}, B = {4, 16, 20, 28}.

Va i) (A B)I = AI BI Am{U (ii) (A B)I = AI BI

5. Oa A = {1, 2, 3}, B = {2, 3, 4, 5}, C = {2, 4, 5, 6} hoU = {1, 2, 3, 4, 5, 6, 7, 8} Mo CngQ> AmhoV (A B)I = AI BI Am{U (A B)I = AI BI.hoVnmgm.

6. Oa A = {2, 3, 5, 7, 11, 13}, B = {5, 7, 9, 11, 15} ho U = {2, 3, 5, 7, 9, 11, 13, 15}. MoCngQ> AmhoV Va S>r-‘m°J©ÝgMo {Z¶‘ Vnmgm.

7. Imbrb CXmhaUm§À¶m ñnï>rH$aUmgmR>r doZ AmH¥$Vr H$mT>m.(i) (A B) (ii) ( A B)I (iii) AI B (iv) A BI

(v) A \ B (vi) A B \ C) (vii) A B C) (viii) C B A)

(ix) C B \ A) (x) A \ B C) (xi) (A \ B) (A \ C) (xii) (A B) \ A C)

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Sets 11

Vwåhr nmhmb H$s,

n(A) Am{U n(B) Mr ~oarO n(A B) Am{U n(A B) À¶m ~oaOoer g‘mZ Amho.

åhUyZ AmnU Ago {bhÿ eH$Vmo H$s, n(A) + n(B) = n(A B) + n(A B)

Oa ¶m g§~§YmVrb H$moUVrhr VrZ nXo ‘mhrV AgVrb Va nXm§Mr AXbm~Xb H$éZ AmnU am{hbobo

nX H$mTy> eH$Vmo. g§~§Y doJdoJù¶m nÜXVrZo Xe©{dÊ¶mMm Aä¶mg H$am.* n(A B) = n(A) + n(B) n(A B)

* n(A B) = n(A) + n(B) n(A B)

* n(A) = n(A B) + n(A B) n(B)

* n(B) = n(A B) + n(A B) n(A)

ho g§~§Y VH$m©Zo gwÜXm {gÜX H$aVm ¶oVmV.H$moUË¶mhr A Am{U B, ¶m XmoZ gQ>mgmR>r n(A B) = n(A) + n(B) – n(A B)ho {gÜX hmoVmo.

A Am{U B gQ> doZ AmH¥$VrZo Xe©{dÊ¶mMm {dMma H$am.

A B

A\B B\A

A B

ho ñnï> Amho H$s, A B hm A \ B, B \ AAm{U A B ¶m§Mm g§¶moJ Amho.

n(A B) = n(A \ B) + n(A B) + n(B \ A)

COì¶m ~mOy‘Ü¶o n(A B) {‘idyZ dOm Ho$bo AgVm, Vwåhmbm {‘iVo.

n(A B) = n(A \ B) + n(A B) + n(B \ A) + n(A B) – n(A B)

JQ> H$éZ,

n(A B) = [n(A \ B) + n(A B)] + [n(B \ A) + n(A B)] – n(A B)

Amnë¶mbm ‘m{hV Amho H$s, n(A \ B) + n(A B) = n(A) Am{U

n(B \ A) + n(A B) = n(B)

¶m qH$‘Vr dmnéZ, Amåhmbm {‘iob.

n(A B) = n(A) + n(B) – n(A B)

{Q>n: Oa A Am{U B ho Ag§~§YrV gQ> AmhoV Va, A B = Am{U n(A B) = 0

n(A B) = n(A) + n(B)

¶mdéZ AmnU Agm {ZîH$f© H$mTy> H$s,

AmR>dm: A\B hm XmoZ gQ>mVrb’$$aH$ Amho OoWo x A

Am{U x A

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Sets 13

åhUyZ JQ>m‘Ü¶o 100 {dÚmWu AmhoV.

darb CXmhaU doZ AmH¥$Vr dmnéZ gwÜXm gmoS>{dVm ¶oVo. Imbrb doZ AmH¥$VrMm Aä¶mg H$am. Ë¶m‘Ü¶o XmoZ

N>oXUmar dVw©io AmhoV, àË¶oH$s EH$ dVw©io Xe©{dVmV. àË¶oH$ {df¶mVrb {dÚmÏ¶mªMr g§»¶m hr XmI{dë¶mà‘mUo

H$mT>Vm ¶oVo.

hm {ZH$mb Imbrb à‘mUo Vnmgbm OmVmo.

JQ>mVrb EHy$U {dÚmÏ¶mªMr g§»¶m.

= 30 + 45 + 25 = 100

CXmhaU 3: Jmdm§Vrb 520 gXñ¶m‘Ü¶o, 360 gXñ¶ OZmdao nmiÊ¶mV Jw§Vbobo AmhoV

Am{U 280 gXñ¶ Hw$ŠHw$Q>nmbZ Am{U 180 OU ì¶dgm¶ H$aVmV. Va {H$Vr gXñ¶

i)H$moUË¶mhr H$m¶m©V Jw§Vbobo ZmhrV. ii) {H$Vr OU ’$³V Hw$Hw$Q >nmbZm‘Ü¶o Jw§Vbobo

AmhoV?

Sol: OZmdao nmiUmao, n(A) = 360 Hw$ŠHw$Q> nmbZ H$aUmao n(B) = 280

XmoÝhr H$aUmao n(A B) = 180

Amåhmbm _m{hV Amho H$mo, n(A B) = n(A) + n(B) – n(A B)

n(A B) = 360 + 280 – 180

n(A B) = 360 + 100 = 460

H$m¶m©V Jw§Vboë¶m gXñ¶m§Mr g§»¶m Amho. 460.

(i) H$moUË¶mhr H$m¶m©V Z Jw§Vboë¶m gXñ¶m§Mr g§»¶m = 520 – 460 = 60.

60 gXñ¶ H$moUË¶mhr H$m¶m©V Jw§Vbobo ZmhrV.

(ii) \$ŠV Hw$ŠHw$Q>nmbZm_Ü o Jw§Vboë`m bmoH$mMr g§»`m = n(B) \ n (A B)

= 280 – 180 = 100

doZAmH¥$VrZo darb CXmhaU Xe©{dVm oVoo

(iii) H$moUË`mM H$m_mV Z Jw§Vboë`m bmoH$m§Mr g§»`m

= 520 (180 + 180 + 100)

= 520 (460) = 60

K SU

360-180=180 =100

520-(180+180+100)=60

280-180

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Sets 15

Properties ofCardinality of sets

De Morgan's

Commutative Associative Distributive

I Law II Law

Disjoint setsn(A B) =

Union of sets Intersectionof sets A B

Union of sets Intersection of sets

gQ>

gQ>mdarb {H«$`m§MoJwUY_©

S>r- _m°J©Ýg {Z`_ gQ>m§Mr àm_w»`V:Cordindity of Sets

gmhM`© JwUY_©

H«$_{danoj JwUY_©

{dVaU JwUY_©

g§~§{YV gQ>n(A B) =

n (A) +n (B)

Ag§~§{YV gQ>n(A B) =n (A) +n

(B)-n (A B)

gQ>m§Mm g§ moJA B = A B

gQ>m§Moo N>oXZ A B =B

I {Z`_ (A B)1= A1 B1

II {Z`_ (A B)1= A1 B1

gQ>m§Mm g§ moJA B C) =(A B) C

gQ>m§Mm N>oXZA B C) =(A B) C

CÎmao

ñdmÜ¶m¶ 1.2I. 1) 12 2) (i) 39 (ii)11 3) 125 4) (i) 26 (ii) 18 (iii) 68 5)73.33%

II. 1) (i)20 (ii) 25 (iii) 36 2) 15

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* H«$‘* A§H$J{UVr H«$‘ Am{U loUr* JwUmH$ma ì¶ñV H«$‘* JwUmoÎma H«$‘ Am{U loUr* JwUmoÎma loUr Am{U A§H$J{UVr

loUrMr ~oarO* A§H$J{UVr, JwUmH$ma, ì¶ñV Am{U JwUmoÎma ‘Ü¶.

2

{bZmS>m} {ngmZmo ~moJmobmo

[1170-1250, BQ>br]

* Vo {\$~moZmgråhUyZ AmoiIbo OmVmVVo g§»`mÀ`m H«$_mgmR>>r bmoH${n«¶ AmhoV

åhUyZ Ë¶m§Zm {’$~mooZmgr H«$‘ AgoåhUVmV. 0, 1, 1, 2, 3, 5,

8,...... ¶oWo H«$‘mVrb n«Ë¶oH$ nX hoË¶mÀ¶m AJmoXaÀ¶m XmoZ g§»¶m§Mr ~oarOAmho. (n{hbo d 2 ao nX gmoSy>Z) amo‘Z

g§»¶mÀ¶m EodOr {h§Xÿ Aao{~H$g§»¶mÀ¶m dmnamÀ¶m àgmambm Ë¶m§Zr

‘XV Ho$br.

* ¶m KQ>H$m‘wio Vwåhmbm ¶m §Mo gwb^rH$aUhmoVo. * A.P. Mr ì¶m»¶m H$aÊ¶mg.

* A.P. Mo gm‘mÝ¶én {bhrÊ¶mg.* A.P. Mo 'a', 'c.d', Am{U 'Tn' emoYÊ¶mg.* ‘¶m©XrV A§H$J{UVr loUrMr ~oarO H$mT>Ê¶mMo gyÌ

aMÊ¶mg.* ‘¶m©XrV A§H$J{UVr loUrMr ~oarO H$mT>Ê¶mg.* JwUmH$ma ì¶ñV (H.P)H«$‘mMr ì¶m»¶m H$aÊ¶mg.* H.P. Mo gm‘mÝ¶ én H$mT>Ê¶mg* JwUmoÎma H«$‘mMr ì¶m»¶m H$aÊ¶mg.* G.P Mo gm‘mÝ¶ én {bhrÊ¶mg.* G.P. Mo gm‘mÝ¶ JwUmoÎma {bhrÊ¶mg.* ‘¶m©XrV / A‘¶m©XrV JwUmoÎma loUrMr ~oarO

H$mT>Ê¶mMo gyÌ aMÊ¶mg.* ‘¶m©XrV /A‘¶m©XrV JwUmoÎma loUrMr ~oarO H$mT>Ê¶mg* A.P, G.P Am{U H.P. À¶m Cn¶moJmMr CXmhaUo

gmoS>{dÊ¶mg.* A§H$J{UVr, JwUmoÎma Am{U JwUmH$ma ì¶ñV ‘Ü¶

H$mT>Ê¶mg.* AM, GM Am{U HM. ‘Yrb g§~§Y ñWm{nV

H$aÊ¶mg.

"Mathematicians have tried in vain to this day todiscover some order in the sequence of prime numbersand we have reason to believe that it is a mysteryinto which human mind will never penetrate."

- Leonard Euler.

H«$‘(Progressions )

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Progressions 17

¶oWo ~wÜXr~im chess. À¶m emoYmMr Hw$VwhbmMrJmoï> Amho Vr dmMm.

nagr¶mMm amOm ~wpÜX~i Ioim{df¶r AË¶§V IwfPmbo hmoVo. Ë¶m§Zr hm Ioi emoYUmè¶m ì¶º$sbm H$moUVmhrnwañH$ma Úm¶bm V¶ma Pmbo. emoYboë¶m ì¶º$sZo ~wpÜX~iIoimÀ¶m n{hë¶m Mm¡H$moZm‘Ü¶o JhÿMm EH$ XmZm (YmÝ¶)R>odÊ¶mg gm§{JVbo, Xþgè¶m Mm¡agm‘Ü¶o 2 JhÿMo XmUoAm{U {Vgè¶m 4 Mm¡agm‘Ü¶o 4, ¶mà‘mUo R>odÊ¶mggm§{JVbo.

emoYH$mMr ‘mJUr EH$X‘ gmYr ({dZ‘«) hmoVr. na§VwamOmbm Vr nyU© H$aVmZm ‘moR>r g‘ñ¶m Pmbr. H$m?

¶m g‘ñ¶oMo CÎma emoYÊ¶mgmR>r, Vwåhmbm J{UVmÀ¶m H$m§hr ¶w³Ë¶m ‘mhrVr Agë¶m nm{hOoV.Ë¶m§Mm AmnU Aä¶mg H$é¶m

H«$‘ (Sequence)

Vwåhmbm g§»¶oÀ¶m Z‘wÝ¶m§Mm n[aM¶ Amho. Imbrb g§»¶m§À¶m gQ>mVrb nw{T>b g§»¶m emoYÊ¶mMmà¶ËZ H$am.

(i) 1, 5, 9, 13, .......... (ii) 1000, 100, 10, 1, ......... ((iii 1, 4, 9, 16, ............

(iv) 0, 7, 26, 63......... (v) 74, 15, 20, 31, 82, .........

Vwåhmbm H$m¶ {XgyZ Ambo? n{hë¶m Mma KQ>Zo‘Ü¶o Vwåhr nwT>rb g§»¶m H$er emoYbr? VwåhmbmnmMì¶m KQ>Zo‘Ü¶o nwT>r>b A§H$ emoYUo e³¶ Pmbo Zmhr H$m?

Ë¶mMo H$maU åhUOo n{hë¶m Mma KQ>Zo‘Ü¶o, nXo EH$m {d{eï> {Z¶‘mZo H«$‘mV ‘m§S>br AmhoV.{Z¶‘ g‘OyZ Kody¶m.

g§»¶m Z‘wZm {Z¶‘

(i) 1, 5, 9, 13, 17 1 + 4 = 5, 5 + 4 = 9, 9 + 4 = 13, 13 + 4 = 17 4 {‘i{dbo

(ii) 1000, 100, 10, 1, 110 1000 × 1

10 =100, 100 × 110 =10, 1

10 Zo JwUbo10 × 1

10 =1, 1 × 110 = 1

10

(iii) 1, 4, 9, 16, 25 12 = 1, 22 = 4, 32 = 9, 42 = 16, 52 = 25 Z¡g{J©H$ g§»`mMm

– 8 × 2 = – 16, –16 × 2 = – 32 dJ©(iv) 0, 7, 26, 63, 124 13 –1 = 0,23 –1 =7,33 –1=26, 43–1 = 63,53– 1= 124 Z¡g{J©H$ g§»`mMm

KZ dOm 1

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Progressions 18

H$m§hr {Z¶‘mZwgma Ho$boë¶m g§»¶m§À¶m ‘m§S>Urbm H«$‘ Ago åhUVmV.H«$‘mVrb àË¶oH$ g§»¶obm H«$‘mMr nXo Ago åhUVmV.

2, 5, 8, 11, .........¶m H«$‘m‘Ü¶on{hbo nX 2

Xþgao nX 5

{Vgao nX 8

ndo nXH«$‘m‘Yrb àË¶oH$ nX {MÝhmZo Xe©{dVmV.

nXo n{hbo Xþgao {Vgao Mm¡Wo .... .... ndo .... ....

{MÝh T1 T2 T3 T4 .... .... Tn .... ....

H«$‘mMr nXo hr AmhoV, T1, T2, T3, T4, .............Tn..........

¶oWo, 'n' H«$‘mVrb nXm§Mr pñWVr (ñWmZ) Xe©{dVo.

‘¶m©XrV H«$‘ Am{U A‘¶m©XrV H«$‘ [Finite sequence and infinite sequence]

H«$‘mVrb nXo ‘moOVm ¶oV AgVrb Va Ë¶mbm ‘¶m©XrV H«$‘ Ago åhUVmV.

CXm :1) 2, 4, 6, 8, 10, 12, 14, 16, 18, 20

2) A = {x : x hm 5 À¶m nQ>rVrb Amho, 1 x 30}

‘¶m©{XV H«$‘mMo gm‘mÝ¶ én, T1, T2, T3, T4, ............. Tn.

H«$‘mVrb nXo ‘moOVm ¶oV ZgVrb Va Ë¶mbm A‘¶m©{XV H«$‘ åhUVmV.CXm: 1) 1, 5, 9, 13, 17, .......... 2) B = {x : x < 0}

A‘¶m©{XV H«$‘mMo gm‘mÝ¶ én, T1, T2, T3, T4, ..........

H«$‘ hr EH$ J{UVr g§H$ënZm Amho Á¶mMm Cn¶moJ AZoH$ X¡Z§{XZ OrdZmVrb KQ>Zm‘Ü¶o hmoVmo.{ZË¶ OrdZmVrb KQ>Zm§À¶m H«$‘mMr H$mhr CXmhaUo Imbrb à‘mUo {Xbobr AmhoV.1) A‘¥VmZo àË¶oH$ ‘{hÝ¶mbm n¡em§Mr ~MV H$am¶bm gwadmV Ho$br. {VZo n{hë¶m ‘{hÝ¶mV <50

Mr ~MV Ho$br, Xþgè¶m ‘{hÝ¶mV <60, {Vgè¶m ‘{hÝ¶mV <70 B.{VMr ~MV H«$‘m‘Ü¶o Aer ì¶º$ H$aVmV H$s,

50, 60, 70, 80,.....2. VéU Am°{’$gbm OmÊ¶mgmR>r [ajmV ~gbm. Ë¶mZo ‘rQ>aMo [aS>tJ nm{hbo. Vo <20 hmoVo.

Ë¶mZ§Va àË¶oH$ 100 ‘r. À¶m àdmgmZ§Va 1< Zo dmT> Pmbr. Va ‘rQ>aMo [aS>tJ (Zm|X).

20, 21, 22, 23, ........

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Progressions 19

3 OrdmUy§À¶m H¥${Ì‘ dmT> H|$Ðm (laboratory) ‘Ü¶o, OrdmUy§Mr g§»¶m àË¶oH$ Vmgmbm XwwßnQ> PmbobrAmT>iyZ Ambr. Oa gwadmVrbm VoWo 50 {OdmUy hmoVo, Va Ë¶m§Mr EHo$H$ VmgmZo g§»¶m Aer dmT>br H$s,50,100, 200,400.........4. EH$m {H$aUmoËgO©H$ ‘wbÐì¶mMm AY©OrdZ H$mi 12 Vmg Amho. Oa {H$aUmoËgO©H$ ‘wbÐì¶mMmgwadmVrMm Z‘wZm 100 J«°‘ Agob Va àË¶oH$s 12 VmgmZ§Va Z‘wZm Agm KQ>Vm o H$s ,100, 50, 25, 12.5, .........

ñnîQ>rH$aUmË_H$ CXmhaUo

1. n do nX 2n + 3 $AgUmè`m H«$_m§Mr n{hbr VrZ nXo H$mT>m.

CH$b : oWo Tn = 2n + 3

T1, emoYÊ`mgmR>r Tn = 2n + 3 _Ü`o n=1 ¿`m

T1= 2(1) + 3 = 2 + 3 = 5

T2, emoYÊ`mgmR>r> n = 2 _Ü` Tn = 2n + 3 _Ü`o n=1 ¿`m

T2 = 2(2) + 3 = 4 + 3 = 7

VgoM T3 = 2(3) + 3 = 6 + 3 = 9

H«$_mMr n{hbr VrZ nXo VgoM 5, 7, 9. hr AmhoV

2. n do nX2

1n

n AgUmè`m H«$_mMr n{hbr 4 nXo H$mT>m.

CH$b : Tn =2

1n

n

T1 =21 1

1 1 2, T2 =

22 42 1 3

, T3 =23 9

3 1 4=

+, T4 =

24 164 1 5

=+

H«$_mMr n{hbr Mma nXo1 4 9 16, , , .2 3 4 5 hr AmhoV

3. Oa Tn = 3n – 10 Va H«$_mMo 20 do nXo H$mT>m.

CH$b : Tn = 3n – 10

T20 = 3(20) – 10 = 60 – 10 T20 = 50

4. Oa Tn = 5n + 2, Va Tn + 1 H$mT>m.

CH$b: Tn = 5n + 2

Tn + 1 = 5(n + 1) + 2 = 5n + 5 + 2 Tn + 1 = 5n + 7

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Progressions 20

5. Oa Tn = n3 – 1 Va Tn = 26. AgVmZm 'n' Mr qH$_V H$mT>m

CH$b: Tn = n3 – 1

26 = n3 – 1, n3 = 26 + 1 = 27, n = 3 27 n = 3

ñdmÜ`m` 2.1

1. Im{bbn¡H$s H$moUVo H«$_ AmhoV ?

(i) 4, 11, 18, 25, ..... (ii) 43, 32, 21, 10, ......

(iii) 27, 19, 40, 70, ...... (iv) 7, 21, 63, 189, ......

2. Imbrb H«$_mMr nwT>rb XmoZ nXo {bhm.

(i) 13, 15, 17, ____, _____ (ii)2 3 4, , , ___,___3 4 5

(iii) 1, 0.1, 0.01, ___, _____ (iv) 6, 12, 24, ____, ___

3. Oa Tn = 5 – 4n, Va n{hbr VrZ nXo H$mT>m

4. Oa Tn = 2n2 + 5, Va (i) T3 Am{U (ii) T10 H$mT>m

5. Oa Tn = n2 – 1, Va (i) Tn – 1 Am{U (ii) Tn + 1 H$mT>m

6. Oa Tn = n2 + 4 Am{U Tn = 200, Va 'n' Mr {H$_V H$mT>m.

A§H$J{UVr H«$‘ (Arithmetic progression)

¶ooWo H$m§hr H«$‘ AmhoV. Ë¶m§Mo {ZarjU H$am.

H«$‘ T2 – T1 T3 – T2 T4 – T3

5, 8, 11, 14, ..... 3 3 33, 13, 23, 33, .... 10 10 101, -1, –3, –5,... –2 –2 –21, 1.5, 2, 2.5, .... 0.5 0.5 0.5

darb H«$‘ XmoZ g‘mB©H$ ~m~tMo à{VnmXZ H$aVmV.

1. àË¶oH$ H«$‘ ghOnUo ¶oUmè¶m g§»¶onmgyZ gwé hmoVmo. 2. ¶mXrVrb nwT>rb g§»¶m hr Ë¶mÀ¶m ‘mJrb g§»¶oV g‘mZ g§»¶m {‘i{dÊ¶mZo ¶oVo.H«$‘mVrb H«$‘dma nXo g‘mZ (gma»¶mM) g§»¶oZo EH$Va H$‘r qH$dm dmT>V AgVrb Va Ë¶mbmA§H$J{UVr H«$‘ åhUVmV.Vo WmoS>³¶mV A.P. Ago {bhrVmV.A§H$J{UVr H«$‘ hm Agm H«$‘ Amho H$s Á¶m‘Ü¶o àË¶oH$ nw{T>b nX ho ‘mJrb nXm‘Ü¶o(n{hbo nX gmoSy>Z) R>am{dH$ g§»¶m {‘i{dë¶mZo {‘iVo.

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Progressions 21

A.P, ‘Ü¶o àË¶oH$ nX Am{U Ë¶mnwTr>b nX ¶m‘Yrb ’$aH$ pñWa AgVmo. Ë¶mbm gm‘mÝ¶ ’$aH$ AgoåhUVmV.Vo WmoS>³¶mV c.d.. Ago {bhrVmV. Am{U Vmo 'd' Zo Xe©{dVmV. A.P, ‘Ü¶o gmYmaU ’$aH$ hm EH$Va YZ g§»¶m. F$Ug§»¶m qH$dm eyÝ¶ ¶mn¡H$s AgVo.Oa T1, T2, T3, T4,....... hr A.P ‘Yrb nXo AmhoV.d = T2 – T1 = T3 – T2 = T4 – T3 = ................. = pñWa

{Q>n: A§H$J{UVr H«$‘m‘Ü¶o gmYmaU ’$aH$ eyÝ¶ Agob Va Ë¶mbm A§H$J{UVr pñWa H«$‘ åhUVmV.CXm : 11, 11, 11, 11, ......... ½, ½, ½, ½, .........

¶oWo A§H$J{UVr H«$‘mMm g‘mdoe Agboë¶m H$m§hr KQ>Zm AmhoV.(i) B‘maVrÀ¶m darb mJmbm a§J bmdÊ¶mgmR>r amOy {eS>rda 12 ’y$Q> C§Mrda

C^m Amho. Amnbo H$m‘ nyU© Pmë¶mZ§Va Vmo {eS>rdéZ Imbr CVaVmo, Ë¶m àË¶oH$doir

Ë¶mMo O‘rZrnmgyZMo A§Va 1.5 ’y$Q>mZo H$‘r hmoV OmVo. Va Ë¶mMo O{‘ZrnmgyZMoA§Va, 12 ’y$Q>, 10.5 ’y$Q>, 9 ’y$Q>, 7.5 ’y$Q>,.....

¶m KQ>ZoV, Vwåhr nm{hbo Amho H$s XmoZ H«$‘dma nXm‘Yrb ’$aH$ 1.5 Amho.

åhUyZ hm H«$‘ A.P. Amho.(ii) ‘wbo Pmonmù¶mda IoiV AmhoV. Ooìhm Vo PmoH$m KoVmV Voìhm gwadmVrÀ¶m ñWmZmnmgyZ

OmñVrV OmñV 50 g|.‘r. A§Vamda nmohmoMVmV. nmR>monmR> Am§XmobZm‘Ü¶o, àË¶oH$

Am§XmobZmVrb A§Va 2 g|.‘r. Zo H$‘r hmoVo. åhUyZ àË¶oH$ A§mXmobZmÀ¶mdoir AmH«${‘boboA§Va, 50, 48, 46, 44, .......

¶oWo gwÜXm Vwåhr nm{hbo Amho H$s XmoZ H«$‘dma g§»¶m‘Yrb gmYmaU ’$aH$ -2 Amho.åhUyZ hm H«$‘ A.P Amho.Ë¶mMà‘mUo, AmnU nmZ 4 da MMm© Ho$bobr KQ>Zm gwÜXm A.P. Xe©{dVo.A‘¥VmMr ~MV, 50, 60, 70, 80, ..... Amho.¶oWo AmnU nmhVmo H$s gmYmaU ’$aH$ 10 Amho.

darb H«$‘ hm A.P. Amho.VéU Zo nm{hboë¶m ‘rQ>a‘Yrb Zm|Xr 20, 21, 22, 23....... AmhoV.¶oWo gmYmaU ’$aH$ 1 Amho.darb H«$‘ hm A.P Amho.

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Progressions 22

darb CXmhaUmdéZ ho CKS> Amho H$s, A.P ‘Ü¶o H$moUË¶mhr nXm‘Ü¶o {ZpíMV g§»¶m åhUOoM gmYmaU’$aH$ {‘i{dbm AgVm nwT>rb nX {‘iVo.Oa 'a' ho n{hbo nX Am{U 'd' hm gmYmaU ’$aH$ Amho, Va

T1 = a

T2 = T1 + d = a + d

T3 = T2 + d = a + d + d = a + 2d

T4 = T3 + d = a + d + d + d = a + 3d Am{U BË¶mXr.n{hbo nX 'a' gmYmaU ’$aH$ 'd' Agboë¶m A.P Mm gmYmaU Z‘wZm

a, a + d, a + 2d, a + 3d,..........

‘¶m©{XV Am{U A‘¶m©{XV A.P ( Finite & Infinite A.P) A§H$J{UVr H«$‘

A.P ‘Yrb nXm§Mr g§»¶m ‘moOVm ¶oV Agob Va Ë¶mbm ‘¶m©{XV A§H$J{UVr H«$‘ Ago åhUVmV.:CXm :1>) 6, 12, 18, 24, 30. 2>) 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

A.P ‘Yrb nXm§Mr g§»¶m ‘moOVm ¶oV Zgob Va Ë¶mbm A‘¶m©{XV A§H$J{UVr H«$‘ Ago åhUVmV.CXm :1>) 3, 7, 11, 15, ....... 2>) 10, 8, 6, 4, 2, 0, –2, .......

A.PMm gm‘mÝ¶ Z‘wZm (General form of an A.P)

pñ‘VmZo ‘mMrgÀ¶m H$mS>çm§Mr ‘m§S>Ur H$éZ Mm¡agmH¥$Vr AmH$ma ~Z{dbo. n{hbm Mm¡ag 4 ‘m{MgÀ¶mH$mS>çmZr V¶ma hmoVmo. {VZo AmUIr 3 H$mS>çm {‘idyZ 2 am Mm¡ag ~Z{dbm Z§Va {VZo AmUIr VrZH$mS>çm {‘idyZ {Vgam Mm¡ag ~Z{dbm Am{U B.Imbr {Xboë¶m gmaUrMo {Z[ajU H$am.àË¶oH$ KQ>Zo‘Ü¶o, V¶ma Pmboë¶m Mm¡agmMm AmH$ma XmI{dbm Amho.àË¶oH$ KQ>ZoV dmnaboë¶m H$mS>çm Am{U g§»¶m§Mr ‘m§S>Ur {Vgè¶m Am{U Mm¡Ï¶m ñV§ mV {Xbobr Amho.nmMì¶mñV§ mV V¶ma Pmboë¶m Mm¡agm§Mr g§»¶m {dMmamV KodyZ g§»¶mMr ‘m§S>Ur nwÝhm {bhrbobr Amho.Mm¡agm§Mr H$mS>çm§Mr g§»¶m ‘m§S>Ur Mm¡agmÀ¶m g§»¶oÀ¶mg§»¶m ñdénmV ì¶º$ H$aUo.

1. 4 4 4

2. 7 4 + 3 4 + 3(2 – 1)

3. 10 4 + 3 + 3 4 + 3(3 – 1)

4. 13 4 + 3 + 3 + 3 4 + 3(4 – 1)

n 4 + 3 (n – 1)

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Progressions 23

gmaUr déZ AmnU Agm {ZîH$f© H$mTy> H$s,(i) 4, 7, 10, 13..... ho A.P ‘Ü¶o AmhoV.

(ii) n{hbo nX 4 Amho i.e., a = 4(iii) gmYmaU ’$aH$ 3 Amho i.e., d = 3(iv) Oa VoWo 'n' Mm¡ag AmhoV, Va V¶ma Pmbobr ‘m§S>Ur 4 + 3(n – 1)(v) Va n do nX, a + d (n – 1) qH$dm Tn = a + d (n – 1)

A.P ‘Ü¶o, Oa n{hbo nX 'a' gmYmaU ’$aH$ 'd' Agob Va n do nX H$mT>Ê¶mMo gyÌ Tn = a + (n – 1)dCXmhaU 1 : 2, 6, 10, 14, .......¶m A.P Mm {dMma H$am.¶oWo T1 = a = 2, d = T2 – T1 = 6 – 2 = 4T1 = 2 = a = a + (1 – 1)dT2 = 6 = 2 + 4 = a + d = a + (2 – 1)dT3 = 10 = 2 + 4 + 4 = a + d + d = a + (3 – 1)dT4 = 16 = 2 + 4 + 4 + 4 = a + d + d + d = a + (4 – 1) d

Tn = a + d + d......+ (n – 1) doim = a + (n – 1)d

Tn = a + (n – 1) d

darb CXmhaUmdéZ AmnU Agm {ZîH$f© H$mTy> H$s,n{hbo nX 'a' gmYmaU ’$aH$ 'd' Agboë¶m A.P Mm gm‘mÝ¶ Z‘wZm, a, a + d, a + 2d,..............., a + (n – 1)d,......AmhoA.P Mo gm‘mÝ¶ nX Tn = a + (n – 1)dCXmhaU 2 : 3, 4, 5, 6,.....¶m H«$‘mMo 1000 do nX emoYm.H«$‘m‘Ü¶o, ‘mJrb nXm‘Ü¶o 1 {‘i{dbo AgVm nwTr>b nX {‘iVo.

hr nÜXV dmnéZ 1000do nX H$mT>VmZm, Amnë¶mbm 1000 nXo {bhmdr bmJVmV. hr nÜXV doiH$mTy> Am{U H§$Q>midmUr Amho.

Oa ¶m H«$‘mMo gm‘mÝ¶ nX aMbo, Va H$moUVohr hdo Agbobo nX ghOnUo H$mT>Vm ¶oVo.

¶m H«$‘m‘Ü¶o gmYmaU ’$aH$ (c.d) 1 Amho.1bo nX T1 = 3 = 3 + 0(1) 2 ao nX T2 = 4 = 3 + 1 = 3 + 1(1)3 ao T3 = 5 = 3 + 1 + 1 = 3 + 2(1)

ndo nX Tn = 3 + 1 + 1 + ....(n – 1) doim = 3 + (n – 1)(1)= 3 + n – 1 = 2 + n

Tn = 2 + n

1000do nX = T1000 = 2 + 1000 = 1002

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Progressions 24

Ogo AmVmM AmnU A.P Mo n do nX emoYbo. AmnU A.P. ‘Ü¶o Zgboë¶m H$m§hr H«$‘mMo ndo nX

gwÜXm H$mTy> eH$Vmo.

CXmhaU 3 :

1) 2, 4, 6, 8, .........¶m H«$‘mMo ndo nX H$mT>m., 2) ¶m H«$‘mMo ndo nX H$mT>m.1 2 3, , ,......2 3 4

T1 = 2 = 2 × 1 T1 =12 =

11 1

T2 = 4 = 2 × 2 T2 =23

=2

2 1T3 = 6 = 2 × 3 T3 =

34

=3

3 1Tn = 2 × n

Tn = 2 × n Tn = 1n

n + Tn = 1

nn +

na§Vw gd© H«$‘m§Mo gm‘mÝ¶ én H$mT>Uo e³¶ Zmhr.

{dMma H$am Am{U JQ> H$éZ MMm© H$am.

(i) 5, 17, 29, 31, 100, 450,....ho MT>Ë¶m H«$‘mV {b{hbo na§Vw ¶mMo gm‘mÝ¶ nX

H$mT>Uo Ae³¶ Amho. (ii) 2, 3, 5, 7, 11, ...... hm ‘yi g§»¶m§Mm H«$‘ Amho. ¶m H«$‘mMo

gm‘mÝ¶ nX H$mT>Uo Ae³¶ Amho.

Amåhmbm ‘mhrV Amho H$s, A.P ‘Ü¶o gmYmaU ’$aH$ 'd'.Amho.

( (i) Tn + d = Tn + 1 ) ( (ii) Tn – d = Tn – 1)Imbrb H$moï>H$mMo {ZarjU H$am.

A.P ‘Ü¶o,

T2 = T1 + d T2 = T1 + (2 – 1)d T2 – T1 = (2 – 1)d d = 2 1

2 1T T

T3 = T1 + d + d T3= T1 + (3 – 1)d T3 – T1 = (3 – 1)d d = 3 1

3 1T T

T5 = T4 + d T5 = T4 + (5 – 4)d T5 – T4 = (5 – 4)d 5 4d5 4

T T

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Progressions 25

gm‘mÝ¶nUo, p qT Td =

p q

Oa Tp = Tn Am{U Tq = T1(ie., a) Va darb g§~§Y

Tn 1n ad Agm {bhrbm OmVmo.

ñnï>rH$aUmË‘H$ CXmhaUo

CXmhaU 1. 13, 19, 25, 31,....... ho A.P. ‘Ü¶o AmhoV H$m Vo Vnmgm.

CH$b : ¶oWo, T1 = 13, T2 = 19, T3 = 25, T4 = 31 Am{U BË¶mXrd = T2 – T1 = 19 – 13 = 6d = T3 – T2 = 25 – 19 = 6d = T4 – T3 = 31 - 25 = 6

gmYmaU ’$aH$ pñWa Amho, {Xbobm H«$‘ A.P. ‘Ü¶o Amho.

CXmhaU 2. Oa a = 3 Am{U c.d = 4, Va A.P. emoYm.

CH$b: a = 3, c.d = 4 Am{U T1 = a = 3 qH$dm.T1 = a = 3T2 = a + d = 3 + 4 = 7 T2 = T1 + d = 3 + 4 = 7T3 = a + 2d = 3 + 2(4) = 3 + 8 = 11 T3 = T2 + d = 7 + 4 = 11T4 = a + 3d = 3 + 3(4) = 3 + 12 = 15 T4 = T3 + d = 11 + 4 = 153, 7, 11, 15, ......hr A.P Amho.

CXmhaU 3 : 12, 19, 26,......¶m A.P Mo Tn Am{U T15 nX H$mT>m.

CH$b : a = 12, d = 19 – 12 = 7, Tn = ?, T15 = ?

Tn = a + (n – 1)dTn = 12 + (n – 1)7 = 12 + 7n – 7Tn = 5 + 7n

g‘Om, Tn = 5 + 7n

T15 = 5 + 7(15) = 5 + 105T15 = 110

CXmhaU 4: 7, 13, 19, ........151 ¶m A.P ‘Yrb nXm§Mr g§»¶m H$mT>m.CH$b : a = 7, d = T2 – T1 = 13 – 7 = 6, Tn = 151, n = ?

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Progressions 26

Tn = a + (n – 1)d151 = 7 + (n – 1) 6 = 7 + 6n – 6151 = 1 + 6n 151 – 1 = 6n 150 = 6n

n =150

25

6 n = 25

CXmhaU 5: A.P Mo 8 do nX 17 Am{U 19 do nX 39 Amho. Va 25 do nX H$mT>m.CH$b : T8 = 17, T19 = 39, T25 = ?

T25, H$mT>Ê¶mgmR>r Amnë¶mbm 'a' Am{U 'd' H$mT>bo nm{hOo. n`m© r nÜXV

d =T Tp q

p q

d = 19 8T T19 8 =

39 1711 =

222

11 d = 2

g‘Om, T8 = 17

a + 7d = 17 [ Tn = a + (n – 1)d]a + 7(2) = 17

a + 14 = 17

a = 3

T25 , Tn = a + (n – 1)d H$mT>VmZm T25 = 3 + (25 – 1)2 = 3 + (24 × 2) = 3 + 48

T25 = 51

CXmhaU 6 : H« _mMo 4Wo nX 17 Am{U 10 do nX 7ì¶m nXmnojm 12Zo OmñV Amho Va A.PemoYm.

CH$b : T4 = 17, T10 = T7 + 12

a + 9d = a + 6d + 12

3d = 12

d = 4

T4 =a + 3d 17 a + 12

a =17 – 12a = 5

5, 5 + 4, 5 + 2(4),......hr Amho. 5, 9, 13, .....

T8 =a + 7d = 17 T19=a +18d = 39(a +18d) - (a +7d) = 39 - 17

11d = 22

d = 2T8 =a + 7× 2 = 17

a +14 = 17a =3

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Progressions 27

CXmhaU 7 : EH$m A.P ‘Ü¶o 7ì¶m nXmMr 7 nQ> hr 11ì¶m nXmÀ¶m 11 nQ>r BVH$s Amho VaA.P Mo 18 do nX H$mT>m.

CH$b : 7 × T7 = 11 × T11

7(a + 6d) = 11(a + 10d)

7a + 42d = 11a + 110d

7a – 11a = 110d – 42d

–4a = 68d a =684

d

a = – 17d

g‘Om, T18 = a + 17d

= – 17d + 17d

T18 = 0

ñdmÜ¶m¶ 2.2

1. Imbrb A.P Mr nwT>rb Mma nXo {bhm.(i) 0, –3, –6, –9, ___, ___, ___, ___,

(ii)1 1 1, , ,6 3 2 ___, ___, ___, ___,

2. H«$‘ H$mT>m Oa,(i) Tn = 2n – 1

3. A.P ‘Ü¶o,(i) Oa a = -7, d =5, Va T12 H$mT>m. (ii) a = -1, d = –3 , VaT50 H$mT>m.

(iii) Oa a = 12, d = 4, Tn = 76, Va n H$mT>m. (iv) Oa a =12, d = –2, T22= –39, Va a H$mT>m.(v) Oa a = 13, T15 = 55, Va 'd' H$mT>m.

4. 100, 96, 92, ......., 12.¶m A.P ‘Yrb nXm§Mr g§»¶m H$mT>m.5. {ÌH$moUmMo H$moZ A.P ‘Ü¶o AmhoV. Oa bhmZ H$moZ 500 Amho Va BVa XmoZ H$moZ emoYm.6. EH$m A.P ‘Ü¶o 50 nXo AmhoV, Ë¶mMo {Vgao nX 12 Am{U eodQ>Mo nX 106 Amho Va 29

do nX H$mT>m.7. A.P ‘Yrb 4Ï¶m Am{U 8 ì¶m nXm§Mr ~oarO 24 Am{U Ë¶mM H«$‘mÀ¶m 6ì¶m Am{U 10

ì¶m nXm§Mr ~oarO 44 Amho. Va n{hbr VrZ nXo H$mT>m.8 EH$m A.P À¶m 7 ì¶m d 3è¶m nXm§Mo JwUmoÎma 12:5 Va 13ì¶m d 4Ï¶m nXm§Mo JwUmoÎma

H$mT>m.

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Progressions 28

9. EH$m H§$nZrZo 2001 ‘Ü¶o 400 ì¶º$s¨Zm H$m‘ {Xbo Am{U àË¶oH$ dfu 35 ì¶º$s¨Mrg§»¶m dmT>{dbr. Va H$moUË¶m dfu H§$nZrV H$m‘Jmam§Mr g§»¶m 785 hmoB©b.?

10. Oa EH$m A.P Mo p do nX q Am{U q do nX p Amho Va ndo nX (p + q – n) Amho ho {gÜXH$am.

11. A.P À¶m Xþgè¶m Am{U {Vgè¶m nXm§Mr ~oarO 22 Am{U n{hë¶m d 4Ï¶m nXm§MmJwUmH$ma 85 Amho. Va A.P Mr Mma nXo H$mT>m.

H«$_mVrb nXm§Mr ~oarO (Sum to n terms of a sequenec)

AOw©ZZo JmoQ>çm§Mr ‘m§S>Ur Imbr XmI{dë¶mà‘mUo Ho$br.Am{U BVa (BË¶mXr)

Oa Ë¶mZo hr ‘m§S>Ur (Z‘wZm) AmUIr 10 nm¶è¶mV Ho$br.

AOw©ZZo ‘m§S>Ur Ho$boë¶m EHy$U JmoQ>çm§Mr g§»¶m {H$Vr?¶m CXmhaUmMo CÎma emoYÊ¶mgmR>r, Amåhmbm àË¶oH$ JQ>mVrb JmoQ>çm§Mr g§»¶m H$mT>mdr bmJob.Oa T1 + T2 + T3 + ...... + Tn hm H«$‘ Amho. Va T1 + T2 + T3 + ...... + Tn À¶m ~oaOobm {Xboë¶m H«$‘mVrb

nXm§Mr ~oarO Ago åhUVmV. Vr 'Sn' Zo Xe©{dVmV.Sn = T1 + T2 + T3 + ...........+Tn

Oa T1 + T2 + T3 + ...... + Tn hm H«$‘ Amho. Va T1 + T2 + T3 + ...... + Tn amerbm loUr åhUVmV.

{Q>n: S1 = T1 S2 = T1 + T2 S3 = T1 + T2 + T3 Am{U BVa


CXmhaU 1.Oa Tn = 2n – 1 Va S3H$mT>m. CXmhaU 2. Oa Tn = n2 + 1, Va S2 H$mT>m.

CH$b : S3 = T1 + T2 + T3 CH$b : S2 = T1 + T2

Tn = 2n – 1 Tn = n2 + 1

T1 = 2(1) – 1 = 2 – 1 = 1 T1 = 12 + 1 = 1 + 1 = 2

T2 = 2(2) – 1 = 4 – 1 = 3 T2 = 22 + 1 = 4 + 1 = 5

T3 = 2(3) – 1 = 6 – 1 = 5 S2 = T1 + T2

S3 = T1 + T2 + T3 = 1 + 3 + 5 S2 = 2 + 5

S3 = 9 S2 = 7

A.P Mo 'n' do nX Am{U n ì¶m nXm§ n¶ªVMr ~oarO ¶m‘Yrb g§~§Y.g‘Om,Tn = 5n – 2

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Progressions 29

T1 = 5(1) – 2 = 5 – 2 = 3

T2 = 5(2) – 2 = 10 – 2 = 8

T3 = 5(3) – 2 = 15 – 2 = 13

T4 = 5(4) – 2 = 20 – 2 = 18

T5 = 5(5) – 2 = 25 – 2 = 23

nmhm H$s,S2 – S1 = 11 – 3 = 8 = T2S3 – S2 = 24 – 11 = 13 = T3S4 – S3 = 42 – 24 = 18 = T4S5 – S4 = 65 – 42 = 23 = T5

Sn – Sn – 1 = Tn

A§H$J{UVr loUr (Arithmetic series)

A§H$J{UVr H«$‘mVrb nXo loUr ‘Ü¶o AgVrb Va Ë¶mbm A§H$J{UVr loUr Ago åhUVmV.

CXm: 2 + 4 + 6 + 8 + 10 = 30 1 + 5 + 9 + 13 + 17 + 20 = 66

{Q>n : A‘¶m©{XV A§H$J{UVr loUrMr ~oarOH$mT>Uo Ae³¶ Amho.

A.P À¶m 'n' nXm§Mr ~oarO [Sum of first 'n' terms of an A.P]

~oaOoMm Ioi Ioiy¶m Ago g‘Oy. Imbrb loUrMr ~oarO H$mT>m.1 + 22 + 5 + 84 + 7 + 10 + 13 + 16_1 + 3 + 5 + 7 + ....... + 101

H$moUË¶m KQ>Zo‘Ü¶o Vwåhr ~oarO ghOnUo H$mT>mb? H$m? MMm© H$am.darb CXmhaUmdéZ, Amåhmbm Ago g‘OVo H$s, Oa A.P ‘Yrb nXm§Mr g§»¶m bhmZ Agob Va AmnUË¶mMr ~oarO ghOnUo H$éZ ~oarO {‘idy eH$Vmo. na§Vw nXm§Mr ~oarO ‘moR>r Agob Va Amnë¶mbm H$R>rUOmVo Am{U OmñV doi bmJVmo.

ObXnUo Am{U ghOnUo ~oarO H$mT>Vm ¶oÊ¶mMr H$moUVrhr gmonr nÜXV Amho H$m?¶oWo J{UVrÀ¶m B{VhmgmVrb EH$ Hw$VwhbmMm àg§J Amho.JmoîQ>rMr gwédmV Aer Pmbr. H$mb© ’|$S>rM Jm°g hm ‘wbJm ImoS>H$a hmoVm Am{U {ejH$m§Zr {Xbob§

CXmhaU ËdarV gmoS>dm¶Mm.EH$ {Xdg {ejH$m§Zr Ë¶mbm 100 n¶ªVÀ¶m Z¡g{J©H$ g§»¶m§Mr ~oarO H$am¶bm gm§{JVbr Am{U

S1 = T1 = 3S2 = T1 + T2 = 3 + 8 = 11S3 = T1 + T2 + T3 = 3 + 8 + 13 = 24S4 = T1 + T2 + T3 + T4 = 3 + 8 + 13 +18 = 42S5 = T1 + T2 + T3 + T4 + T5 = 3 + 8 + 13 + 18 + 23 = 65

darb CXmhaUmdéZ, AmnU Agm {ZîH$f© H$mTy> H$s,S2 – S1 = T2S3 – S2 = T3S4 – S3 = T4S5 – S4 = T5

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Progressions 30

Ë¶mbm H$m‘mV Jw§VdÊ¶mMm à¶ËZ Ho$bm. na§Vw Jm°gZo H$mhr doimVM ~oarO 5050 ¶oVo Ago CÎma {Xbo.AmVm AmnU AJXr ghOnUo CÎma H$go H$mT>bo Vo nmhÿ¶m.Ë¶mZo hr ~oarO Aer {bhrbr H$s,

S100 = 1 + 2 + 3 + 4 + .......... + 100

S100 = 100 + 99 + 98 + ........ + 2 + 1 (CbQ> H$éZ)

2S = 101 +101 + 101 +....................... + 101 (~oarO H$éZ)

2S = 101 × 100 [ 101 ho 100 doim {‘i{dbobo Amho]

S =101 100´

50

2S100 = 5050

AmnU VoM V§Ì dmnéZ A.P À¶m 'n' nXm§Mr ~oarO H$mTy> eH$Vmo.Oa A.P Mo 'a' ho n{hbo nX d 'd' hm gmYmaU ’$aH$ Am{U Sn hr A.P. À¶m 'n' nXm§Mr ~oarO Xe©{dVo.Va

Sn = T1 + T2 + T3 + ......... + TnSn = a + (a + d) + (a + 2d) + .......... + [a + (n– 1)d].

Sn = [a +(n – 1)d] + [a + (n – 2)d] +................................... + a (CbQ >H$éZ)

2Sn =[2a + (n – 1)d] + [2a + (n – 1)d] + ..........................................+[2a + (n – 1)d] (~oarOH$éZ)

2Sn =[2a + (n – 1)d] × n [ 2a + (n – 1)d] bm 'n' doim {‘idy. )

[ ]+ -nn

S 2 a (n 1 )d2

{Q>n : Oa 1 + 2 + 3 + 4 + ...........+n, hr A.P n{hë¶m 'n' (Z¡g{J©H$ g§»¶m§Mr ~oarO) Amho.¶oWo, a = 1, d = T2 – T1 = 2 – 1 = 1, n = n

Sn = 2 ( 1)2n a n d

H$mb© ’«|$S´>rM Jm°g [1777-1855, O‘©Zr] hm J{UVmMm amOHw$‘ma Amho Ë¶m§ZrAmnbr JUUoMr eº$s AË¶§V ~mbnUr d¶mÀ¶m 3è¶m dfu àX{e©V Ho$br. Ë¶m§Zrd¶mÀ¶m 24ì¶m dfu e³¶Vmo AmVmn¶ªVMo J{UVmVrb ewÜX nwñVH$ g§»¶memómMoAm¡nMmarH$ boI à{gÜX Ho$bo. ¶wp³bS>À¶m A§H$J{UVm‘Yrb ‘yb yV à‘o¶mMrg§nyU© {gÜXVm H$aUmao Vo n{hbo AmhoV.

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Progressions 31

Sn = 2.1 ( 1)12n n = 2 1

2n n

( 1)2n

n nS qH$dm n(n +1)n =2

n{hë¶m 'n' Z¡g{J©H$ g§»¶m§Mr ~oarO =( 1)

2n n

{Q>n : Amnë¶mbm ‘m{hV Amho H$s, Sn = [2 ( 1) ]2n a n d

ho Agohr {bhrbo OmVo H$s, Sn = ( 1)2n a a n d

n nnS = a + T2 [ Tn = a + (n – 1)d]

¶oWo, a n{hbo nX, Tn ndo nXT

2na hr A.P. À`m n{hë`m Am{U eodQ>À`m nXm§Mr gamgar Amho.

`mMm AW© A.P. À`m n{hë`m 'n' nXm§Mr ~oarO hr 'n' doim n{hbo Am{U eodQ>Mo nX m§Mr gamgar AmhoOa AmnU darb gyÌ n{hë¶m 'n' Z¡g{J©H$ g§»¶m§Mr ~oarO H$mT>Ê¶mgmR>r dmnabo OmVo.

ñnï>rH$aUmË‘H$ CXmhaUoCXmhaU 1: Oa Tn = 5n – 2, Va S4H$mT>m.CH$b : g‘Om,

Tn = 5n – 2T1 = 5(1) – 2 = 5 – 2 = 3 S4 = T1 + T2 + T3 + T4T2 = 5(2) – 2 = 10 – 2 = 8` S4 = 3 + 8 + 13 + 18T3 = 5(3) – 2 = 15 – 2 = 13 S4 = 42T4 = 5(4) – 2 = 20 – 2 = 18

CXmhaU 2 : 1 + 2 + 3 + ..... ¶m loUrÀ¶m n{hë¶m 20 nXm§Mr ~oarO H$mT>m.CH$b : {Xbobr loUr hr Z¡g{J©H$ g§»¶m§Mr Amho. åhUyZ AmnU g§~§Y dmnéZ ~oarO H$mTy>eH$Vmo.

n =( 1)

2n n + ¶oWo, n = 20

S20=20(20 1)

2+

S20 =20

1021

2S20 = 210

(AmnU {MÝh dmnéZ n Z¡g{J©H$ g§»¶m§Mr~oarO H$mTy> eH$Vmo. Vo Ago Xe©{dVmV.)

n

1

1 + 2 + 3 + ........+na = 1, Tn = n, n = nSn =

T2 nn a

Sn = 12n n =

( 1)2

n n n

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Progressions 32

CXmhaU 3 : 5, 8, 11, 14, ... ¶m A.P À¶m n{hë¶m 15 nXm§Mr ~oarO H$mT>m.

CH$b :a = 5, n = 15, c.d = 8 – 5 = 3

Sn = 2 ( 1)2n a n d

S15 =15 2(5) (15 1)32 =

15 10 (14)(3)2 S15 = 15

252´

26

S15 = 390 S15 = 390

CXmhaU 4 : 200 Am{U 300 ‘Yrb nyU©nUo 6 Zo ^mJ OmUmè¶m gd© Z¡g{J©H$ g§»¶m§Mr

~oarO H$mT>m.

CH$b :

204 + 210 + 216 + .......... + 294 hr A.P Amho.

a= 204, d = 210 – 204 = 6, Tn = 294, Sn = ?

Sn, H$mT>Ê¶mgmR>r àW‘ Amåhmbm 'n'Mr qH$‘V H$mT>mdr bmJob.

Tn = 294 'n' Mr qH$_V H$mT>Ê`mMr n`m© r nÜXV

a + (n – 1)d = 294 Tn=a + (n-1) d Mm {dMma H$am 'n' À`m ñdê$nmV Amho.

204 + (n – 1) 6= 294 gyÌ nwÝhm {bhm

204 + 6n – 6= 294T 1n an

d198 + 6n = 294 294 204

16

n-

\ = +90 16

6n= 294 – 198 = 96 n = 15 + 1 = 16

n = 96/6

n= 16

Sn = T2 nn a

S16 = [ ]16204 294

2+ =

168

2498

S16 = 3984

CXmhaU 5 : a‘oebm «‘U ¶§Ì (Mobile Phone) IaoXr H$amd¶mMm Amho. Vmo Ë¶mbm <15000 amoI^éZ qH$dm 12 g‘mZ ‘m{gH$ háo H$éZ n{hë¶m ‘{hÝ¶mV < 1800 d Xþgè¶m ‘{hÝ¶mV <1750,{Vgè¶m ‘{hÝ¶mV < 1700 Am{U B. éZ IaoXr H$aVmo. Oa Ë¶m§Zo a¸$‘ hßË¶m§Zr abr Va (i)12 hßË¶m‘Ü¶o ^abobr EHy$U a¸$‘ (ii) amoI a¸$‘onojm Ë¶mbm {H$Vr OmñVrMr a¸$‘ ^amdrbmJob Vo H$mT>m.

à`ËZ H$am,Sn= n ( a +Tn)

hm g§~§Y dmnê$Z gmoS>dm

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Progressions 33

CH$b : ¶oWo,

1800 + 1750 + 1700 + ......hr A.P Amho.a = 1800, d = T2 – T1 = 1750 – 1800 = – 50, n = 12, S12 = ?

Sn = 2 ( 1)2n a n d

S12 =12

6

22(1800) (12 1) ( 50)

S12 = 6[3600 + 11(–50)] = 6[3600 – 550]S12 = 6 × 3050S12 = 18,300

12 hßË¶m‘Ü¶o abobr EHy$U a¸$‘ = < 18,300

OmXm amdr bmJUmar a¸$‘ = 18,300 -15000 =3300

CXmhaU 6: A.P _Yrb n{hbo VrZ YZ nyUmªH$ H$mT>m Aem [aVrZo H$s Ë`m§Mr ~oarO 24 Am{U

JwUmH$ma 480 Amho.

CH$b: a-d, a, a+d hr VrZ nXo AmhoV Ago _m§Zy

~oarO =24

a - d + a + a + d = 24

3 a = 24

Oa a - d + a + a + d = 24

3 a = 24

a=8

Oa a = 8, d=2 Va H«$_mVrb nXo a - d = 8 - 2 = 6, a = 8, a+ d = 8 + 2 = 10

H«$_mVrb nXo 6,8,10 hr Amho

CXmhaU 7: A _Ü`mnmgyZ gwadmV H$ê$Z EH$mZ§Va EH$ gVV AY© dV©wio KoodyZ Ë`mMm _Ü` A Z§Va

B KodyZ doQ>moio ~Z{dbo Ë`m§Mr pÌÁ¶m AmH¥$VrV XmI{dë`m à_mUo 0.5 go. _r, 1go._r,1.5

go._r,2 go._r ......B Amho Va 13 H«$_dma AY©dVw©io OmoSy>Z ~Z{dboë`m doQ>moù`mMr bm§~r

H$mT>m? ( 22 ¿`m)

à`ËZ H$am: Ooìhm

a =8, d=2

Agob Va nXo H$mT>m

Vwåhmbm H$m` {XgyZ ooVo ?

JwUmH$ma =480

(a-d) a (a+d) =480

(8-d) 8(8+d)=480 ( a = 8)

(8-d) (8+d)=480/8=60

64-d2= 60

d2= 64 - 60

d=±2

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Progressions 34

r1 = 0.5 cm, r2 = 1cm, r3 = 1.5 cm,.......... `m {ÌÁ¶oÀ`m

AY©dVw©im§Mr bm§~r AZwH«$_o l1, l2 l3 Amho.

dV©wimMm n[aK = 2 r

AY©dVw©imMm n[aK = r

=l1 = r1 = × 0.5 =2

,l2 = r2 = × 1 = = 22

V©wimMm n[aK l13 = r13 =132 = 13

2

doQ>moù`mMr EHy$U bm§~r= l1 + l2 + ........ + l13

= 2 3 ..... 132 2 2 2 =

2[1 + 2 + 3 + .......+13] =

13

12n

=13(13 1)

2 2 =13 14

2

7

2 =

12

2211

713 7 = 143 g|.‘r.

ñdmÜ¶m¶ 2.3

1. Oa Tn = 2n + 3, Va S3 H$mT>m.2. Imbrb loUtMr ~oarO H$mT>m.

(i) 3 + 7 + 11 + ........ 25 ì¶m nXmn¶ªV (ii) -3, 1, 5 ........ 17 ì¶m nXmn¶ªV(iii) 3a, a, -a ........ a ì¶m nXmn¶ªV (iv) p, o, -p ........ p ì¶m nXmn¶ªV

3. A.P À¶m 111 nXm§Mr ~oarO H$mT>m Á¶mMo 56 do nX 537Amho.

4. Z¡g{J©H$ g§»`mÀ`m H«$_mgmR>r a) (i) 20 (ii) S50 - S 40 (iii) S30+ S15 H$mT>m.

b) 'n' H$mT>m Oa (i) Sn= 55 (ii) Sn =15

5. n{hë¶m 'n' {df‘ Z¡g{J©H$ g§»¶m§Mr ~oarO H$mT>m.

6. 1 Am{U 201 ‘Yrb 5 Zo mJ OmUmè¶m gd© Z¡g{J©H$ g§»¶m§Mr ~oarO H$mT>m.

7. EH$m loUrÀ¶m n nXm§Mr ~oarO 1 7 -12

n n Agob Va n{hbr Mma nXo H$mT>m.

8. 1, 4, 7, ...... ¶m A.P ‘Yrb {H$Vr nXm§Mr ~oarO 51 hmoB©b?

9. AP ‘Yrb VrZ nXo H$mT>m. Oa Ë¶m§Mr ~oarO Am{U JwUmH$ma AZwH«$‘o (i) 21 Am{U 231(ii) 36 Am{U 1620 Amho.

l5 l3

l1

l2l4

A B

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Progressions 35

10.A.P À¶m 6 nXm§Mr ~oarO 345 Amho n{hë`m d eodQ>À`m nXm_Yrb \$aH$ 55 Amho Va Vr nXoH$mT>m .

(1/4) Amho. Va T20 = –112 ho XmIdm Am{U S20 H$mT>m.

11.EH$m AP Mo n{hbo nX 2 Amho, n{hë¶m nmM nXm§Mr ~oarO hr nwT>rb nmM nXm§À¶m EH$ MVwWmªe

(1/4) Amho. Va T20 = –112 ho XmIdm Am{U S20 H$mT>m.

12. EH$ A.P Mo {Vgao nX 8 Am{U A.P Mo 9 do nX ho 3è¶m nXmÀ¶m 3 nQ>rnojm 2 Zo A{YH$ Amho.

Va Ë¶mÀ¶m n{hë¶m 19 nXm§Mr ~oarO H$mT>m.

JwUmH$ma ì¶ñV H«$‘ [HARMONIC PROGRESSION]

Imbrb H$moï>H$mMo {ZarjU H$am.

H«$‘ ì¶ñV

1 1 1 1, , , ,......2 5 8 11

2, 5, 8, 11,.....

1 1 1 1, , , ,......30 28 26 24

30, 28, 26, 24, ......

2 2 21, , , ,......3 4 5

3 4 51, , , ,......2 2 2

Aem H«$_mbm JwUmH$ma ì`ñV H«$_ åhUVmV d Vmo WmoS>Š`mV H.P Ago {bhrVmVH«$‘mVrb àË¶oH$ nXmMm JwUmH$ma ì¶ñV KoVë¶mg V¶ma hmoUmam H«$‘ A§H$J{UVr H«$‘Agob Va Ë¶mbm JwUmH$ma ì¶ñV H«$‘ Ago åhUVmV.

H.P Mo gm‘mÝ¶ én [General form of H.P.]

Oa A.P Mo n{hbo nX 'a' Am{U gmYmaU ’$aH$ 'd' Amho Va Ë¶mMo ñdén.a, a + d, a + 2d, ........., a + (n – 1)d.

Amnë¶mbm ‘m{hV Amho ¶m nXm§Mm JwUmH$ma ì¶ñV H.P hmoVmo.

Ë¶mMm Z‘wZm,

1 1 1 1, , ,..........2 ( 1)a a d a d a n d hm Amho.

H.P. Mm gm‘mÝ¶ Z‘wZm

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Progressions 36

hm Amho

{Q>n : H.P.À¶m 'n' ì¶m nXm§Mr ~oarO H$mT>Ê¶mMo gyÌ Zmhr.


CXmhaU 1 : 1 1, ,1, 1,.......5 3 ¶m H.P Mo T10 H$mT>m.

CH$b :{Xboë¶m HP Mm JwUmH$ma ì¶ñV A.PAgVmo.5, 3, 1, –1......... ho A.P ‘Ü¶o AmhoV.a = 5, d = T2 – T1 = 3 – 5 = – 2, T10 = ?

Tn = a + (n – 1)dT10 = 5 + (10 – 1) (–2) = 5 + 9(–2) = 5 – 18 =-13.T10 = –13

A.P Mo 10 do nX T10 = –13 Amho.

g§JV H.P Mo T10 ho1

13 ho Amho.

CXmhaU 2 : H.P ‘Ü¶o T3 =1

7 Am{U T7 =1

5 . Va T15H$mT>m.

CH$b : AP Mo g§JV T3 Am{U T7 ho T3 = 7, T7 = 5 AmhoV.

d =T Tp q

p q = 7 3T T7 3

d =5 7

4 =

24

=12

d =1

2

g§JV AP Mo T3=7 g‘Oy.

a + 2d= 7 2a 12 = 7

a – 1 = 7a = 7 + 1a = 8 T15 H$mT>VmZm, Tn = a + (n – 1)d

T15= 8 + (15 – 1)1

2

T15= 8 147 1

2

gyÌ dmnê$Z gmoS>{dÊ`mMr n`m© r nÜXV

Tn =1

a+(n-1)d

= 1a + 2d 7

= a+2d

a+2d=7

VgoM a + 6 d = 5 (-) (-) _

_ 4d =2d =2/-4 =- 1/2

g‘rH$aU‹ (1) dê$Z a+2d=7

a + 2 (- 1 ) =7 2

a = 7 + 1=8

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Progressions 37

g ñdmÜ¶m¶ 2.4

1. Imbrbn¡H$s H$moUVo JwUmH$ma ì¶ñV H«$‘mV AmhoV?

(i)1 1 11, , , ,......4 7 10 (ii)

2 1 21, , , ,......3 2 5 (iii)

1 1 1, , ,......2 6 18

(iv)1 1 1, , ,......3 7 11

(v) 6, 4, 3, ..... (vi)1 11, , ,......2 4

2. (i) Tn H$mT>m. (ii) 1 1, ,1,......7 4

H«$‘mMo T10 H$mT>m.

3. H.P ‘Ü¶o T5 =1

12Am{U T11 =

115

Va T25 H$mT>m.

4. H.P ‘Ü¶o T4 =111

Am{U T14 =3 ,23

Va (i) T7 (ii) T19 H$mT>m.

JwUmoÎma H«$‘ (Geometric progression)

Vwåhmbm H|${Ð¶ {dKQ>Z {H«$¶oMr ‘m{hVr Amho.

Ooìhm U - 235 da Ý¶yQ´>mMm ‘mao Ho$bm Va Ë¶m§Mo

{dKQ>Z hmodyZ 2 Ý¶yQ´>m°Z ‘wº$ hmoVmV. ho XmoZ

Ý¶yQ´>m°Z {dKQ>Z {H«$¶m nwT>o Mmby R>odVmV Am{U

nwT>rb nm¶arV 4 Ý¶yQ´>m°Z ‘wº$ hmoVmV Am{U ¶m

nw{T>b nm¶arV 8 Ý¶yQ´>m°Z Am{U hr {H«$¶m nwT>o

Mmby amhVo.

àË¶oH$ nm¶arV Ý¶yQ>m°ZÀ¶m g§»¶oMo {ZarjU H$am.

nm¶ar I II III IV V VI .......

Ý¶yQ>m°ZMr g§»¶m 1 2 4 8 16 32 ........

1, 2, 4, 8, 16, ...... hm H«$‘ {‘iVmo.

¶oWo àË¶oH$ nX ho ‘mJrb nXmÀ¶m XþßnQ> Amho åhUOoM.‘mJrb nXmbm 2 Zo JwUbo AgVm nwT>rb nX

{‘iVo..

AmUIr H$mhr CXmhaUo nmhm.

H «$‘ {Z¶‘3, 9, 27, 81, ..... 3 Zo JwUbo.1000, 100, 10, 1, ..... 1

10 Zo JwUbo.

5, 25, 125, 625, ..... 5 Zo JwUbo.

1 1 1, , ,......2 4 6

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Progressions 38

Aem àH$maÀ¶m H«$‘mbm JwUmoÎma H«$‘ åhUVmV. Vmo WmoS>³¶mV G.P. Ago {bhrVmV.‘mJrb nXmbm EH$m R>am{dH$ g§»¶oZo JwUbo AgVm nwT>rb g§»¶m ¶oUmè¶m H«$‘mbm JwUmoÎma

H«$‘ Ago åhUVmV.G.P, ‘Ü¶o nXm§Mo JwUmoÎma Am{U Ë¶m§Mo ‘mJrb nX ho eyÝ¶ ahrV pñWa AgVo. ¶m pñWam§H$mbm gmYmaUJwUmoÎma åhUVmV. Am{U Vmo 'r' [c.r] Zo Xe©{dVmV.

H«$‘ 2

1

TT

3

2

TT

4

3

TT

3, 9, 27, 81,..... 3 3 3

1000, 100, 10, 1, ..... 110

110

110

5, 25, 125, 625, ..... 5 5 5

Oa T1, T2, T3, T4, ........ hr G.P, Mr nXo AmhoV Var = 2 3 4

1 2 3

T T T ..........T T T pñWa

H«$‘mMm g‘mdoe Agboë¶m {ZË¶ OrdZmVrb H$m§hr KQ>Zm§Mr ¶mXr AmnU nmZ Z§.16 da Ho$bobr Amho.¶oWo,KQ>Zm 3 _Yrb àË oH$ Vmgmbm OrdmUyMr g§»`m XwßnQ> hmoVo Am{U åhUyZ Ë`mMm H«$_ ......... 50,

100, 200, 400, r =100 200 400 ......... 250 100 200

pñWaÁ¶mAWu Amåhmbm gm‘mB©H$ JwUmoÎma {‘iVo Ë¶mAWu Vmo H«$‘ G.P. Amho.KQ>Zm 4 ‘Ü¶o,{H$aUmoËgO©H$ Z‘wZm 100, 50, 25, 12.5, ......‘Ü¶o KQ>Vmo.

r =50 25 12.5 1......

100 50 25 2pñWa

Á¶mAWu gm‘mB©H$ (gmYmaU) JwUmoÎma pñWa Amho Ë¶mAWu Vmo H«$‘ G.P. Amho.G.P Mm gm‘mÝ¶ Z‘wZm General form of a G.PJ§Jmbm g§Xoe Amnë¶m ‘¡{ÌUrbm nmR>dm¶Mm Amho. Á¶mAWu Ë¶m§Mm JQ> ‘moR>m Amho åhUyZ Vmo àË¶oH$mn¶ªVnmohMy eH$V Zmhr. Ë¶m‘wio {VZo R>a{dbo H$s, hm g§Xoe {VKr n¶ªV nmohMdm¶Mm, Ë¶mZ§Va àË¶oH$sZo{VKm§Zm nmR>dm`Mm Amho Am{U BË`mXr. ¶m KQ>ZoV V`ma hmoUma g§»¶mMm H«$_ H$moUVm Amho ?Imbrb AmH¥$Vr nmhÿZ g§»¶mMm H«$‘ Aä¶mg.................

T1 = 1 = 1 × 30 = 1 × 31 – 1 = aT2 = 3 = 1 × 3 = 1 × 31 = 1 × 32–1 = arT3 = 9 = 1 × 3 × 3 = 1 × 32 = 1 × 33 – 1 = ar2

T4 = 27 = 1 × 3 × 3 × 3 = 1 × 33 = 1 × 34 – 1 = ar3

Tn = 1 × 3 × 3 .....(n – 1) doim = 1 × 3n – 1 = a rn–1

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Progressions 39


CXmhaU 1: Oa a = 4 Am{U r = 2 AgVmZm G.P À¶m n{hbr 3 nXo H$mT>m.CH$b :T1 = a = 4T2 = ar = 4 × 2 = 8

T3 = ar2 = 4 × 22 = 16

G.P Mr n{hbr VrZ nXo 4, 8, 16. hr AmhoV.

CXmhaU 2 : 3 3 3, , ,.......2 4 8 ho G.P ‘Ü¶o AmhoV Va nmMdo nX H$mT>m.

CH$b. a = 32 , r = 2

1

3T 343T

2 4 2

23

12 , T5 = ?

Tn = arn – 1

T5 =5 13 1

2 2 =

43 12 2

=3 12 16

= T5 =332

CXmhaU 3: 2,2 2,4,...... ¶m G. P Mo H$moUVo nX 32 Amho?

CH$b : a = 2, r = 2

1

T 2T

22

2 , Tn = 32, n = ?

Tn = arn – 1

32 = 12 2

n

3216

2= 1

2n 16 = 1

2n

24 = 12

n = (2½)n – 1 ½2 2

24 =1

22n

4 =1

2n

G.P, ‘Ü¶o Oa n{hbo nX 'a' gmYmaU JwUmoÎma 'r' Amho. Va G.P Mm Z‘wZm,a, ar, ar2, ar3 ............ar n – 1

G.P : Mo gmYmaU nX Tn = ar n – 1

{Q>n : gmYmaU JwUmoÎma 'r' Agboë¶m G.P ‘Ü¶o.

* nT>rb nX {‘i{dÊ¶mgmR>r 'r' Zo JwUUo Tn + 1 = Tn × r

* ‘mJrb nX {‘i{dÊ¶mgmR>r 'r' Zo mJUo. Tn–1 = Tn r

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Progressions 40

8 = n – 1 n = 8 + 1 = 9

G.P.Mo 9 do nX 32 Amho.CXmhaU 4:Mo G.P. n{hbo nX 25 Am{U 6do nX 800 Amho Va gmVdo nX H$mT>m?

CH$b : a = 25, T6 = 800, T7 = ?

T7, H$mT>V AgVmZm àW‘ Amnë¶mbm n{hbo nX r H$mT>mdo bmJob.

T6 = 800 ar5 = 800 Amåhmbm ‘mhrV Amho H$s, Tn + 1 = Tn × r

25 × r5 = 800 T7 = T6 × r

r5 =80025 = 32 T7 = 800 × 2

r5 = 25

r = 2 T7 = 1600

ñdmÜ¶m¶ 2 .5

1. Imbrb G.P ‘Yrb gmYmaU JwUmoÎma H$mT>m.(i) –5, 1,

1,5

......... (ii) 3,3,3 3,........ .

2. gyMZoà‘mUo H$am.

(i) Oa a = 1 Am{U r = 23 Va (a) Tn (b) T4 H$mT>m. (ii) 729, 243, 81,...... ¶m G.P Mo T7

H$mT>m.

3. 5do nX 64 Am{U gmYmaU JwUmoÎma 2 AgUmè¶m G.P Mo 12 do nX H$mT>m.

4. Imbrbm§Mr qH$‘V H$mT>m.

i) 256, 128, 64, .....¶m G.P.Mo 10 do 16 do nX H$mT>m .

ii) 81, -27, 9, ¶m G.P. Mo 8 do Am{U 12 do nX.

iii) 0.008, 0.04, 0.2 ..... ¶m G.P. Mo 4Wo Am{U 8do nX.

5. Imbrb loUtMo eodQ>Mo nX H$mT>m.

i) 2, 4, 8 ..... 9ì¶m nXmn¶ªV. ii)4, 42, 43 ..... 2n ì¶m nXmn¶ªV

iii) 2, 3, 412

..... 6ì¶m nXmn¶ªV iv) x , 1, 20 ì¶m nXmn¶ªV

6. Oa T5 : T10 = 32 : 1 Am{U T7 = 132Va G.P H$mT>m.

1 . . . .x

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Progressions 41

7. H$m§hr {H$aUmoËgO©H$ nXmWmªMm AY© OrdZ H$mi 1 Vmg Amho. Oa nXmWm©À¶m gwadmVrÀ¶m Z‘wÝ¶mModñVw‘mZ 500 J«°‘ Amho. Va 5ì¶m VmgmZ§Va {e„H$ Agbobo Z‘wÝ¶mMo dñVw‘mZ {H$Vr?

8. 3, 6, 12, ..... ¶m H«$‘mMo H$moUVo nX 1536 hmoB©b?

9. EH$m Mo 4Wo Am{U 8do nX AZwH«$‘o 24 Am{U 384 Amho Va n{hbo nX Am{U gmYmaU JwUmoÎmaH$mT>m.

10. Imbrb G.P. H$mT>m.

i) 10do nX 320 Am{U 6do nX 20 Amho.

ii) 2ao nX 6 Am{U 6 do nX 9 6 Amho.

JwUmoÎma loUr (Geometric Series.)

loUrMr nXo JwUmoÎma H«$‘mV AgVrb Va {Vbm JwUmoÎma loUr Ago åhUVmV.

CXm : 1 + 3 + 9 + 27 = 401 1 152 12 4 4

G.P À¶m n{hë¶m n nXm§Mr ~oarO (Sum of the first n terms of a G.P)

10 {‘Ì EH${ÌVnUo JmoQ>çm§Mm Ioi IoiV AmhoV. n{hë¶m ì¶º$sH$S>o 2 JmoQ>çm AmhoV. Xþgè¶m ì¶º$sH$S>on{hë¶m ì¶º$sH$S>o Agboë¶m JmoQ>çm§À¶m XþßnQ> AmhoV. {Vgè¶m ì¶º$sH$S>o Xþgè¶m Odi Agboë¶mÀ¶mXþßnQ> AmhoV Am{U BË¶mXr.

Ë¶m§À¶m Odi Agboë¶m EHy$U JmoQ>çm Mr g»¶m {H$Vr?

hr loUr Aer {bhrbr OmVo H$s,

2 + 4 + 8 + ......... + T10

21 + 22 + 23 + ............. + 210

n«Ë¶oH$mMr qH$‘V H$mTy>Z ~oarO H$aÊ¶mMr nÜXV doi H$mTy> Am{U H§$Q>midmUr Amho.

Vr gmoß¶m àH$mao Imbrb à‘mUo Ho$br OmD$ eH$Vo

a + ar + ar2 + ........ + arn – 1 hr n{hbo nX 'a' Am{U gmYmaU JwUmoÎma 'r' Agboë¶m 'n' nXm§Mr~oarO Amho Ago ‘m§Zy.

Sn= a + ar + ar2 + .................. + arn – 1

'r' Zo JwUbo.

r Sn= ar + ar2 + ar3 + ............ + arn

Sn a ar 2ar 1.......... narSnr ar 2ar 1........... nar

( ) ( ) ( ) ( ) ( ) ( )

S .S

n

nn n

ar

r a ar hm g§~§Y nwÝhm Agm {bhrbm OmVmo.

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Progressions 42

Sn – r. Sn = a – arn

Sn (1 – r) = a (1– rn)

Oa r=1Amåhr XmoÝhr ~mOwZm 1 – r Zo

^mJ {Xë¶mg Amåhmbm {‘iVo

n

na(1 - r )S =

1 - r Ooìhm < 1n

na r -1S =

r -1 Ooìhm r > 1

Oa r < 1, 1-r < 0 AmUr darb à‘mUo Amåhr gyÌ dmnaVmo.{Q>n: Oa r =1 Va Sn = a + a + a....n doim = na.

hm g§~§Y H$mT>ë¶mZ§Va, nmZ Z§. 14 (¶m KQ>H$mVrb) Vwåhr dmMbobo Amho, amOmbm emoYH$mMr‘mJUr nwa{dUo H${R>U Jobo ¶mMo H$maU Vwåhmbm emoYUo e³¶ hmoB©b.

~wÜXr~i Ioim‘Ü¶o 64 Mm¡ag AmhoV.

JhÿÀ¶m YmÝ¶mMr Mm¡agm‘Ü¶o R>odÊ¶mMr g§»¶m G.P. ~Z{dVo.1, 2, 4, 8, ..............263

a = 1, r =2

1

T 2=T 1 = 2, n = 64

OmUyZ ¿¶m

Sn =( 1)

1

na rr

S64= 18, 446, 744, 073, 709, 551, 615

S64 =64 641 2 1 1 2 1

2 1 1S64 = 264 – 1

ho YmÝ` _moOÊ`mg {H$Vr doi bmJob ? emoYH$mMr ‘mJUr ¶mo½¶ hmoVr H$m?A‘¶m©{XV JwUmoÎma loUrÀ¶m nXm§Mr ~oarO H$mT>Uo [ Ooìhm r < 1]‘¶m©{XV JwUmoÎma loUrMr ~oarO H$mT>Ê¶mMm Vwåhr Aä¶mg Ho$bobm Amho.Imbrb CXmhaUmMm {dMma H$am.

1 1 11 ......2 4 8 S ¶m G.P Mm {dMma H$é.

¶oWo, a = 1. r =1 121 2

g§~§YmnmgyZ gwadmV H$é Ago ‘m§Zy.

Sn =[1 ]1

na rr

Oa r > 1 V§oìhm 1-r < 0 Am{U A§e Am{U N>oX¶mZm -1 Zo JwUmH$ma H$ê$Z darb g§~§Y nwÝhmImbrb à‘mUo {bhrVm ¶oVmo.

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Progressions 43

Sn =11 1 211 2

n

=11 1 2

12

n

=11 2

12

n

¶oWo 'n' Mr qH$‘V dmT>Vo Voìhm 12

nMo H$m¶ KS>Vo Vo nmhÿ.

n = 1,11

2 = 12 = 0.5

n = 2,21

2 = 14 = 0.25

n = 331

2 = 18 = 0.125

n = 4,41

2 = 116 = 0.0625

n = 8,81

2 = 1256 = 0.00391

n = 10,101

2 = 11024 = 0.00098

'n' ho Odi Amho, Va 'arn' Mr qH$‘V eyÝ¶mOdi ¶oVo. (ho H$Yrhr eyÝ¶ ZgVo) åhUyZ qH$‘V

(AË¶§V H$‘r) AgVo.darb CXmhaUmgmR>r S hrS =

11 212

=

12 Xþb©{jV Ho$bobo Amho

S = 2

n¶m©¶r nÜXV :

A E1 unit 1 unit B

1u

nit

1un

it

F

D C

G ½ unit H

I K1/4 unit

L N1/8

O Q

J

M

P

R S

1/16

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Progressions 44

ABCD hm 2 EH$H$ ~mOyMm Mm¡ag Amho. A‘¶m©{XVo n¶ªV aoIm§{H$V mJmÀ¶m joÌ’$im§Mr ~oarOH$mT>Ê¶mMm à¶ËZ H$é¶m.

{Z[ajU H$am H$s, aoIm§H$sV ^mJ Mm¡ag AmhoV.

ñ‘aU H$am H$s, Mm¡agmMo joÌ’$i = a2

[AEGF + GHIJ + IKLM + .........] Mo joÌ’$i

=2 2 2

2 1 1 11 .........2 4 8

n¶ªV

=1 1 11 ........4 16 64 n¶ªV

n{hë¶m, "n" Mm¡agm§À¶m joÌ’$im§Mr ~oarO H$mTy>.

¶oWo, a = 1, r =1

4 11 4

n{hë¶m, "n" Mm¡agm§À¶m joÌ’$im§Mr ~oarO = a + ar + ar2 + ........ + arn – 1

Sn =[1 ]1

na rr

=11 1 411 4

n

=11 4

43

n

{ZarjU H$am H$s "n" OgOgo ‘moR>‘moR>o hmoVo Vg Vgo 14

n bhmZ bhmZ ~ZV OmVo.

14 H$amho BVHo$ bmhmZ hmho H$s Vo Odi Odi eyÝ¶ hmoVo.

åhUyZ aoIm§H$sV mJmMo joÌ’$i,1-0 1

4 4

43 3 3

S

VgoM OoÁhm r < 1 Am{U "n" AË¶§V ‘moR>o AgVo Voìhm r AË¶§V bhmZ hmoVo.( Odi Odi eyÝ¶)

1-0

1-S

a

r =

a=1 - r

S

MMm© H$am : r > 1. Va A‘¶m©{XV JwUmoÎma loUtMr ~oarO H$mT>Uo Ae³¶ Amho. JQ>mZo MMm© H$am.

¶oWo, H$m§hr G.P r < 1 À¶m A‘¶m©{XoVoÀ¶m ~oaOoMo ‘‚moera Cn¶moJ AmhoV.

CXmhaU : 0.6666...... hr g§»¶m n[a‘o¶ g§»¶oÀ¶m ñdénmV ì¶º$ H$am.

CH$b : ho AmnU 0.6 + 0.06 + 0.006 + 0.0006 + .......Agohr {bhÿ eH$Vmo.

11 212

n

3 4

4=

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Progressions 45

a = 0.6, r =0.060.6

= 0.1

S =a

1 r-S =

0.61 0.1

=0.60.9

S = 23

23 hr n[a‘o¶ g§»¶m Amho Ooìhm hr 0.6 ¶m Xem§em‘Ü¶o Xe©{dbobr AgVo.


CXmhaU 1 : 3, 6, 12, ....... ¶m G.P À¶m n{hë¶m 6 nXm§Mr ~oarO H$mT>m.

CH$b :¶oWo a = 3, r =2

1

T 6 2T 3 , n = 6

Sn =( 1)

1

na rr

[ r > 1]

S6 =63[2 1]

2 1=

3[64 1]1

= 3 × 63

S6 = 189

CXmhaU 2 : 1 + 4 + 16 + ..... ¶m loUrÀ¶m {H$Vr nXm§Mr ~oarO 1365 hmoB©b?

CH$b : ¶oWo, a = 1, r = 41 = 4, Sn = 1,365, n = ?

Sn =( 1)

1

na rr

[ r > 1]

1365 =1 4 1

4 1

n

=1

3

n

1365 × 3 = 4n – 14 095 = 4n – 1

40 95 + 1 = 4n

40 96 = 4n

46 = 4n

n = 6

CXmhaU 3 : 2 22 + + + ..........3 9 ¶m G.P À¶m nXm§Mr ~oarO n¶ªV H$mT>m..

CH$b : ¶oWo, a = 2, r =2 232

13 2

13

, S = ?

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Progressions 46

S = 1a

r

S = 2113

=2

23

S = 2 32

S = 3

CXmhaU4 : Oa G.P Mo {Vgao nX 12 Am{U Ë¶mMo 6do nX 96 Amho Va 9 nXm§Mr ~oarO H$mT>m.CH$b : T3 = 12 Am{U T6 = 96

T3 = 12ar2 = 12 ......(1) ar5 = 96 ......(2)

(2) bm (1) Zo ^mJ Úm.5

2

arar

=9612

r3 = 8r3 = 23

r 2=

g‘Om, ar2= 12a(2)2 = 12a × 4 = 12 a =

3a

AmVm AmnU S9.H$mTy>.

Sn =( 1)

1

na rr

S9 =93(2 1)

2 1=

3(512 1)1 = 3(511)

9 1533S

CXmhaU 5: G.P À¶m VrZ nXm§Mr ~oarO 31 Am{U Ë¶m§Mm JwUmH$ma 125 Amho. Va Ë¶mg§»¶m H$mT>m.

CH$b : be ,ar

a, ar hr G.P Mr VrZ nXo AmhoV Ago ‘m§Zy.

VrZ nXm§Mm JwUmH$ma = 125

{Q>n :

Oap>q

TT

p

q = rp-q

{Q>n S n =rn+1

Sn

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Progressions 47

ar

a a r = 125

a3 = 125

a = 3 125 a = 5

VrZ nXm§Mr ~oarO = 31

a a arr = 31

5 5 5rr = 31

5 5rr = 31 – 5 Þ

25 5rr

= 26

5 + 5r2 = 26r5r2 – 26r + 5 = 0

5r2 – 25r – r + 5 = 05r(r – 5) – 1(r – 5 = 0

(r – 5)(5r – 1) = 0

5r – 1 = 0 qH$dm r – 5 = 0

r = 15 qH$dm r = 5

Oa a = 5 Am{U r = 15 , Va Ë¶mMr nXo

ar

=5 5

51 15 = 25

a = 5

ar =155

= 1

Oa a = 5 Am{U r = 5 Va Ë¶m§Mr nXo.ar =

51

5= a = 5 ar = 5 × 5 = 25

3 nXo 25, 5, 1 qH$dm 1,5,25, hr AmhoV. ñdmÜ¶m¶ 2.6

1. Imbrb JwUmoÎma loUtMr ~oarO H$mT>m.(i) 1 + 2 + 4 + .... 10ì¶m nXmn¶ªV

(ii)1 1

1 ........3 9

+ + + n¶ªV

1 24

3

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Progressions 48

2. r = 2 Am{U S8 = 510 AgVmZm G.P Mo n{hbo nX H$mT>m.3. G.P Mr ~oarO H$mT>m (i)1 + 2 + 4 + ...... + 512 (ii) 10

1 1 1.........2 4 2

4. 2 + 4 + 8 + ...... ¶m loUrÀ¶m {H$Vr nXm§Mr ~oarO 1022 hmoB©b?

5. emoYm, (i) 5 + 10 + 20 + .......¶m loUrMr S2 : S4

(ii) 4 + 12 + 36 + .....¶m loUrMr S4 : S8

6. G.P H$mT>m Oa, (i) S6 : S3 = 126 : 1Am{U T4 = 125 (ii) S10 : S5 = 33 : 32 Am{U T5 = 64

7. A‘¶m©{XV JwUmoÎma loUrMo n{hbo nX 6 Am{U ~oarO 8 Amho Va G.P. H$mT>m.

8. G.PMr VrZ nXo H$mT>m, Oa Ë¶m§Mr ~oarO Am{U JwUmH$ma AZwH«$‘o (i) 7 Am{U 8

(ii) 21 Am{U 216 (iii) 19 Am{U216 AmhoV.

9. EH$m ì`ŠVr Zo _mJrb dfm©À`m ~MVrÀ`m AYu aŠH$_ àË`oH$ dfu ~MV Ho$br Oa nmM dfm©_Ü`oË`mZo Ho$bobr ~MV 19,375 Amho Va Ë`mZo n{hë`m dfu Ho$bobr ~MV {H$Vr?

A§H$J{UVr ‘Ü¶ (Arithmetic Mean (A.M.))Imbr {Xbobo H$moï>H$ nmhm.

A.P ‘Ü¶ {ZarjU H$am1, 6, 11

1 6 11 186

3 3+ +

= =1 11 12

62 2

+= =

2, 8, 142 8 14 24

83 3

+ += =

2 14 168

2 2+

= =

7, 12, 177 12 17 36

123 3

+ += =

7 17 2412

2 2+

= =

AmnU nm{hbo H$s, Oa VrZ nXo AP ‘Ü¶o AmhoV Va ‘Ybo nX ho BVa XmoZ nXm§Mr gamgar Amho.A{U Ë¶mbm BVa XmoZ g§»¶mMm A§H$J{UVr ‘Ü¶ åhUVmV.

a,A, b ho A.P ‘Ü¶o AmhoV Va A = 2a b+

a, A, b ho A.P ‘Ü¶o AmhoVVa,A – a = b – A [ d = T2 – T1 = T3 – T2]A + A = a + b

2A = a + b

XmoZ g§»¶m§Mm A§H$J{UVr ‘Ü¶ hm XmoZ g§»¶m§À¶m ~oaOoÀ¶m {Zå‘m AgVmo.

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Progressions 49

ñnï>rH$aUmË‘H$ CXmhaUoCXmhaU 1 : 7 Am{U 13 Mm A§H$J{UVr ‘Ü¶ H$mT>m.CH$b : a = 7, b = 13

A.M =7 13

2 2a b+ +

= =202 = 10

AM = 10

CXmhaU 2 : Oa (p – q), x, (p + q) ho A.P.‘Ü¶o AmhoV Va 'x' H$mT>m.CH$b : a = p – q, AM = x, b = p + q

A.M = 2a b+

x =p q- p q+ +

2=

22p

x = p

JwUmH$ma ì¶ñV ‘Ü¶ (H.M) (Harmonic Mean)

Oa VrZ nXo H.P ‘Ü¶o AmhoV, Va ‘Yë¶m nXmbm BVa XmoZ nXm§Mm JwUmH$ma ì¶ñV ‘Ü¶ åhUVmV.Oa a,H,b ho H.P ‘Ü¶o AmhoV.

Va 1 1 1, ,

a H b ho A.P.‘Ü¶o AmhoV.

1 1H a =

1 1Hb [ d = T2 – T1 = T3 – T2]

1 1H H =

1 1a b

+

2H =

b aab+

2ab = H(a + b)

XmoZ g§»¶m§Mm JwUmH$ma ì¶ñV ‘Ü¶ hm Ë¶m§À¶m JwUmH$mamÀ¶m XþßnQ>rbm ~oaOoZo mJ {Xbm AgVm Ë¶m§À¶merg‘mZ AgVmo.

ñnï>rH$aUmË‘H$ CXmhaUoCXmhaU 1 : 1 Am{U 9 ‘Yrb JwUmH$ma ì¶ñV ‘Ü¶ H$mT>m.CH$b : a = 1 Am{U b = 9

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Progressions 50

4M =2ab

a b+ =2 1 9 18

1 9 10HM = 1.8

JwUmoÎma ‘Ü¶ (G.M)

Oa H$moUVrhr VrZ nXo G.P ‘Ü¶o AmhoV, Va ‘Yrb nXmbm BVa XmoZ nXm§Mo JwUmoÎma ‘Ü¶ åhUVmV.CXm : G.P Mean Observe

1, 3, 9 3 1 9 9 3= =4, 8, 16 8 4 16 64 8= =1000, 100, 10 100 1000 10 10000 100´ = =

Oa a, G, b ho G.P,‘Ü¶o AmhoV Va G = a b

Oa a, G, b ho G.P,‘Ü¶o AmhoV.

GGb

a= 2 3

1 2

T TT T

G2 = a × b

G a b

H$moUË¶mhr XmoZ g§»¶mMm JwUmoÎma ‘Ü¶ hm Ë¶m§À¶m JwUmH$mamMo dJ©‘yi AgVo.

ñnï>rH$aUmËm‘H$ CXmhaUoCXmhaU 1:4 Am{U 36 ‘Yrb G.M H$mT>m.CH$b :a = 4, b = 36G = a b = 4 36 144=

G = 12 $

A.M, G.M Am{U H.M ‘Yrb g§~§Y(I) Oa 'a' Am{U 'b' ¶m XmoZ YZ g§»¶m§Mm A, G, H ho AM, GM Am{U HM AmhoV Va A, G H ho

G.P. ‘Ü¶o AmhoV. 'a' Am{U 'b' ¶m XmoZ YZ g§»¶m§gmR>r Amnë¶mbm ‘m{hV Amho H$s,

CXmhaU 2 : Oa 6, x + 2, 54 hr G.P

‘Ü¶o AmhoV Va x H$mT>m.CH$b : a = 6, G = x + 2, b = 54

G = a b x + 2 = =6 54 324x + 2 = 18

x = 18 – 2 = 16x = 16

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Progressions 51

A. M A = 2a b+

G.M G = ab H.M H =2ab

a b+

g‘Om, , A × H =a b+

22

´ab

a b+ = ab

A H= ab

A H = G.

G hm A Am{U H Mm JwUmoÎma ‘Ü¶ Amho.A, G, H ho G.P ‘Ü¶o AmhoV.

CXmhaU 1 : .Oa, a = 2 Am{U b = 32, Va A.M., G.M. Am{U H.M. H$mT>m.

CH$b : A.M =2 32 34

172 2 2

a b+ += = =

G.M = 2 32 64 8ab = = =

H. M =2 2 2 32 128 64

2 32 34 17ab

a b= = =

+ +Vmim : G = A H

G = 176417 = 64

G = 8

CXmhaU 2 : a Am{U b À¶m {Xboë¶m qH$‘Vrbm AM, GM Am{U HM H$mT>m.CH$b : (i) Oa a = 4, b = 16

AM =4 16 20

102 2 2

a b+ += = =

GM = 4 16 64 8ab = = =

HM =2 2 4 16 128

6.44 16 20

aba b

= = =+ +

(ii) a = 3, b = 27

AM =3 27 30

152 2 2

a b+ += = =

GM = 3 27 81 9ab = = =

HM =2 2 3 27 2 3 27

5.43 27 30

aba b

= = =+ +

(iii) a = 2, b = 50

AM =2 50 52

262 2 2

a b+ += = =

GM = 2 50 100 10ab = = =

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Progressions 52

HM =2 2 2 50 2 2 50

3.82 50 52

aba b

= = =+ +

darb H$moï>H$mdéZ AmnU Agm {ZîH$f© H$mTy> eH$Vmo H$s, a Am{U b, ¶m H$moUË¶mhr XmoZ YZg§»¶mgmR>r.

a b A G H4 16 10 8 6.43 27 15 9 5.42 50 26 10 3.8

A > G > H ..... (1)

nmhm,(i) Oa a = 3, b = 3 Va

AM =3 3 6

32 2 2

a b+ += = =

GM = == 3 3 3a b

HM =2 2 3 3 2 3

3 3a b

a b= =

+ +3

63=

(ii) Oa a = 17, b = 17 Va

AM =17 17 34 17

2 2 2a b

GM = 17 17 17a b

HM =2 2 17 17 2 17

17 17ab

a b17

3417

darb CXmhaUm dê$Z Amåhr Agm {ZîH$f© H$mTy> H$s, Oa a = b VaA.M = G.M = H.M .......(2)1 Am{U 2, À¶m {ZH$bmdéZ, Amnë¶mbm ‘m{hV Amho H$s, 'a' A{U'b', ¶m H$moUË¶mhr XmoZ YZ

g§»¶mgmR>r A.M G.M H.M

ñdmÜ¶m¶ 2.7

1. Imbrb g§»¶m§‘Yrb A.M. GM d H.M H$mT>m.

(i) 12 Am{U 30 (ii) 12 Am{U 1

8 (iii)-8 Am{U -42 ( iv) 9 Am{U 18

2. Oa 5, 8, x ho H.P ‘Ü¶o AmhoV Va x H$mT>m.

3. Imbrb nXo A.P AmhoV Va x H$mT>m.

> >

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Progressions 53

(i) 5, (x – 1), 0 (ii) (a + b)2, x, (a – b)2

4. XmoZ g§»¶m§Mm JwUmH$ma 119 Am{U Ë¶m§Mm A§H$J{UVr ‘Ü¶ 12 Amho Va Ë¶m g§»¶m H$mT>m.

5 Oa 2, x,12 ho G.P ‘Ü¶o AmhoV Va x H$mT>m.

6. XmoZ g§»¶m§Mm A§H$J{UVr ‘Ü¶ 17 Am{U Ë¶m§Mm JwUmoÎma ‘Ü¶ 15 Amho Va Ë¶m g§»¶m H$mT>m.

7. XmoZ g§»¶m§Mm A§H$J{UVr ‘Ü¶ 132 Am{U Ë¶m§Mm JwUmoÎma ‘Ü¶ 6 Amho. Va Ë¶m§Mm JwUmoÎma ì¶ñV ‘Ü¶

H$mT>m..

H«$_H«$_

Arithmeticprogression

Progressions

T = a+(n–1)dn

Arithmeticseries

Harmonicprogression

Geometricprogression

11nT

a (n )d

Geometricseries

T = a.rn n–1

nnS = [2a+(n-1)d]2

When r>11

1

n

na(r )

Sr

Finite geometricseries

Infinite GeometricSeries

When r<111

n

na( r )

Sr

11

aS ,whenr

r

AM, GM & HM

AM=a+b2

GM = ba

AM GM HM

HM=2ab+a b

G A H

First n naturalnumbers

12n

n(n )S =

Ooìhm

H«$_

A§H$J{UVr H«$‘ JwUmoÎma H«$‘

A§H$J{UVr loUr JwUmoÎma loUr

‘¶m©{XV JwUmoÎma loUrZ¡g{J©H$ g§»¶m§Mr ~oarOA‘¶m©{XV JwUmoÎmaloUr S

Ooìhm

JwUmH$ma ì`ñV

H«$‘

Ooìhm

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Progressions 54

ñdmÜ¶m¶ 2.1

1. (i), (ii), (iv) ho H«$‘ AmhoV 2. ( ii )19,21 (ii) 6 75 6, (iii) 0.001, 0.0001 (iv) 48,96 3)

.1,3, -7 4) (i) T3 = 23 T10 = 205, 5) (i) n2 - 2n (ii) n2+2n(6) n=14ñdmÜ¶m¶ 2.2

1. (i)-12, 15, -18, - 21. (ii) 2, 5, 1, 7 (iii) a-5b, a-7b, a-9b, a-11b.3 6 6

2) (i) 1,3,5 .........(ii) 6, 11,16 .....3) (i) 48 (ii) - 148 (iii) 17 (iv) 3 (v) 3.4) 23 5) 600 Am{U 700 6) 64

7) - 13, - 8, -3. 8) 10:3 9) df© 2012 10) 5.9,13, 17. qH$dm 17,13,9,5ñdmÜ¶m¶ : 2.3

1. 21 2. (i) 1275 (ii) 493 (iii) 4a2-a3 (vi) P2/2(3-P)

3. 15 4. (a) (i) 210 (ii)455 (iii) 585 (b) (i) 10 (ii) 5 5. n2 6. 4100

7. 3, 10, 17, 24. 8) 6

9. (i) 3, 7, 11 qH$dm 11,7,3 (ii) 9, 12, 15. qH$dm 15,12,9 10. 30, 41, 52, 63, 74, 85.

11. -1100 12. 551

ñdmÜ¶m¶ 2.41. (i), (ii), (iv), (v) 2(i) 1 (ii) -1 3. 1 4. (i) 1 (ii) 1

2n 20 22 10 6

. ñdmÜ¶m¶ 2.51. (i) -1

5(ii) 3 2. (i) (a) (2/3) n-1 (b) 8/27 (ii) 1. 3. 8192

4.(i) T10 = 1/2. T16 = 1/128 (ii) T8 = -1/27, T12 = - 1/2187

(iii) T4 = 1, T8 = 625.

5. (i) 512 (ii) 42n (iii) 243/16 (iv) 1/x28,

6. 2, 1, 1/2, ...... 7. 15.625 J«°‘.

8. 10 do nX 9. r = 2, a = 3 10. (i) 5/8, 5/4, 5/2 .... (ii) 2, 6, 3 2

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Progressions 55

ñdmÜ¶m¶ 2.61. (i) 1023 (ii) 3/2 .2 3. (i) 1023 (ii) 1023/1024

4. 9, 5 (i) 1:5 (ii) 1: 82 6. (i) 1, 5, 25,..... (ii) 1024, 512, 256......

7. 6, 3/2, 3/8, ..... 8. (i) 1, 2, 4. qH$dm 4,2,1 (ii) 3, 6, 12 qH$dm 12,6,3 (iii) 4, 6, 9. qH$dm

9, 6,4 9. < 10,000

ñdmÜ¶m¶ 2.7

1. (i) AM = 21, GM= 6 10 , HM =120

7(ii) AM =

516

, GM =14

, HM =15

(iii) AM = 25, GM = 4 21 , HM =33625 (iv) AM =

272 , GM = 9 2 , HM = 12

2.20 3. (i)72 (ii) a2 + b2 4. 7 Am{U 17 5. 1

6. 9 Am{U 25 7.7213

********

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56

¶m KQ>H$m‘wio Vwåhmbm Imbrb JmoîQ>rMo gwb^rH$aU hmoVo.

* ¶wp³bS>À¶m boå‘mMo ñnï>rH$aU H$aÊ¶mg

* ¶wp³bS>Mm ^mJmH$ma AëJmoarW‘ ( H$m¶m©dbr) dmnéZ

‘.gm.{d. H$mT>Ê¶mgmR>r,

* A§H$J{UVm ‘Yrb ‘wb yV à‘o¶ ì¶º$ H$aÊ¶mgmR>r.

* Cnà‘o¶ {gÜX H$aÊ¶mg,

* 2, 3, 5 ¶m An[a‘o¶ g§»¶m AmhoV ho

{gÜX H$aÊ¶mg,

.

* ¶wp³bS>Mm boå‘m

* A§H$J{UVm ‘Yrb ‘yb yV à‘o¶

* Cnà‘o¶

* g§»¶m§Mm ‘.gm.{d

3

Mathematics is the queen of sciences,and Arithmetic is the queen of Mathematics.

-C.F. Gauss

Abo³Pm§S>r¶mMo ¶y³brS>, 300 B.g. nyd©. Ë¶ m §Z r Abo³Pm §S ´ > r¶m‘Yrb

{díd{dÚmb¶m‘Ü¶o J{UVmMr emim H$mT>br.A‘¶m©{XV AZoH$ ‘yi g§»¶m AmhoV ho {gÜXH$aUmao Vo n{hbo AmhoV.‘.gm.{d. H$mT>Ê¶mgmR>r¶wp³bS>À¶m AëJmoarW‘Mr ¶moOZm Ë¶m§Zr Ho$br.Ë¶m§Zr ’$º$ 5 ßboQ>m°{ZH$ KZ (Platonic Sol-ids) Agë¶mMo {gÜX Ho$bo. ""X E{b‘|Q>'' ¶m13 ^mJ Agboë¶m J«§Wm§À¶m ì¶mnH$ (Com-prehensive) J{UVr nwñVH$mMo boIH$ AmhoV.AZoH$doim Ë¶m§Zm " w{‘VrMo OZH$' åhUyZ g§~mo{YboOmVo.

dmñVd g§»¶m(Real Numbers)

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Real Numbers 57

Vwåhmbm doJdoJù¶m àH$maÀ¶m g§»¶m Am{U Ë¶m‘Yrb g§~§Y ¶m{df¶r ‘m{hVr Amho. Imbrb AmH¥$VrMo {ZarjU H$am.R

QZ

WN

Q

¶m g§»¶mdarb ‘yb^yV {H«$¶m Am{U Ë¶m§Mo ‘hËdnyU© JwUY‘m©{df¶rMm Aä¶mg Vwåhr Ho$bobm Amho.J{UVm‘Yrb Am{U BVa {df¶m‘Yrb AZoH$ g‘ñ¶m gmoS>{dÊ¶mgmR>r ¶m JwUY‘mªMr ‘XV hmoVo.

AmVm AmnU dmñVd g§»¶m, Ë¶m§Mo JwUY‘© Am{U Cn¶moJ ¶m{df¶r A{YH$ Aä¶mg H$é¶m.àW‘, AmnU g§»¶mdarb à{gÜX H$moS>çmMm {dMma H$é,EH$m PmonS>r‘Ü¶o 120 nojm OmñV ‘|T>ao(Sheep) Agy eH$V ZmhrV.Oa XmoZÀ¶m nQ>rV ‘moObr Va EH$ {e„H$ amhVo.Oa VrZÀ¶m nQ>rV ‘moObr Va XmoZ {e„H$ amhVmV.Oa MmaÀ¶m nQ>rV ‘moObr Va VrZ {e„H$ amhVmV.Oa nmMÀ¶m nQ>rV ‘moObr Va Mma {e„H$ amhVmV.Oa ghmÀ¶m nQ>rV ‘moObr Va nmM {e„H$ amhVmV.Oa gmVÀ¶m nQ>rV ‘moObr Va H$m§hr {e„H$ amhmV ZmhrV.Va PmonS>r‘Ü¶o Agboë¶m ‘|T>am§Mr g§»¶m {H$Vr ?nS>Vmim H$aÊ¶mgmR>r g§»¶m§Mm H$moUVm JwUY‘© dmnabm nm{hOo?àW‘ JwUY‘mªMm Aä¶mg H$é¶m Am{U Z§Va H$moS>o gmoS>dy¶m..

nyUmªH$mMr {d^mÁ¶Vm (Divisibility of Integers)Amåhmbm ^mJmH$mamÀ¶m ‘yb yV {H«$¶m§Mr ‘m{hVr Amho.

Amåhr åhUVmo H$s b bm a Zo ^mJ OmVmo Oa H$moUË¶mhr k ¶m Z¡g{J©H$ g§»¶ogmR>r b=KaOoìhm ^mÁ¶mbm ^mOH$mZo ^mJ {Xbm OmVmo, Voìhm Amåhmbm g§JV ^mJmH$ma Am{U ~mH$s {‘iVo ¶m Mma

nXm‘Yrb g§~§Y ^mÁ¶ =(^mOH$ × ^mJmH$ma) + ~mH$s‘a‘ hm ^mÁ¶, ‘b‘ hm ^mOH$,

‘q‘ hm ^mJmH$ma Am{U ‘r‘ hr ~mH$s Amho. Ago g‘Oy.¶m nXm‘Yrb g§~§Y a= (b x q) + r Zo Xe©{dVmV

‘m{hV Agboë¶m ¶m g§~§Ymbm ^y{‘VrÀ¶m OZH$mMo Zm§d XoÊ¶mV Ambo Amho, ¶mbm "¶wp³bS>Mm ^mJmH$maboå‘m" ( EUCLID'S DIVISION LEMMA) Ago åhUVmV.

boå‘m: hm EH$ gmYm {ZH$mb Amho, Á¶mMm ‘w»¶ CÔoe à‘o¶ {gÜX H$aÊ¶mg ‘XV H$aUo hm Amho. boå‘m ho à‘o¶{gÜX H$aÊ¶mÀ`m ‘mJm©Vrb AmYma yV gmYZ Amho. åhUyZ ho {gÜX Ho$bobo {dYmZ Amho Á¶mMm Cn¶moJ BVa{dYmZo {gÜX H$aVmZm hmoVmo.

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Real Numbers 58

'a' Am{U 'b' ¶m YZ nyUmªH$mgmR>r'q' Am{U 'r' ho doJio (A{ÛVr¶) nyUmªH$ Aem [aVrZo AmT>iVmVH$s, a = bq + r, 0 r < b

hm ‘m{hV Agbobm {ZH$mb Amho, Omo ¶wp³bS>À¶m E{b‘|Q> nwñVH$mÀ¶m VII ^mJmV g§J«hrV Amho¶m boå‘m Mo {H$Ë¶oH$ Cn¶moJ AmhoV, XmoZ YZ nyUmªH$mMm ‘.gm.{d. gwb^nUo H$mT>bm OmVmo ¶m nÜXVrbm

¶wp³bS>Mm ^mJmH$ma AëJmo[aW‘ åhUVmV.ñ‘aU H$am H$s, ‘a’ Am{U ‘b’ ¶m XmoZ YZ nyUmªH$mMm ‘.gm.{d. hm ‘moR>çmV ‘moR>m YZnyUmªH$ ‘c’ Amho. Á¶mZo ‘a’ d ‘b’ ¶m XmoÝhtZm ^mJ OmVmo.H$moUË¶mhr XmoZ g§»¶m§À¶m‘.gm.{d bm ‘hÎm‘ gmYmaU {d^mOH$ (GCD) åhUVmV.40A{U 735 ¶m§Mm ‘.gm.{d H$mT>Ê¶mÀ¶m CXmhaUmMm {dMma H$am Amåhr ¶mnydr©MAd¶d nÜXVrZo ‘.gm.{d. H$mT>Ê¶mMm Aä¶mg Ho$bobm AmhoAmVm AmnUm XmoZ g§»`mMm _.gm.{d. wŠbrS>À`m ^mJmH$ma AëJmoar`_Zo H$mTy>

¶m nm¶ar‘Ü¶o Á¶mAµWr© ~mH$s eyÝ¶ Amh|, _gm{d hm nm`ar 4 ^mOH$ Amho.

i,e 40 Am{U 735 ¶m§Mm ‘.gm.{d 5 Amho.

à¶ËZ H$am: 735 Am{U 40 ¶m§À¶m Ad¶dm§Mr ¶mXr H$ê$Z hm {ZH$mb Vwåhr nS>VmiyZ nmhÿ eH$Vm

CXmhaU 2: 513 Am{U 783 ¶m§Mm ‘. gm.{d. H$mT>m.

nm¶ar 1: a= 783 A{U b=513 ¿¶m nm¶ar 2: a= 513 A{U b=270 ¿¶m

wŠbrS>Mm ^mJmH$ma boå_m Imbrb à_mUo Xe©{dVmV

{Q>n:(a,b.) åhUOoa Am{U b Mm_.gm.{d

a=15 A{U b=10 Mm {dMma H$am.

¶oWo, q = 1, r = 5

¶w[³bS>À¶m boå‘mZwgmaa = (b x q) + r15 = (10 x 1) + 5

5 ) 10 ( 2 -10 00

¶oWo, q = 2, r = 0

¶w[³bS>À¶m boå‘mZwgmaa = (b x q) + r10 = (10x 1) + 5

nm¶ar 3 :

10 )15( 1 -10 5

nm¶ar 4: a=10 A{U b=5 ¿¶m

513 ) 783 ( 1

-513

270

¶oWo, q = 1, r = 270,

¶wp³bS>À¶m boå‘mZwgma10 By Euclid's lemma,a = (b × q) + r

40 = (15 × 2) + 10

270 ) 513 ( 1

- 270

243

¶oWo, q = 1, r = 243,


40 = (15 × 2) + 10513 = (270 x 1) + 243

AëJmoarW‘ hr ñnï> ì¶m»¶m Ho$boë¶mnm¶è¶m§Mr loUr Amho, Or CXmhaUogmoS>{dÊ¶mMr nÜXV XoVo.

nm¶ar 1: 40) 735(18 -40 335 -320 15

¶oWo, q = 18, r = 15


15 )40( 2 -30 10

¶oWo, q = 2, r = 10


a=40 Am{U b=15 Mm {dMma H$amnm¶ar 2 :40 Am{U 735 ¶m§Mm {dMma H$am.

783 = (513 x 1)+ 270

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Real Numbers 59

~mH$s eyÝ¶ ¶oB©n¶ªV hr {H«$¶m nwT>o Mmby R>odm. ¶m nm¶ar‘Ü¶o ^mOH$ (eyÝ¶ ahrV~mH$s).hm {Xboë¶m YZ nyUmªH$mMm ‘.gm.{d. Amho.

27 ) 243 ( 9

-243

00

nm¶ar 3: a=270 A{U b=243 ¿¶m nm¶ar 4: a= 243 A{U b=27¿¶m

243 ) 270 ( 1

-243

27

¶oWo, q = 1, r = 27,


40 = (15 × 2) + 10

¶oWo, q = 9, r = 0,

eodQ>Mr ewÝ¶ahrV ~mH$s 27 Amho783 Am{U 513 Mm ‘.gm.{d.

27 Amho270 = (243 x 1) + 27

nm¶ar 1 : 'x ' Am{U 'y ' bm ¶wp³bS>Mm boå‘m bmJyHo$ë¶mZo

x = (y × q) + r, 0 r < y

OoWo q hm mJmH$ma Am{U 'r' hr ~mH$s Amho.

nm¶ar 2 :Oa r = 0, Va y hm x Am{U y Mm ‘.gm.{d. Amho.

Oa r 0, Va 'y ' Am{U 'r ' bm ¶wp³bS>Mm boå‘m bmJy H$am.

nm¶ar 3 :

ñnîQ>rH$aUmË‘H$ CXmhaUo

CXmhaUo:1: ^mJmH$ma AëJ°marW‘À¶m ‘XVrZo 455 Am{U 42 bm ^mJ OmUmar ‘moR¶mV ‘m oR >r g§»¶m H$mT>m.

CH$b :i) 455 ¶m ‘moR²>¶m nyUmªH$mnmgyZ gwadmV H$am

ii 42 35 Mm {dMma H$am. iii 35 7 Mm {dMma H$am.

35)42(135

7

7)35(535

00

42 = (35 × 1) + 7 35 = (7 × 5) + 0

{dMma H$am& XmoZ g§»¶mÀ¶m ‘.gm.{d {df¶r H$m¶KS>Vo Oa* H$moUmVrhr EH$ g§»`m eyÝ` AgVo* XmoÝhr g§»`m eyÝ` AgVmV

42 )455 (10 420 3542)455 (10 quotient

^mJmH$ma

~mH$s

42)455 (10 quotient

gd© gmYmaUnUo, ypŠbS>M ^mJmH$ma AëJmoarW_ dmnê$Z x > y Agboë`m x Am{U y `m XmoZ YZ nyUmªH$mMm_.gm.{d Imbrb nm¶è¶m‘Ü¶o {_i{dVm oVmo.±

{Q>n: Oa ^mOH$ hm ^mÁ¶mMm Ad¶d AgobVa eodQ>Mr ~mH$s eyÝ¶ ¶oVo. eodQ>À¶m nm¶arnydu¶oUmar eyÝ¶ ahrV ~mH$s hr ‘.gm.{d. AgVo.

455 = (42 × 10) + 35 (¶wp³bS>À¶m boå‘mZwgma)

à¶ËZ H$am : ¶m KQH$mÀ¶m gwadmVrbmMMm© Ho$bobo ‘|T>am§À¶m g§»¶oMo H$moS>ogmoS>dm.

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Real Numbers 60

[455,42] Mm ‘.gm.{d = [ 42,35 ] Mm ‘.gm.{d = [ 35, 7] Mm ‘.gm.{d.=7455 Am{U 42 bm ^mJ OmUmar ‘moR>çmV ‘moR>r g§»¶m 7 Amho.

CXmhaU 2: àË¶ oH $ YZ g‘ n yUm ªH $ 2q ñdénmV AgVm o Am{U àË¶oH $ YZ {df‘ n yUm ªH$2q + 1 ñdénmV AgVmo ho XmIdm; OoWo q hr nyU© g§»¶m Amho.

CH$b : (i) a hr g‘ YZ nyUmªH$ Amho Ago ‘m§Zy a Am{U b gmR>r ^mJmH$ma AëJm°arW‘ bmJy H$am, OoW o b=2 Amho.

a = (2 × q) + r OoW o 0 r < 2

a = 2q + r OoW o r = 0 qH$dm r = 1

Á¶mAWu 'a' hm g‘ YZ nyUmªH$, Amho Ë¶mAWu ‘a’. bm 2 Zo ^mJ OmVmor = 0 a = 2q + 0 = 2q

åhUyZ a = 2q Ooìhm ‘a’ hm g‘ YZ nyUmªH$ Amho.ii) 'a' hm {df‘ YZ nyUmªH$ Amho Ago ‘m§Zy. a Am{U b gmR>r ^mJmH$ma AëJm°arW‘ bmJy H$é, OoW o b = 2 Amho.

a = (2 × q) + r OoW o 0 r < 2

a = 2q + r OoW o r = 0 qH$dm r = 1

¶oWo, r 0 ( a hr g‘ Zmhr)r = 1

a = 2q + 1

åhUyZ, a = 2q + 1 Ooìhm ‘a’ hm {df‘ nyUmªH$ Amho.CXmhaU 3$ : H$m§hr 'm' nyUmªH$mgmR>r H$moUË¶mhr YZ nyUmªH$mMm KZ hm EH$ Va 9m, 9m + 1 qH$dm9m + 8 ¶mn¡H$s EH$ AgVmo ho ¶wp³bS>Mm ^mJmH$ma boå‘m dmnéZ XmIdm.CH$b : a Am{U b ho XmoZ YZ nyUmªH$ AmhoV Ago ‘m§Zy Am{U a > b

a = (b × q) + r OoWo q Am{U r YZ nyUmªH$ Am{U 0 r < b

b = 3 ‘m§Zy. ( Oa 9 bm 3 Zo JwUbo Va nyU© YZ g§»¶m {‘iVo.) a = 3q + r OoWo 0 r < 3

(i) Oa r = 0, a = 3q (ii) Oa r = 1, a = 3q + 1 (iii) Oa r = 2, a = 3q + 2

¶m KQ>Zm§À¶m KZm§Mm {dMma H$am.KQ>Zm : (i) a = 3q

a3 = (3q)3 = 27q3 = 9(3q3) = 9m OoW o 'm' hm nyUmªH$ Amho Am{U m=3q3

KQ>Zm : (ii) a = 3q + 1a3 = (3q + 1)3 [(a + b)3 = a3 + b3 + 3a2b + 3ab2]

= 27q3 + 1 + 27q2 + 9q = 27q3 + 27q2 + 9q + 1

= 9(3q3 + 3q2 + q) + 1 = 9m + 1,

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Real Numbers 61

OoW o 'm' hm nyUmªH$ Am{U m=3q3 + 3q2 + q Amho.KQ>Zm : (iii) a = 3q + 2

a3 = (3q + 2)3 = 27q3 + 8 + 54q2 + 36q

= 27q3 + 54q2 + 36q + 8 = 9 (3q3 + 6q2 + 4q) + 8

= 9m + 8

OoW o m hm nyUmªH$ Am{U m = 3q3 + 6q2 + 4q Amho.H$moUË`mhr YZ nyUmªH$mMm KZ hm 'm' nyUmªH$mgmR>r EH$ Va 9m, 9m + 1 qH$dm 9m + 8 `mn¡H$s

EH$ AgVmoCXmhaU 4 : Oa x Am{U y ho YZ nyUmªH$ AmhoV Va x2 + y2 ho g‘ Amho na§Vw 4 Zo ^mJ OmV Zmhr

ho {gÜX H$am.

CH$b : Amåhmbm ‘m{hV Amho H$s, H$m§hr nyUmªH$ q gmR>r, H$moUVmhr {df‘ YZ nyUmªH$ 2q + 1 ñdénmVAmho.

m Am{U n ¶m H$m§hr nyUmªH$mgmR>r x = 2m + 1 Am{U y = 2n + 1 ‘m§Zy.

Amnë¶mH$S>o Amho x2 + y2

x2 + y2 = (2m + 1)2 + (2n + 1)2

x2 + y2 = 4m2 + 1 + 4m + 4n2 + 1 + 4n = 4m2 + 4n2 + 4m + 4n + 2

x2 + y2 = 4(m2 + n2) + 4(m + n) + 2 = 4{(m2 + n2) + (m + n)} + 2

x2 + y2 = 4q + 2, Ooìhm q = (m2 + n2)+(m + n)

x2 + y2 bm 4Zo ^mJ {Xbm AgVm Vr g‘ AgyZ ~mH$s 2 amhVo.x2 + y2 hr g‘ Amho na§Vw 4Zo ^mJ OmV Zmhr.

CXmhaU 5: EH$m nwñVH$ {dH«o$Ë¶mH$S> o 28 H$ÞS> Am{U 72 B§p½be nwñVHo$ AmhoV. nwñVHo$ EH$mMAmH$mamMr AmhoV. hr nwñVHo$ doJdoJù¶m JÇ>çm ‘Ü¶o ~m§Ybr nm{hOoV Am{U àË¶oH$ JÇ`>`mVrbnwñVH$m§Mr g§»¶m g‘mZ Agmdr. Va V¶ma Ho$boë¶m JÇ`m§Mr H$‘rVH$‘r g§»¶m {H$Vr Am{U àË¶oH$JÇ`mVrb n wñVH$m §Mr g§»¶m gwÜXm H$mT >m.CH$b : H$ÞS> nwñVH$m§Mr g§»¶m = 28 B§p½be nwñVH$m§Mr g§»¶m = 72

AmnU ¶wp³bS>À¶m boå‘mZo 28 Am{U 72 Mm ‘.gm.{d. H$mT>bm nm{hOo.72 = (28 × 2) + 16

28 = (16 × 1) + 12

16 = (12 × 1) + 04

12 = (4 × 3) + 0

28 Am{U 72 ¶m§Mm ‘.gm.{d. = 4 àË¶oH$ JÇ`m‘Ü¶o 4 nwñVHo$ AmhoV.

H$ÞS> nwñVH$m§À¶m JÇ`m§Mr g§»¶m =284 = 7

B§p½be nwñVH$m§À¶m JÇ`m§Mr g§»¶m =724

= 18

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Real Numbers 62

ñdmÜ¶m¶ 3.11. Imbrb g§»¶m§Mm ‘.gm.{d. H$mT>Ê¶mgmR>r ¶wp³bS>Mm ^mJmH$ma AëJm°[aW‘ dmnam.

(i) 65 Am{U 117 (ii) 237 Am{U 81 (iii) 55 Am{U 210 (iv) 305 Am{U 793.

2. H$moUVmhr YZ, g‘ nyUmªH$ hm 4q qH$dm 4q + 2 ñdénmV Amho ho XmIdm, OoWo q hr nyU© g§»¶m Amho.

3. ¶wp³bS>Mm ^mJmH$ma boå‘m dmnéZ H$m§hr nyUmªH$ 'm' gmR>r, H$moUË¶mhr YZ nyUmªH$mMm dJ© 3m qH$dm

3m + 1 ¶mn¡H$s EH$m ñdénmV AgVm o, na§Vw 3m+2. ñdénmV ZgVmo. ho XmIdm

4. VrZ H«$‘dma YZ nyUmªH$mÀ¶m JwUmH$mambm 6 Zo ^mJ OmVmo ho XmIdm.

5. 75 Jwbm~ Am{U 45 H$‘i ’w$bo AmhoV hr XmoÝhr ’w$bo Agbobo JwbXñVo (bouquets) V¶ma

H$amd¶mMo AmhoV. gd© ~Z{dboë¶m JwbXñË¶mVrb ’w$bmMr g§»`m gmaIrM Amho Va OmñVrV OmñV ’w$bo

Agboë¶m JwbXñË¶mMr g§»¶m Am{U Ë¶m‘Yrb ’w$bm§Mr g§»¶m H$mT>m.

6. EH$m Am¶VmH¥$Vr eoVmMr bm§~r Am{U é§Xr AZwH«$‘o 110‘r. Am{U 30 ‘r. Amho eoVmMr AJXr ~amo~a

bm§~r é§Xr ‘moOUmè¶m ‘moR>çmV ‘moR>çm girMr (rod) bm§~r H$mT>m.

A§H$J{UVm‘Yrb ‘yb^yV à‘o¶ (Fundamental Theorem of Arithmatic)Amåhmbm ‘mhrV Amho H$s, H$moUVrhr Z¡g{J©H$ g§»¶m hr Ë¶m§À¶m ‘yi Ad¶dmÀ¶m JwUmH$mamV {b{hbr OmVo.

CXmhaUmW© : 2 1 = 3 × 7

8 8 = 2 × 2 × 2 × 11

3 82 5 = 3 × 3 × 5 × 5 × 17

darb {dYmZmMm ì¶Ë¶mggwÜXm gË¶ Amho. Vmo åhUOo,

Oa Amåhr YZ ‘yi g§»¶m§Mm JwUmH$ma Ho$bm Va Amåhmbm

Z¡g{J©H$ g§»¶m qH$dm YZ nyUmªH$ {‘iVmo.

CXmhaUmW© : 3 × 11 = 33

11 × 2 × 2 = 44 3× 3 × 3 × 7= 189

Aem àH$mao, Oa Amåhr gd© e³¶ _yi A{d^mÁ¶ g§»¶m KoVë¶m Va Amnë¶mbm gd© _ywi g§»¶m Am{U gd© ‘wi

g§»¶mMm JwUmH$ma Agboë¶m A‘¶m©{XV g§»¶m§Mm g‘yh {‘iVmo.

¶m g‘yhm‘Ü¶o g§¶wº$ g§»¶m gwÜXm AmhoV H$m?

‘yi g§»¶m§Mm JwUmH$ma Zgboë¶m g§¶wº$ g§»¶m AmhoV H$m?

¶mMo CÎma emoYÊ¶mMm à¶ËZ H$é. g§¶wº$ g§»¶oMm {dMma H$am Am{U Ë¶m§À¶m Ad¶dmMr d¥jmH¥$Vr‘Ü¶o ¶mXr H$am.

‘wi g§»¶m :'P' ¶m YZ nyUmªH$mbm ‘wi g§»`m g‘OVmV Oa i) P>1 Am{Uii) P bm 1 Am{U P gmoS>yZ BVa Ad¶d ZmhrV.

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Real Numbers 63

31,416

2 15,708

2 7,854

2 3,927

3 1,309

7 187

11 17

¶mbm ‘yi Ad¶drH$aU qH$dm A{YH¥$V (canonical) Ad¶drH$aU åhUVmV.

H$m§hr g§¶wº$ g§»¶m ¿¶m Am{U ‘yi Ad¶drH$aU H$am.

Vwåhmbm AmT>iyZ ¶oB©b H$s, àË¶oH$ g§¶wº$ g§»¶m hr Ë¶mÀ¶m ‘yi Ad¶dm§À¶m JwUmH$mamV ì¶º$ H$aVmV. ¶mbm

A§H$J{UVmMm ‘yb^yV à‘o¶ åhUVmV.

A§H$J{UVmÀ¶m ‘yb yV à¶o‘mMo ñnï>rH$aU nwT>rb à‘mUo Amho.

àË¶oH$ g§¶wº$ g§»¶m hr ‘yi Ad¶dm§À¶m JwUmH$mamV ì¶º$ H$aVmV Am{U ¶oUmè¶m ‘yi Ad¶dmMm

H«$‘ gmoSy>Z ho Ad¶drH$aU A{ÛVr¶ (doJio) Amho.

bj Úm H$s, H$moUË¶mhr {Xboë¶m g§¶wº$ g§»¶ogmR>r Ë¶m g§»¶oÀ`m ‘yi Ad¶dmMm JwUmH$ma {b{hÊ¶mMr ’$º$ EH$

Am{U EH$M nÜXV Amho Ë¶mÀ¶m Ad¶dmÀ¶m H«$‘mH$S>o {deof bj {Xbo OmV Zmhr.

{Xboë¶m g§»¶oÀ¶m ‘yi Ad¶dm§Mr ¶mXr H$éZ, Ë¶m§Mm ‘.gm.{d Am{U b.gm.{d. ghOnUo H$mT>bm OmD$eH$Vmo.

ñnîQ >rH$aUmË‘H$ CXmhaUo

CXmhaU 1 :‘yi Ad¶drH$aU nÜXVrZo 18 Am{U 45 ¶m§Mm ‘.gm.{d Am{U b.gm.{d. H$mT >m.CH$b : 18 Am{U 45 ho ‘yi Ad¶dm§À¶m JwUmH$mamV ì¶º$ H$am.

18= 2 × 3 × 3 = 2 × 345= 3 × 3 × 5 = 3 × 5

2

2

18 d 45 Mm ‘.gm.{d = 3 × 3 = 9

18 d 45 Mm b.gm.{d = 3 × 3 × 2 × 5 = 90

ñ‘aU H$am : H$moUË¶mhr XmoZ YZ nyUmªH$mMm ‘.gm.{d. hm g§»¶mÀ¶m g‘mB©H$ ‘yi Ad¶dmÀ¶m bhmZmV bhmZ

g§¶wº$ g§»¶m :Or g§»¶m 1 nojm ‘moR>r Amho Am{U ‘yig§»¶m Zmhr Vr åhUOoM g§¶wº$ g§»¶m

31,416 = 2 × 2 × 2 × 3 × 7 × 11 × 1731,416 = 23 × 3 × 7 × 11 × 17

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Real Numbers 64

KmVm§H$mMm JwUmH$ma AgVmo.H$moUË¶mhr XmoZ YZ, nyUmªH$mMm b.gm.{d. hm g§»¶m§À¶m àË¶oH$ ‘yi Ad¶dm§À¶m ‘moR>çmV ‘moR>m KmVm§H$mMmJwUmH$ma AgVmo.CXmhaU 2: ‘yi Ad¶drH$aU nÜXVrZo åhUOoM A§H$J{UVmÀ¶m ‘yb^yV à‘o¶mZo 42 Am{U 72 Mm‘.gm.{d Am{U b.gm.{d H$mT>m.

CH$b : 42 = 2 × 3 × 7 72 = 2 × 2 × 2 × 3 × 3 = 23 × 32

(42, 72) Mm ‘.gm.{d = 2 × 3 = 6

(42, 72) Mm b.gm.{d = 23 × 32 × 71 = 504

{ZarjU H$am H$s 42 × 72 = 3,024

(42, 72) Mm ‘.gm.{d. × (42, 72) Mm b.gm.{d = 6 × 504 = 3,024

42 × 72 = ‘.gm.{d.(42, 72) × b.gm.{d (42, 72)

a Am{U b, ¶m H$moUË¶mhr XmoZ YZ nyUmªH$mgmR>r

(a, b) Mm ‘.gm.{d × (a, b) Mm b.gm.{d = a × b

Oa ‘.gm.{d H$mT>bobm Agob Va hm g§~§Y XmoZ YZ nyUmªH$mMm b.gm.{d H$mT>Ê¶mgmR>r dmnaVmV.

CXmhaU :3 ‘yi Ad¶drH$aU nÜXVrZo 344 Am{U 60 Mm ‘.gm.{d Am{U b.gm.{d gwÜXm H$mT>m.CH $b : 344 = 2 × 2 × 2 × 43 = 23 × 43

60 = 2 × 2 × 3 × 5 = 22 × 3 × 5 (344, 60)Mm ‘.gm.{d= 22

Amåhmbm ‘mhrV Amho H$s (a, b) Mm ‘.gm.{d × (a, b) Mm b.gm.{d = a × b

(a, b) Mm b.gm.{d =HCF( , )

a ba b

(344, 60) Mm b.gm.{d =344 60

3015

2 2

(344, 60) Mm b.gm.{d = 5160CXmhaU 4 : 150, 187 Am{U 203 bm ^mJë¶m Z§Va AZwH«$‘o 6,7 Am{U 11 ~mH$s amhUmar

‘moR>çmV ‘moR>r YZ nyUmªH$ g§»¶m H$mT>m.CH$b : {Xbobo Amho H$s 150 bm Amdí¶H$ g§»¶oZo ^mJ {Xbm Va ~mH$s 6 amhVo.

150 6 = 144144 bm Amdí¶H$ g§»¶oZo Zo‘H$m (nyU©nUo) ^mJ OmVmo.Amdí¶H$ g§»¶m hr 144 Mm Ad¶d Amho.Ë¶mMà‘mUo, nm{hOo Agbobm YZ nyUmªH$ hm187 7=180 Am{U 203 11 =192. Mm Ad¶d Amho.

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Real Numbers 65

nm{hOo Agbobm YZ nyUmªH$ hm 144, 180 d 192 Mm ‘.gm.{d Amho.

‘yi Ad¶drH$aUmZwgma,

144 = 24 × 32 180 = 22 × 32 × 5 192 = 26 × 3

(144, 180, 192) Mm ‘.gm.{d = 22 × 3 = 12

åhUyZ nm{hOo Agbobm YZ nyUmªH$ 12 Amho.

150, 187 Am{U 203 bm ^mJ {Xë¶m Z§Va AZwH«$‘o 6,7 Am{U 11 ~mH$s amhUmam ‘moR>çmV

‘moR>m YZ nyUmªH$ 12 Amho.CXmhaU 5 : Ooìhm 35, 56 Am{U 91 bm ^mJ {Xbm Va àË¶oH$ KQ>ZoV 7 ~mH$s amhmV

Agob Va Vr bhmZmV bhmZ g§»¶m H$mT> m.

CH$b : Oa 35, 56 Am{U 91 bm g§»¶oZo ^mJ OmV Agob Va Vr g§»¶m ¶m ‘yi g§»¶m§Mm

b.gm.{d Amho.

‘yi Ad¶drH$aUm§Zo b.gm.{d emoYm.

35 = 5 × 7 56 = 23 × 7 91 = 7 × 13

35, 56,Am{U 91 Mm b.gm.{d = 23 × 5 ×7 × 13 = 3,640

35,56 Am{U 91 bm ^mJ OmUmar bhmZmV bhmZ g§»¶m 3,640 Amho.

Á¶mAWu ~mH$s 7 amhVo, Ë¶mAWu nm{hOo Agbobr g§»¶m

3 ,640 + 7 = 3,647 Amho.

CXmhaU 6 : IoimÀ¶m ‘¡XmZm^modVr dVw©imH$ma ‘mJ© Amho. {ebmbm Ë¶m^modVr EH$ ’o$ar Mmb{dÊ¶mgmR>r

36 {‘{ZQ> o bmJVmV Va JrVmbm Ë¶mM H$m‘mgmR>r 32 {‘{ZQ> o bmJVmV. Oa XmoKtZr EH$mM q~XÿVyZ

gwadmV Ho$br Am{U EH$mM doir Am{U EH$mM {XeoZ o, Va {H$Vr {‘{ZQ>mZ §Va Ë¶m n wÝhm EH$Ì

gwadmVrÀ¶m q~X ÿda ^oQ>Vrb?

CH$b : ho emoYÊ¶mgmR>r Amnë¶mbm 36 Am{U 32 Mm b.gm.{d. H$mT>bm nm{hOo. (b.gm.{d. H$m?)

36 = 2 × 2 × 3 × 3 = 22 × 32 32 = 2 × 2 × 2 × 2 × 2 = 25

b.gm.{d. (36, 32) = 25 × 32 = 32 × 9

b.gm.{d. (36, 32) = 288

{ebm Am{U JrVm nwÝhm gwadmVrÀ¶m q~Xÿ‘Ü¶o 288 {‘{ZQ>mZ§Va oQ>VmV.

CXmhaU 7: (7 × 11 × 13 + 13) hr g§¶wº$ g§»¶m Amho ho {dYmZ ñnï> H$am.

CH$b: (7 × 11 × 13) + 13 = 13 [(7 × 11) + 1] = 13[77 + 1] = 13 × 78

hr g§¶wº$ g§»¶m Amho.

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Real Numbers 66

ñdmÜ¶m¶ 3.21. Imbrb àË¶oH$ g§»¶m hr ‘yi Ad¶dm§Mm JwUmH$ma Amho ho ì¶º$ H$am.

(i) 120 (ii) 3825 (iii) 6762 (iv) 32844

2. Oa 25025 = 2 31 41 2 3 4. . .x xx xp p p p Va p1, p2, p3, p4 Am{U x1, x2, x3, x4 Mr qH$‘V H$mT>m.

3. Imbrb nwUmªH$mZm ‘yi g§»¶m§À¶m JwUmH$mamV ì¶º$ H$éZ Ë¶m§Mm ‘.gm.{d. Am{U b.gm.{dH$mT>m.(i) 12, 15 Am{U 30 (ii)18, 81 Am{U 108

4. nyUmªH$mÀ¶m OmoS>rMm ‘.gm.{d. Am{U b.gm.{d H$mT>m Am{U (a, b) b.gm.{d.× (a, b) Mm‘.gm.{d.= a × b ho nS>VmiyZ nmhm.(i) 16 Am{U 80 (ii) 125 Am{U 55

5. Oa 52 Am{U 182 Mm ‘.gm.{d 26 Amho Va Ë¶m§Mm b.gm.{d H$mT>m.6. ‘yi Ad¶drH$aU nÜXVrZo 105 Am{U 1515 ¶m§Mm ‘.gm.{d. H$mT>m Am{U Ë¶m§Mm b.gm.{d

gwÜXm H$mT>m.7. Oa bhmZmV bhmZ g§»¶m 17 Zo dmT>{dbr Va {Vbm 520 Am{U 468 ¶m XmoÝhtZr nyU©nUo ^mJ

OmVmo Va Vr g§»¶m H$mT>m.8. EH$m Am¶VmH$ma ImobrMr bm§~r 18‘r. 72 g|.‘r. Am{U é§Xr 13 ‘r. 20 g|.‘r. Amho. Ë¶mbm

gma»¶mM AmH$mamÀ¶m Mm¡agmH¥$Vr ’$aem ~gdm¶À¶m AmhoV Va e³¶ Agboë¶m bhmZmV bhmZ’$aem§Mr g§»¶m H$mT>m.

9. emioVrb 8dr, 9dr Am{U 10drÀ¶m dJm©Vrb {dÚmÏ¶mªMr g§»¶m AZwH«$‘o 48, 42 Am{U 60Amho. 8dr, 9dr Am{U 10dr À¶m {dÚmÏ¶mªZm g‘mZarË¶m nwñVHo$ dmQ>md¶mMr AmhoV. Va hdrAgboë¶m nwñVH$m§Mr H$‘rV H$‘r g§»¶m H$mT>m.

10 x, y Am{U z Zr EH$mM doir EH$mM {XeoZo dVw©imH$ma {H«$S>m§JUm^modVr Ymdm¶bm gwadmV Ho$br.gdmªZr EH$mM q~XÿVyZ gwadmV H$éZ x Zo 126 goH§$Xm‘Ü¶o, y Zo 154 goH§$Xm‘Ü¶o Am{U z Zo 231goH§$Xm‘Ü¶o EH$ ’o$ar nyU© Ho$br. {H$Vr doimZ§Va Vo gd©OU gwadmVrÀ¶m q~XÿV ^oQ>Vrb? Ë¶mMdoioV x,y Am{U z Zr {H$Vr ’o$è¶m nyU© Ho$ë¶m AgVrb?

An[a‘o¶ g§»¶m [ Irrational Numbers]Vwåhmbm IX dr À¶m dJm©‘Ü¶o An[a‘o¶ g§»¶mMm n[aM¶ Pmbm Amho.ñ‘aU H$am H$s, g§»¶m§Zm An[a‘o¶ g§»¶m Ago åhUVmV, Oa Ë¶m

pq ñdénmV {bhrVm ¶oV ZmhrV, OoW o

p Am{U q nyUmªH$ AmhoV Am{U q 0

CXm : 2, 3 , BË¶mXr.AmVm 2, 3, 5 An[a‘o¶ AmhoV ho {gÜX H$é, åhUOoM gd© gmYmaUnUo p hr An[a‘o¶Amho, OoW o p hr ‘yi g§»¶m Amho.¶m g§»¶m§Mo An[a‘o¶Ëd Amåhmbm {gÜX H$aÊ¶mnydu, Amnë¶mbm Aä¶mg H$éZ à‘o¶ {gÜX H$aUoJaOoMo Amho H$maU Ë¶mMr {gÜXVm A§H$J{UVmÀ¶m ‘yb yV à‘o¶mda AmYmarV Amho

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Real Numbers 67

à‘o¶ : Oa ‘yi g§»¶m p Z o a2 ^mJ OmVmo, Va a bm p Zo ^mJ OmVmo, OoW o 'a' hm YZ nyUmªH$

Amh o.

{gÜXVm : a = p1 × p2 × ........pn ‘m§Zy OoWo p1, p2.....pn ho ‘a’ Mo ‘yi Ad¶d AmhoV

XmoÝhr ~mOy§Mm dJ© H$éZ

a2= (p1 × p2 × ......× pn)2

na§Vw, p Zo a2 bm ^mJ OmVmo [ J¥hrV]

p hm p12 × p2

2 × .......pn2 Mm Ad¶d Amho.

A§H$J{UVmÀ¶m ‘yb yV à‘o¶mZwgma

p12 × p2

2 × p32 × .........×pn

2 À¶m Ad¶drH$aUm‘Ü¶o ‘yi ho A{ÛVr¶ (doJio) Amho.

p1, p2 .......pn‘Yrb p hm EH$ ‘yi Ad¶d Amho.Oa p = pk , k À¶m 1 nmgyZ n n¶ªVÀ¶m H$m§hr qH$‘VrgmR>r.

p1 × p2 × p3 × .....pn bm pk ^mJ OmVmo.

pk hm a bm ^mJVmo.p hm a bm ^mJVmo [ pk = p]

AmVm 2 hr An[a‘o¶ g§»¶m Amho ho {gÜX H$é.

{gÜXVm : 2 hr n[a‘o¶ g§»¶m Amho Ago g‘Oy.VoWo p Am{U q ho nyUmªH$ AemàH$mao AmT>iVmV H$s,

2 = pq OoW o, p Am{U q ho EH$‘oH$mMo gh-‘wi AmhoV.

2 q = p

XmoÝhr ~mOy§Mm dJ© H$é, 2q2 = p2

2 ho p2 bm ^mJVo (^mJ OmVmo)

2 Zo p2 bm ^mJ OmVmo (2 hm p2 bm ^mJVm o p hm g‘ Amho.

p = 2k ‘m§Zy OoW o k hm nyUmªH$ Amho.AmVm, 2q2 = (2k)2

2q2 = 4k2

q2 = 2k2 q hm g‘ Amho.

p Am{U q ho XmoÝhr g‘ AmhoV.p Am{U q Mm 2 hm gm‘mB©H$ Ad¶d Amho, ho Am‘À¶m J¥hrVmÀ¶m {déÜX Amho. åhUOoM

p Am{U q ho gh-A{d^mÁ¶ AmhoV.

Am‘Mo J¥hrV 2 hr n[a‘o¶ g§»¶m Amho MyH$ Amho.

2 hr An[a‘o¶ g§»¶m Amho.

3 hr An[a‘o¶ Agë¶mMm

{gÜX H$aÊ¶mMm à¶ËZH$am

gô - A{d^mÁ¶

‘ a‘ Am{U ‘ b‘ ¶m XmoZ g§»¶m gh A{d^mÁ¶ AmhoV. Oa 'a' A{U "b‘ Mm ’$³V gm‘mB©H$^mOH$ 1 Amho

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Real Numbers 68

Cn{gÜXm§V 1: Oa 'a' h o bc bm ^mJVo Am{U (a, b) Mm ‘.gm.{d = 1 Va a h o c bm ^mJVo.§{gÜXVm : a ho bc bm 'k' doim (k Z ) ^mJVo Ago ‘m§Zy. ak= bc H$m§hr x, y Z,

Cn{gÜXm§V 2 : Oa XmoZ nyUmªH$mbm JwUmH$mamZo A{d^mÁ¶Zo (‘yi) ^mJ OmVmo. Va{Vbm H$‘rVH$‘r EH$mZo Var ^mJ OmVm o (^mJVm o)

CXm.: Oa p hr A{d^mÁ¶ Am{U p/ab Va p/a qH$dm p/b .

ñnï>rH$aUmË‘H$ CXmhaU o

CXmhaU 1. 5 3- hr An[a‘o¶ g§»¶m Amho ho {gÜX H$am

CH$b . 5 3 hr n[a‘o¶ g§»¶m Amho Ago g‘Oy.

5 3 = pq , OoWoo p, q z, a = 0

5 – pq = 3 5q p

q = 3

3 hr nar‘o¶ g§»¶m Amh o5q p

q hr nar‘o Amho

na§Vw 3 hr n[a‘o¶ g§»¶m Zmhr. ho Amnë¶mbm {damoYm^mg Xe©{dVo.Amnbr g‘OyV 5 3 hr n[a‘o¶ g§»¶m Amho ho MyH$ Amho.5 3 hr An[a‘o¶ g§»¶m Amho.

CXmhaU 2. 3 2 hr An[a‘o¶ g§»¶m Amho ho {gÜX H$am.CHCH$b : 3 2 hr n[a‘o¶ g§»¶m Amho Ago g‘Oy.

3 2 =pq , OoWo p, q, z , Am{U q 0

3 = 2pq

XmoÝhr ~mOy§Mm dJ© H$é, 23 =

2

2pq

3 =2

2 2. 2. 2p pqq

2 2.pq =

2

2 2 3pq

2 2.pq =

2

2 1pq

{Q>np/ab ho ab bm p Zo ^mJ OmVmo.p/a a bm p Zo ^mJ OmVmo.p/b b bm p Zo ^mJ OmVmo.p/a a bm p Zo ^mJ OmV Zmhr AgodmMVmV.

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Real Numbers 69

2 hr n[a‘o¶ g§»¶m Amh o2 2

2p q

pq hr n[a‘o¶ Amho.

na§V w 2 hr n[a‘o¶ g§»¶m Zmhr. ¶oWo Amnë¶mbm {damoYm^mg XoVo

Amnbr g‘OyV 3 2 hr n[a‘o¶ g§»¶m Amho MyH$sMo Amho.

3 2 hr An[a‘o¶ g§»¶m Amho.

ñdmÜ¶m¶ 3.3

1. 5 hr An[a‘o¶ g§»¶m Amho ho {gÜX H$am.

2. Imbrb g§»¶m An[a‘o¶ g§»¶m AmhoV ho {gÜX H$am.

(i) 2 3 (ii)74

(iii) 3 5+ (iv) 2 5+ (v) 2 3 4-

2 2pq =

2 2

2

1p qq

2 =2 21

2p q q

q p

2 =2 212

p qpq

(vi) 52 + (v) 3-42

An[a‘o¶ g§»¶m

A§H$J{UVmMm ‘yb yV à‘o¶dmnê$Z ‘.gm. {d Am {Ub.gm.{d. H$mT>Uo

¶wp³bS>Mm ^mJmH$ma AëJmo[aW_

a = (b × q) + r, o< r < b

A§H$J{UVmMo ‘yb^yVà‘o¶

à‘o¶: Oa p hr A{d^mÁ¶ g§»¶mAmho Am{U Am{U Amho

¶yp³bS>Mm ^mJmH$ma AëJom[aW‘dmnê$Z XmoZ nyU©H$mMm ‘.gm.{dH$mT>Uo

dmñVd g§»¶m

( 2 , 3 ............)H$m¶m©dbr

p/a2 p/a

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Real Numbers 70

ñdmÜ¶m¶ 3.1 1 (i) 13 (ii) 3 (iii) 5 (iv) 61 6. JwbXñË¶m§Mr g§»¶m = 15, àË¶oH$ JwbXñË¶m‘Yrb µ’w$bm§Mr g§»¶m = (5+3 ) =8 7. 10 ‘r ñdmÜ¶m¶ 3.2 1. (i) 23x 3 x 5 (ii) 3

2x 5

2x 17

(iii) 2 x 3x 72x 23 (iv) 2

2x 3 x 7 x 17 x 23

2. P1=5, P2=7, P3=11, P4=13, x1=2, x2=1, x3=1, x4=1

3. (i) ‘.gm.{d =3, b.gm.{d = 60, ‘.gm.{d =9, b.gm.{d = 324

5. 364 6. 15 ; 10, 605

7. 4,663 8. 4,290

9. 1,680

10. 1,386 go§H$XmZ§Va x 11 ’o$è`m, y 9 ’oè¶m, Z 6 ’o$è¶m

*****

CÎma o

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¶m KQ>H$m‘wio Vwåhmbm ¶m §Mo gwb^rH$aU hmoVo.

* ‘moOÊ¶mMo ‘yb yV VËd ñnï> H$aUo.

* H«$‘doe Am{U g§doemMr ì¶m»¶m H$aUo.

* H«$‘M¶ {MÝh dmnaUo.

* H«$‘doem§À¶m g§»¶oMo gyÌ aMU o (nPr).* g§doemÀ¶m (nCr) g§»¶oMo gyÌ aMUo.

* H«$‘doe Am{U g§doem‘Yrb ’$aH$ H$mT>Uo.

* nPr Am{U nCr.dmnéZ CXmhaUo gmoS>{dUo.

* ‘yb yV ‘moOÊ¶mMo VËd.

* H«$‘doe.

* H«$‘M¶ {MÝh.

*!

!n

rnP

n r

* g§doe

*

4

OH$m°~ ~aZm¡ëbr(1654 - 1705 A. D.)

OH$m°~ ~aZm¡ëbr ho ñdrg J{UVVk

AmhoV. Ë¶m§Zr Ag© H§$O|³Q>§³S>r

(’$m°ñWw‘m¡Obr 1713 A. D ‘Ü¶o à{gÜX)

¶m Amnë¶m nwñVH$m‘Ü¶o g§doe Am{U

H«$‘doe darb g§nyU© dV©dUyH$ {Xbr.

In mathematics the art of proposing a questionmust be held of higher value than solving it.

- George Cantor

H«$‘doe Am{U g§doePermutations and Combinations

!! !

nr

nCr n r! !r n r r

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72 UNIT-4

¶m KQ>H$mMr gwadmV OrdZmVrb gË¶ àg§JmdéZ H$é Aä¶mgm Am{U JQ> H$éZ MMm© H$am.ñnï>rH$aU 1 :AmnU añË¶mdéZ AZoH$ àH$maMr dmhZo OmVmZm nmhVmo. àË¶oH$ dmhZmda Zm|XUr g§»¶m Agbobr Z§~a ßboQ>AgVo.AmR>dU H$am, H$s àË¶oH$ Z§~a ßboQ>da Imbrb ‘m{hVr AgVo.

1) amÁ¶mMo Zm§d (CXm : KA - H$Zm©Q>H$mgmR>r)2) àmXo{eH$ Q´>m§gnmoQ>© Am°{’$g g§Ho$V g§»¶m.

(05 - ~|Jbmoa - O¶ZJa RTO)3) B§J«Or ‘wimjao.4) g§»¶m AgVo (1 A§H$s Vo 4 A§H$s g§»¶m)

Vwåhmbm ‘m{hV Amho H$s, EH$m loUr‘Ü¶o {H$Vr dmhZm§Mr Zm|X Ho$br OmVo.? g‘Oë¶mZ§Va Vwåhr M{H$Vìhmb H$s 9999 dmhZm§Mr Zmo§X Ho$br OmVo.

Vwåhr ¶mMm H$Yr {dMma Ho$bmV H$m? H$s ¶m g§»¶m R.T.O H$Sy>Z H$em {Xë¶m OmVmV? ¶m JUVr‘Ü¶oJ{UVmMo H$moUVo VËd Amho?ñnï>rH$aU 2 :g‘Om H$s Amnë¶mH$S>o g§»¶m§Mm Hw$byn Agbobr gwQ>Ho$g Amho. g§»¶m Hw$bwnmda VrZ MmHo$ AgyZ àË¶oH$MmH$mda 0 nmgyZ 9 Aem 10 A§H$m§Mr Zm§do AgVmV. Hw$bwn CKS>Vmo, Oa VrZ {deof A§H$ R>am{dH$ ‘m§S>UrVaMbo Va qH$dm nwÝhm-Z KoVm H«$‘mV KoVbo Va. g‘Om Vwåhr g§»¶m§Mm H«$‘ {dgabobm Amho Ago ‘m§Zy. hmHw$bwn CKS>m¶Mm Agob Va {H$Vr VrZ A§H$s g§»¶m§Mo H«$‘ Vwåhr Vnmgmdo? ¶m àíZmMo CÎma XoÊ¶mgmR>rAmåhmbm 3 A§H$s g§»¶m§Mr gd© ¶mXr H$amdr bmJob. na§Vw ¶mgmR>r ^anya doi bmJVmo. ¶mH$arVm H$moUVrhrJ{UVr ¶wº$s Amho H$m Or e³¶ Agboë¶m g§»¶m§Mr ‘m{hVr {‘i{dÊ¶mg ‘XV H$aVo.

¶m KQ>H$mV AmnU H$mhr ‘yb yV ‘moOÊ¶mMo V§Ì Aä¶mgy Oo Amnë¶mbm darb àg§JmVrb CXmhaUm§MrCH$b H$mT>Ê¶mg ‘XV H$aVmV.

n{hbr nm¶ar åhUOo Amåhr VËdm§Mo H$miOrnyd©H$ narjU Ho$bo nm{hOo Oo AË¶§V ‘yb^yV Aem‘moOÊ¶mÀ¶m Aä¶mgmMo V§Ì Amho. ‘moOUo hm ¶m KQ>H$mMm ‘w»¶ {df¶ Amho. dñVy§Mm gQ> {Xbobm Amho. H$m§hr{deofVo:À¶m AmYmao CngQ>mV ‘m§S>Uo qH$dm H$m§hr {deofVoÀ¶m AmYmao CngQ>m§Mr {ZdS> H$aÊ¶mMo H$m¶© {Xbo Amho?‘yb^yV ‘moOÊ¶mMo VËd : (Fundamental Principle of Counting)Imbrb ñnï>rH$aUo Aä¶mgm.CXmhaU 1 : AmH¥$VrV XmI{dë¶mà‘mUo VrZ ~mhþë¶m AmhoV.

Ë¶mn¡H$s XmoZ aH$mÝ¶m‘Ü¶o R>odm¶À¶m AmhoV.Va {H$Vr àH$mao Ë¶m§Mr ‘m§S>Ur Ho$br OmD$ eH$Vo? ho AmnU H$go ‘moOy eH$Vmo?n{hbr Am{U Xþgar ~mhþbtMr {ZdS> H$aÊ¶mMr d¥jmH¥$Vr Xe©{dbobr Amho Vrnmhm.

KA05EV4007

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Permutations and Combinations 73

¶m AmH¥$VrdéZ AmnU Agm {ZîH$f© H$mTy> H$s, n{hë¶m ~mhþbrMr {ZdS> 3 doJdoJù¶m àH$mao Am{U àË¶oH$n{hë¶m {ZdS>rbm Xþgar ~mhþbr XmoZ doJdoJù¶m àH$mao {ZdS>br OmVo.Ë¶m‘yio EHy$U 6 àH$ma AmhoV

VrZ n¡H$s XmoZ ~mhþë¶m§À¶m ‘m§S>UrÀ¶m EHy$U àH$mam§Mr g§»¶m 3 × 2 =6 àH$mao ho 6 doJdoJio àH$maImbrbà‘mUo Xe©{dbobo AmhoV.

CXmhaU 2 : EH$m ‘wbmH$S>o 2 n±Q> Am{U 4 eQ>© AmhoV. Vmo n±Q> Am{U eQ>© ¶m§À¶m OmoS>çm {H$Vr àH$mao n[aYmZ(Kmby) H$é eHo$b?CH$b : n±QMr {ZdS> H$aÊ¶mMo XmoZ àH$ma AmhoV Am{U ¶m àË¶oH$ {ZdS>rgmR>r eQ>© {ZdS>Ê¶mMo Mma àH$ma AmhoV.XmoZ n±Q> P1 , P2 Am{U 4 eQ>© S1, S2, S3 , S4 Ago Xe©{dbobo AmhoV. n±Q> Am{U eQ>© ¶m§À¶m OmoS>çmH$aÊ¶mMo doJdoJio àH$ma d¥jmH¥$Vr‘Ü¶o Imbrb à‘mUo XmI{dbobo AmhoV.

eQ>©²g

Pants

P1

P2

S ––– P S1 1 1

S ––– P S2 1 2

S ––– P S3 1 3S ––– P S4 1 4

S ––– P S1 2 1

S ––– P S2 2 2

S ––– P S3 2 3S ––– P S4 2 4VoWo EHy$U 8 àH$ma AmhoV.

Xþgè¶m eãXmV 2 × 4 = 8 àH$mao n±Q> Am{U eQ>©Mr OmoS>r H$é eH$Vmo.CXmhaU 3 : g§O¶ hm°Q>ob‘Ü¶o Zmï>m H$aÊ¶mgmR>r OmVmo.nXmWm©À¶m ¶mXr‘Ü¶o Imbrb nXmW© Agë¶mMo Xe©{dbo Amho.

Zmï>m JmoS> Ja‘ no¶BS>br (T1) {eam (S1) H$m°’$s (H1)

Xmogm (T2) Om‘yZ (S2) Mhm (H1)

nwar (T3) ~Xm‘ XþY (H1)

nwbmd (T4)àË¶oH$ àH$maMm EH$ nXmW© {ZdS>Ê¶mMo EHy$U {H$Vr àH$ma AmhoV?Zmï>m 4 àH$mao {ZdS>bm OmD$ eH$Vmo. ZmîQçmZ§Va JmoS>nXmW© 2 doJdoJù¶m àH$mao {ZdS>bo OmD$ eH$VmV.

4 × 2 = 8 àH$mao Zmï>m Am{U JmoS> nXmW© {ZdS>bo OmD$ eH$VmV.darb àË¶oH$ 8 àH$mamgmR>r Ja‘ no¶ 3 doJdoJù¶m àH$mao {ZdS>bo OmD$ eH$Vo.

8 × 3 = 24 àH$mao g§O¶ Amnbo 3 àH$ma {ZdSy> eH$Vmo.EHy$U àH$mam§Mr g§»¶m = 4 × 2 × 3 = 24

n±Q²>g

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74 UNIT-4

Oa AmnU Zmï>m T1, T2, T3 Am{U T4, d JmoS> S1, S2, Am{U Ja‘ no¶ H1, H2 Am{U H3, Ago Xe©{dboVa 24 e³¶ Agbobo àH$ma Imbrb H$moï>H$mV XmI{dbo AmhoV.

T1 S1 H1 T2 S1 H1 T3 S1 H1 T4 S1 H1

T1 S2 H1 T2 S2 H1 T3 S2 H1 T4 S2 H1

T1 S1 H2 T2 S1 H2 T3 S1 H2 T4 S1 H2

T1 S2 H2 T2 S2 H2 T3 S2 H2 T4 S2 H2

T1 S1 H3 T2 S1 H3 T3 S1 H3 T4 S1 H3

T1 S2 H3 T2 S2 H3 T3 S2 H3 T4 S2 H3

darb VrZ CXmhaUmVrb ‘m{hVr Imbrb H$moï>H$mV ^abobr Amho. H$moîQ>H$ Aä¶mgm Am{U JQ>m‘Ü¶o MMm©H$éZ ‘moOÊ¶mMo VËd gwMdm. CXmhaU àH$mam§Mr g§»¶m EHy$U àH$mam§Mr g§»¶m

H¥$Vr1 H¥$Vr 2 H¥$Vr 31 3 2 -- 3 × 2 = 62 4 2 – 4 × 2 = 83 4 2 3 4 × 2 × 3 = 24

AmnU {ZîH$f© H$mTy> Am{U ‘moOÊ¶mMo ‘yb yV VËd (FPC) qH$dm JwUmH$mamMo VËd Imbrb à‘mUo ñnï>H$é eH$Vmo.

Oa EImXr H¥$Vr 'm' doJdooJù¶m àH$mao Ho$br OmVo Am{U n{hë¶m H¥$VrÀ¶m àË¶oH$ g§JV àH$mamer Xþgar H¥$Vr

(n{hë¶m H¥$VrnmgyZ ñdV§Ì) 'n' doJdoJù¶m àH$mao Ho$br OmD$ eH$Vo Va XmoÝhr H¥$Ë¶m EH$mZ§Va EH$ ¶mà‘mUo

(m × n) àH$mam§À¶m g§»¶oZo Ho$ë¶m OmVmV.darb VËd H$moUË¶mhr ‘¶m©{XV g§»¶oÀ¶m KQ>ZogmR>r aMbo OmD$ eH$Vo.

VrZ H¥$VrgmR>r FPC Mo ñnï>rH$aU Ago Ho$bo OmD$ eH$Vo.Oa EH$ H¥$Vr 'm' g§»¶oZo doJdoJù¶m àH$mao Ho$br OmVo. 'm' À¶m àË¶oH$ doJdoJù¶m àH$mamgmR>r Xþgar

H¥$Vr 'n' g§»¶oZo doJdoJù¶m àH$mao Ho$br OmVo Am{U ¶m àË¶oH$ H¥$VrgmR>r {Vgar H¥$Vr 'p' àH$mao Ho$br OmVo Vadarb {VÝhr H¥$Vr EH$mZ§Va EH$ ¶m à‘mUo (m × n × p) àH$mao Ho$ë¶m OmVmV.

AmVm AmnU ghOnUo XmoZ nojm OmñV Ho$boë¶m H¥$VrMr, ¶mXr, ‘moOUo qH$dm d¥jmH¥$Vr Z H$mT>Vm EHy$UàH$mam§Mr g§»¶m H$mTy> eH$Vmo. ¶m FPC Mo H$m§hr CXmhaUo KodyZ ñnï>rH$aU H$é¶m.

CXmhaU 1 : {1, 2, 3, 4, 5} ho A§H$ nwÝhm-nwÝhm Z KoVm {H$Vr XmoZ-A§H$s g§»¶m V¶ma hmoVmV?CH$b : {ZdS>rÀ¶m gQ>mVrb A§H$ {1, 2, 3, 4, 5} ho AmhoV.

Á¶mAWu Amnë¶mbm XmoZ A§H$s g§»¶m V¶ma H$am¶Mr Amho Voìhm XmoZ noQ>çm H$mT>m EH$m‘Ü¶o gwQ>o ñWmZAm{U Xþgè¶m‘Ü¶o XeH$ ñWmZ ¿¶m.

AmnU àW‘ EH$‘ ñWmZ Am{U Z§Va Xe‘ ñWmZ ^é.


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1, 2, 3, 4 ho A§H$ dmnéZ EH$‘ ñWmZ 5 doJdoJù¶m àH$mao noQ>rV (box) âbo

OmVo..

EH$‘ ñWmZ âë¶mZ§Va Amnë¶mH$S>o 4 A§H$ {e„H$ amhVmV. A§H$ nwÝhm-nwÝhm KoVbo

OmV ZmhrV Ë¶m‘wio Xe‘ ñWmZm Mr noQ>r 4 doJdoJù¶m àH$ma o âbr OmVo.

‘yb yV ‘moOÊ¶mMo VËd dmnéZ 5 × 4 = 20 àH$ma {‘iVmV.

1, 2, 3, 4, 5 ho A§H$ nwÝhm-nwÝhm Z KoVm 2 A§H$s g§»¶m 20 àH$mao V¶ma hmoVmV.

Oa 1, 2, 3, 4, 5 ho A§H$ nwÝhm-nwÝhm KoVbo Va Xe‘ ñWmZ gwÜXm 5 doJdoJù¶m àH$mao âbo OmVo.

1,2,3,4,5 ho A§H$ nwÝhm-nwÝhm KodyZ 2 - A§H$s g§»¶m V¶ma H$aÊ¶mÀ¶m EHy$U àH$mam§Mr g§»¶m =

5 × 5 = 25 àH$mao.

{Q>n : JQ> H$éZ MMm© H$am Am{U n«Ë¶jnUo 2 A§H$s g§»¶mMr ¶mXr H$éZ VnmgyZ nmhm.

CXmhaU 2 nwÝhm-nwÝhm Z KoVm 5 ñda dmnéZ {H$Vr 3- Ajam§Mo g§Ho$V V¶ma hmoVmV?

CH$b : 5 ñda a, e, i, o Am{U u ho AmhoV.

Amnë¶mbm VrZ Ajam§Mm g§Ho$V V¶ma H$am¶Mm Amho n{hbo-Aja, 5 àH$mao {ZdS>Vm ¶oVo. Xþgao Aja 4

àH$mao {ZdS>Vm ¶oVo (nwÝhm -nwÝhm Z KoVm) {Vgao Aja 3 àH$mao {ZdS>Vm ¶oVo (nwÝhm nwÝhm Z KoVm) 3 Ajam§Mo

g§Ho$V V¶ma H$aÊ¶mÀ¶m EHy$U àH$mam§Mr g§»¶m = 5 × 4 × 3 = 60 àH$mao.

CXmhaU 3 : 0, 1, 2, 3 Am{U 4 ho A§H$ nwÝhm Z KoVm {H$Vr 3 A§H$s g§»¶m V¶ma H$aVm ¶oVmV?

CH$b : 3-A§H$s g§»¶o‘Ü¶o VrZ ñWmZo AmhoV-eV‘², Xe‘² Am{U EH$‘² {ZdS>rÀ¶m gQ>mVrb A§H$

{0, 1, 2, 3, 4} ho AmhoV.

eV‘² ñWmZ : eyÝ¶ ho eV‘² ñWmZr ¶oV Zmhr H$maU Ë¶m‘wio 2 A§H$s g§»¶m hmoVo. åhUyZ 1, 2, 3, 4.¶m A§H$mZr eV‘² ñWmZ 4 àH$mao âbo OmVo.

Xe‘² ñWmZ : nwÝhm-nwÝhm KoD$Z Am{U eyÝ¶ dmnéZ Xe‘² ñWmZ 5 àH$mao âbo OmVo.

EH$‘² ñWmZ : ho gd© A§H$ dmnéZ 5 àH$mao âbo OmVo.

0, 1, 2, 3, 4, 5 ho A§H$ dmnéZ 3-A§H$s g§»¶m V¶ma H$aÊ¶mÀ¶m EHy$U àH$mam§Mr g§»¶m =4 ×5 × 5 = 100 àH$mao.

CXmhaU 4 : ‘yb yV ‘moOÊ¶mMo V§Ì dmnéZ gwQ>Ho$gMo CXmhaU (ñnï>rH$aU

2 nmZ 2 darb) gmoS>{dÊ¶mMm à¶ËZ H$é.

CH$b : gwQ>Ho$gÀ¶m g§»¶m Hw$bynm‘Ü¶o VrZ A§H$ AmhoV.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ho A§H$ dmnéZ àË¶oH$

g§»¶m 10 àH$mao V¶ma Ho$br OmVo.

6 2 07 3 18 4 2

T U

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76 UNIT-4

¶oWo A§H$ nwÝhm nwÝhm gwÜXm KoVbo OmD$ eH$VmV.

g§»¶m Hw$bwnmMm g§Ho$V V¶ma H$aÊ¶mÀ¶m EHy$U àH$mam§Mr g§»¶m 10 × 10 × 10 = 1000 àH$mao.Ë¶m‘wio gwQ>Ho$gMo Hw$bwn CKS>VmZm 1000 doJdoJù¶m àH$mao Vnmgbo nm{hOo.

1. H$moUVmhr A§H$ nwÝhm Z KoVm 1, 2, 3, 4, 5, 6 ho A§H$ dmnéZ {H$Vr 3-A§H$s g§»¶m V¶ma H$aVm ¶oVmV?

2. 3, 5, 7, 8, 9, ho A§H$ nwÝhm Z KoVm ho A§H$ dmnéZ {H$Vr 3-A§H$s g‘ g§»¶m V¶ma H$aVm ¶oVmV?

3. n{hbr 10 B§p½be ‘wimjao nwÝhm-nwÝhm Z KoVm {H$Vr 3-Ajam§Mo g§Ho$V V¶ma H$aVm ¶oVmV?

4. 0 Vo 9 ho A§H$ dmnéZ Oa àË¶oH$ H«$‘m§H$ 65 nmgyZ gwé hmoVmo Am{U H$moUVmhr A§H$ EH$ nojm OmñV doim

Z KoVm {H$Vr 5-A§H$s XþaÜdZr H«$‘m§H$ V¶ma H$aVm ¶oVmV?

5. Oa EH$ CÎm‘ ZmUo 3 doim CS>{dbo Va ¶oUmè¶m KQ>Zm§Mr g§»¶m H$mT>m.

6. doJdoJù¶m a§JmMo 5 P|S>o {Xbobo AmhoV. Oa àË¶oH$ Bemè¶m‘Ü¶o EH$‘oH$mda EH$ Ago XmoZ P|S>o dmnabo Va

{H$Vr àH$mao doJdoJio Bemao V¶ma H$aVm ¶oVmV?

H«$‘doe Am{U g§doe (Permutations and Combinations)‘wb yV ‘moOÊ¶mÀ¶m VËdmMm Aä¶mg H$aVmZm AmnU nm{hbo Amho H$s, darb gd© {Xboë¶m CXmhaUm‘Ü¶o H$m§hr

dñVy {Xboë¶m AmhoV. Ë¶mn¡H$s H$m§hr dñVy§Mr {ZdS> Ho$br Amho. AmVm, AmnU dñVy§Mr {ZdS> Am{U ‘m§S>Ur {df¶r

A{YH$ Aä¶mg H$é. H$moï>H$mV {Xbobr CXmhaUo Aä¶mgm.

ñV§ 'P' ‘Yrb n{hbo CXmhaU Am{U ñV§ 'C' ‘Yrb n{hbo CXmhaU dmMm. AmVm Ë¶m§Mr VwbZm H$am. hrM

nÜXV am{hboë¶m CXmhaUmgmR>r nwÝhm H$am.

A.g§. ñV§ P ñV§ C

1. AÜ¶j Am{U g{Md ¶m nXmgmR>r 3 bmoH$ 2 nXmgmR>r 3 C‘oXdma ñnYo©V AmhoV Va Vr nXo {H$Vr

ñnYo©V AmhoV. Ë¶m§Mr {ZdS> {H$Vr àH$mao àH$mao ^abr OmD$ eH$VmV.

Ho$br OmVo?

2. EH$m ‘wbrH$S>o 8 ’«$m°H$ Am{U 4 ~wQ>m§À¶m EH$m ~mJo‘Ü¶o 8 nm§T>ao Am{U 4 bmb Jwbm~ AmhoV. 2

OmoS>çm AmhoV. Va Vr ‘wbJr ñdV…Mm bmb Agbobr 4 ’w$bo CMbÊ¶mMo àH$ma {H$Vr?

nmofmH$ {H$Vr àH$mao H$ê$ eHo$b?

3. TEACH ¶m eãXmVrb gd© Ajam§Mr ‘m§S>Ur MEANS ¶m eãXmVrb gd© Ajam§Mr {ZdS> {H$Vr àH$mao

{H$Vr àH$mao H$aVm ¶oVo? H$aVm ¶oVo?

àË¶oH$ ñV§ mVrb CXmhaUm‘Ü¶o H$moUVrhr g‘mZVm Vwåhr n{hbr Amho H$m? 'P' ñV§ mVrb CXmhaUmgmR>r

Amåhmbm ‘m§S>UrMr g§»¶m H$mT>mdr bmJVo. 'C' ñV§ mVrb CXmhaUmgmR>r Amåhmbm {ZdS>rMr g§»¶m H$mT>mdr

bmJVo.

ñdmÜ¶m¶ 4.1

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darb àH$maMr CXmhaUo gmoS>{dVmZm àW‘ AmnU Vr ‘m§S>Ur Amho qH$dm {ZdS> Amho ho H$miOrnyd©H$

AmoiIbo nm{hOo. CXmhaUmW© darb CXmhaUmVrb H$moï>H$m‘Yrb n{hbo CXmhaU g‘OyZ ¿¶m.

darb MM}déZ ho gwÜXm ñnîQ> hmoVo, P ñV§ mVrb CXmhaUm‘Ü¶o ‘m§S>Ur qH$dm dñVy§Mr {ZdS> qH$dm ì¶º$s¨Mr{ZdS> H«$‘mZo Ho$br Amho. C ñV§ mVrb CXmhaUm‘Ü¶o dñVy§Mr, nXmWmªMr qH$dm ì¶º$s¨Mr {ZdS> H«$‘mZo Ho$br Zmhr.

Aem H«$‘mZo Ho$boë¶m ‘m§S>Urbm H«$‘doe Ago åhUVmV Am{U H«$‘mMm {dMma Z H$aVm Ho$boë¶m {ZdS>rbmg§doe Ago åhUVmV.H«$‘doe (Permutation) : {ZpíMV H«$‘m‘Ü¶o Ho$boë¶m dñVy§À¶m JQ>mÀ¶m ‘m§S>Urbm H«$‘ Ago åhUVmV. hrH«$‘mbm ‘hËd XodyZ Ho$bobr dñVy§Mr ‘m§S>Ur Amho.

EH$mMdoir n dñVyn¡H$s EH$mMdoir r dñVy§Mm H«$‘doe hm H«$‘mZo H$moUË¶mhr n KQ>H$m§À¶m gQ>m‘YyZ rKQ>H$m§Mm CngQ> KoUo Agm Amho.

'n' doJdoJù¶m dñVy‘YyZ EH$mMdoir 'r' dñVyÀ¶m H«$‘doemMr g§»¶m nPr Zo Xe©{dVmV r ng§doe (Combination) : H$moUË¶mhr H«$‘m{dZm dñVy§À¶m gQ>m§Mr {ZdS> H$aUo ¶mbm g§doe Ago åhUVmV.H$moUË¶mhr H«$‘mMm g‘mdoe Zgboë¶m dñVy§À¶m {ZdS>rMm àH$ma Amho.EH$mMdoir n dñVyn¡H$s EH$mMdoir r dñVy§Mm g§doe hm H$moUË¶mhr H«$‘m{dZm n KQ>H$m§À¶m gQ>m‘YyZ rKQ>H$m§Mm CngQ> KoUo Agm hmoVmo.* bjmV ¿¶m {H$ gQ>m‘Ü¶o KQ>H$m§À¶m H«$‘mbm ‘hËd Zm§hr. CXm: (1,3,2) = (2,1,3)

'n' doJdoJù¶m dñVy‘YyZ EH$mMdoir 'r' g§doemMr g§»¶m, Vr nCr, Zo Xe©{dVmV OoWo r n.

A, B Am{U C. ¶m VrZ ì¶º$s AmhoV Ago ‘m§Zy. noQ>r{MÝh dmné¶m Am{U gd© e³¶ Agbobo n¶m©¶ {bhÿ.

nXm§gmR>r n¶m©¶ àH$ma

A.Z§ nX-1 nX-2g§»¶m

1 A B

2 A C

3 B C

KQ>Zm 1 :Oa 'A' Mr {ZdS> EH$m nXm§gmR>r Ho$br Va XþgaonX B {H$§dm C bm {Xbo OmVo.KQ>Zm 2 : Oa B Mr {ZdS> EH$m nXmgmR>r Ho$br Va XþgaonX C bm {Xbo OmVo na§Vw A bm Zmhr H$maU AB Am{UBA hr {ZdS> EH$M Amho. Ë¶m‘wio ho nX ^aÊ¶mMo VrZàH$ma AmhoV.

A, B Am{U C ¶m VrZ ì¶º$s AmhoV Ago ‘m§Zy. VoWoAÜ¶j Am{U g{Md hr XmoZ nXo AmhoV. noQ>r(box)

{MÝh dmné¶m Am{U gd© e³¶ àH$ma {bhÿ.

KQ>Zm 1: Oa 'A' hm AÜ¶j Amho, Va 'B' ho g{MdqH$dm 'C' ho g{Md Amho.KQ>Zm 2 : Oa 'B' hm AÜ¶j Amho Va 'A' hm g{MdqH$dm 'C' hm g{Md Agy eH$Vmo.KQ>Zm 3 : Oa 'C' hm AÜ¶j Amho Va 'A' hm g{MdqH$dm B hm g{Md Agy eH$Vmo.åhUyZ {ZdS>rMo e³¶ Agbobo 6 àH$ma AmhoV.

A.Z§§§§§§§§§§§§. AÜ¶j g{Md1 A B2 A C3 B A4. B C5 C A6 C B

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I. Imbr ‘m§S>Ur Am{U {ZdS>rMr CXmhaUo {Xbobr AmhoV. Ë¶m§Mo dJuH$aU H«$‘doe Am{U g§doemÀ¶mCXmhaUm‘Ü¶o H$am.

1. 12 bmoH$m§À¶m JQ>m‘YyZ 5 gXñ¶ Agbobr g{‘Vr {ZdS>m.2. aH$mÊ¶m‘Ü¶o nmM doJdoJù¶m {df¶mÀ¶m nwñVH$m§Mr ‘m§S>Ur H$aUo.3. 8 IwÀ¶m© AmhoV Am{U Ë¶mda 8 bmoH$ ~gbobo AmhoV.4. 7 ì¶º$s¨À¶m g{‘Vr‘YyZ, EH$ AÜ¶j, EH$ g{Md Am{U EH$m I{OZXmamMr {ZdS> H$am¶Mr

Amho.5. ‘wbm§À¶m H$nS>çmÀ¶m XþH$mZ ‘mbH$mH$S>o 10 H$bmHw$ga Ho$bobo ’«$m°H$ AmhoV ¶mn¡H$s VrZ ’«$m°H$ nwT>À¶m

{IS>H$sda {XImì¶mgmR>r R>odm¶Mo AmhoV.6. 'ARITHMETIC' ¶m eãXmnmgyZ 3 Ajam§Mo eãX V¶ma H$am¶Mo AmhoV.7. EH$m àíZn{ÌHo$‘Ü¶o 12 àíZ AmhoV. {dÚmÏ¶mªZr n{hë¶m XmoZ àíZm§Mr CÎmao {Xbr nm{hOoV na§Vw am{hboë¶m‘YyZ

8 àíZm§Mr {ZdS> Ho$br nm{hOo.8. EH$m ~aUr‘Ü¶o 5 H$mio Am{U 7 nm§T>ao M|Sy> AmhoV. Ë¶mn¡H$s 3 M|Sy> Ago CMbbo nm{hOoV H$s Ë¶m‘Ü¶o

2 H$mio, 1 nm§T>am M|Sy> Amho.9. 1,3,5,7,9 ¶mn¡H$s H$moUVmhr A§H$ nwÝhm Z KoVm {H$Vr 3 A§H$s g§»¶m V¶ma hmoVmV.

10. EH$m dVw©imH$ma H$S>çm‘Ü¶o 5 Mmì¶m§Mr ‘m§S>Ur Ho$bobr Amho.11. EH$m àVbm‘Ü¶o 7 q~Xÿ AmhoV Am{U 3 q~Xÿ Z¡H$aofr¶ AmhoV Va VrZ Z¡H$aofr¶ q~Xÿ OmoSy>Z {H$Vr {ÌH$moU

V¶ma hmoVmV.12. 10 IoiÊ¶m§Mm gQ> 2 ‘wbm§À¶m‘Ü¶o g‘mZ[aË¶m {d^mJbm.

darb àË¶oH$ CXmhaU ho H«$‘doemMo qH$dm g§doemMo H$m Amho? Ë¶mMo H$maU Úm.

à¶ËZ H$am: H«$‘doe Am{U g§doemÀ¶m 5 CXmhaUm§Mr ¶mXr H$am Am{U dJm©V MMm© H$am.

r dr Mm¡H$Q> (noQ>r) âÊ¶mÀ¶m àH$mamÀ¶m g§»¶oMo gm‘mÝ¶ gyÌ H$mT>Uo.FPC dmnéZ AmnU àË¶oH$ H«$‘doemMr amer Imbrà‘mUo {bhÿ eH$Vmo :-àË¶oH$ H«$‘doem‘Ü¶o eodQ>Mr Mm¡H$Q> âÊ¶mÀ¶m g§»¶mMo àH$ma H$miOrnyd©H$ nmhm. gd© H«$‘doem‘Ü¶o H$m§hrgm‘mB©H$ Z‘wZo {Xbobo AmhoV. VoWo n, r À¶m {H§$‘Vr‘Ü¶o Am{U eodQ>Mr Mm¡H$Q> âÊ¶mÀ¶m g§»¶m§À¶m àH$mam‘Ü¶og§~§Y Amho.

hm g§~§Y Imbrbn«‘mUo Amho.'r' Mm¡H$Q>r (noQ>çm) âÊ¶mgmR>r 'n' ‘YyZ 'r' dOm H$am Am{U Ë¶m‘Ü¶o 1

{‘i{dbm Va Amåhmbm g§»¶m§Mo àH$ma {‘iVmV.i.e., (n – r + 1) bjÚm H$s 'r' dr Mm¡H$Q> âÊ¶mgmR>r 'r' hmH$‘rVH$‘r 1 AgVmo Ë¶m‘yio r > 0

ñdmÜ¶m¶ 4.2

55P 5 4 3 2 1

54P 5 4 3 2

53P 5 4 3

52P 5 4

51P 5

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Imbrb CXmhaUo Aä¶mgm.

1) 5P5 - Mm {dMma H$am, Amåhmbm ‘mhrV Amho H$s, eodQ>Mr Mm¡H$Q> åhUOoM 5dr Mm¡H$Q> âÊ¶mMm EH$MàH$ma Amho. ho {‘i{dÊ¶mgmR>r AmnU 5 ‘YyZ 5 dOm H$éZ Ë¶m‘Ü¶o EH$ {‘idm .i.e., (5 – 5 + 1) = 0 + 1 = 1 àH$mao.

2) 5P4 ¿¶m, ¶oWo gwÜXm Amåhmbm ‘mhrV Amho H$s, eodQ>Mr Mm¡H$Q> åhUOo 4 Wr Mm¡H$Q> aÊ¶mMo 2àH$ma AmhoV. ho {‘i{dÊ¶mgmR>r AmnU 5 ‘YyZ 4 dOm H$éZ Ë¶m‘Ü¶o EH$ {‘idm.i.e., (5 – 4 + 1) = 1 + 1 = 2 àH$mao.

3) 5P3 ¿¶m, ¶oWo gwÜXm Amåhmbm ‘mhrV Amho H$s, darb H$moï>H$m‘Ü¶o eodQ>Mr Mm¡H$Q> 3 n«H$mao aVm¶oVo. ho {‘i{dÊ¶mgmR>r AmnU 5 ‘YyZ 3 dOm H$éZ Ë¶m‘Ü¶o EH$ {‘ibm.

i.e., (5 – 3 + 1) = 2 + 1 = 3 àH$mao.Amåhr Agm {ZîH$f© H$mTy> H$s, nPr ‘Ü¶o r dr Mm¡H$Q> (noQ>r) (n – r + 1) àH$mao âbr OmD$ eH$Vo.

nwÝhm-nwÝhm Z KoVm 'n' ñnï> dñVyn¡H$s EH$mMdoir 'r' dñVy§À¶m H«$‘doem§Mr g§»¶m H$mT>Uo, OoW o 0 r n.'n' dñVyn¡H$s 'r' dñVy§À¶m H«$‘doem§Mr g§»¶m hr 'n' {Xboë¶m dñVyn¡H$s Mm¡H$Q>r‘Yrb (noQ>rVrb) 'r' OmJm âÊ¶m

BVH$sM AgVo.

doJdoJù¶m 'n' dñVy Am{U 'r' ‘moH$ù¶m Mm¡H$Q>r Imbrb à‘mUo XmI{dboë¶m AmhoV.

1 2 3 r –1 r

nways

( 1)waysn ( 2)

waysn [ – 2)]n r [ –n r 1)]

ways ways

n{hbr Mm¡H$Q> 'n' doJdoJù¶m {Xboë¶m dñVy KodyZ 'n' g§»¶oÀ¶m àH$mao âVmV.n{hbr noQ>r âÊ¶mgmR>r 'n' doJdoJio àH$ma AmhoV.

n{hbr noQ>r 'n' n¡H$s EH$m dñVyZo âë¶mZ§Va Amnë¶mH$S>o (n – 1) dñVy AmhoV. Á¶m nwÝhm-nwÝhm KoVm¶oV ZmhrV.

Xþgar noQ>r âÊ¶mgmR>r (n – 1) doJdoJio àH$ma AmhoV.n{hë¶m XmoZ noQ>çm (Mm¡H$Q>r) âÊ¶mgmR>r n (n – 1) àH$ma AmhoV (FPC)

n{hë¶m XmoZ noQ>çm âë¶mZ§Va Amnë¶mH$S>o (n – 2) dñV {e„H$ amhVmV.{Vgar noQ>r (n – 2) g§»¶oÀ¶m àH$mamZo âbr OmD$ eH$Vo.

nwÝhm F.P.C dmnéZ n{hë¶m VrZ Mm¡H$Q>r n(n – 1) (n – 2) àH$mao âë¶m OmVmV. {Z[ajU H$am H$sàË¶oH$ Mm¡H$Q> âë¶mZ§Va ZdrZ KQ>H$mMm n[aM¶ hmoVmo Am{U H$moUË¶mhr doir KQ>H$m§Mr g§»¶m hr âboë¶mMm¡H$Q>r (noQ>r) À¶m g§»¶oer g‘mZ AgVo.

FPC dmnéZ H«$‘dma EH$ 'r' Mm¡H$Q>r âÊ¶mMo g§»¶oMo àH$ma Imbrbà‘mUo {Xbobo AmhoV.n (n – 1) (n – 2) ............................. [n – (r – 1)]

àH$mao àH$mao, ..... àH$mao àH$mao, .....

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80 UNIT-4

'n' dñVyn¡H$s EH$mMdoir 'r' dñVy§À¶m H«$‘doemMo àH$ma nPr Zo {Xbo OmVmV. nPr = n (n – 1) (n – 2) ............................. [n – (r – 1)]

Cn{gÜXm§V : 'n' dñVyn¡H$s EH$mMdoir gd© dñVy KoÊ¶mÀ¶m H«$‘doemMr g§»¶m hr.nPn = n (n – 1) (n – 2) ............................. [n – (n – 1)]nPn =n (n – 1) (n – 2) ............................. 1

qH$dm, nPn = n (n – 1) (n – 2) ............................. 3 2 1 ('n' Ad¶d)

H«$‘M¶ {MÝh : (Factorial notation)nmhm H$s darb amerMr COdr ~mOy hr n{hë¶m 'n' Z¡g{J©H$ g§»¶m§Àmm JwUmH$ma Amho. ho n ‘o Xe©{dVmV H«$‘M¶n! ho n Ago dmMVmV (n!)

n (n – 1) (n – 2) ............................. 3 2 1 = n!n! ho n{hë¶m n Z¡g{J©H$ g§»¶m§Mm JwUmH$ma Xe©{dVo.

4! = 1 × 2 × 3 × 4 qH$dm 4 × 3 × 2 × 13! = 1 × 2 × 3 qH$dm 3 × 2 × 12! = 1 × 2 qH$dm 2 × 1n! = 1 × 2 × 3 ..... × n qH$dm n × (n – 1) (n – 2) × .... 3 × 2 × 1

ì¶m»¶odê$Z 0! ho 1 KoVbo Amho. {Q>n : Oa n hr F$U g§»¶m qH$dm Xem§e, Agob Va n! Mr ì¶m»¶m hmoV Zmhr.

bjmV R>odm :n! hm n{hë¶m 'n' Z¡g{J©H$ g§»¶m§Mm JwUmH$ma Amho. n! = n (n – 1) (n – 2) × ...... × 3 × 2 × 1

1

n

n hr n{hë¶m 'n' Z¡g{J©H$ g§»¶m§Mr ~oarO.1

n

n = 1 + 2 + 3 + ........... + n

Imbr {Xboë¶m H«$‘M¶ {MÝhmVrb amer Aä¶mgm.n! = n (n – 1) (n – 2) × ...... × 3 × 2 × 1

(n – r)! = (n – r) (n – r – 1) (n – r – 2) × ...... × 3 × 2 × 1

(n – r + 1)! = (n – r + 1) (n – r) (n – r – 1) × ...... × 3 × 2 × 1

(n – r + 1)! = (n – r + 1) (n – r – 2) (n – r – 3) × ...... × 3 × 2 × 1

(r – 1)! = (r – 1) (r – 2) (r – 3) × ...... × 3 × 2 × 1

gd©gmYmaUnUo n! = n (n – 1) (n – 2) × ...... × 3 × 2 × 1n! = n [(n – 1) (n – 2) × ...... × 3 × 2 × 1]n! = n [(n – 1)!]

qH$dm n! = n (n – 1) (n – 2)!qH$dm n! = n (n – 1) (n – 2) (n – 3) !........ BË¶mXr

CXmhaUmW©, 5! g‘Om.5! = 5 × 4 × 3 × 2 × 15! = 5 × 4!5! = 5 × 4 × 3 !5! = 5 × 4 × 3 × 2!

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ñdmÜ¶m¶ 4.3

1. Imbrb JwUmH$mam§Mo énm§Va H«$‘M¶ ‘Ü¶o H$am. (i) 1 × 2 × 3 × 4 × 5 × 6 × 7 (ii) 18 × 17 ........ 3 × 2 × 1

(iii) 6 × 7 × 8 × 9 (iv) 2 × 4 × 6 × 8

2. qH$‘V H$mT>m: (i) 6! (ii) 9! (iii) 8! – 5!

(iv)7!5!

(v)12!

9! 3! (vi)30!28!

3. qH$‘V H$mT>m : (i)!

!n

n r (ii)!! !

nn r r Ooìhm n = 15 Am{U r = 12 Amho

4. 4!, 5!, 6! ¶m§Mm b.gm.{d. H$mT>m. 5. (i) Oa (n +1)! = 12 (n – 1)! Va n Mr qH$‘V H$mT>m

H«$‘M¶ {MÝh dmnê$Z npr Mo gyÌ V¶ma H$aUo

‘moOÊ¶mÀ¶m ‘yb yV VËdmdéZ (FPC)nPr = n (n – 1) (n – 2) × .......... × (n – r + 1)

g‘rH$aUmMr COdr ~mOy (RHS) ¿¶m.n (n – 1) (n – 2) ........ (n – r + 1)

RHS ‘Ü¶o Z¡g{J©H$ g§»¶mMm JwUH$maCOì¶m ~mOyMo eodQ>Mo nX (n – r + 1) Amho.Oa AmnU hm JwUmH$ma 1 n¶ªV Ho$ë¶mg Amnë¶mbm H$m¶ {‘iVo? Amnë¶mbm n! ho {‘iVo. n! {‘i{dÊ¶mgmR>rH$moUVm Á¶mXmMm Ad¶d Amnë¶mbm {bhmdm bmJob?[(n – r ) (n – r + 1) ........ 3 × 2 × 1] ho (n – r)! hmoVo.nPr = n (n – 1) (n – 2) ........ (n – r + 1)

1 2 ..... 1n

r

n n n n r n rP

.... 3 2 1

1n r n r .... 3 2 1

nPr =n!

n r ! nPr =

n!n r ! [[Oa r > 0]]

n dñVyn¡H$s H«$‘mdoemZo EH$mModoir r dñVy KoÊ¶mÀ¶m àH$maMr g§»¶m.!

!n

rnP

n r( OoW o 0 r n )

darb gyÌ nPr gmR>r bmJy nS>Vo OoWo nwZamdVuV hmoV ZmhrV.

'n' nwÝhm-nwÝhm Z KoVm EH$mM doir 'n' doJdoJù¶m dñVyn¡H$s 'r' dñVyÀ¶m H«$‘doemMr g§»¶m.n P r Amho.

AmVm nPr À¶m {deof KQ>Zm§Mm Aä¶mg H$é.KQ>Zm 1 : Ooìhm r = 0 AgVmZm H$m¶ KS>Vo? Vo AmVm AmnU nmhÿ. Oa r = 0 Va n n¡H$s 0 dñVy§Mr ‘m§S>Ur AmnU

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82 UNIT-4

H$emàH$mao H$ê$ eH$Vmo. H$moUË¶mhr dñVyMr ‘m§S>Ur Z H$aVm gd© dñVy Oemg Vem nmR>r‘mJo amhVmV Am{U

Amåhmbm ‘mhrV Amho H$s, Ago H$aÊ¶mMr EH$M nÜXV Amho.

0!0 !

n nn

P =!!

nn B

0 1nP = CXm 100P0 = 1, 200P0 = 1, 958P0 = 1,

KQ>Zm 2 : r = nnPr = n (n – 1) (n – 2 ) × .................... × (n – r + 1)nPn = n (n – 1) (n – 2 ) × .................... × (n – n + 1)nPn = n (n – 1) (n – 2 ) × .................... × 1nPn = n (n – 1) (n – 2 ) × .................... × 3 × 2 × 1nPn = n!

gwÜXm! ! !

! 0!n

nn n n

n nP (0! = 1 hr ì¶m»¶m AmR>dm.)

CXmhaU : 1009P1009 = 1009 !, 10P10 = 10 !, 1400P1400 = 1400 !KQ>Zm 3 : r = 1

!!

nr

nn r

P 1!1 !

n nPn

1

1 !n

n nP

1 !n nP1 = n

CXmhaUo 100P1 = 100, 457P1 = 457

MMm© H$am : 5P6 ho AW©hrZ Amho. H$m?

bjmV R>odm :

nPr

!!

nn r

Oa r = 0, nP0 = 1 Va r = n Oa nPn = n! Oa r = 1 Va nP1 = n

CXmhaU 1 . qH$‘V H$mT>m : (i) 7P3 (ii) 8P5

CH$b : (i) 73

7! 7! 7 6 5 4!7 3 ! 4!

P4!

7 6 5 210

(ii) 85

8! 8! 8 7 6 5 4 3!8 5 ! 3!

P3!

8 7 6 5 4 6720

ñnï >rH$aUmË‘H$ CXmhaUo

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CXmhaU 2 : Oa 5. 4Pr = 6 .5Pr – 1 Va 'r ' H$mT>m.

CH$b :4! 5!5 6

4 ! 5 1 !r r

5! 6 5!4 !r 6 !r

14 !r

66 5 4 !r r r

(6 – r) (5 – r) = 6 r2 – 11r +24 = 0 (r – 8) (r – 3) = 0

r = {8,3} r =8 ho AW©{hZ Amho H$maU r >>>n.>

CXmhaU 3 : n! + (n + 1)! = n! (n + 2) ho {gÜX H$am.CH$b : S>mdr ~mOy LHS: n! + (n + 1)! = n! + (n + 1) n! = n! (1 + n + 1)

= n! (n + 2) = RHS (COdr ~mOy)

CXmhaU 4 : Oa 41

4

53

n

n

PP Va 'n' H$mT>m.

CH$b : S>mdr ~mOy =( 1)n n 2n 3n

( 1)n n 2n 3n5

4 34n

nn

5(n – 4) = 3n 5n – 20 = 3n 2n = 20n = 10

CXmhaU 5 : Oa 2n + 1Pn – 1 : 2n + 1Pn = 3 : 5 Va n. H$mT>m.

CH$b :2 1

12 1

35

nn

nn

PP

i.e., 5 × 2n + 1Pn – 1 = 3 × 2n – 1Pn

5. 2 1 ! 3. 2 1 !2 1 !2 1 1 !

n nn nn n

5 2 1 2 2 1 ! 3 2 1 !2 ! 1 !

n n n nn n

10 2 1 32 1 1 ! 1 !

n nn n n n n

10 (2n + 1) = 3(n + 2) (n + 1)

3n2 – 11n – 4 = 0 (3n + 1) (n – 4) = 013

n or n = 4.

Á¶mAWu n hm YZ nyUmªH$ Amho, n = 4.

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84 UNIT-4

CXmhaU 6 : (i) Oa nPn = 5040, Va 'n' H$mT>m.

CH$b : nPn = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040

n! = 7! n = 7

(ii) Oa nP2 = 90 Va 'n' H$mT>m.

CH$b : nP2 = 90 $¶oWo AmnU XmoZ H«$‘dma g§»¶mMm JwUmH$ma 90 emoYy

n(n – 1) = 10 × 9 n = 10

(iii) Oa 11Pr = 990 Va 'r' H$mT>m.

CH$b : 11Pr = 990 11Pr = 11 × 10 × 9 r = 3

I. {H$‘Vr H$mT>m.

i) 12P4 ii)75P2 iii) 8P8 iv) 15P1 v) 38P0

2. (i) Oa nP4 = 20 nP2 Va 'n' H$mT>m

(ii) Oa 5Pr = 2 6Pr -1 Va 'r' H$mT>m

3. 1.Oa nP4 : nP5 = 1 : 2 Va 'n' H$mT>m

4. 1. Oa 9P5 + 5 9P4 = 10Pr Va 'r' H$mT>m.

H«$‘doemdarb àm¶mo{JH$ CXmhaUoCXmhaU 1 : aH$mdÝ¶m‘Ü¶o 7 doJdoJir nwñVHo$ {H$Vr àH$mao ‘m§S>br OmD$ eH$VmV? {H$Vr àH$mao VrZ nwñVHo$ EH$Ì¶oVmV?CH$b : 7 nwñVH$m§Mr ‘m§S>Ur H$am¶Mr Amho.

7P7 = 7!= 7 × 6 × 5 × 4 × 3 × 2 × 1= 5040 àH$mao.

Á¶mAWu VrZ nwñVHo$ EH$Ì ¶oVmV, Ë¶mAWu EH${ÌV Agbobr VrZ nwñVHo$ ho EH$ nwñVH$ Ago g‘OboOmVo.AmVm Amnë¶mH$S>o 7 nwñVHo$ - (3 EH${ÌV nwñVHo$) = 4 nwñVHo$.Am{U 4 nwñVHo$ + (1 VrZ EH${ÌV nwñVHo$ ) = 5 nwñVHo$.5 nwñVH$m§Mr ‘m§S>Ur Aer Ho$br OmVo H$s,

ñdmÜ¶m¶ 4.4

11 99010 90

9

1 5040

2 5040

3 2520

4 840

5 210

6 42

7

Á¶mAWu bhmZmV bhmZ Ad¶d 1 Amho.

Ë¶mAWu AmnU 1Zo ^KJ {Xbm nm{hOo.

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5P5 = 5! = 5 × 4 × 3 × 2 × 1 = 120 àH$mao.¶m 120 àË¶oH$ àH$mam‘Ü¶o, 3 R>am{dH$ nwñVH$mMr ‘m§S>Ur 3! = 3 × 2 × 1 = 6 àH$mao hmoVo.

{ (B1 B2 B3)(B1 B3 B2), (B2 B1 B3), (B2 B3 B1), (B3 B1 B2)(B3 B2 B1) }

FPC déZ EHy$U ‘m§S>UrMr g§»¶m 5P5 × 3! = 120 × 6 = 720 àH$mao.

7 nwñVH$m§À¶m ‘m§S>UrMo àH$ma Aem [aVrZo H$s VrZ R>am{dH$ nwñVHo$ H|$ìhmhr EH${ÌV ¶oV ZmhrV

= 5040 – 720 = 4320

CXmhaU 2 : {2, 3, 5, 7, 9} ho A§H$ dmnéZ 30,000 Vo 90,000 ¶m‘Ü¶o {H$Vr 5 A§H$s g§»¶m V¶ma H$aVm¶oVrb.

CH$b :

.

hì¶m Agboë¶m g§»¶m 30,000 nojm ‘moR>r Am{U Vo 90,000 nojm bhmZ ¶m‘Ü¶o AmhoV.åhUyZ Xeghó ñWmZr ’$º$ ho {3, 5 qH$dm 7} A§H$ KoVbo OmD$ eH$VmV.Ë¶m‘wio Xe ghó ñWmZ ’$º$ VrZ àH$mao ^abo OmD$ eH$Vo. am{hbobr 4 ñWmZo am{hboë¶m 4 A§H$mZr ^abrOmD$ eH$VmV. ho 4! àH$mao Ho$bo OmD$ eH$Vo.

30,000 Vo 90,000 ¶m‘Ü¶o ¶oUmè¶m EHy$U g§»¶m 3 × 4! { FPC} déZ= 3 × 4 × 3 × 2 × 1 = 72

CXmhmaU 3 : DNA aoUy‘Ü¶o Zm¶Q´>moOZMm aoUw hm Vi Amho Á¶mbm doJdoJio Vi AgboboA, G, T qH$dm C ho gd© OmoS>bobo AmhoV? Va nwÝhm-nwÝhm Z KoVm 3 doJdoJio AmYma (Vi) Agbob o{H$Vr àH$ma AmhoV?

CH$b : ¶oWo, n = 4 Am{U r = 3

!!

nr

nPn r

43

4! 4 3 2 1 24 ways4 3 ! 1

P àH$mao

1.'JOULE' ¶m eãXmVrb EH$ Aja EH$XmM dmnéZ gd© Ajao dmnéZ {S>³eZar AW© Z dmnaVm AW©nyU© {H$VreãX V¶ma hmoVmV?

2. 5 ‘mUgo Am{U 4 {ó¶m Aem àH$mao AmoirV ~gdm¶Mo Amho H$s g‘ ñWmZr {ó¶m AmhoV. Va Aem e³¶Agboë¶m {H$Vr ‘m§S>Ê¶m AmhoV?

3. 6 {ó¶m gd© 6 {dhrar ‘YyZ {H$Vr àH$mao nmUr Cngy (H$mTy>) eH$VmV.

Xe ghó ghó eV‘² Xe‘² gwQ>o

ñWmZr ñWmZr ñWmZr ñWmZr ñWmZr

3 àH$mao 4 àH$mao 3 àH$mao 2 àH$mao 1 àH$mao

(3,5 qH$dm 7

A = AS>oZmB©Z C = gm¶Q>mogmB©Z G = {OZmZmB©Z T = Wm¶‘mB©Z

ñdmÜ¶m¶ 4.5

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86 UNIT-4

4. EH$m ñnY}V 8 {dÚmÏ¶mªZr ^mJ KoVbm Amho. Va n{hbr VrZ ~{jgo {H$Vr àH$mao qOH$br OmD$ eH$VmV.5. XmoZ A§H$s g§»¶m§À¶m EHy$U g§»¶m H$mT>m.6. 1, 2, 3, 7, 8, 9 ho A§H$ nwÝhm Z KoVm ho A§H$ dmnéZ {H$Vr 4 A§H$s g§»¶m ~Z{dë¶m OmD$ eH$VmV.

(a) Ë¶mn¡H$s {H$Vr 6000 nojm bhmZ AmhoV?(b) Ë¶mn¡H$s {H$Vr g‘ AmhoV?(c) Ë¶mn¡H$s {H$VtMm eodQ> 7 Zo hmoVmo?7. XmoZ eham‘YyZ 15 ~gJmS>çm YmdVmV. EH$ ì¶º$s EH$m ehambm OmD$Z naV doJù¶m ~gJmS>rZo ¶oVo Va

Vr ho {H$Vr àH$mao H$é eH$Vo.'n' ñnï> dñVyn¡H$s EH$mMdoir 'r' dñVy§À¶m g§doemMr g§»¶m H$mT>Uo OoWo 0 r n.a, b, c Am{U d AjamnmgyZ Ë¶m§Mr ‘m§S>Ur Am{U {ZdS> H$aÊ¶mMr Imbrb CXmhaUo {dMmamV ¿¶m.

g§doe Combinations H«$‘doe Permutationsn = 4, r = 1 n = 4, r = 1a, b, c, d a, b, c, d4C1 = 4 4P1 = 44C1 Mm àË¶oH$ g§doe 1! H«$‘doe XoVmo. r = 2 ab,ac,ad,bc,bd,cd

ab,ac,ad,ba,bc,bd ca,cb,cd,da,db,dc4C2 = 6 4C2 Mm àË¶oH$ g§doe 2! H«$‘doe XoVm o 4P2 = 12r = 3 abc,abd,acd,bcd abc abd bcd acd

4C2 = 6 acb adb bdc adcbac bad cbd cadbca bda cdb cdacab dab dbc daccoa dba dcb dca

4P3 = 24 àH$mao4C3 Mm àË¶oH$ g§doe 3! H«$‘À¶ XoVmo.r = 4 abcd abcd bacd cabd dabc

4C4= 1 abdc badc cadb dacbacbd bcad cdba dbacadcb bdac cdab dcababdc bdca cbda dcba

4P4: 244C4 Mm àË¶oH$ g§doe 4! H«$‘doe XoVmV.

gd©gmYmaUnUo,nCr Mm àË¶oH$ g§doe r! H«$‘doe XoVmVnCr Mo gd© g§doe = ?

EHy$U H«$‘doem§Mr g§»¶m = r ! × nCrnPr = r ! × nCr

nCr = !

nrP

r nCr =

!! !

nn r r

'n' ñnï> dñVyn¡H$s EH$mMdoir 'r' g§doemMr g§»¶m

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nr

n!Cr! n - r !

{Q>n nPr dnCr ‘Yrb g§~§Y nCr =

!

nrP

r Zo {Xbm OmVmo.

nCr À¶m doJdoJù¶m {dñVmam§Mm Aä¶mg H$é:

1. r

!

nn

rPCr

2.!

! !n

rnC

r n r

3.1 2 ............... 3 2 1

! !n

r

n n nC

r n r

nCr À¶m H$m§hr {deof KQ>Zm§Mm Aä¶mg H$é¶m

KQ>Zm (i) : r = 0 ‘m§Zy nC0=1

Oa r = 0 Va, 0! ! 1

0! 0 1 !n n nC

n nnC0=1

CXmhaU 100C0 = 1, 500C0 = 1, 1000C0 = 1KQ>Zm (ii): r = 1 ‘m§Zy

Oa r = 1 Va, 1!

1! 1 !n nC

n = 1

1n

n nC

!

1 1n ! nC1 = n

CXmhaUo : 100C1 = 100, 200C1 = 200, 357C1 = 357

KQ>Zm (iii): r = n

Oa r = n Va,! !

! ! !0!n

nn nC

n n n n

! 1! 1n

n

nCn =

CXmhaUoo, 100C100 = 1, 789C789 = 1, 1497C1497 = 1

bjmV R>odm:

!! !

nr

nCn r r

nC0 = 1, r = 1, nC1 = n, nCn = 1

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88 UNIT-4

CXmhaU 1 : Oa 6Pr = 360 Am{U 6Cr = 15, Va 'r' H$mT>m.

CH$b : 6Pr = nCr × r !6Pr = 15 × r ! 360 = 15 × r !

360! 2415

r = 1 × 2 × 3 × 4r = 4

CXmhaU 2: nCr = nCn – r ho {gÜX H$am.

CH$b : Amåhmbm ‘m{hV Amho H$s, !C! !

nr

nr n r

.........(i)

g‘r (i) ‘Ü¶o r EodOr (n – r) KoVbo AgVm Amåhmbm {‘iob

!C! !

nn r

nn r n n r

!C!

nn r

nn r n n !r

!C! !

nn r

nn r r .........(ii)

g‘r (i) Am{U (ii) Mr VwbZm H$éZ Amåhr gm§Jy eH$Vmo H$s, nCr = nCn – r

CXmhaU 3 : Oa nC8 = nC12, Va n H$mT>m.

CH$b nC8 = nC12

nC8 = nCn –12n n

r n rC C

8 = n–12 n = 12 + 8 = 20.

1. qH$‘Vr H$mT>m (i) 10C3 (ii) 60C60 (iii) 100C97

2. (i) Oa nC4 = nC7 Va n H$mT>m.(ii) Oa nPr = 840, nCr = 35, Va n H$mT>m.

3. Oa 2nC3 : nC3 = 11:1, Va n H$mT>m.

4. 8C4 + 8C5 = 9C4 ho Vnmgm.


ñdmÜ¶m¶ 4.6

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5. 11

CC

nr

nr

nr Ooìhm 1 .r n ho {gÕ H$am.

CXmhaU 1: EH$m ‘mUgmMo 6 {‘Ì AmhoV. EH$ qH$dm EH$m nojm OmñV OUm§Zm Vmo g‘ma§ m‘Ü¶o {H$Vr àH$maoAm‘§ÌU XoVmo?CH$b : 6 {‘Ìm§Zm Am‘§{ÌV H$aÊ¶mÀ¶m doJdoJù¶m g§doemMo àH$ma.(~amo~a 1......~amo~a2.....~amo~a......6)

ho 6 6 6 6 6 61 2 3 4 5 6C C C C C C Vo1 qH$dm, 2 qH$dm 6 ¶m§Zm Am‘§{ÌV H$ê$ eH$VmV.

6 6 6 6 6 61 2 3 4 5 6C C C C C C

6 + 15 + 20 + 15 + 6 + 16 6 6 6 6 6

1 2 3 2 1 0C C C C C C

6 + 15 + 20 + 15 + 6 + 1 = 63 àH$mao.CXmhaU 2 : MyH$ H$s ~amo~a ¶m 5 àíZm§À¶m gQ>m§Mr gd© ~amo~a CÎmao H$moUË¶mhr {dÚmÏ¶m©Zo {Xbr ZmhrV Am{UH$moUË¶mhr XmoZ {dÚmÏ¶mªZr gma»¶mM CÎmam§Mm H«$‘ {Xbobm Zmhr. ho e³¶ hmoÊ¶mgmR>r dJm©Vrb OmñVrV OmñV{dÚmÏ¶mªMr g§»¶m {H$Vr Agob? àË¶oH$ àíZ MyH$ qH$dm ~amo~a ¶m XmoZ àH$mao àíZmMr CÎma XoÊ¶mMo àH$ma

CH$b : EHy$U àH$mam§Mr g§»¶m = 2 × 2 × 2 × 2 × 2 = 32¶m 32 àH$mamn¡H$s AJXr EH$ àH$ma ~amo~a Amho.Á¶mAWu dJm©Vrb OmñVrV OmñV {dÚmÏ¶mªMr g§»¶m = 32 – 1 = 31.

CXmhaU 3 : 4 {‘Ì EH$‘oH$mbm hmV {‘i{dVmV. Va hmV {‘i{dÊ¶mMr g§»¶m H$mT>m.CH$b : Vo {‘Ì A, B, C Am{U D AmhoV Ago ‘m§Zy.

hmV {‘i{dÊ¶mgmR>r XmoZ ì¶º$s¨Mm g‘mdoe Agbm nm{hOoV. gwadmVrbm 'A'ì¶º$s Zo 'B' ì¶º$ser hmV {‘i{dbm Ago g‘Oy. ho 'B' ì¶º$sZo 'A' ì¶º$ser hmV {‘i{dÊ¶mer gmaIoM Amho.

AB Am{U BA ho EH$M AmhohmV {‘i{dÊ¶mMr g§»¶m H$mT>VmZm AmnU nmhVmo H$s H«$‘mbm ‘hËd Zmhr.hmV {‘i{dÊ¶mMr g§»¶m Zoh‘r

nC2, Am{U nC2 =1

2n n

.

¶m CXmhaUm‘Ü¶ o n = 4.hmV {‘i{dÊ¶mMr g§»¶m 4C2 Amho.

24

24! 4C

4 2 !2!3 2 1

2 1 26

1 àH$mao.

CXmhaU 4 : g‘ma§^mVrb àË¶oH$ OU EH$‘oH$mbm hmV {‘i{dVmV. Oa EHy$U hmV {‘i{dÊ¶mMr g§»¶m 45Amho. Va g‘ma§^mVrb ì¶º$s¨Mr g§»¶m H$mT>m.CH$b : g‘ma§ mVrb ì¶º$s¨Mr g§»¶m n ‘mZy. hmV {‘i{dÊ¶mg Amdí¶H$ ì¶ÎH$s¨Mr g§»¶m = 2

nC2 = 45

B

CD

A

{AB, AC, AD}

{BC, BD}

C D

{CD}

C

DB

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90 UNIT-4

! 452 !2!

nn

145

2n n

n(n – 1) = 90n(n – 1) = 10 × 9n(n – 1) = 10(10 – 1)

n = 10g‘ma§ mVrb EHy$U ì¶º$s¨Mr g§»¶m = 10.

CXmhaU 5 : AÜ¶jmgh {Xboë¶m 12 ì¶º$s‘YyZ 5 OUm§À¶m g{‘VrMr {ZdS> {H$Vr àH$mao Ho$br OmVo? CH$b : AÜ¶jmMr {ZdS> 12 àH$mao Ho$br OmVo Am{U g{‘Vr‘Yrb BVa 4 OUm§Mr {ZdS> 11C4 àH$mao Ho$brOmVo.

Aem e³¶ Agboë¶m g{‘Ë¶m§Mr g§»¶m = 12 × 11C4

= 12 × 330 = 3960. àH$mao

CXmhaU 6 : 8 Z¡H$aofr¶ q~Xÿ AmhoV. ho gd© q~Xÿ OmoSy>Z {H$Vr gai aofm H$mT>Vm ¶oVrb?CH$b : XmoZ q~Xÿ OmoS>bo AgVm gai aofm {‘iVo. Vo q~Xÿ A Am{U B ‘mZy.

EH$Va A q~Xÿ B q~Xÿbm qH$dm B q~Xÿ A bm OmoS>bm AgVm gai aofm {‘iVo. AB Am{U BA ho EH$M AmhoV.ho g§doemdarb CXmhaU Amho.'n' Z¡H$aofr¶ q~XÿVyZ H$mT>ë¶m OmUmè¶m gai aofm§Mr g§»¶m = nC2

Amåhmbm ‘m{hV Amho H$s, 21 12

nC n n

¶oWo, n = 8, r = 2

8C2 =1 8 8 12

8 Z¡H$aofr¶ q~Xÿ OmoSy>Z 28 gai aofm H$mT>Vm ¶oVmV.

CXmhaU 7 :10 Z¡H$aofr¶ q~Xÿ AmhoV. VrZ q~Xÿ OmoSy>Z {H$Vr {ÌH$moU aMVm ¶oVmV?CH$b : 3 Z¡H$aofr¶ q~Xÿ OmoS>bo AgVm {ÌH$moU {‘iVmo. ho q~Xÿ H$moUË¶m H«$‘mZo OmoS>bo AmhoV ho {dMmamVKoVbo OmV Zmhr.

'n' Z¡H$aofr¶ q~XÿnmgyZ aMë¶m OmUmè¶m {ÌH$moUm§Mr g§»¶m = nC3.

¶oW o, n = 10, r = 3,!! !

nr

nCn r r

o

A

B C

A B

82

4

!! !

8!8 2 !2!

8

nr

nCn r r

C

Verification :

7 6!6!

82

2 28C

V mi m

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103

10! 10!10 3 !3! 7! 3!

C 10 93

84

7!7! 3 2 1 = 120

10 Z¡H$aofr¶ q~Xÿ KodyZ 3 q~Xÿ OmoS>bo AgVm 120 {ÌH$moU aMVm ¶oVmV.

n¶m©¶r nÜXV :

3!

! !n nC

n r r!

3 !3!n

n1 2 3 !n n n n

3 !n 3 2 13

1 26

n n n nC

Oa, n = 10

3

10 10 1 10 26

nC310 9 8

4

6 3= 120

CXmhaU 8: fï> ~hþ^yOmH¥$Vr ‘Ü¶o {H$Vr H$U© H$mT>Vm ¶oVmV?CH$b : fï>~hþ yOmH¥$Vr‘Ü¶o 6 {eamoq~Xÿ AmhoV n = 6

{déÜX {eamoq~XÿÀ¶m OmoS>çm gm§YyZ H$U© {‘iVmV.EHy$U aofm Am{U H$UmªMr g§»¶m = 6C2

2

12

n n nC 6

2

6 6 12

C = 15

15 aofm‘Ü¶o 6 ~mOy AmhoV. H$UmªMr g§»¶m = 15 – 6 = 9.

n¶m©¶r nÜXV :

H$Um©Mr g§»¶m = aMboë¶m gai aofm§Mr g§»¶m – ~hþ yOmH¥$VrÀ¶m ~mOy§Mr g§»¶m.

= nC2 – n

=1 12

n n n

2 22

n n n

2 32

n n 32

n n

n ~mOy§À¶m ~hy yOmH¥$VrÀ¶m H$UmªMr g§»¶m =3

2n n

fï>~hþ wOmH¥$Vr‘Ü¶ o, n = 6

H$UmªMr g§»¶m3

2n n 6 6 3

2 = 9

CXmhaU 9 : ~hþ^wOmH¥$Vr‘Yrb H$UmªMr OmñVrV OmñV (H$‘mb) g§»¶m 14 Amho Va ~mOw§Mr g§»¶m H$mT>m.CH$b : Amåhmbm ‘m{hV Amho H$s,number of diagonals 3

2n n

3141 2

n nn(n – 3) = 14 × 2 n(n – 3) = 7 × 4

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92 UNIT-4

n(n – 3) = 7(7 – 4) n = 7n¶m©¶r nÜXV :

3141 2

n n

n(n – 3) = 14 × 2 n2 – 3n – 28 = 0 n2 – 7n + 4n – 28 = 0n(n – 7) + 4(n – 7) = 0 (n – 7) (n + 4) = 0

n – 7 = 0 qH$dm n + 4 = 0n = 7 qH$dm n = – 4

'n' ho F$U Agy eH$V Zmhr.

n = 7 (gá ~hþ wOmH¥$Vr)

bjmV R> odm :

Ooìhm nmVirVrb Z¡H$aofr¶ q~XÿMr g§»¶m {Xbobr Agob Va

* gai aofm§Mr g§»¶m = 2

12

n n nC

* {ÌH$moUm§Mr g§»¶m = 3

1 26

n n n nC

* ~hþ wOmH¥$Vr‘Yrb H$UmªMr g§»¶m = 2

32

n n nC n

nmñH$bMm {ÌH$moU [Pascal Triangle]nmñH$b ãbmB©g (1623 - 1662) à{gÜX ’«|$M J{UVVk, g§ mì¶VmVk g§doemVk,, ^m¡{VH$Vk Am{UVËdkmZr AmhoV.nmñH$bZr emoYboë¶m {ÌH$moU ‘m§S>UrMo {Z[ajU H$am.XmoZ {ÌH$moUmVrb g§JV KQ>H$m§Mr VwbZm H$am.COì¶m ~mOyÀ¶m {ÌH$moUmbm nmñH$bMm {ÌH$moU Ago åhUVmV.

1C0

1C1

2C2

6C6

5C5

3C3

4C4

6C5

5C4

4C3

3C2

2C1

1C0

2C0

3C1

4C2

5C3

6C4

3C0

4C1

5C2

6C3

4C0

5C1

6C26C1

5C0

6C0

1

1

1

1

1

6

5

4

3

2

1

1

3

6

10

15

1

4

10

20

1

5

156

1

1

1

1

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1. 7 ì¶§OZ d 4 ñda ¶mn¡H$s 3 ì¶§OZ Am{U 2 ñda Agbobo {H$Vr eãX aMbo OmD$ eH$VmV? 2. 10 OUm§À¶m JQ>m‘YyZ 5 IoimSy>§Mr {ZdS> {H$Vr àH$mao Ho$br OmD$ eH$Vo?3. 17 IoimSy>‘YyZ 11 IoimSy>§Mr {ZdS> H$éZ {H«$Ho$Q> g§K V¶ma H$é eH$VmV, Á¶m‘Ü¶o 5 IoimSy>

Jmob§XmOr H$é eH$VmV? àË¶oH$ g§KmV 2 Jmob§XmO Agbo nm{hOoV?4. dVw©imdarb 8 q~XÿVyZ {H$Vr (i) gai aofm (ii) {ÌH$moU H$mT>Vm ¶oVmV?5. (i) Xe ~hþ yOmH¥$Vr (ii) {de§‘~hþ^wOmH¥$Vr, {H$Vr H$U© H$mT>bo OmD$ eH$VmV?6. 44 H$U© Agboë¶m ~hþ wOmH¥$Vr‘Yrb ~mOy§Mr g§»¶m H$mT>m.

ñdmÜ¶m¶ 4.7

Oa r = 0,nP0 = 1

H«$‘doe Am{U g§doe

‘wb^yV ‘moOÊ¶mMoVËd (FPC)

g§doenCr

H«$‘doenPr

H«$‘M¶ gm§Ho${VH$ {MÝh n!nPr =

!( )!

nn r

OoWo 0 r nnCr =

!!( )!

nr n r

Oar = n,nPn = n!

Oa r = 1,nP1 = n

Oa r = 0,nC0 = 1

Oa r = n,nCn = 1

Oa r = 1,nC1 = n

nPr = nCr×r!

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94 UNIT-4

CÎmao

ñdmÜ¶m¶ 4.11) 120 2) 12 3) 720 4) 336 5) 8 6) 20

ñdmÜ¶m¶ 4.3 1) (i) 7! (ii) 18! (iii)

9!5!

(iv) 16 × 4 !

2) (i) 720 (ii) 362880 (iii) 40200 (iv) 42 (v) 220 (vi) 870 3) (i) 210 (ii) 105

4) 720 5) 3ñdmÜ¶m¶ 4.4

1) (i) 11880 (ii) 5550 (iii) 40320 (iv) 15 (v) 12) (i) 7 (ii) 3 3) 6 4) 5

ñdmÜ¶m¶ 4.5 1) 120 2) 2880 3) 720 4) 336 5) 90

6) (a) 180 (b) 120 (c) 60 7) 210ñdmÜ¶m¶ 4.6

1) (i) 120 (ii) 1 (iii) 1,61,700 2) (i) 11 (ii) 7 3) 6ñdmÜ¶m¶ 4.7

1) 25200 2) 252 3) 2200 4) (i) 28 (ii) 56 5) (i) 35 (ii) 170

6) 11 7) 720 8) (i) 186 (ii) 186 9) 3136 10) 117

*****·

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Probability 95

¶m KQ>H$m‘wio Vwåhmbm ¶m§Mo gwb^rH$aU hmoVo.

* ghO KS>Umè¶m ¶mÑpÀN>H$ à¶moJmÀ¶m AWm©Moñnï>rH$aU.

* Z‘wZm AdH$me Am{U KQ>Zm ¶m§Mr ì¶m»¶m H$aUo.

* g‘mZ g§ dzZr¶ KQ>Zm§À¶m AWm©Mo ñnï>rH$aU.

* {Xboë¶m KQ>ZoMo Z‘wZm AdH$me Am{U àmW{‘H$KQ>Zm emoYUo.

* doJdoJù¶m àH$maÀ¶m KQ>Zm§À¶m AWm©Mo ñnï>rH$aU.

* KQ>ZoÀ¶m g§ mì¶VoMr ì¶m»¶m H$aUo.l

* P(E) = 1 – P(E) ho {gÕ H$aUo.

* g§ mì¶VoÀ¶m ~oaOoMm {Z¶‘ {gÕ H$aUo.P(E1 E2) = P(E1) + P(E2)

* {Xboë¶m KQ>Zm§Mr g§ mì¶Vm H$mT>Uo.

5 g§^mì¶Vm(Probability)

Probability theory is nothing butcommon sense reduced tocalculation.

- Pierre de Laplace

* ghO KS>Umao à¶moJ (¶mÑpÀN>H$)

* Z‘wZm AdH$me

* g‘mZ g§^d{Z¶ KQ>Zm

* KQ>Zm§Mo àH$ma

* g§^mì¶VoMm AW©

* H$moQ>r KQ>Zm§Mr g§ mì¶Vm

* nañnamer Ho$di ({deof) KQ>Zm§Mrg§ mì¶Vm.

{nAar {S> b°nbogPierre de Laplace

(1749-1827, ’«$mÝg)

1812 ‘Ü¶o b°nbogZr gm§p»¶H$s‘YrbAZoH$ ‘wb yV {ZH$mb ñWm{nV Ho$bo.AmhoV. Ë¶m§Zr g§ mì¶Voda AmWmarVJ{UVr AZw‘m{ZV ¶w{º$dmX ‘m§S>bo.Ë¶m§Zr g§ mì¶VoÀ¶m AZoH$ VËdm§Mmn[aM¶ H$ê$Z {Xbm. g§ mì¶Vm hrAZwHy$b (nmofH$) KQ>Zm§Mo Ë¶m§À¶mere³¶ Agboë¶m EHy$U KQ>Zm§Mo JwUmoÎmaAmho.

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96 UNIT-5

nyduÀ¶m B¶ÎmoV AmnU Aä¶mgbobo Amho H$s X¡Z§{XZ OrdZmVrb Odi Odi gd© H$mhr Oo AmnU nmhVmo qH$dmH$aVmo Vo e³¶Vm ¶m {df¶mer g§~§{YV Amho. AMwH$nUm Am{U ^{dî¶dmUr hr gwÕm gma»¶mM pñWVrV KS>UmarCXmhaUo AmhoV.

Aem KQ>Zm§Mm {dMma H$am. Imbrb {dYmZo Aä¶mgm.1. 2730k ‘Ü¶o nmè¶mMr KZVm 13,590 gm/c.c. Amho.2. gai CS>{dbobm M|Sy> da OmÊ¶mgmR>r Am{U Imbr ¶oÊ¶mgmR>r gmaImM doi KoVmo.3. g‘wÐ gnmQ>rda nmÊ¶mMm CËH$bZ q~Xÿ 100°C Amho.4. ^maVr¶ {H«$Ho$Q> g§Kmbm dëS>© H$n qOH$Ê¶mMr CÎm‘ g§Yr Amho.5. ¶mdoir AmnU gy¶© J«hUmMo CÎm‘ Xe©Z Kody eH$Vmo.6. åh¡gyê$Mo AmajU {V{H$Q> H$Xm{MV ‘bm {‘iUma Zmhr.7. AmO nmD$g ¶oÊ¶mMr OmñV g§ mì¶Vm Amho.

n{hë¶m VrZ {dYmZmnmgyZ eodQ>Mr Mma CXmhaUo H$er doJir AmhoV.n{hbr VrZ ñnï> Am{U {ZpíMV Va eodQ>Mr Mma g§{X½Y AmhoV.{dYmZo 1, 2, 3 hr ImÌrnyU© {dYmZo AmhoV Am{U 4, 5, 6, 7 hr {dYmZo ImÌrnyU© ZmhrV. (A{dídmgmW©

AmhoV)l {dYmZ EH$‘Ü¶o {Xbobo à¶moJ {H$Vrhr doim Ho$bo Var gmaImM {ZH$mb {‘iVmo qH$dm hm {dídmgmh© {ZH$mb

Amho.gmaImM {ddmX {dYmZ 2 Am{U 3 gmR>r bmJy nS>Vmo.

l Xþgè¶m eãXmV gy¶© J«hU nmhUo ho e³¶ qH$dm Ae³¶ Amho. {H«$Ho$Q> hm {dídmg Z R>odÊ¶mMm Ioi Amho.EH$ CÎm‘ ’$b§XmO n{hë¶mM M|Sy>V ~mX hmoD$ eH$Vmo qH$dm EH$m CÎm‘ Jmob§XmOmbm EH$hr JS>r ~mX H$aVmAmbm Zmhr. åhUyZ g§K gm‘Zm qOHo$b qH$dm Zmhr.gmaImM {ddmX {dYmZ 6 Am{U 7 gmR>r bmJy nS>Vmo.eãX Ogo H$s "g§Yr', "g§^mì¶Vm', "e³¶Vm' ~hþYm BË¶mXr EH$m R>am{dH$ KQ>Zo~Ôb A{ZpíMVVm Xe©{dVmV.

A{ZpíMVVoMo AJXr ¶mo½¶ ‘moO‘mn Zmhr, na§Vw J{UVm‘Ü¶o H$mhr pñWVrV Amnë¶mH$S>o KQ>ZoÀ¶m {ZpíMVVoMmgm§p»¶H$ qH$‘VrMr JUZm H$aÊ¶mMr nÕV Amho. J{UVmÀ¶m Á¶m emIo‘Ü¶o A{ZpíMVVoMr ‘mno nwa{dbrOmVmV {Vbm g§^mì¶VoMm {gÕm§V Ago åhUVmV. ¶m {gÕm§VmMm ‘hËdmMm Cn¶moJ åhUOo gm§p»¶H$s‘Yrb EH$‘yb yV gmYZ Amho Am{U ¶m§À¶m Cn¶moJmMr ‘moR>r ì¶már "^m¡{VH$ {dkmZ, A{^¶m§{ÌH$s, O¡{dH$ {dkmZ, d¡Ú{H$¶emó, dm{UÁ¶ hdm‘mZmMr ^{dî¶dmUr BË¶mXr ‘Ü¶o hmoVmo.

OmUyZ ¿¶m …g§^mì¶VoMm B{Vhmg … g§ mì¶VoÀ¶m g§H$ënZoMm {dH$mg EH$m AZmo»¶m nÕVrZo Pmbm. 1654 ‘Ü¶o17 ì¶m eVH$mV Ho$d{b¶a {S> ‘oao hm OwJmar H$mhr ’$mem§À¶m g‘ñ¶m KodyZ à{gÕ ’«|$M VËdkmZrAm{U J{UVVk ãbo¶go nmñH$b ¶m§À¶mH$S>o Ambm. nmñH$b ¶mZr ¶m§Mr MMm© AmUIr EH$m ’«|$MJ{UVVk {nAar S>r ’$a‘°Q>arer Ho$br Am{U ’$mem§À¶m g‘ñ¶oda VmoS>Jm emoYyZ H$mT>bm. ho H$m¶©åhUOoM "g§^mì¶Vm {gÕm§VmMr' gwadmV hmoVr. Oar g§ mì¶VoMr gwadmV OwJmamZo Pmbr, Var AmVmhr J{UVmÀ¶m emIoVrb gd© {dkmZ Am{U X¡Z§{XZ OrdZmVrb KQ>Zm‘Ü¶o AË¶§V ‘hËdmMm Cn¶moJ Amho.

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Probability 97

BQ>brMo J{UVVk Oo H$mS>©Z (1501-1576) ¶m {df¶mda nwñVH$ {b{hbo Ë¶mMo Zm§d "~wH$ Am°Z JoågAm°’$ MmÝg' ho Amho. Oo ~aZmobr, nr. b°nbmg, E.‘maH$moìh Am{U E.EZ².H$mob‘moJmoaìh ¶m J{UVVkm§Zrg§ mì¶VoÀ¶m {gÕm§VmÀ¶m {dH$mgmgmR>r ¶moJXmZ {Xbo Amho. a{e¶Z J{UVVk E.EZ².H$mob‘moJmoaìhZr gQ>m§Mm{gÕm§V dmnê$Z g§ mì¶VoMm AmYw{ZH$ ‘mJ© gwM{dbm.

¶m KQ>H$m‘Ü¶o Amåhr g§ mì¶VoÀ¶m {gÕm§Vm{df¶r A{YH$ Aä¶mg H$ê$¶m.

KQ>Zo‘Ü¶o KS>Umè¶m g§YrMo Ooìhm n[aUm‘mË‘H$ n¥W…H$aU Ho$bo OmVo Voìhm ¶mbm g§ mì¶Vm Ago åhUVmV.

hr EH$ g§H$ënZm Amho Or A{ZpíMVVoMr g§»¶mË‘H$ ‘mno XoVo Am{U åhUyZ KQ>ZoV KSy>Z ¶oUmè¶m{ZpíMVVm Amho. g§ mÊ¶mVoMo XmoZ ^mJ AmhoV.

g§ mì¶Vm

àm¶mo{JH$ g§^mì¶Vm VmpËdH$ g§ mì¶VmqH$dm qH$dm

àm¶mo{JH$ VËd g§^mì¶Vm emór¶ g§ mì¶VmOo àm¶mo{JH$ AZw^mda AmYmarV Amho. d¡km{ZH$ (emór¶) ¶w³Ë¶mda AmYmarV Amho.

¶m KQ>H$m‘Ü¶o AmnU g§ mìVoÀ¶m VmpËdH$VoMm Aä¶mg H$ê$¶m.g§^mì¶VoMr VmpËdH$Vm … (Theoretical approach to probability)

g§^mì¶VoÀ¶m VmpËdH$Vo‘Ü¶o, AmnU àË¶jnUo à¶moJ Z H$aVm ^{dî¶ dV©{dÊ¶mMm à¶ËZ H$aVmo. Agonm{hbo Jobo Amho H$s Oa à¶moJmVrb narjm§Mr g§»¶m OmñV Agob Va KQ>ZoVrb à¶moJmMr g§ mì¶Vm hr Ë¶mÀ¶mVmpËdH$ g§ mì¶Vo Odirb AgVo.

XmoÝhr àg§JmV AmnU KSy>Z ¶oUmè¶m {ZînÎmrer g§~§{YV Amho. à¶moJ ¶m eãXmMm AW© hr {H«$¶m Amho.Omo CÎm‘ ì¶m»¶m Ho$boVr {ZînVr qH$dm {ZH$mb XoVmo. AmnU à¶moJmMo dJuH$aU XmoZ àH$mamV H$ê$ eH$Vmo.

à¶moJ

{ZpíMV à¶moJ ghO KS>Umao à¶moJqH$dm qH$dm

^mH$sV à¶moJ g§^mdrV à¶moJ

Ooìhm gma»¶mM pñWVrV ho à¶moJ nwÝhm-nwÝhm Ho$bo AgVm VoM {ZH$mb qH$dm{ZînÎmr {ZH$mb {‘iVmo.

Ooìhm gma»¶mM pñWVrV ho à¶moJ nwÝhm-nwÝhm Ho$bo AgVm VmoM {ZH$mb qH$dm {ZînÎmr{‘iV ZmhrV.

CXmhaUo CXmhaUo

* ZmÊ¶mMm ì¶mg ‘moOUo * ZmUo CS>{dUo* nmÊ¶mMm CËH$bZ q~Xÿ H$mT>Uo * ’$mgm ’o$H$Uo* {dÚwV ‘§S>imVyZ dmhUmar * IoimÀ¶m nÎ¶m‘YyZ EH$

{dÚwV Ymam ‘moOUo nÎmm H$mT>Uo

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98 UNIT-5

{dkmZ Am{U A{^¶m§{ÌH$s‘Ü¶o gma»¶mM pñWVr‘Ü¶o à¶moJ nwÝhm-nwÝhm Ho$bo AgVm, Amnë¶mbm àË¶oH$doirgmaIoM {ZH$mb {‘iVmV. Ë¶mbm {ZpíMV à¶moJ åhUVmV. J{UVm‘Ü¶o Amåhr ¶mÑpÀN>H$ à¶moJmer g§~{YVAmho.

¶mÑpÀN>H$ (g§ d) à¶moJ … Z¸$s {ZînÎmrMr ^{dî¶dmUr Z H$aVm ¶oUmam g§^d à¶moJ Amho. Var nU¶mÑpÀN>H$ à¶moJmVrb e³¶ Agboë¶m gd© {ZînÎmtMr ¶mXr Ho$br OmD$ eH$Vo.

hm EH$ à¶moJ Amho. Omo gma»¶mM pñWVr‘Ü¶o nwÝhm-nwÝhm Ho$bm Va àË¶oH$doir VrM {ZînÎmr qH$dm {ZH$mb{‘iV Zmhr. na§Vw MmMUrVrb {ZînÎmr hr e³¶ Agboë¶m AZoH$ {ZînÎmrn¡H$s EH$ Amho.

à¶moJ hm ¶mÑpÀN>H$ à¶moJ AmhMo, Oa Vmo Imbrb AQ>r nS>VmiVmo.* Ë¶mbm EH$ nojm OmñV e³¶ {ZînÎmr AmhoV.* AJmoXaM e³¶ Agboë¶m {ZînÎmtMr ^{dî¶dmUr H$aVm ¶oV Zmhr.

¶m nyU© KQ>H$m‘Ü¶o Amåhr ’$º$ KSy>Z ¶oUmè¶m à¶moJm{df¶r MMm© H$aUma AmhmoV åhUyZ à¶moJ ho nX¶mÑpÀN>H$ à¶moJ Xe©{dVo.

¶oWo ghO KS>Umè¶m à¶moJmer g§~§{YV H$mhr nXo AmhoV. hr nXo Amåhmbm g§ mì¶VoMr VmpËdH$ g§H$ënZmg‘OyZ KoÊ¶mg ‘XV H$aVmV. AmVm AmnU ¶mÑpÀN>H$ à¶moJmÀ¶m nXm§Zm Aä¶mgUma AmhmoV.

Imbrb Vº$m Aä¶mgm Am{U àË¶oH$ eãX CXmhaUmgh g‘OyZ KoÊ¶mMm à¶ËZ H$am.

nX AW© CXmhaUo

MmMUr (Trial)

{ZînÎmr

(Outcome)

AdH$me Z‘wZm

(Sample space)

Z‘wZm q~Xÿ(Sample point)

KQ>Zm (Event)

ghO KS>Umao à¶moJ H$aUo ¶mbm MmMUrAgo åhUVmV.

MmMUr‘Ü¶o Oo da {‘iVo Ë¶mbm {ZînÎmrqH$dm {ZH$mb åhUVmV. ¶mÑpÀN>H$à¶moJmVrb {ZH$mbm§Zm {ZînÎmr AgoåhUVmV.

ghO KS>Umè¶m à¶m oJm‘Ü¶o e³¶AgUmè¶m gd© {ZînÎmrZm Z‘wZm AdH$meåhUVmV. ho gm‘mÝ¶nUo 'S' Zo Xe©{dVmV.

Z‘wZm AdH$memVrb àË¶oH$ KQ>H$ qH$dmgXñ¶m§Zm Z‘wZm q~Xÿ åhUVmV.

¶mÑpÀN>H$ à¶moJm‘Ü¶o H$moUVrhr e³¶{ZînÎmr qH$dm {ZînÎmtMm g§¶moJ ¶mbmKQ>Zm åhUVmV. åhUOoM AdH$meZ‘wÝ¶mVrb àË¶oH$ CngQ>mbm KQ>ZmåhUVmV. ¶m KQ>Zm A, B, C, D ZoXe©{dVmV.

(i) ZmUo ’o$H$Uo qH$dm CS>{dUo

(ii) ’$mgm ’o$H$Uo

(i) N>mn (H), H$mQ>m (T)(ii) 1, 2, 3, 4, 5, 6

(i) S = {H, T}(ii) S = {1, 2, 3, 4, 5, 6}

(i) H Am{U T ho Z‘wZm q~Xÿ AmhoV.

(ii) 1, 2, 3, 4, 5d 6 ho Z‘wZm q~Xþ AmhoV.

(i) N>mn {‘i{dUo A = {H} H$mQ>m {‘i{dUo B = {T}4 {‘i{dUo A = {4}

(ii) g‘ g§»¶m {‘i{dUo B = {2,4, 6}‘wi g§»¶m {‘i{dUo C = {2, 3, 5}Z¡g{J©H$ g§»¶m {‘i{dUo

D = {1, 2, 3, 4, 5, 6}

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Probability 99

Amåhr Aä¶mgbobo Amho H$s, KQ>Zm hr AdH$me Z‘wÝ¶mMm CngQ> Amho. Omo H$m§hr nmiVmo. EH$mM doirXmoZ ZmUr CS>{dÊ¶mMo (qH$dm EH${ÌV) CXmhaU {dMmamV ¿¶m. d¥jm H¥$Vr aMyZ ¶m ¶mÑpÀN>H$ à¶moJmVrb Z‘wZmAdH$me ghOnUo {b{hVm ¶oVmV.

Imbrb d¥jmH¥$VrMo {ZarjU H$am.

H1

T1

H – H H

T – H T

H – T H

T – T T

2 1 2

2 1 2

2 1 2

2 1 2

AmVm AmnU, ghO KS>Umè¶m à¶moJmVrb {‘iUmè¶m KQ>Zm {bhy.

* XmoÝhr ZmÊ¶mda N>mn ¶oVmo. E1 = { HH }* XmoÝhr ZmÊ¶mda H$mQ>m ¶oVo. E2 = { TT }* n{hë¶m ZmÊ¶mda N>mn Am{U E3 = { HT }

Xþgè¶m ZmÊ¶mda H$mQ>m ¶oVo.* n{hë¶m ZmÊ¶mda H$mQ>m Am{U E4 = { TH }

Xþgè¶m ZmÊ¶mda N>mn ¶oVmo.

darb CXmhaUm‘Ü¶o, XmoÝhr ZmUr EH${ÌV CS>{dÊ¶mMm ghO KS>Umam à¶moJ Amho. Z‘wZm AdH$meS = { HH, HT, TH, TT }

ghO KS>Umè¶m à¶moJmVrb àË¶oH$ {ZînÎmrbm àmW{‘H$ (gmonr) KQ>Zm åhUVmV.

XmoZ ZmUr EH${ÌV CS>{dÊ¶mÀ¶m ¶mÑpÀN>H$ à¶moJm‘Ü¶o HH, HT, TH, TT ¶m àmW{‘H$ KQ>Zm AmhoV.

XmoZ ZmUr EH${ÌV CS>{dÊ¶mÀ¶m ¶mÑpÀN>H$ à¶moJm‘Ü¶o àmW{‘H$ KQ>Zm {edm¶ BVa H$mhr KQ>Zm gwÕm e³¶AmhoV.

* AJXr ~amo~a N>mn {‘iVmo E5 = {HT, TH}* H$‘rVH$‘r EH$ H$mQ>m {‘iVmo E6 = {HH, HT, TH}

* OmñVrV OmñV XmoZ amÊ¶m {‘iVmV. E7 = {HT, TH, TT}

¶oWo AmnU nmhVmo H$s ¶mÑpÀN>H$ à¶moJmer g§~§YrV XmoZ qH$dm A{YH$ àmW{‘H$ KQ>Zm§Zm OmoSy>ZAmnë¶mbm KQ>Zm {‘iVmV. Aem KQ>Zm§Zm g§¶wº$ KQ>Zm Ago åhUVmV.

¶mÑpÀN>H$ à¶moJmer g§~§{YV KQ>Zm hr g§¶wº$ KQ>Zm Amho, ghO KS>Umè¶m à¶moJmer g§~§{YV XmoZ qH$dmA{YH$ àmW{‘H$ KQ>Zm§Zm OmoS>ë¶mZo hr {‘iVo.

AmnU KQ>Zm KS>Vo Ago Ho$ìhm åhUVmo?{Z:njnmVr ’$mgm CS>{dÊ¶mÀ¶m ¶mÑpÀN>H$ à¶moJmMm {dMma H$am.Z‘wZm AdH$me S = {1, 2, 3, 4, 5, 6}

g‘ g§»¶m {‘iUmar KQ>Zm 'E' Zo Xe©{dVmV Ago ‘mZy.

¶m KQ>Zoer g§~§{YV ‘wb yV KQ>Zm 2, 4, 6 ¶m AmhoV.

MmMUrVrb {ZînÎmr 4 Amho Ago g‘Oy AmnU åhUVmo 'E' hr KQ>Zm KS>Vo Ë¶m MmMUr‘Yrb {ZînÎmr 3

AdH$me Z‘wZm,

S = { H1H2, H1T2, T1H2, T1T2 }

S = { HH, HT, TH, TT }n{hbo ZmUo

Xþgaoo ZmUo

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100 UNIT-5

AmhoV Voìhm AmnU åhUVmo H$s E hr KQ>Zm KS>V Zmhr.¶m CXmhaUmdê$Z, AmnU Agm {ZîH$f© H$mTy> eH$Vmo H$s KQ>Zm KS>Ê¶mMm AW© Imbrbà‘mUo Amho.¶mÑpÀN>H$ à¶moJmer g§~§{YV E hr KQ>Zm KS>Vo Ago åhUVmo, Oa E hr AZwHy$b Amho Ago åhUVmo, H$maU

àË¶oH$ àmW{‘H$ KQ>Zm hr g§¶wº$ KQ>ZoÀ¶m ì¶m»¶oMo g‘mYmZ H$aVmV. Vo åhUOo E KQ>ZoV g‘ g§»¶m KS>VmV Am{U2, 4, 6 ¶m g‘ g§»¶oÀ¶m ì¶m»¶oMo g‘mYmZ H$aVmV.

¶m MM}À¶m {R>H$mUr, EH$m Hw$VwhbmÀ¶m Jmoï>rH$S>o AmnU bj {Xbo nm{hOo. Á¶m-Á¶mdoir AmnU¶mÑpÀN>H$ à¶moJ H$aVmo Va g§~§YrV KS>Umè¶m g§Yr g‘mZ AgVrb qH$dm ZgVrb.CXmhaUmW© …

(i) ZmUo CS>{dÊ¶mÀ¶m à¶moJm‘Ü¶o N>mn Am{U H$mQ>m {‘iÊ¶mÀ¶m g§Yr g‘mZ KS>VmV.(ii) Ooìhm ’$mgm ’o$H$bm OmVmo, Voìhm 1, 2, 3, 4, 5, 6 ho {‘iÊ¶mÀ¶m g‘mZ g§Yr Amho.

(iii) Ooìhm 4 bmb Am{U 1 {Zim M|Sy> {nedr‘YyZ M|Sy> CMbÊ¶mÀ¶m g‘mZ g§Yr AmhoV.na§Vw bmb M|Sy> CMbÊ¶mÀ¶m e³¶Vm {Zù¶m M|Sy>nojm OmñV AmhoV.

CXmhaU (i) Am{U (ii) ‘Ü¶o àË¶oH$ à¶moJm‘Ü¶o g‘mZ g§Yr ¶oÊ¶mMr e³¶Vm Amho. Xþgè¶m eãXmV, àË¶oH$àmW{‘H$ KQ>Zm hr g‘mZ[aË¶m g§ dZr¶ KQ>Zm Amho. CXmhaU (iii) ‘Yrb a§JrV M|Sy> CMbÊ¶mÀ¶m g‘mZ[aË¶mg§ dZr¶ KQ>Zm AmhoV. VarnU bmb qH$dm {Zim M|Sy> CMbÊ¶mÀ¶m g‘mZ[aË¶m g§ dZr¶ KQ>Zm e³¶ ZmhrV.åhUyZ gd© ghO KS>Umao à¶moJ ¶m‘Ü¶o g‘mZ g§ dZr¶ {ZînÎmr AgVrbM Ago Zmhr. AmVm AmnU g‘mZ[aË¶mg§ dZr¶ KQ>Zm§Mr ì¶m»¶m H$ê$¶m.

¶mÑpÀN>H$ à¶moJmVrb XmoZ qH$dm A{YH$ KQ>Zm ¶m g‘mZ[aË¶m g§^dZr¶ KQ>Zm AmhoV Oa Ë¶mn¡H$sàË¶oH$mbm {ZînÎmrÀ¶m g‘mZ g§Yr AmhoV.

bj Úm … ¶m KQ>H$m‘Ü¶o, Amåhr Ago J¥hrV Yê$ H$s, gd© à¶moJm‘Ü¶o g‘mZ[aË¶m g§ dZr¶ KQ>Zm qH$dm{ZînÎmr AmhoV.

AmVm n¶ªV AmnU ghO KS>Umè¶m à¶moJmer g§~§{YV nXm§Mm Aä¶mg Ho$bobm Amho. AmVm AmnU KQ>ZoÀ¶mg§^mì¶VoMr ì¶m»¶m H$ê$¶m.KQ>ZoMr g§ mì¶Vm (Probability of an event)

Amåhr MMm© Ho$bobr Amho H$s, KQ>ZoMr g§ mì¶Vm hr KS>Umè¶m g§YrMr KQ>Zm n[aUm‘H$maH$nUo ì¶º$ H$aUohr Amho.

AmVm ¶mÀ¶mer g§~§{YV H$m§hr KQ>Zm§Mm {dMma H$am Am{U ¶m KQ>Zm KS>Ê¶mMr g§Yr qH$dm ¶m KQ>Zm§Mrg§ mì¶Vm H$mT>m.

ghO KS>Umao à¶moJ Ogo H$s ZmUo CS>{dUo qH$dm ’$mgm ’o$H$Uo ¶m§Mr Vwåhmbm H$ënZm Amho. Imbrb¶mÔpÀN>H$ à¶moJ Aä¶mgm.CXmhaU 1 …

* MH$Vr (V~H$S>r) da ~mU ‘maUo5 g‘mZ mJ Ho$boë¶m V~H$S>rMm {dMma H$am, Á¶mMo mJ a§J{dbobo AmhoV, 2 bmb,2 {Zio Am{U 1 {hadm Amho.Ooìhm hr V~H$S>r {’$adyZ ~mU ‘mabm OmVmo Am{U Vmo nmMn¡H$s EH$m ^mJmbmbmJVmo d V~H$S>r Wm§~Vo. àË¶oH$ doir V~H$S>r {’$a{dbr AgVm ~mUmÀ¶mpñWVrMr ^{dî¶dmUr H$aUo e³¶ Zmhr.S = {r, r, b, b, g}

e³¶ Agboë¶m {ZînÎmtMr g§»¶m 5 Amho.ho n(S) = 5 Ago Xe©{dVmV.

rb

rg

b

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Probability 101

H$m§hr KQ>Zm ¶m,* Q>moH$ bmb a§JmÀ¶m ^mJmda Amho. E1 = {r, r}* Q>moH$ {haì¶m a§JmÀ¶m ^mJmda Amho. E2 = {g}* Q>moH$ {Zù¶m a§JmÀ¶m ^mJmda Amho. E3 = {b, b}* Q>moH$ {Zù¶m a§JmÀ¶m ^mJmda Zmhr. E4 = {r, r, g}

CXmhaU 2 … * XmoZ {Z:njnmVr ’$mgo EH$mMdoir CS>{dUo. Ooìhm XmoZ ’$mgo ’o$H$bo OmVmV Voìhm Ë¶m§Mm Z‘wZm AdH$me,

S = {(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)(2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)(3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)(4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)(5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)(6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)}

n(S) = 6 × 6 = 36H$m§hr KQ>Zm ¶m,

* XmoZ g‘mZ A§H$ {‘iUmè¶m KQ>Zm A = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}

* XmoZ A§H$mMr ~oarO 5 ¶oUmè¶m KQ>Zm. B = P(1, 4), (2, 3), (3, 2), (4, 1)}* XmoZ A§H$mMr ~oarO 3 nojm H$‘r ¶oUmè¶m KQ>Zm. C = {(1, 1)}

AmVm AmnU Ago H$mhr ¶mÑpÀN>H$ à¶moJ Am{U KQ>Zm§Mm {dMma H$ê$¶m. ¶m KQ>Zm§Mm Z‘wZm AdH$me, AZwHy$b{ZînÎmtMr g§»¶m Am{U KQ>Zm KS>Umè¶m g§YrMr Zm|X H$moï>H$m‘Ü¶o Ho$br Amho. àË¶oH$ CXmhaUmMm Aä¶mg H$am.

1. ZmUo CS>{dUo S = {H, T} N>mn da ¶oUo A = {H}, 2 n¡H$s 112

n(S) = 2 n(A) = 12. ’$mgm ’o$H$Uo S = {1, 2, 3, 4, 5, 6} g‘ g§»¶m A = {2, 4, 6} 6 n¡H$s 3 3

6n(S) = 6 da ¶oVmV n(A) = 35 nojm ‘moR>r g§»¶m A = {6} 6 n¡H$s 1

16

da ¶oVo n(A) = 13. {’$aË¶m V~H$S>rda S = {r, r, b, b, g} Q>moH$ bmb a§JmÀ¶m A = {r, r} 5 n¡H$s 2 2

5~mU ‘maUo n(S) = 5 ^mJmda Wm§~Uo n(A) = 2Q>moH$ bmb a§Jmda A = {b, b, g} 5 n¡H$s 3

35

Wm§~V Zmhr n(A) = 34. XmoZ ZmUr EH$mM AJmoXaÀ¶m nmZmda 102 g‘mZ A§H$ A = {(1,1), (2,2), 36 n¡H$s 6 6

36doir CS>{dbr da nmhm n(S) = 36 da ¶oUo (3,3), (4,4) (5,5)(6,6)}n(A) = 6

~oarO 3 nojm A = {(1,1)} 36 n¡H$s 1136

H$‘r ¶oVo. n(A) = 1

ghO KS>Umaoà¶moJ

CXm.g§.

Z‘wZm AdH$meS

KQ>Zm AKQ>Zoer AZwHy$b

{ZînÎmr AKQ>Zm KS>Ê¶mMr

g§Yr

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102 UNIT-5

H$moï>H$mVrb eodQ>À¶m ñV§ mMo {ZarjU H$am. Vo KQ>ZoV KS>Umè¶m g§Yr qH$dm KQ>ZoMr g§ mì¶Vm Xe©{dVmV.g§»¶mË‘H$ qH$‘Vr H$moUË¶m ñdê$nmV {b{hboë¶m AmhoV? Ë¶m AnyUmªH$s ñdê$nmV AmhoV. Á¶m‘Ü¶o A§eKQ>ZoÀ¶m AZwHy$b {ZînÎmtMr g§»¶m Am{U N>oX e³¶ Agboë¶m {ZînÎmtMr g§»¶m Xe©{dVmV.

gd© gmYmaUnUo E hr H$moUVrhr EH$ KQ>Zm Agob Va,E ¶m KQ>Zoer AZwHy$b àmW{‘H$ KQ>Zm§Mr g§»¶m E = n(E),Z‘wZm AdH$memVrb EHy$U àmW{‘H$ KQ>Zm§Mr g§»¶m S=n(S),E ¶m KQ>ZoMr g§ mì¶Vm = P(E),

KQ>ZoMr g§^mì¶Vm =KQ>Zoer AZwHy$b àmW{‘H$ KQ>Zm§Mr g§»¶m

Z‘wZm AdH$memVrb EHy$U àmW{‘H$ KQ>Zm§Mr g§»¶m P(E) =

AnyUmªH$ hm JwUmoÎma ñdê$nmV gwÕm {b{hbm OmVmo. AmnU KQ>ZoÀ¶m g§ mì¶VoMr ì¶m»¶m Imbrbà‘mUoH$ê$ eH$Vmo.

KQ>ZoMr g§ mì¶Vm hr E ¶m KQ>Zoer AZwHy$b Agboë¶m àmW{‘H$ KQ>Zm§Mr g§»¶m Am{U Z‘wZm AdH$memVrbEHy$U àmW{‘H$ KQ>Zm§Mr g§»¶m ¶m§Mo JwUmoÎma Amho.

Xþgè¶m eãXmV, Oa ¶mÑpÀN>H$ Z‘wZm AdH$me‘Ü¶o 'n' {ZînÎmr Am{U E ¶m KQ>Zobm AZwHy$b 'm'AmhoV, Va E ¶m KQ>ZoMr g§^mì¶VoMr ì¶m»¶m m Mo n er JwUmoÎma ho Amho.

P(E) = m : n =(E)(S)

m nn n

AmVmn¶ªV AmnU MMm© Ho$br Amho H$s, KQ>ZoMr g§ mì¶Vm hr AnyUmªH$s ñdê$nmVrb g§»¶mË‘H$ qH$‘VAmho.

ho gd© KQ>Zm§Zm gË¶ Amho H$m? g§ mì¶Vm AnyUmªH$s Zgboë¶m KQ>Zm AmhoV H$m? H$m§hr CXmhaUm§Mm {dMmaH$am.AmnU ’$mgm ’o$H$bm Ago g‘Oy

(i) Z¡g{J©H$ g§»¶m ¶oÊ¶mMr g§ mì¶Vm {H$Vr?

¶oWo, S = {1, 2, 3, 4, 5, 6} n(S) = 6

E = {Z¡g{J©H$ g§»¶m {‘i{dUo}E = {1, 2, 3, 4, 5, 6} n(E) = 6

P(E) =(E)(S)

nn =

66

= 1

(ii) 7 nojm bhmZ g§»¶m {‘i{dÊ¶mMr g§ mì¶Vm {H$Vr?n(S) = 6

E = {7 nojm bhmZ g§»¶m {‘iUo}E = {1, 2, 3, 4, 5, 6} n(E) = 6

P(E) =(E) 6(S) 6

nn = 1

darb XmoZ àg§Jm‘Ü¶o, Vwåhr nmhmb H$s, E = S Am{U n(E) = n(S) Am{U P(E) = 1, E ¶m KQ>ZoMr

n(E)n(S)

Q>rn … g§^mì¶VoMr emór¶ì¶m»¶m Cn¶moJr nS>Vo. Oa e³¶Agboë¶m {ZînÎmtMr g§»¶m‘¶m©{XV Am{U {ZînÎmr g‘mZ[aË¶mg§ dZr¶ AmhoV.

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Probability 103

g§ mì¶Vm 1 nyU© g§»¶m Amho. Aem KQ>Zm§Zm Z³H$s KQ>Zm Ago åhUVmV. Ooìhm Z‘wZm AdH$memVrb gd© KQ>H$KQ>Zoer AZwHw$b AmhoV.

¶mÑpÀN>H$ à¶moJmVrb Z‘wZm AdH$membm Z³H$s qH$dm ImÌrMr KQ>Zm åhUVmV. Oa H$moUË¶mhr MmMUrà¶moJm‘Ü¶o H$moUVmVar EH$ KQ>H$ Z³H$s AmT>iVmo.

Z³H$s KQ>ZoMr g§ mì¶Vm 1 Amho.

{dMma H$am !Ooìhm ZmUo CS>{dbo Va N>mn qOH$Vmo Am{U H$mQ>m Ambr Va haVmo. hr H$moUË¶m àH$maMr KQ>Zm Amho?

Ë¶mMr g§ mì¶Vm {H$Vr?(iii) 7 nojm ‘moR>r g§»¶m {‘i{dÊ¶mMr g§ mì¶Vm {H$Vr?

n(S) = 6

A = {7 nojm ‘moR>r g§»¶m}

A = { } = n(A) = 0

P(A) =(A) 0(S) 6

nn = 0

(iv) 1 nojm bhmZ g§»¶m {‘i{dÊ¶mMr g§ mì¶Vm {H$Vr?

n(s) = 6

A = {1 nojm bhmZ g§»¶m}

A = { } n(A) = 0

P(A) =(A) 0(S) 6

nn = 0

darb XmoZ àg§Jm‘Ü¶o Vwåhr nmhmb H$s,

A = Am{U n(A) = 0, P(A) = 0 i.e., 'A' ¶m KQ>ZoMr g§ mì¶Vm '0' Amho. hr EH$ nyU© g§»¶m Amho.Aem KQ>Zm§Zm Ae³¶ KQ>Zm Ago åhUVmV.

H$moUË¶mhr MmMUr à¶moJm‘Ü¶o Or KQ>Zm H$moUË¶mhr pñWVrV KSy>Z ¶oV Zmhr {Vbm Ae³¶ KQ>Zm AgoåhUVmV.

Ae³¶ KQ>ZoMr g§ mì¶Vm '0' AgVo.

darb MM}dê$Z AmnU Agm {ZîH$f© H$mTy> eH$Vmo H$s EImÚm KQ>ZoÀ¶m g§ mì¶Vo{df¶r Imbrb ~m~r ¶oVmV.

* Oa A = , Va n(A) = 0 P(Ae³¶ KQ>Zm) = 0

* Oa A = S, Va n(A) = n(S) P(ImÌrMr KQ>Zm) = 1

* A hr Aer KQ>Zm Amho, Á¶m‘Ü¶o Ae³¶ KQ>Zm Am{U Z³H$s KQ>Zm gmoSy>Z BVa AmhoV. n(A) > 0 Am{U n(A) < n(S)

P(A) > 0 Am{U P(A) < 1

0 P(A) 1 Ae³¶ KQ>Zogh H$moUË¶mhr A KQ>ZogmR>r Am{U Z¸$s KQ>ZogmR>r 0 P(A) 1

EImÚm KQ>ZoMr g§ mì¶Vm hr 0 Am{U 1 ‘Yrb g§»¶m Amho Am{U Ë¶m‘Ü¶o 0 Am{U 1 AmhoV, ’$º$ Oa

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104 UNIT-5

Ë¶m‘Ü¶o AZwH«$‘o Ae³¶ KQ>Zm Am{U ImÌrÀ¶m KQ>Zm AmhoV.AmVm, AmnU KQ>ZoÀ¶m g§ mì¶VoMo ñnï>rH$aU H$ê$.

* Zoh‘r 0 Am{U 1 ¶m XmoZ nyU© g§»¶m§À¶m‘Ü¶o AgVo.* hm 0 nojm ‘moR>m Am{U 1 noojm bhmZ AnyUmªH$ Amho.* KQ>ZoMr g§ mì¶Vm 0 qH$dm 1 Agy eH$Vo qH$dm 0 Am{U 1 ‘Yrb H$moUVmhr AnyUmªH$ Agy eH$Vmo.

darb ¶w³Ë¶mÀ¶r g§^mì¶Vm Imbrbà‘mUo à‘mUmV Xe©{dVm ¶oVo.0.5

0 ½Impossible Certain

EImÚm KQ>ZoMr g§ mì¶Vm H$mT>Ê¶mÀ¶m H$m§hr CXmhaUm§Mo ñnï>rH$aU Imbrbà‘mUo {Xbobo Amho. Ë¶m§MmAä¶mg H$am.


CXmhaU 1 … ’$mgm ’o$H$bm. Imbrb KQ>Zm {‘i{dÊ¶mMr g§ mì¶Vm H$mT>m.(i) 5 hr g§»¶m ¶oÊ¶mMr (ii) {df‘ g§»¶m ¶oÊ¶mMr(iii) 2 nojm ‘moR>r g§»¶m ¶oÊ¶mMr (iv) 6 Mo ‘yi Ad¶d ¶oÊ¶mMr

CH$b … ’$mgm ’o$H$ë¶mda ¶oUmao, Z‘wZm AdH$me S = {1, 2, 3, 4, 5, 6} n(S) = 6

(i) 5 hr g§»¶m ¶oÊ¶mMr 'A' hr KQ>Zm Amho Ago ‘mZy.

A = { 5 } n(A) = 1

P(A) =(A) 1(S) 6

nn

(ii) {df‘ g§»¶m {‘i{dÊ¶mMr 'B' hr KQ>Zm Amho Ago ‘mZy.

B = {1, 3, 5} n(B) = 3

P(B) =(B) 3 1(S) 6 2

nn

(iii) 2 nojm ‘moR>r g§»¶m {‘i{dÊ¶mMr 'C' hr EH$ KQ>Zm Amho Ago ‘mZy.C = {3, 4, 5, 6} n(C) = 4

P(C) =(C) 4 2(S) 6 3

nn

(iv) 6 Mo ‘yi Ad¶d {‘i{dÊ¶mMr 'D' hr KQ>Zm Amho Ago g‘Oy.

D = {2, 3} n(D) = 2

P(D) =(D) 2 1(S) 6 3

nn

CXmhaU 2 … EH$ Mm§Jbo ZmUo XmoZ doim CS>{dbo Va Imbrb KQ>Zm {‘i{dÊ¶mMr g§ mì¶Vm H$mT>m.(i) XmoZ N>mn (ii) H$‘rV-H$‘r EH$ N>mn(iii) N>mn Zgbobr (iv) Zo‘H$m EH$ H$mQ>m

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Probability 105

CH$b … Oa EH$ CÎm‘ ZmUo XmoZ doim da CS>{dbo Va Z‘wZm AdH$me S = {HH, TT, HT, TH} n(S) = 4

(i) A hr XmoZ N>mn {‘i{dÊ¶mMr KQ>Zm ‘mZy.

A = {HH) n(A) = 1 P(A) =(A) 3(S) 4

nn =(C) 1(S) 4

(ii) B hr H$‘rV H$‘r EH$ N>mn {‘i{dÊ¶mMr KQ>Zm ‘mZy.

B = {HH, HT, TH) n(B) = 3

P(B) =(B) 3(S) 4

nn

(iii) C hr KQ>Zm N>mn Z ¶oÊ¶mMr Amho Ago g‘Oy

C = {TT) n(C) = 1

P(C) =(C) 1(S) 4

nn

(vi) D hr Zo‘H$s EH$ H$mQ>m ¶oÊ¶mMr KQ>Zm ‘mZy.

D = { HT, TH) n(D) = 2

P(D) =(D) 2(S) 4

nn

CXmhaU 3 … VrZ CÎm‘ ZmUr EH$mM doir EH${ÌV CS>{dbr Va Imbrb KQ>Zm {‘iÊ¶mMr g§^mì¶Vm H$mT>m.(i) gd© H$mQ>m (ii) H$‘rV H$‘r EH$ H$mQ>m(iii) OmñVrV OmñV EH$ H$mQ>m (iv) OmñVrV OmñV XmoZ N>mn

CH$b … VrZ CÎm‘ ZmUr EH$mM doir EH${ÌV CS>{dbr, Va Z‘wZm AdH$me

S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} n(S) = 8

(i) A hr gd© H$mQ>m {‘i{dÊ¶mMr KQ>Zm Amho.

A = { TTT } n(A) = 1

P(A)=(A) 1(S) 8

nn

(ii) B hr H$‘rV H$‘r EH$ H$mQ>m Agbobr KQ>Zm Amho.

B = {HHT, HTH, THH, HTT, THT, TTH, TTT}

n(B) = 7 P(B) =(B) 7(S) 8

nn

(iii) C hr OmñVrV OmñV EH$ H$mQ>m {‘i{dÊ¶mMr KQ>Zm Amho.

C = {HHH, HHT, HTH, THH}

n(C) = 4 P(C)=(C) 4(S) 8

nn

(iv) OmñVrV OmñV XmoZ N>mn ¶oÊ¶mMr D hr KQ>Zm Amho.

D = {HHT, HHT, HTH, THH, HTT, THT, TTH, TTT}

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106 UNIT-5

n(D) = 7 P(D) =(D) 7(S) 8

nn

CXmhaU 4 … XmoZ {Z…njnmVr ’$mgo EH$XmM {’$a{dbo Va Imbrb KQ>Zm {‘iÊ¶mMr g§^mì¶Vm {H$Vr?

(i) XmoÝhr A§H$ gmaIoM (ii) A§H$mOr ~oarO 7 (iii) A§H$mMr ~oarO 10 nojm H$‘r

CH$b … Amnë¶mbm ‘m{hV Amho H$s, Ooìhm XmoZ {Z…njnmVr ’$mgo EH$XmM {’$a>{dbr, Va e³¶ Agboë¶m EHy$U{ZînÎmtMr g§»¶m 6 × 6 = 36 Amho.

n(S) = 36

(i) A hr XmoÝhr A§H$ gmaIoM Agbobr KQ>Zm ‘mZy.

A= {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}

n(A) = 6 P(A) =(A) 6(S) 36

nn

(ii) B hr XmoZ A§H$mMr ~oarO 7 ¶oUmar KQ>Zm ‘mZy.

B= {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}

n(B) = 6 P(B) =(B) 6(S) 36

nn

(iii) C hr XmoZ A§H$mMr ~oarO 10 nojm H$‘r AgUmar KQ>Zm ‘mZy.

C = {(1, 6), (1, 5), (1, 4), (1, 3), (1, 2), (1, 1), (1, 2), (2, 2), (2, 3), (2, 4), (2, 5),(2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (5,1), (5, 2), (5, 3), (5, 4), (6, 1), (6, 2), (6, 3)}

n(C) = 30

P(C)=(C) 30(S) 36

nn

CXmhaU 5 … EH$m dJm©‘Ü¶o 30 ‘wbo Am{U 25 ‘wbr AmhoV. {dÚmÏ¶mªMr {ZdS> ghOnUo Ho$br. {ZdS>boë¶m{dÚmÏ¶mªMr g§ mì¶Vm H$mT>m. (i) ‘wbo (ii) ‘wbr

CH$b … EHy$U {dÚmÏ¶mªMr g§»¶m = 30 + 25 = 55

n(S) = 55(i) A hr ‘wbm§Mr {ZdS> H$aÊ¶mMr KQ>Zm ‘mZy. n(A) = 30

(ii) B hr ‘wbrMr {ZdS> H$aÊ¶mMr KQ>Zm ‘mZy. n(B) = 25

CXmhaU 6 … {ZdS>boë¶m {bn dfm©‘Ü¶o 53 a{ddma AmhoV Va Ë¶mMr g§^mì¶Vm {H$Vr?CH$b … {bn dfm©‘Ü¶o Agbobo {Xdg = 366 {Xdg

366 {Xdg = 52 AmR>dS>o Am{U 2 {Xdgam{hbobo XmoZ {Xdg ho(i) a{ddma Am{U gmo‘dma (ii)gmo‘dma Am{U ‘§Jidma

(iii) ‘§Jidma Am{U ~wYdma (iv) ~wYdma Am{U Jwédma(v) Jwédma Am{U ewH«$dma (vi) ewH«$dma Am{U e{Zdma

(vii) e{Zdma Am{U a{ddma53 a{ddma Agboë¶m {bn dfm©‘Ü¶o eodQ>Mo XmoZ {Xdg a{ddma Am{U gmo‘dma qH$dm e{Zdma Am{U a{ddma¶mn¡H$s EH$ AmhoV.

P(A) =(A) 30(S) 55

nn

P(B) =(B) 25(S) 55

nn

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Probability 107

Aem AZwHy$b {ZînÎmtMr g§»¶m = n(A) = 2

Z‘wZm AdH$mem‘Ü¶o EHy$U e³¶ Agboë¶m {ZînÎmtMr g§»¶m = n(S) = 7

P(A) =(A)(S)

nn P(53 a{ddma Agbobo {bn df©)=

27

CXmhaU 7 … EH$m {nedr‘Ü¶o 6 bmb M|Sy> Am{U H$mhr {Zio M|Sy> AmhoV. {Zim M|Sy> H$mT>Ê¶mMr g§ mì¶Vm hr bmbM|Sy> H$mT>Ê¶mÀ¶m g§ mì¶VoÀ¶m XþßnQ> Amho. Va {nedrVrb {Zù¶m M|Sy>Mr g§»¶m H$mT>m.

CH$b … {Zù¶m M|Sy>Mr g§»¶m = x ‘mZy.EHy$U M|Sy>Mr g§»¶m = (6 + x)

P(bmb M|Sy> H$mT>Ê¶mMr g§ mì¶Vm) =6

6 x

P({Zim M|Sy> H$mT>Ê¶mMr g§^mì¶Vm)= 6x

x 6x

x = 626 x 6

xx =

126 x

12(6 + x) = x(6 +x) 72 + 12x = 6x + x2

x2 – 6x – 72 = 0 (x – 12)(x + 6) = 0 x – 12 = 0 or x + 6 = 0 x = 12 or x = – 6

{Zù¶m M|Sy>Mr g§»¶m = 12

CXmhaU 8 … CÎm‘ {ngboë¶m 52 nÎ¶m§À¶m gQ>mVyZ EH$ nÎmm ghOnUo H$mT>bm Va Imbrb KQ>Zm§Mr g§^mì¶VmH$mT>m.(i) amUr (ii) bmb amOm (iii) Mm¡H$Q> (iv) ~Xm‘Mm¡H$Q> ~Xm‘ BpñnH$ {H$bmda

A A A A

CH$b … {H$bmda Am{U BpñnH$Mo nÎmo H$mù¶m a§JmMo AmhoV. ~Xm‘ Am{U Mm¡H$Q>Mo nÎmo bmb AmhoV. àË¶oH$ g§MmMo 13 nÎmoAmhoV. Ë¶m‘Ü¶o 3 à{Vð>rV qH$dm Xe©Zr n¥ð> mJmMo nÎmo AmhoV. Vo åhUOo amOo (K), amÊ¶m (Q) Am{U Jwbm‘ (J)

CÎm‘ {ngboë¶m nÎ¶m§À¶m gQ>mVyZ g‘mZarË¶m g§ dZr¶ {ZînÎmr {‘iVrb. 52 Ioim§À¶m nÎ¶m§Mo Imbrbà‘mUo dJuH$aU Ho$bo Amho.

BpñnH$ A 2 3 4 5 6 7 8 9 10 J Q K 13

~Xm‘ A 2 3 4 5 6 7 8 9 10 J Q K 13

{H$bmda A 2 3 4 5 6 7 8 9 10 J Q K 13

Mm¡H$Q> A 2 3 4 5 6 7 8 9 10 J Q K 13

CÎm‘ {ngboë¶m 52 nÎ¶m§n¡H$s EH$ nÎmm 52C1 àH$mao {ZdS>Vm ¶oVmo. n(S) = 52

(i) ¶oÊ¶mMr g§^mì¶Vm n(Q) = 4

P(Q) =(Q) 4(S) 52

nn

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108 UNIT-5

(ii) bmb amOm ¶oÊ¶mMr g§^mì¶Vm n(RK) = 2

P(RK) =2

52(S)nn(RK)

(iii) Mm¡H$Q> ¶oÊ¶mMr g§^mì¶Vm n(D) = 13

P(D) =1852(S)n

n(D)

(iv) ~Xm‘ ¶oÊ¶mMr g§^mì¶Vm n(H) = 13

P(H) =1352(S)n

n(H)

CXmhaU 9 … XmoZ ghO KS>Umao à¶moJ {dMmamV ¿¶m.

(i) CÎm‘ ZmUo EH$Xm CS>{dbo. (ii) CÎm‘ ’$mgm EH$Xm ’o$H$bm.

àË¶oH$ àg§JmVrb àË¶oH$ àmW{‘H$ KQ>ZoMr g§ mì¶Vm H$mT>m Am{U ~oarO H$am. Ë¶mdê$Z Vwåhr H$moUVm

{ZîH$f© H$mTy> eH$Vm?

CH$b … (i) EH$ ZmUo CS>{dUo

S = {H, T} n(S) = 2

Ë¶mÀ¶m àmW{‘H$ KQ>Zm,

E1 = { H } E2 = { T } P(E1) =12 P(E2) =

12

P(E1) Am{U P(E2) Mr ~oarO Ho$br AgVm.

P(E1) + P(E2) =1 12 2

= 1

(ii) ’$mgm ’o$H$Uo

S = {1, 2, 3, 4, 5, 6} n(S) = 6

Ë¶mÀ¶m àmW{‘H$ KQ>Zm,

E1 = { 1 }, E2 = { 2 }, E3 = { 3 }, E4 = { 4 }, E5 = { 5 }, E6 = { 6 }

P(E1) = P(E2) = P(E3) = P(E4) = P(E5) = P(E6) =16

darb gd© àmW{‘H$ KQ>Zm§Mr ~oarO Ho$br AgVm Amnë¶mbm {‘iob.

P(E1) + P(E2) + P(E3) + P(E4) + P(E5) + P(E6)

=1 1 1 1 16 6 6 6 6 =

66 = 1

darb CXmhaUm‘Ü¶o AmnU nmhVmo H$s, à¶moJmVrb gd© àmW{‘H$ KQ>Zm§À¶m g§ mì¶VoMr ~oarO 1 Amho. gd©

gmYmaUnUo ho gdmªgmR>r gË¶ Amho.

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Probability 109

ñdmÜ¶m¶ 5.1

1. EH$ ’$mgm CS>{dbm. Va Imbrb KQ>Zm {‘i{dÊ¶mMr g§ mì¶Vm H$mT>m.

(i) 4 hr g§»¶m (ii) dJ© g§»¶m

(iii) KZ g§»¶m (iv) 1 nojm ‘moR>r g§»¶m

2. XmoZ ZmUr EH${ÌV da CS>{dbr. Va Imbrb KQ>Zm {‘i{dÊ¶mMr g§ mì¶Vm H$mT>m.

(i) H$mQ>m Zmhr (ii) OmñVrV OmñV XmoZ H$mQ>m

(iii) Z³H$s EH$ N>mn (iv) H$‘rV H$‘r EH$ H$mQ>m

3. VrZ ZmUr EH${ÌVnUo da CS>{dbr Va Imbrb KQ>Zm {‘i{dÊ¶mMr g§ mì¶Vm H$mT>m.

(i) H$‘rV H$‘r EH$ N>mn (ii) OmñVrV OmñV XmoZ N>mn

(iii) N>mn Zmhr (iv) gd© N>mn

4. XmoZ ’$mgo EH${ÌVnUo da CS>{dbo Va Imbrb KQ>Zm {‘i{dÊ¶mMr g§ mì¶Vm H$mT>m.

(i) A§H$mMr ~oarO 8 (ii) g§»¶m§Mr ~oarO 12 nojm H$‘r

(iii) ~oaOobm 4 Zo ^mJ OmVmo (iv) JwUmH$ma 12

(v) JwUmH$ma 20 nojm H$‘r (vi) JwUmH$mambm 5 Zo ^mJ OmVmo.

5. 1 nmgyZ 50 n¶ªVÀ¶m g§»¶oMr {ZdS> ghOnUo Ho$br. Va Ë¶m§Mr g§ mì¶Vm {H$Vr?

(i) ‘yi g§»¶m (ii) nyU© KZ Zmhr

(iii)nyU© dJ© Amho (iv) {ÌH$moUmH¥$Vr g§»¶m Amho

(v) 6 À¶m nQ>rVrb (vi) 2 À¶m nQ>rVrb Zmhr.

6. 52 nÎmo {ngyZ 1 nÎmm ghOnUo CMbbm. da Imbrbn¡H$s EH$ nÎmm CMbÊ¶mMr g§ mì¶Vm H$mT>m.

(i) bmb a§JmMm nÎmm (ii) H$mù¶m a§JmVm nÎmm Zmhr

(iii) E³H$m Zmhr (iv) Mm¡H$Q> Zmhr

(v)H$mim amOm (vi) 10 nojm H$‘r g§»¶m Agbobm ~Xm‘

7. 2, 5, 7 ho A§H$ nwÝhm-nwÝhm Z KoVm ¶m g§»¶mnmgyZ 2 A§H$s g§»¶m ~ZVmV. Imbrb g§»¶m ~ZVmV Va

Ë¶m§Mr g§ mì¶Vm H$mT>m.

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110 UNIT-5

(i) dJ© g§»¶m (ii) 3 Zo ^mJ OmUmè¶m

(iii)52 nojm ‘moR>r (iv) 57 nojm bhmZ

(v) 25 nojm bhmZ (vi) nyU© g§»¶m

8. 30 CÎm‘ Am§ã¶m‘Ü¶o 9 Hw$Obobo Am§~o {‘gibo. ghOnUo EH$ Am§~m {ZdS>bm. Va

(i) CÎm‘ Am§~m (ii) Hw$Obobm Am§~m ho {ZdS>Ê¶mMr g§ mì¶Vm {H$Vr?

9. hmoirÀ¶m gUmbm gmoZmbrZo bmb, {Zim, {hadm, Jwbm~r, {ndim, Om§ im Am{U Zma§Jr a§J dmnê$Z 7 ~mQ>ë¶m a§JrV

nmÊ¶mZo ^aë¶m. EH$ ~mQ>br ghOnUo {ZdS>br. Imbrb a§JmMr ~mQ>br {ZdS>Ê¶mMr g§ mì¶Vm {H$Vr?

(i) Zma§Jr a§J (ii) {ndim a§J Zmhr

(iii) ZmernwS>r a§J (VnH$sar a§J) (iv) bmb qH$dm {hadm ¶mn¡H$s EH$

(v) {ndim qH$dm Jwbm~r Zmhr

10. EH$m noQ>rV 144 noZ AmhoV Ë¶mn¡H$s 20 Iam~ Am{U am{hbobo CÎm‘ AmhoV. Oa EH$m ì¶º$sZo Mm§Jbo noZ

IaoXr Ho$bo nU Iam~ IaoXr Ho$bo ZmhrV. XþH$mZXmamZo noQ>rVyZ ghOnUo EH$ noZ H$mTy>Z Ë¶m ì¶º$sbm {Xbo.

Va Vmo ì¶º$s

(i) Vo IaoXr H$aVmo (ii) Vo IaoXr H$aV Zmhr ¶mMr g§ mì¶Vm {H$Vr?

KQ>Zm (Events)

AmVmn¶ªV AmnU KQ>Zm {df¶r Aä¶mg Ho$bobm Amho. KQ>Zmda AmYmarV Ë¶m§Mo dJuH$aU doJdoJù¶màH$mamV Ho$bo OmVo.

KQ>Zm

Z³H$s KQ>Zm gmÜ¶m KQ>Zm g§¶wº$ KQ>ZmAm{U qH$dm

Ae³¶ KQ>Zm àmW{‘H$ KQ>Zm

AmVm AmnU AmUIr H$m§hr KQ>Zm§Mm Aä¶mg H$ê$¶m Á¶m gmÜ¶m qH$dm g§¶w³V KQ>Zm AmhoV.

H$moQ>r KQ>Zm (Complementary events)

AmnU ’$mgm EH$Xm ’o$H$bm Ago g‘Oy. Ë¶mÀ¶m XmoZ KQ>Zm§Mm {dMma H$am.

(i) g‘ g§»¶m {‘i{dUo - E1 E1 = {2, 4, 6}

(ii) {df‘ g§»¶m {‘i{dUo - E2 E2 = {1, 3, 5}

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Probability 111

AmVm AmUIr EH$ ""g‘ g§»¶m Z {‘iUo'' hr KQ>Zm ¿¶m. = {1, 3, 5}

XmoZ KQ>Zm§Mr VwbZm H$am.

""{df‘ g§»¶m {‘i{dUo'' Am{U ""g‘ g§»¶m Z {‘iUo''

Ë¶m XmoÝhr g‘mZ AgyZ ~amo~a {1, 3, 5} Amho.

E2 hr E Zmhrer g‘mZ Amho, AmnU nmhVmo H$s E1 hr KQ>Zm KS>Vo ’$º$ Ooìhm E2 hr KS>V Zmhr Am{UCbQ>njr

E1 Am{U E2 ¶m KQ>Zm§Zm H$moQ>r KQ>Zm Ago åhUVmV.

åhUOoM "KQ>Zm E' hr Zgbobr "KQ>Zm E' er H$moQ>r Am{U "Zgbobr KQ>Zm E' hr "KQ>Zm E'Mr H$moQ>rAmho.E KQ>ZoMr H$moQ>r E Zo Xe©{dVmV.

E KQ>ZoÀ¶m H$moQ>rbm "E Zgbobr KQ>Zm' gwÕm åhUVmV. KQ>ZoÀ¶m H$moQ>rbm "KQ>ZoMm ZH$ma' Ago gwÕm åhUVmV.

AmVm, AmnU KQ>Zm E Am{U {VMr H$moQ>r KQ>Zm E ¶m‘Yrb g§~§Y H$mTy>.

Imbrb H$moï>H$ Aä¶mgm. H$m§hr KQ>Zm, Ë¶mÀ¶m H$moQ>r KQ>Zm Am{U Ë¶m§Mr g§ mì¶Vm {Xbobr Amho.

P(E) Am{U P(E ) {df¶r AmnU H$moUVm {ZîH$f© H$mTy> eH$Vmo?

ghO KS>Umao KQ>Zm E H$moQ>r KQ>Zm qH$dm P(E) P(E ) P(E) + P(E ) à¶moJ Zgbobr KQ>Zm E

ZmUo CS>{dUo N>mn {‘i{dUo H$mQ>m {‘i{dUo

S {H, T} qH$dm

n(S) = 2 E = {H} N>mn Z {‘i{dUo12

12

12 +

12 = 1

E = {T}

2) ’$mgm ’o$H$Uo 3 nojm ‘moR>r 3 nojm ‘moR>r Zgbobr

g§»¶m {‘i{dUo g§»¶m {‘i{dUo36

36

36

+36

= 1

S = {1,2,3,4,5,6} E = {4, 5, 6} E = {1, 2, 3}n(S) = 6

VrZÀ¶m nQ>rVrb VrZÀ¶m nQ>rV

g§»¶m {‘i{dUo Zgbobr g§»¶m {‘i{dUo26

46

26 +

46 = 1

E = {3, 6} E = {1, 2, 4, 5}

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112 UNIT-5

darb H$moï>H$mdê$Z, Amåhr Agm {ZîH$f© H$mTy> H$s,P(E) + P(E ) = 1 P(E ) = 1 – P(E)

Xþgè¶m eãXmV, P(E) + P(E Zmhr) = 1P(E Zmhr) = 1 – P(E) P(E) = 1 – P(E Zmhr)

AmnU gQ>m§Mr ^mfm dmnê$Z gwÕm darb g§~§Y {gÕ H$ê$ eH$Vmo.Amåhmbm ‘m{hV Amho, E E = S Am{U E E =

n(E) + n(E ) = n(S)n(S) Zo gdm©Zm ^mJbo.

(E)(S)

nn +

(E)(S)

nn =

(S)(S)

nn P(E) + P( E ) = 1 P(E) = 1 – P( E )

Q>rn … Oa A hr KQ>Zm 'm' doim KS>Vo Am{U 'n' doim KS>V Zmhr, ¶m gd© g‘mZ[aË¶m g§ dZr¶ KS>VmV Va

* KQ>Zm KS>Ê¶mMr g§ mì¶Vm = mm n

* KQ>Zm Z KS>Ê¶mMr g§ mì¶Vm =n

m n


CXmhaU 1 … Oa gm‘Zm qOH$Ê¶mMr g§^mì¶Vm 0.3 Amho. Va gm‘Zm J‘{dÊ¶mMr g§ mì¶Vm {H$Vr?CH$b … A hr gm‘Zm qOH$Ê¶mMr KQ>Zm Amho. Va A hr gm‘Zm haÊ¶mMr KQ>Zm Amho.

{Xbobo. P(A) = 0.3.Amåhmbm ‘m{hV Amho H$s, P(A) + P( A ) = 1

P( A ) = 1 - P(A) P( A ) = 1 - 0.3 = 0.7gm‘Zm J‘{dÊ¶mMr g§ mì¶Vm 0.7.

CXmhaU 2 … Oa A hr ¶mÑpÀN>H$ à¶moJmMr KQ>Zm Aem[aVrZo Amho H$s,

P(A) : P (A ) = 5:11, Va (i) P(A) Am{U P (A ) H$mT>m (ii) P(A) + P(A ) = 1 ho VnmgyZ nmhm.

CH$b … (i) {Xbobo P(A) : P( A ) = 5:11P(A)P(A)

511 11 P(A = 5 P( A )

11 P(A) = 5 [1-P( A )] ( P(A) + P( A ) = 1)

11 P(A) = 5 – 5 P( A ) 11 P(A) + 5 P(A)=5 6 P(A = 5

P(A) =5

16P( A ) = 1 – P(A)

P( A ) = 1 –5

16P( A ) =

1116

{dMma H$am …

H$moUË¶mhr ghO KS>Umè¶mà¶moJm‘Ü¶o Z³H$s KQ>ZoMrH$moQ>r hr Ae³¶ KQ>Zm AmhoH$m? ¶m KQ>ZmgmR>r darbMMm© Ho$bobm g§~§Y bmJy nS>VmoH$m? JQ> H$ê$Z MMm© H$am.

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Probability 113

(ii) P(A) + P( A ) = 1

S>mdr ~mOy (LHS) = P(A) + P( A )

=5

16 +1116 =

1616 = 1 = COdr ~mOy (RHS)

nañna§mer Ho$di KQ>Zm (Mutually exclusive events)’$mgm ’o$H$Ê¶mÀ¶m ghO KS>Umè¶m à¶moJmMm {dMma H$ê$¶m. S = {1, 2, 3, 4, 5, 6}Aem KQ>Zm ‘mZy,

E1 = 3 nojm bhmZ {‘iUmar g§»¶m E1 = {1, 2}E2 = 2 nojm ‘moR>r {‘iUmar g§»¶m E2 = {5, 6}E1 Am{U E2 ¶m H$moQ>r KQ>Zm ZmhrV. E1 E2 S

Va ¶m KQ>Zm H$moUË¶m àH$maÀ¶m AmhoV?bj Úm H$s E1 Am{U E2 ¶m EH${ÌV qH$dm EH$mMdoir KS>V ZmhrV. Ooìhm E2 hr KS>V Zgob Va ’$º$

E1 KS>Vo Am{U CbQ>njr.Voìhm AmnU E1 Am{U E2 ¶m KQ>Zm nañnamer Ho$di AmhoV Ago åhUy. åhUOoM EH$ KQ>Zm KS>V Agob

Va Xþgè¶m KQ>ZoÀ¶m KS>Ê¶mg ZmH$maVo.XmoZ qH$dm A{YH$ KQ>Zm nañnamer Ho$di AgVmV. Oa EH$ KQ>Zm KS>V Agob Va Xþgar KQ>Zm KS>Ê¶mgWm§~[dVmV qH$dm ZmH$maVmV.

Oa E1 Am{U E2 ¶m nañnamer Ho$di KQ>Zm AmhoV Va, E1 E2 = .

gQ> {gÕm§VmÀ¶m ^mfo‘Ü¶o E1 Am{U E2 ho Ag§~§YrV gQ> AmhoV.Ë¶mZm gm‘mB©H$ {ZînÎmr qH$dm àmW{‘H$ KQ>Zm ZmhrV. n(E1 E2) = n(E1) + n(E2)

XmoÝhr ~mOybm n(S) Zo ^mJ {Xë¶mgAmåhmbm {‘iob.

1 2(E E )(S)

nn = 21 (E )(E )

(S) (S)nn

n n P(E1 E2) = P(E1) + P(E2)

¶m {ZH$mbmbm g§^mì¶VoÀ¶m ~oaOoMm {Z¶‘ åhUVmV.

bjmV R>odm …

• XmoZ nañnam§er Ho$di KQ>Zm§Mr g§ mì¶Vm H$mT>Ê¶mgmR>r hm {Z¶‘ dmnaVmV.• XmoZ nojm OmñV nañnamer Ho$di KQ>Zm§gmR>rgwÕm hm {Z¶‘ dmT>{dbm OmD$ eH$Vmo.

If E1, E2, E3 ...............En ¶m nañnamer Ho$di KQ>Zm AmhoV Va,P(E1 E2 E3 ...........En) = P(E1) + P(E2) + P(E3) ...........P(En)


CXmhaU 1 … XmoZ ZmUr EH$mMdoir CS>{dbr Va Z³H$s XmoZ N>mn qH$dm H$‘rV H$‘r EH$ H$mhm da ¶oÊ¶mMrg§^mì¶Vm H$mT>m.

CH$b … S ={HH, HT, TH, TT} n(S) = 4

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114 UNIT-5

10 nojm OmñV ~oarO qH$dm 5 nojm H$‘r ~oarO ¶oÊ¶mMr g§^mì¶Vm9

36 qH$dm14

CXmhaU 3 … CÎm‘[aË¶m {ngboë¶m 52 IoimÀ¶m nÎ¶m§À¶m gQ>mVyZ ghOnUo nÎmo H$mTbo. Va bmb nÎmm qH$dmH$mim amOm qH$dm g‘ g§»¶oMm {H$bmda nÎmm ¶oÊ¶mMr g§^mì¶Vm {H$Vr Vo H$mT>m?CH$b … A hr bmb nÎmm {‘iÊ¶mMr KQ>Zm Amho Ago ‘mZy.

n(A) = 13 Mm¡H$Q> + 13 ~Xm‘ n(A) = 26B hr KQ>Zm H$mim amOm {‘iÊ¶mMr Amho Ago ‘mZy.n(B) = 1 (BpñnH$ amOm) + 1 ({H$bmda amOm) n(B) = 2C hr KQ>Zm {H$bmdaMr g‘ g§»¶m Amho Ago ‘mZy

C = (2, 4, 6, 8, 10 ho {H$bmdaMm nÎmo) n(C) = 5

A ¶m KQ>Zo‘Ü¶o Z³H$s XmoZ N>mn {‘iVmV Ago ‘mZy.

A = {HH} Am{U n(A) = 1 P(A) =(A) 1(S) 4

nn

B ¶m KQ>Zo‘Ü¶o H$‘rV H$‘r EH$ H$mQ>m {‘iVo Ago ‘mZy.

B = {HT, TH, TT} Am{U n(B) = 3 P(B) =(B) 3(S) 4

nn

AmVm A Am{U B ¶m KQ>Zm nañnamer Ho$di AmhoV. P(A B) = P(A) + P(B) ( g§^mì¶VoMm ~oaOoMm {Z¶‘)

P(A B) =14

+34

=1

Z³H$s XmoZ N>mn qH$dm H$‘rV H$‘r EH$ H$mQ>m {‘i{dÊ¶mMr g§^mì¶Vm 1 Amho.

CXmhaU 2 … EH${ÌVnUo XmoZ ’$mgo ’o$H$bo. Va ~oarO 10 nojm OmñV qH$dm ~oarO 5 nojm H$‘r ¶oÊ¶mMrg§^mì¶Vm {H$Vr?

CH$b … Amåhmbm ‘m{hV Amho H$s, Ooìhm XmoZ ZmUr ’o$H$br Jobr Voìhm n(S) = 36.A hr KQ>Zm 10 nojm OmñV ~oarO ¶oUmar Amho Ago ‘mZy.Va A = {(5,6), (6,5), (6,6)}

n(A) = 3

P(A) =3

36(S)nn(A)

B hr KQ>Zm 5 nojm H$‘r ~oarO ¶oUmar Amho Ago ‘mZy.

B = {(1,1), (1,2), (1,3), (2,1), (2,2), (3,1)} n(B)=6

P(B) =(B)(S)

nn =

636

AmVm, A Am{U B ¶m KQ>Zm nañnamer Ho$di AmhoV. P(A B) = P(A) + P(B) ( g§^mì¶VoMm ~oaOoMm {Z¶‘)

P(A B) =7 636 36 P(A B) =

936 =

14

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Probability 115

P(A) =2652

P(B) =252 P(C) =

552

AmVm A, B Am{U C ¶m nañnamer Ho$di KQ>Zm AmhoV.

P(A B C) = P(A) + P(B) + P(C)

P(A B C) =2652

+252

+552

P(A B C) =26 2 5

52 =3352

bmb nÎmm qH$dm H$mim amOm qH$dm {H$bmdaMo g‘ g§»¶m§Mo nÎmo {‘iÊ¶mMr g§ mì¶Vm3352

CXmhaU 4 … MATHEMATICIAN ¶m eãXmVrb Ajao ghOnUo {ZdS>br Va M qH$dm A Aja {ZdS>Ê¶mMrg§^mì¶Vm H$mT>m.

CH$b … MATHEMATICIAN ¶m eãXm‘Ü¶o 13 Ajao AmhoV.n(S) = 13E hr KQ>Zm M ho Aja {‘i{dÊ¶mMr Amho.

n(E) = 2

P(E) =(E)(S)

nn =

213

F hr KQ>Zm A ho Aja ¶oÊ¶mMr Amho. n(F) = 3

P(F) =(F)(S)

nn =

313

AmVm E Am{U F ¶m nañnamer Ho$di KQ>Zm AmhoV.P(E F) = P(E) + P(F)

P(E F) =2

13 +3

13 =5

13CXmhaU 5 … dJm©Vrb 40% {dÚmWu EH$mo (Eco) g§KmMo gXñ¶ AmhoV Am{U 25% {dÚmWu J{UV g§KmMogXñ¶ AmhoV Am{U VoWo EH$mM g§KmMo gXñ¶ Agy eH$VmV. dJm©Vrb {dÚmÏ¶mªMr {ZdS> ghOnUo Ho$bobrAmho. H$moUË¶mhr g§KmMo gXñ¶ Zgboë¶m {dÚmÏ¶mªMr g§^mì¶Vm {H$Vr?

CH$b … A hr KQ>Zm EH$mo n(s) =100 ‘mZy (Eco) g§KmVrb gXñ¶m§À¶m {ZdS>rMr Amho Ago ‘mZy.

P(A) =40

100 =

25

B hr KQ>Zm J{UV g§KmVrb gXñ¶m§À¶m {ZdS>rMr Amho Ago ‘mZy. P(B) =25

100 =

14

A Am{U B ¶m nañnamer Ho$di KQ>Zm AmhoV.P(A B) = P(A) + P(B)

P(A B) =25

+14

=1320

EH$mo (Eco) g§K qH$dm J{UV g§KmVrb {dÚmWu gXñ¶m§Mr {ZdS> H$aÊ¶mMr g§ mì¶Vm1320

Amho.AmVm, C hr KQ>Zm H$moUË¶mhr g§KmMo gXñ¶ Zgboë¶m {dÚmÏ¶mªMr {ZdS> H$aÊ¶mMr Amho Vr (A B) Mr

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116 UNIT-5

H$moQ>r KQ>Zm Amho.P(A B) + P(C) = 1 P(C) = 1 - P(A B)

P(C) = 1 –1320 =

720 = 35%

H$moUË¶mhr g§KmMo gXñ¶ Zgboë¶m {dÚmÏ¶mªMr {ZdS> H$aÊ¶mMr g§^mì¶Vm720 qH$dm 35% Amho.

CXmhaU 6 …¶mÑ[ÀN>H$ à¶moJm‘Ü¶o A, B Am{U C n¡H$s ’$º$ VrZ OmoS>çm nañnamer Ho$di KQ>Zm AmhoV. OaP(A) = 3

2, P(B) Am{U P(C) =

12 P(A), Va P(B) H$mT>m.

CH$b … P(B) = a ‘mZy

AmVm P(A) = 32

P(B) = 32

a P(C) =12 P(A) =

12 × 1

2 a =34 a

A, B Am{U C ¶m nañnamer Ho$di KQ>Zm AmhoV Ago {Xbobo Amho.

P(A B C) = P(A) + P(B) + P(C)

Á¶mAWu Amåhmbm {‘iVo

P(S) = 1 P(A) + P(B) + P(C) =132

a + a +34

a = 1 46a + 4a + 3a = 1

= 1 13a = 4 a =4

13 P(B) =

413

ñdmÜ¶m¶ 5.2

1. EH$m {d{eð> {Xder 0.64 BVH$m nmD$g nS>Ê¶mMr g§ mì¶Vm Amho. Ë¶m {Xder nmD$g Z nS>Ê¶mMrg§ mì¶Vm {H$Vr?

2. Z‘wÝ¶m‘YyZ Iam~ Zgboë¶m dñVy CMbÊ¶mMr g§ mì¶Vm 7/12 Amho. Va Iam~ dñVy CMbÊ¶mMr g§ mì¶Vm {H$Vr.

3. A hr ghO KS>Umè¶m à¶moJmMr g§ mì¶Vm Aem[aVrZo Amho H$sP(A) : P( A ) = 6:15, Va i) P(A) ii) P( A ) H$mT>m.

4. Oa A Am{U B ¶m nañnamer Ho$di KQ>Zm Aem[aVrZo Amho H$s P(A) =35

Am{U P(B) =27

, Va P(AUB) H$mT>m.5. XmoZ ZmUr EH$X‘ ’o$H$br. Va EH$Va XmoÝhr amOm (N>mnm) qH$dm XmoÝhr amUr (H$mQ>m) ¶oÊ¶mMr g§ mì¶Vm {H$Vr?6. Ooìhm ’$mgm ’o$H$bm OmVmo, Voìhm EH$Va {df‘ g§»¶m qH$dm MmaÀ¶m nQ>rVrb g§»¶m ¶mn¡H$s EH$ ¶oÊ¶mMr

g§ mì¶Vm H$mT>m.7. 1 Vo 25 n¶ªVÀ¶m g§»¶m§À¶m H$mS>m©‘YyZ EH$ g§»¶m H$mS>© ghOnUo {ZdS>bo. Va Ë¶mbm 3 qH$dm 11 Zo ^mJ

OmUmè¶m g§»¶oMr g§ mì¶Vm H$mT>m.8. EH$X‘ XmoZ ’$mgo ’o$H$bo. Va n¥ð>^mJmdarb g§»¶m§À¶m ~oaOobm 4 qH$dm 5 ¶mn¡H$s EH$mZohr ^mJ Z OmÊ¶mMr

g§ mì¶Vm H$mT>m.H«$‘doe Am{U g§doemÀ¶m JUZooMr g§ mì¶Voer g§~§YrV CXmhaUo …

Amåhmbr ‘m{hV Amho H$s, KQ>ZoMr g§ mì¶Vm Imbrb VrZ ‘w»¶ nm¶è¶m§‘YyZ emoYbr OmVo.(i) n (S) H$mT>Uo (ii) n (A) H$mT>Uo

(iii) P(A) =(A)

(S)

n

n gyÌ dmnê$Z P(A) H$mT>Uo

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Probability 117

¶m KQ>H$mV AmVmn¶ªV Ho$boë¶m ñnï>rH$aUmË‘H$ MM}dê$Z AmnU àmW{‘H$ KQ>Zm§Mr ¶mXr H$ê$Z, d¥jmH¥$VraMyZ qH$dm H$moï>H$mdê$Z n(A) Am{U n(S) AmnU H$mT>bo Amho. Imbrb CXmhaUo Aä¶mgm

na§Vw ¶oWo H$mhr CXmhaUo AmhoV, Á¶m‘Ü¶o Amnë¶mOdi àË¶jnUo H«$‘doe Am{U JUZm H$ê$Z n(A) Am{U n(S)

{‘iVo Imbrb CXmhaUm§Mm Aä¶mg H$am


CXmhaU 1 … 1 Vo 25 g§»¶m Agbobo nÎmo AmhoV. EH$mZ§Va EH$ ¶mà‘mUo XmoZ nÎmo H$mT>bo. Va EH$m âË¶mdarb darb g§»¶mhr7À¶m nQ>rVrb Am{U Xþgè¶m nÎ¶mdarb g§»¶m 11À¶m nQ>rVrb ¶oÊ¶mMr g§ mì¶Vm {H$Vr?

CH$b … 7À¶m nQ>rVrb nÎmo {7, 14, 21} Am{U 11À¶m nQ>rVrb g§»¶m§Mo nÎmo {11, 22} ho AmhoV.

n(s) = 25C2 n(A)=3C12C1

g§ mì¶Vm = 1 1

2

2C C

C

3

25= 3 2

300=

150

7 À¶m nQ>rVrb Am{U 11 À¶m nQ>rVrb g§»¶m Agbobo XmoZ nÎmo H$mT>Ê¶mMr g§ mì¶Vm150

Amho.

CXmhaU 2 … 52 nÎ¶m§À¶m gQ>m‘YyZ CÎm‘[aË¶m {ngyZ ghOnUo Mma nÎmo CMbbo Va

i) gd© Mm¡H$Q> nÎmo ¶oÊ¶mMr g§^mì¶Vm H$mT>m.

ii) XmoZ nÎmo ~Xm‘ Am{U XmoZ nÎmo BpñnH$Mo ¶oÊ¶mMr g§ mì¶Vm H$mT>m.

CH$b … (i) n(A) = 13C4 n(S) = 52C4

P(A) = 4

225

13CC

=13 12 11 1052 51 50 49 =

114165 P(A) =

26820825

(i) n(B) = 13C2 13C2 n(S) = 52C4

P(B) =4

52C2

13C 213C

=

13 12 13 122

4 3 2 15 51 50 49 =

46820825

XmoZ ~Xm‘ Am{U XmoZ BpñnH$Mo nÎmo ¶oÊ¶mMr g§ mì¶Vm268

20825Amho.

CXmhaU 3 … EH$m noQ>r‘Ü¶o 4 bmb Am{U 3 H$mù¶m JmoQ>çm AmhoV. ghOnUo 4 JmoQ>çm CMbë¶m Va ImbrbKQ>Zm§Mr g§^mì¶Vm H$mT>m.

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118 UNIT-5

(a) A = XmoZ JmoQ>çm bmb AmhoV. (b) B = gd© JmoQ>çm bmb AmhoV.

(c) C = gd© JmoQ>çm H$mù¶m AmhoV.

CH$b … EHy$U 7 JmoQ>çm AmhoV, Ë¶mn¡H$s 4 JmoQ>çm CMbÊ¶mMo àH$ma7C4 = 35 àH$mao n(S) = 35

(a) 4 bmb JmoQ>çm‘YyZ 2 bmb JmoQ>çm CMbÊ¶mMo àH$ma 4C2 = 6 àH$mao

am{hboë¶m XmoZ JmoQ>çm H$mù¶m AmhoV, Ë¶m CMbÊ¶mMo àH$ma 3C2 = 3 àH$mao

n(A) = 4C23C2 = 6 3 = 18 àH$mao P(A) =

(A)(S)

nn =

1835

(b) 4 bmb JmoQ>çmnmgyZ 4 bmb JmoQ>çm CMbÊ¶mMo àH$ma 4C4 = 1 àH$mao n(B) = 1 P(B) =135

(c) C hr Ae³¶ KQ>Zm Amho.

3 H$mù¶m JmoQ>çmnmgyZ 4 H$mù¶m JmoQ>çm CMbÊ¶mMo àH$ma e³¶ ZmhrV. P(C) = 0

CXmhaU 4 … EH$m noQ>rV 6 bmb, 7 nm§T>aoo Am{U 7 H$mio M|Sy> AmhoV. XmoZ M|Sy> ghOnUo CMbbo. gd© bmbqH$dm gd© H$mio M|Sy> ¶oÊ¶mMr g§^mì¶Vm {H$Vr?

CH$b … 20 M|Sy>‘YyZ 2 M|Sy> CMbÊ¶mMo àH$ma 20C2 =20 191 2

= 190 àH$mao n(S) = 190

6 bmb M|Sy>‘YyZ 2 bmb M|Sy> CMbÊ¶mMo àH$ma 6C2 = 15 àH$mao

n(A) = 15 P(2 bmb) =15190

7 H$mù¶m M|Sy>‘YyZ 2 H$mio M|Sy> CMbÊ¶mMo àH$ma 7C2 = 21 àH$mao

n(B) = 21 P(H$mio) =21

190

darb XmoZ KQ>Zm ¶m nañnamer Ho$di AmhoV.

P(bmb qH$dm H$mio) =15190

+21

190 =

36190

=15190

XmoZ M|Sy> H$mio qH$dm {Zio ¶oÊ¶mMr g§ mì¶Vm =1895 .

CXmhaU 5… ^maV, Zonmi, nm{H$ñVmZ, lrb§H$m ^yVmZ Am{U ¶m§Zr ìhm°br~m°b ñnY}V ^mJ KoVbm Amho.qOH$boë¶m§Zm Ë¶m§À¶m àJVrÀ¶m H«$‘mZo I, II Am{U III ~{jg {‘iVo. ^maVmbm H$moUVohr EH$~{jg {‘iÊ¶mMr g§^mì¶Vm {H$Vr?

CH$b … Ooìhm ^maVmZo n{hbo ~{jg qOH$bo, am{hbobr XmoZ ~{jgo 4P2 àH$mao ‘m§S>Vm ¶oVmV. åhUyZ ^maVmbmn{hbo ~{jg {‘iÊ¶mÀ¶m H«$‘doem§Mr g§»¶m = 4P2 Amho.

^maV qOH$Ê¶mÀ¶m ‘m§S>UrMr g§»¶m = 3 × 4P2

25

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Probability 119

na§Vy n (S) = 5P3

^maVmbm EImXo ~{jg {‘iÊ¶mMr g§ mì¶Vm =4

25

3

3 PP =

3 4 35 4 3 =

35

ñdmÜ¶m¶ 5.3

1. EH$m Q>monbr‘Ü¶o 2 bmb Am{U 2 {ndir ’w$bo AmhoV. EH$m ‘wbmZo ghOnUo 3 ’w$bo CMbbr. Va XmoÝhr’w$bo {ndir CMbÊ¶mMr g§ mì¶Vm {H$Vr?

2. 5 ì¶º$s¨À¶m JQ>m‘Ü¶o eoIa EH$ gXñ¶ Amho. Oa 5 n¡H$s 3 ì¶º$s¨Mr {ZdS> Ho$br Amho. g{‘Vr‘Ü¶o eoIaAgÊ¶mMr g§ mì¶Vm H$mT>m.

3. 52 nÎ¶m§À¶m g§Mm‘YyZ ghOnUo 3 nÎmo CMbbo. gd© VrZ nÎmo amOmMo ¶oÊ¶mMr g§ mì¶Vm {H$Vr?

4. 4 nwéf Am{U 3 {ó¶m ¶m‘YyZ 5 OUm§À¶m g{‘VrMr {ZdS> H$am¶Mr Amho. g{‘Vr‘Ü¶o Imbrb OUAgÊ¶mMr g§ mì¶Vm {H$Vr? (i) EH$ nwéf (ii) 2 nwéf (iii) 2 {ó¶m (iv)H$‘rV H$‘r XmoZ nwéf

Probability

Empirical probability

Deterministicexperiments

Randomexperiments

Sample Space

Event ACS

Probability of an

event P(E) = n(E)n(S)

Theoretical probability

P(E)= No. of trials in which event happensTotal No. of trials

P(E) = 1

P(E) = 0

P(E E ) =P(E )+P(E )1 2 1 2

P(E) = 1–P(E)–

Certain Event

ImpossibleEvent

Complementaryevents (E)

Mutuallyexclusive events

g§§ mì¶Vm

àm¶mo{JH$ g§§ mì¶Vm VmpËdH$ g§§ mì¶Vm

KQ>ZoV KSy>Z ¶oUmè¶m MmMUtMr g§»¶m

EHy$U MmMUtMr g§»¶m

{ZpíMV à¶moJ ¶mÑpÀN>H$ à¶moJ Z¸$s KQ>Zm

Z‘wZm AdH$meAe³¶ KQ>Zm

H$moQ>r KQ>Zm (E)KQ>ZoMr g§ mì¶Vm

KQ>Zm

nañnamer Ho$di KQ>Zm

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120 UNIT-5

CÎmao

ñdmÜ¶m¶ 5.1

1. (i)16

(ii)26

(iii)16

(iv)56

2. (i)14

(ii)44

(iii)24

(iv)34

3. (i)78

( ii)78

(iii)18

(iv)18

4. (i)536

(ii)3536

(iii)936

(iv)436

(v)79

(vi)1136

5. (i) (ii) (iii) (iv) (v) (vi)

6. (i) (ii) (iii) (iv) (v) (vi)

1550

4750

750

950

850

2550

2652

2652

3952

4852

252

852

2836

7. (i)16

(ii) (iii) (iv) (v) 0 (vi) 1

8. (i) (ii)

9. (i) (ii) (iii) (iv) (v)

10. (i) (ii)

4 6

3 6

3 6

3 0 39

9 39

1 7

6 7

1 7

2 7 7

5

124 144

2 0 144

ñdmÜ¶m¶ 5.2

1. 0.36 2. 3. (i) (ii) 4.

5. 6. 7. 8.

ñdmÜ¶m¶ 5.3

1. 2. 3.

4. (i) 0. (ii) (iii) (iv) 1

24

61 0

15525

27

47

125 2

757

12

46

1 02 5

3 43 6

3 13 5

*******

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¶m KQ>H$m‘wio Vwåhm§bm ¶m§Mo gwb^rH$aU hmoVo.

* à‘m{UV {dMbZmÀ¶m AWm©Mo ñnï>rH$aU.

* à‘m{UV {dMbZ H$mT>Ê¶mMo gyÌ V¶ma H$aUo.

* àË¶j Iam ‘Ü¶, J¥hrV ‘Ü¶ Am{U nm¶ar {dMbZ nÕVrZo

AdJuH¥$V gm‘J«rMm ì¶Ë¶mg Am{U à‘m{UV {dMbZ ¶m§Mr

JUZm ({hemo~) H$aUo.

* MbZ JwUH$mMm Cn¶moJ Am{U AWm©Mo ñnï>rH$aU.

* MbZ JwUH$ H$mT>Uo Am{U {ZH$mbmMo {díbofU H$aUo.

* {ÌÁ¶m§Va I§S>mboI aMUo.

* {ÌÁ¶m§Va I§S>mboImMo {díbofU H$aUo.

* à‘m{UV {dMbZmMm AW©

* à‘m{UV {dMbZ H$mT>Ê¶mMo gyÌ

* AdJuH¥$V Am{U dJuH¥$V gm‘J«rMm à‘m{UV

{dMbU H$mT>Uo.

* àË¶j nÕV

* Iam ‘Ü¶ nÕV

* J¥hrV ‘Ü¶ nÕV

* nm¶ar {dMbZ nÕV

* MbZ JwUH$

* {ÌÁ¶m§Va I§S>mboI aMUo

6 g§§»¶m emó (gm§p»¶H$s)(Statistics)

Statistical thinking will one day be as necessaryfor efficient citizenship, as the ability to readand write.

-H.G. Wells

H$mb© {n¶agZ (Karl Pearson)

(1857-1936, B§½b§S>)

H$mb© {nAagZ ho {~«{Q>e g§»¶memók AmhoV.

Vo g§»¶memómÀ¶m AmYw{ZH$ joÌmMm emoY

bmdÊ¶mV AmKmS>rda AmhoV. Ë¶m§Zr J{UV g§»¶m

emóm‘Ü¶o {eñV ñWm{nV Ho$br. H$mb© Jm°gZr

dmnaboë¶m "‘Ü¶ MwH$' ¶m eãXmÀ¶m {R>H$mUr

H$mb© {nAagZZr 1894 ‘Ü¶o à‘m{UV {dMbZ

hm eãX àW‘ dmnabm.

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122 UNIT-6

¶m nyduÀ¶m dJm©V Amåhr Aä¶mgbobo Amho H$s H$moUË¶mhr gm‘mÝ¶ JwUm§H$mÀ¶m {dVaUm‘Ü¶o àË¶oH$ JwUm§H$mMm H$b hm

{dVaUm‘Ü¶o ‘Ü¶m^modVr g‘yhmZo {dIyabobo AgVmV. ‘Ü¶, ‘Ü¶m§H$ Am{U ~hþbH$ hr H|$Ðr¶ H$b ‘mnZmMr ‘mno AmhoV. Amåhr

¶mMmhr Aä¶mg Ho$bobm Amho H$s H|$Ðr¶ H$b ‘mnZmMr ‘m{hVr {dVaUm {df¶r nyU© ‘m{hVr XoV Zmhr. CXmhaUmW© EH$m ‘m{bHo$‘Ü¶o XmoZ

{H«$Ho$Q>nÅz>Zr Ho$boë¶m Ymdm§Mm {dMma H$am.

A 59, 65, 73, 61, 67B 83, 120, 40, 22, 60

XmoÝhr gQ>mVrb JwUm§H$mMm ‘Ü¶ 65 hm gmaImM Amho. åhUOoM ¶ = 65

A, ¶m gQ>mVrb gd© JwUm§H$ 65 ¶m ‘Ü¶mOdi AmhoV. Am{U B gQ>mVrb JwUm§H$ 65 ¶m ‘Ü¶mnmgyZ ‘moR>çm à‘mUmV {dIyabobo

AmhoV.

åhUyZ H|$Ðr¶ H$b ‘mnZ {dVaUm{df¶r ñnï> {MÌ XoV Zmhr. JwUm§H$ ‘Ü¶mnmgyZ H$em[aVrZo {dIwabobo AmhoV ¶mMo ‘mnZ

H$aÊ¶mMr Amåhmbm JaO Amho.

{dVaUmVrb {dIwaboë¶m gm‘J«r{df¶r ‘m{hVr XoUmè¶m ‘mnZmbm {djonU åhUVmV. Vo {dVaUmVrb JwUm§H$mÀ¶m ~Xbm§Mr {Xem

Am{U à‘mU ¶m§Mo ‘mnZ H$aVo. {djonUmÀ¶m Mma àH$mamn¡H$s n„m MVwW©H$ {dMbZ Am{U ‘Ü¶ {dMbZ ¶m§Mm Am{U Ë¶m§À¶m ‘¶m©XoMm

Aä¶mg AmnU AmYr (nydu) Ho$bobmM Amho.

Amåhmbm ‘m{hV Amho H$s n„m XmoZ A§Ë¶ JwUm§H$mda Adb§~yZ AgVmo. Am{U MVwW©H$ {dMbZ ’$º$ 50% JwUm§H$mMm {dMma

H$aVo. ‘Ü¶ {dMbZ ho n„m Am{U MVwW©H$ {dMbZ ¶m ‘mnZmnojm CÎm‘ g‘Obo OmVo. ‘Ü¶ {dMbZm‘Ü¶o AmnU nm{hbo Amho H$s,

JwUm§H$mMo {dMbZ ‘Ü¶mnmgyZ YZ qH$dm F$U AgVo. na§Vw JUZm ({hemo~) H$aVmZm AmnU F$U qH$‘V gmoSy>Z XoVmo Am{U {dMbZm§Mr

’$º Iar {H$§‘V KoVmo.

’$º$ YZ qH$‘Vr KoÊ¶mgmR>r J{UVm‘Ü¶o H$moUVmhr CÎm‘ àH$ma Amho H$m?

darb à‘mUo MMm© Ho$bobo CXmhaU {dMmamV ¿¶m.

Scores ( ) 59 65 73 61 67Deviation( ) 6 0 8 4 2

xA

d

Iar qH$‘V d = –6 hr|d| = +6 Am{U d = –4 hr |d| = +4, Aer KoÊ¶m EodOr AmnU Ë¶m§Mm dJ© H$ê$ Am{U YZ

qH$‘V {‘idy.

d2 = (–6)2 = +36d2 = (–4)2 = +16

AmUIr EH$m KQ>ZoMr MMm© H$am Or {dMbZmMm dJ© Ho$bm AgVm {djonUmMo ‘mnZ H$aÊ¶mg ‘XV H$aVo. AË¶§V gm‘mÝ¶

{dVaU Xe©{dÊ¶mè¶m gQ>m§À¶m gm‘J«rMm {dMma H$am.

12 13 14 15 162 1 0 1 2

xd = 14

Amåhr darb XmoZ KQ>Zm§Mo {ZarjU Ho$bo AgVm {dMbZmMr gamgar 0(eyÝ¶) AmT>iVo. åhUyZ {djonUmMr ‘mnZmMr nwT>rb

JUZm gwÕm (0)eyÝ¶ Amho? Vo ‘Ü¶mnmgyZ H$moUË¶mhr JwUm§H$mMo {dMbZ Pmbo Zmhr Ago gm§JVo H$m? ho gË¶ Zmhr. åhUyZ JwUm§H$mMo

F$U {MÝh gmoSy>Z XoÊ¶mMr JaO Amho. ho {dMbZmMm dJ© H$ê$Z Ho$bo OmVo.

JwUm§H$

{dMbZ d=x-x

x

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Statistics 123

hr {djonU ‘mnZmMr CÎm‘ nÕV Amho. Á¶m ‘Ü¶o àË¶oH$ JwUm§H$m‘Yrb ’$aH$ Am{U ‘Ü¶ ¶m§À¶m gamgarMm nydu dJ© Ho$bm

OmVmo.

¶m {djonU ‘mnZmÀ¶m nÕVrbm ì¶Ë¶mg Ago åhUVmV. ì¶Ë¶mg hm Zoh‘r YZ AgVmo. Oa Amåhr ì¶Ë¶mgmÀ¶m YZ

qH$‘VrMo dJ©‘wi KoVbo Va Amåhmbm YZ Am{U F$U qH$‘Vr {‘iVmV. ì¶Ë¶mgmÀ¶m YZ dJ©‘wimbm à‘m{UV {dMbZ Ago

åhUVmV. åhUyZ à‘m{UV {dMbZmMr ì¶m»¶m Imbrbà‘mUo H$aVm ¶oVo.

A§H$J{UVr ‘Ü¶mnmgyZ {dMbZmÀ¶m dJm©À¶m gamgar dJ©‘wimbm qH$dm à‘m{UV [dMbZ ‘Ü¶ {dMbZmÀ¶m

dJ©‘yimbm Ago åhUVmV.

à‘m{UV {dMbZ ' ' ¶m {MÝhm§Zo Xe©{dVmV. ho {g½‘m Ago dmMVmV. {g½‘m hm J«rH$ eãX Amho. Vo {dVaUmÀ¶m ‘Ü¶m^modVr

{dIyaboë¶m àË¶oH$ JwUm§H$mMo ‘mnZ H$aVo. Vo Zoh‘r gm‘J«r ¶m EH$H$m‘Ü¶o ì¶º$ H$aVmV. Vo ‘Ü¶mnmgyZ {H$Vr {dMbrV Pmbo Amho

Vo XmI{dVo. {Z[ajUmMm Cƒ arË¶m EH$gmaIonUm Am{U g‘mZVm Xe©{dVmo. {dMbZmMm {ZƒV‘ Z‘wZm Xe©{dVmo H$s JwUm§H$ ho

‘Ü¶mda AmhoV Ë¶mM à‘mUo {dMbZmMm CƒÎm‘ Z‘wZm gm‘J«r ‘moR>çm në¶mÀ¶m qH$‘VrV {dIyabobr AmhoV Var gwÕm ‘Ü¶ AJXr

{dê$Õ AgVmo.

AmVm, AmnU {djonU ‘mnZ H$em àH$mao H$mT>VmV Vo Aä¶mgy. Vo åhUOo à‘m{UV {dMbZ.

à‘m{UV {dMbZ H$mT>Ê¶mMr nÕV (Method of calculating standard deviation)

Amåhmbm ‘m{hV Amho H$s {dVaUmMr gm‘J«r g‘mZarË¶m {dIwabobr Amho H$s Zmhr, JwUm§H$ ho nyUmªH$ qH$dm nyU© qH$dm

AnyUmªH$ AmhoV Am{U ‘Ü¶mMr qH$‘V AnyUmªH$s g§»¶m qH$dm nyUmªH$s g§»¶m Amho. {ZpíMV gyÌ dmnê$Z doJdoJù¶m nÕVrZo

gm‘J«rÀ¶m ñdénmda AmYmarV à‘m[UV {dMbZ H$mTy> eH$Vmo.

AdJuH¥$V gm‘wJ«rMo à‘m{UV {dMbZ àË¶oH$ nÕV Aä¶mgy¶m Am{U à‘m{UV {dMbZ H$mT>Ê¶mMr nÕV g‘OyZ Kody.

¶m nÕVr AdJuH¥$V Am{U dJuH¥$V gm‘J«rbm bmJy nS>Vmo. àW‘ AmnU AdJuH¥$V gm‘J«r Kody Am{U Ë¶mÀ¶m doJdoJù¶m nÕVr

g‘OyZ Kody Am{U Ë¶m dJuH¥$V gm‘J«rn¶ªV dmT>dy¶m ì¶Ë¶mg Am{U à‘m{UV {dMbZ Imbrb nÕVr dmnê$Z H$mT>bo OmVo.

AdJuH¥$V gm‘J«rMo à‘m{UV {dMbZ Am{U ì¶Ë¶mg Imbrb nÕVr dmnê$Z à‘m{UV {dMbZ H$mT>bo OmVo.

i) àË¶j (WoQ>) nÕV (Direct method) gai nÕV

ii) dmñV{dH$ (Iar) ‘Ü¶ nÕV (Actual mean method)

iii) J¥hrV ‘Ü¶ nÕV (Assumed mean method)

iv) nm¶ar {dMbZ nÕV (Step deviation method.)

(i) àË¶j (WoQ>) nÕVrZo ì¶Ë¶mg d à‘m{UV {dMbZ H$mT>Uo (Variance and Standard deviation by direct

method)

àW‘ àË¶j (WoQ>) nÕVrZo à‘m{UV {dMbZ H$mT>Ê¶mMo gyÌ V¶ma H$ê$.

ì¶m»¶odê$Z à‘m{UV {dMbZ ho ‘Ü¶mnmgyZ {dMbZmÀ¶m dJm©Mo YZ gamgar dJ©‘wi Amho.

x1, x2, x3 ...........xn ho JwUm§H$ Am{U x hm ‘Ü¶ Amho Ago ‘m§Zy

1 2, .......... nx x x x x x ‘Ü¶mnmgyZMo {dMbZ Amho.

{dMbZmMr ~oarO = x x 2

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124 UNIT-6

‘Ü¶mnmgyZ {dMbZmMm dJ© 1 2, ..........x x x x x x2 ,1 2, ..........x x x x x x2 AmhoV., .......... nx x x x x x 2

{dMbZmÀ¶m gamgar =x xn

ì¶Ë¶mg qH$dm 2 =2

x xn

à‘m{UV {dMbZ = =2

x xn

(ì¶m»¶odê$Z)

=22 2x xx x

n

=22 2. .x x x x

n n nx x

n

=22

22x nxx

n n =

22 2

2x x xn

Amåhmbm ‘m{hV Amho H$s x2 = x2+x2...........n doim = nx2

=2

2x xn

=2x x

n nxx

n

=22

22x nxx

n n =

22 2

2x x xn

22

S.D or x x=n n

à‘m{UV {dMbZ ho dJ© qH$‘VtÀ¶m ~oaOoMr gamgar Am{U gd© qH$‘VrÀ¶m dJmªÀ¶m ~oaOoMr gamgar ¶m§Mr dOm~mH$s Amho.

ho àË¶j (WoQ>) nÕVrZo à‘m{UV {dMbZ H$mT>Ê¶mÀ¶m nÕVrMo gyÌ Amho.

Amnë¶mbm ‘m{hV Amho H$s, à‘m{UV {dMbZ ho ì¶Ë¶mgmMo YZ dJ©‘wi Amho. ì¶Ë¶mg hm à‘m{UV {dMbZmMm dJ© Amho.ì¶Ë¶mg = ( )2

, Mo gyÌ {bhrbo AgVm Amnë¶mbm ì¶Ë¶mg {‘iob.

ì¶Ë¶mg =

222x x

n n=

22x xn n

à‘m{UV {dMbZ Am{U ì¶Ë¶mg ¶m§À¶m XmoÝhr gyÌm‘Ü¶o, 'x' åhUOo JwUm§H$ Am{U 'n' åhUOo EHy$U JwUm§H$mMr g§»¶m. ¶mMmAW© Agm hmoVmo H$s ì¶Ë¶mg H$mT>VmZm JwUm§H$mMo ‘Ü¶mnmgyZMo {dMbZmMr Adí¶H$Vm Zmhr.

Aem [aVrZo à‘m{UV {dMbZ H$mT>Ê¶mÀ¶m nÜXVrbm àË¶j nÜXV åhUVmV.

( {dMbZmÀ¶m dJm©Mr

gamgar = ì¶Ë¶mg Am{U 2 Zo Xe©{dVmV)

qH$dm

qH$dm

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Statistics 125

àË¶j (WoQ>) nÕVrZ o à‘m{UV {dMbZ H$mT>Ê¶mÀ¶m nm¶è¶m :

1. x2 H$mT>Uo

2. x Am{U x2 H$mT>Uo

3. x, x2 Am{U n ¶m§À¶m qH$‘Vr gyÌm‘Ü¶o dmnaUo.

ì¶Ë¶mg =22x x

n nqH$dm =

22x xn n

Ooìhm JwUm§H$mMm dJ© ghOnUo H$mT>bm OmVmo Voìhm àË¶j nÕV dmnaVmV.

CXmhaUm§À¶m gmhmæ¶mZo àË¶j nÕV g‘OyZ Kody ¶m.

CXmhaU 1: df© ^am‘Ü¶o B-8 drÀ¶m {dÚmÏ¶m©Zr bmdboë¶m amonQ>¶m§Mr g§»¶m 2, 6, 12, 5, 9, 10, 7, 4 ¶mgm‘J«rMo à‘m{UV {dMbZ H$mT>m.

CH$b : àW‘ AmnU ¶m JwUm§H$m§Zm Ë¶m§À¶m MT>Ë¶m H«$‘mV ‘m§Sy> Am{U dJ© H$ê$.

{Q>n: ¶m nÕVr‘Ü¶o JwUm§H$mZm Ë¶m§À¶m MT>Ë¶m H«$‘mV ‘m§S>boM nm{hOo Ago Zmhr. na§Vw gm§p»¶H$s‘Ü¶o gmoS>{dÊ¶mnyduJwUm§H$mZm Ë¶m§À¶m H«$‘mV ‘m§S>Ê¶mMm gamd Agbm nm{hOoV.

x 2 4 5 6 7 9 10 12 x = 55

x2 4 16 25 36 49 81 100 144 x2 = 455

¶oWo n = 8, x = 55 Am{U x2 =455

=22x x

n n=

2455 558 8

= 256.88 (6.88) = 56.88 47.33 = 9.55 3.09

gm‘mÝ¶nUo àË¶oH$ JwUm§H$ ‘Ü¶mnmgyZ 3.09 Zo {dMbrV hmoVmo.

àË¶j nÕVrÀ¶m ‘¶m©Xm (Limitations of direct method)

àË¶j nÕVr‘Ü¶o ì¶Ë¶mg Am{U à‘m{UV {dMbZ H$mT>Ê¶mgmR>r x, x2 Am{U n Mr JaO Amho. Ë¶mZ§VaAmnU x Am{U x2, H$mT>Vmo. ¶m qH$‘Vr gyÌm‘Ü¶o dmnê$Z ì¶Ë¶mg H$mTy> eH$Vmo. ¶m nÕVr‘Yrb ‘hËdmMrn{hbr JUZm åhUOo JwUm§H$mMm dJ© H$aUo.

Ooìhm JwUm§H$ bhmZ g§»¶oMo Am{U bhmZ nyUmªH$s AgVmV Voìhm hr nÕV ¶mo½¶ Amho.

nyUmªH$s J¥hrV g§»¶oMm dJ© H$aUo gmono Zmhr.

Ë¶m‘wio àË¶j nÕV hr ¶mo½¶ Zmhr, Ooìhm

* gm‘J«r ‘moR>r AgVo.

* JwUm§H$ nyUmªH$ ZgVrb Va.

3.09

3 9.553 9

609 55005481

19

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126 UNIT-6

Ë¶m‘wio à‘m{UV {dMbZ H$mT>Ê¶mMr n¶m©¶r nÕV JaOoMr Amho.

¶mH$S>o gwÕm bj Úm H$s à‘m{UV [dMbZ H$mT>V AgVmZm AmnU JwUm§H$mMo {dMbZ dmnaboboZmhr.

à‘m{UV {dMbZ H$mT>VmZm ‘Ü¶mnmgyZMo {dMbZ dmnabobr Xþgar nÕV Amho H$m?

AmVm AmnU Xþgè¶m nÕVrMm Aä¶mg H$ê$¶m.Á¶m‘Ü¶o à‘m{UV {dMbZ H$mT>VmZm JwUmH$mMo ‘Ü¶mnmgyZ{dMbZ dmnabo OmVo.

(ii) dmñV{dH$ (Iar) ‘Ü¶ nÕV : (Actual mean method)

Iè¶m ‘Ü¶ nÜXVrZo à‘m{UV {dMbZ H$mT>Ê¶mÀ¶m nm¶è¶m.

1 gm‘J«rMm dmñV{dH$ ‘Ü¶ H$mT>Uo. (A§H$J{UVr ‘Ü¶)

2 dmñV{dH$ ‘Ü¶mnmgyZ àË¶oH$ JwUm§H$mMm {dMbZ H$mT>Uoo.

3 {dMbZmMm dJ© H$aUoo.

4 {dMbZmÀ¶m dJm©Mr ~oarO H$mT>Uoo.

5 {dMbZmÀ¶m dJm©Mr ~oarO H$mT>Uo (ì¶Ë¶mg)

6 {dMbZmÀ¶m dJm©À¶m gamgarMo dJ©‘wi H$mT>Uo.

Amåhmbm à‘m{UV {dMbZ {‘iVo.

CXmhaU 2: 10 doJdoJù¶m XdmImÝ¶m‘Ü¶o OÝ‘ KoVboë¶m ‘wbm§Mr g§»¶m,

9, 12, 15, 18, 20, 22, 23, 24, 26, 31,

hr Amho Va à‘m{UV {dMbZ H$mT>m.

CH$b : x d x x d2

9 –11 121

12 –8 64

15 –5 25

18 –2 4

20 0 0

22 2 4

23 3 9

24 4 16

26 6 36

31 11 121

x = 200 d2 = 400

x

d = x x

d2 =2

x x

d2

2dn

2dn

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Statistics 127

A§H$J{UVr ‘Ü¶ =200

2010

xn

Amnë¶mbm ‘m{hV Amho H$s ' ' ho {dMbZmÀ¶m gamgarMo dJ©‘yi Amho.

=2x x d

n n

=400 40 6.3210

Omo n¶ªV ‘Ü¶ qH$‘V nyUmªH$ Amho Vmo n¶ªV hr nÕV ¶mo½¶ Amho. Oa ‘Ü¶ nyUmªH$ Agob Va {dMbZ gwÕmnyUmªH$s AgVo Ë¶m‘wio Ë¶m§Mm dJ© H$aUo gmono OmVo.

gm‘J«rMm ‘Ü¶ x hm nyUmªH$ Zmhr. na§Vw Xem§e g§»¶m Amho Ago g‘Oy. Ë¶mdoir x x hr gd© {dMbZoXem§e g§»¶m hmoVmV Am{U Xem§emMm dJ© H$aUo ho H§$Q>midmUo H$m‘ Amho.

Aem KQ>Zm‘Ü¶o, OoWo ‘Ü¶ nyUmªH$s g§»¶m AmhoV VoWo AmnU à‘m{UV {dMbZ H$mT>Ê¶mgmR>r AmUIr EH$n¶m©¶r nÕV dmnaVmo {Vbm J¥hrV ‘Ü¶ nÕV åhUVmV.

¶oWo dmñV{dH$ ‘Ü¶ H$mT>bo OmV Zmhr. Ë¶mEodOr AmnU gm‘J«rnmgyZ 'A' ¶m JwUm§H$mMr {ZdS> H$aVmo Aem[aVrZo H$s (x – A) hm ’$aH$ gd© bmhmZ g§»¶m e³¶Vmo YZ AgVmV.

(iii) J¥hrV ‘Ü¶ nÕV (Assumed mean method)

¶m nÕVr‘Ü¶o Zmdm§‘Ü¶o gwM{dë¶m à‘mUo, AmnU

* {dVaUmMm H$moUVmhr JwUm§H$ J¥hrV ‘mZbm OmVmo (e³¶Vmo H$ê$Z ‘Ü¶mdarb JwUm§H$)

* J¥hrV Yaboë¶m JwUm§H$mnmgyZ àË¶oH$ JwUm§H$mMo {dMbZ H$mT>bo OmVo.

' ' H$mT>Ê¶mÀ¶m J¥hrV ‘Ü¶ nÕVrÀ¶m nm¶è¶m

1. JwUm§H$ hm ‘Ü¶ åhUyZ J¥hrV Yam. A2. J¥hrV Yaboë¶m ‘Ü¶mnmgyZ àË¶oH$ JwUm§H$mMo {dMbZ H$mT>m. d = x – A3. {dMbZmMm dJ© H$mT>m. d2 or (x – A)2

4. {dMbZmÀ¶m dJm©Mr ~oarO H$mT>m. d2

5. {dMbZmMr ~oarO H$mT>m. d6. gyÌ dmnam. 22d d

n n7. d2 Am{U d Mr qH$‘Vr dmnam.

8. à‘m{UV {dMbZ H$mT>m.

CXmhaU 3 : EH$m AmR>dS>çm‘Ü¶o àË¶oH$ {Xder EH$m XdmImÝ¶m‘YyZ ~mhoa nS>boë¶m AmOmarbmoH$m§Mr g§»¶m Imbrbà‘mUo {Xbobr Amho.

50, 56, 59, 60, 63, 67, 68 Va à‘m{UV {dMbZ H$mTm.

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128 UNIT-6

CH$b : {Xboë¶m JwUm§H$mMm J¥hrV ‘Ü¶ A = 60 Amho Ago ‘mZy.

¶m ‘mZboë¶m ‘Ü¶mnmgyZ àË¶oH$ JwUm§H$mMo {dMbZ H$mTy>.i.e., d = x – A

x d = x – A d2

50 –10 10056 –4 1659 –1 160 0 063 3 0967 7 4968 +8 64

n = 7 d = +3 d2 = 239

iv ) nm¶ar {dMbZ nÕV: (step deviation method)

CXmhaU 4 : ‘{hÝ¶mVrb n{hë¶m Xhm {Xdgm‘Ü¶o emioVrb dmMZmb¶m‘YyZ {Xboë¶m nwñVH$m§Mrg§»¶m Imbrb à‘mUo {Xbobr Amho.

20, 30, 40, 60, 80, 90, 110, 120, 130, 140

Va à‘m{UV {dMbZ H$mT>m.

CH$b :{Xboë¶m JwUm§H$mgmR>r A = 90 ho J¥hrV ‘mZbobo ‘Ü¶ g‘Oy.

JwUm§H$mMm {dMma H$am. H$moUVr {deofV: AmT>iVo H$m?

bj Úm H$s gd© JwUm§H$mMm 10 hm g‘mB©H$ Ad¶d Amho. Vo gd© 10 À¶m nQ>rVrb AmhoV.

åhUyZ XmoZ JwUm§H$mMm ’$aH$ Am{U J¥hrV ‘Ü¶ ¶mbm gmYmaU Ad¶d 10 Zo ^mJ Úm Am{U à‘m{UV{dMbZ H$mT>m. Ë¶m‘yio {hemo~ gmono OmVo.

d =x A

c. ¶m {dMbZmbm nm¶ar {dMbZ Ago åhUVmV.

Ooìhm gm‘J«rMm AmH$ma ’$ma ‘moR>m AgVmo Am{U Ë¶mbm gmYmaU Ad¶d AgVmo Voìhm AmnU A, ho J¥hrV

‘Ü¶ KodyZx Ad

Cdmnê$Z OoW o C hm JwUm§H$mMm gmYmaU Ad¶d Amho {dMbZ H$mTy> eH$Vmo.

=22d d C

n nho gyÌ dmnê$Z à‘m{UV {dMbZ H$mT>Vmo.

¶m nÕVrbm nm¶ar {dMbZ nÕV Ago åhUVmV.

22d dn n

2239 37 7

34.14 (0.18)

33.965.83

15

18

A

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Statistics 129

CXmhaU 4 ‘Ü¶o H$moUVo gmono Amho?

* gmYmaU Ad¶d C H$mT>Ê¶mnydu d Mm dJ© H$mT>Uo.

* gmYmaU Ad¶d C H$mTy>Z d Mm dJ© H$mT>Uo.

{dMbZmMm gmYmaU Ad¶d H$mT>ë¶mZo Vo à‘m{UV {dMbZ H$mT>Ê¶mgmR>r H$emàH$mao ‘XV H$aVo ?

nm¶ar {dMbZ nÕVrZo à‘m{UV {dMbZ H$mT>Ê¶mÀ¶m nm¶è¶m

1 1..EImÚmMm JwUm§H$ ‘Ü¶ J¥hrV ‘mZUo A

2. àË¶oH$ JwUm§H$ Am{U J¥hrV ‘Ü¶ ¶m‘Yrb ’$aH$ H$mT>Uo. d = x – A

3. JwUm§H$mMm gmYmaU Ad¶d AmoiIUo. C

4. ’$aH$mbm gmYmaU Ad¶dmZo ^mJ XoUo.x Ad

C

5. {dMbZmMm dJ© H$aUo. d2

6. {dMbZmÀ¶m dJm©Mr ~oarO H$mT>Uo. d2

22d dC

n n

x d = x – A nm¶ar {dMbZ d2

x Adc

20 –70 –7 4930 –60 –6 3640 –50 –5 2560 –30 –3 980 –10 –1 190 0 0 0110 20 2 4120 30 3 9130 40 4 16140 50 5 25

n = 10 d = –8 d2 = 174

=22d d

n nC =

2174 810 10

10 = 17.4 6.4 × 10 = 40.9

AmVm n¶ªV AmnU Aä¶mgbobo Amho H$s ì¶Ë¶mg Am{U à‘m{UV {dMbZ ho darb MmanÕVrn¡H$sH$moUË¶mhr EH$m nÕVrZo H$mT>bo OmD$ eH$Vo. J¥hrV ‘Ü¶ nÕV Am{U nm¶ar {dMbZ nÕV.

22

14

7. gyÌ dmnê$Z à‘m{UV {dMbZ H$mT>Uo.

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130 UNIT-6

Anojoà‘mUo doJdoJir nÕV gma»¶mmM gm‘J«rgmR>r Mo doJdoJio CÎma XoV Zmhr. Ë¶m‘wio gm‘J«rbm ¶mo½¶Aer H$moUVrhr EH$ nÕV dmnabr OmD$ eH$Vo.

{Q>n : J¥hrV ‘Ü¶ nÕV Am{U nm¶ar {dMbZ nÕV ¶m àË¶j nÕVrMo g§{já (gwYmarV) ê$n Amho.

Amnë¶m> gmo¶rgmR>r, ì¶Ë¶mg Am{U à‘m{UV {dMbZ H$mT>Ê¶mgmR>r AmnU nm¶ar {dMbZ nÜXVrMm

Cn¶moJ H$ê$.

AmVm n¶ªV AmnU AdJuH¥$V gm‘J«rMm ì¶Ë¶mg Am{U à‘m{UV {dMbZ H$mT>Ê¶mMm Aä¶mg Ho$bobm Amho.

dJuH¥$V gm‘J«r gmR>r gwÕm hrM nÕV bmJy nS>Vo H$m?

dmnaboë¶m gyÌm‘Ü¶o H$m§hr ~Xb Ho$bo nm{hOoV H$m? Ë¶m§Mm Aä¶mg H$ê$.

dJuH¥$V gm‘J«rMo à‘m{UV {dMbZ :- (Standard deviation of grouped data)

Amåhmbm ‘m{hV Amho H$s dJuH¥$V gm‘J«rMr ‘m§S>Ur d¡¶{º$H$ JwUm§H$ qH$dm g§ mJ loUrÀ¶m H«$‘m‘Ü¶o

Ho$br OmD$ eH$Vo. àË¶oH$ d¡¶{º$H$ JwUm§H$ qH$dm loUrbm dma§dmaVm (f) AgVo. ‘Ü¶mMr JUZm ({hemo~) H$aVmZm,

Amnë¶mb m JwUm§H$mMr ~oarO H$mT>mdr b mJVo. Ë¶mgmR>r Amåhmb m àË¶oH$ JwUm§H$ qH$dm g§mJ - loUrÀ¶m

‘Ü¶mbm dma§dmaVoZo JwUbo nm{hOo åhUOoM. f(x).

¶mM àH$maoo dJuH¥$V gm‘J«rMo à‘m{UV {dMbZ H$mT>VmZm, {dMbZmMr ~oarO H$mT>mdr bmJVo. Ë¶mgmR>r

Amåhmbm àË¶oH$ JwU§mH$mZo [H$§dm g§ mJ loUrÀ¶m ‘Ü¶ dma§dmaVoZo JwUbo nm{hOo åhUOoM. fd.

åhUyZ fd Am{U fd2 H$mT>bo nm{hOoV.

dJuH¥$V gm‘J«rMo à‘m{UV {dMbZ H$mT>Ê¶mMo gyÌ Imbrb à‘mUo {Xbobo Amho.

=22fd fd C

n n

AmVm AmnU dJuH¥$V gm‘J«rMo à‘m{UV {dMbZ, nm¶ar {dMbZ nÕVrZo H$mT>y>.

CXmhaU 5 : doJdoJù¶m 5 {OëømVrb 6 {XdgmVrb nmdgmMr Zm|X Imbrb à‘mUo {Xbobr Amho.

nmD$g [‘.‘r. ‘Ü¶o 35 40 45 50 55

{R >H$ mUm §Mr g §»¶m 6 8 12 5 9

nm¶ar {dMbZ nÕVrZo à‘m{UV {dMbZ H$mT> m.

CH$b : CXmhaUm‘Ü¶o [‘.‘r. ‘Yrb nmD$g JwUm§H$ (x) Am{U {R>H$mUm§Mr g§»¶m dma§dmaVm (f) Zo Xe©{dVmV.

à‘m{UV {dMbZ gd© Mma nÕVr‘Ü¶o H$mTy>.

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Statistics 131

x f nm¶ar {dMbZ fd d2 fd2

x AdC

35 6 –2 –12 4 2440 8 –1 –8 1 845 12 0 0 0 050 5 +1 +5 1 555 9 +2 +18 4 36

n=40 fd = +3 fd2 = 73

J¥hrV ‘Ü¶ A = 45 ¿¶m.

JwUm§H$mMm gmYmaU Ad¶d C = 5 Amho.

22fd fd Cd n

273 35 1.82 5 6.7

40 40

CXmhaU 6 : {dÚmÏ¶mªÀ¶m JQ>mbm J{UV‘Yrb CXmhaUo gm oS >dm¶bm {Xbr Ë¶m§Zr KoVbobr doi

(goH§$X‘Ü¶o) {Xbobr Amho Va gm‘J«rMo à‘m{UV {dMbZ H$mT> m.

C-I 0-10 10-20 20-30 30-40 40-50

f 7 10 15 8 10

x f x nm¶ar {dMbZ d2 fd fd2

x AdC

0–10 7 5 –4 16 –28 112

10–20 10 15 –2 4 –20 40

20–30 15 25 0 0 0 0

30–40 8 35 +2 4 +16 32

40–50 10 45 +4 16 +40 160

n=50 fd = +8 fd2 = 344

2 22 344 8 550 50

fd fd Cn n

= 6.88 0.03 5 6.85 5 13.1

¶oVo EH$ Hw$VwhbmMr Jmoï> åhUOo, Ooìhm dJuH¥$V gm‘J«r {Xbobr Agob nm¶ar {dMbZ nÕV WmoS>çm

20

23

48

56

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132 UNIT-6

doJù¶m ñdê$nmV bJoM dmnabr nm{hOo. g§ mJloUr ‘m§S>Ur MT>Ë¶m qH$dm CVaË¶m H«$‘mV Ho$br OmVo.¶m KQ>ZoV gmÜ¶m pñWVrV d ¶m ñV§ m‘Ü¶o J¥hrV ‘Ü¶ KodyZ O {bhrbo OmVo. Am{U O À¶m da -1, - 2.....Am{U Imbr +1,+2 ...... {bhrbo OmVo.Vwåhr Vnmgm H$s ¶m qH$‘Vr åhUOoM

x Adi

, ¶oWo i hr g§ mJloUrMr ì¶már Amho.am{hboë¶m nm¶è¶m d2, fd, fd2, fd Am{U fd2 ¶m nyduà‘mUoM H$mT>m. à‘m{UV {dMbZ H$mT>Ê¶mgmR>rVoM gyÌ Oam ~Xb H$ê$Z dmnabo nm{hOo.

22fd fdn n

i.

¶m nÕVrZo H$mTy>.

C I f xx A

di

d2 fd fd2

0–10 7 5 –2 4 –14 28

10–20 10 15 –1 1 –10 10

20–30 15 25 0 0 0 0

30–40 8 35 +1 1 +8 8

40–50 10 45 +2 4 +20 40

n=50 fd = +4 fd2 = 86

22 86 4 1050 50

fd fd in n

= 21.72 (0.08) 10 = 13.1


CXmhaU 1 : 68, 72, 80, 84, 92, 100 ¶m JwUm§H$mMm ì¶Ë¶mg Am{U à‘m{UV {dMbZ H$mT>m.

CH$b : gd© JwUm§H$m‘Ü¶o 4 hm gmYmaUAd¶d Amho ¶mMo {ZarjU H$am.

A = 84 hm J¥hrV ‘Ü¶ KoVbm.

‘Ü¶ =496

82.66

fxn

24

28

8

6

JwUm§H$ {dMbZ d2

x84

4x

d

68 –4 16

72 –3 9

80 –1 1

84 0 0

92 +2 4

100 +4 16

x = 496 d = –2 d2 = 46

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Statistics 133

Á¶mAWu gm‘Jr«bm gmYmaU Ad¶d 4 Amho Am{U ‘Ü¶ 82.6 hm nyUmªH$ Zmhr. AmnU nm¶ar {dMbZ nÕVdmnê$Z ì¶Ë¶mg Am{U à‘m{UV {dMbZ H$mTy> eH$Vmo.

d =84

4x A x

C

i.e.,68 84 16

44 4

Am{U BË¶mXr

(i) ì¶Ë¶mg 2=2 22 46 2

6 6d dn n

= 7.7 – 0.1

2 = 7.6

(ii) = Variance 7.6 = 2.76

CXmhaU 2: EH$m MmMUr n[aj‘Ü¶o 60 {dÚmÏ¶mªZr {‘i{dbobo JwU {Xbo AmhoV.

JwU 5-15 15-25 25-35 35-45 45-55 55-65

{dÚmÏ¶mªMr g§§»¶m 8 12 20 10 7 3

{dVaUmMo ‘Ü¶ à‘m{UV {dMbZ H$mT>m. {ZH$mbmMo ñnï>rH$aU gwÕm H$am.

CH$b : g§ mJ loUrÀ¶m ‘Ü¶ q~Xÿbm 10 hm gmYmaU Ad¶d Amho. Am{U ‘Ü¶ nyUmªH$ Zmhr.

nm¶ar {dMbZ nÕVrMm Cn¶moJ H$ê$.

C-I f x fx nm¶ar {dMbZ d2 fd fd2

x AdC

5-15 8 10 80 –2 4 –16 32

15-25 12 20 240 –1 1 –12 12

25-35 20 30 600 0 0 0 0

35-45 10 40 400 +1 1 +10 10

45-55 7 50 350 +2 4 +14 28

55-65 3 60 180 +3 9 +9 27

n = 60 fx = fd = +5 fd2 = 109

1850

(i) ‘Ü¶ =1850 30.8

60fxn

(nyUmªH$ Zmhr)

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134 UNIT-6

(ii) à‘m{UV {dMbZ =2 22 109 5 10

60 60fd fd Cn n

= 1.81 10 1.34 10

= 13.4

¶mMm AW© àË¶oH$ JwUm§H$ ‘Ü¶mnmgyZ (30.8) gamgar 13.4 Zo {dMbrV hmoVmo.

CXmhaU 3 : 30 JwUm§H$mMm ‘Ü¶ 18 Am{U à‘m{UV {dMbZ 3 Amho. Va gd© JwUm§H$mMr ~oarO H$mT>m Am{U

gd© JwUm§H$mÀ¶m dJm©Mr ~oarO H$mT>m.

CH$b : JwUm§H$mMr g§»¶m = n = 30

JwUm§H$mMr ‘Ü¶ = x = 18

Amnë¶mbm ‘m{hV Amho H$s JwUm§H$mMr ~oarO = x = 18 × 30 = 540xx

ngd© JwUm§H$mMr ~oarO = 540

30 JwUm§H$mMo à‘m{UV {dMbZ = = 3

2 =22x x

n n

2218

30x

= 9

2

32430x

+ 9

x2 – 33 × 30 = 9990

gd© JwUm§H$mMm dJm©Mr ~oarO = 9990

{Q>n : Amåhr Aä¶mgbo Amho H$s nm¶ar {dMbZ nÜXV hr AË¶§Z gmonr Amho Am{U

à‘m{UV {dMbZ gm oS > {dÊ¶mgmR>r hr AJXr ¶m o½¶ Amho. Ë¶m‘yio à‘m{UV {dMbZ

gmoS> {dÊ¶mgmR> r BVa VrZ nÜXVr EodOr hr nÜXV AmnU dmnaVmo

ñdmÜ¶m¶ 6.1

1. Imbrb gm‘J«rMo àm‘{UV {dMbZ nm¶ar {dMbZ nÜXVrZo H$mT>m.x 3 8 13 18 23f 7 10 15 10 8

2. nwñVH$mÀ¶m àXe©ZmVyZ 200 {dÚmWm©Zr IaoXr Ho$boë¶m nwñVH$m§Mr g§»¶m Imbrbà‘mUo {Xbobr Amho.

nwñVH$m§Mr g§»¶m 0 1 2 3 4

{dÚmÏ¶mªMr g§»¶m 35 64 68 18 15

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Statistics 135

ì¶Ë¶mg Am{U à‘m{UV {dMbZ H$mT>m.

3. [dkmZ narjo‘Ü¶o 60 {dÚmÏ¶m©Zr {‘i{dbobo JwU Imbrb à‘mUo {Xbobo AmhoV Va ‘Ü¶, ì¶Ë¶mg Am{U

à‘m{UV {dMbZ H$mT>m.

JwUm§H$ (x) 10 20 30 40 50 60

{dÚmÏ¶mªMr g§»¶m(f ) 8 12 20 10 7 3

4. EH$m H$maImÝ¶mVrb 40 H$m‘JmamMr amoOMr (X¡Z§{XZ) ‘Owar Imbrb H$moï>H$mV {Xbr Amho. Va (i) ‘Ü¶

(ii) ì¶Ë¶mg (iii) ‘OwarMo à‘m{UV {dMbZ H$mT>m Am{U H$mT>boë¶m qH$‘VrMo ñnï>rH$aU H$am.

‘Owar (ê$n¶m‘Ü¶o) 30-34 34-38 38-42 42-46 46-50 50-54

H$m‘Jmam§Mr g§»¶m 4 7 9 11 6 3

5. 100 dñVy§Mm ‘Ü¶ 48 Am{U Ë¶m§Mo à‘m{UV {dMbZ 10 Amho. gd© dñVy§Mr ~oarO Am{U gd© dñVy§À¶m

dJmªMr ~oarO H$mT>m.

6. EH$m Jm§dmVrb ‘Yy‘oh ê$½Um§À¶m Aä¶mgmdê$Z Imbrb Jmoï>tMo {ZarjU H$am.

d¶ dfm©‘Ü¶o 10-20 20-30 30-40 40-50 50-60 60-70

ê$½Um§Mr g§»¶m 2 5 12 19 9 3

‘Ü¶ Am{U à‘m{UV {dMbZ H$mT>m. {ZH$mbm§Mo ñnï>rH$aU gwÕm H$am.

MbZ JwUH$ (Coefficient of Variation)

A Am{U B ¶m XmoZ dó CÚmoJm‘Ü¶o V¶ma eQ>m©Mo CËnmXZ Ho$bo. XmoÝhr H$maImÝ¶m‘Ü¶o 6 H«$‘dma {Xdgm‘Ü¶o

{ebmB© Ho$boë¶m eQ>mªMr g§»¶m Imbrb H$moï>H$m‘Ü¶o {Xbobr Amho.A 85 102 37 60 72 106B 44 60 55 70 63 56

eQ>mªMr g§»¶m ~Z{dÊ¶mV H$moUË¶m H$maImÝ¶mMr gamgar OmñV Amho?

eQ>mªMr g§»¶oMr gamgar H$mT>m.

A - ¶m H$maImÝ¶mMm ‘Ü¶ =462

776

xn

B - ¶m H$maImÝ¶mMm ‘Ü¶ =348 586

xn

{ZarjU H$am H$s A ¶m H$maImÝ¶mZo ~Z{dboë¶m eQ>mªÀ¶m g§»¶oMr gamgar hr B nojm OmñV Amho.

H$nS>çmÀ¶m KmD$H$ {dH«o$Ë¶mbm {Z¶{‘V eQ>© nwa{dÊ¶mgmR>r H$maImÝ¶mbm Amkm Km¶Mr Amho Ago g‘Om.‘mbH$mbm OmñV MbVm (nardV©ZerbVm) ZgVmZm nwadR>çm‘Ü¶o gmVË¶ (gagVm) hdo Amho.

Va A Am{U B ¶m H$maImÝ¶mn¡H$s ‘mbH$mZo H$moUmMr {ZdS> H$amdr?

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136 UNIT-6

CËnmXZmV gmVË¶ Am{U CÎm‘ nwadR>m ho ’$º$ Cƒ gamgar qH$‘Vr nojm CÎm‘ AgUmam H$maImZm ‘m{hVrMmAgVmo.

Amnë¶m X¡Z§{XZ OrdZm‘Ü¶o Ago AZoH$ àg§J ¶oVmV OooWo H$m¶© gmVË¶ Am{U CËnmXZm‘Ü¶o gmVË¶ ho ’$º$‘moR>çm g§»¶oZo ‘mnmnojm OmñV gmo¶rñH$a AgVo.

CXmhaUmW© :

* {H«$Ho$Q> gm‘mÝ¶m‘Ü¶o ’$b§XmO qH$dm Jmob§XmO {ZdS>Uo.

* amï´>r¶ ñVamdarb narjm ñnY}gmR>r {dÚmÏ¶mªMr {ZdS> H$aUo.

* Am°qb{nH$gmR>r YmdnÅz>§Mr {ZdS> H$aUo.

* nmbo mÁ¶m qH$dm ’w$bmMm {dH«o$Vm {ZdS>Uo.

gagVm (gmVË¶) Am{U MbVm (n[adV©ZerbVm) H$emàH$mao H$mT>br OmVo.

¶m H$maUmgmR>r gm§p»¶H$s‘Ü¶o AmnU MbZ JwUH$ (C.V) H$mT>Vmo.

MbZ JwUH$ ho {djonU ‘mnZmer g§~§{YV Amho. ho dma§dmaVm {dVaUmMm A§H$J{UVr ‘Ü¶Am{U à‘m{UV {dMbZmda AmYmarV Amho. ¶mbm g §~ §{YV à‘m{UV {dMbZ Ago gwÕmåhUVmV.

MbZ JwUH$mMr ì¶m»¶m

MbZ JwUH$ =C.V = 100à‘m{UV {dMbZA§H$J{UVr ‘Ü¶

Aer H$aVmV.

C.V = 100x

MbZ JwUH$mdê$Z Ago AmT>iVo H$s.

* ho Q> o$dmar (eoH$S>odmar) V ì¶º$ H$aVmV.

* ho EH$H$mnmgyZ ñdV§Ì AmhoV.

* ho ‘Ü¶ x Am{U à‘m{UV {dMbZmdê$Z ( ) emoYVmV.

* ho {dVaÊmmMo gmVË¶ Am{U MbVm emoYVo.H$maImZm A

JwUm§H$ x d = x – A d2

37 –48 230460 –25 62572 –13 16985 0 0102 +17 289106 +21 441n=6 d = 48 d2=3828

H$maImZm B

JwUm§H$ x d = x – A d2

44 –16 25655 –5 2556 –4 1660 0 063 +3 970 +10 100

n=6 d = –12 d2=406

25

13

86

38

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Statistics 137

na§Vw MbZ JwUH$ dmnê$Z gmVË¶ Am{U MbVm H$er {ZpíMV Ho$br OmVo? C.V H$er H$mT>br OmVo.

nwÝhm EH$Xm H$maImZm A Am{U B ¶m§Mm {dMma H$ê$.

Amnë¶mbm Ë¶m§Mr ‘Ü¶ qH$‘V ‘m{hV Amho, AmVm AmnU à‘m{UV {dMbZ H$mTy> Am{U Z§Va MbZ JwUH$

(C.V) H$mTy>.

(1) Am{U (2), ¶m§Mr VwbZm Ho$br AgVm Amnë¶mbm Ago {XgyZ ¶oVo H$s H$maImZm B Mm MbZ JwUH$

hm H$maImZm A À¶m MbZ JwUH$mnojm H$‘r Amho.

A = 60 ‘mZy

=22fd fd

n n

=

2406 126 6

= 67.67 4

= 63.67

= 7.98

C.V = 100x

=7.98 10058

C.V = 13.76

H$maImZm B Zo ~Z{dboë¶m eQ>mªMm MbZ JwUH$

13.76 Amho. .....(2)

A = 85 ‘mZy

=22fd fd

n n

=23829 48

6 6

= 638.17 64

= 574.17

= 23.96

C.V = 100x

=23.96

10077

C.V = 31.12

H$maImZm A Zo ~Z{dboë¶m eQ>mªMm MbZ JwUH$

31.12 Amho ...(1)

Ooìhm MbZ JwUH$

OmñV AgVmo

H$‘r AgVm o

{Xbobr gm‘J«r H$‘r(gag) gmVË¶nyU© Amho.

{Xbobr gm‘J«r OmñVgag gmVË¶nyU © Amho

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138 UNIT-6

åhUyZ AmnU åhUy H$s eQ>© ~Z{dÊ¶m‘Ü¶o H$maImZm B Mo gmVË¶ H$maImZm A nojm OmñV Amho.

darb MM}dê$Z AmnU Agm {ZîH$f© H$mTy> H$s XmoZ qH$dm A{YH$ g‘whmÀ¶m gm‘J«rMm MbZ JwUH$

gagVm (gmVË¶) VwbZm H$aÊ¶mg Amnë¶mbm ‘XV H$aVo.


CXmhaU 1 :‹ Aê$U Am{U âV ¶m XmoZ {H«$Ho$Q> IoimSy>Zr 15 S>mdm§‘Ü¶o 1050 Am{U 900 Ymdm

Ho$ë¶m AmhoV Ë¶m §Mm MbZ JwUH$ AZwH «$‘o 4.2 Am{U 3.0 Amho. Va Ymdm O‘{dÊ¶mV H$m oU

Mm §Jbm Amh o? Hw$UmMm Ioi A{YH$ gag (gmVË¶nyU©) Amho?

CH$b : Ioiboë¶m gm‘Ý¶m§Mr g§»¶m n = 15

Aê$UÀ¶m JwUm§H$mMr (IoimMr) gamgar1050 7015

x

âVÀ¶m JwUm§H$mMr (IoimMr) gamgar900 6015

x

x Am{U dmnê$Z MbZ JwUH$ H$mTy>.

IoimSy> ‘Ü¶ à‘m{UV {dMbZ C.V = 100x

Aê$U 70 4.24.2 100 6.070

âV 60 3.03.0

100 5.060

(i) Aê$UMm gamgar JwUm§H$ hm âVÀ¶m gamgar JwUm§H$mnojm OmñV Amho, Ë¶m‘wio Aê$U Ymdm

O‘{dÊ¶mV Mm §Jbm Amho.

(ii) âVMm MbZ JwUH$ hm Aê$UÀ¶m MbZ JwUH$mnojm H$‘r Amho. åhUyZ âV hm OmñV gag

(gmVË¶nyU©) Amho.

CXmhaU 2 : Imbrb {dVaUmMm à‘m{UV {dMbZ Am{U MbZ JwUH$ H$mT>m.

JwU (x) 10 20 30 40 50

{dÚmÏ¶mªMr g§»¶m (f) 4 3 6 5 2

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Statistics 139

JwUm§H$ dma§dmaVm fx ‘Ü¶mnmgyZMo {dMbZ d2 fd2

x f d x x

10 4 40 –19 361 1444

20 3 60 –9 81 243

30 6 180 +1 1 6

40 5 200 +11 121 605

50 2 100 +21 441 882

n = 20 fx = 580 fd2 = 3180

(i) A§H$J{UVr ‘Ü¶ =580

2920

fxxn ‘Ü¶ = 29x =

(ii) =2 3180 159 12.61

20fdn

= 12.61

(iii) C.V =12.61

100 10029x

= 1261 43.4829

C.V = 43.48

ñdmÜ¶m¶ 6.2

1. Imbrb gm‘J«rMm MbZ JwUH$ H$mT>m. 40, 36, 64, 48, 52.

2. gm‘J«rÀ¶m g‘whmMm MbZ JwUH$ 45 Am{U à‘m{UV {dMbZ 2.5 Amho Va ‘Ü¶ H$mT>m.

3. ^aVrgmR>r Amboë¶m JQ>mVrb 100 C‘oXdmam§Mr emararH$ MmMUr KoÊ¶mV Ambr. Ë¶m§Mr gamgar C§Mr163.8 g|.‘r. d MbZ JwUH$ 3.2 Amho. Va Ë¶m§À¶m C§MrMo à‘m{UV {dMbZ {H$Vr?

4. Oa n = 10, 12x = Am{U x2 = 1530, Va MbZ JwUH$ H$mT>m.

5. XmoZ ‘m{bH$m§Mm MbZ JwUH$ 58 Am{U 69 Amho. Ë¶m§Mo à‘m{UV {dMbZ 21.2 Am{U 51.6 Amho. VaË¶m§Mm A§H$J{UVr ‘Ü¶ H$mT>m.

6. A ¶m ’$b§XmOmÀ¶m àË¶oH$ S>mdmVrb gamgar Ymdm 64 Am{U Ë¶mMo à‘m{UV {dMbZ 18 Ymdm Amho.VgoM B ’$b§XmOmMr Ymdm§Mr gamgar 43 Am{U à‘m{UV {dMbZ 9 ho gma»¶mM S>mdm‘Ü¶o AmhoV. Va’$b§XmOmMo gm‘W© Am{U gagVm (gmVË¶) ¶mMr MMm© H$am.

7. A Am{U B, XmoZ ~m§YH$m‘ H§$nZrMr gamgar ‘Owar ê$n¶m‘Ü¶o Am{U à‘m{UV {dMbZ Imbrb à‘mUo{Xbobo Amho.

H§$nZr gamgar ‘Owar ‘OwarMo à‘m{UV {dMbZ

ê$n¶m‘Ü¶o ê$n¶m‘Ü¶o

A 3450 6.21

B 2850 4.56

d¡¶{º$H$ ‘Owar‘Ü¶o H$moUVr H§$nZr Mb Amho Vo emoYm.

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140 UNIT-6

{ÌÁ¶m§Va I§S>mboI (Pie charts)Amåhr Aä¶mgbobo Amho H$s g§»¶mË‘H$ gm‘J«r {MÌ qH$dm AmboI ñdê$nmV Xe©{dbr OmVo. ñ‘aU H$am

H$s g§»¶mË‘H$ gm‘J«r {MÌmH¥$Vr, aofm AmboI, ñV§ mboI, {hñQ>moJ«m‘ Am{U {ÌÁ¶m§Va I§S>mboImZo Xe©{dVmV.{ÌÁ¶m§Va I§S>mboIm‘Ü¶o ‘m{hVr dVw©im‘Ü¶o {d^mJbr OmVo. åhUyZ {ÌÁ¶m§Va I§S>mboImbm dVw©i AmboI

Agohr åhUVmV. {ÌÁ¶m§Va I§S>mboI hm H$mhr ^mJm§Mr dmQ>Ur qH$dm {d^mJUrMm àH$ma Amho.{ÌÁ¶m§Va I§S>mboIm‘Ü¶o dVw©i g§nyU© gm‘J«r (‘m{hVr) Xe©{dVo, Z§Va àË¶oH$ ^mJmMr ‘m{hVr hr dVw©i

‘Ü¶mnmgyZ R>am{dH$ H$moZ H$aVo åhUyZ àË¶oH$ ^mJmMr ‘m{hVr hr H|$Ðr¶ H$moZm‘Ü¶o EH$ ^mJ Xe©{dVo. åhUyZ{ÌÁ¶m§Va I§S>mboImbm I§S>m AmboI (sector Graph) Agohr åhUVmV.

g§»¶mË‘H$ gm‘J«rMo a¸$‘, ‘VXmZ, ‘m{hVr, ~OoQ>, hdm‘mZ bmoH$g§»¶m BË¶mXtMo n¥W:H$aU H$aÊ¶mgmR>r{ÌÁ¶m§Va I§S>mboImMm Cn¶moJ hmoVmo.

Imbrb CXmhaUmMm Aä¶mg H$am.CXmhaU 1: dJm©‘Ü¶o 36 {dÚmWu AmhoV.

Imbrb H$moï>H$ Vo emiobm H$go ¶oVmV ho XmI{dVo.MmbV gm¶H$b ~g H$maJmS>r emim ìh°Z12 8 3 4 9

{ÌÁ¶m§Va I§S>mboI‘Ü¶o hr ‘m{hVr H$er Xe©{dVmV Ë¶mMm Aä¶mg H$ê$.gd© 36 {dÚmÏ¶mªMm JQ> dVw©im‘Ü¶o Xe©dm¶Mm Amho.Amnë¶mbm ‘m{hV Amho H$s dVw©imÀ¶m H|$Ðm‘Ü¶o 360° Mm H$moZ AgVmo.åhUyZ 36 {dÚmÏ¶m©Zm 360° ‘Ü¶o Xe©{dbo nm{hOo. àË¶oH$ {dÚmWu hm Ë¶mZo ~Z{dboë¶m H|$Ðr¶ H$moZmÀ¶mà‘mUmV AgVmo.Oa 36 {dÚmÏ¶mªMm g§nyU© JQ> 360° er g§~§YrV AgVmo Va àË¶oH$ {dÚmÏ¶mªMm g§~§YrV H$moZ

36036

= 10° 12 {dÚmWu MmbV ¶oVmV Ë¶m§Mm H$moZ 12 × 10° = 120°. ¶mMm AW© 12 MmbV ¶oUmao {dÚmWu

H|$ÐmnmgyZ 120° Mm H$moZ dVw©im‘Ü¶o Xe©{dVmV.¶mMàH$mao doJdoJù¶m nÕVrZo ¶oUmè¶m {dÚmWmªMr g§»¶m Am{U Ë¶m§Zr H|$Ðmer Ho$bobm H$moZ Imbrb H$moï>H$mV{Xbobm Amho.

emiobm ¶oUmè¶m {dÚmÏ¶mªMr g§»¶m {dÚmÏ¶mªMm H|${Ð¶ H$moZ

{dÚmÏ¶mªMo àH$ma

MmbV 121236 × 360º =120°

gm¶H$b 8836

× 360º = 80°

~g 3336

×360º = 30°

H$maJmS>r 4436

×360º = 40°

emim ìh°Z 9936

×360º = 90°

36 360°

Car

30° Bicycle40°

Van80°

Bus

Walk

120°

emim ìh°Z

90°MmbV

gm¶H$b

H$maJmS>r

gm¶H$b

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Statistics 141

darb à‘mUo dV©wimÀ¶m H$moZm§Mr aMZm H$ê$Z hr ‘m{hVr {ÌÁ¶m§Va I§S>mboIm‘Ü¶o Xe©dy eH$Vmo.

{ÌÁ¶m §Va I§S >mb oI aMÊ¶mÀ¶m nm¶è¶m.

nm¶ar 1 : EHy$U ^mJm§Mr g§»¶m ({dÚmWu) 12 + 8 + 3 + 4 + 9 = 36

nm¶ar 2 : àË¶oH$ ^mJmMm H|$ÐñW H$moZ H$mT>m. (darb H$moï>H$mV Xe©{dë¶m à‘mUo)

nm¶ar 3 : R>am{dH$ {ÌÁ¶oMo dVw©i H$mT>m.

nm¶ar 4 : H§$nmg dmnê$Z dVw©imÀ¶m H|$Ðmer H|$ÐñW H$moZ H$mT>m.

nm¶ar 5 : H$moZ H$mT>m Am{U àË¶oH$ ^mJmÀ¶m KQ>H$mMo Zm§d {bhm.

CXmhaU 2 : O§Jbm‘Ü¶o EH$ Mm¡ag {H$bmo‘rQ>a joÌ’$imÀ¶m ^mJmV AmT>iUmè¶m Mma ‘hËdmÀ¶m dZñnVr Imbrb

H$moï>H$mV {Xboë¶m AmhoV. {ÌÁ¶m§Va I§S>mboI H$mT>m.

gmJdmUr bmHy$S> amP bmHy$S> XodXma bmHy$S> {Zb{Jar bmHy$S>

360 300 385 135

CH$b : {ÌÁ¶m§Va I§S>mboI H$mT>Ê¶mÀ¶m nm¶è¶m dmnam.

nm¶ar 1 : EHy$U dZñnVtMr g§»¶m ¿¶m 360 + 300 + 285 + 135 = 1080

nm¶ar 2 : àË¶oH$ àH$maÀ¶m dZñnVtMm H|$ÐñW H$moZ H$mT>m.

dZñnVrMm àH$ma dZñnVtMr g§»¶m H|$ÐñW H$moZ

gmJdmUr bmHy$S> 360360

1080 × 360º = 120°

amoP bmHy$S> 300300

1080× 360º = 100°

XodXma bmHy$S> 285285

1080× 360º = 95°

{Zb{Jar bmHy$S> 1351351080

× 360º = 45°

1080 360°

nm¶ar 3 : R>am{dH$ {ÌÁ¶oMo dVw©i H$mT>m.

nm¶ar 4 : H$moï>H$m‘Ü¶o H$mT>boë¶m H$moZmZwgma H|$ÐñW H$moZ aMm.

nm¶ar 5 : H$moZmV {d^mJm Am{U àË¶oH$ ^mJm‘Ü¶o dZñnVrMo àH$ma {bhm.

AmVm n¶ªV AmnU {Xboë¶m g§»¶mË‘H$ gm‘J«rMm {ÌÁ¶m§Va I§S>mboI

aMÊ¶mMm Aä¶mg Ho$bobm Amho.

AmVm AmnU {Xboë¶m {ÌÁ¶m§Va I§S>mboImbm dmMyZ ‘m{hVrMo ñnï>rH$aU

H$ê$.

Rose wood

Teak

woo

d

Eucalyp

tus

Devadaru

120° 100°

45° 95°

gmJd

mUr b

mHy$S>

amoP bmHy$S>

[ZbJra

r bmHy$S

>

XodXma

bmHy$S>

45°

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142 UNIT-6

{ÌÁ¶m§Va I§S>mboImMo dmMZ A{U {díbofU:

CXmhaU 3 : df©^amVrb Hw$Qw>§~mMm IM© Am{U ~MV Xe©{dUmam {ÌÁ¶m§Va I§S>mboI Imbrb à‘mUo XmI{dbobm Amho.

{ÌÁ¶m§Va I§S>mboImMm Aä¶mg H$ê$Z Imbrb àíZm§Mr CÎmao Úm.

(i) Oa Hw$Q>w>§~mMo EHy$U CËnÞ <75,000 Amho. Va ‘wbm§À¶m {ejUmgmR>r Ho$bobm IM© {H$Vr?

(ii) H$nS>çmgmR>r CËnÞmÀ¶m {H$Vr Q> o$ a¸$‘ IM© Ho$br?

(iii) Hw$Qw>§~mÀ¶m CËnÞmMr ~MV {H$Vr?

(iv) AmhmamgmR>r H$obobm IM© hm KamgmR>r Ho$boë¶m IMm©nojm {H$VrZo OmñV Amho?

(v) Ka> IM© Am{U XiUdiU IM© ¶m‘Yrb ’$aH$ {H$Vr Amho?

CH$b : (i) {Xbobo Hw$Q>w>§~mMo EHy$U CËnÞ = < 75,000

‘wbm§À¶m {ejUmgmR>r Ho$bobm IM© CËnÞmÀ¶m 12% Amho.

‘wbm§À¶m {ejUmgmR>r Ho$bobm IM© =12100 × 75,000 = < 9,000

< 75,000, n¡H$s ‘wbm§À¶m {ejUmgmR>r Ho$bobm IM© < 9,000 Amho.

(ii) H$nSçmgmR>r Ho$bobm IM© CËnÞmÀ¶m 10% Amho.

H$nSçmgmR>r Ho$bobm IM© =10100

× 75000 = <7500

H$nSçmgmR>r Ho$bobm IM© = < 7500

(iii) {Xbobo, ~MV Ho$bobo CËnÞ 15% Amho.

~MV Ho$bobr a¸$‘ =15100

× 75,000 = 11,250

75,000, CËnÞmn¡H$s Ho$bobr ~MV <11,250 Amho.

(iv) Amhmamda Ho$bobm IM© 23% Am{U Ka IM© 15% {Xbobm Amho.

AmhmamgmR>r Ho$bobm IM© =23

100× < 75,000 = < 17,250

KamgmR>r Ho$bobm IM© =15100 × < 75,000 = < 11,250

Amhmamda Ho$bobm IM© - KamgmR>r Ho$bobm IM© = <17,250 – <11,250 = < 6,000

Savings15% O

thers20%

Transport20%12%

childreneducation

Housing15%

Clothing10%

Food 23%Amhma

23%~MV

BVa 20%

XiU diU 20%‘wbm§Mo

{ejU 12%

KamgmR> r

15%

H$nS>çm§gmR>r 10%

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Statistics 143

AmhmamgmR>r Ho$bobm IM© ê$.6000 Zo KamgmR>r Ho$boë¶m IMm©nojm OmñV Amho.

(v) KamgmR>r Ho$bobm IM© = <11,250

XiUdiUmgmR>r Ho$bobm IM © =5

100× < 75,000 = < 3,750

XiUdiU IM© Am{U Ka IM© ¶m‘Yrb ’$aH$

<11,250 – `< 3,750 = `< 7,500

CXmhaU 4 : Mma eham§Mr bmoH$g§»¶m Xe©{dbobm {ÌÁ¶m§Va I§S>mboI Imbrb à‘mUo XmI{dbm Amho. {ÌÁ¶m§Va

I§S>mboI dmMm Am{U 'S' ehamMr bmoH$g§»¶m H$mT>m.

CH$b : 'S' ¶m ehamÀ¶m {d^mJmMm H|$ÐñW H$moZ x° ‘m§Zy.

Voìhm 120° + 96° + 60° + x° = 360°

276° + x° = 360°

x° = 360° - 276° = 84°

120° À¶m H$moZmÀ¶m ^mJmer =100 bmI g§~§YrV bmoH$g§»¶m

84° À¶m H$moZmÀ¶m ^mJmer g§~§YrV bmoH$g§»¶mA sector of 84° corresponds 5

= 84 × lakh = 70 lakh6

S ¶m ehamMr bmoH$g§»¶m = 70 bmI

ñdmÜ¶m¶ 6.3

I. Imbrb gm‘J«rMm {ÌÁ¶m§Va IS>mboI H$mT>m.

1. Amnë¶m AmdS>rÀ¶m Ioim‘Ü¶o gm‘rb hmoUmè¶m {dÚmÏ¶mªMr g§»¶m {Xbr Amho.

IoimMo Zm§d ’y$Q>~m°b Q>o{Zg ìhm°br ~m°b hm°H$s ~mñHo$Q> ~m°b BVa

{dÚmÏ¶mªMr g§»¶m 35 14 10 6 5 2

2. ghbr‘Ü¶o {R>H$mUm§Zm oQ> XoÊ¶mgmR>r dJm©‘Ü¶o gd}jU KoÊ¶mV Ambo Am{U àË¶oH$ {R>H$mUm§Mr {ZdS>

Ho$boë¶m {dÚmÏ¶mªMr g§»¶m {Xbr Amho.

{R>H$mU åh¡gê$ {dO¶nya JmoH$U© {MÌXþJ©

{dÚmÏ¶mªMr g§»¶m 14 6 2 18

?60°

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SQ

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100 lakhbmI

}

bmI bmI

à¶ËZ H$am:

Q A{U R ehamMr bmoH$g§»¶m H$mT>m.

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144 UNIT-6

3. EH$m IoS>çmVrb bmoH$m§Zr doJdoJù¶m àH$maMo gm~U dmnaboë¶m§Mo gd}jUmZo Aä¶mgbo.

gm~UmMm àH$ma A B C BVa

IoS>çmVrb bmoH$m§Mr Q> o$dmar 50% 30% 15% 5%

II. Imbr {Xboë¶m {ÌÁ¶m§Va I§S>mboImMm Aä¶mg H$ê$Z àË¶oH$ KQ>ZoV {Xboë¶m àíZm§Mr CÎmao Úm.

1. EH$m {R>H$mUÀ¶m eoVmVrb dm{f©H$ CËnÞmMm {ÌÁ¶m§Va I§S>mboI {Xbobm

Amho

Oa EHy$U CËnmXZ 8100 Q>Z Amho

Imbrb àíZm§Mr CÎmao Úm.

(i) ^mV, ZmMUm, D$g Am{U BVam§Mo Q>Zm‘Ü¶o CËnmXZ {H$Vr?

(ii) ZmMÊ¶mMo CËnmXZ ^mVmnojm {H$Vr Q>¸o$ A{YH$ Amho?

(iii) doJdoJù¶m dfm©Vrb D$gmMo CËnmXZ 2400 Q>Z Agob Va ^mVmMo

CËnmXZ {H$Vr?

2. doJdoJù¶m ñVamVrb bmoH$m§Mr ‘wbmIV KoÊ¶mV Ambr Am{U H$moUVo XÿaXe©Z H$m¶©H«$‘ gdm©V OmñV

AmdS>VmV Ago {dMmaÊ¶mV Ambo. Ë¶mMm {ZH$mb {ÌÁ¶m§Va I§S>mboImV

XmI{dbobm Amho.

àíZm§Mr CÎmao {bhm.

(i) ‘wbmIV KoVboë¶m ‘Ü¶o nmhÊmmè¶m§Mm AnyUmªH$ {H$Vr ?

(a) M°Zob 3 (b) M°Zob 5

(c) M°Zob 1 (d) M°Zob 2

(e) M°Zob 9

(ii) Oa VoWo 200 bmoH$ AgVrb Va {H$Vr bmoH$mZr àË¶oH$ M°Zob nm{hbo?

******

Ragi100°

140°

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40°80°

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el1

Channel

9

90°

Channel2

45°Channel

3

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5

30°

M°Zob 1

108

0 M°Zob 9

M°ZobM°Zob 72

0

M°Zob

D$g ZmMUm

^mVBVa

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Statistics 145

Statistics

Standard deviationv

Variance2v

Coef?cient ofvariation Pie charts

Ungrouped data Grouped data

22x xn n

22fx fxn n

2dn

2fdn

22fd fdn n

22fd fdn n

22fd fd Cn n

Step deviationmethod

22fd fd Cn n

. . 100C Vx

Constructingpie charts

Interpretationof pie charts

C.V. is more C.V. is less

Data is lessconsistent

Data is moreconsistent

Coefficient ofvariation

-

g§»¶m emó

à‘m{UV {dMbZ ì¶Ë¶mg MbZ JwUH$ {ÌÁ¶m§Va I§S>mboI

Ad{J©H¥$V gm‘J«r dJuH¥$V gm‘J«r{ÌÁ¶m§Va I§S>mboI

aMUo{ÌÁ¶mZ§Va I§S>mboImMo

{ddaU H$aUo

àË¶j nÕV OmñV Agob Va H$‘r Agob Va

dmñV{dH$ ‘Ü¶ nÕV

‘m{hVr ApñWa(Mb)AgVo

‘m{hVr [ñWa(gag)AgVo

J¥{hV ‘Ü¶ nÕV

nm¶ar {dMbZ nÕV

ñdmÜ¶m¶ 6.11) 6.32 2) v = 1.23, = 1.107 3) v = 0.179, = 13.4

4) ‘Ü¶ =41.7, V = 1.94 = 5.58 5) x = 4800, x2 = 2,40,4000 6) ‘Ü¶ = 42.84, = 11. 93

ñdmÜ¶m¶ 6.2

1) C.V = 20.41 2) X = 5.55 3) = 5.2 4)C.V = 25 5) X = 36.55 Am{U 74.646) A = 28.125, B = 20.93 7) A = 0.18, B = 0.16

ñdmÜ¶m¶ 6.3

II 1) (i) ZmMUm = 2250 Q>Z, D$g = 1800 Q>Z, Vm§Xÿi = 900 Q>Z, BVa = 3150 Q>Z

(ii) ZmMÊ¶mMr Q>¸o$dmar = 16.66%, (iii) 1,200

2. (i) ( a)18 , (b)

340

, (c)310 (d)

14 , (e)

14

(ii) (a) 25 (b) 15 (c) 60 (d) 50 (e) 50

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¶m KQ>H$mÀ¶m Aä¶mgmZo ¶m§Mo gwb^rH$aU hmoVo.

* gOmVr¶ Am{U {dOmVr¶ H$aUr AmoiIUo.

* {ÛnXr H$aUrMr ì¶m»¶m

* gOmVr¶ Am{U {dOmVr¶ H$aUrVrb ’$aH$.

* gOmVr¶ H$aUrMr ~oarO Am{U dOm~mH$s.

* {Xboë¶m H$aUrMm JwUmH$ma H$mT>Uo.

* H$aUrer g§~§{YV ~oarO, dOm~mH$s Am{U JwUmH$mamMr

CXmhaUo gmoS>{dUo.

* {ÛnXr H$aUtMm JwUmH$ma H$mT>Uo.

* H$aUr, {ZamgH$ H$aÊ¶mÀ¶m nÜXVrMo dU©Z H$aUo.

* {Xboë¶m H$aUrMm H$aUr {ZamgH$ H$aUo.

* A§emMm qH$dm N>oXmMm H$aUr {Zamg H$éZ gai én

XoUo.

* H$aUrMo àH$ma* {ÛnXr H$aUr

* H$aUrdarb {H«$¶m * {ÛnXr H$aUtMm JwUmH$ma* H$aUr {Zamg* N>oXmMm H$aUr {ZamgH$* A§emMm H$aUr {ZamgH$

7

Mathematics is concerned only with theenumeration and comparision of relations.

-Carl Friedrich Gauss

H$aUr (Surds)

Ab- ImodmarP‘r

Ab-IodmarP‘rZr n [a‘o¶g§»¶mZm A°mS>r~b (EoHy$ ¶oUmao) åhQ>boAm{U Z§Va Ë¶m§Zm H$aUr (‘wHo$,EoHy$ Z¶ oUma o) Ago Z § md {Xbo.h oM eãX{’$~moZm³H$sZr gwÜXm dmnabo. na§Vw hodJ©‘yi Zgboë¶m g§»¶ogmR>r dmnabo.

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147

gOmVr¶ d {dOmVr¶ H$aUr ( like and unlike surds )

Amåhr ¶mnyduM H$aUr~Ôb Aä¶mgbobo Amho. Imbrb gmaUrVrb g§{jßV énmVrb H$aUrMo H$aUr H«$‘ dH$aUrnX nmhm.

A.Z§ H$aUr g§{jßV én H$aUr H«$‘ H$aUrnX

1. 3 3 2 3

2. 12 2 3 2 3

3. 75 5 3 2 3

4. 3 27 9 3 2 3

5. 22 3x 2 3x 2 3

da {Xboë¶m gd© g§{jßV énmVrb H$aUr‘Ü¶o H$aUr H«$‘ d H$aUrnX gmaIoM Amho. Aem àH$maÀ¶mH$aUrZm gOmVr¶ H$aUr Ago åhUVmV.

g§{jßV énmVrb EH$mM H«$‘mÀ¶m Am{U gmaIrM H$aUrnXo AgUmè¶m H$aUrÀ¶m JQ>mg gOmVr¶ H$aUr(like surds)Ago åhUVmV.

AmVm Imbr {Xboë¶m gmaUrVrb H$aUtMm JQ> nmhm Am{U gmaUr nyU© H$am.

JQ> H$aUr g§{jßV én H$aUr H«$‘ H$aUrnX gmaIm doJdoJim gmaIm doJdoJim

1. 8, 12, 20, 54 2. 3 450, 54, 32

3. 43 518, 24, 64, 192

da {Xboë¶m gmaUr‘Ü¶o AmnU nmhÿ eH$Vmo H$s

JQ> 1 ‘Ü¶o H$aUr H«$‘ gmaIm Amho, H$aUrnX doJdoJio Amho.

JQ> 2 ‘Ü¶o H$aUr H«$‘ doJdoJio AmhoV, H$aUrnX gmaIo Amho.

JQ> 3 ‘Ü¶o H$aUr H«$‘ doJdoJio AmhoV, H$aUrnX doJdoJio AmhoV.

Aem àH$maÀ¶m JQ>mVrb H$aUtZm {dOmVr¶ H$aUr (unlike surds>) Ago åhUVmV.

g§{jßV énmVrb doJdoJiçm H«$‘mÀ¶m qH$dm doJdoJir H$aUrnXo AgUmè¶m qH$dm XmoÝhr doJdoJio AgUmè¶mH$aUtÀ¶m JQ>mbm {dOmVr¶ H$aUr (unlike surds) Ago åhUVmV.

Imbrb CXmhaUm§Mm Aä¶mg H$am:1. 3 5+ XmoZ H$aUtMr ~oarO

2. 7 3- XmoZ H$aUtMr ~Om~mH$s

{Xboë¶m g§{jßV énmVrb H$aUrMm H$aUr H«$‘ Am{U H$aUrnX ¶m§Mr VwbZm H$am.

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× ×

3. 6 3 5+ EH$nXr H$aUr Am{U n[a‘o¶ g§»¶m ¶m§Mr ~oarO

4. 6 5+x y XmoZ H$aUtMr ~Om~mH$s

darb àË¶oH$ CXmhaUm‘Ü¶o XmoZ H$aUtMr ~oarO qH$dm dOm~mH$s AWdm EH$nXr H$aUr d EH$ n[a‘o¶g§»¶m Amho. Aem àH$maÀ¶m H$aUtZm {ÛnXr H$aUr Ago åhUVmV. (binomial surds.)H$aUtMr ~oarO Am{U dOm~mH$s : (Addition and substraction of surds )

H$aUtMo gOmVr¶ d {dOmVr¶ H$aUr ¶m XmoZ àH$mamV dJuH$aU H$aVmV. ¶m XmoZ àH$maÀ¶m H$aUtgmR>r~oarO Am{U dOm~mH$s H$aUo e³¶ Amho H$m?

¶m àûZmMo CÎma XoÊ¶mgmR>r gOmVr¶ Am{U {dOmVr¶ ~¡{OH$ nXm§Mr ~oarO Am{U dOm~mH$s AmR>dm.

Amåhmbm ‘m{hV Amho H$s ’$³V gOmVr¶ nXm§Mr ~oarO Am{U dOm~mH$s H$aVm ¶oVo, Ë¶mMà‘mUo ’$³VgOmVr¶ H$aUtMr ~oarO Am{U dOm~mH$s H$é eH$Vmo.

åhUOoM g§{jßV énmVrb Á¶m H$aUtMm H«$‘ d H$aUrnX gmaIo Amho ’$³V Ë¶m§MrM ~oarO Am{U dOm~mH$sH$é eH$Vmo.

åhUyZ H$aUtMr ~oarO d dOm~mH$s H$aÊ¶mgmR>r H$aUtZm g§{jßV én XoD$Z Ë¶m§À¶m ghJwUH$m§Mr ~oarO qH$dmdOm~mH$s {dVaU {Z¶‘mMm Cn¶moJ H$éZ H$aVm ¶oVo.

gmoS>{dbobr CXmhaUo

CXmhaU 1 : 2 3 2 5 2+ + Mr qH$‘V H$mT>m.

CH$b: 2 3 2 5 2+ + (H$aUr g§{jßV énmV AmhoV.)(Ë¶m gOmVr¶ H$aUr AmhoV)

= (1 + 3 + 5 ) ({dVaU JwUY‘m©Mm Cn¶moJ H$éZ)

=9 2

2 3 2 5 2+ + = 9 2

CXmhaU 2 : 4 63 5 7 8 28+ - gwb^én Úm.

CH$b: 4 63 5 7 8 28+ -

= 4 9 7 5 7 8 4 7+ - = 12 7 5 7 16 7+ -

= (12 + 5 – 16) 7 = 7 ({dVaU JwUY‘© dmnéZ)

= 4 63 5 7 8 28 7+ - =

CXmhaU 3 : gwb^ én Úm 3 3 3 32 16 81 128 192+ - +

CH$b : àW‘ H$aUr Ë¶mÀ¶m g§{jßV énmV ‘m§S>m

32 16 = 3 3 32 8 2 2 2 2 4 2= = 3 81 = 33 27 3 3 3 =× ×

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3 128 = 3 364 2 4 2 = 3 192 = 3 364 3 4 3 =3 3 3 32 16 81 128 192+ - + = 3 33 34 2 3 3 4 2 4 3+ - +

= 34 2 34 2- 3 33 3 4 3+ + (gOmVr¶ H$aUtMr ‘m§S>Ur H$éZ)

= (3 + 4) 3 3 = 37 33 3 3 32 16 81 128 192+ - + = 37 3

CXmhaU 4 : ( )4 6+x y Am{U ( )3 4 3-x y Mr ~oarO H$am.

CH$b: ( ) ( )4 6 3 4 3+ + -x y x y

= 4 6 12 9+ + -x y x y (H§$g gmoS>dyZ)

= 4 12 6 9+ + -x x y y (gOmVr¶ H$aUtMm JQ> H$éZ)

= (4 12) (6 9)+ + -x y (gOmVr¶ H$aUtMr ~oarO d dOm~mH$s)

= 16 3-x y

( ) ( )4 6 3 4 3+ + -x y x y = 16 3-x y

ñdmÜ¶m¶ 7.1

I. Imbrb H$aUtZm gaién Úm. :

1. 75 108 192+ - 2. 4 12 50 7 48- -

3. 45 3 20 3 5- + 4. 2 2 3 8 2+ -a a a

5. 3 33 3 2 9+ -x x x x 6. 12 50 5 3 147 32+ + - -

7. 4 7 3 252 5 343- + 8.1 1 1 1

50 75 18 38 6 8 3

+ - -

II. Imbrb H$aUtMr ~oarO H$am.

1. ( ,2 ,4 )x y x y x y

2. 3 3 35 p, 3 p, 2 p

3. 3 3, , 3 , 4x x y y x y

4. ( 12 20), (3 3 2 5), ( 45 90)+ + -

5. ( 3 2), (2 2 3 3), (4 2 3 3)+ + -

6. ( 2 ), (2 3 ), (3 )+ - +x y x y x y

III. 1. 5 x ho 9 x ‘YyZ dOm H$am Am{U CÎma KmVm§H$ énmV {bhm.

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2. 3 p ho 10 p ‘YyZ dOm H$am.

3. 3 a ho 4 a d 2 a ¶m§À¶m ~oaOo‘YyZ dOm H$am.

4. 2 3+x y ho 5 -x y ‘YyZ dOm H$am.

H$aUtMm JwUmH$ma (Multiplication of surds )

Amåhr H$aUrMr ~oarO d dOm~mH$s Aä¶mgbobr Amho. ¶oWo AQ> Aer H$s$ g§{jßV énmVrb H$aUtMm H«$‘Am{U H$aUrnX gmaIo Agbo nm{hOo.

H$aUrMm JwUmH$ma H$aVmZm hrM AQ> bmJy nS>Vo H$m¶?

H$aUr KmVm§H$ énmV ‘m§S>Vm ¶oVmV ho Amåhr Aä¶mgbobo Amho. KmVm§H$mÀ¶m {Z¶‘mZwgma Amåhmbm ‘m{hVAmho H$s am × an = am + n . hm {Z¶‘ O|ìhm KmVm§H$mMm AmYmam§H$ gmaIm AgVmo. V|ìhm bmJy nS>Vmo. O|ìhmAmYmam§H$ gmaIm ZgVmo V|ìhm JwUmH$ma H$gm H$aUma?

CXm. :1na ×

1nb = ?

darb CXmhaU AmnU H$aUrénmV KmVm§H$m§Mm IV {Z¶‘mZo Ago {bhÿ eH$Vmo. = =n n n na b a ×× b ab

¶mdéZ AmnU Agm {ZîH$f© H$mTy> eH$Vmo H$s$, H$aUrMm JwUmH$ma H$aVmZm H$aUrMm H«$‘ gmaIm Agbmnm{hOo. H$aUr nX gmaIo qH$dm doJio Agy eH$Vo.

AmVm AmnU H$mhr H$aUrMm JwUmH$ma {eHy$¶m.

KQ>Zm 1: g‘mZ H«$‘mÀ¶m H$aUtMm JwUmH$ma

Imbrb CXmhaUm§Mo {ZarjU H$am.

CXmhaU 1 : gmoS>dm 5 3

CH$b: 5 3 = 5 3 = 15

CXm 2 : 57 JwUmH$ma H$am.

CH$b: 57 = 7 5 = 35

CXmhaU 3 : ( )3 5 6+ gaién Úm.

CH$b: ( )3 5 6+ = 3 5 3 6+ ({dVaU JwUY‘m©Mm Cn¶moJ H$éZ)

= 3 5 3 6+ =n n na b ab hm {Z¶‘ dmnéZ)

= 15 18 15 9 2 15 3 2+ = + = +

3( 5 6) 15 3 2+ = +

×

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CXmhaU 4 : 33 32 3 3 4 4 2 Mr qH$‘V H$mT>m.

CH$b: 33 32 3 3 4 4 2 (gmaIm H«$‘ )

= 32 3 4 3 4 2 n n n na b c abc hm {Z¶‘ dmnéZ )

= 3 3 3324 24 24 3 2 24 2 3= = = 348 3

33 332 3 3 4 4 2 48 3´ ´ =

nmhmb H$s darb gd© CXmhaUm‘Ü¶o H$aUtMm H«$‘ gmaIm Amho.

KQ>Zm 2 : {^ÝZ H«$‘mÀ¶m H$aUrMm JwUmH$ma

Oa H$aUr gma»¶mM H«$‘mÀ¶m ZgVrb Va Amåhr AJmoXaÀ¶m ^mJmV MMm© Ho$boë¶m JwUmH$mamÀ¶m nÜXVrMmdmna H$é eH$V Zmhr.

Va, AmVm {^ÝZ H«$‘mÀ¶m H$aUtMm JwUmH$ma H$gm H$am¶Mm? Imbrb CXmhaUo nmhm.

Imbrb CXmhaUo AÜ¶mgm

JwUmH$ma H$am 5 Am{U 3 2

H$aUrMm JwUmH$ma H$aVmZm, H$aUrMm H«$‘ gmaIm Agbm nm{hOo. Ë¶mgmR>r ¶m {Xboë¶m H$aUtMm H«$‘ gmaImH$éZ KoD$.

{Xboë¶m H$aUrMm H«$‘ H$gm gmaIm H$am¶Mm? H$aUrMm H«$‘ gmaIm H$aÊ¶mÀ¶m Imbrb nm¶è¶m nhm.

nm¶ar 1 : {Xboë¶m H$aUr KmVm§H$ énmV ‘m§S>m ½ 1/335 2 5 2

¶m KmVm§H$ énmVrb {bhrboë¶m H$aUtMm KmVm§H$ hm n[a‘o¶ énmV Amho.

¶oWo KmVm§H$mVrb N>oX hm H$aUtMm H«$‘ Xe©{dVmo. åhUyZ Oa Amåhmbm H$aUtMm H«$‘ gmaIm H$am¶MmAgob Va n[a‘o¶ én ho g‘mB©H$ N>oXm‘Ü¶o énm§VarV H$é.

g‘mB©H$ N>oXm‘Ü¶o ho H$go ê$nm§VarV Ho$bo OmVo ?

AmVm {Xboë¶m H$aUrVrb H«$‘m§Mm b.gm.{d H$mT>Ê¶mMr nÜXV Amåhr AmR>dy. åhUyZ Amåhr H$aUtÀ¶mH«$‘mMm b.gm.{d. H$mTyZ gma»¶mM H$aUr H«$‘mV énm§Va H$é.

nm¶ar 2: 2 Am{U 3 Mm b.gm.{d. H$mT>m 2 Am{U 3 ¶m§Mm b.gm.{d 6 Amho.

nm¶ar 3:àË¶oH$ KmVm§H$mZm b.gm.{d.Zo JwUyZ×

×

×125

63 16 32

62

6

gaiê$n Úm. ×× ×1 1

3 26 65 2

nm¶ar 4: naV H$aUrénmV ‘m§S>m ×6 63 25 2 = × 66125 4

nm¶ar 5: ´ =n n na b ab hm {Z¶‘ dmnam 66125 4 500=

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635 2 500´ =

{^ÝZ H$aUr H«$‘mMm H$aUtMm JwUmH$ma H$aVmZm dmnaë¶m OmUmè¶m nm¶è¶m :

1. {Xboë¶m H$aUtÀ¶m H«$‘mMm b.gm.{d. H$mT>m.

2. àË¶oH$ H$aUr gma»¶mM H$aUr H«$‘mV ~Xbm.

3. ´ =n n na b ab ¶m {Z¶‘mMm Cn¶moJ H$éZ JwUmH$ma H$am.

gmoS>{dbobr CXmhaUo

CXmhaU 1 : 3 2 Am{U 4 3 ¶m§Mm JwUmH$ma H$am.

CH$b: 3 Am{U 4 ho doJdoJio H«$‘ AgyZ 3 Am{U 4 ¶m§Mm b.gm.{d. 12 Amho.

43 2 3 = 1 13 42 3 (H$aUrMo KmVm§H$ én)

=132

124 112 43

123

12 (àË¶oH$ KmVm§H$mbm b.gm.{d. Zo JwUmH$ma d ^mJmH$ma H$am.)

=4 3

12 122 3 =

= 1212 4 32 3 (KmVm§H$ én H$aUr énmV nwÝhm {bhm.)

= 12 4 32 3 = 1216 27 ( =n n na b ab ¶m {Z¶‘mMm Cn¶moJ H$éZ)

= 12 432

1243 2 3 432=

darb XmoZ CXmhaUmdéZ Amåhr Agm {ZîH$f© H$mTy> eH$Vmo H$s XmoZ H$aUr Á¶m§Mm H«$‘ gmaIm Zmhr Ë¶m§MmH«$‘ gmaIm H$amdm bmJVmo. H«$‘ b.gm.{d. BVH$m H$amdm bmJVmo. hm H«$‘ H$aUtÀ¶m H«$‘m§Mm b.gm.{d.hmo¶.åhUOoM {Xboë¶m H$aUr XmoZ H$aU§rÀ¶m b.gm.{d. ê$nmV ‘m§S>VmV.

CXmhaU 2 : 3 Am{U 3 5 ¶m§Mm JwUmH$ma H$am.

CH$b: n¶m©¶r nÜXV :

H$aUrÀ¶m H«$‘mMm b.gm.{d. = 6

3 Am{U 3 5 Mm H«$‘ 6 énm§VarV H$am.

3 ¶m H$aUrÀ¶m H«$‘mbm åhUOoM 2, bm 3 Zo JwUm.

Oa ho Ho$bo Va H$m¶ AmR>dVo Vo nmhm.

2 3 63 3 3Þ = nU, 3 3¹ 63 3¹

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¶oWo Amåhr nmhVmo H$s H$aUrMr qH$‘V ~XbVo. Amåhmbm H$m¶ H$amdo bmJob H$s OoUoH$éZ Ë¶mMr nyduMrqH$‘V A~m{YV am{hb?

Imbrb à‘mUo nmhm.

32 3 33 3 3=1 12 3

123 3= = =

ho Ago gwM{dVo H$s Amåhr {Xboë¶m H$aUrÀ¶m ‘yi H«$‘mMm JwUmH$ma H$aUrMm H«$‘mVm ~XbVmZm H$aVmZmË¶mMr ‘yi qH$‘V VerM R>odyZ H$aUr¶ nXm§Mm gmaImM KmVm§H$ KoVmo.

Ë¶mMà‘mUo, 3 5 = 23 2 25 5=

1 13 2 1 335 5=

33 5 = 2 3 3 23 23 5

= 6 627 25 = 66 27 25 675= = 6 675

33 5 = 6 675

KQ>Zm 3: {ÛnXr H$aUm§Mm JwUmH$ma (Multiplication of Binomial surds)

CXmhaU 1: ( )6 2+ bm ( )6 2+ Zo JwUm.

CH$b: ( )6 2+ ( )6 2+

= ( )26 2+ ho (a+b)2 À¶m ñdê$nmV Amho.

= ( ) ( )2 26 2 6 2 2+ + (a+b)2 = a2 + 2ab + b2

= 6 +2 12 + 2

= 8 + 2 12

CXmhaU 2: ( )2 3x + bm ( )3 3x + Zo JwUm

CH$b ( ) ( )2 3 3 3x x+ + ho (x+a) (x+b) À¶m ñdê$nmV Amho.

= ( )2 2 3 3 3 2 3 3 3x x+ + + ´ (x+a) (x+b) = x2 + (a + b) x + ab

= x2 + 5 3 + 18

CXmhaU 3: ( ) ( )3 18 2 12 by 50 57+ - Zo JwUm

CH$b: ( ) ( )3 18 2 12 50 57+ -

= ( ) ( )9 2 4 3 5 2 3 3+ - àË¶oH$ H$aUr Ë¶mÀ¶m g§{já énmV ~Xbm.

bm

×

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= 9 2 ( ) ( )5 2 3 3 +4 3 5 2 3 3- -

= 45 × 2 – 27 6 + 20 6 – (12 × 3)

= 90 – 7 6 36-

= 54 – 7 6

ñdmÜ¶m¶ 7.2

I. gaiê$n Úm:

1. 3 7 2. 3 34 5 3. 4 44 6 4. 55 10 115. 66 2 5 6. n nx y 7. 8.

II. Imbrb H$aUtMm JwUmH$ma H$am.

1. 2 Am{U 3 4 2. 3 3 Am{U 4 2 3. Am{U 4. 3 Am{U 4 5

5. 5 Am{U 3 3 6. 3 4 Am{U 5 2 7. 3 5 Am{U 4 4 8. 3 2 Am{U 6 5

III. gwb^ê$n Úm.

1. ( ) ( )3 2 2 3 2 3 4 2+ - 2. ( ) ( )75 45 20 12- +

3. ( ) ( )3 2 3 2x y y x+ - 4. ( ) ( )6 5 6 5a b a b- +

5. ( ) ( )6 2 7 3 6 2 7 3- - 6. ( ) ( )3 27 5 9 3 7+ +

H$aUr {ZamgH$ Ad¶d Am{U H$aUr {ZamgH$ (Rationalising factor (R.F) and Rationalisation ofsurds)

Imbr {Xboë¶m gmaUr‘Ü¶o H$aUr, Ë¶m§Mm JwUmH$ma Am{U {ZH$mb {Xbobm Amho. Ë¶m§Mm Aä¶mg H$am.

A.Z§. H$aUr JwUmH$ma {ZH$mb

1. 7 , 7 7 × 7 = 7 n[a‘o¶ g§»¶m

2. 5 x , x 5 x x = 5x n[a‘o¶ g§»¶m

3. +x y , +x y +x y x y+ = +x y n[a‘o¶ g§»¶m

4. ab , ab ab ab = ab n[a‘o¶ g§»¶m

5. ( ) ( )6 3 5 6 3 5+ - ( )2 26 3 – 5 =108 – 25 = 83 n[a‘o¶ g§»¶m

6. ( ) ( )8 8x y x y+ - ( ) ( )2 28 – 64x y x y= - n[a‘o¶ g§»¶m

3 32 37 4 18 27 128

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155

darb gd© CXmhaUm‘Ü¶o XmoZ H$aUtMm JwUmH$ma hr n[a‘o¶ g§»¶m Amho.åhUyZ, 7 Mm H$maUr {ZamgH$ Ad¶d 7 Amho.

+x y Mm H$maUr {ZamgH$ Ad¶d +x y Amho.

( )6 3 5+ Mm H$aUr {ZamgH$ Ad¶d ( 6 3 5- ) Amho.

EH$m H$aUrbm Xþgè¶m H$aUrZo JwUbo AgVm n[a‘o¶ g§»¶m {‘iVo ¶m nÕVrbm H$aUr {ZamgH$ AgoåhUVmV.

Oa XmoZ {ÛnXr H$aUtMm JwUmH$ma hm n[a‘o¶ g§»¶m AgoV Va àË¶oH$ H$aUr hr Xþgè¶m H$aUrbm H$aUr {ZamgH$Ad¶d åhUVmV. pìXnXr H$aUrÀ¶m H$aUr {ZamgH$ Ad¶dmbm AZw~ÜX H$aUr Agohr åhUVmV.

gmoS>{dbobr CXmhaUo

CXmhaU 1 : 3 2+ ¶m§Mr AZw~ÜX H$aUr ( 3 2- ) Amho ho nS>Vmim.

CH$b: 3 2+ = ( 3 2)( 3 2)+ -

= 2 2( 3) ( 2)- = 3 2 = 1 [(a + b) (a b) = a2 b2 {ZË¶ g‘rH$aU dmnéZ]

( 3 2)+ Mr AZw~ÜX H$aUr ( 3 2)- Amho.

CXmhaU 2 : (5 3 )-x y Mm H$aUr {ZamgH$ H$mT>m.

CH$b: (5 3 )-x y = (5 3 )-x y ( )5 3+x y

= ( ) ( )2 25 3-x y ((a – b)(a + b) = a2 – b2 Mm Cn¶moJ H$éZ)

= 25x – 9y

( )25 3-x y bm Ë¶mÀ¶m AZw~ÜX H$aUrMm ( )5 3-x y Mm Cn¶moJ H$éZ H$aUr [Zamg Ho$bo Amho.

CXmhaU 3 : 3 5- + x ¶mMr AZw~ÜX H$aUr 3 5+ + x Amho ho nS>VmiyZ nhm.

CH$b: Oa ( )3 5- + x ¶mMr AZw~ÜX H$aUr ( )3 5+ + x Va ¶mMm JwUmH$ma n[a‘o¶ g§»¶m Agbrnm{hOo.

( )3 5- + x ( )3 5- + x = ( )223 5- + x = 9 – (5 + x) = 9 – 5 – x = 4 – x

(4 - x) hr n[a‘o¶ g§»¶m Amho.

( )3 5+ + x Mr AZw~ÜX H$aUr ( )3 5- + x Amho.

-

+

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- -

-

CXmhaU 4 :133 1

33- Mm H$aUr {ZamgH$ H$mT>m.

CH$b: g‘Om a =133 b = 1

33-

Va, a3 = ( )3133 = 3 Am{U b3 = ( )31

33- = 1 13

3- =

a3 - b3 =1 9 1 8

33 3 3

-- = =

nU a3 – b3 = (a – b)(a2 + ab + b2)

( ) ( ) ( )2 21 1 1 1 1 13 3 3 3 3 33 3 3 3 3 3- + + =

83

( )( )1 1 2 1 1 23 3 3 3 3 33 3 3 3 3- + ´ =

83

( )( )1 1 2 23 3 3 33 3 3 3 1- + + =

83

na§Vy83 n[a‘o¶ g§»¶m Amho, ( )1 1

3 33 3- Mm H$aUr {ZamgH$ Ad¶d ( )2 23 33 3 1- + hm Amho.

ñdmÜ¶m¶ 7.3

I. Imbrb H$aUrMm H$aUr {ZamgH$ Ad¶d {bhm.

(1) a (2) 2 x (3) 7 y (4) xy (5) 4 +p q (6) 8 -x y (7)

1p

2 (8) a ab(9) x mn (10) 5p +a b

II. Imbrb {ÛnXr H$aUr|Mr AZw~ÜX H$aUr {bhm.

(1) +a b (2) 2-x y (3) 3 2-p q (4) 3+x y

(5) 10 2 3 5+ (6) 5 3+ (7) 8 5- (8) 3 7 7 3+

(9)1

22

+ (10)1 12 2

+x y (11) x a y b+ (12) xy z yz x+

III. Imbrb {ÛnXr H$aUrMm H$aUr {ZamgH$ Ad¶d emoYm.

(1)1 13 32 2

-+ (2)

1 13 35 5

-+ (3) 1 1+ - -y y

N>oXmMm H$aUr {Zamg H$éZ gaién XoUo. N>oXmMm H$aUr {Zamg H$aVmZm N>oXmÀ¶m {ZamgH$ Ad¶dmZo A§e Am{U N>oXmbm JwUmdo.

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157

gmoS>{dbobr CXmhaUo

CXmhaU 1 : CXm (1)35 ¶m N>oXmMm H$aUr {Zamg H$éZ gaién Úm.

CH$b:35 =

35 5 Mm {ZamgH$ Ad¶d 5 Amho.

35 =

3 55 5

´ (A§e Am{U NoXmbm N>oXmÀ¶m AZw~ÜX H$aUr> Zo JwUm)

=3 55 5

´´ =

155

CXmhaU 2 :68 N>oXmMm H$aUr {Zamg H$éZ gai én Úm.

CH$b:6 88 8

=6 8 6 4 2 12

8 8= =

32

8 2

322

=

6 88 8

´6 3 2

28=

CXmhaU 3 :bca N>oXmMm H$aUr {Zamg H$éZ gai én Úm.

CH$b:bca =

bca ×

aa

( a Mm H$aUr {ZamgH$ a Amho.)

=bc a

a =

abca

bca

abca

CXmhaU 4 :3

5 3- N>oXmMm H$aUr {Zamg H$éZ gai én Úm.

CH$b:3

5 3- 5 3- hm NoX Amho.>

5 3- Mm H$aUr {ZamgH$ 5 3+ Amho.

35 3- =

3 5 35 3 5 3

+- +

=( )

( ) ( )2 2

3 5 3

5 3

+

- =

( ) ( )3 5 3 3 5 35 3 2

+ +=

-

×

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35 3- =

( )3 5 32+

CXmhaU 5 :6 36 3

+-

N> oXmMm H$aUr {Zamg H$éZ gai én Úm.

CH$b:6 36 3

+- 6 3- Mr AZw~ÜX H$aUr 6 3+ Amho.

=6 3 6 36 3 6 3

+ +- +

=( )

( ) ( )

2

2 2

6 3

6 3

+

-( (a – b) (a + b) = a2 – b2 dmnéZ)

=( ) ( )2 2

6 3 2 6 36 3

+ +-

(a + b)2 = a2 + 2ab + b2) dmnéZ]

=6 3 2 18 9 6 2 3

3 3+ + +

= =( )3 2 2

3+

= 3 2 2+

6 33 2 2

6 3+

= ++

CXmhaU 6 :5 3

5 2 5 2-

+ -gaiê$n Úm.

CH$b:5 3

5 2 5 2-

+ -

5 2+ H$aUr {ZamgH$ 5 2- Am{U 5 2- H$aUr {ZamgH$ 5 2+ Amho.

5 35 2 5 2

-+ -

=5 5 2 3 5 2

5 2 5 2 5 2 5 2- +

-+ - - +

(A§e Am{U N>oXmbm Ë¶m§À¶m

H$aUr {ZamgH$mZo)

=( )

( )( )

( )2 22 2

5 5 2 3 5 2

5 2 5 2

- +-

- -

=( )5 2 5 15 2 3

5 4 5 4- +

-- -

= 5 2 5 15 2 3- - -

5 35 2 5 2

-+ -

= 5 2 5 15 2 3- - -

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159

CXmhaU 7 :1 1

4 483 2

+ gai ê$n Úm.

CHb$:1 1

4 483 2

+ =4 48 4 4 3

2 23 3+ = + 3 Mm H$aUr {ZamgH$ 3 Amho.

4 4 323

+ =4 3 4 3

23 3´ + =

4 3 4 33 2

+

8 3 12 36

+= =

2010

36 3

=10 3

31 1 10 3

4 483 2 3

+ =

gmoS>{dbobr CXmhaUo 7.4

I. N>oXmMm H$aUr {Zamg H$éZ gaién Úm.

A. (1)83 (2)

32 x (3)

52y (4)

1 22 5

a

B. (1)2

3 2+ (2) -x

x y (3)10

5 3+(4)

3 56 3-

(5)-ab

a b

C. (1)3 23 2

+-

(2)5 2 33 2 5

--

(3)4 3 2

3 2+

+(4)

3 63 6+

+

II. gaién Úm.

(1)2 5

3 2 3 2+

- + (2)5 3

5 3 5 3-

- +

(3)6 25 5 2

++

(4)7 3 2 56 3 8 2

-- +

(5)21 2 5

3 7 21 5+

+ +(6) Oa x = 2 6 5+ Va

1+x

x H$mT>m.

III. x Mr qH$‘V H$mT>m. 1) 3x-23 +2x

=2+3x - 42

2) x-1x +1

=4+x - 1

2

3 56

(5)

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160

a

H$aUr

H$aUrMm AW© H$aUrnX a H$aUrdarb {H«$¶m H$aUr {ZamgH$

H$aUtMo àH$ma H$aUrMm H«$‘ n H$aUtMr ~oarO {ZamgH$ Ad¶d

gOmVr¶ H$aUr H$aUtMr dOm~mH$s {ÛnXrH$aUtMr

{dOmVr¶ H$aUr H$aUtMm JwUmH$maAZw~Õ H$aUr

{Û‘mZ H$aUr gmaImM H«$‘ doJdoJimH«$‘ N>oXmMm/A§emMm H$aUr {Zamg H$ê$Z gai ê$n XoUo

n a

CÎmao

ñdmÜ¶m¶ 7.1

I. (1) 3 3 (2) 5( 2 4 3 )- - (3) 0 (4) 7 2a (5) 0

(6) 2 (7) 21 7 (8)2 34 2

+

II. (1) 7x y (2) 310 P (3) 4 5x x y y+ (4) 5 3 7 5 3 10+ -

(5) 3 7 2+ (6) 6 x

III. (1) ( )124 x (2) 7 P (3) 3 a (4) 3 4x y-

ñdmÜ¶m¶ 7.2

I. (1) 21 (2) 3 20 (3) 4 24 (4) 5 110

(5) 610 (6) n xy (7) 36 28 (8) 144 3

II. (1) 6 128 (2) 12 648 (3) 12 200 (4) 4 45

(5) 6 1125 (6) 15 8192 (7) 12 40,000 (8) 6 20

III. (1) 6 + 6)–2( (2) 4 15 (3) 5 6 6xy x y- + (4) 36a – 25b

(5) 219 – 84 6 (6) 278 + 108 3

ñdmÜ¶m¶ 7.3

I. (a) a (b) x (c) y (d) xy (e) p q+

(f) x y- (g) p (h) ab (i) mn (j) a b+

II. (a) a b- (b) 2x y+ (c) 3 2p q+ (d) 3x y- (e) 10 32 5-(f) 5 3- (g) 8 5+ (h) 3 7 7 3- (i)

12

2- (j)

1 12 2

x y-

(k) x a y b- (l) xy z yz x-

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III. (a)2 23 3[2 2 1]-

+ - (b)2 23 3[5 [

1 5-- - (c) 1 1y y+ + -

ñdmÜ¶m¶ 7.4

I. A. 5)4)33

81) 3) 102y

y 1010

a 302

32xx2) B.1) 2)

))

))x y-2 3 - 2 x x y+

a-b

) ) )3) 5)4)52

2 - 5 ab6+ a+30 C. 1510-31) 2)5+213

6 -6 30 + 5

)

4)2 - 2 63) 10-3 116

-

II. 30+10

) )31015 -5 -1) 2) 3)2+ 15

6 +2

4 ))

4)3

21 2 + 1 10-6) 10III. 1) 12 2) 81

***********

*******

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~hþnXr

* ~hþnXrM H$moQ>r

* ~hþnXrMo ewÝ¶

* ~hþnXrMr ^mJmH$ma H$m¶m©dbr(Algorithm)

* eof {gÜXm§V

* Ad¶d {gÜXm§V

* g§íbofH$ ^mJmH$ma

8

Paolo Ruffin(1765-1822, Italy)

An elegant way of dividing apolynomial by a l inearpolynomial was introduced byPaolo Ruffin in 1809. Hismethod is known as Syntheticdivision, which facilitates thedivision of a polynomial by alinear polynomial or binomialof the form (x-a) with the helpof the co-efficients involved.

[Polynomials]

¶m KQ>H$mÀ¶m Aä¶mgmZ§Va ho gwb^ hmoB©b.

* ~hþnXrMr H$moQ>r AmoiIUo.

* ~hþnXrMo ewÝ¶ emoYUo.

* ~hþnXrMo ^mJmH$ma H$m¶m©dbr ñnï> H$aUo.

* ^mÁ¶, ^mOH$, ^mJmH$ma, ~mH$s emoYÊ¶mgmR>r

^mJmH$ma H$m¶m©dbrMm Cn¶moJ H$aUo.

* eof {gÜXm§V ñnï> H$aUo.

* Ad¶d {gÜXm§V ñnï> H$aUo.

* ^mJmH$ma Z H$aVm eof {gÜXm§MmMm Cn¶moJ H$éZ ~mH$s emoYUo.

* Ad¶d {gÜXm§VmMm Cn¶moJ H$éZ, {Xbobr ~hþnXr, Xþgè¶m ~hþnXrMm Ad¶d Amho H$m, Vo VnmgUo.

* ^mJmH$ma Am{U ~mH$s emoYÊ¶mgmR>r g§íbofH$ ^mJmH$ma nÜXVrMm Cn¶moJ H$aUo.

~rOJ{UV ho Ago ~m¡pÜXH$ gmYZ Amho Á¶mMr {Z{‘©VrOJmVrb n[a‘mUdmMH$ Ñï>rH$moZmMo g‘W©Z H$aÊ¶mgmR>rPmbr Amho.

-Aë’«o$S> Zm°W© ìhmB©Q>hoS>

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Polynomials 163

Vw‘À¶m ‘mJrb dJm©‘Ü¶o Vwåhr ~hþnXr Am{U Ë¶mdarb {H«$¶m ¶m§Mm Aä¶mg Ho$bobm Amho. Ë¶mMr AmR>dU H$éZCOiUr H$é. 'x ' ho MbnX, 'n' hm YZ nyUmªH$ Am{U a1, a2, a3.....an hr pñWa nXo (dmñVd g§»¶m).Va, p(x) = a0 + a1x + a2x2 + a3x3 + ....... + anxn, ¶m Z‘wÝ¶mVrb ~¡{OH$ amer Á¶m‘Ü¶o MbnXmMmKmVm§H$ ’$º$ YZ nyUmªH$ Agob Va Ë¶m ~¡{OH$ amerbm x ‘Yrb ~hþnXr Ago åhUVmVbjmV ¿¶m H$s, x

0 1 22 ........ n

np(x) x + aa + a ax + + ~hþnXr‘Ü¶o

* an 0* a0, a1x1, a2x2, a3x3......anxn ¶m§Zm ~hþnXrMr nXo åhUVmV.* a0, a1, a2 .......an – 1, an ho AZwH«$‘o x0, x1, x2,......xn – 1, xn ¶m§Mo ghJwUH$ AmhoV.

CXmhaUmW©,p(x) = 4x – 3 hr x. ‘Yrb ~hþnXr Amho.g(y) = 2y2 + 3y – 10 hr y ‘Yrb ~hþnXr Amho.

f(u) =14

u3 + 6u2 + u + 8 hr u ‘Yrb ~hþnXr Amho.

Imbrb ~¡{OH$ ametMo {ZarjU H$am.

* 3x2 –2 5x * x3 – 2x2 + 3 x + 10 * 2

14 6x x

¶m ~hþnXr AmhoV H$m ? H$maU gm§Jm.~hþnXrMr H$moQ>r : (Degree of a Ploynomial)

Imbrb H$moï>H$mV {Xboë¶m ~hþnXr, Ë¶m§Mo MbnX Am{U KmVm§H$mMo {ZarjU H$am.~hþnXr MbnX ‘moR>çmV ‘moR>m KmVm§H$5x + 4 x 13y2 + 4y + 1 y 2

3 21 5 63

m m m m 3

2u4 + u3 – 2u + 8 u 4

* ~hþnXr‘Yrb MbnXmÀ¶m ‘moR>çmV ‘moR>çm KmVm§H$mbm Ë¶m ~hþnXrMr H$moQ>r Ago åhUVmV.~hþnXr p(x) = 5x + 4 Mr H$moQ>r 1 Amho.~hþnXr p(y) = 3y2 + 4y + 1 Mr H$moQ>r 2 Amho.

~hþnXr p(y) = 3 21 5 63

m m m Mr H$moQ>r 3 Amho.

~hþnXr p(y) = 2u4 + u3 – 2u + 8 Mr H$moQ>r 4 Amho.

CXmhaUmW©,

p(y) = 3y2 + y – 6 ¶m y ‘Yrb ~hþnXrMr H$moQ>r 2 Amho.

p(x) = 7x2 + x – 6 ¶m x ‘Yrb ~hþnXrMr H$moQ>r 2 Amho.

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Polynomials 164

pñWa ~hþnXr (Constant Polynomial)Imbr {Xboë¶m ~hþnXtMo {ZarjU H$am.

f(x) = 10 p(x) = – 2 g(x) =17

h(y) =34

¶mZm pñWa ~hþnXr Ago åhUVmV.

pñWa ~hþnXr 0 qH$dm f(x) = 0 bm ewÝ¶ ~hþnXr Ago åhUVmV.

~hþnXtMo àH$ma [Types of Polynomials]AmVm AmnU doJdoJù¶m H$moQ>tÀ¶m ~hþnXr Am{U Ë¶m§À¶m Zmdm§Mm {dMma H$é.

Imbrb gmaUrMm Aä¶mg H$am.

~hþnXr H$ m oQ >r gm‘mÝ¶ Z‘wZm CXmhaU o

aofr¶ ~hþnXr 1 ax + b 4x + 5OoWo a 0 3y + ½

dJ© ~hþnXr 2 ax2 + bx + c 2x2 – 6x + 5

OoWo a 0 2 13 42

x x

KZ ~hþnXr 3 ax3 + bx2 + cx + d x3 + 2x2 – 4x + 7

OoWo a 0 3 12 83

x x

darb gd© KQ>Zm‘Ü¶o a, b, c, d, e.... dmñVd g§»¶m qH$dm dmñVd ghJwUH$ AmhoV.

~hþnXtMr qH$‘V [Value of a Polynomial]

CXmhaU 1 : f(x) = 4x + 5. ¶m aofr¶ ~hþnXrMm {dMma H$é. Oa x = 1, Va ~hþnXr f(x) Mr qH$‘V H$m¶?

CH$b: ~hþnXr f(x) ‘Ü¶o x Mr qH$‘V dmnê$Z gmoS>dm. f(x) = 4(x) + 5f(1) = 4(1) + 5 = 4 + 5 = 9

f(x) = 9 Ooìhm x = 1

f(x) = 4(x) + 5 Mr qH$‘V 9 Amho, Ooìhm x = 1.

CXmhaU 2 : dJ© ~hþnXr g(y) = y2 + 3y + 5. Mm {dMma H$é. ~hþnXr g(y),Mr qH$‘V {H$Vr? Ooìhmy = 2.

CH$b : g(y) = y2 + 3y + 5

g(2) = (2)2 + 3(2) + 5 = 4 + 6 + 5 = 15

g(y) = y2 + 3y + 5 Mr qH$‘V 15 Amho, Ooìhm y = 2.

{dMma H$am! ewÝ¶ ~hþnXrÀ¶m H$moQ>rMr

ì¶m»¶m H$aVm ¶oV Zmhr.

H$m ?

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Polynomials 165

darb CXmhaUmnmgyZ Amåhr {ZîH$f© H$mTy> eH$Vmo H$s,

Oa f(x) hr 'x' ‘Yrb ~hþnXr Am{U 'k' hr dmñVd g§»¶m AgyZ, f(x) ‘Ü¶o 'x' À¶m EodOr 'k'

KodyZ {‘iUmè¶m dmñVd g§»¶obm f(x), Mr qH$‘V Ago åhUVmV. Am{U ho f(k) Ago Xe©{dVmV.

KZr¶ ~hþnXrMr {H$‘V :

f(x) = x3 + 2x2 – 3x + 4 ¶m KZ ~hþnXrMr qH$‘V, x = 1 Am{U x = – 1 AgVmZm Imbrbà‘mUo Amho,

f(1) = 13 + 2(1)2 – 3(1) + 4 = 1 + 2 – 3 + 4 = 4

f(–1) = (–1)3 + 2(–1)2 – 3(–1) + 4 = –1 + 2 + 3 + 4 = 8

~hþnXrMo ewÝ¶ (Zero of a polynomial)

AmVm AmnU H$m§hr ~hþnXr KodyZ Xe©{dë¶mà‘mUo Ë¶m§Mr qH$‘V H$mTy>.

CXmhaU 1 : f(x)= x2 – x – 2 OoWo x = 2

f(2)= 22 – 2 – 2 = 4 – 4 = 0 f(2) = 0

MbnXmÀ¶m {Xboë¶m {H$§‘VrgmR>r darb gd© CXmhaUm‘Ü¶o ~hþnXrMr qH$‘V ~amo~a 0 (ewÝ¶) Amho.

MbnXmÀ¶m ¶m qH$‘Vrbm "~hþnXrMo ewÝ¶o' Ago åhUVmV.

2 ho f(x) = x2 – x – 2 ¶m ~hþnXrMo ewÝ¶ Amho.

– 1 ho p(x) = x2 – 3x – 4 ¶m ~hþnXrMo ewÝ¶ Amho.

darb MM}VyZ Amåhr Agm {ZîH$f© H$mTy> eH$Vmo H$s, Oa p(x) hr ~hþnXr Am{U k hr H$moUVrhr dmñVd

g§»¶m Aer Amho H$s, p(k) = o, Va k bm ~hþnXr p(x) Mo ewÝ¶ Ago åhUVmV.

CXmhaUmW© :

f(x) = x2 - 5x + 6 ¶m ~hþnXrMr ewÝ¶o 2 Am{U 3 AmhoV.H$maU f(2) = 0 Am{U f (3) = 0.

H¥$Vr : H$m§hr aofr¶, dJ© Am{U KZ~hþnXtMm {dMma H$am. Ë¶m ~hþnXtMo ewÝ¶ emoYm. Imbrb {dYmZo nS>VmiyZ

nmhm.

1. aofr¶ ~hþnXrbm OmñVrV OmñV EH$ ewÝ¶o AgVmo.

2. dJ© ~hþnXrbm OmñVrV OmñV XmoZ ewÝ¶o AgVmV.

3. KZ ~hþnXrbm OmñVrV OmñV VrZ ewÝ¶o AgVmV.

gm‘mÝ¶nUo 'n' H$moQ>rÀ¶m ~hþnXrbm OmñVrV OmñV 'n' ewÝ¶o AgVmV.

CXmhaU 2 : p(x)= x2 – 3x – 4 OoWo x = – 1

p(–1)= (–1)2 – 3(–1) – 4 = 1 + 3 – 4 = 0 p(–1) = 0

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Polynomials 166

ho OmUyZ ¿¶m !^m¡{‘{VH$ÑîQ>çm H$moUË¶mhr ~hþnXrMm ewÝ¶ åhUOo Ogm ~hþnXrMm AmboI Am{U x - Aj EH$‘oH$mg N>oXVAgVrb Va Ë¶m§À¶m N>oXZ q~XþMm x - gh{ZX}eH$ hmo¶.

{dMma H$am !

~hþnXrMo H$moUVohr ewÝ¶ gd© dmñVd g§»¶m‘YyZ AgboM Ago Zmhr.

CXmhaU, p(x) = x2 + 1 ¶m ~hþnXrMo ewÝ¶o dmñVd g§»¶m n¡H$s Zmhr. åhUOoM k ho dmñVd Zmhr Ogo H$s

p(k) = 0. H$m ?

gmoS>{dbobr CXmhaUo

CXmhaU 1 : x 2 + 14x + 48 ¶m dJ© ~hþnXrMo ewÝ¶ emoYm. Am{U nS>VmiyZ nmhm.

CH$b : {Xbobr ~hþnXr x 2 + 14x + 48.

dJ© ~hþnXrMo Ad¶d nmSy>Z,Amåhmbm {‘iob.

x2 + 14x + 48 = (x + 8) (x + 6)

åhUOoM x2 + 14 x + 48 Mr {H$‘§V eyÝ¶ Ooìhm x + 8 = 0 qH$dm x + 6 =0

ho Ooìhm x = -8 qH$dm x=-6

x2 + 14x + 48 ¶m ~hþnXrMo ewÝ¶o -8 d -6 AmhoV.

AmVm AmnU ~hþnXr‘Ü¶o {H$§‘Vr dmnê$Z nS>VmiyZ nmhÿ.

p(x) = x 2 + 14x + 48=(8)2+14-(-8)+48=64-112+48=0 p( 8) = 0

p( 6) = (-6)2+14 (-6)+48=36-84+48 p( 6)

XI X

YI

Y

O 1 2 3 4–1–2

–2

–3

–1

5

4

3

1

2

–3–4

y = +1x2

(0,1)

y = -5+6x x2

XI X

YI

Y

O 1 2 3 4–1–2

–2

–3

–1

5

4

3

1

2

–3–4(3, 0)(2, 0)

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Polynomials 167

CXmhaU 2. x 2 - 3 ¶m ~hþnXrMo ewÝ¶o emoYm Am{U nS>VmiyZ nmhm.

CH$b : p(x) = x 2 – 3

x 2 – 3 Mo Ad¶d nmSy>Z x 2 – 3 = x2 – ( 3 )2 = (x + 3 ) (x – 3 )

(x 2 – 3) Mr qH$‘V ewÝ¶o Amho Oa x = 3 qH$dm x = – 3

(x 2 – 3) Mo ewÝ¶o 3 Am{U (– 3 ) AmhoV.nS>Vmim :

(x 2 – 3) Mo ewÝ¶o 3 Am{U – 3 AmhoV.p(x) = x2 – 3p( 3 ) = ( 3 )2 – 3 = 3 – 3 = 0

VgoM p(– 3 ) = (– 3 )2 – 3 = 3 – 3 = 0

CXmhaU g§J«h 8.1

1. Imbrb ~hþnXtMr H$moQ>r (KmVm§H$) emoYm.(i) x2 – 9x + 20 (ii) 2x + 4 + 6x2

(iii) x3 + 2x2 – 5x – 6 (iv) x3 + 17x – 21 – x2 (v) 3 x3 + 19x + 14

2. Oa f(x) = 2x3 + 3x2 – 11x + 6, Va emoYm.(i) f(0) (ii) f(1) (iii) f(–1) (iv) f(2) (v) f(–3)

3. Imbrb ~hþnXtMr qH$‘V H$mT>m.(i) g(x) = 7x2 + 2x + 14, Ooìhm x = 1 (ii) p(x) = – x3 + x2 – 6x + 5, Ooìhm x = 2

(iii) p(x) = 2x2 +14

x + 13, Ooìhm x = – 1

(iv) (px) = 2x4 – 3x3 – 3x2 + 6x – 2, Ooìhm x = – 2

4. Imbr Xe©{dboë¶m g§»¶m {Xboë¶m ~hþnXtMo ewÝ¶o AmhoV H$m? Vnmgm.

(i) f(x) = 3x + 1, x = –13

(ii) p(x) = x2 – 4, x = 2, x = –2

(iii) g(x) = 5x – 8, x =45

(iv) p(x) = 3x3 – 5x2 – 11x – 3, x = 3, x = –1 Am{U x = –13

5. Imbrb dJ© ~hþnXtMo ewÝ¶ emoYm. A{U nS>VmiyZ nmhm.

(i) x2 + 4x + 4 (ii) x2 – 2x – 15 (iii) 4a2 – 49 (vi)2a2 – 2 2 a+ 1

6. Oa x = 1 ho f(x) = x3 – 2x2 + 4x + k, ¶m ~hþnXrMo ewÝ¶o Agob Va k Mr qH$‘V emoYm.

7. k À¶m H$moUË¶m qH$‘Vrg –4 ho x2 – x – (2k + 2) ¶m ~hþnXrMo ewÝ¶ hmoB©b?

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Polynomials 168

~hþnXrgmR>r ^mJmH$ma H$m¶m©dbr [Division algorithm for polynomials]

Vwåhr ‘mJrb dJm©‘Ü¶o EH$m ~hþnXrbm Xþgè¶m ~hþnXrZo, [XK© ^mJmH$ma nÜXVrZo ^mJUo ho {eH$bmAmhmV.AmVmAmnU Ë¶m nÜXVrMr COiUr H$é.

CXmhaU 1 : (5x + x2 + 14 + 2x3) bm (x + 2) Zo ^mJm.

nm¶ar 0 :

¶m CXmhaUm‘Ü¶o ^mÁ¶ p(x) = 2x3 + x2 + 5x + 14

^mOH$ g(x) = x + 2

^mJmH$ma q(x) = 2x2 – 3x + 11

~mH$s r(x) = – 8

darb gd© CXmhaUm‘Ü¶o, Ooìhm ~hþnXr p(x) bm Xþgar ~hþnXr g(x), Zo ^mJbo OmVo Voìhm ^mJmH$ma q(x)

Am{U ~mH$s r(x). Imbrb gmaUrV 3 CXmhaUmMr g{dñVa ‘m{hVr {Xbr Amho.

CXmhaU ^mÁ¶ ^mOH$ ^mJmH$ma ~mH$sp(x) g(x) q(x) r(x)

1. 2x3 + x2 + 5x + 14 x + 2 2x2 – 3x + 11 –8

2. 9x4 – 4x2 + 4 3x2 + x – 1 3x2 – x –x + 4

3. 3x4 + 5x3 – 7x2 + 2x + 2 x2 + 3x + 1 3x2 – 4x + 2 0

^mÁ¶, ^mOH$, ^mJmH$ma Am{U ~mH$s ¶m‘Yrb ñnï> g§~§Ym~Ôb Amåhr ¶mnyduM {eH$bmo AmhmoV..

¶wp³bS>À¶m ^mJmH$ma {gÜXm§VmZwgma, (boå‘mZwgma)

^mÁ¶= ^mOH$ × ^mJmH$ma + ~mH$s

åhUOoM , Oa a Am{U b ho H$moUVohr XmoZ nyUmªH$ AgVrb Va, a = bq + r,

OoWo 0 r b.

hm g§~§Y ~hþnXtÀ¶m ^mJmH$mamgmR>r gwÜXm bmJy nS>Vmo.

åhUOoM ~hþnXr p(x) bm Xþgar ~hþnXr g(x), Zo ^mJë¶mg ¶oUmam ^mJmH$ma q(x) d ~mH$s r(x), Agob,

Va p(x) = g(x) × q(x) + r(x), OoWo g(x) ~amo~a ewÝ¶ Zmhr. Am{U r(x) ~amo~a ewÝ¶ qH$dm r(x) Mr H$moQ>rg(x) À¶m H$moQ>rnojm bhmZ Agob.

2

23

3

2

2

4

4

5

56

8

3

33

11

1111

14

1422

22

2

x

xx

x

xx

x

x

xx

xx

x-

-

-

-

(+) (+)

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Polynomials 169

hr H$m¶m©dbr ¶wp³bS>À¶m ^mJmH$ma {gÜXm§Vmbm g‘én AgyZ ¶mbmM "~hþnXtMr ^mJmH$ma H$m¶m©dbr'Ago åhUVmV.

~hþnXtÀ¶m ^mJmH$ma H$m¶m ©dbrZwgma,

Oa p(x) Am{U g(x) ¶m H$moUË¶mhr XmoZ ~hþnXr AgyZ g(x) 0, Agob Va Amåhr q(x)Am{U r(x) ¶m ~hþnXr AemàH$mao emoYy eH$Vmo H$s,

p(x) = g(x) × q(x) + r(x), OoWo r(x) = 0 qH$dm r(x) Mr H$moQ>r g(x).À¶m H$moQ>r.nojm bhmZAmh o.

¶mbmM "¶wp³bS>Mm ~hþnXrgmR>r {gÜXm §V' Agohr åhUVmV.

AmVm AmnU MMm© Ho$boë¶m darb gmaUrVrb VrZ CXmhaUmgmR>r "~hþnXr§Mr ^mJmH$ma H$m¶m©dbr'nS>VmiyZ nmhÿ¶m.

CXmhaU 1 : 5x + x2 + 14 + 2x3 bm x + 2 Zo ^mJm.

CH$b : {XK© ^mJmH$mamZ§Va Amåhmbm {‘iVo

^mJmH$ma q(x) = 2x2 – 3x + 11 Am{U

~mH$s r(x) = – 8

~hþnXtÀ¶m ^mJmH$ma H$m¶m©dbrZwgma,

p(x) = g(x) × q(x) + r(x)

AmVm AmnU g‘rH$aUmMr COdr ~mOy Kody.

g(x) × q(x) + r(x)

^mOH$ × ^mJmH$ma + ~mH$s

(x + 2) (2x2 – 3x + 11) + (–8)

2x3 – 3x2 + 11x + 4x2 – 6x + 22 – 8

2x3 + x2 + 5x + 14 = ^mÁ¶

^mJmH$ma H$m¶m©dbr bmJy nS>Vo. (pgÜX hmoVo)

CXmhaU 2 : 9x4 – 4x2 + 4 bm 3x2 + x – 1 Zo ^mJm.

CH$b: ¶m CXmhaUm‘Ü¶o,

^mÁ¶ p (x) = 9x4 – 4x2 + 4, ^mOH$ g (x) = 3x2 – x -1

^mJmH$ma q (x) = 3x2 – x , ~mH$s r(x) = – x + 4

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Polynomials 170

~hþnXtÀ¶m ^mJmH$ma H$m¶m©dbrZwgma,

p(x)= g(x) × q(x) + r(x)

RHS= g(x) × q(x) + r(x)

= (3x2 + x – 1)(3x2 – x) + (–x + 4)

= 4 39 3x x 2 33 3x x 2x x x 4

= 9x4 – 4x2 + 4 = ^mÁ¶

^mJmH$ma H$m¶m©dbr bmJy nS>Vo. ({gÜX hmoVo)

[Q>n : AemM àH$mao {Vgè¶m CXmhaUmgmR>r ^mJmH$ma H$m¶m©dbr nS>VmiyZ nmhm.

Vwåhmbm ~hþnXtMm ^mJmH$ma ‘m{hV Amho. VgoM H$moUË¶mhr XmoZ {Xboë¶m ~hþnXr ^mÁ¶ d ^mOH$ XrK©^mJmH$ma nÜXVrZo ^mJmH$ma d ~mH$s H$mTy> eH$Vm. ^mJmH$ma H$m¶m©dbrMm Cn¶moJ H$éZ ¶oUmam n[aUm‘ nS>VmiyeH$Vm. ¶m{R>H$mUr ‘hËdmMm Agm àíZ {Z‘m©U hmoVmo H$s,

^mJmH$ma Am{U ~mH$s {Xë¶mg ^mÁ¶ Am{U ^mOH$ emoYUo e³¶ Amho H$m?

hmo¶, ""~hþnXtMr ^mJmH$ma H$m¶m©dbr dmnéZ Amåhr ^mÁ¶ d ^mOH$ emoYy eH$Vmo.''

AmVm, Amåhr ^mJmH$ma H$m¶m©dbrMm Cn¶moJ ñnï> H$aÊ¶mg H$mhr CXmhaUo Kody¶m.

(i) ^mJmH$ma H$m¶m©dbrMm Cn¶moJ H$éZ ^mÁ¶ d ^mOH$ emoYUo.

CXmhaU 3 : 3x3 + x2 + 2x + 5 bm ~hþnXr g(x), Zo mJë¶mg ¶oUmam mJmH$ma d ~mH$s AZwH«$‘o (3x – 5) Am{U(9x + 10) Amho Va g(x) emoYm.

CH$b : p(x) = g(x) × q(x) + r(x) [~hþnXtMr ^mJmH$ma H$m¶m©dbr]

Amåhmbm g(x). emoYm¶Mo Amho. åhUyZ darb g‘rH$aU nwÝhm {bhÿ¶m.

g(x) × q(x) + r(x) = p(x) ~mOy ~XbyZ

g(x) × q(x)= p(x) – r(x)

g(x) =( ) ( )

( )p x r x

q x =3 2(3 2 5) (9 10)

3 5x x x x

x

=3 23 2 5 9 10

3 5x x x x

x

g(x) =3 23 7 5

3 5x x x

x

AmVm g(x) [XK© ^mJmH$ma nÜXVrZo emoYy.

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Polynomials 171

2

2 3

2 13 5 3

x xx x 2

37 5

3x x

x 2

( ) ( )

2

5

6

x

x2

76x

( ) ( )10

3

x

x

( )

53x

( )5

0

g(x) = x2 + 2x + 1

^mOH$ x2 + 2x + 1hm Amho.

CXmhaU 2: ~hþnXr p(x) bm (2x – 1). Zo ^mJbo. ^mJmH$ma Am{U ~mH$s AZwH«$‘o (7x2 + x + 5) Am{U 4 AmhoV.Va p(x). emoYm.CH$b : p(x) = g(x) × q(x) + r(x) [~hþnXtMr ^mJmH$ma H$m¶m©dbr]

p(x) = (2x – 1)(7x2 + x + 5) + 4

p(x) = 14x3 + 2x2 + 10x – 7x2 – x – 5 + 4

p(x) = 14x3 – 5x2 – 9x – 1

^mÁ¶ p(x) = 14x3 – 5x2 – 9x – 1

(ii) àË¶j ^mJmH$ma Z H$aVm ^mJmH$ma H$m¶m©dbrMm Cn¶moJ H$éZ ^mJmH$ma d ~mH$s emoYUo.

CXmhaU 3 : p(x) = x3 – 6x2 + 15x – 8 bm g(x) = x – 2 Zo ^mJ XodyZ ^mJmH$ma d ~mH$s emoYm.

CH$b : p(x) = x3 – 6x2 + 15x – 8 p(x) Mr H$moQ>r 3 Amho.

g(x) = x – 2 g(x) Mr H$moQ>r 1Amho.

^mJmH$ma q(x) Mr H$moQ>r = 3 – 1 = 2 ~mH$s r(x) Mr H$moQ>r ewÝ¶ Amho.

AmVm q(x) = ax2 + bx + c (2 H$moQ>tMr ~hþnXr)

r(x) = k (pñWa ~hþnXr)

^mJmH$ma H$m¶m©dbrMm Cn¶moJ H$éZ,p(x) = g(x) × q(x) + r(x)

x3 – 6x2 + 15x – 8 = (x – 2)(ax2 + bx + c) + k= ax3 + bx2 + cx – 2ax2 – 2bx – 2c + k

x3 – 6x2 + 15x – 8 = ax3 + (b – 2a)x2 + (c – 2b) x – 2c + k

g‘rH$aUmÀ¶m XmoÝhr ~mOybm KZ ~hþnXr AmhoV.

a, b, c.À¶m qH$‘Vr {‘i{dÊ¶mgmR>r x3, x2, x Am{U k À¶m ghJwUH$m§Mr VwbZm Ho$ë¶mg,

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Polynomials 172

1 = a XmoÝhr ~mOybm x3 Mm ghJwUH$

–6 = b – 2a XmoÝhr ~mOybm x2 Mm ghJwUH$

15 = c – 2b XmoÝhr ~mOybm x ghJwUH$

–8 = – 2c + k XmoÝhr ~mOybm pñWa nX

AmVm b, c Am{U k Mr qH$‘V {‘i{dÊ¶mgmR>r Imbrb g‘rH$aUo gmoS>dy.b – 2a = – 6; b = – 6 + 2a = – 6 + 2(1) = – 4c – 2b = 15; c = 15 + 2b = 15 + 2(–4) = 7

–2c + k = –8; k = – 8 + 2c = – 8 + 2(7) = 6

q(x)= ax2 + bx + c = (1)x2 + (–4)x + 7 = x2 – 4x + 7 Am{U r(x) = k = 6

^mJmH$ma = x2 – 4x + 7 Am{U ~mH$s 6.

^mJmH$ma H$m¶m©dbrMo AmUIr H$m§hr Cn¶moJ Imbrb CXmhaUmV ñnï> Ho$bobo AmhoV.

CXmhaU 4 : 6x4+13x3+30x +20. ‘YyZ H$m¶ dOm Ho$ë¶mg ¶oUmè¶m ~hþnXrbm 3x2+2x+5 Zo AMyH$ ^mJOmB©b?

CH$b : p(x)= g(x) × q(x) + r(x) [~hþnXtMr ^mJmH$ma H$m¶m©dbr]

= p(x) – r(x) = g(x) × q(x)

ho ñnï> Amho H$s, darb g‘rH$aUmÀ¶m COì¶m ~mOybm ^mOH$ g(x) Zo ^mJ OmVmo.

S>mì¶m ~mOybm gwÜXm ^mOH$mZo ^mJ OmVmo.

åhUyZ, ^mÁ¶ p(x) ‘YyZ ~mH$s r(x) dOm Ho$br Va Ë¶mbm ^mOH$ g(x) Zo AMyH$nUo ^mJ OmB©b,

AmVm AmnU 6x4 + 13x3 + 13x2 + 30x + 20 by 3x2 + 2x+5Zo mJ XoD$.

2

2

2 3 13 2 5 6

x xx x x 3 2

4

13 13 30 20

6

x x x

x 3 2

( ) ( ) ( )

3

4 10

9

x x

x 2

33 30

9x x

x 2

( ) ( ) ( )

2

6 15

3

x x

x2

15 203

xx

( ) ( ) ( )2 5

17 25

x

x

^mJmH$ma q(x) = 2x2 + 3x – 1

~mH$s r(x) = 17x + 25

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Polynomials 173

17x + 25 ho 6x4 + 13x3 + 13x2 + 30x + 20, ‘YyZ dOm Ho$ë¶mg ¶oUmè¶m ~hþnXrbm by (3x2 +

2x + 5)Zo ^mJ OmB©b.

[Q>rn : p(x) ‘YyZ 17x + 25 dOm H$éZ Am{U Z§Va ¶oUmè¶m ~hþnXrbm 3x2 + 2x + 5 Zo ^mJ XodyZ Vw‘MoCÎma nS>VmiyZ nmhm ]

CXmhaU 5:

p(x) = x4 + 2x3 – 2x2 + x – 1 ¶m ~hþnXr‘Ü¶o H$m¶ {‘i{dë¶mg {‘iUmè¶m ~hþnXrbm x2 + 2x – 3 ZonyU© ^mJ OmB©b?

CH$b : p(x) = g(x) × q(x) + r(x) ^mJmH$ma H$m¶m©dbr.

= p(x) – r(x) = g(x) × q(x)= p(x) + {–r(x)} = g(x) × q(x)

ho ñnï> hmoVo H$r COì¶m ~mOybm g(x) Zo ^mJ OmVmo.

S>mì¶m ~mOybm gwÜXm g(x) Zo ^mJ OmVmo.

åhUyZ, Oa p(x) ‘Ü¶o –r(x) {‘i{dbo Va {‘iUmè¶m ~hþnXrbm g(x) Zo ^mJ OmVmo.

AmVm AmnU ~mH$s r(x) emoYÊ¶mgmR>r p(x) bm g(x) Zo ^mJ XoD$.

2

2

12 3

xx x x 3 2

4

2 2 1x x x

x 3 2

( ) ( ) ( )

2

2 3x x

x2

1xx 2

( ) ( ) ( )2 3

2

x

x

r (x) = -x + 2

- { r(x)} = x - 2

åhUyZ p(x) ‘Ü¶o (x – 2) {‘i{dë¶mg {‘iUmè¶m ~hþnXrbm g(x) Zo nyU© ^mJ OmB©b.

CXmhaU g§J«h : 8.2

1. Imbrb àË¶oH$ àH$mam‘Ü¶o p(x) bm g(x) Zo ^mJm. Am{U ^mJmH$ma H$m¶m©dbr nS>VmiyZ nmhm.

(i) p(x) = x2 + 4x + 4 g(x) = x + 2

(ii) p(x) = 2x2 – 9x + 9 g(x) = x – 3

(iii) p(x) = x3 + 4x2 – 5x + 6 g(x) = x + 1

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Polynomials 174

(iv) p(x) = x4 – 3x2 – 4 g(x) = x + 2

(v) p(x) = x3 – 1 g(x) = x – 1

(vi) p(x) = x4 – 4x2 + 12x + 9 g(x) = x2 + 2x – 3

2.^mOH$ g(x) emoYm, Ooìhm ~hþnXr p(x) = 4x3 + 2x2 – 10x + 2 bm g(x) Zo ^mJbo Am{U ^mJmH$ma d ~mH$sho AZwH«$‘o (2x2 + 4x + 1) Am{U 5 AmhoV.

3. p(x) = x3 – 3x2 + x + 2 ¶m ~hþnXrbm g(x) Zo ^mJbo AgVm ^mJmH$ma d ~mH$s AZwH«$‘o (x – 2) Am{U

(–2x + 4) ¶oV AgVrb Va g(x) ~hþnXr emoYm.

4. ~hþnXr p(x) bm g(x),Zo ^mJë¶mg {‘iUmam ^mJmH$ma q(x) Am{U ~mH$s r(x) Imbrb H$moï>H$mV {XboboAmho. àË¶oH$ àH$mamV p(x) emoYm.

A.Z§ p(x) g(x) q(x) r(x)i ? x – 2 x2 - x + 1 4i i ? x + 3 2x2 + x + 5 3x + 1ii i ? 2x + 1 x3 + 3x2 – x + 1 0iv ? x – 1 x3 – x2 – x – 1 2x – 4v ? x2 + 2x + 1 x4 – 2x2 + 5x – 7 4x + 12

5. àË¶j ^mJmH$ma Z H$aVm p(x) bm g(x) Zo ^mJm. Am{U ^mJmH$ma d ~mH$s emoYyZ H$mT>m.

(i) p(x) = x2 + 7x + 10 g(x) = x - 2

(ii) p(x) = x3 + 4x2 - 6x + 2 g(x) = x - 3

6. (x3 + 5x2 + 5x + 8) ‘YyZ H$m¶ dOm H$amdo åhUOo {‘iUmar ~hþnXr (x2 + 3x - 2) Zo nyU© mJ OmUmarAgob?

7. (x4 - 1) ‘Ü¶o H$m¶ {‘idmdo åhUOo Ë¶mbm (x2 + 2x + 1) Zo AMyH$nUo ^mJ OmB©b?

eof {gÜXm§V [Remainder Theorem]

[XK© ^mJmH$ma nÜXVrZo VgoM ^mJmH$ma H$m¶m©dbrZo ~hþnXtMm ^mJmH$ma H$aUo Vwåhmbm ‘m{hV Amho.

AmVm AmnU H$m§hr CXmhaUo nmhy OoWo p(x) ¶m ~hþnXrbm (x-a) Z‘wÝ¶mVrb aofr¶ ~hþnXr g(x )Zo ^mJOmVmo.

Imbrb H$moîQ>H$mV {Xboë¶m g{dñVa ‘m{hVrMm Aä¶mg H$am.

CXm. ^mÁ¶ p(x) ^mOH$^ ^mJmH$ma ~mH$sg(x) = x – a q(x) r(x)

1. 4x2 – 7x + 9 x – 2 4x + 1 +11

2 8x3 – 18x2 + 23x + 12 x – 3 8x2 + 6x + 41 +135

3. x5 + a5 x – a x4 + ax3 + a2x2 + a3x + a4 2a5

{ZarjU H$am H$s, ~mH$s pñWa nX Amho.

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Polynomials 175

Á¶m AWu g(x) = x – a hr aofr¶ ~hþnXr Amho, {VMr H$moQ>r ~amo~a 1 Amho.

Amåhr OmUVmo H$s, r(x) Mr H$moQ>r g(x) Mr H$moQ>r.

r(x) Mr H$moQ>r ewÝ¶ Amho.

AmVm AmnU Xþgao CXmhaU Kody.

(x6+4x5-9x2+15) bm x-2 ^mJë¶mg ~mH$s H$m¶ {‘iob?

~mH$s emoYÊ¶mgmR>r Amåhr {XK© ^mJmH$ma nÕV dmnaVmo. hr nÕV ’$maM ‘moR>r Am{U doi bmJUmar Amho.

Xþgè¶m EImÚm bKw Am{U gmoß¶m nÜXVrZo ~mH$s emoYZo e³¶ Amho H$m?

AmVm AmnU Vr nÜXV {eHy$.

¶m AJmoXa MMm© Hoo$bobo, OoWo ~mH$s emoYÊ¶mgmR>r XrK© ^mJmH$ma nÜXVrMm Cn¶moJ Ho$bobo CXmhaU Kody.

CXmhaU 1: 4x2 – 7x + 9 bm x – 2 Zo ^mJm. ¶oWo ^mÁ¶ p(x)= 4x2-7x+9

CH$b : ^mOH$ g(x)= x-a= x-2

Oa x – 2=0 Va x= 2

x = 2 KoD$Z p(x) Mr H$s¨‘V H$mT>m.

p(x)= 4(2)2 -7(2)+9 = 4x4-14+9

p(x) = 16-14 + 9

p(x) = 11

bjmV ¿¶m H$s p(2) Mr {‘imbobr qH$‘V {XK© ^mJmH$ma nÕVrZo {‘iUmè¶m ~mH$sBVH$s Amho.

CXmhaU 2: 8x3 – 18x2 + 23x + 12 bm x – 3 Zo ^mJm.

CH$b :

¶oWo p(x)= 8x3 – 18x2 + 23x + 12

g(x) = x – 3

Oa x-3= 0, x=3

x=3 KoD$Z p(x) Mr qH$‘V H$mT>y.

p(3) = 8x(3)3 -18(3)2 + 23(3)+12

=216-162 + 69+12=135

p(3) = 135

{XK© ^mJmH$ma nÕVrZo {‘iUmè¶m ~mH$ser VwbZm H$am.

CXmhaU 3 : x5 + a5 bm x – a Zo ^mJm. ¶oWo p(x)= x5+a5 Am{U g(x)= x--a

Oa x--a=0 Va x-=a

x-=a KoD$Z p(x) Mr qH$‘V H$mT>ë¶mg Amåhmbm {‘iVo.

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Polynomials 176

p(a)= a5+a5=2a5

{XK© ^mJmH$ma nÕVrZo {‘iUmè¶m ~mH$ser p(a) $À¶m qH$‘VrMr VwbZm H$am.

darb àË¶oH$ CXmhaUm‘Ü¶o, Ooìhm mÁ¶ p(x) bm ^mOH$ g(x) = x – a Zo ^mJ {Xbm Voìhm ^mJmH$ma g(x)Am{U ~mH$s r(x) {‘imbr.

darb CXmhaUo g{dñVa Imbrb H$moï>H$mV ‘m§S>m. eodQ>À¶m XmoZ aH$mÝ¶m§Mr VwbZm H$am.

CXm. ^mÁ¶ p(x) ^mOH $ p(a)Mr XrK© ^mJmH$ma nÜXXVrZog(x) = x – a qH$‘V {‘imbobr ~mH$s

1. 4x2 – 7x + 9 x – 2 p(2) = +11 +11

2 8x3 – 18x2 + 23x + 12 x – 3 p(3) = +135 +135

3. x5 + a5 x – a p(a) = +2a5 +2a5

darb H$moï>H$mdéZ ho ñnï> hmoVo H$s, p(a) Mr qH$‘V Am{U ~mH$s gma»¶mM AmhoV.

Amåhr {ZîH$f© H$mTy> eH$Vmo H$s,

Oa ~hþnXr p(x) bm aofr¶ ~hþnXr (x – a) Zo ^mJbo Va ~mH$s p(a) Amho.

¶mbmM eof {gÜXm§V Ago åhUVmV.

AmVm, AmnU (x6 + 4x5 – 9x2 + 15) bm aofr¶ {ÛnXr (x – 2) Zo ^mJë¶mZo {‘iUmar ~mH$s H$mTy>.

p(x) = x6 + 4x5 – 9x2 + 15

g(x) = x – 2 Oa x – 2=0 Va x = 2

p(2) = 26 + 4.25 – 9.22 + 15 = 64 + 128 – 36 + 15 = 207-36=171

p(2) = 171

àË¶j ^mJmH$ma Z H$aVm (x6 + 4x5 – 9x2 + 15) bm (x – 2) Zo ^mJë¶mg {‘iUmar ~mH$s Amåhr emoYbr Amho.

[[Q>n : [XK© ^mJmH$ma nÜXVrZo CÎma nS>VmiyZ nmhm]

{ZarjU Ho$ë¶mg ho ñnï> hmoVo H$s, Ooìhm ^mOH$ hm {ÛnXr Am{U (x – a) ¶m aofr¶ Z‘wÝ¶mV AgVmo VoìhmMeof {gÜXm§V Cn¶wº$ Amho.

Amåhr ho gwÜXm Z‘yX H$é H$s,

* Ooìhm ^mOH$ (x – a), Z‘wÝ¶mV AgVmo V|ìhm ~mH$s p(a) AgVo.* Ooìhm ^mOH$ (x + a), Z‘wÝ¶mV AgVmo V|ìhm ~mH$s p(–a). AgVo.

H$maU (x + a) hoM {x – (–a)} Agohr {bhrVm ¶oVo.AmVm AmnU eof {gÜXm§VmMo {dYmZ {gÜX H$é¶m.eof {gÜXm§V : Oa ~hþnXr p(x) bm (x – a)Zo ^mJbo Va ~mH$s p(a) BVH$s AgVo.{gÜXVm : Ooìhm ~hþnXr p(x) bm (x – a) Zo ^mJbo OmVo Voìhm ^mJmH$ma H$m¶m©dbrZwgma {‘iUmam ^mJmH$ma q(x)Am{U pñWa ~mH$s r Aer AgVo H$s,p(x) = (x – a). q(x) + r

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Polynomials 177

p(a) = (a – a). q(a) + r = 0.q(a) + r

p(a) = r åhUyZ ~mH$s p(a) Amho.

Oa p(x) bm (x + a) Zo ^mJbo Va ~mH$s p(–a) AgVo.

Oa p (x) bm (ax + b) Zo ^mJbo Va ~mH$sbp

a AgVo.

gm oS >{dbobr CXmhaUo

CXmhaU 1: p(x) = x3 – 4x2 + 3x + 1 ¶m ~hþnXrbm Ooìhm (x–1) Zo ^mJbo AgVm ~mH$s H$mT>m.

CH$b :eof {gÜXm§VmZo Amdí¶H$ AgUmar ~mH$s p(1) er g‘mZ Amho.

p(x) = x3 – 4x2 + 3x + 1p(1) = 13 – 4(1)2 + 3(1) + 1 = 1 – 4 + 3 + 1 = 1

Amdí¶H$ ~mH$s = p(1) = 1

CXmhaU 2: p(x) = x3 – 6x2 + 2x – 4 ¶m ~hþnXrbm, Ooìhm g(x) = 3x – 1 Zo ^mJbo AgVm ~mH$sH$mT>m.

CH$b :¶oWo, g(x) = 3x – 1. eof {gÜXm§V bmJy H$aÊ¶mgmR>r (dmnaÊ¶mgmR>r) (3x-1) bm (x - a) ¶m ñdénmVénm§VarV Ho$bo nm{hOo.

3x – 113 (3x – 1) x –

13 g(x) = 3

13

x

eof {gÜXm§VmZo, r(x) = p(a) = p13

AmVm, p(x) =x3 – 6x2 + 2x – 4 p13 =

313 – 6

213 + 2

13 – 4 =

127

–69

+23

– 4

=1 18 18 108

27 =

10727

~mH$s p 13

=10727

CXmhaU 3: (ax3 + 3x2 – 13) Am{U (2x3 – 4x + a) ¶m ~hþnXtZm (x–3)Zo ^mJbo. Oa XmoÝhràH$mam‘Ü¶o ~mH$s g‘mZ Agob Va a Mr qH$‘V emoYm.

CH$b :p(x) = ax3 + 3x2 – 13g(x) = 2x3 – 4x + a

eof {gÜXm§VmZwgma, p(3) Am{U g(3) ¶m XmoZ ~mH$s hmo¶.

{Xboë¶m AQ>rZwgma, p(3) = g(3)

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Polynomials 178

p(3) = a. 33 + 3.32 – 13 = 27a + 27 – 13 = 27a + 14

g(3) = 2.33 – 4.3 + a= 54 – 12 + a = 42 + a

na§Vw p(3) = g(3), åhUyZ

27a + 14 = 42 + a

26a = 28

a =2826

=1413

CXmhaU 4: XmoZ ~hþnXr (2x3 + x2 – 6ax + 7) Am{U (x3 + 2ax2 – 12x + 4) ¶m§Zm AZwH«$‘o (x+1)

Am{U (x–1) Zo ^mJbo. Oa R1 Am{U R2 ¶m ~mH$s AgyZ 2R1 + 3R2 = 27, Agob

Va a Mr qH$‘V emoYm.

CH$b : p(x) = 2x3 + x2 – 6ax + 7

f(x) =x3 + 2ax2 – 12x + 4

Oa, p(x) bm (x + 1) Zo ^mJbo Va ~mH$s R1 Amho.

R1 = p(–1)

R1 = 2(–1)3 + (–1)2 – 6a(–1) + 7 = –2 + 1 + 6a + 7

R1 = 6a + 6

Am{U,

Oa f(x) bm (x-1) Zo ^mJbo Va ~mH$s R2 Amho.

R2 = p(1)

R2 = 13 + 2a(1)2 – 12(1) + 4 = 1 + 2a – 12 + 4

R2 = 2a – 7

R1 Am{U R2 À¶m qH$‘Vr 2R1 + 3R2 = 27, ‘Ü¶o dmnê$

2 (6a + 6) + 3(2a – 7) = 27

12a + 12 + 6a – 21 = 27

18a – 9 = 27 18a = 27 + 9 = 36 a =3618

a = 2

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Polynomials 179

ñdmÜ¶m¶ 8.3

1. Imbrb àË¶oH$ CXmhaUmbm eof {gÜXm§VmMm dmna H$am Am{U Ooìhm p(x) bm g(x). Zo ^mJbo AgVm ~mH$sH$mT>m.

àË¶j ^mJmH$mamZo CÎma nS>Vmim.

(i) p(x) = x3 + 3x2 – 5x + 8 g(x) = x – 3

(ii) p(x) = 4x3 – 10x2 + 12x – 3 g(x) = x + 1

(iii) p(x) = 2x4 – 5x2 + 15x – 6 g(x) = x – 2

(iv) p(x) = 4x3 – 12x2 + 14x – 3 g(x) = 2x – 1

(v) p(x) = 7x3 – x2 + 2x – 1 g(x) = 1 – 2x

2. Oa (2x3 + ax2 + 3x – 5) Am{U (x3 + x2 – 4x – a) ¶m ~hþnXrZm Ooìhm (x-1) Zo mJbo AgVm gmaIrM(g‘mZ) ~mH$s amhVo Va a.Mr qH$‘V H$mT>m.

3. (2x3 – 5x2 + x + a) Am{U (ax 3+ 2x2 -3) ¶m ~hþnXrZm Ooìhm (x -2) ^mJbo AgVm AZwH«$‘o R1 Am{UR2 ~mH$s amhVo. Va àË¶oH$ CXmhaUmVrb a Mr qH$‘V H$mT>m.

Oa

(i) R1=R2 (ii) 2R1+ R2=0 (iii) R1+R2 =0 (iv) R1- 2R2=0

Ad¶d à‘o¶ (Factor Theorem)

Amåhr ho Aä¶mgbo Amho H$s, Oa ~hþnXr p(x) bm (x–a) ¶m ñdê$nmVrb {ÛnXrZo ^mJbo AgVm ~mH$s p(a){‘iVo. Amåhr ho gwÕm nm{hbo Amho H$s p(a) À¶m qH$‘VrÀ¶m XmoZ e³¶Vm AmhoV. Ë¶m åhUOo p(a)=0 qH$dmp(a)=0

^mÁ¶ Am{U ^mOH$mer g§~§YrV Agbobm H$moUË¶mhr àH$maMm JwUY‘© p(a) Mr qH$‘V {ZYm©arV H$aVmo H$m?Oa p(a)=0 Mr H$moUVrhr {deofVm: Amho H$m?

eof {gÕm§VmMm {dñVma hm AmUIr EH$m ~hþnXrMm Hw$VwhbmË‘H$ JwUY‘© XoVmo. Ë¶m JwUY‘m©Mm Aä¶mg H$ê$.

A§H$J{UVm‘Yrb CXmhaUo KodyZ àW‘V: JwUY‘©m§À¶m ñnï>rH$aUmMm Aä¶mg H$ê$.

CXmhaU 1:

¶oWo 6 bm 2 Zo nyU© nUo ^mJ OmVmo Am{U ~mH$s eyÝ¶ amhVo.

2x3=6

^mOH$ 2 Am{U ^mJmH$ma 3 ho 6 Mo Ad¶d AmhoV.

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Polynomials 180

CXmhaU 2 :

¶oWo, 6 Zo 14 bm nyU©nUo ^mJ OmV Zmhr Am{U ~mH$s hr eyÝ¶mBVH$s Zmhr

(eyÝ¶mBVH$s Zmhr. (ewÝ¶mer g‘mZ).

^mOH$ 6 hm 14 Ad¶d Zmhr.

2x3=6

^mOH$ 2 Am{U ^mJmH$ma 3 ho 6 Mo Ad¶d AmhoV.

darb XmoZ CXmhaUmdéZ Amåhr Agm {ZîH$f© H$mT>Vmo H$s

* Oa ~mH$s hr eyÝ¶mBVH$s (g‘mZ) Agob Va ^mOH$ hm ^mÁ¶mMm Ad¶d AgVmo.

* Oa ~mH$s hr eyÝ¶mBVH$s (g‘mZ) Zgob Va ^mOH$ hm ^mÁ¶mMm Ad¶d ZgVmo.

* Oa ^mOH$ hm ^mÁ¶mMm Ad¶d Agob Va ^mJmH$ma gwÜXm ^mÁ¶mMm Ad¶d AgVmo.

åhUyZ,

Oa ~mH$s hr eyÝ¶mBVH$s (eyÝ¶m g‘mZ) Agob Va ^mOH$ Am{U ^mJmH$ma ho XmoÝhr ^mÁ¶mMo Ad¶dAgVmV.

Oa ~mH$s hr eyÝ¶mBVH$s (eyÝ¶mg‘mZ) Zgob Va ^mOH$ hm ^mÁ¶mMm Ad¶d ZgVmo Am{U ^mJmH$ma hm^mÁ¶mMm Ad¶d Agy eHo$b qH$dm AgUma gwÜXm Zmhr.

hm JwUY‘© ~hþnXrZmgwÜXm bmJw nS>Vmo. Oa EH$m ~hþnXr p(x) bm Xþgè¶m ~hþnXr g(x) Zo ^mJbo AgVm ~mH$seyÝ¶ Agob Va ^mOH$ g(x) hm p(x) Mm Ad¶d AgVmo

AmVm hm JwUY‘© ñnîQ> H$aÊ¶mgmR>r eof {gÜXm§VmMm dmna H$é.

Oa p(a) ho eyÝ¶ Amho ¶mMm AW© Agm H$s ~mH$s eyÝ¶ Amho. Am{U (x – a) hm ~hþnXr p(x) MmAd¶d AgVmo.

{dYmZmMm ì¶Ë¶mg (CbQ> AW©) XoIrb gË¶ Amho.

Oa (x – a) hm ~hþnXr p(x) Mm Ad¶d Amho. ¶mMm AW© Agm H$s ~mH$s eyÝ¶ Amho Am{U p(a) hoe yÝ¶mBVHo$ Amho.

Amåhr Ago {gÕ H$ê$ H$s

* Oa p(a) = 0, Va (x – a) hm p(x) Mm Ad¶d Amho.

* Ooìhm (x – a) hm p(x), Mm Ad¶d AgVmo Voìhm p(a) = 0

¶bm ~hþnXrMm Ad¶d à‘o¶ åhUVmV.

Ad¶d à‘o¶ hm eof à‘o¶mMm AmUIr EH$ Cn‘o¶ Amho.

Ooìhm ~hþnXr p(x) bm (x – a) ¶m ñdénmVrb aofr¶ ~hþnXrZo ^mJ OmVmo ’$º$ VoìhmM Ad¶d {gÜXm§V dmnabm OmVmo. (bmJy Ho$bm OmVmo) ho EH$ AË¶§V ‘hËdmMo Zm|X H$aÊ¶mgmaIo Amho.

26 14

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Polynomials 181

Oa aofr¶ ~hþnXr hr (x – a) ¶m ñdénmV Zgob Va Vr nwÝhm à‘m{UV Z‘wÝ¶mV {bhm Am{U Z§Va Ad¶d à‘o¶ bmJy H$am (dmnam.)

ho Aä¶mgÊ¶mgmR>r Imbrb {dYmZm§Mo {ZarjU H$am.

* Oa p(a) = 0 Va (x – a) hm p(x) ~hþnXrMm Ad¶d Amho.

* Oa p(–a) = 0 Va x + a hm p(x) ~hþnXrMm Ad¶d Amho.

* Oa pba = 0 Va (ax – b) hm ~hþnXr p(x) Mm Ad¶d Amho.

* Oa pb

a = 0 Va (ax + b) hm ~hþnXr p(x) Mm Ad¶d Amho.

* Oa p(a) = 0 Am{U p(b) = 0 Va (x – a) (x – b) ho ~hþnXr p(x) Mo Ad¶d AmhoV.AmVm Ad¶d à‘o¶mMm Cn¶moJ H$éZ H$m§hr CXmhaUo gmoS>dy.

gmoS>{dbobr CXmhaUo

CXmhaU 1: (x + 2) hm (x3 – 4x2 + 2x + 20)¶m ~hþnXrMm Ad¶d Amho Ago XmIdm.

CH$b : p(x) = x3 – 4x2 – 2x + 20 ‘mZy

Ad¶d à‘o¶mZo (x + 2) hm p(x) Mm Ad¶d Amho Oa p(–2) = 0.

(x + 2) hm p(x) Mm Ad¶d Amho ho Xe©{dUo nwaogo Amho.

AmVm, p(x) = x3 – 4x2 – 2x + 20

p(–2) = (–2)3 – 4(–2)2 – 2(–2) + 20

= –8 –16 + 4 + 20 = 0

(x + 2 hm p(x) = x3 – 4x2 – 2x + 20 Mm Ad¶d Amho.

CXmhaU 2: (x – 1) hm (xn – 1). Mm Ad¶d Amho ho XmIdm.

CH$b: p(x) = xn – 1 ‘mZy.

(x – 1) hm (xn – 1), Mm Ad¶d Amho ho Xe©{dÊ¶mgmR>r p(1) = 0.ho XmI{dUo nwaogo Amho.

AmVm, p(x) = xn – 1

p(1) = 1n – 1 = 1 – 1 = 0

(x + 1) hm (xn-1) Mm Ad¶d Amho.

CXmhaU 3: Oa (x - a) hm (x3 – a2x + x + 2).Mm Ad¶d Agob. Va a Mr qH$‘V H$mT>m.

CH$b : Oa p(a) = 0 Va Ad¶d à‘o¶mZo (x – a) hm p(x) Mm Ad¶d Amho.

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Polynomials 182

p(a) = a3 - a2 x a + a + 2 = 0

a3 - a3 + a + 2 = 0 a + 2 = 0 a = -2

CXmhaU 4: àË¶j ^mJmH$ma Z H$aVm (x4 - 4x2 + 12x - 9) bm (x2 + 2x - 3) Zo nyU©nUo ^mJ OmVmo ho {gÜXH$am.

CH$b : p(x) = x4 – 4x2 + 12x – 9 g(x) =x2 + 2x – 3

g(x) = (x + 3) (x – 1)

åhUyZ, (x + 3) Am{U (x – 1) ho g(x) Mo Ad¶d AmhoV.

p(x) bm g(x) Zo nyU©nUo mJ OmVmo ho {gÜX H$aÊ¶mgmR>r p(x) bm (x + 3) Am{U (x – 1) Zo mJ OmVmoho Xe©{dUo nwaogo Amho.

(x + 3) Am{U (x – 1) ho p(x) Mo Ad¶d AmhoV ho XmIdy.

AmVm, p(x) = x4 – 4x2 + 12x – 9

p(-3) = (-3)4 – 4(-3)2 + 12(–3)–9

= 81 – 36 – 36 – 9 = 81 – 81 = 0

p(-3) = 0

p(1) = (1)4 – 4(1)2 + 12(1) – 9

= 1 – 4 + 12 – 9 = 13 – 13 = 0

p(1) = 0

(x + 3) Am{U (x – 1) ho p(x) Mo Ad¶d AmhoV.

g(x) = (x + 3) (x – 1) ho gwÜXm p(x) Mo Ad¶d AmhoV.

åhUyZ, p(x) bm g(x) Zo nyU©nUo ^mJ OmVmo.

(x4 – 4x2 + 12x – 9) bm (x2 + 2x – 3) Zo nyU©nUo ^mJ OmVmo

ñdmÜ¶m¶ 8.4

1. Imbrb àË¶oH$ CXmhaUm‘Ü¶o Ad¶d à‘o¶mZo g(x) hm ~hþnXr p(x) Mm Ad¶d Amho qH$dm Zmhr Vo emoYm

(i) p(x) = x3 – 3x2 + 6x – 20 g(x) = x – 2

(ii) p(x) = 2x4 + x3 + 4x2 – x – 7 g(x) = x + 2

(iii) p(x) = 3x4 + 3x2 - 4x – 11 g(x) = x –12

(iv) p(x) = 3x3 + x2 – 20x + 12 g(x) = 3x – 2

(v) p(x) = 2x4 + 3x3 – 2x2 – 9x – 12 g(x) = x2 – 3

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Polynomials 183

2.Oa (x - 5) hm (x3 – 3x2 + ax – 10) Mm Ad¶d Agob Va a Mr qH$‘V H$mT>m.3. Oa (x3 + ax2 -bx +10) bm (x2 - 3x +2) Zo ^mJ OmV Agob Va a Am{U b Mr qH$‘V H$mT>m.

4.Oa (x –2) Am{U12

x ho XmoÝhr (ax2 + 5x + b), Mo Ad¶d AmhoV Va a = b Ago XmIdm.

g§íbofH$ ^mJmH$ma : [Synthetic Division]‘mJrb {d^mJmV Amåhr eof {gÜXm§VmÀ¶m Cn¶moJm§Mm Aä¶mg Ho$bobm Amho AmVm AmnU ^mJmH$ma Am{U ~mH$sA{YH$ gmoß¶m [aVrZo H$mT>Ê¶mÀ¶m eof {gÜXm§VmÀ¶m Xþgè¶m Cn¶moJmMm Aä¶mg H$é.^mJmH$ma Am{U ~mH$s hr {XK© ^mJmH$ma nÜXVrZo H$mT>Uo ho Amåhmbm ‘m{hV Amho. hr {XK© nÜXV Amho ho AmåhrAmoiIbo Amho.

Ë¶mZ§Va àË¶j ^mJmH$ma Z H$aVm eof {gÜXm§VmMm Cn¶moJ H$éZ ~mH$s H$mT>Uo ho Amåhr Aä¶mgbo Amho.eof {gÜXm§VmZo ~mH$s gmoß¶m Am{U ObX[aË¶m H$mT>Vm ¶oVo ¶mV e§H$m Zmhr.~mH$s Zìho Va ^mJmH$ma gwÜXm H$mT>Ê¶mMr X þgar ’$ma gmonr Am{U ObX nÜXV Amho.¶m eof {gÜXm§VmMm dmna Ho$boë¶m AË¶§V gmoß¶m nÜXVrbm g§íbofH$ ^mJmH$ma Ago åhUVmV.

AmVm ^mJmH$ma Am{U ~mH$s H$mT>Ê¶mÀ¶m g§íbofH$ ^mJmH$ma nÜXVrMo AÜ¶¶Z H$é. Ooìhm ~hþnXrbm(x – a) ¶m {ÛnXrZo ^mJ OmVmo. OoWo a hm pñWam§H$ Amho.

Imbrb CXmhaUo Aä¶mgm.

CXmhaU 1 : p(x) = x4 – 3x + 5 ¶m ~hþnXrMm {dMma H$am.

p(x) bm (x – 4) ¶m ñdénmVrb aofr¶ ~hþnXrZo ({ÛnXr) ^mJ OmVmo Ago g‘Om.

x4 – 3x + 5 bm 4 Zo ^mJm.

x4 – 3x + 5 bm 4 Zo ^mJÊ¶mgmR>r dmnaboë¶m g§íbofH$ ^mJmH$mamÀ¶m nm¶è¶m§Mo {ZarjU H$am.

nm¶ar 1 : g§íbofH$ ^mJmH$mamMr ‘m§S>Ur H$am.

ho H$aÊ¶mMm gmonm ‘mJ© åhUOo àW‘ Vwåhr {XK© ^mJmH$mamMr Am{U Z§Va g§íbofH$ ^mJmH$mamgmR>r ‘m§S>Ur H$am.

{XK© ^mJmH$ma nÜXVr‘Ü¶o ^mÁ¶ ho Mm¡H$Q>rÀ¶m AmV Am{U ^mOH$ ho Mm¡H$Q>rÀ¶m ~mhoa AgVo.

^mÁ¶ {bhrVmZm Ë¶mMr nXo KmVm§H$mÀ¶m CVaË¶m H«$‘mV {bhrë¶mMr ImÌr H$éZ ¿¶mdr. Oa EImXo nXJmibobo Agë¶mg Vo eyÝ¶ ghJwUH$mZo {bhmdo.

{Xbobm ^mÁ¶ x4 – 3x + 5 Amho.

ho nwÝhm x4 + 0.x3 + 0.x2 – 3x + 5 Ago {bhrVmV.

XrK© ^mJmH$ma ¶mgmaIr {XgVo.

4 3 24 0. 0. 3 5x x x x x

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Polynomials 184

^mÁ¶mVrb àË¶oH$ nXm§À¶m ghJwUH$m§Mo {ZarjU H$am.

Vo 1, 0, 0, –3, 5 AmhoV.

^mOH$ x – 4. ‘Yrb a À¶m qH$‘VrMo [ZarjU H$am a Mr qH$‘V 4 Amho

a À¶m Ë¶m qH$‘Vrbm JwUH$ qH$dm {H«$¶mË‘H$ Ad¶d Ago åhUmV.

AmVm g§íbofH$ ^mJmH$mamMr ‘m§S>Ur H$é. ho Imbrb à‘mUo {XgVo.

^mJmH$mamMo ghJwUH$

‘a’ hm (x - 4) Mm ^mOH$ 1 0 0 3 54

Vwåhr nwÝhm EH$Xm {XK© ^mJmH$ma Am{U g§íbofH$ ^mJmH$mamÀ¶m ‘m§S>UrMr VwbZm H$am.

{XK© ^mJmH$ma g§íbofH$ ^mJmH$ma

4 3 24 0. 0. 3 5x x x x x 1 0 0 3 54

^mÁ¶mVrb nXm§À¶m CVaË¶m H«$‘mVrb KmVm§H$mMo ghJwUH$ ho g§íbofH$

^mJmH$mamÀ¶m n{hë¶m n§º$sV {bhrbobo AmhoV ¶mMo {Z[ajU

H$am.

x – 4 ¶m ^mOH$mVrb a Mr qH$‘V 4 hr Mm¡H$Q>rÀ¶m ~mhoa

{bhrbobr Amho.

nwÝhm g§íbofH$ ^mJmH$mamMr ‘m§S>Ur H$éZ Amåhr nÜXV Mmby R>ody.

nm¶ar 2 : n{hbm ghJwUH$ Imbr AmUm.* g§íbofH$ ^mJmH$mamMr ‘m§S>Ur H$aÊ¶mgmR>r AmUIr XmoZ n§º$s {Z‘m©U H$am.

* n{hë¶m n§º$sVrb n{hbo nX ho {Vgè¶m n§º$sVrb n{hbo nX g§~moYyZ Imbr AmUm.nm¶ar 3 : {Vgè¶m n§º$sVrb qH$‘VrMo JwUH$mer JwUmH$ma H$am.

* {Vgè¶m n§º$sV {bhrboë¶m qH$‘VrÀ¶m n{hë¶m nXmbm 4 ZoJwUm (JwUH$mZo)

* ^mÁ¶mVrb Xþgè¶m ghJwUH$mÀ¶m Imbr JwUmH$ma {bhm.ho Xþgè¶m n§º$rVrb n{hbo nX {bhm.nm¶ar 4 : Xþgam ghJwUH$ Am{U JwUmH$ma {‘idm.

* n{hë¶m n§º$sVrb Xwgao nX Am{U Xþgè¶m n§º$sVrb n{hbo nX {‘idm.åhUOo., 0 + 4 = 4

4 hr ~oarO {Vgè¶m n§º$sV {bhm.

4 1 0 0 3 5

14

n{hbr n§º$sXþgar n§º$s{Vgar n§º$s

1 row 4 1 0 0 3 52 row3 row 1


1 row 4 1 0 0 3 52 row3 row


st

nd

rd

1 row 4 1 0 0 3 52 row 4 16

1 43 row


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Polynomials 185

nm¶ar 5: ZdrZ ~oaOoMm JwUH$mer JwUmH$ma H$am.* {Vgè¶m n§º$sVrb Xþgè¶m nXmMm JwUH$mer JwUmH$ma H$am.

åhUOo, 4 × 4 = 16

* n{hë¶m n§º$sÀ¶m {Vgè¶m nXmImbr JwUmH$ma {bhm.nm¶ar 6 : ZdrZ JwUmH$ma Am{U {Vgè¶m ghJwUH$ {‘idm.

* n{hë¶m n§º$sMo {Vgao nX Am{U Xþgè¶m n§º$sMo Xþgao nX {‘idm.* hr ~oarO {Vgè¶m n§º$sMo {Vgao nX åhUyZ {bhm.

nm¶ar 7:* nm¶ar 3 Am{U 4 à‘mUoM nyU© ^mJmH$ma hmoB©n¶ªV nÜXV nwÝhm H$am.

nm¶ar 8: CÎma {bhm.eodQ>À¶m n§º$sVrb g§»¶m ({Vgar n§º$s) hr ^mJmH$ma VgoM ~mH$sMo ghJwUH$ ~ZVo.~hþnXr p(x) Mm KmVm§H$ 4 Am{U ^mOH$mMm KmVm§H$ EH$ AgyZ Vr aofr¶ ~hþnXr Amho. 1.

^mJmH$mamMm KmVm§H$ hm ^mÁ¶mÀ¶m KmVm§H$mnojm EH$ (1) Zo H$‘r AgVmo. ¶oWo ^mJmH$ma hr 3 KmVm§H$ AgUmar~hþnXr KoVbr Va.n{hbr g§»¶m (1) hr x3 Mm ghJwUH$ Amho.Xþgar g§»¶m (4) hr x2 Mm ghJwUH$ Amho.{Vgar g§»¶m (16) hr x Mm ghJwUH$ Amho.Mm¡Wr g§»¶m (61) hr pñWa Amho.nmMdr g§»¶m qH$dm eodQ>Mr g§»¶m (249) hr ~mH$s Amho.^mJmH$ma x3+4x2+16x +61 Am{U ~mH$s 249 Amho.AmVm AmnU {XK© ^mJmH$ma nÜXV Am{U g§íbofH$ ^mJmH$ma nÜXVrMr VwbZm H$ê$. XmoÝhr nÜXVrVrb g‘mZVH$S>o

1 0 0 3 544 16 64 244

1 4 16 61 249

––

3

3

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2

2

2

2

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5

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3x64x

4 16 61x xx x x x

xx

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. .

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1 row 4 1 0 0 3 52 row 4 16

1 4 163 row


nmhm.

st

nd

rd

1 row 4 1 0 0 3 52 row 4 16 64 244

1 4 16 61 2493 row


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Polynomials 186

nmhm H$s1.g§íbofH$ ^mJmH$mamVrb n{hë¶m AmoirVrb A§H$ ho ^mÁ¶mMo ghJwUH$ AmhoV.2.{XK© ^mJmH$mamVrb Mm¡H$Q>r‘Yrb ghJwUH$ ho g§ûbofU ^mJmH$mamMr Xþgar Amoi Amho.3.{XK© ^mJmH$mamVrb dVw©i Ho$bobo A§H$ ho g§ûbofU ^mJmH$mamMr {Vgar Amoi Amho.4.^mOH$mMr {H$§‘V ‹(4) hr JwUmH$ma H$ê$Z g§ûbofU ^mJmH$mamÀ¶m Mm¡H$Q>r ~mhoa {b{hbobr Amho.5.^mJmH$mamMo ghJwUH$ ho {Vgè¶m AmoirVrb eodQ>Mr g§»¶m gmoSy>Z ¶oVmV.6.S>mdr H$Sy>Z {Vgè¶m AmoirVrb eodQ>Mr g§»¶m hr ~mH$s Amho.g§íbofH$ ^mJmH$mamdarb AmUIr H$m§hr CXmhaUo Kody.

gmoS>{dbobr CXmhaUo

CXmhaU 1: 3x3 + 11x2 + 34x + 106 bm x – 3 Zo ^mJm.

CH$b : g§íbofH$ ^mJmH$mamMr ‘m§S>Ur H$am Am{U nÕV dmnam.

3 3 11 34 1069 60 282

3 20 94 388^mJmH$mamMoghJwUH$ ~mH$s$

^mJmH$ma 3x2 + 20x + 94 Am{U ~mH$s 388 Amho.

({Q>n : ^mJmH$ma Am{U ~mH$s {bhrVmZm COì¶m ~mOyH$Sy>Z {Vgè¶m n§º$sVrb g§»¶mMm gwÕm Amåhr {dMma H$aVmo.)

n{hbr g§»¶m (388) hr ~mH$s Amho.

Xþgar g§»¶m (94) hr ^mJmH$mamMr pñWa g§»¶m Amho.

x Mm ghJwUH$ (20) hr {Vgar g§»¶m Amho.

Mm¡Wr g§»¶m (3) hr x2 Mm ghJwUH$ Amho.

^mJmH$ma 3x2 + 20x + 94 Am{U ~mH$s 388 Amho.

CXmhaU 2: x6 – 2x5 – x + 2 bm x – 2 Zo ^mJm.

CH$b : ^mÁ¶ ho nwÝhm Ago {bhrVmV.

x6 – 2x5 + 0 . x4 + 0.x3 + 0.x2 – x + 2

AmVm g§íbofH$ ^mJmH$mamZo1 2 0 0 0 1 2 2 0 0 0 0 2

1 0 0 0 0 1 0

– –

––

2

^mJmH$ma x5 – 1 Am{U ~mH$s 0 Amho.

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Polynomials 187

ñdmÜ¶m¶ 8.5

1. g§íbofH$ ^mJmH$mamMm Cn¶moJ H$ê$Z ^mJmH$ma Am{U ~mH$s H$mT>m.

(i) (x3 + x2 – 3x + 5) (x – 1)

(ii) (3x3 – 2x2 + 7x – 5) (x + 3)

(iii) (4x3 – 16x2 – 9x – 36) (x + 2)

(iv) (6x4 – 29x3 + 40x2 – 12) (x – 3)

(v) (8x4 – 27x2 + 6x + 9) (x + 1)

(vi) (3x3 – 4x2 – 10x + 6) (3x – 2)

(vii) (8x4 – 2x2 + 6x – 5) (4x + 1)

(viii) (2x4 – 7x3 – 13x2 + 63x – 48) (2x – 1)

2. (x4 + 10x3 + 35x2 + 50x + 29) bm (x + 4) Zo mJbo AgVm (x3 - ax2 + bx + 6), ^mJmH$ma {‘iVmo Vaa,b Am{U ~mH$s gwÕm H$mT>m.

3. (8x4 - 2x2 + 6x - 7) bm (2x + 1) Z o ^mJbo AgVm(4x3 + px2 - qx + 3) ^mJmH$ma {‘iVmo Va p, q Am{U ~mH$s gwÕm H$mT>m.

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Polynomials 188

{XK©

^mJ

mH$ma

nÕ

Ve

of {g

Õm§V

g§íb

ofH$ ^

mJmH

$ma

g(x)

=x-

ag(

x)=

(2x-

1)

=

x-

= 0

x = p

(x) =

2x5 -

3x4 +

8x3 -

2x2 +

x+2

p=

2-3

+8-2

+2

=2

+8-2

-+2

=-

-1-

-+2

= = ==

~mH$

s~m

H$s

~hþnX

tMm ^m

JmH$ma

VrZ

doJdoJ

ù¶m

[aV

rZo H

$mT>Ê¶

mMr nÕ

V A

måhr

{eH$bmo A

mho.

Imb

r {Xboë¶

m gm

aUr‘

Ü¶o V

oM C

Xmha

U V

rZ d

oJdoJù

¶m n

ÕVr

Zogm

oS>{db

obo A

mho.

Ë¶mM

r Vwb

Zm H

$am.

~mH$

s

19

24

107

22

7 29

12

54

[[

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Polynomials 189

~hþnXr

~hþnXrMrH$moQ>r

~hþnXrMrqH$‘V

~hþnXrMoeyÝ¶

~hþnXrÀ¶m ^mJmH$mamMrH$m¶m©dbrp(x) = q(x) x g(x )+r(x)

OoWo r(x)=0 qH$dmr(x) Mr H$moQ>r g(x) Mr H$moQ>r

eof{gÕm§Vp(x)(x-a)

Va r(x)=p(a)

^mÁ¶ Am{U ^mOH$ H$mT>Uo

àË¶j ^mJmH$ma Z H$aVm^mJmH$ma Am{U ~mH$sH$mT>Uo.

g§íbofH$^mJmH$ma

Oa p(a)=0Va (x-a) hmp(x) MmAd¶d Amhoo.

Oa (x-a) hmp(x) Mm Ad¶dAmho Va p(a)=0

Ad¶d à‘o¶

CÎma o

ñdmÜ¶m¶ : 8.1

1) (i) 2 (ii) 2 (iii) 3 (iv) 3 (v) 3 2) (i) 6 (ii) 0 (iii)18 (iv)12 (v)12

3) 23 (ii)-11 (iii) 594 (iv) 30 4) (i) 1 0

3=f (- ) hmo¶ (ii)p 2 =0- ,hmo¶

(iii) g -4 45 = , Zm§hr (iv) p (3)=0, hmo¶ ; p (-1)=0, hmo¶; 1 0

3=-p , hmo¶

5) (i) x=-2 (ii) x=-2 qH$dm x=5 (iii) -72=a qH$dm 7

2=a (iv) 21a =

6) k = -3 7) k = 9

ñdmÜ¶m¶ : 8.2

1) (i) q (x) = x + 2, r (x) = 0 (ii) q (x) = x + 3, r (x) = 0

(iii) q (x) = x2 + 3 x -8, r (x) = 14 (iv) q (x) = x3+ 3 x2 + x-2, r (x) = 0

(v) q (x) = x2 + x + 1, r (x) = 0

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Polynomials 190

2) g (x) 2 x - 3 3) g (x) = x2 - x + 1 4) (i) p(x) = x3- 3x2 + 3x +2

(ii) p(x) = x3+ 7x2 + 11x +16 (iii) p(x) = x4+ 7x3 + x2 + x +1

(iv) p(x) = x4+ 2x3 + 2x -3 (v) p(x) = x6+ 2x5 - x4 + x3 + x2 -5x+5

5) (i) q (x) = x + 9, r (x) = 28 (ii) q (x) = x2 + 7x + 15, r (x) = 47

6) (x + 12 ) 7.) 4 x + 4

ñdmÜ¶m¶ : 8.3

1) (i) r(x ) =47 (ii) r(x ) = - 29 (iii) r(x ) =36 (iv) 32r (x)= (v) 5

8r (x)=

2) (i) a=-1 3)(i) a=-1 (ii) -1 10a= (iii) -4

5a=

ñdmÜ¶m¶ : 8.4

2) a = -8 3) a=2, b = 13

ñdmÜ¶m¶ : 8.5

1) (i) q (x) = x2 + 2 x -1, r (x) = 4 (ii) q (x) = 3 x2 -11x + 40, r (x) = -125

(iii) q (x) = 4x2 -24x + 39, r (x) = -114 (iv) q (x) = 6x3 -11x2 + 7x +21, r (x) = 51

(v) q (x) = 8x3 -8x2 -19x +25, r (x) = -16 (vi) 34 149 9

23

2q (x) =x - x - , r (x)= -

(vii) r (x)x + 3 51 -2118 32 32

12

3 2q (x) = x -2 x - , =

(viii) 3 r(x)x + 55 -418 2 23 2q (x) = x - x - , =

2) a=- 6, b=11, r(x) = 5 3) p==2, q=0, r (x) = -10

*********

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¶m KQ>H$m‘wio Vwåhmbm ¶mMo gwb^rH$aU hmoVo.

* dJ© g‘rH$aU, dJ© Ed dJ© Am{U AdJ© Ed dJ©

g‘rH$aUm§Mr ì¶m»¶m H$aUo.

* dJ© Ed dJ© g‘rH$aUo gmoS>{dUo.

AdJ© Ed dJ© g‘rH$aUo Ad¶dr H$aUmZo gmoS>{dU.

* nyU© dJ© nÕVrZo dJ© g‘rH$aUo gmoS>{dU.

* dJ© g‘rH$aUm§Mr ~rOo H$mT>Ê¶mMo gyÌ aMUo.

* gyÌmMm Cn¶moJ H$ê$Z dJ© g‘rH$aUo gmoS>{dUo.

* dJ© g‘rH$aUm§Mo AmboI H$mT>mUo.

* AmboImZo dJ© g‘rH$aUo gmoS>{dUo.

* dJ© g‘rH$aUm§Mr ~rOo Am{U ghJwUH$ ¶m‘Yrb

g§~§Y àñWm{nV H$aUo.

* dJ© g‘rH$aUo aMUo.

* dJ© g‘rH$aUmMo {ddoMH$ H$mT>mUo Am{U ~rOm§Mo

ñdê$n ñnï> H$aUo.

dJ© g‘rH$aUo

* dJ© amer Am{U dJ© g‘rH$aU* dJ© Ed dJ© Am{U AdJ© Ed dJ©

g‘rH$aUo* dJ© g‘rH$aUmMr CH$b

(dJ© g‘rH$aUo gmoS>{dUo)* Ad¶drH$aU nÕV* nyU© dJ© nÕV* gyÌ nÕV* AmboI nÕV* ~rOo Am{U ghJwUH$ ¶m‘Yrb g§~§Y* dJ© g‘rH$aUo aMUo (V¶ma H$aUo)

9

Brahmagupta(A.D 598-665, India)

Solving of quadratic equationsin general form is often creditedto ancient Indianmathematicians. Brahmaguptagave an explicit formula tosolve a quadratic equationof the form ax2 + bx = c. LaterSridharacharya (A.D. 1025)derived a formula, now knownas the quadratic formula, forsolving a quadratic equation bythe method of completing thesquare.

An equation means nothing tome, unless it expresses a

thought of God.- Srinivas Ramanujan

(Quadratic Equations)

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192 UNIT-9

g§»¶mÀ¶m Ioim~Ôb Amåhr n[a{MV AmhmoV. AmVm AmnU Aer XmoZ CXmhaUo {dMmamV KoD$.

CXmhaU 1 CXmhaU 2

* eyÝ¶ ahrV nyU© g§»¶m ¿¶m. * eyÝ¶ ahrV nyU© g§»¶m ¿¶m.* ¶m‘Ü¶o 7 {‘idm. * Ë¶m‘Ü¶o 7 {‘idm.* ho 12 er g‘mZ {bhm. * ~oaOobm Ë¶mM nyU© g§»¶oZo JwUm.* Vr g§»¶m H$moUVr Amho? * ho 12 er g‘mZ {bhm.

* Vr g§»¶m H$moUVr Amho?

g§»¶m H$em H$mT>mb? Vr eyÝ¶ ahrV nyU© g§»¶m 'x' ‘mZy.Oa Amåhr nm¶è¶m§Mo AZwH$aU Ho$ë¶mg Amåhmbm {‘iVo.CXmhaU 1 ‘Ü¶o, x, x + 7 Am{U x + 7 = 12........(i)

CXmhaU 2 ‘Ü¶o, x x + 7 Am{U x(x + 7) = 12........(ii)

g‘rH$aU (i), x + 7 = 12 ¿¶m.¶oWo x ho MbnX Amho Á¶mMm KmVm§H$ 1 Amho.Vo aofr¶ g‘rH$aU Amho. Ë¶mbm ’$º$ EH$ ~rO AgVo.åhUOoM x + 7 = 12 x = 12 – 7 x = 5

g‘rH$aU (ii) ¿¶m, x (x + 7) = 12 x2 + 7x = 12¶oWo 'x' ho MbnX Amho Á¶mMm KmVm§H$ 2 Amho. ho dJ© g‘rH$aU Amho.Vo H$go gmoS>dmb? dJ© g‘rH$aUmbm {H$Vr ~rOo AmhoV?ho {eH$Uo AË¶§V OéarMo Amho H$maU J{UVmÀ¶m BVa emIo‘Ü¶o BVa {df¶mV Am{U {ZË¶ OrdZmVgwÜX

dJ© g‘rH$aUmMo {dñV¥V Ago Cn¶moJ AmhoV.CXmhaUmW© g‘Om EH$m KamVrb d¥Õ {dídñVmZo 300 Mm¡.‘r. joÌ’$imÀ¶m O{‘Zrda EH$ àmW©Zm g^mJ¥h

~m§YÊ¶mMm {ZYm©a Ho$bobm Amho. Á¶mMr bm§~r Ë¶mÀ¶m é§XrÀ¶m XþßnQ>rnojm 1 ‘rQ>aZo A{YH$ Amho Va Ë¶mg^mJ¥hmMr bm§~r Am{U é§Xr {H$Vr Agob?

é§Xr 'x' ‘rQ>a ‘mZy. Va Ë¶mMr bm§~r (2x + 1) ‘rQ>a hmoB©b.Ë¶mMo joÌ’$i = x(2x + 1) Mm¡.‘r. x(2x + 1) = 300 ( {Xbobo)

hr ‘m{hVr AmH¥$VrZo Imbrbà‘mUo Xe©{dbr OmVo.

Amåhr joÌ’$i = x(2x + 1) = (2x2 + x) Mm¡.‘r. 2 1 300x x Mm¡. ‘r.

åhUyZ 2x2 + x = 300. ho dJ© g‘rH$aU Amho. (2x + 1) ‘rQ>a

Imbr H$m§hr A{YH$ boIr {dYmZmÀ¶m ñdê$nmVrb CXmhaUo {Xbobr AmhoV. Ë¶mMo ê$nm§Va g‘rH$aUmÀ¶ménmV Ho$bo AgVm dJ© g‘rH$aU ñdê$n {‘iVo.

{dYmZm§Mm Aä¶mg H$am. àË¶oH$ {dYmZ g‘rH$aUmVrb ñdê$nmV Xe©{dÊ¶mMm à¶ËZ H$am.1. ~|Jiya Am{U åh¡gya ‘Yrb 132 {H$.‘r. A§Va OmÊ¶mgmR>r qH$dm H$mnÊ¶mgmR>r ObX aoëdo hr àdmgr aoëdonojm

1 Vmg H$‘r KoVo. Va ObX aoëdoMm gamgar doJ àdmgr aoëdo nojm 11 {H$.‘r.Xa VmgmZo A{YH$ Agob Va

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Quadratic Equations 193

XmoÝhr aoëd|Mm gamgar doJ {H$Vr?

2. EH$ bmH$S>r H$maImZm EH$m {Xdgmbm H$m§hr bmH$S>r IoiÊ¶mM§ CËnmXZ H$aVmo. àË¶oH$ IoiÊ¶mÀ¶m CËnmXZmMrqH$‘V (én¶m‘Ü¶o) hr V¶ma Ho$boë¶m IoiÊ¶mÀ¶m g§»¶oÀ¶m 55 Zo H$‘r Agë¶mMo AmT>iVo. EH$m {d{eï>{XdgmÀ¶m CËnmXZmMr EHy$U qH$‘V 750 é. hmoVr. Va Ë¶m {Xder V¶ma Ho$boë¶m IoiÊ¶m§Mr g§»¶m {H$Vr?

3. g§W nmÊ¶mVrb EH$m ‘moQ>ma ~moQ>rMm doJ 18 {H$.‘r./Vmg AgyZ àdmhmÀ¶m {Xeonojm àdmhmÀ¶m {déÕ {XeoZo

24 {H$.‘r. OmD$Z Ë¶mM {R>H$mUr naV ¶oÊ¶mgmR>r 1 Vmg OmñV doi bmJVmo. Va àdmhmMm doJ H$mT>m.

4. XmoZ ZimZr EH${ÌVnUo EH$ hm¡X 938

VmgmV âVmo. bhmZ ì¶mgmÀ¶m Zimnojm ‘moR>çm ì¶mgmÀ¶m Zimbm

ñdV§ÌnUo Q>mH$s âÊ¶mg 10 Vmg H$‘r bmJVmV Va àË¶oH$ ZimZo ñdV§ÌnUo Vmo hm¡X {H$Vr doioV âob Vo

H$mT>m.

hr CXmhaUo H$er gmoS>dmb?

OmUyZ ¿¶m … ~o~rbmoZrAÝg ¶m§Zr gd© àW‘ dJ© g‘rH$aUo gmoS>{dbr Ago ‘mZbo OmVo. J«rH$ J{UV Vk ¶w³brS>¶mZo ^m¡{‘ÎmrH$ Ñï>rH$moZmMm {dH$mg H$ê$Z bm§~r emoYyZ H$mT>br. ho Xþgao H$m§hr ZgyZ dJ© g‘rH$aUo gmoS>{dÊ¶mMrCH$b Amho.

gm‘mÝ¶ ñdê$nmVrb dJ© g‘rH$aUo gmoS>{dÊ¶mMo lo¶ Zoh‘r àmMrZ J{UV Vk ~«÷Jwá (B.g. 598-665)Am{U lrYamMm¶© (B.g. 1025) ¶m§Zm OmVo. Aa~ J{UV Vk Ab-»dm[aPZr (gw‘mao B.g. 800) ¶mZr gwÕmdJ© g‘rH$aUm§Mm doJdoJù¶m àH$mamZo Aä¶mg Ho$bm Amho.

Aa~ J{UV Vk Ab »dm[aPZr (Al-Khwarizni) (gw‘mao B.g.800) doJdoJù¶m àH$maMr dJ© g‘rH$aUoAä¶mgbr.

"A~«mh‘ ~ma {hæ¶m' hm-Zmgr' ¶mZr B.g. 1145 ‘Ü¶o ¶wamonmV àH$m{eV Ho$boë¶m Ë¶m§À¶m "{b~aEå~mS>m}a‘' "Liber embardorum" ¶m nwñVH$mV doJdoJù¶m dJ© g‘rH$aUm§Mr g§nyU© CH$br {Xboë¶m AmhoV.

¶m KQ>H$mV AmnU dJ© g‘rH$aUo, Ë¶m§Mr ~rOo H$mT>Ê¶mÀ¶m {d{dY nÕVr Am{U dJ© g‘rH$aUm§Mo Cn¶moJ¶m§Mm gwÕm Aä¶mg H$ê$.

dJ© g‘rH$aUo (Quadratic equation)

KQ>H$ 8 ‘Ü¶o Amåhr dJ© ~hþnXr~ÕbMm Aä¶mg Ho$bobm Amho Vo AmR>dm.

ax2 + bx + c, ¶m ñdê$nmVrb ~hþnXr OoWo a 0 hr dJ© ~hþnXr Amho qH$dm 2 KmVm§H$mMr MbnX 'x'

‘Yrb amer Amho.

Oa dJ© amer ax2 + bx + c hr eyÝ¶mÀ¶m g‘mZ Ho$br Va Vo dJ© g‘rH$aU ~ZVo.

Imbr H$m§hr boIr {dYmZo {Xbobr AmhoV. Ooìhm Vr dJ© amerV ê$nm§VarV Ho$br Am{U nwT>o ewÝ¶mer g‘mZHo$ë¶mg dJ© g‘rH$aU ~ZVo. CXmhaUm§Mm Aä¶mg H$am.

åhUyZ Vo Ago gwM{dVo H$s Oa P(x) hr dJ© ~hþnXr Amho Va P(x) = 0 ho dJ© g‘rH$aU Amho.

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194 UNIT-9

ho OmUyZ ¿¶m.b°Q>rZ eãX "³dmS´>°Q>‘' ¶m eãXmnmgyZ dJ©hm eãX KoVbobm Amho Á¶mMm AW©Mm¡agmH¥$Vr Agm hmoVmo.

AWm©V P(x) = 0 ¶m ñdê$nmVrb H$moUVohr g‘rH$aU OoWo P(x) hr KmVm§H$ 2 ‘Yrb ~hþnXr AgyZ Vo dJ©g‘rH$aU Amho {OMm à‘m{UV Z‘wZm

ax2 + bx + c = 0, a 0.

dJ © g‘rH$ aU h o 'x' MbnXmM o ax2 + bx + c = 0 ¶m ñdê$nmV rb g‘rH $aU Amh o.O oWo a, b, c, ¶m dmñVd g§»¶m Amh oV Am{U a 0.

dJ© g‘rH$aUm§Mo JwUY‘© AmhoV.

• ho EH$ MbnX AgUmao g‘rH$aU Amho.

• ho EH$ Ago g‘rH$aU Amho H$s Á¶mÀ¶m EH$mM MbnXmMm KmVm§H$ 2 AgVmo.

• dJ© g‘rH$aUmMm à‘m{UV Z‘wZm ax2 + bx + c = 0 hm Amho.

¶oWo x2 Mm ghJwUH$ a Amho.

x Mm ghJwUH$ b Amho.

c ho pñWa nX Amho.

a, b, c ¶m dmñVd g§»¶m AmhoV Am{U a 0.

• MbnXo hr KmVm§H$mÀ¶m CVaË¶m H«$‘mV {bhrbobr AmhoV.

dJ©©g‘rH$aUm‘Ü¶o a 0 ho H$m Amho?

Oa a = 0 AgVmZm dJ© g‘rH$aUmbm H$m¶ hmoB ©b?

dJm©V MMm© H$am.

dJ© g‘rH$aUm‘Ü¶o a 0 ¶mMr Amåhr MMm© Ho$bobr Amho.

Ooìhm b qH$dm c qH$dm XmoÝhr b Am{U c ho eyÝ¶merg‘mZ AgVrb Va dJ© g‘rH$aUmÀ¶m à‘m{UV Z‘wÝ¶mMo

A.Z§. boIr {dYmZo dJ© amer dJ© g‘rH$aU

1. EH$ g§»¶m Am{U {VÀ¶m dJm©Mr 5x2 + x 5x2 + x =0nmM nQ> ¶m§Mr ~oarO

2. 12 g|.‘r. bm§~rMr EH$ Vma dmH$dyZH$mQ>H$moZ {ÌH$moUmÀ¶m H$mQ>H$moZ H$aUmè¶m~mOy ~Z{dë¶m AmhoV. Ë¶mn¡H$s EH$ ~mOyXþgè¶m ~mOynojm 2 g|.‘r. Zo A{YH$ AmhoVa Ë¶mMo joÌ’$i {H$Vr Agob?

3. {H$H«o$Q> g§KmZo nQ>H$m{dboë¶m Ymdm ¶mn{hë¶m ’$b§XmOmZo nQ>H$m{dboë¶m Ymdm x2 – x – 6 x2 – x – 6 =0Am{U Ë¶mZo nQ>H$m{dboë¶m Ymdm§À¶m dJm©À¶m’$aH$mnojm 6 Zo H$‘r AmhoV.

1 ( 2)2

x x 21 ( 2 ) 02

x x

21 ( 2 )2

x x 21 ( 2 ) 02

x x

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H$m¶ hmoB©b?

Imbr {Xboë¶m gmaUrMo {ZarjU H$am.

A.Z§. b Mr qH$‘V c Mr qH$‘V n[aUm‘ ({ZH$mb)

1. b = 0 c 0 ax2 + c = 0

2. b 0 c = 0 ax2 + bx = 0

3 b = 0 c = 0 ax2 = 0

4. b 0 c 0 ax2 + bx + c = 0

darb gd© KQ>Zo‘Ü¶o g‘rH$aU ho dJ© g‘rH$aUM amhVo ¶mMo {ZarjU H$am.

gmoS>{dbobr CXmhaUo

CXmhaU 1 … Imbrb dJ© g‘rH$aUo Agë¶mMo Vnmgm.

(i) 2x + x2 + 1 = 0 (ii) 6x3 + x2 = x

(iii)3

( 8) 10 04

x x x (iv) x(x + 1) + 8 = (x + 2) (x – 2)

CH$b(i) 2x + x2 + 1 = 0nXo KmVm§H$mÀ¶m CVaË¶m H«$‘mV ‘m§S>m

x2 + 2x + 1 = 0

ho ax2 + bx +c = 0 ¶m à‘m{UV Z‘wÝ¶mV Amho.

{Xbobo g‘rH$aU ho dJ© g‘rH$aU Amho.

(ii) 6x3 + x2 = x

nXm§Mr nwÝhm ‘m§S>Ur Ho$ë¶mg 6x3 + x2 – x = 0 Amåhmbm {‘iVo.

MbnXm§Mm gdm©V ‘moR>m KmVm§H$ 3 Amho.

{Xbobo g‘rH$aU ho dJ© g‘rH$aU Zmhr.

(iii)3 ( 8) 10 04

x x x

g‘rH$aUmbm gaiê$n {Xë¶mZo Amåhmbm {‘iVo.2 23 248 10 0

4 4x x x x

4x2 – 32x + 3x2 – 24x + 40 = 0

x2 – 56x + 40 = 0. ho ax2 + bx + c = 0 ¶m ñdê$nmV Amho.

{Xbobo g‘rH$aU ho dJ© g‘rH$aU Amho.(iv) x (x + 1) + 8 = (x + 2)(x – 2)

g‘rH$aUmbm gaiê$n {Xë¶mZo Amåhmbm {‘iVo.

x2 + x + 8 = x2 – 4 2x 2x + x + 8 + 4 = 0 x + 12= 0

ho ax2 + bx + c = 0 ¶m ñdê$nmV Zmhr.MbnX x ho ’$º$ EH$ KmVm§H$mMo Amho.

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196 UNIT-9

{Xbobo g‘rH$aU ho dJ© g‘rH$aU Zmhr.

ñdmÜ¶m¶ 9.1

1. Imbrb dJ© g‘rH$aUo Agë¶mMo Vnmgm

(i) x2 – x = 0 (ii) x2 = 8 (iii) 2 1 02

x x (iv) 3x – 10 = 0

(v) 2 29 5 04

x x (vi) 225 65

x x (vii) 22 3 0x x (viii)22313

x

(ix) x3 – 10x + 74 = 0 (x) x2 – y 2 = 0

2. Imbrb g‘rH$aUm§Zm gaiê$n Úm Am{U Vr dJ© g‘rH$aUo Agë¶mMo Vnmgm.(i) x(x + 6) = 0 (ii) (x - 4)(2x – 3) = 0

(iii) (x + 2)(x – 7) = 5 (iv) (x + 1)2 = 2(x – 3)(v) (2x – 1)(x – 3) = (x + 5)(x – 1) (vi) (x + 2)3 = 2x(x2 – 1) = (x – 2)3

3. dJ© g‘rH$aUmÀ¶m ñdê$nmV Imbrb Xe©dm.(i) XmoZ H«$‘dma nyUmªH$mMm JwUmH$ma 306 Amho.

(ii) EH$m Am¶VmH$ma CÚmZmMr bm§~r (‘rQ>a‘Ü¶o) hr é§XrÀ¶m XþßnQ>rÀ¶m 1 ‘rQ>a Zo A{YH$ Amho Am{UË¶mMo joÌ’$i 528 Mm¡.‘r. Amho.

(iii) EH$ AmJJmS>r 480 {H$.‘r. A§Va EH$g‘mZ doJmZo OmVo. Oa AmJJmS>rMm doJ 8 {H$.‘r./Vmg Zo H$‘rHo$ë¶mg VoM A§Va OmÊ¶mgmR>r 3 Vmg OmXm doi bmJVmo.

{Xboë¶m gmaUrVrb H$m§hr A{YH$ CXmhaUm§Mo {ZarjU H$am.

A B

x2 = 144 x2 – 7x + 10 = 0

x2 – 81 = 0 17y – y2 + 30 = 0

2y2 = 0 p2 – 20 = – 8p

Amåhmbm Ago {XgyZ ¶oVo H$s• A Am{U B ‘Yrb gd© g‘rH$aUo hr dJ© g‘rH$aUo AmhoV.• A ñV§ mVrb gd© dJ© g‘rH$aUmÀ¶m MbnXmMm KmVm§H$ ’$º$ XmoZ Amho. Ë¶m§Zm dJ© Ed dJ© g‘rH$aUo Ago åhUVmV.• B ñV§ mVrb gd© dJ© g‘rH$aUmÀ¶m MbnXmMm XmoZ Am{U EH$ Agm XmoÝhr KmVm§H$ Amho Ë¶m§Zm AdJ© Ed

dJ© g‘rH$aUo Ago åhUVmV.dJ© g‘rH$aUm§M CH$b (gmoB{dUo)…x2 – 25 = 0 ¶m dJ© Ed dJ© g‘rH$aUmMm {dMma H$am.AmnU 'x' Mr dmñVd qH$‘V KoD$ Am{U g‘rH$aUm‘Ü¶o ‘m§Sy>.

‘mZy x = 1 ‘mZy x = 5S>mdr ~mOy = x2 – 25 = 12 – 25 = – 24 S>mdr ~mOy = x2 – 25 = 52 – 25 = 0

LHS RHS S>mdr ~mOy = COdr ~mOy

‘mZy x = – 2 ‘mZy x = – 5

S>mdr ~mOy = x2 – 25 =(–2)2 – 25 = – 21 S>mdr ~mOy = x2 – 25 = (–5)2 – 25 = 0

S>mdr ~mOy COdr ~mOy S>mdr ~mOy = COdr ~mOy

ñV§ A Am{U B ‘Yrbg‘rH$aUm§Mr VwbZm H$am.

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ho 'x' À¶m doJdoJù¶m qH$‘VrgmR>r à¶ËZ H$am.

Amåhmbm Ago {XgyZ ¶oVo H$s x = 5 Am{U x = – 5 ¶m 'x' À¶m ’$º$ XmoZ qH$‘VrgmR>r S>mdr ~mOy = COdr ~mOy Amho.

Amåhr Ago gm§Jy H$s +5 Am{U –5 ho x2 – 25 = 0 ¶m g‘rH$aUmMr nyV©Vm H$aVo.

+5 Am{U –5 Zm dJ© g‘rH$aUmMr ~rOo Ago åhUVmV.

+5 Am{U –5 ¶mZm x2 – 25 ¶m ~hþnXtMr eyÝ¶o Ago åhUVmV.

dJ© g‘rH$aUmMr ~rOo g‘rH$aUmMo g‘mYmZ H$aVmV. ¶mMmM AW© S>mdr ~mOy = COdr ~mOy.

darb MMm© hr AdJ© Ed dJ© g‘rH$aUm§Zm gwÕm bmJy nS>Vo.

x2 – 3x – 10 = 0 ¶m AdJ© Ed dJ© g‘rH$aUmMm {dMma H$am.

‘mZy x = 1 ‘mZy x = –3

S>mdr ~mOy LHS= x2 – 3x – 10 = 12 – 3(1) – 10 S>mdr ~mOy = x2 – 3x – 10 = (–3)2 – 3(–3) –10

= – 12 = 8

S>mdr ~mOy COdr ~mOy S>mdr ~mOy COdr ~mOy

‘mZy x = 5 ‘mZy x = – 2

S>mdr ~mOy = x2 – 3x – 10 = 52 – 3(5) – 10 S>mdr ~mOy = x2 – 3x – 10 = (–2)2 – 3(–2) – 10

= 0 = 0

S>mdr ~mOy = COdr ~mOy S>mdr ~mOy = COdr ~mOy

Amåhmbm Ago {XgyZ ¶oVo H$s x2 – 3x – 10 = 0 ¶m dJ© g‘rH$aUmMr nyV©Vm ’$º$ x = 5 Am{U x = – 2

¶m qH$‘VrgmR>r hmoVo.

5 Am{U –2 hr x2 – 3x – 10 = 0 ¶m dJ© g‘rH$aUmMr ~rOo AmhoV.

5 Am{U –2 hr x2 – 3x – 10 = 0 ¶m dJ© ~hþnXtMr eyÝ¶o AmhoV.

gm‘mÝ¶nUo ax2 + bx + c = 0 ¶m dJ© g‘rH$aUmMr dmñVd g§»¶m 'k' ho ~rO Amho. a 0 Oaak2 + bk + c = 0.

Amåhr Ago gwÕm gm§Jy H$s x = k hm dJ© g‘rH$aUmMo CH$b Amho, qH$dm k dJ© g‘rH$aUmMr nyV©Vm H$aVo.

ax2 + bx + c ¶m dJ© ~hþnXrMr eyÝ¶o Am{U ax2 + bx + c ¶m dJ© g‘rH$aUmMr ~rOo g‘mZ AgVmV hobjmV R>odm.

Amåhmbm ‘m{hV Amho H$s dJ© ~hþnXrbm OmñVrV OmñV XmoZ eyÝ¶o AgVmV.

H$moUË¶mhr dJ© g‘rH$aUmbm OmñVrV OmñV XmoZ ~rOo AgVmV.

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198 UNIT-9

dJ© g‘rH$aU gmoS>{dUo åhUOo dJ© g‘rH$aUmMr ~rOo H$mT>Uo hmo¶. dJ© g‘rH$aUm‘Ü¶o qH$‘Vr ‘m§Sy>Z~rOm§Mm nS>Vmim H$aVm ¶oVmo Am{U g‘rH$aUmMr nyV©Vm hmoVo H$s Zmhr ho VnmgVm ¶oVo. ~rOo hr dJ©g‘rH$aUmMmCH$b g§M ~Z{dVmV.

dJ© Ed dJ© g‘rH$aUmMr CH$b … (Solution of a pure quadratic equation)

1. x2 – 225 = 0 gmoS>dm.

CH$b … x2 – 225 = 0 x2 = 225 x = 225 (XmoÝhr ~mOy§Mo dJ©‘yi H$mT>ë¶mZo)

x = 15 x = +15 qH$dm x = – 15

2. gmoS>dm … 143 = t2 – 1

CH$b … 143 = t2 – 1

g‘rH$aU à‘m{UV Z‘wÝ¶mV nwÝhm {bhm.

t2 = 143 + 1 t2 = 144 t = + 144 t = +12

t = +12 qH$dm t = – 12

3. Oa V = r2h Va 'r' H$Vm© H$am Am{U Ooìhm V = 176 Am{U h = 14 Va 'r' Mr qH$‘V H$mT>m.

CH$b … {Xbobo V = r2h

r2h = V r2 =Vh

r =Vh

Oa V = 176 Am{U h = 14 Amåhmbm {‘iVo

r =176

14(qH$‘Vr ‘m§Sy>Z) r =

176 7 17622 14

8 4

7

22 142 4 = 2

r = +2 qH$dm r = – 2

4. ídoVmZo Mm¡ag AmH$mamMr 625 Mm¡.‘r. O{‘Z {dH$V KoVbr Amho. {Vbm Ë¶m O{‘Zrbm H$mQ>oar VmaoMoHw§$nU Kmbm¶Mo Amho. g^modVmbr 4 ’o$ao KmbÊ¶mgmR>r {H$Vr bm§~rMr Vma bmJob?

CH$b … Mm¡agmMr bm§~r x ‘rQ>a ‘mZy.

Mm¡agmMo joÌ’$i = x2

625 = x2 x2 = 625 x = 625 = 25 x =+25 qH$dm x = – 25

bm§~r F$U AgV Zmhr Amåhr x = +25 KoD$.

Mm¡agmÀ¶m EH$m ~mOyMr bm§~r = +25 ‘rQ>a

Mm¡agmMr n[a{‘Vr = 4x ‘rQ>a

Mma ’o$è¶mgmR>r bmJUmar Vma = 4 × 4x = 16x ‘rQ>a

AmR>dmYZ dmñVd g§»¶mZm XmoZ ~rOo AgVmV.EH$ YZ Am{U Xþgao F$U AgVo.

1 1 & 0 0Am{U1 1 & 0 0

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bmJUmè¶m VmaoMr bm§~r = 16 × 25 = 400 ‘rQ>a

ñdmÜ¶m¶ 9.2

1. dJ© Ed dJ© g‘rH$aU Am{U AdJ© Ed dJ© g‘rH$aUm‘Ü¶o Imbrb g‘rH$aUmMo dJuH$aU H$am.(i) x2 = 100 (ii) x2 + 6 = 6 (iii) p(p – 3) = 1 (iv) x2 + 3 = 2x

(v)(x + 9)(x – 9) (vi) 2x2 = 72 (vii) x2 – x = 0 (viii) 7x =35x

(ix)1 5xx

(x)814xx

(xi) (2x – 5) 2 = 81 (xii)2( 4) 2

18 9x

2. dJ© g‘rH$aUo gmoS>dm.

(i) x2 – 196 = 0 (ii) 5x2 = 625 (iii) x2 – 1 = 101 (iv) 7x =647x

(v) (x + 8)2 – 5 = 31 (vi)2 3 17

2 4 4x

(vii) –4x2 + 324 = 0 (viii) –37.5x2 = –37.5

3. Imbrb àË¶oH$mgmR>r {Xboë¶m 'x' À¶m qH$‘Vr dJ© g‘rH$aUmMr CH$b Amho qH$dm Zmhr Vo R>adm.

(i) x2 + 14x + 13 = 0 ; x = –1, x = – 13 (ii) 7x2 – 12x = 0 ; x =13

(iii) 2m2 – 6m + 3 = 0 ; m=12

(iv) 2 2 4 0y y ; y = 2 2

(v)1

2 2x

x ; x = 2 Am{U x = 1 (vi) 6x2 – x – 2 = 0 ; x =12 Am{U x =

23

4. (i) Oa A = r2 Va 'r' H$Vm© H$am Am{U Oa A = 77 Am{U =227

Va 'r' Mr qH$‘V H$mT>m.

(ii) Oa r2 = l2 + d2 Va 'd' H$Vm© H$am Am{U Oa r = 5 Am{U l = 4 Va 'd' Mr qH$‘V H$mT>m.

(iii) Oa c2 = a2 + b2 Va 'b' H$Vm© H$am Am{U Oa a = 8 Am{U c = 17 Va 'b' Mr qH$‘V H$mT>m.

(iv) Oa A =23

4a Va 'a' H$Vm© H$am Am{U Oa A = 16 3 Va 'a' Mr qH$‘V H$mT>m.

(v) Oa k =12

mv2 Va 'v' H$Vm© H$am Am{U Oa k = 100 Am{U m = 2 Va 'v' Mr qH$‘V H$mT>m.

(vi) Oa v2 = u2 + 2as Va 'v' H$Vm© H$am Am{U Oa u = 0, a = 2, s = 100 Va 'v' Mr qH$‘V H$mT>m.

AdJ© Ed dJ© g‘rH$aUmMr CH$b …

Amåhmbm ‘mhrV Amho H$s AdJ© Ed dJ© g‘rH$aUmMm gm‘mÝ¶ Z‘wZm ax2 + bx + c = 0 Amho. a 0, ho g‘rH$aU

ax2 + b = 0, ax2 + c = 0 Am{U ax2 = 0 Aem doJdoJù¶m ñdê$nmV XoIrb AmT>iVo.

ax2 = 0 Am{U ax2 + c = 0 hr dJ© Ed dJ© g‘rH$aUo AmhoV Am{U ‘mJrb ^mJmV Vr gmoS>{dÊ¶mMr nÕV

Amåhr {eH$bobr (Aä¶mgbobr) Amho.

AdJ© Ed dJ© g‘rH$aU H$go gmoS>dmdo?

ax2 + bx + c = 0 qH$dm ax2 + b = 0 øm Ë¶m§À¶m Z‘wÝ¶mda AmYmarV AdJ© Ed dJ© g‘rH$aUo

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200 UNIT-9

gmoS>{dÊ¶mÀ¶m nwîH$i nÕVr AmhoV.

Ë¶mMm AmnU Aä¶mg H$é¶m.

(A) Ad¶drH$aU nÕVrZo dJ© g‘rH$aUmMr CH$b …dJ© ~hþnXrMo Ad¶d Ë¶m§À¶m ‘Yë¶m nXmMr ’$moS> H$ê$Z H$mT>Uo ho Amåhr {eH$bobo Amho. ¶m kmZmMm

Cn¶moJ H$ê$Z Amåhr dJ© g‘rH$aUmMr ~rOo H$mTy> eH$Vmo.

Ooìhm dJ© g‘rH$aUmMo XmoZ aofr¶ Ad¶d nmS>bo OmVmV Voìhm Ad¶drH$aU nÕV dmnaVmV. Ad¶drH$aUmZ§VaË¶mÀ¶m XmoZ aofr¶ Ad¶dmV dJ© g‘rH$aU ì¶º$ H$aVmV Am{U Vo eyÝ¶mÀ¶m g‘mZ ‘m§S>VmV.

åhUOoM ax2 + bx + c = 0 (x m) (x n) = 0Z§Va Amåhr eyÝ¶ JwUmH$ma {Z¶‘ bmJy H$aVmo Am{U àË¶oH$ Ad¶d eyÝ¶mÀ¶m g‘mZ ‘m§S>Vmo Am{U

Aì¶º$mgmR>r gmoS>{dVmo.x m = 0 x = mx n = 0 x = n

åhUyZ ax2 + bx + c = 0 ¶m dJ© g‘rH$aUmMr ~rOo mAm{U n AmhoV.

x2 + 5x + 6 = 0 ho dJ© g‘rH$aU {dMmamV ¿¶m.‘Ybo nX 5x Am{U Ë¶mMm ghJwUH$ 5 Amho.AmnU 5 Mr ’$moS> Aer H$ê$ H$s m + n = 5 Am{U mn = 6 hmoB©b.x2 + 3x + 2x + 6 = 0 x(x + 3) + 2(x + 3) = 0 (x + 3)(x + 2) = 0

AmVm x2 + 5x + 6 = 0 Mo XmoZ aofr¶ Ad¶d (x + 3) Am{U (x + 2) ho nmS>m.eyÝ¶ JwUmH$ma {Z¶‘mMm Cn¶moJ Ho$ë¶mZo Amåhmbm (x + 3)(x + 2) = 0 {‘iVo.x + 3= 0 qH$dm x + 2 = 0 x + 3= 0 ho XoVo x = – 3 x + 2 = 0 ho XoVo x = – 2

CH$b g§M {–3, –2} Amho.Aem[aVrZo -3 Am{U -2 hr XmoZ ~rOo x2 + 5x + 6 = 0 ¶m dJ© g‘rH$aUmMr AmhoV.~¡{OH$ ’$aí¶m§Mm Cn¶moJ H$ê$Z ho AmH¥$VrÛmao gwÕm Imbrbà‘mUo Xe©{dVm ¶oVo.

x

x

x2 + x x x x x x x x x x +

1 1 1 1 1

+ 5 + 6x x2

11 11

11 11

11 11

11 11

11 11

11 11

gd© ’$aí¶m§Mr ‘m§S>Ur Ho$br AgVm Amåhmbm AmH¥$Vr {‘iVo.

{ZarjU H$am H$s bm§~r = x + 3 Am{U é§Xr = x + 2 Am{U Ë¶mMojoÌ’$i x2 + 5x + 6 Amho.

Am¶VmMo joÌ’$i A = l × b

x2 + 5x + 6 = (x + 3)(x + 2)

¶mdê$Z Amåhr Agm {ZîH$f© H$mT>Vmo H$s

eyÝ¶ JwUmH$ma {Z¶‘a Am{U b øm H$moUË¶mhr XmoZ dmñVd g§»¶mqH$dm Ad¶d AmhoV. Oa a × b = 0 Va a= 0 qH$dm b = 0 XmoÝhr a = 0 Am{U b = 0

x x x x x

x

x2

x x 1 1 1

1 1 11 1 1

11

( + 2)x

( + 3)x

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Oa dJ© ~hþnXr ax2 + bx + c hr Mm¡ag qH$dm Am¶VmMo joÌ’$i Xe©{dV Agob Va Ë¶mMo XmoZ Ad¶d hobm§~r Am{U é§Xr Xe©{dVmV.

Ad¶drH$aU nÕVrZo dJ© g‘rH$aUmdarb gmoS>{dboë¶m CXmhaUm§Mm Aä¶mg H$am.

gmoS>{dbobr CXmhaUo

CXmhaU 1) gmoS>dm x2 – 3x + 2 = 0CH$b … {Xbobo x2 – 3x + 2 = 0

x2 – 2x – 1x + 2 = 0 (amerMo n¥W…H$aU)x(x – 2) –1(x – 2) = 0 (g‘mB©H$ Ad¶d KoD$Z) (x – 2)(x – 1) = 0 (g‘mB©H$ Ad¶d KoD$Z)

x – 2 = 0 or x – 1= 0 (àË¶oH$ Ad¶d eyÝ¶mÀ¶m g‘mZ ‘m§S>m)x = + 2 qH$dm x = 1

–2 Am{U + 1 hr (x2 – 3x + 2) Mr ~rOo AmhoV.CXmhaU 2) gmoS>dm … 6x2 – x – 2 = 0

CH$b … {Xbobo … 6x2 – x – 2 = 0

6x2 + 3x – 4x – 2 = 0 3x(2x + 1) – 2(2x + 1) = 0 (2x + 1)(3x – 2) = 0

2x + 1 = 0 qH$dm 3x – 2 = 0 2x = – 1 qH$dm 3x = 2 x =12 qH$dm x =

231

2Am{U

23

hr 6x2 – x – 2 = 0 Mr ~rOo AmhoV.

CXmhaU 3) 3x2 – 2 6x + 2 = 0 ¶m dJ© g‘rH$aUmMr ~rOo H$mT>m.

CH$b … {Xbobo … 23 2 6 2x x = 0

23 6 6 2x x x = 0 3 3 2 2 3 2x x x = 0

3 2 3 2x x = 0 3 2 0x 3 2x = 0

3x = 2 qH$dm 3x = 2 , x =23

, x =23

åhUyZ àË¶oH$ nwZamd¥ÎmrV Ad¶dmgmR>r 3 2x EH$ ¶mà‘mUo ~rOm§Mr XmoZXm nwZamd¥Îmr hmoVo.23 2 6 2 0x x Mr XmoZ g‘mZ ~rOo 2 2,

3 3hr AmhoV.

CXmhaU 4) 24 10x = 3 – 4x

CH$b … {Xbobo … 24 10x = 3 – 4 x

XmoÝhr ~mOy§Mm dJ© Ho$ë¶mg Amåhmbm {‘iVo.2

24 10x = (3 – 4x)2 24 – 10x = 9 – 24x + 16x2 16x2 – 24x + 10x + 9 – 24 = 0

16x2 – 24x + 10x – 15 = 0 8x(2x – 3) + 5(2x – 3) = 0

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202 UNIT-9

(2x – 3)(8x + 5) = 0 2x – 3 = 0 qH$dm 8x + 5 = 0

2x = 3 qH$dm 8x = – 5, x =32

x =58

24 10x = 3 – 4x Mr ~rOo32

Am{U58

hr AmhoV.

Ad¶drH$aU nÕVrZo dJ© g‘rH$aUmMr ~rOo H$mT>Ê¶mÀ¶m nm¶è¶m.1. {Xbobo g‘rH$aU dJ© g‘rH$aUmÀ¶m à‘m{UV Z‘wÝ¶mV ax2 + bx + c = 0 ‘Ü¶o {bhm.2. dJ© amer Ë¶mÀ¶m ‘Yë¶m nXmMr ’$moS> nmSy>Z (S>mdr ~mOy) ‘m§S>m.3. g‘mB©H$ Ad¶d ¿¶m Am{U XmoZ aofr¶ Ad¶d {‘idm.4. àË¶oH$ Ad¶d eyÝ¶mMr g‘mZ ‘m§S>m.

5. àË¶oH$ aofr¶ g‘rH$aUmbm gaiê$n Úm Am{U Aì¶º$mMr qH$‘V H$mT>m.

ñdmÜ¶m¶ 9.3

Ad¶drH$aU nÕVrZo dJ© g‘rH$aU gmoS>dm.

1. x2 + 15x + 50 = 0 2 .6 – p2 = p 3. 100x2 – 20x + 1 = 0

4. 22 7 5 2 0x x 5. x2 + 4kx + 4k2 = 0 6.7 6mm

7. 0.2t2 – 0.04 t = 0.03 8. 25 2 3 5x x 9.1 34

1 15x x

x x10.

1 3 132 4 3

x xx x

11. a2b2x2–(a2+b2)x +1= 0 12. (2x – 3) = 22 2 21x x

nyU© dJ© nÕVrZo nyU© H$aUo …

x2 + 5x + 5 = 0 ¶m dJ© g‘rH$aUmMm AmnU {dMma H$é.

¶oWo 5x ¶m ‘Yë¶m nXmMr m + n = 5 Am{U mn = 5 Aer ’$moS> H$aVm ¶oV Zmhr. ¶mMmM AW© AgmH$s XmoZ Ad¶dmÀ¶m JwUmH$mamV g‘rH$aUmMr ‘m§S>Ur AmåhmbmH$aVm ¶oV Zmhr. Am{U Ad¶drH$aU nÕVrZo Vo gmoS>{dVm ¶oVZmhr. dJ© g‘rH$aUo gmoS>{dÊ¶mgmR>r Ad¶drH$aU nÕVrbm hr‘¶m©Xm Amho. ‘Yë¶m nXmMr ’$moS> H$aUo Am{U {Xboë¶m dJ©~hþnXrMo Ad¶d nmS>Uo ho Ooìhm e³¶ Amho ’$º$ VoìhmM hrnÕV dmnaVmV.

Va OoWo dJ© ~hþnXrMo Ad¶d nmS>Vm ¶oV ZmhrV Vr dJ©g‘rH$aUo H$er gmoS>dmdrVmV?

¶mMo Ad¶d H$m nmS>Vm ¶oV ZmhrV ho OmUyZ KoÊ¶mgmR>r Amåhr {Xbobo dJ© g‘rH$aU x2 + 5x + 5= 0 hoAmH¥$VrÛmao gwÕm Xe©dy eH$Vmo.

x2 + 5x + 5 x x x

x2

x x 1 1 1

1 1

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Imbrb AmH¥$VrMo {ZarjU H$am.

Amåhmbm Ago {XgyZ ¶oVo H$s x2 + 5x + 5 Xe©{dUmar AmH¥$Vr hr nyU© Mm¡ag qH$dm Am¶V Zmhr. ¶mMm AW©Agm H$s Oa AmH¥$Vr nyU© Mm¡ag qH$dm Am¶V Ho$br Va Vo Amåhr gmoS>dy eH$Vmo.Z§Va AmnU Ad¶d nÜXVrZ gmoS>dyeH$Vmo.

Ago nyU© Mm¡ag qH$dm Am¶V ¶m§Mo nyU© Mm¡ag qH$dm nyU© Am¶V H$aVmZm Ë¶m‘Ü¶o H$m§hr ^mJ {‘i{dboOmVmV. Á¶m nÜXVr‘Ü¶o H$m§hr ^mJ {‘idyZ nyU© Mm¡ag qH$dm nyU© Am¶Vm ~Z{dbo OmVmV. Ë¶mbm nyU© dJ© nÜVåhUVmV.

B. dJ© g‘rH$aU nyU© dJ© nÕVrZo gmoS>{dUo …

da MMm© Ho$boë¶m x2 + 5x + 5 ¶m g‘rH$aUmMm {dMma H$am. AmH¥$Vrdê$ZAmåhmbm Ago {XgyZ ¶oVo H$s x2 + 5x + 5 ho 12 qH$dm nyU© Mm¡ag qH$dm Am¶V V¶mahmoÊ¶mgmR>r 1 Zo H$‘r Amho.

Xþgè¶m eãXmV {(x2 + 5x + 5) + 1} Zo nyU© Mm¡ag V¶ma hmoB©b Am{U Ë¶mMo Ad¶dnmS>Vm ¶oVmV.

åhUOoM x2 + 5x + 5 + 1 = 0 x2 + 5x + 6 = 0 (x + 3)(x + 2) = 0

x = –3 qH$dm x = –2

åhUyZ x2 + 5 x – 5 ho nyU© Mm¡ag Xe©{dV Zmhr.

nyU© Mm¡ag ~Z{dÊ¶mgmR>r {Xboë¶m g‘rH$aUm‘Ü¶o H$m¶ {‘idmdo qH$dm dOm H$amdo?

nyU© Mm¡ag Xe©{dUmao {ZË¶ g‘rH$aU AmR>dm.

a2 + 2ab + b2 = (a + b)(a + b) ho Amåhmbm ‘m{hVAmho.Oa Vo (a2 + 2ab + b2) ¶m ñdénmV Agob Va ho AgoXe©{dVo H$s H$moUË¶mhr dJ© ~hþnXrMo aofr¶ Ad¶d nmS>Vm¶oVmV.

a2 + 2ab + b2 ‘Ü¶o ‘Ybo nX2ab Amho OoWo 2 ho pñWam§H$ Amho.a ho n{hë¶m nXmMo dJ©‘yi Amho.b ho eodQ>À¶m nXmMo dJ©‘yi Amho.AmVm x2 + 5x – 5 = 0 ¿¶m. ‘Ybo nX 4x Amho.5x Mr VwbZm 2ab er H$am.

2ab =5x 2 × x × b = 5x

b =52

xx = 2

x ¶m ghJwUH$mÀ¶m {Zå‘o b =52 Amho ¶mMo {ZarjU H$am.

x x x x2

x x 1 1 1

1 1 1

a2 + 2ab + b2

a2 ab

ab

a

b

b

b

a

a

a

(+

)a

b

( + )a b

©

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204 UNIT-9

b Mm dJ© Ho$ë¶mZo, b2 =52

2( ) =254

dJ© g‘rH$aU x2 + 5 x – 5 = 0 À¶m g‘rH$aUmÀ¶m XmoÝhr ~mOybm254 {‘idyZ qH$dm S>mì¶m ~mOyV

254 {‘idyZ nyU© dJm©V énm§Va Ho$bo OmVo. g‘rH$aUmMr ‘yi qH$‘V VrM amhrb Aem[aVrZo Amåhr XmoÝhr

~mOybm {‘i{dVmo.

nwÝhm x2 + 5 x + 5 = 0 ho g‘rH$aU ¿¶m. x2 + 5 x = 5

x2 +5 x +254 =– 5 +

254

(x +52 )

2 =

54

5 52 2

+ = =x

5 52 2 + =x + Am{U 5 5

2 2 + =x -5 52 2 =x - 5 5

2 2 =x --( )5 5

2 =x - ( )5 52 + =x -

5 52- Am{U ( )5 5

2 +- hr x2 + 5 x + 5 = 0 ¶m g‘rH$aUmMr ~rOo AmhoV.

åhUyZ Amåhr x2 + 5 x + 5 = 0 ho g‘rH$aU nyU© dJ© nÜXVrZo gmoS>{dbo Amho. ¶mbm nyU© dJ© H$ê$Z

gm oS{dUo Ago åhUVmV.


CXmhaU 1) nyU© dJ© H$ê$Z x2 + 6x – 7 = 0 ho dJ© g‘rH$aU gmoS>dm.

CH$b … {Xbobo … x2 + 6x – 7 = 0 2 × a × b = 6x

x2 + 6x + 9 – 9 – 7 = 0 2 × x × b = 6x ( a = x)

x2 + 3x + 3x + 9 = + 16 b =62

xx

x(x + 3) + 3(x + 3) = 16 b = 3

(x + 3)(x + 3) = 16 b2 = 9

(x + 3)2 = 42 XmoÝhr ~mOyMo dJ©‘yi KoVë¶mg

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x + 3 = ± 4 (x À¶m ghJwUH$mÀ¶m {Zå‘o =62 b = 3)

Oa x + 3 = 4 x + 3 = – 4 x = 4 – 3 x = – 4 – 3 x = 1 x = – 7

1 Am{U –7 hr x2 + 6x – 7 = 0 Mr ~rOo AmhoV.

CXmhaU 2) nyU© dJ© nÕVrZo 3x2 – 5x + 2 = 0 gmoS>dm.

CH$b … {Xbobo … 3x2 – 5x + 2 = 0

¶oWo x2 Mm ghJwUH$ 3 Amho Am{U Vmo nyU© dJ© Zmhr. Aer dJ© g‘rH$aUo XmoZ àH$mao gmoS>{dVm ¶oVmV. AmnUË¶m XmoÝhr àH$mamZo gmoS>dy.

(i) g§nyU© g‘rH$aUmbm 3 Zo JwU.

(3x2 – 5x + 2 = 0) × 3 9x2 – 15x + 6 = 0

AmVm x À¶m ghJwUH$mMo {Zå‘o52 Amho. b =

52 Am{U b2 =

252

åhUyZ 9x2 – 15x +2 25 5 6

2 2= 0 (3x)2 – 15x +

252

=25

2 – 6

2532

x =25 64

=14

3x –52

=12

(XmoÝhr ~mOy§Mo dJ©‘yi KoVë¶mZo)

3x –52 = +

12 qH$dm 3x –

52 = –

12

3x =12

+52

3x =52

–12

3x = 62

= 3 3x =42

= 2

x = 1 x =23

(ii) g§nyU© g‘rH$aUmbm 3 Zo ^mJm.

(3x2 – 5x + 2 = 0) 3 x2 –5 23 3

x = 0

AmVm nyduà‘mUoM AmnU nwT>o OmD$2

21 5 5 52 3 6 6

b b

åhUyZ2 2

2 5 5 5 23 6 6 3

x x = 025

6x =

25 26 3

256

x =25 2 136 3 36

XmoÝhr ~mOy§Mo dJ© ‘yi KoD$.

56

x =16

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206 UNIT-9

56

x = +16

, x =5 16 6

=6 16

x = 1 qH$dm

x –56 = –

16 , x =

5 1 4 26 6 6 3 x =

23

1 Am{U23

hr 3x2 – 5x + 2 = 0 ¶m dJ© g‘rH$aUmMr ~rOo AmhoV.

ñdmÜ¶m¶ 9.4

Imbrb dJ© g‘rH$aUo nyU© dJ© H$ê$Z gmoS>dm.

(i) 4x2 – 20x + 9 = 0 (ii) 4x2 + x – 5 = 0 (iii) 2x2 + 5x – 3 = 0(iv) x2 +16x – 9 = 0 (v) x2 – 3x + 1 =0 (vi) t2 + 3t = 7

J{UVmV {heo~ Am{U g‘ñ¶m gmoS>{dUo ho gyÌm§À¶m Cn¶moJmZo gmono Pmbobo Amho. ho Amåhmbm ‘mhrV Amho.Ë¶mMà‘mUo gwÌmMm Cn¶moJ H$ê$Z dJ© g‘rH$aUo gmoß¶m [aVrZo gmoS>{dVm ¶oVmV. dJ© gyÌ Oo Ë¶mMr ~rOoH$mT>Ê¶mgmR>r A{Ve¶ Cn¶wº$ Amho Vo nyU© dJ© nÕVrMm Cn¶moJ H$ê$Z {gÕ H$aVm ¶oVo. AmnU dJ© gyÌ {gÕ H$ê$.Am{U dJ© g‘rH$aUm§Mr ~rOo H$mT>Ê¶mgmR>r Ë¶mMm H$gm dmna H$amdm ho {eHy$¶m.

C) gyÌ nÕVrZo dJ© g‘rH$aUmMr CH$b …

dJ© g‘rH$aU ax2 + bx + c = 0, a 0 ho {dMmamV ¿¶m.

g‘rH$aUmbm a Zo ^mJm (x2 Mm ghJwUH$)b cx xa a

= 0

x À¶m ghJwUH$mÀ¶m {Zå‘o H$mT>m Am{U Ë¶mMm dJ© H$am.

12

ba = 2

ba

2

2ba

=2

24ba

ca Mo COì¶m ~mOyg njm§Va H$am 2 bx x

a = –ca

2

2ba g‘rH$aUmÀ¶m XmoÝhr ~mOyg {‘idm.

2 22

2 2b b b cx xa a a a

S>mì¶m ~mOyMo Ad¶d nmS>m Am{U COì¶m ~mOybm gaiê$n Úm.2 2 2

2 2

42 4 4b b c b acxa a a a

g‘rH$aUmÀ¶m XmoÝhr ~mOyMo dJ©‘yi ¿¶m.

2 2

2

4 42 4 2b b ac b acxa a a

2 42 2b b acxa a

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2 42

b b acxa

ax2 + bx + c = 0 ¶m dJ© g‘rH$aUmMr ~rOo

x =2 4

2b b ac

aAm{U x =

2 42

b b aca

AmhoV.

x =2 4

2b b ac

a¶mbm dJ© gyÌ Ago åhUVmV.

darb {gÕVo‘Ü¶o nyU© dJ© Zgbobm x2 Mm ghJwUH$ g‘rH$aUmbm 'a' Zo ^mJyZ Amåhr H$mTy>Z Q>mH$bm.VWm{n g‘rH$aUmbm '4a' Zo JwUyZ gwÕm gmoS>{dVm ¶oVo. ¶m nÕVrbm "lrYamMm¶m©Mr nÕV' Ago åhUVmV.lrYamMm¶© (1025 B.g.) ¶mZm nyU© dJ© nÕVrZo dJ© g‘rH$aUmMo gyÌ {gÕ H$aÊ¶mMo lo¶ OmVo. àmMrZ ^maVr¶J{UV Vk lrYamMm¶© ¶mZr {dH$grV Ho$boë¶m dJ© gyÌ {gÕVoMm Aä¶mg H$am.

dJ© g‘rH$aUmÀ¶m à‘m{UV Z‘wÝ¶mMm ax2 + bx + c = 0, a 0 Mm {dMma H$am.

XmoÝhr ~mOybm 4a Zo JwUm

{(ax2 + bx + c) = 0} × 4a 4a2x2 + 4abx + 4ac = 0 4a2x2 + 4abx = –4ac

XmoÝhr ~mOybm b2 {‘idm.

4a2x2 + 4abx + b2 = b2 – 4ac (2ax)2 + 4abx + b2 = b2 – 4ac (2ax + b)2 = b2 – 4ac

2ax + b = 2 4b ac 2ax = 2 4b b ac x = 2 4b b ac

x =2 4

2b b ac

aAm{U x =

2 42

b b aca

gmoS>{dbobr CXmhaUo

CXmhaU 1) x2 – 7x + 12 = 0 gyÌ nÕVrZo gmoS>dm.

CH$b … {Xbobo x2 – 7x + 12 = 0

ho ax2 + bx + c = 0 ¶m Z‘wÝ¶mV Amho. OoWo a =1, b = –7 Am{U c = 12

dJ© gyÌ x =2 4

2b b ac

aAmho.

qH$‘Vr ‘m§S>ë¶mZo Amåhmbm {‘iVo.

x =2( 7) ( 7) 4 1 12

2 1 x =

7 49 482

x =7 1

2x =

7 12

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208 UNIT-9

Oa x =7 1 8

2 2 = 4 x =

7 1 62 2

= 3

4 Am{U 3 hr x2 – 7x + 12 = 0 Mr ~rOo AmhoV.

CXmhaU 2) m2 = 2 + 2m gmoS>dm

CH$b … {Xbobo m2 = 2 + 2m

g‘rH$aU nwÝhm à‘m{UV Z‘wÝ¶mV åhUOoM m2 – 2m – 2 = 0 ‘m§S>m.

ho ax2 + bx + c = 0 ¶m ñdénmV Amho. OoWo a = 1, b = –2, c = –2

x =2 4

2b b ac

a(dJ© gyÌ) x =

2( 2) ( 2) 4(1)( 2)2 1

x =2 4 8

2 x =

2 122

x = 2 2 3 2 1 3 1 32 2

x = 1 3 Am{U 1 3 hr dJ© g‘rH$aU m2 = 2 + 2m Mr ~rOo AmhoV.

CXmhaU 3) gmoS>dm 1 2 41 2 4x x x

CH$b … {Xbobo1 2 4

1 2 4x x x

g‘rH$aUmbm gaiê$n Úm.

11x

=4 2

4 2x x1

1x=

2 124 2x x

11x

=2( 2) 1( 4)2

( 4)( 2)x xx x

11x

=2 4 42( 4)( 2)x xx x

11x =

2( 4)( 2)

xx x (x + 4)(x + 2)= 2x(x + 1) x2 + 6x + 8 = 2x2 + 2x

x2 – 4x – 8 = 0 ho dJ© g‘rH$aU Amho.

¶oWo a = 1, b = –4, c = –8

x =2 4

2b b ac

ax =

2( 4) ( 4) 4 (1)( 8)2(1)

x =4 48

2

x =4 4 3

2 = 4 1 3 2 1 3

2

x = 2 1 3 Am{U x = 2 1 3

2 1 3 Am{U 2 1 3 hr {Xboë¶m dJ© g‘rH$aUmMr ~rOo AmhoV.

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CXmhaU 4) x2 + x – (a + 2)(a + 1) = 0 gyÌmMm Cn¶moJ H$ê$Z gmoS>dm.

CH$b … {Xbobo x2 + x – (a + 2)(a + 1) = 0 ¶oWo a = 1, b = 1, c = –(a + 2)(a + 1)

x =2 4

2b b ac

a=

21 1 4 ( 2)( 1)2(1)

a a =

21 1 4 ( 3 2)2a a

x =21 1 4 12 8

2a a

=21 4 12 92

a a =

21 (2 3)2a

=1 (2 3)

2a

x =1 2 3

2a

Am{U x =1 2 3

2a

x =2 2

2a

Am{U x =2 4

2a

x = a + 1 x = –(a + 2)

x2 + x – (a + 2)(a + 1)= 0 Mr ~rOo (a + 2) Am{U –(a + 2) hr AmhoV.

dJ© gyÌmMm Cn¶moJ H$ê$Z dJ© g‘rH$aU gmoS>{dÊ¶mgmR>r Imbrb nm¶è¶m§Mm Adb§~ H$amdm.

nm¶ar 1 … g‘rH$aU ax2 + bx + c = 0 ¶m à‘m{UV Z‘wÝ¶mV {bhm qH$dm à‘m{UV Z‘wÝ¶mV AmUm (‘m§S>m).nm¶ar 2 … à‘m{UV Z‘wÝ¶mer {Xboë¶m g‘rH$aUmMr VwbZm H$am Am{U a, b, c À¶m qH$‘Vr AmoiIm.

nm¶ar 3 … dJ© gyÌ2 4

2b b acx

a{bhm.

nm¶ar 4 … gyÌm‘Ü¶o a, b Am{U c À¶m qH$‘Vr ‘m§S>m (R>odm).nm¶ar 5 … gaiê$n Úm Am{U XmoZ ~rOo {‘idm.

ñdmÜ¶m¶ 9.5

gyÌmMm Cn¶moJ H$ê$Z Imbrb dJ© g‘rH$aUo gmoS>dm.

1. x2 – 4x + 2 = 0 2. x2 – 2x + 4 = 0

3. 2y2 + 6y = 3 4. 15m2 – 11m + 2 = 0

5. 8r2 = r + 2 6. (2x + 3)(3x – 2) + 2 = 0

7. a(x2 + 1) = x(a2 + 1) 8. x2 + 8x + 6 = 0

{Xboë¶m dJ© g‘rH$aUmMr ~rOo doJdoJù¶m nÕVrZo H$mT>Ê¶mMo Amåhr {eH$bobmo AmhmoV. AmåhmbmAgo {XgyZ ¶oVo H$s ~rOo hr dmñVd g§»¶m AmhoV. øm ~rOm§Mo ñdê$n H$m¶ Amho? ~rOm§Mo ñdê$n H$emZo{ZYm©arV H$aVmV?

{Xboë¶m dJ© g‘rH$aUmÀ¶m ~rOm§Mo ñdê$n àË¶j Z gmoS>{dVm {ZYm©arV H$aUo e³¶ Amho H$m¶? dJm©VMMm© H$am Am{U øm àým§Mr CÎmao XoÊ¶mMm à¶ËZ H$am.

AmVm AmnU Ë¶m~Ôb {eHy$¶m.

dJ©g‘rH$aUm§À¶m ~rOm§Mo ñdê$n …

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210 UNIT-9

Imbrb CXmhaUm§Mm Aä¶mg H$am.

1. x2 – 2x + 1 = 0 ¶m g‘rH$aUmMm {dMma H$am.

ho ax2 + bx + c = 0 ¶m Z‘wÝ¶mV Amho.

ghJwUH$ a = 1, b = –2, c = 1 AmhoV.

x =2 4

2b b ac

a x =

2( 2) (2) 4.1 12 1

x =2 4 4

2x =

2 02

x =2 0

2 qH$dm x =2 0

2 x = 1 qH$dm x = 1 ~rOo g‘mZ AmhoV.

2. x2 – 2x – 3 = 0 ¶m g‘rH$aUmMm {dMma H$am.


ghJwUH$ a = 1, b = –2, c = –3 AmhoV.

x=2 4

2b b ac

ax =

2( 2) ( (-2) )4

.

112

21

+b = x =2 42

x =2 4

2

qH$dm x =2 4

2

x=62 qH$dm x =

22 x = 3 qH$dm x = -1 hr ñnï> ~rOo AmhoV.

3. x2 – 2x + 3 = 0 ¶m g‘rH$aUmMm {dMma H$am.


ghJwUH$ a = 1, b = –2, c = 3 AmhoV.

x =2 4

2b b ac

ax =

2 2( 2) ( 2) 4(1)(3)2 1

x =2 4 12

2x =

2 82

x=2 2 2

2x = 2 1 2 1 2

2

1 2x qH$dm 1 2 ~rOo g§{H$U© AmhoV.

darb CXmhaUmdê$Z ho ÑT> hmoVo H$s dJ© g‘rH$aUmMr ~rOo dmñVd Am{U g‘mZ, dmñVd Am{U ñnï>g§{H$U© qH$dm H$mën{ZH$ AmhoV.

b2 – 4ac À¶m qH$‘Vr ~rOm§Mo ñdê$n {ZYm©arV H$aVmV ¶mMo gwÕm {ZarjU H$am. Amåhr Ago gm§Jy H$sb2 – 4ac À¶m qH$‘Vrda ~rOm§Mo ñdê$n Adb§~yZ AgVo.

ax2 + bx + c = 0 ¶mÀ¶m ~rOm§Mo ñdê$n b2 – 4ac À¶m amerÀ¶m qH$‘VrZo emoYVm ¶oVo Am{U åhUyZ Ë¶mbmdJ© g‘rH$aUmMo {ddoMH$ Ago åhUVmV. Vo Ë¶mbm dJ© g‘rH$aUmMo {ddoMH$ Ago åhUVmV. ¶m g§koZo

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Xe©{dVmV Am{U S>oëQ>m Ago dmMVmV.

gm‘mÝ¶nUo ax2 + bx + c = 0 ¶m dJ© g‘rH$aUmMr ~rOo x =2 4

2b b ac

a

• Oa b2 – 4ac = 0 Va g‘rH$aUmMr XmoZ ~rOo g‘mZ AgyZ dmñVd AgVmV. AemàH$mao x = 2ba

• Oa b2 – 4ac > 0 Va g‘rH$aUmMr XmoZ ~rOo ñnï> AgyZ dmñVd AgVmV.

AemàH$mao x =2 4

2b b ac

a, x =

2 42

b b aca

• Oa b2 – 4ac < 0 Va g‘rH$aUmMr ~rOo dmñVd ZmhrV.

Á¶mAWu 2( 4 )b ac ho AmT>iV Zmhr Am{U Amåhr Ago gm§Jy H$s Vo g§{H$U© Amho.

darb {ZîH$f© ho Imbr {Xboë¶m gmaUrV Xe©{dbobo AmhoV.{ddoMH$ ~rOm§Mo ñdê$n

= 0 dmñVd Am{U g‘mZ > 0 dmñVd Am{U ñnï> < 0 dmñVd ~rOo ZmhrV (g§{H$U© ~rOo)

gmoS>{dbobr CXmhaUo

CXmhaU 1) 2x2 – 5x – 1 = 0 ¶m g‘rH$aUmÀ¶m ~rOm§Mo ñdê$n AmoiIm.

CH$b … ho ax2 + bx + c = 0 ¶m Z‘wÝ¶mV Amho. ghJwUH$ a = 2, b = –5, c = –1 AmhoV. = b2 – 4ac = (–5)2 –4(2) (–1) = 25 + 8 = 33

> 0 åhUyZ ~rOo dmñVd Am{U ñnï> AmhoV.

CXmhaU 2) 4x2 – 4x + 1 = 0 ¶m g‘rH$aUmÀ¶m ~rOm§Mo ñdê$n AmoiIm.

CH$b … 4x2 – 4x + 1 = 0 ¶m g‘rH$aUmMm {dMma H$am.

ho ax2 + bx + c = 0 ¶m Z‘wÝ¶mV Amho. a = 4, b = –4, c = 1 ho ghJwUH$ AmhoV. = b2 – 4ac = (–4)2 – 4(4) (1) = 16 – 16

= 0 ~rOo dmñVd Am{U g‘mZ AmhoV.

CXmhaU 3) 'm' À¶m H$moUË¶m qH$‘Vrg x2 + mx + 4 = 0 Mr ~rOo (i) g‘mZ (ii) ñnï> AmhoV.

CH$b … x2 + mx + 4 = 0 ¶m g‘rH$aUmMm {dMma H$am.

ho ax2 + bx + c = 0 ¶m Z‘wÝ¶mV Amho. a = 1, b = m, c = 4 ho ghJwUH$ AmhoV. = b2 – 4ac = m2 – 4(1)(4) = m2 – 16

(i) Oa ~rOo g‘mZ AgVrb Va = 0 m2– 16 = 0 m2 = 16 m = 16 m = 4

(ii) Oa ~rOo ñnï> AgVrb Va 0 m2– 16 0 m2 16 m 16 m 4

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212 UNIT-9

ñdmÜ¶m¶ 9.6

A. Imbrb g‘rH$aUm§À¶m ~rOm§À¶m ñdê$nm§Mr MMm© H$am. (~rOm§Mo ñdê$n AmoiIm)i) y2 - 7y + 2 = 0 ii) x2 - 2x + 3 = 0 iii) 2n2 + 5n - 1 = 0iv) a2 + 4a + 4 = 0 v) x2 + 3x - 4 = 0 vi) 3d2 - 2d + 1 = 0

B. ‘m’ À¶m H$moUË¶m KZ qH$‘Vrg Imbrb g‘rH$aUmMr ~rOo

1) g‘mZ 2) ñnï 3) g§{H$U© AgVmVi) a2-ma+1=0 ii) x2-mx+9=0 iii) r2-(m+1) r+4=0 iv) mk2-3k+1=0

C. dJ© g‘rH$aUm§Mr ~rOo g‘mZ AgVrb Va ‘P’ Mr qH$‘V H$mT>m.i) x2-px+9 = 0 ii) 2a2+3a+p = 0 iii) pk2-12k+9 = 0 iv) 2y2-py+1 = 0v) (p+1) n2+2 (p+3)n+(p+8)=0 vi) (3p+1)c2+2(p+1)c+p=0

dJ© g‘rH$aUmÀ¶m nXm§Mo ghJwUH$ Am{U ~rOo ¶m‘Yrb g§~§Y …

Oa ax2+bx+c=0 ¶m dJ© g‘rH$aUmMr ~rOo ‘m’ Am{U ‘n’ AgVrb Va

m =2 4

2b b ac

aAm{U n =

2 42

b b aca

m+n =2 4

2b b ac

a +

2 42

b b aca

=2 24 4

2b b ac b b ac

a =

22

ba

m + n =b

a

mn =2 4

2b b ac

a

2 42

b b aca =

22 2

2

4

4

b b ac

a

mn =2 2

2

44

b b aca

=2 2

2 2

4 44 4

b b ac aca a

mn =ca

Oa ax2 + bx + c = 0 ¶m dJ© g‘rH$aUmMr ~rOo m d n AgVrb Va

~rOm§Mr ~oarO = m+ n=b

a ~rOm§Mm JwUmH$ma =m n=c

a


CXmhaU 1) x2 + 2x + 1 = 0 ¶m g‘rH$aUmÀ¶m ~rOm§Mr ~oarO Am{U JwUmH$ma H$mT>m.

CH$b … ho ax2 + bx + c = 0 ¶m ñdê$nmV Amho. a =1, b = 2, c=1 ho ghJwUH$ AmhoV.

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~rOo m Am{U n ‘mZy.

i) ~rOm§Mr ~oarO m+n =b

a =2

1 2m n

ii) ~rOm§Mm JwUmH$ma mn =ca

=11 1mn

CXmhaU 2) 3x2 + 5 = 0 ¶m g‘rH$aUmÀ¶m ~rOm§Mr ~oarO Am{U JwUmH$ma H$mT>m.

CH$b … ho ax2 + bx + c = 0 ¶m Z‘wÝ¶mV Amho. a = 3, b = 0, c = 5

~rOo p Am{U q AmhoV Ago ‘mZy.

i) ~rOm§Mr ~oarO p+q =b

a =

03

p q 0+

ii) ~rOm§Mm JwUmH$ma pq =ca

=53

5pq3

ñdmÜ¶m¶ 9.7

g‘rH$aUmÀ¶m ~rOm§Mr ~oarO Am{U JwUmH$ma H$mT>m.

1. x2 – 5x + 8 = 0 2. 3a2 – 10a – 5 = 0 3. 8m2 – m = 2

4. 6k2 – 3 = 0 5. pr2 = r – 5 6. x2 + (ab) x + (a + b) = 0

dJ© g‘rH$aU aMUo …

{Xboë¶m dJ© g‘rH$aUmMr ~rOo Am{U Ë¶m§Mr ~oarO Am{U JwUmH$ma gwÕm H$mT>Uo ¶m~Ôb Amåhmbm ‘m{hVAmho.

~rOm§Mr ~oarO Am{U JwUmH$ma {Xbo AgVm dJ© g‘rH$aU aMUo e³¶ Amho H$m¶?

Oa 'm' Am{U 'n' hr ~rOo AgVrb Va g‘rH$aUmMm gm‘mÝ¶ Z‘wZm x2 – (~rOm§Mr ~oarO) x+ (~rOm§§Mm JwUmH$ma) = 0 Amho. x2 – (m + n) x + mn = 0.


CXmhaU 1) 2 Am{U 3 ~rOo AgUmao dJ© g‘rH$aU aMm.CH$b … 'm' Am{U 'n' hr ~rOo ‘mZm m = 2, n = 3

~rOm§Mr ~oarO = m + n = 2 + 3 m + n = 5~rOm§Mm JwUmH$ma = mn = (2) (3) mn = 6BpÀN>V g‘rH$aU = x2 – (m + n)x + mn = 0 x2 – (5)x + (6) = 0

2 5 6 0x xCXmhaU 2) 3 2 5 Am{U 3 2 5 ~rOo AgUmao dJ© g‘rH$aU aMm.

CH$b … 'm' Am{U 'n' ‘mZy m = 3 2 5 and n = 3 2 5

~rOm§Mr ~oarO = m + n = 3 2 5 3 2 5 6m n

~rOm§Mm JwUmH$ma = mn = 3 2 5 3 2 5 = (3)2 –2

2 5 = 9-n(5)= 9 – 20 11mn

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214 UNIT-9

x2 – (m + n) x + mn = 0 x2 – 6x – 11 = 0CXmhaU 3) Oa x2 – 3x + 1 = 0 ¶m g‘rH$aUmMr ~rOo 'm' Am{U 'n' AgVrb Va

(i)m2n + mn2 (ii) 1 1m n

Mr qH$‘V H$mT>m

CH$b … x2 – 3x + 1 = 0 ¶m g‘rH$aUmMm {dMma H$am.

ho ax2 + bx + c = 0 ¶m Z‘wÝ¶mV Amho. a = 1, b = –3, c = 1 ho ghJwUH$ AmhoV.

(i) ~rOm§Mr ~oarO m + n =( 3) 31

ba 3m n

(ii) ~rOm§Mm JwUmH$ma mn =ca =

11 1mn

åhUyZ

(i) m2n + mn2 = mn (m + n) = 1 × 3 = 3

(ii)1 1m n

=31

n m m nmn mn

= 31 1 3m n

CXmhaU 4) 'm' Am{U 'n' hr x2 – 3x + 4 = 0 ¶m g‘rH$aUm§Mr ~rOo AgVrb Va m2 d n2 ~rOo AgUmao

dJ© g‘rH$aU aMm.

CH$b … x2 – 3x + 4 =0 ¶m g‘rH$aUmMm {dMma H$am. a = 1, b = –3, c = 4 ho ghJwUH$ Amho.

(i) ~rOm§Mr ~oarO = m + n =( 3)1

ba 3m n

(ii) ~rOm§Mm JwUmH$ma= mn =ca =

41 mn = 4

Oa m2 Am{U n2 hr ~rOo AgVrb Va

~rOm§Mr ~oarO m2 + n2 = (m + n)2 – 2mn = (3)2 – 2(4) = 9 – 8 m2 + n2 = 1

~rOm§Mm JwUmH$ma m2n2 = (mn)2 = 42 m2n2 = 16

x2 – (m2 + n2)x + m2n2 = 0 x2 – (1)x + (16) = 0 2 16 0x x

CXmhaU 5) x2 – 6x + q = 0 ¶m g‘rH$aUmMo EH$ ~rO Xþgè¶m ~rOmÀ¶m XþßnQ> Amho Va 'q' Mr qH$‘V H$mT>m.

CH$b … x2 – 6x + q = 0 ¶m g‘rH$aUmMm {dMma H$am. OoWo a = 1, b = –6, c = q

(i) ~rOm§Mr ~oarO m + n =( 6)1

ba 6m n

(ii) ~rOm§Mm JwUmH$ma mn = 1c qa mn = q

Oa EH$ ~rO m Agob Va XþßnQ> ~rO 2m Amho. m = m Am{U n = 2m

m + n = 6 m + 2m = 6 3m = 6 m =63

= 2

Amåhmbm ‘mhrV Amho H$s q = mn q = m(2m) 2m2 = 2(2)2 = 8 q = 8

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CXmhaU 6) x2 – 2x + (k + 3) = 0 ¶m g‘rH$aUmMo EH$ ~rO eyÝ¶ Amho Va k Mr qH$‘V H$mT>m.

CH$b … x2 – 2x + (k + 3) = 0 ¶m g‘rH$aUmMm {dMma H$am. a = 1, b = –2, c = k + 3 ho ghJwUH$ AmhoV.

~rOm§Mm JwUmH$ma = mn

mn =ca mn =

31

kmn = k + 3

åhUyZ 'm' Am{U 'n' hr ~rOo AmhoV Am{U EH$ ~rO eyÝ¶ Amho Va

m = m Am{U n = 0, mn = k + 3

m(0) = k + 3 0 = k + 3 k = –3.

ñdmÜ¶m¶ 9.8

A. Imbrbà‘mUo ~rOo AgUmao dJ© g‘rH$aU aMm.

i) 3 Am{U 5 ii) 6 Am{U –5 iii) –3 Am{U32 iv)

23 Am{U

32

v) 2 3 Am{U 2 3 vi) 3 2 5 Am{U 3 2 5

B.1. Oa 'm' Am{U 'n' hr x2 – 6x + 2 = 0 ¶m g‘rH$aUmMr ~rOo AgVrb Va qH$‘Vr H$mT>m.

(i) (m + n) mn (ii)1 1m n (iii) 3 2 3 2m n n m (iv)

1 1n m

2. Oa 'a' Am{U 'b' hr 3m2 = 6m + 5 ¶m g‘rH$aUmMr ~rOo AmhoV Va qH$‘Vr H$mT>m.

(i)a bb a (ii) (a + 2b)(2a + b)

3. Oa 'p' Am{U 'q' hr 2a2 – 4a + 1 = 0 ¶m g‘rH$aUmMr ~rOo AmhoV Va qH$‘Vr H$mT>m.

(i) (p + q)2 + 4pq (ii) p3 + q3

4.pq Am{U

qp ~rOo AgUmao dJ© g‘rH$aU aMm.

5. x2 + 4x + (k + 2) = 0 ¶m g‘rH$aUmMo EH$ ~rO eyÝ¶ Amho Va 'k' Mr qH$‘V H$mT>m.

6. 2x2 – 3qx + 5q = 0 ¶m g‘rH$aUmMo EH$ ~rO Xþgè¶m ~rOmÀ¶m XþßnQ> Amho Va 'q' Mr qH$‘V H$mT>m.

7. 4x2 – 8px + 9 = 0 ¶m g‘rH$aUmÀ¶m ~rOm‘Yrb ’$aH$ 4 Amho Va 'p' Mr qH$‘V H$mT>m.

8. Oa x2 + px + q = 0 ¶m g‘rH$aUmMo EH$ ~rO Xþgè¶m ~rOmÀ¶m {VßnQ> Agob Va 3p2 = 16q Ago {gÕ H$am.

D. AmboI nÕVrZo dJ© g‘rH$aUmMr CH$b …

aofr¶ g‘rH$aUmMo AmboI H$mT>Uo Am{U g‘¶m{‘H$ aofr¶ g‘rH$aUo gwÕm AmboImZo gmoS>{dUo ¶m~Ôb

Vwåhr n[a{MV AmhmV.

CXmhaU 1) 0 = x2 ¶m dJ© g‘rH$aUmMm {dMma H$am. Amåhmbm q~XÿÀ¶m gh{ZX}eH$mMm AmboI H$mT>Ê¶mMr Oê$ar Amho.

AmnU y = x2 ¶m g‘rH$aUmÀ¶m 'x' Am{U 'y' qH$‘VrMm Vº$m y=x ¶m g‘rH$aUmgmR>r V¶ma H$é.

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216 UNIT-9

2 1 0 1 24 1 0 1 4

xy

A(–2, 4), B(–1, 1), C(0, 0), D(1, 1)

Am{U E(2, 4) ho q~Xÿ ñWm{nV H$am.

q~Xÿ dH«$ H§$gmZo gm§Ym.

y = x2 Mm AmboI dH«$ aofm Amho.

XmoZ A{YH$ dJ© g‘rH$aUm§À¶m AmboIm§Mo {ZarjU H$am.

CXmhaU 2) 2x2 =0 CXmhaU 3) 21 02

x

2 1 0 1 28 2 0 2 8

xy

2 1 0 1 22 ½ 0 ½ 2

xy

AemàH$mao Amåhmbm Ago {XgyZ ¶oVo H$s dJ© g‘rH$aUo 0 = x2, 0 = 2x2 Am{U 0 = 212

x AmboI dH«$aofm

AmhoV. dJ© g‘rH$aUo Xe©{dUmè¶m ¶m dH«$ aofobm "nadb¶' Ago åhUVmV.

dJ© g‘rH$aUm§À¶m øm AmboImdê$Z Amåhmbm Ago AmT>iVo H$s ({XgyZ ¶oVo H$s).

• AmboImMo nadb¶ ho g§~§{YV Ajmer à‘mU~Õ AmhoV.

• dH«$Vm OoWo gdm©V ‘moR>r AgVo Ë¶m q~Xÿbm {eamoq~Xÿ Ago åhUVmV. nadb¶ ¶m q~Xÿda diU KoVmo.

AmmVm AmnU nadb¶ ¶m~Ôb A{YH$ {eHy$.(x2 – 4x) Am{U (–x2 + 2x + 5) ¶m amer {dMmamV ¿¶m.

1. x2 – 4x 2. –x2 + 2x + 5¶oWo 'a' ho eyÝ¶mnojm ‘moR>o Amho. ¶oWo a hr eyÝ¶mnojm H$‘r Amho.y = x2 – 4x ‘mZm. y = – x2 + 2x + 5 ‘mZm.

• qH$‘VrMm Vº$m V¶ma H$am. •AmnU qH$‘VrMm Vº$m V¶ma H$ê$.

X O X

Y

654321

1 2 3–1–2–3

Y

E(2, 4)A(–2, 4)

B(—1, 1) D(1, 1)C(0, 0)

x - axis : 1cm = 1unit y - axis : 1 cm =1unit

X¢–2–3–4–5

98

76543

21

X O 12345–1

Y

Y¢

E(2,8)A(-2,8)

B(–1,2) D(1,2)

C(0,0)

x - axis : 1cm = 1unit y - axis : 1 cm =0.5unit

X I

–2–3

2.5

2

1.5

1

0.5

X O 1 2 3–1

Y

YI

P(-2,2)

R(0,0)

T(2,2)

S(1, 0.5)Q(-1,0.5)

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–x2 + 2x + 5• nadb¶ Imbrb {Xeobm {demb hmoV OmVmo.• H$‘rV H$‘r q~Xÿ JmR>on¶ªV x Mr qH$‘V OgOer

dmT>V OmVo Ver y Mr gwÕm qH$‘V dmT>V OmVo.• y Mr ‘moR>çmV ‘moR>r qH$‘V 6 Amho.

x2 – 4x• nadb¶ darb {Xeobm {demb hmoV OmVmo.• y Mr {H$‘mZ qH$‘V JmR>on¶ªV OgOer x Mr

qH$‘V dmT>V OmVo VgVer y Mr qH$‘V H$‘rhmoV OmVo.

• y Mr gdm©V H$‘r qH$‘V –4 Amho.

Y

X O X¢

Y¢

54321

–1–2

–3–4–5

1 2 3 4 5–1–2–3–4–5

D(3,–3)B(1, 3)

A(–1, 5)F(5, 5)

E(4,0)

C(2 –4)


Y

X OX

Y

654321

–1–2–3–4–5

1 2 3 4 5–1–2–3–4–5

D(1,6)

C(0, 5) E(2, 5)

F(3,2)B(–1,2)

A(–2,–3)


1 0 1 2 3 4 55 0 3 4 3 0 5

xy

-- - -

2 1 0 1 2 33 2 5 6 5 2

xy

• àË¶oH$ H«${‘V OmoS>r (x, y) Mo q~Xÿ ñWmnZ H$am. • àË¶oH$ H«${‘V OmoS>r (x, y) Mo q~Xÿ ñWmnZ H$am.

• ñWmnZ Ho$boë¶m q~XÿVyZ dH«$ H§$g H$mT>m. • ñWmnZ Ho$boë¶m q~XÿVyZ dH«$ H§$g H$mT>m.

darb XmoÝhr AmboImdê$Z Amåhr Imbrb {ZarjUo ‘m§Sy>.

ñdmÜ¶m¶ 9.9

I. Imbrb dJ© g‘rH$aUm§Mo AmboI H$mT>m.

i) y = –x2 ii) y = 3x2 iii) y x = - 2212 iv) y = x2 – 2x

v) y = x2 – 8x + 7 vi) y = (x + 2)(2 – x) vii) y = x2 + x – 6 viii) y = x2 – 2x + 5

dJ© g‘rH$aUm§À¶m AmboIr¶ CH$br …

¶m ^mJmV AmnU dJ©g‘rH$aUo gmoS>{dÊ¶mÀ¶m AmboIr¶ nÕVrMm Aä¶mg H$ê$.

gmoS>{dbobr CXmhaUo

CXmhaU 1) y = x2 – x – 2 ho g‘rH$aUMm AmboI H$mT>m Am{U ~rOo H$mT>m.

CH$b … AmnU ¶m g‘rH$aUMm AmboI H$mTy> Am{U ~rOo AmboImZo H$mTy>

nm¶ar 1 … y = x2 – x – 2 À¶m qH$‘VrMm Vº$m V¶ma H$am.

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218 UNIT-9

2 1 0 1 24 0 2 2 0

xy

nadb¶ AmboI H$mT>m. x –Ajmbm nadb¶ AmboI N>oXUmao gh{ZXo}eH${bhm

g{‘H$aUmMr ~rOo x = – 1Am{U x =2 hr AmhoV.

nmÜXV 2 … x2 – x –2= x2 = x + 2

y = x2 ‘mZy ..............(1) Am{U y = x +2-2........(2)

2 1 00

11

24 40

xy

2101

22 4 0

xy

g‘r 1 Am{U 2 Mm nadb¶ Am{U aofm AmboI H$mT>m.gai aofm

nadb¶ AmboImbm N>oXUmè¶m q~XÿMo gh {ZXo}eH$ {bhm Vo B (-1,0) Am{U

E (2,0) ho AmhV.

g‘r. Mr ~rOo x =-1 Am{U x = 2 hr AmhoV.

CXmhaU 2) x2 -10x +25 ho dJ© g‘rH$aU AmboImZo gmoS>mdm.

CH$b: y = x2 -10x +25 ¶m g‘rH$aUmÀ¶m qH$‘VrMm VÎH$m V¶maH$am.

1 2 3 4 5 616 9 4 1 0 1

xy

gd© q~Xÿ AmboImda ñWmnZ H$am Am{U ñWm{nV q~XÿVyZ dH«$aofm H$mT>m.x- AjmÀ¶m Am{U dH«$ aofoÀ¶m N>oXZ q~XÿÀ¶m IwUm H$am.

dH«$ aofm x- Ajmbm ’$º$ (5,0) ¶m EH$ q~XÿV N>oXVo.g‘rH$aUmMr ~rOo 5 Am{U 5 hr AmhoV.

CXmhaU 3 … y = x2 + 2 Mm AmboI aMm Am{U ~rOo H$mT>m.

CH$b … nm¶ar 1 … y = x2 + 2 ¶m g‘rH$aUmÀ¶m qH$‘VrMm Vº$m V¶maH$am.

2 1 0 1 26 3 2 3 6

xy

Y

X O X¢

Y¢

54321

–1–2–3–4–5

1 2 3 4 5–1–2–3–4–5

D(1,–2)

B(–1, 0)

A(–2, 4)

E(2,0)

C(0 –2) - interceptx - interceptx


Y

X O X¢

Y¢

252015105

1 2 3 4 5 6–1–2–3–4

A(1,16)

= –10 +25y x x2

x - intercept

B(2,9)

C(3,4) D(4

,1)

E(5,

0)F(

6,0)

Y

X O X¢

Y¢

87654

321

1 2 3 4 5–1–2–3–4–5

A(–2, 6)

B(–1, 3) D(1, 3)

E(2, 6)

C(0, 2)

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dH«$ aofm x- Ajmbm N>oXZ Zmhr.VoWo x2 + 2x = 0 gmR>r x Mr dmñVd qH$‘V Zmhr.

åhUyZ VoWo dmñVd ~rOo ZmhrV.

darb {VÝhr CXmhaUmMm Vnerb AmVm AmnU V³Ë¶mV Zm|X H$ê$. Ë¶mMm Aä¶mg H$am.

x2 – x – 2 x2 – 10x + 25 x2 + 2Ë¶mbm XmoZ N>oXZ Ë¶mbm EH$ N>oXZ Ë¶mbm N>oXZ

q~oXÿ AmhoV. q~oXÿ Amho. q~oXÿ ZmhrV.

X X

2 solutions

X X

1 solutions

X X

No solutionsax2 + bx + c = 0 bm ax2 + bx + c = 0 bm ax2 + bx + c = 0 bmXmoZ dmñVd Am{U XmoZ dmñVd Am{U dmñVd ~rOo ZmhrV

Ag‘mZ ~rOo AmhoV nwZamd{V©V ~rOo AmhoVb2 – 4ac > 0 b2 – 4ac = 0 b2 – 4ac < 0

CXmhaU 4) y = (2 – x)(4 + x) ho gd© g‘rH$aU AmboImZo gmoS>dm.

CH$b … y = (2 – x)(4 + x) Amho.

COì¶m ~mOybm gaiê$n Úm Am{U ax2 + bx + c = 0 à‘m{UV Z‘wÝ¶mV AmUm.

y = (2 – x)(4 + x)

y = –x2 – 2x + 8

4 3 2 1 0 1 20 5 8 9 8 5 0

xy

x - Ajmbm nadb¶ AmboI NoXZmè¶m> q~XÿZm IwUm H$am. NoXZmao q~Xy (-4,0 ) Am{U (2,0) AmhoV.

åhUyZ 2 Am{U -4 hr ~rOo AmhoV.

CXmhaU 5 … y = 2x2 Mm AmboI H$mT>m Am{U AmboImMm Cn¶moJH$ê$Z 5 Mr qH$‘V H$mT>m.

CH$b … nm¶ar 1 … y = 2x2 ¶m g‘rH$aUmMrnyV©Vm H$aUmè¶m x Am{U y À¶m g§JVqH$‘VrMm Vº$m aMm.

0 1 1 2 2 50 2 2 8 8 10

xy

Y

X O X¢

Y¢

987654321

1 2 3 4 5–1–2–3–4–5

C(–2, 8) E(0, 8)

F(1, 5)B(–3, 5)

D(–1, 9)

A(–4, 0) G(2, 0)

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220 UNIT-9

nm¶ar 2 … gd© q~Xy AmboI nÌmda ñWm{nV H$am Am{Unadb¶ H$mT>m.

nm¶ar 3 … Ooìhm 5 5x 102

y= = =2x-Ajmbm g‘m§Va AgUmar y = 10 hr gaiaofm H$mT>m.nm¶ar 4 … nadb¶ Am{U gaiaofm N> oXZ q~X ÿV yZx-Ajmda b§~ H$mT>m.nm¶ar 5 … x- Ajmdarb q~Xÿ‘Ü¶o {‘iUmao b§~ hoM

5 Mr qH$‘V Amho.

x = 2.3

5 2 3 = .

ñdmÜ¶m¶ 9.10I. Imbrb dJ© g‘rH$aUmMo AmboI H$mT>m.

i) x2–4x=0 ii)x2 +x – 12 =0 iii) x2 – x – 2 =0 iv) x2 – 5x + 6 = 0

II.1. y = x2 Mm AmboI H$mT>m Am{U 3 Mr qH$‘V H$mT>m.

2. y = x2 Mm AmboI H$mT>m Am{U 7 Mr qH$‘V H$mT>m.

3. y =12 x2 Mm AmboI H$mT>m Am{U 10 Mr qH$‘V H$mT>m.

dJ© g‘rH$aUmda AmYmarV CXmhaUo gmoS>{dUo (Solving problems based on quadratic equations)

Amnë¶m X¡Z§{XZ OrdZmV Ago nwîH$i àg§J ¶oVmV H$s OoWo Amåhr dJ© g‘rH$aUo gmoS>{dÊ¶mÀ¶m nÕVrMmdmna H$ê$Z Amåhr Vr CXmhaUo gmoS>{dVmo. AmnU H$mhr CXmhaUm§Mm {dMma H$ê$

gmoS>{dbobr CXmhaUo

CXmhaU 1 … XmoZ H«$‘dma YZ {df‘ g§»¶mMm JwUmH$ma 195 Amho. Va Ë¶m g§»¶m H$mT>m.

CH$b … nm¶ar 1 … g‘rH$aU aMUo

EH$ {df‘ YZ g§»¶m x ‘mZm Am{U Xþgar {df‘ YZ g§»¶m (x + 2) hmoB©b.

g§»¶m§Mm JwUmH$ma x(x + 2) = 195 Amho.

nm¶ar 2 … g‘rH$aU gmoS>{dUo

Y

X OX

1110987654321

1 2 3 4 5–1–2–3–4–5

=10y

D(–2, 8) C(2, 8)

A(1, 2)B(–1, 2)

= –2.3x = 2.3x

Y

5 = ± 2.3

AMyH$ CH$b A§XmOo CH$b

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x(x + 2) = 195

x2 + 2x = 195

x2 + 2x – 195 = 0

x2 + 15x – 13x – 195 = 0

x + 15 = 0 or x – 13 = 0

(x + 15)(x – 13) = 0

x + 15 = 0 qH$dm x – 13 = 0

x = – 15 qH$dm x = 13

nm¶ar 3 … ñnï>rH$aU H$aUo Am{U CH$b H$mT>Uo Ho$ìhmhr g§»¶m YZ Agmdr. x = – 15 ho KoVm ¶oV Zmhr.

x = 13 Am{U x + 2 = 13 + 2 = 15

XmoZ H«$‘dma {df‘ YZ g§»¶m 13 Am{U 15 AmhoV.

CXmhaU 2 … AÛ¡VZo 60 é. Zm H$mhr nwñVHo$ {dH$V AmUbr (KoVbr) Oa Ë¶mZo {VV³¶mM a¸$‘oV 5 nwñVHo$A{YH$ KoVbr AgVr Va àË¶oH$ nwñVH$mMr qH$‘V 1 Zo H$‘r Pmbr AgVr. Va AÛ¡VZo IaoXr Ho$boë¶mnwñVH$m§Mr g§»¶m {H$Vr Am{U àË¶oH$ nwñVH$mMr qH$‘V {H$Vr.CH$b … nm¶ar 1 … nwñVH$m§Mr g§»¶m = x ‘mZy.

nwñVH$m§Mr EHy$U qH$‘V = 60 é. àË¶oH$ nwñVH$mMr qH$‘V =60x

é.

Oa nwñVH$mMr g§»¶m (x + 5) Agob Va àË¶oH$ nwñVH$mMr qH$‘V =60

5xé. hmoB©b

IaoXr qH$‘VrVrb =

’$aH$ 1 é. Amho.

CH$b:60 60

5x x= 1

60( 5) 60( 5)

x xx x = 1

2

60 300 605

x xx x

= 1 2

3005x x

=11

x2 + 5x = 300 x2 + 5x – 300 = 0

x2 + 20x – 15x – 300 = 0 x(x + 20) – 15 (x + 20) = 0

(x + 20)(x – 15) = 0

EH§$Xa x + 20 = 0 qH$dm x – 15 = 0

x = – 20 qH$dm x = 15

nwñVH$m§Mr g§»¶m = x = 15

nwñVH$m§Mr qH$‘V Ho$ìhmhr F$U ZgVo åhUyZ -15 ho dJibo Amho.àË¶oH$ nwñVH$mMr qH$‘V =

60 6015x

= ` 4 ê$.

nwñVH$mMr g§»¶m AgVmZmàË¶oH$ nwñVH$mMr qH$‘V

x nwñVH$mMr g§»¶m AgVmZmàË¶oH$ nwñVH$mMr qH$‘V

(x+5)

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222 UNIT-9

ñdmÜ¶m¶ 9.11

1. XmoZ H«$‘dma {df‘ YZ g§»¶m Aem H$mT>m H$s Ë¶m§À¶m dJmªMr ~oarO 130 hmoB©b.2. EH$m g§»¶oÀ¶m dJm©À¶m {VßnQ>rhÿZ Ë¶mM g§»¶oMr Mma nQ> dOm Ho$ë¶mg g§»¶m 15 ¶oVo Va Vr nyU© g§»¶m

H$mT>m.3. XmoZ Z¡g{J©H$ g§»¶mMr ~oarO 8 Amho. Oa Ë¶m§À¶m ì¶ñVm§Mr ~oarO

8.

15Amho Va Ë¶m g§»¶m AmoiIm.

4. EH$ XmoZ A§H$s g§»¶m Aer Amho H$s Ë¶m§À¶m A§H$mMm JwUmH$ma 12 Amho. Ooìhm Ë¶m g§»¶oV 36{‘i{dë¶mg A§H$mÀ¶m ñWmZm§Mr AXbm~Xb hmoVo Va Ë¶m g§»¶m AmoiIm.

5. VrZ H«$‘dma YZ nyUmªH$mMr ~oarO Aer Amho H$s n{hë¶m g§»¶oMm dJ© Am{U BVa XmoZ g§»¶mMm JwUmH$ma¶m§Mr ~oarO 154 Amho Va nyUmªH$ H$mT>m.

6. gwOb Am{U ì¶§H$Q> ¶m§Mr d¶o 11 df} Am{U 14 df} AmhoV. {H$Vr dfm©V Ë¶m§À¶m d¶m§Mm JwUmH$ma 304hmoB©b.

7. d¶ {VÀ¶m ‘wbmÀ¶m d¶mÀ¶m dJm©À¶m XþßnQ>r BVHo$ Amho. 8 dfm©Z§Va Ë¶m AmB©Mo d¶ Ë¶mÀ¶m ‘wbmÀ¶md¶mÀ¶m {VßnQ>rnojm 4 dfm©Zo A{YH$ Amho. Va Ë¶m§Mr AmVmMr d¶o H$mT>m.

8. Am¶VmMo joÌ’$i 56 Mm¡.g|.‘r. Amho. Oa Ë¶mÀ¶m nm¶mMo ‘mn (x + 5) Am{U Ë¶mMr C§Mr (x – 5) Amho.Va Am¶VmMr ‘mno H$mT>m.

9. {ÌH$moUmMr b§~ C§Mr Ë¶mÀ¶m nm¶mnojm 6 g|.‘r.Zo A{YH$ Amho. Ë¶mMo joÌ’$i 108 Mm¡.g|.‘r. Amho. VaË¶mMm nm¶m H$mT>m.

10. ABCD g‘^wO Mm¡H$moZmMo AC BCAm{U H$U© E ‘Ü¶o N>oXVmV. Oa AE = x, BE = x + 7 Am{U

AB = x + 8 Va AC Am{U BD H$UmªMr bm§~r H$mT>m.

11. ABC g‘Ûr wO {ÌH$moUmV AB BC Am{U BD hm AC nm¶mda H$mT>bobm b§~ Amho. Oa DC = x,BD = 2x – 1 Am{U BC = 2x + 1 Va {ÌH$moUmÀ¶m {VÝhr ~mOy§Mr bm§~r H$mT>m.

12. EH$m ¶m§{ÌH$ ~moQ>tMm g§W nmÊ¶mVrb doJ 15 {H$.‘r./Vmg AgyZ àdmhmÀ¶m {XeoZo Am{U naV ‘mJo 30{H$.‘r. A§Va 4 Vmg 30 {‘{ZQ>mV OmVo. Va àdmhmMm doJ AmoiIm.

13. EH$m ì¶mnmè¶mZo EH$ dñVy é.24 bm {dH$ë¶mZo Ë¶mÀ¶m IaoXr qH$‘Vr EdT>m eoH$S>m Z’$m hmoVmo. Va Ë¶mdñVwMr IaoXr qH$‘V H$mT>m.

14. AmpídZrbm EH$ H$m‘ nyU© H$aÊ¶mg YZlr nojm 6 {Xdg H$‘r bmJVmV. Va VoM H$m‘ XmoKr AmpídZr

Am{U YZlr {‘iyZ 4 {XdgmV VoM H$m‘ nyU© H$aV AgVrb Va YZlr EH$Q>rbm Vo H$m‘ nyU© H$aÊ¶mg{H$Vr {Xdg bmJVrb?

* * *

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CÎmao

ñdmÜ¶m¶ 9.2

2. (i) 14 (ii) 5 5 (iii) 10 (iv)8

7(v) –2 {H$dm –14 (vi) 4 (vii) 9 (viii)

1

4. (i) r = 27

(ii) d = 3 (iii) b = 15 (iv) a = 8 (v) v = 10 (vi) v = 20

ñdmÜ¶m¶ 9.3

1. –10, –5 2. –3, 2 3.1

10,

1

104. - 2,

2

5

dJ© g‘rH$aUo

dJ© Ed dJ© g‘rH$aUax2 + c = 0

A dJ© Ed dJ©g‘rH$aU

ax2 + bx + c = 0

~rOo Am{U ghJwUH$¶m‘Yrb g§~§Y

~rOm§Mo ñdê$n

dJ© g‘rH$aUo gmoS>{dUo~rOm§Mr ~oarO

m + n =-ba

{ddoMH$= b2 – 4ac

~rOo dmñVdd g‘mZ

~rOo dmñVdd ñnï>

~rOo g§H$sU©

~rOm§Mm JwUmH$ma

mn =ca

dJ© g‘rH$aUo aMUo2x (m n)x mn 0

Ad¶d nÕV

nyU© dJ© nÕV

gyÌ nÕV

AmboI nÕV

= 0

> 0

< 0

2 , 3 .... nadb¶MmCn¶moJ H$ê$Z emoYUo

nadb¶ nÕVrZodJ©g‘rH$aUo gmoS>{dUo

nadb¶ aMUo

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224 UNIT-9

5. –2k, 6. 7, –1 7.10 2

-3 1, 8.

3– 5

5,

9.5 3

– ,2 2 10.52 , 5 11.

1 1,b2 2a

12. 6, –1

ñdmÜ¶m¶ 9.4

1. (i)9 1

,2 2

(ii) 1,-5

4(iii) –3,

1

2(iv) –8 ± 73 (v) 3± 5

2(vi) -3± 37

2

ñdmÜ¶m¶ 9.5

1. 2 2 2. 1 -3 3.15–3

24.

2 1

5 3, 5.

651

16

6.1 -4

2 3, 7. a,

1

a8. - 4 10

ñdmÜ¶m¶ 9.6

C. (i) 6 (ii)9

8(iii) 4 (iv) 22 (v)

1

3(vi) 1,

1

2ñdmÜ¶m¶ 9.7

1. 5, 8 2.10 -5

3 3, 3.

1 -1

8 4, 4. 0,

-1

25.

1 5

p p, 6. –ab, (a+b)

ñdmÜ¶m¶ 9.8

B 1. (i) 12 (ii) 3 (iii) 24 (iv) 7

2. (i)-22

5(ii)

19

33. (i) 6 (ii) 5 5. –2 6. 5 7.

5

2

ñdmÜ¶m¶ 9.11

1. 7, 9 2. 3 3. 3, 5 {H$dm 5, 3 4. 26 5. 8, 9, 10 6. 5 7. 32, d 4 df}

8. 14 go.‘r , 4go.‘r 9. 12go.‘r, 18go.‘r 10. 10go.‘r, 24 go.‘r 11. 5go.‘r, 8 go.‘r

17 go.‘r, 17 go.‘r, 16 go.‘r 12. 5 {H$.‘r./ Vmg 13. `20 14. 12 {Xdg

*****

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Circle-Tangent Properties 225

* {Xboë¶m ‘mnmMr Ordm aMUo.

* g‘mZ Ordm dVw©i‘Ü¶mnmgyZ g‘mZ A§VamdaAgVmV.

* dVw©iI§S>mVrb H$moZ

* EH$H|$Ðr¶ d EH$ê$n dVw©io

* dVw©imdarb q~XwVyZ ñn{e©Ho$Mr aMZm.* {ÌÁ¶m‘Yrb H$moZ {Xbm AgVm ñn{e©H$m§Mr

aMZm* ~møq~XwVyZ ñn{e©H$m aMZm Am{U {gÕVm.* ñne© H$aUmar dVw©io, AW©, aMZm Am{U

{gÕVm.* g‘mB©H$ ñn{e©H$m DCT Am{U TCT* g‘mB©H$ ñn{e©H$m g‘mB©H$ ~møñn{e©H$m

Am{U Am§Vañn{e©H$m

o.

* MmH$mÀ¶m emoYmZo ‘mZdr OrdZmV ‘hmZ ~XbKSy>Z Ambo AmhoV. MmH$mÀ¶m A^mdmZodmhVyH$, H$maImZo g§nH©$ CnH$aUo BË¶mXtMoH$m¶ KS>bo AgVo ¶m§Mr H$ënZm H$am.

* J{UVm‘Ü¶o dVy©im§À¶m gmhmæ¶mZo ^y{‘VrMmAm{U JVrdmT> H$aUmè¶m J{UVmMm {dH$mghmoÊ¶mg ‘XV Pmbr Amho.


* R>am{dH$ {ÌÁ¶oÀ¶m dVw©imV R>am{dH$ bm§~rMr Ordm aMUo.

* {dYmZ H$aUo d nS>VmiyZ nmhUo.

(a) g‘mZ Ordm dVw©i‘Ü¶mnmgyZ g‘mZ A§VamdaAgVmV.

(b) EH$mM dVw©iI§S>mVrb H$moZ g‘mZ AgVmV.

(c) {demb dVw©iI§S>mVrb H$moZ bKwH$moZ AgVmo.

(d) bKw dVw©iI§S>mVrb H$moZ {demb H$moZ AgVmo.

(e) AY©dVw©i I§S>mVrb H$moZ H$mQ>H$moZ AgVmo.

* EH$H|$Ðr¶ d EH$ê$n dVw©io ¶m‘Yrb ’$aH$AmoiIUo.

* N>o{XH$m, Ordm Am{U ñn{e©H$m ¶m‘Yrb ’$aH$.* ñn{e©Ho$À¶m JwUY‘m©Mo {dYmZ H$aUo Am{U nS>VmiUo.* darb {dYmZmÀ¶m ì¶Ë¶mgmMo {dYmZ.* ñne© dVw©im§À¶m AWmªMo ñnï>rH$aU, ~møñne© Am{U

Am§Vañne© H$aVmZm.* ñne© H$aUmè¶m dVw©im§À¶m d, R Am{U r ‘Yrb g§~§Y

AmoiIUo.* VH©$ewÕ nÕVrZo à‘o¶ {gÕ H$aUo.* à‘o¶ Am{U Ë¶m§Mo ì¶Ë¶mg ¶mda AmYmarV g§»¶mË‘H$

g‘ñ¶m Am{U Cnà‘o¶ gmoS>{dUo.* dVw©imdarb q~XwVyZ ñn{e©H$m aMUo.

* dVw©imbm ~mô¶ q~XþVyZ ñn{e©H$m H$mT>Uo.

* DCT Am{U TCT À¶m JwUY‘mªMo {dYmZ H$aUo.

* {Xboë¶m XmoZ dVw©imZm DCT Am{U TCT aMUo.

* DCT Am{U TCT À¶m bm§~rMo ‘mnZ H$ê$Z nS>VmiyZnhmUo.

The ancient Indian symbol of a circle, with adot in the middle, known as 'bindu', wasprobably instrumental in the use of circle asa representation of the concept zero.

10 dVw©io ( Circles )

dVw©io ( Circles )

dVw©io ( Circles )

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226 UNIT-10

OrdoMo JwUY‘©

Vwåhr dVw©img§~§Yr ì¶m»¶m§Mm (terms) Aä¶mg Ho$bm AmhmV. Imbrb AmH¥$VtMo {ZarjU H$am. nwÝhm ñ‘aU H$amd ì¶m»¶m AmoiIm.

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{Xboë¶m dVw©imV {Xboë¶m ‘mnmMr Ordm aMZo

CXm. 1 … 3 g|.‘r. {ÌOoÀ¶m dVw©imV 5 g|.‘r.bm§~rMr Ordm H$mT>m.

CH$b: nm¶ar 1 … H$mJXmda 'O' dVw©i‘Ü¶{ZpíMV H$am. H§$nmg‘Ü¶o 3 g|.‘r. {ÌOmKodyZ 'O' ‘Ü¶ H$ënyZ dVw©i H$mT>m.dVw©imÀ¶m n[aKmda 'P' q~Xÿ {ZpíMVH$am.

nm¶ar 2 : 'P' ‘Ü¶ H$ënyZ 5 g|.‘r.{ÌÁ¶oZo 'Q' ‘Ü¶o N>oXUmam Mmn ‘mam.P d Q gm§Ym.

åhUyZ 'PQ' hr {Xboë¶m dVw©imVrb 5g|.‘r. bm§~rMr B©ï>> Ordm hmo¶.

CXm. 2 : 2.5 g|.‘r {ÌÁ¶oÀ¶m dVw ©imV4 g|.‘r. bm§~rMr Ordm aMm. Am{U OrdoModVw©i‘Ü¶mnmgyZ A§Va ‘moOm.

n{hbr 1: 'O' dVw©i‘Ü¶ Agbobo 2.5g|.‘r. {ÌÁ¶oMo dVw©i H$mT>m. Ë¶m‘Ü¶o'LM' = 4 g|.‘r.

Xþgar 2 : 'LM' ¶m OrdoMm ‘Ü¶q~Xÿ'P' KodyZ 'OP' gm§Ym Am{U nÅ>rZo ‘moOm.

'O' dVw©i‘Ü¶ d Ordm 'LM' ¶m‘YrbA§Va OP = 1.5 cm.

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5cm

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1.5 cm

4 cm

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ñdmÜ¶m¶ 10.1

1. 3.5 g|.‘r. {ÌÁ¶oÀ¶m dV©wimV 6 g|.‘r. bm§~rMr Ordm aMm. dVw©i‘Ü¶ d Ordo‘Yrb A§Va ‘moOm.2. 4.5 g|.‘r. {ÌÁ¶oÀ¶m dVw©imV 6 g|.‘r. Am{U 8 g|.‘r. bm§~rÀ¶m XmoZ Ordm dVw©i‘Ü¶mÀ¶m EH$mM

~mOyg H$mT>m. Ord|Mo dVw©i‘Ü¶mnmgyZMo A§Va ‘moOm.3. 4.5 g|.‘r. {ÌÁ¶oÀ¶m dVw©imV 6.5 g|.‘r. bm§~rÀ¶m XmoZ Ordm dVw©i‘Ü¶mÀ¶m XmoÝhr ~mOyg H$mT>m.

Ord|Mo dVw©i ‘Ü¶mnmgyZ A§Va ‘moOm.4. 5 g|.‘r. {ÌÁ¶oÀ¶m dVw©imV 9 g|.‘r. Am{U 7 g|.‘r. bm§~rÀ¶m XmoZ Ordm dVw©i‘Ü¶mÀ¶m XmoÝhr

~mOyg H$mT>m OrdoMo dVw©i‘Ü¶mnmgyZ A§Va ‘moOm.

dVw©imÀ¶m Ordoer g§~§{YV JwUY‘©

Imbrb AmH¥$Ë¶m§Mo {ZarjU H$am Am{U àË¶oH$ AmH¥$VrÀ¶m ~m~VrV OrdoMr bm§~r, dVw©i ‘Ü¶mnmgyZ OrdoMoA§Va H$mT>m.

AmH¥$Vr 1 AmH¥$Vr 2 AmH¥$Vr 3

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OM

KJ

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Ordm AB = Ordm EF = Ordm IJ =

Ordm CD = Ordm GH = Ordm KL =

Ordm MN =

b§~m§Va b§~m§Va b§~m§Va

OP = OR = OT =

OQ = OS = OU =

OV =

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228 UNIT-10

àË¶oH$ KQ>ZoVrb Ord|Mr bm§~r d dVw©i‘Ü¶mnmgyZMo b§~m§Va ¶m§Mr VwbZm H$am. Amnbm AZw‘mZ H$m¶?

• EH$mM dVw©imVrb g‘mZ Ordm dVw©i‘Ü¶mnmgyZ g‘mZ A§Vamda AgVmV.

dVw©im‘Ü¶o AZoH$ Ordm H$mT>Vm ¶oVmV. Ë¶m OrdoMo A§Va g‘mZ Agob Va Ë¶m Ordm dVw©i‘Ü¶mnmgyZ g‘mZA§Vamda AgVmV. hm ì¶Ë¶mg Iam Amho.

• Oa EImÚm dVw©imV dVw©i‘Ü¶mnmgyZ g‘mZ A§Vamda H$mT>boë¶m Ordm g‘mZ AgVmV.

AmVm Ordm d dVw©i‘Ü¶ ¶m‘Yrb b§~m§Va dmT>Vo qH$dm H$‘r hmoVo ¶mMm Aä¶mg H$ê$.

• Imbrb AmH¥$VtMo {ZarjU H$am Ordm Am[U dV©wi‘Ü¶mnmgyZMo A§Va ¶m§À¶m qH$‘VrMm Vº$m nyU© H$am.

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OrdoMr bm§~r dVw©i‘Ü¶mnmgyZMo b§~m§Va

1. DC = OQ =

2. EF = OR =

3. AB = OP =

4. XY =

darb V³Ë¶mVrb ‘m{hVrdê$Z Ago {gÕ hmoVo H$s,

(a) Oer OrdoMr bm§~r dmT>Vo Vgo dVw©i‘Ü¶mnmgyZMo A§Va..............

(b) Oer OrdoMr bm§~r H$‘r Vgo dVw©i‘Ü¶mnmgyZMo A§Va..............

(c) dVw©imVrb gdm©V ‘moR>r Ordm Am{U dVw©i‘Ü¶ ¶m‘Yrb A§Va..............

åhUyZ Amåhr Agm {ZîH$f© H$mT>Vmo H$s,

Oa OrdoMr bm§~r OgOer OmñV hmoVo Vgo dVw©i‘Ü¶mnmgyZMo A§Va H$‘r hmoVo.

Oa OrdoMr bm§~r OgOer H$‘r hmoVo Vgo dVw©i‘Ü¶mnmgyZMo A§Va OmñV hmooVo.

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dVw©im§Mo I§S> (SEGMENT OF CIRCLES)

Imbrb AmH¥$VtMo {Z[ajU H$am Am{U Ë¶mImbrb àým§Mr CÎma Úm.

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SAmH¥$Vr -1 AmH¥$Vr -2

1. Am.Z§. 1 ‘Ü¶o … bKwH§$g d {dembH§$g AmoiIm d Zm§do gm§Jm.

2. Am.Z§. 2 ‘Ü¶o … PRQ H$moU Amho?

3. darb AmH¥$Vr‘Ü¶o bKwdVw©iI§S>, {demb dVw©iI§S> d AY©dVw©i I§S>m§Mr Zm§do {bhm.

dVw©iI§S>mVrb H$moZ (Angles in a segment)

~mOyÀ¶m AmH¥$VrMo {ZarjU H$am.

'O' dVw©i‘Ü¶ Agboë¶m dVw©imV 'XY' hr Ordm. XZYX hm bKwdVw©i I§S>Am{U XAYX hm {dembdVw©i I§S> Amho.

XPYÐ Am{U XQYÐ ¶mMo {ZarjU H$am.

Voìhm Ago gm§JVm ¶oB©b H$s, XPYÐ hm AYX {dembdVw©i I§S>mVrb H$moZd XQYÐ hm XZYX bKwdVw©i I§S>mVrb H$moZ AmhoV.

¶mdê$Z dVw©iI§S>mVrb H$moZm§Mo JwUY‘© {gÕ H$aVm ¶oVrb.

H¥$Vr 1 … Imbrb AmH¥$Ë¶m§‘Yrb bKwdVw©i I§S>mVrb H$moZm§Mo {ZarjU H$am, H$moZ ‘moOm d H$moZm§Mr ‘mno ImbrbaH$mÝ¶mV ^am.

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230 UNIT-10

AmH¥$Vr bKwdVw©i I§S>mVrb H$moZ H$moZmMm àH$ma

1. Ð =ADC

2. Ð =ABC

darb ‘mnmdê$Z Ago {gÕ hmoVo H$s,

H$moUË¶mhr dVw©im‘Ü¶o bKwdVw©i I§S>mVrb H$moZ {dembH$moZ AgVmo.

H¥$Vr 2 … Imbrb AmH¥$Ë¶m§Mo {ZarjU H$am, H$moZ ‘moOm Am{U H$moZm§Mr ‘mno aH$mÝ¶mV ^am.

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AmH¥$Vr 1 AmH¥$Vr2

AmH¥$Vr {dembdVw©i I§S>mVrb H$moZ H$moZmMm àH$ma

1. RPQ =

2. LKN =


H$moUË¶mhr dVw©im‘Ü¶o {dembdVw©i I§S>mVrb H$moZ bKwH$moZ AgVmo.

H¥$Vr 3 … Imbrb AmH¥$Ë¶m§Mo {ZarjU H$am, H$moZ ‘moOm Am{U H$moZm§Mr ‘mno aH$mÝ¶mV ^am.

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AmH¥$Vr -1 AmH¥$Vr -2

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AmH¥$Vr AY© dVw©iI§S>mVrb H$moZ H$moZmMm àH$ma

1 XYZÐ =

2 PSQÐ =


AY© dVw©iI§S>mVrb H$moZ H$mQ>H$moZ AgVmo.

H¥$Vr 4 … H$moZ ‘moOm Am{U H$moZm§Mr ‘mno Imbr {Xboë¶m aH$mÝ¶mV ^am.

Imbr {Xboë¶m VrZ dVw©imMo {ZarjU H$am.

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HIG

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(i) (ii) (iii)

AmH¥$Vr EH$mM dVw©iI§S>mVrb H$moZ {ZîH$f©

i PQTÐ = PRTÐ = PSTÐ =

i i ACBÐ = ADBÐ = AEBÐ =

ii i XGYÐ = XHYÐ = XIYÐ =

darb V³Ë¶mdê$Z Ago {gÕ hmoVo H$s, EH$mM dVw©iI§S>mVrb H$moZ g‘mZ AgVmV.

* bKw dVw©iI§S>mVrb gd© {dembH$moZ g‘mZ AgVmV

* {demb dVw©iI§S>mVrb gd© bKwH$moZ g‘mZ AgVmV

* AY© dVw©iI§S>mVrb gd© H$mQ>H$moZ g‘mZ AgVmV

Agm {ZîH$f© {ZKVmo H$s,

EH$mM dVw©iI§S>mVrb gd© H$moZ Zoh‘r g‘mZ AgVmV.

EH$H|$Ðr¶ qH$dm g‘H|$Ðr¶ dVy©io (CONCENTRIC CIRCLES) :

Imbrb AmH¥$VrMo {ZarjU H$am d Imbrb àým§Mr CÎmao {bhm.


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232 UNIT-10

O A

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CD

C1

C2

C3

C4

1. C1, C2, C3 d C4 ¶m AmH¥$Ë¶m H$m¶ AmhoV?

2. 'O' dVw©iH|$Ðm~m~V Vw‘Mm {ZîH$f© H$m¶?

3. OA, OB, OC, OD ‘moOm? d Ë¶m g‘mZ AmhoV H$m?

darb àým§À¶m CÎmamdê$Z Ago {gÕ hmoVo H$s,-

doJdoJù¶m {ÌÁ¶m nU dVw©iH|$Ð EH$M AmhoV. Vr EH$H|$Ðr¶ qH$dm g‘H|$Ðr¶ dVw©io AmhoV.

{ÌÁ¶m {^Þ na§Vw g‘mZ H|$Ð AgUmè¶m dVw©imZm EH$H|$Ðr¶ qH$dm g‘H|$Ðr¶ dVw©io Ago åhUVmV.

EH$ê$n dVw©io (Congruent circles) :

Imbrb AmH¥$VtMo {ZarjU H$am. C1, C2 Am{U C3 ¶m dVw©im§À¶m {ÌÁ¶m ‘moOm, Ë¶m g‘mZ AmhoV, nU dVw©i H|$Ð‘mÌ {^Þ AmhoV.

A B

C1

C D

C2

E F

C3

{ÌÁ¶m g‘mZ na§Vw {^Þ H|$Ð AgUmè¶m dVy©im§Zm EH$ê$n dVy©i åhUVmV.

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ñdmÜ¶m¶ 10.2

1. 'O' dVw©i‘Ü¶ KodyZ AZwH«$‘o 1.5 g|.‘r., 2.5 g|.‘r. d 3.5 g|.‘r. {ÌÁ¶oMr 3 EH$H|$Ðr¶ dVw©io H$mT>m.

2. O1 d O2 dVw©i‘Ü¶ AgUmar Am{U 3 g|.‘r. g‘mZ {ÌÁ¶oMr dVw©io H$mT>m. XmoÝhr dVw©i‘Ü¶mVrb A§Va 5g|.‘r. Amho.

3. AB = 8 g|.‘r. aofmI§S> H$mTy>Z 'C' hm Ë¶mMm ‘Ü¶q~Xÿ ¿¶m. 2 g|.‘r. {ÌÁ¶m KodyZ AZwH«$‘o A, B Am{U Cho dVw©i‘Ü¶ AgUmar VrZ dVw©io H$mT>m. 'C' dVy©i‘Ü¶ H$ënyZ 4 g|.‘r.Mo AmUIr EH$ dVw©i H$mT>m. ¶m‘Ü¶oEH$H|$Ðr¶ d EH$ê$n dVw©io AmoiIm.

4. 4 g|.‘r. {ÌÁ¶oÀ¶m dVw©imV 6 g|.‘r. bm§~rMr Ordm H$mT>m. XmoÝhr dVw©iI§S>mV 2 bKwH$moZ d 2 {dembH$moZaMm. Am{U {demb dVw©iI§S>mVrb H$moZ bKwH$moZ Am{U bKwdVy©i I§S>mVrb H$moZ {dembH$moZ AgVmo ho‘moOyZ nS>Vmim.

5. 3.5 g|.‘r. {ÌÁ¶oÀ¶m ‘O’ dVw©i‘Ü¶ AgUmè¶m dVw©imMm AOB hm ì¶mg H$mT>m. ì¶mgmÀ¶m XmoÝhr ~mOybmH$moZ Am§VarV H$am Am{U AY© dVw©iI§S>mVrb H$moZ H$mQ>H$moZ AgVmV ho ‘moOm Am{U nS>Vmim.

****

Circle - chord properties

Construction of chords ofgiven length

Verification ofproperties Concentric

CirclesCongruent

Circles

Chord propertyConstruction

acute anglesEqual chords areequidistant from

the centre Minor segment

Semi circle

obtuse angles

Equal

dVw©io - Á¶m- JwUY‘©

{Xboë¶m bm§~rMr Á¶m aMZo JwUY‘mªMm nS>Vmimg‘H|$Ð dVw©io

aMZmÁ¶m JwUY‘©

g‘mZ bm§~rÀ¶m Á¶mdVw©i ‘Ü¶mnmgyZ g‘mZ

A§Vamda AgVmV

{demb I§S>

bKw I§S>

I§S>mVrb H$moZ

AY© dVw©i

g‘mZ

H$mQ>H$moZ

{demb H$moZ

bKwH$moZ

EH$ê$n dVw©io

g‘mZ dVw©iI§S>

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234 UNIT-10

ñn{e©Ho$Mo JwUY‘©

‘mJrb àH$aUm‘Ü¶o Vwåhr dVw©imÀ¶m OrdoÀ¶m JwUY‘m©{df¶r Aä¶mg Ho$bobm AmhmV. ¶m àH$aUm‘Ü¶o AmnUdVw©imÀ¶m H$m§hr A{YH$ JwUY‘mªMm Aä¶mg H$ê$.

Oa dVw©i Am{U gaiaofm EH$mM nmVirV H$mT>boë¶m AgVrb Va Imbr Xe©{dë¶mà‘mUo VrZ e³¶VmAmT>iVmV.


N>o{XH$m : dVw©imbm XmoZ doJdoJù¶m q~XÿV N>oXUmè¶m gaiaofobm N>o{XH$m Ago åhUVmV.

ñn{e©H$m : dVw©imbm ’$º$ EH$mM q~XÿV ñne© H$aUmè¶m gaiaofobm ñn{e©H$m Ago åhUVmV.

Imbrb AmH¥$Ë¶m§Mo {ZarjU H$am. dVw©imbm ’$º$ EH$mM q~XÿV ñn{e©H$m ñne© H$aVo.

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'O' ‘Ü¶ Agboë¶m dVw©imZm AB , CD , EF Am{U GH ¶m H$mT>boë¶m ñn{e©H$m AmhoV. P, Q, R

Am{U S ho ñne©q~Xÿ qH$dm ñn{e©Ho$Mo q~Xÿ AmhoV.

ñne©q~Xÿ : Á¶m q~XÿV ñn{e©H$m dVw©imbm ñne© H$aVo Ë¶mbm ñne©q~Xÿ Ago åhUVmV.

* RQ gaiaofm dVw©imbm ñne© H$aVZmhr.

* RQ hr dVw©imbm Z N>oXUmar aofmAmho.

* RQ gaiaofm dVw©imbm A Am{UB ‘Ü¶o N>oXVo.

* RQ bm dVw©imMr N>o{XH$m AgoåhUVmV.

* RQ gaiaofm hr dVw©imbm P ¶mEH$ Am{U ’$º$ EH$mM q~XÿV ñne©H$aVo.

* RQ bm dVw©imMr P Vrb ñn{e©H$m

Ago åhUVmV.

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Imbrb CXmhaUmMo {ZarjU H$am. dVw©io Am{U Ë¶m§À¶m ñn{e©H$m Xe©{dUmè¶m X¡Z§{XZOrdZmVrb dñVy {Xboë¶m AmhoV.

CXmhaU 1 : ¶oWo H$ßnr (pulley) dVw©i Xe©{dVo.AB Am{U CD ¶m Vmam H$ßnrÀ¶m ñn{e©H$m Xe©{dVmV.

CXmhaU 2 : O{‘Zrdarb EHo$ar gm¶H$b MmH$mMo {ZarjU H$am. gm¶H$brMo MmH$ hodVw©i Xe©{dVo Am{U O{‘Zrdarb gm¶H$brMm {’$aÊ¶mMm ‘mJ© ho MmH$mMr ñn{e©H$m Xe©{dVo.

ñn{e©Ho$Mo JwUY‘© :AmVm AmnU dVw©imÀ¶m ‘hËdmÀ¶m ñn{e©Ho$À¶m JwUY‘m©~Ôb {eHy$.

H¥$Vr 1 : {Xboë¶m AmH¥$VtMo {ZarjU H$am Am{U Imbrb gmaUr nyU© H$am.


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1. {ÌÁ¶m

2. ñn{e©H$m

3. ñne©q~Xÿ

4.{ÌÁ¶m Am{U ñn{e©H$m ¶m‘Yrb H$moZ(H$moZ ‘mnH$mÀ¶m ghmæ¶mZo H$moZ ‘moOm.)

darb {ZarjUmdê$Z Amåhr Agm {ZîH$f© H$mT>Vmo H$s,

H$moUË¶mhr dVw©imV ñne©q~XþVyZ H$mT>bobr {ÌÁ¶m ñn{e©Ho$bm b§~ AgVo.

H¥$Vr 2 … Imbrb AmH¥$VtMo {ZarjU H$am. H$moZ ‘moOm Am{U {Xboë¶m gmaUrV {bhm.

A D

CB


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236 UNIT-10

Imbrb àým§Mr CÎmao gm§Jm …

1. H$moUË¶m AmH¥$VrVrb OP {ÌÁ¶m hr AB bm b§~Amho?

2. {Vgè¶m AmH¥$VrVrb AB bm H$m¶ åhUVmV?

3. H$moUË¶m AmH¥$VrVrb AB ñn{e©H$m Amho? H$m?

H¥$Vr 2 dê$Z Amåhr Agm {ZîH$f© H$mT>Vmo H$s dVw©im‘Ü¶o {ÌÁ¶oÀ¶m H|$Ð Zgboë¶m A§Ë¶ q~XÿVH$mT>bobm b§~ hr dVw©imMr ñn{e©H$m AgVo.

ho ¶m nwdu Z‘yX Ho$boë¶m ñn{e©Ho$À¶m JwUY‘m©À¶m ì¶Ë¶mgmMo {dYmZ Amho.

dVw©imdarb (narKmdarb) q~XþVyZ ñn{e©H$m aMUo.

CXmhaU 1 … 3 g|.‘r. {ÌÁ¶oÀ¶m dVw©imdarb P bm H$moUË¶mhr q~XþVyZ ñn{e©H$m H$mT>m.

nm¶ar 1 … 3 g|.‘r. {ÌÁ¶oMo 'O' ‘Ü¶AgUmao dVw©i H$mT>m. dVw©imÀ¶m narKmdaq~Xÿ P Mr IwU H$am. OP gm§Ym.

nm¶ar 2 … {ÌÁ¶oÀ¶m g‘mZ AgUmamOP H§$g H$mT>bm AgVm OP bm O Am{UdVw©imbm A ‘Ü¶o N>oXVmo.

nm¶ar 3 … H§$gmda B hr IwUAer H$am H$s OA = ABhmoB©b.Ë¶M {ÌOoZo ‘mZyZ XmoZ N>oXUmaoMmn H$mT>.

nm¶ar 4 … PC gm §Ym Am {UAmdí¶H$ ñn{e©H$m {‘i{dÊ¶mgmR>rdmT>dm.

H$moZ AmH¥$Vr 1 AmH¥$Vr 2 AmH¥$Vr 3

1. OPA ........... ........... ........... 2. OPB ........... ........... ...........

g |. ‘ r§.3O P

A

g |. ‘ r§.3O P

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CXmhaU 2 … 2 g|.‘r. {ÌÁ¶oMo dVw©i H$mT>m Am{U XmoZ {ÌÁ¶o§À¶m dVw©i‘Ü¶ Zgboë¶m A§Ë¶q~XÿVyZñn{e©H$m§À¶m OmoS>rMr Aer aMZm H$am H$s XmoÝhr {ÌÁ¶m§‘Yrb H$moZ 100° hmoB©b. ñn{e©H$m§‘Yrb H$moZ ‘moOm.

{Xbobo … dVw©imMr {ÌÁ¶m = r = 2 g|.‘r. ñn{e©H$m‘Yrb H$moZ = 100°

nm¶ar 1 … 2 g|.‘r. {ÌÁ¶oMo O ‘Ü¶o nm¶ar 2 … P ‘YyZ ñn{e©H$m H$mT>m.AgUmao dVw©i H$mT>m. Ë¶mdaP Mr IwU H$am Am{U OP gm§Ym.

nm¶ar 3 … O ‘Ü¶o POQ =100° H$mT>m. nm¶ar 4 … 'Q' ‘Ü¶o Xþgar ñn{e©H$m H$mT>m.ñn{e©Ho$‘Yrb H$moZ ‘moOm.

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ñn{e©H$o‘Yrb H$moZ 80º Amho.

{ZarjU H$am H$s dVw©im‘Ü¶o {ÌÁ¶m§‘Yrb Am{U ñn{e©H$m§‘Yrb H$moZ nyaH$ AgVmV.

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CXmhaU 3 … 2.5 g|.‘r. {ÌÁ¶oÀ¶m dVw©imV 4 g|.‘r. bm§~rMr Ordm H$mT>m. OrdoÀ¶m A§Ë¶ q~XyVyZ ñn{e©H$m H$mT>m.

{Xbobo … r = 2.5 g|.‘r., OrdoMr bm§~r = 4 g|.‘r.

nm¶ar 1 … 2.5 g|.‘r. {ÌÁ¶oMo nm¶ar 2 … 4 g|.‘r. bm§~rMr AB Ordm H$mT>m.'O' ‘Ü¶ AgUmao dVw©i H$mT>m.

nm¶ar 3 … OA Am{U OB.gm§Km nm¶ar 4 … A Am{U B ‘YyZ dVw©imbmñn{e©H$m H$mT>m.

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Cn{gÕm§V …

1. ñne© q~XþVyZ ñn{e©Ho$bm H$mT>bobm b§~ hm dVw©i‘Ü¶mVyZ OmVmo.

2. dVw©imdarb H$moUË¶mhr q~XþVyZ ’$º$ EH$M ñn{e©H$m H$mT>Vm ¶oVo.

3. ì¶mgmÀ¶m A§Ë¶q~XþVyZ H$mT>boë¶m ñn{e©H$m EH$‘oH$m§Zm g‘m§Va AgVmV.

darb VrZ {dYmZm§Mr dJm©‘Ü¶o gH$maU MMm© H$am Am{U ñnï>rH$aU Úm.

ñdmÜ¶m¶ 10.3

1. 4 g|.‘r. {ÌÁ¶oMo dVw©i H$mT>m Am{U dVw©imdarb H$moUË¶mhr q~XþVyZ ñn{e©H$mH$mT>m.

2. 7 g|.‘r. ì¶mgmMo dVw©i H$mT>m Am{U ì¶mgmÀ¶m A§Ë¶ q~XþVyZ ñn{e©H$m H$mT>m.

3. 3.5 g|.‘r. {ÌÁ¶oÀ¶m dVw©imV nañnam§er b§~ AgUmao XmoZ ì¶mg H$mT>m. ì¶mgm§À¶m A§Ë¶ q~XwVyZ ñn{e©H$mH$mT>m.

4. 4.5 g|.‘r. {ÌÁ¶oÀ¶m dVw©imV XmoZ {ÌÁ¶m Aem H$mT>m H$s, Ë¶m‘Yrb H$moZ 70° Mm hmoB©b. {ÌÁ¶oÀ¶m dVw©iH|$Ð Zgboë¶m A§Ë¶ q~XþVyZ ñn{e©H$m H$mT>m.

5. 3 g|.‘r. {ÌÁ¶oMo dVw ©i H$mT> m Am{U Ë¶mbm Xm oZ ñn{e©H$m Aem H$mT>m H$s, Ë¶m‘Yrb H$m oZ40° Mm hmoB©b.

6. 4.5 g|.‘r {ÌÁ¶oMo dVw©i H$mT>m Am{U Ë¶mV 7 g|.‘r. bm§~rMr PQ Ordm H$mT>m. P ‘YyZ ñn{e©H$m H$mT>m.

7. 5 g|.‘r {ÌÁ¶oÀ¶m dVw©imV 8 g|.‘r. bm§~rMr Ordm H$mT>m. OrdoÀ¶m A§Ë¶ q~XþVyZ ñn{e©H$m H$mT>m.

8. 4 g|.‘r. {ÌÁ¶oMo dVw©i H$mT>m Am{U Ë¶mV 6 g|.‘r. bm§~rMr Ordm H$mT>m. dVw©i‘Ü¶mVyZ Ordobm b§~AgUmar {ÌÁ¶m H$mT>m. OrdoÀ¶m Am{U {ÌÁ¶oÀ¶m A§Ë¶ q~XþVyZ ñn{e©H$m H$mT>m.

9. 3.5 g|.‘r. {ÌÁ¶oMo dVw ©i H$mT>m Am{U Ë¶mM H§$gmZo Am§VarV Ho$bobm 80° Mm H|$ÐñW H$moZ H$mT>m.dVw©imdarb q~XþVyZ ñn{e©H$m H$mT>m. ñn{e©H$m EH$‘oH$m§Zm N>oXVrb Aem[aVrZo nwT>o dmT>dm. Vwåhr H$m¶{ZarjU H$amb?

10. 4.5 g|.‘r. {ÌÁ¶oÀ¶m dVw©imV 5 g|.‘r. bm§~rÀ¶m XmoZ g‘mZ Ordm dVw©i‘Ü¶mÀ¶m XmoÝhr ~mOyg H$mT>m.OrdoÀ¶m A§Ë¶q~XþVyZ ñn{e©H$m H$mT>m.

doJdoJù¶m pñWVrVrb q~XþVyZ dVw©imbm ñn{e©H$m H$mT>Uo …1. dVw©imdarb q~Xy …

H¥$Vr 1 … 3 g|.‘r. {ÌÁ¶oMo dVw©i H$mT>m Am{U Ë¶mda q~Xÿ P hr IwU

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H$am. P ‘YyZ dVw©imbm ñn{e©H$m H$mT>m. P ‘YyZ Xþgar ñn{e©H$mH$mT>Ê¶mMm à¶ËZ H$am. ho e³¶ Amho? H$m? dVw©imdarb H$moUË¶mhrq~XþVyZ dVw©imbm ’$º$ EH$M ñn{e©H$m H$mT>Vm ¶oVo ho Vwåhr ¶mnyduAä¶mgbm AmhmV.

2. dVw©im‘Yrb q~Xÿ …

H¥$Vr 2 … 3 g|.‘r. {ÌÁ¶oMo Xþgao EH$ dVw©i H$mT>m. dVw©im‘Ü¶o q~Xÿ Phr IwU H$am. ¶m q~XþVyZ dVw©imbm ñn{e©H$m H$mT>Uo e³¶ Amho? Vwåhmbm Ago {XgyZ ¶oVo H$s øm q~XþVyZ H$mT>bobr àË¶oH$ aofmdVw©imbm XmoZ doJdoJù¶m q~XÿV N>oXVo. øm gd© gaiaofm dVw©imÀ¶mN>o{XH$m AmhoV.

3. dVw©im~mhoarb q~Xÿ …

H¥$Vr 3 … 3 g|.‘r. {ÌÁ¶oMo dVw©i H$mT>m. dVw©im~mhoaq~Xÿ P Mr IwU H$am. øm q~XþVyZ dVw©imH$S>o {H$aUH$mT>m. Ë¶m§Mo {ZarjU H$am. P ‘YyZ OmUmè¶m {H$VrN>o{XH$m dVw©imbm H$mT>Vm ¶oVmV?

P ¶m ~mø q~XþVyZ {H$Vr ñn{e©H$m dVw©imbm H$mT>Vm ¶oVmV?

AmH¥$Vrdê$Z Vwåhmbm Ago {XgyZ ¶oVo H$s ’$º$ PQAm{U PR ho dVw©imbm ñne© H$aVmV. dVw©imbm P ‘YyZH$mT>boë¶m øm ñn{e©H$m AmhoV.

~møq~XþVyZ dVw©imbm ’$º$ XmoZ ñn{e©H$m H$mT>Vm ¶oVmV.

~mø q~XþVyZ dVw©imbm H$mT>boë¶m XmoZ ñn{e©Ho$Zm Img AgoJwUY‘© AgVmV. Imbrb H¥$VrÛmao Ë¶m JwUY‘m©Zm Vwåhr emoYy eH$mb.

Imbr [Xboë¶m AmH¥$VtMo {ZarjU H$am Am{U Imbrb àým§MrCÎmao gm§Jm. (Úm)

(a) 'P' ¶m ~møq~XþVyZ dVw©imbm H$mT>boë¶m ñn{e©Ho$Mr Zm§do {bhm.

(b) PA Am{U PB ‘moOm. Vw‘Mo AZw‘mZ H$m¶ Amho?

(c) H$moZ ‘mnH$mMm Cn¶moJ H$ê$Z AOP Am{U BOP ‘moOm.

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(d) APO Am{U BPO ‘moOm. H$moUË¶m {R>H$mUr H$moZ V¶ma Pmbobo AmhoV?{Xboë¶m gmaUr‘Ü¶o ‘moO‘mnm§Mr Zm|X H$am.

AmH¥$Vr I II III

1. PA = ....... AOP = ....... AOP = ......

PB = ....... BOP = ....... BOP = ......

2. PA = ....... AOP = ....... AOP = ......

PB = ....... BOP = ....... BOP = ......

darb ‘m{hVrÀ¶m {ZarjUmZ§Va Imbrb {dYmZo nyU© H$am.

1. ~mø q~XþVyZ dVw©imbm H$mT>boë¶m ñn{e©H$m ........ g‘mZ AgVmV.

2. ~mø q~XþVyZ dVw©imbm H$mT>boë¶m ñn{e©H$m dVw©i H|$Ðmer ........ H$moZ Am§VarV H$aVmV.

3. dVw©imbm ~møq~XþVyZ H$mT>boë¶m ñn{e©H$m dVw©i‘Ü¶ Am{U ~møq~Xÿ ¶m§Zm gm§YUmè¶m aofoer ........ H$moZAm§VarV H$aVmV.

darb H¥$Vrdê$Z Vwåhr Agm {ZîH$f© H$mT>mb H$s, dVw©imbm ~møq~XþVyZ H$mT>boë¶m ñn{e©H$m

(a) g‘mZ AgVmV.

(b) dVw©i‘Ü¶mer g‘mZ H$moZ Am§VarV H$aVmV.

(c) dVw©i‘Ü¶ Am{U ~mø [~§Xÿ ¶m§Zm gmYUmè¶m aofoer g‘mZ H$moZ Am§VarV H$aVmV.

dVw©imbm ~mø q~XþVyZ H$mT>boë¶m ñn{e©Ho$Mo JwUY‘© Amåhr àm¶mo{JH$ aMZm Am{U ‘moO‘mnmZo Aä¶mgbmoAmhmoV. AmVm AmnU øm JwUY‘m©À¶m Vm{H©$H$ {gÕVoMr MMm© H$ê$. nU Vm{H©$H$ {gÕVobm OmÊ¶mnydu Imbrbàým§Mr CÎmao {bhm. dJm©‘Ü¶o MMm© H$am. ¶m§Mr Vwåhmbm à‘o¶ {gÕ H$aÊ¶mg ‘XV hmoB©b.

1. OPB Am{U OQB ho H$moUË¶m àH$maMo H$moZ AmhoV?

2. dVw©imMm {dMma Ho$ë¶mg OP Am{U OQ ho H$m¶ AmhoV?

3. OPB Am{U OQB ho H$moUË¶m àH$maMo {ÌH$moU AmhoV?

4. OPB Am{U OQB Mm {dMma Ho$ë¶mg OB hr H$m¶ Amho?

5. OPB Am{U OQB ho EH$‘oH$m§er EH$ê$n AmhoV?

6. H$moUË¶m J¥hrVH$mZwgma OPB Am{U OQB ho EH$ê$n AmhoV?

AmVm AmnU Vm{H©$H$ ÑîQ>çm à‘o¶ {gÕ H$ê$.

à‘o¶ … dVw©imbm ~møq~XþVyZ H$mT>boë¶m ñn{e©H$m.

(a) g‘mZ AgVmV.

(b) dVw©i‘Ü¶mer g‘mZ H$moZ Am§VarV H$aVmV.

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(c) dVw©i‘Ü¶ Am{U ~møq~Xÿ gm§YUmè¶m aofoer g‘mZ H$moZ Am§VarV H$aVmV.

J¥hrV : A hm dVw©imMm ‘Ü¶ Amho, B hm ~møq~Xÿ Amho.

BP Am{U BQ øm ñn{e©H$m AmhoV.

AP , AQ Am{U AB OmoS>bobo AmhoV.

gmÜ¶ : (a) BP = BQ

(b) PAB = QAB(c) PBA = QBA

{gÕVm : {dYmZ H$maU

APB Am{U AQB ‘Ü¶o

AP = AQ (EH$mM dVw©imÀ¶m {ÌÁ¶m)

APB = AQB = 90° (ñne© q~XþVyZ dVw©imbm H$mT>bobr

{ÌÁ¶m ñn{e©Ho$bm b§~ AgVo.)

H$U© AB = H$U© AB (g‘mB©H$ ~mOy)

APB AQB (H$m H$ ~m à‘o¶) (R.H.S) (a) PB = QB CPCT

(b) PAB = QAB

(c) PBA = QBA


CXmhaU 1 … AmH¥$Vr‘Ü¶o ABCD Mm¡H$moZmV dVw©i Am§Va{bIrV Ho$bobo Amho. B = 90°. Oa AD = 23 g|.‘r,AB = 29 g|.‘r. Am{U DS = 5 g|.‘r. Va dVw©imMr {ÌÁ¶m H$mT>m.

CH$b…AmH¥$Vr‘Ü¶o AB, BC, CD Am{U DA øm dVw©imbm AZwH«$‘o Q, P, S Am{U R ‘YyZ H$mT>boë¶mñn{e©H$m AmhoV.

DS = DR dVw©imbm 'D' ¶m ~møq~XþVyZ H$mT>boë¶m ñn{e©H$m AmhoV.

na§Vw DS = 5 g|.‘r. ({Xbobo)

DR = 5 g|.‘r.

AmH¥$Vr‘Ü¶o AD = 23 g|.‘r. ({Xbobo)

AR = AD – DR = 23 – 5 = 18 g|.‘r.

na§Vw AR = AQ (dVw©imbm A ¶m ~møq~XþVyZH$mT>boë¶m ñn{e©H$m AmhoV.)

AQ = 18 g|.‘r.

Oa AQ = 18 g|.‘r. Va ( AB = 29 g|.‘r. {Xbobo Amho.)

BQ = AB – AQ = 29 – 18 = 11 g|.‘r.

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BQOP Mm¡H$moZm‘Ü¶o

BQ = BP (B ¶m ~møq~XþVyZ H$mT>boë¶m ñn{e©H$m)

OQ = OP (EH$mM dVw©imÀ¶m {ÌÁ¶m)

QBP = QOP = 90° {Xbobo

OQB = OPB = 90° (ñn{e©Ho$Mm ñne© q~Xÿ Am{U {ÌÁ¶m ¶m‘Yrb H$moZ)

BQOP hm Mm¡ag Amho.

dVw©imMr {ÌÁ¶m = OQ = 11 g|.‘r.

CXmhaU 2 … AmH¥$Vr‘Ü¶o P ¶m ~møq~XþVyZ H$mT>boë¶m AP Am{U BP øm ñn{e©H$m AmhoV. AOB Am{UAPB nyaH$ H$moZ AmhoV ho {gÕ H$am.

CH$b … J¥hrV … 'O' hm dVw©imMm ‘Ü¶ Amho. 'P' ¶m ~møq~XþVyZ H$mT>boë¶m AP Am{U BP øm ñn{e©H$m AmhoV.

gmÜ¶ … AOB + APB = 180°

{gÕVm … Mm¡H$moZm‘Ü¶o AOBP

AOB + OBP + BPA + PAO = 360°

nU OBP = 900 Am{U OAP= 90°

AOB + 900 + BPA + 900 = 360°

AOB + BPA + 1800 = 360°

AOB + BPA = 360° – 180°

AOB + BPA = 1800

AOB Am{U BPA ho nyaH$ H$moZ AmhoV.

CXmhaU 3 …AmH¥$Vr‘Ü¶o 'O' ‘Ü¶ Agboë¶m dV©wimbm PQ Am{U PR ¶m ñn{e©H$m AmhoV.

Oa P =45

O Va O Am{U P H$mT>m.

CH$b … AmH¥$Vr‘Ü¶oQPR + QOR = 180°

( {dê$Õ H$moZ nyaH$ AgVmV.)

na§Vw, QPR =45

QOR

45

QOR + QOR = 180°

4 QOR + 5 QOR5

= 180°

9 QOR = 5 × 180°

QOR=5 180

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9 90 = 100°

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Oa QOR = 100° Va QPR = 180° – QOR = 180° – 100° = 80°

QPR = 80°

CXmhaU 4 … AmH¥$Vr‘Ü¶o 'O' hm dVw©imMm ‘Ü¶ Amho. B Am{U D ‘Yrb ñn{e©H$m EH$‘oH$m§Zm P q~XÿV N>oXVmV.Oa AB hr CD bm g‘m§Va Amho Am{U ABC = 55° Va (i) BOD (ii) BPD H$mT>m.

CH$b … AmH¥$Vr‘Ü¶o AB || CD {Xbobo

ABC = BCD ( ì¶wËH«$‘ H$moZ

na§Vw ABC = 55° ( {Xbobo)

BCD = 55°

AmH¥$Vr‘Ü¶o BOD = 2 BCD (H|$ÐñW H$moZ hm n[aKñW H$moZmÀ¶mXþßnQ> Amho.

BOD = 2 × 55° = 110°.

na§Vw BOD + BPD = 180°

BPD = 180° – BOD = 180° – 110° = 70

CXmhaU 5 … AmH¥$Vr‘Ü¶o ABC Mr n[a{‘Vr = 2 (AP + BQ + CR) AgoXmIdm.

CH$b … ABC Mr n[a{‘Vr = AB + BC + AC

= AP + PB + BQ + QC + CR + RA

nU AP = AR (dVw©imbm A ‘YyZ H$mT>boë¶m ñn{e©H$m)

PB = BQ (dVw©imbm B ‘YyZ H$mT>boë¶m ñn{e©H$m)

CQ = CR (dVw©imbm C ‘YyZ H$mT>boë¶m ñn{e©H$m)

ABC Mr n[a{‘Vr = AP + BQ + BQ + CR + CR + AP

= 2AP + 2BQ + 2CR = 2(AP + BQ + CR)

ñdmÜ¶m¶ 10.4

A. ñn{e©Ho$À¶m JwUY‘m©da AmYm[aV g§»¶mË‘H$ g‘ñ¶m gmoS>dm.

1. AmH¥$Vr‘Ü¶o PQ, PR Am{U BC øm dVw©imÀ¶m ñn{e©H$m AmhoV. BCñn{e©H$m dVw©imbm 'X' ‘Ü¶o ñne© H$aVo. Oa PQ = 7 g|.‘r. VaPBC Mr n[a{‘Vr H$mT>m.

2. 13 g|.‘r. Am{U 5 g|.‘r. {ÌÁ¶oMr XmoZ g‘H|$Ðr¶ dVw©io H$mT>m.AmVrb dVw©imbm ñne© H$aUmè¶m ~mhoarb dVw©imÀ¶m OrdoMr bm§~rH$mT>m.

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3. ABC ‘Ü¶o AB = 12 g|.‘r., BC = 8 g|.‘r. Am{U AC = 10 g|.‘r.Va AF, BD Am{U CE H$mT>m.

4. ABCD Mm ¡H$moZm‘Ü¶o BC = 38 g|.‘r., QB = 27 g|.‘r.DC = 25 g|.‘r. Am{U AD DC Va dVw©imMr {ÌÁ¶m H$mT>m.

5. {Xboë¶m AmH¥$Vr‘Ü¶o AB = BC, ABC = 680. O ‘Ü¶oAgboë¶m dVw©imÀ¶m DA Am{U DB ¶m ñn{e©H$m AmhoV. Va(i) ACB (ii) AOB Am{U (iii) ADB Mr ‘mno H$mT>m.

B. ñn{e©H$m§À¶m JwUY‘m©da AmYm[aV Cnà‘o¶

1. ABCD Mm¡H$moZmV dVw©i Am§Va{bIrV Ho$bobo Amho Va {gÕ H$am H$sAB + CD = AD + BC.

2. 'O' ‘Ü¶ Agboë¶m dVw©imbm AP Am{U AQ øm ñn{e©H$m A ¶m ~møq~XþVyZ H$mT>boë¶m AmhoV. Va {gÕH$am H$s PAQ = 2. OPQ.

3. AmH¥$Vr‘Ü¶o XmoZ dVw©io EH$‘oH$m§Zm 'P' ‘Ü¶o ~mø ñne©H$aVmV. øm dVw©imMr AB hr g‘mB©H$ ~mø ñn{e©H$m Amho.Va {gÕ H$am H$s(a) P ‘Yrb ñn{e©H$m AB bm Q ‘Ü¶o Xþ mJVo.(b) APB = 90°.

4. b§~ ñn{e©Ho$Mr EH$ OmoS>r dVw©imbm ~mø q~XþVyZ H$mT>bobr Amho Va {gÕ H$am H$s àË¶oH$ ñn{e©Ho$Mr bm§~rhr dVw©imÀ¶m {ÌÁ¶oBVH$s AgVo.

5. Oa g‘m§Va wO Mm¡H$moZmÀ¶m ~mOy dVw©imbm ñne© H$aVmV Va {gÕ H$am H$s g‘m§Va^wO Mm¡H$moZ hm g‘ wOMm¡H$moZ Amho.

6. AmH¥$Vr‘Ü¶o Oa AB = AC Va {gÕ H$am H$s BQ = QC.

dVw©imbm ~mø q~XþVyZ ñn{e©H$m H$mT>Uo.

CXmhaU … 1.5 g|.‘r. {ÌÁ¶oMo dVw©i H$mT>m. dVw©i‘Ü¶mnmgyZ 2.5 g|.‘r. darb ~møq~XþVyZ XmoZ ñn{e©H$m H$mT>m.ñn{e©Ho$Mr bm§~r ‘moOm Am{U nS>VmiyZ nhm.

{Xbobo … r = 1.5 g|.‘r. d = 2.5 g|.‘r.

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A BM

A BM

Q

P

A BM

Q

P

nm¶ar 1 … AB aofmI§S> 2 g|.‘r. AgUmam (BVH$m) nm¶ar 2 … AB Mm b§~ Xþ^mOH$ H$mT>m Am{UAB Mm ‘Ü¶q~Xÿ M Aer IwU H$am.H$mT>m. A ‘Ü¶ ‘mZyZ 1.5 g|.‘r. {ÌÁ¶oMo dVw©i H$mT>m.

A B

nm¶ar 3 … MA qH$dm MB {ÌÁ¶oMo AJmoXa H$mT>boë¶m

dVw©imbm N>oXUmao dVw©i H$mT>m Am{U N>oXZ q~XÿÀ¶m

IwUm H$ê$Z P Am{U Q hr Zm§d o{bhm.

nm¶ar 4 … BP Am{U BQ gm§Ym. BP Am{U BQ ømAno{jV ñn{e©H$m AmhoV.

nm¶ar 5 …‘moO‘mnmZo ñn{e©Ho$Mr bm§~r BP = BQ =2g|.‘r. {heo~mZo nS>VmiUo.

t = 2 2d r = 2 22 5 6 25(( 1 25 25)) = 4 = 2 g|.‘r.

ñdmÜ¶m¶ 10.5

1. 6 g|.‘r. {ÌÁ¶oMo dVw©i H$mT>m Am{U dVw©i‘Ü¶mnmgyZ 10 g|.‘r. darb ~mø q~XþVyZ ñn{e©H$m H$mT>m.ñn{e©Ho$Mr bm§~r ‘moOm Am{U nS>Vmim.

2. 3.5 g|.‘r. {ÌÁ¶oÀ¶m dVw©imbm dVw©imnmgyZ (narKmnmgyZ) 3.5 g|.‘r. darb q~XþVyZ XmoZ ñn{e©H$m H$mT>m. 3. 5.5 g|.‘r. {ÌÁ¶oÀ¶m dVw©imbm dVw©imnmgyZ 3.5 g|.‘r. darb q~XþVyZ EH$ ñn{e©H$m H$mT>m.

4. 5 g|.‘r. bm§~rMr b§~ AgUmar ñn{e©Ho$Mr EH$ OmoS>r dVw©imbm H$mT>m.5. 2 g|.‘r. Am{U 4 g|.‘r. {ÌÁ¶o§À¶m XmoZ g‘H|$Ðr¶ dVw©imZm dVw©i‘Ü¶mnmgyZ 8 g|.‘r. A§Vamdarb q~XþVyZ

ñn{e©H$m H$mT>m.

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ñne© dVw©io … Imbr {Xboë¶m dVw©im§À¶m OmoS>rMo {ZarjU H$am.


A B C D E FP

A

B

AmH¥$Vr 1 ‘Ü¶o A Am{U B ‘Ü¶ AgUmar XmoZ dVw©io EH$‘oH$mnmgyZ Xÿada AmhoV. Ë¶m§Zm H$moUVohr g‘mB©H$q~Xÿ ZmhrV.

AmH¥$Vr 2 ‘Ü¶o C Am{U D ‘Ü¶ AgUmar XmoZ dVw©io A Am{U B ‘Ü¶o N>oXVmV. Ë¶m§Zm A Am{U B ho XmoZg‘mB©H$ q~Xÿ AmhoV.

AmH¥$Vr 3 ‘Ü¶o E Am{U F ‘Ü¶ AgUmè¶m XmoZ dVw©imMm g‘mB©H$ q~Xÿ P Amho. Ë¶mZm ñne© dVw©io AgoåhUVmV.ñne© dVw©io … XmoZ dVw©imZm ’$º$ EH$ g‘mB©H$ ñne©q~Xÿ Agob Va Ë¶m§Zm ñne© dVw©io Ago åhUVmV.


O O1 OC2

C1

PA BC2

C1

AmH¥$Vr 4 ‘Ü¶o O Am{U O1 ‘Ü¶ AgUmè¶m XmoZ dVw©imZm g‘mB©H$ q~Xÿ Zmhr. Ë¶m§Zm doJdoJio dVw©i‘Ü¶ AmhoV.

AmH¥$Vr 5 ‘Ü¶o C1 Am{U C2 ¶m XmoZ dVw©im§Zm g‘mB©H$ dVw©i H|$Ð Amho nU Ë¶mZm dVw©imda g‘mB©H$ q~Xÿ

Zmhr. Aem dVw©imZm g‘H|$Ðr¶ dVw©io Ago åhUVmV.

AmH¥$Vr 6 ‘Ü¶o A Am{U B ‘Ü¶ AgUmè¶m C1 Am{U C2 ¶m dVw©imZm g‘mB©H$ q~Xÿ Amho. hr gwÕm ñne© dVw©io AmhoV.

ñne© dVw©io … XmoZ dVw©imZm g‘mB©H$ ñne©q~Xÿ Agob Va Ë¶mZm ñne© dVw©io Ago åhUVmV.

g‘mB©H$ ñne© [~§Xÿ AgUmao EH$ dVw©i Xþgè¶m dVw©imÀ¶m ~mhoa Agob Va Ë¶m dVw©imZm ~mø ñne© H$aUmar

dVw©io Ago åhUVmV. (qH$dm) Oa ñne© dVw©im§Mo ‘Ü¶ ñne©q~XÿÀ¶m XmoÝhr ~mOyg AgVrb Va Ë¶mZm ~møñne©

H$aUmar dVw©io åhUVmV.

g‘mB©H$ ñne© q~Xÿ AgUmao EH$ dVw©i Xþgè¶m dVw©imÀ¶m AmV Agob Va Ë¶m dVw©imZm Am§Vañne© H$aUmar

dVw©io åhUVmV. (qH$dm) Oa ñne© dVw©im§Mo ‘Ü¶ ñn{e©q~XÿÀ¶m EH$mM ~mOyg AgVrb Va Ë¶m§§Zm Am§Vañne©

H$aUmar dVw©io åhUVmV.

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~mø ñne© dVw©io Am§Vañne© dVw©io

AmH¥$Vr 1 AmH¥$Vr 2

ñne© H$aUmè¶m dVw©im§À¶m dVw©i‘Ü¶mVrb A§Va Am{U Ë¶m§À¶m {ÌÁ¶m ¶m‘Yrb g§~§Y …H¥$Vr 1 : Imbrb AmH¥$VtMo {ZarjU H$am, ‘moOm Am{U Imbr {Xboë¶m gmaUr‘Ü¶o ‘m{hVr Z‘yX H$am.

A BP A BP

AmH¥$VrAP(R) BP(r) AB(d)AP, BP Am{U AB ¶m‘Yrb

g§~§Y åhUOo R, r Am{U d

1

2

darb {ZarjUmdê$Z Vwåhr Agm {ZîH$f© H$mT>mb H$s(i) Oa XmoZ dVw©io EH$‘oH$m§Zm ~mø ñne© H$aV AgVrb Va Ë¶m§À¶m dVw©i‘Ü¶mVrb A§Va ho {ÌÁ¶oÀ¶m

~oaOoBVHo$ AgVo. [d = R + r](ii) Oa XmoZ dVw©io EH$‘oH$m§Zm Am§Vañne© H$aV AgVrb Va Ë¶m§À¶m dVw©i‘Ü¶mVrb A§Va ho {ÌÁ¶oÀ¶m

dOm~mH$s BVHo$ AgVo. [d = R – r]H¥$Vr 2 … dVw©i‘Ü¶mVrb A§Va 8 g|.‘r. AgUmar 5 g|.‘r. Am{U 3 g|.‘r. {ÌÁ¶|Mr XmoZ dVw©io H$mT>m.

{ÌÁ¶|Mr ~oarO Am{U dVw©i‘Ü¶mVrb A§Va ¶m‘Ü¶o Vwåhmbm H$moUVm g§~§Y AmT>iVmo.H¥$Vr 3 … dVw©i‘Ü¶mVrb A§Va 2 g|.‘r. AgUmar 5 g|.‘r. Am{U 3 g|.‘r {ÌÁ¶|Mr XmoZ dVw©io H$mT>m.

{ÌÁ¶|Mr dOm~mH$s Am{U dVw©i‘Ü¶mVrb A§Va ¶m‘Ü¶o Vwåhmbm H$moUVm g§~§Y AmT>iVmo?

ñne© H$aUmè¶m dVw©im§Mo dVw©i‘Ü¶ Am{U ñne© q~Xÿer g§~§{YV JwUY‘© …

A BP A BP

A BP A B P

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darb ñne© H$aUmè¶m dVw©imÀ¶m g§X^m©dê$Z,

1. Zm§do qbhm (a) ñne© q~Xÿ (b) Ë¶m§Mo dVw©i‘Ü¶

2. AmH¥$Vr 1 ‘Yrb APB Am{U AmH¥$Vr 2 ‘Yrb ABP H$moUË¶m àH$maMm H$moZ Amho?

3. AmH¥$Vr 1 ‘Yrb A, P, B Am{U AmH¥$Vr 2 ‘Yrb A, B, P ho q~Xÿ H$go AmhoV? H$m?

darb H¥$Vrdê$Z Amåhr Agm {ZîH$f© H$mT>Vmo H$s,

Oa XmoZ dVw©io EH$‘oH$m§Zm ñne© H$aV AgVrb Va Ë¶m§Mo dVw©i‘Ü¶ Am{U ñne© q~Xÿ EH$aofr¶ AgVmV.

AmVm AmnU Vm{H©$H$ÑîQ>çm à‘o¶ {gÕ H$ê$.

à‘o¶ … Oa XmoZ dVw©io EH$‘oH$m§Zm ñne© H$aV AgVrb Va Ë¶m§Mo dVw©i‘Ü¶ Am{Uñne© q~Xÿ EH$aofr¶ AgVmV.

KQ>Zm 1 … Oa XmoZ dVw©io EH$‘oH$m§Zm ~møñne© H$aV AgVrb Va Ë¶m§Mo dVw©i‘Ü¶Am{U ñne© q~Xÿ EH$aofr¶ AgVmV.

J¥hrV … ñne© H$aUmè¶m dVw©im§Mo A Am{U B ho dVw©i‘Ü¶ AmhoV.P hm ñne©q~Xÿ Amho.

gmÜ¶ … A, P Am{U B ho EH$aofr¶ AmhoV.

aMZm … XPY ñn{e©H$m H$mT>m.


AmH¥$Vr‘Ü¶o

APX = 90° .....(1) ñne© q~XþVyZ H$mT>bobr {ÌÁ¶m ñn{e©Ho$bm b§~ AgVo.

BPX = 90° .....(2)

APX + BPX = 90° + 90° (1) Am{U (2) {‘i{dë¶mZo

APB = 180° APB hm gaiH$moZ Amho.

APB hr gaiaofm Amho.

A, P Am{U B ho EH$aofr¶ AmhoV.

KQ>Zm 2 … Oa XmoZ dVw©io EH$‘oH$m§Zm Am§Vañne© H$aV AgVrb Va Ë¶m§Mo dVw©i‘Ü¶ Am{Uñne©q~Xÿ EH$aofr¶ AgVmV.

J¥hrV … A Am{U B ho ñne© dVw©imMo dVw©iH|$Ð AmhoV P hm ñne©q~XÿAmho.

gmÜ¶ … A, B Am{U P ho EH$aofr¶ AmhoV.

aMZm … XPY ñn{e©H$m H$mT>m. AP Am{U BP gm§Ym.

{gÕVm …

AmH¥$Vr‘Ü¶o

X

Y

A BP

X

Y

A B P

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APX = 90°..... (1) ñne© q~XþVyZ H$mT>bobr {ÌÁ¶m ñn{e©Ho$bm b§~ AgVo.

BPX = 90° ..... (2) ñne© q~XþVyZ H$mT>bobr {ÌÁ¶m ñn{e©Ho$bm b§~ AgVo.

APX = BPX = 90°

AP Am{U BP ho EH$mM aofoda AmhoV. A, B Am{U P ho EH$aofr¶ AmhoV.


CXmhaU 1 … VrZ dVw©io EH$‘oH$m§Zm ~møñne© H$aVmV. Oa dVw©imMo dVw©i‘Ü¶gm§YyZ ~Zboë¶m {ÌH$moUmÀ¶m ~mOy AZwH«$‘o 10 g|.‘r., 14 g|.‘r. Am{U 16g|.‘r. AmhoV Va àË¶oH$ dVw©imMr {ÌÁ¶m H$mT>m.

CH$b … A, B Am{U C ‘Ü¶ AgUmar dVw©io AZwH«$‘o P, Q Am{U R ‘Ü¶oAmH¥$VrV Xe©{dë¶mà‘mUo EH$‘oH$m§Zm ~møñne© H$aVmV.

g‘Om AP = x PB = BQ = 10 – x

RC = CQ = 14 – x

na§Vw, CQ + BQ = 16

14 – x + 10 – x = 16

24 – 2x = 16

24 – 16 = 2x

2x = 8

x =8

2 = 4

CXmhaU 2 … AmH¥$Vr‘Ü¶o 9 g|.‘r. Am{U 2 g|.‘r. {ÌÁ¶oÀ¶m dVw©im§Mo AZwH«$‘o P Am{U Q ho dVw©i‘Ü¶ AmhoV.Oa PRQ = 90° Am{U PQ = 17 g|.‘r. Va R ‘Ü¶ AgUmè¶m dVw©imMr {ÌÁ¶m H$mT>m.

CH$b … R ‘Ü¶ AgUmè¶m dVw©imMr {ÌÁ¶m = x EH$H$ ‘mZy.

PQR ‘Ü¶o PQ = 17 g|.‘r.

PR = (x + 9) g|.‘r.

QR = (x + 2) g|.‘r. Am{U PRQ = 90°

PQ2 = PR2 + QR2 (nm¶W°Jmoag à‘o¶) 172 = (x + 9)2 + (x + 2)2

289 = x2 + 81 + 18x + x2 + 4 + 4x2x2 – 22x + 85 – 289 = 0

2x2 + 22x – 204 = 0 2

AR = AP = 4 g|.‘r A ‘Ü¶ Agboë¶m dVw©imMr {ÌÁ¶m

BQ = 10 – 4 = 6 g|.‘r B ‘Ü¶ Agboë¶mdVw©imMr {ÌÁ¶m

CR = 14 – 4 = 10 g|.‘r C ‘Ü¶ Agboë¶m dVw©imMr {ÌÁ¶m

PQ

R x x

2 9

A

B

C

xy

z

x

yz

P

Q

R

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x2 + 11x – 102 = 0 (x + 17)(x – 6) = 0

x + 17 = 0 qH$dm x – 6 = 0

x = –17 qH$dm x = 6

R ‘Ü¶ AgUmè¶m dVw©imMr {ÌÁ¶m = 6 g|.‘r.

CXmhaU 3 … A Am{U B ‘Ü¶ AgUmar dVw©io EH$‘oH$m§Zm Am§Vañne© H$aVmV.P hm ñne© q~Xÿ Amho. Va {gÕ H$am H$s, AR || BQ.

CH$b … BPQ = xº ‘mZm.

PBQ ‘Ü¶o

BP = BQ (EH$mM dVw©imÀ¶m {ÌÁ¶m)

BQP = BPQ (g‘mZ ~mOy g‘moarb H$moZ)

BQP = x° ......(1) ( BPQ = x°)

Ë¶mMà‘mUo PAR ‘Ü¶o

AP = AR (EH$mM dVw©imÀ¶m {ÌÁ¶m)

ARP = APR (g‘mZ ~mOy g‘moarb H$moZ)

ARP = xº ......(2) ( APR = x°

(1) d (2) H$ê$Z

BQP = ARP (gma»¶mM H$moZmer g‘mZ AgUmao H$moZg‘mZ AgVmV.

g§JV H$moZ g‘mZ AmhoV.

AR || BQ åhUyZ {gÕ Pmbo.

ñdmÜ¶m¶ 10.6

A. ñne© dVw©imdarb g§»¶mË‘H$ CXmhaUo …

1. VrZ dVw©io EH$‘oH$m§Zm ~møñne© H$aVmV. Va dVw©imMo dVw©i‘Ü¶ gm§YyZ ~Zboë¶m {ÌH$moUmÀ¶m ~mOy AZwH«$‘o7 g|.‘r., 8 g|.‘r. Am{U 9 g|.‘r. AmhoV Va àË¶oH$ dVw©imMr {ÌÁ¶m H$mT>m.

2. AmH¥$Vr‘Ü¶o A, B Am{U C ‘Ü¶ AgUmar VrZ dVw©io AmH¥$VrV Xe©{dë¶mà‘mUoñne© H$aVmV. Oa øm dVw©imMr {ÌÁ¶m AZwH«$‘o 8 g|.‘r., 3 g|.‘r. Am{U 2g|.‘r. Amho Va ABC Mr n[a{‘Vr H$mT>m.

3. AmH¥$Vr‘Ü¶o AB = 10 g|.‘r., AC = 6 g|.‘r. Am{U bhmZ dVw©imMr {ÌÁ¶m 'X'g|.‘r. Amho Va 'X' H$mT>m.

A B P

RQ

A B

P Q

CO

x

AB

C

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252 UNIT-10

A

P Q

C

Ox

M D B

R

2

A

B

P

C1

C2

B. ñne© dVw©imda AmYmarV Cnà‘o¶ …

1. A Am{U B ‘Ü¶ Agboë¶m XmoZ dVw©im§À¶m ñne© q~XþVyZ H$mT>bobr gaiaofm dVw©im§Zm AZwH«$‘o P Am{U Q‘Ü¶o N>oXVo. AP Am{U BQ ho g‘m§Va AmhoV Ago XmIdm.

2. X Am{U Y ‘Ü¶ AgUmar XmoZ dVw©io EH$‘oH$m§Zm P ‘Ü¶o ~møñne© H$aVmV. àË¶oH$ dVw©im‘Ü¶o EH$ ¶mà‘mUoH$mT>bobo AB Am{U CD ho ì¶mg nañnam§Zm g‘m§Va AmhoV Va {gÕ H$am H$s B, P Am{U C ho EH$aofr¶AmhoV.

3. 'O' ‘Ü¶ Agboë¶m dVw©imV ì¶mg AB Am{U Ordm AD H$mT>bobr Amho. OA ì¶mg AgUmao AD bm

C ‘Ü¶o N>oXUmao Xþgao EH$ dVw©i H$mT>bobo Amho. Va {gÕ H$am H$s BD = 2OC .

4. {Xboë¶m AmH¥$Vr‘Ü¶o AB = 8 g|.‘r. M hm AB Mm ‘Ü¶q~Xÿ Amho.

AB da AY©dVw©io H$mT>bobr AmhoV. AM Am{U MB ho ì¶mg AmhoV.'O' ‘Ü¶ AgUmao dVw©i gd© {VÝhr AY© dVw©im§Zm AmH¥$VrV Xe©{dë¶mà‘mUo

ñne© H$aVo. Va øm dVw©imMr {ÌÁ¶m16 AB Amho Ago {gÕ H$am.

ñne© dVw©im§Mr aMZm

CXmhaU 1 … 2 g|.‘r. Am{U 1 g|.‘r. {ÌÁ¶|Mr Am§Vañne© H$aUmar XmoZdVw©io H$mT>m Am{U ñne© q~XÿMr IwU H$am.

{Xbobo ..… C1 = R = 2 g|.‘r., C2 = r = 1 g|.‘r.

nm¶ar 1 … 'd' H$mT>md = R – r = 2 – 1 = 1 g|.‘r.

nm¶ar 2 … AB = 1 g|.‘r. H$mT>m. A ‘Ü¶ AgUmao 2 g|.‘r.BV³¶m {ÌÁ¶oMo dVw©i H$mT>m.

nm¶ar 3 … 1 g|.‘r. BV³¶m {ÌÁ¶oMo B ‘Ü¶ AgUmao dVw©i H$mT>m.

nm¶ar 4 … AB nwT>o dmT>{dë¶mg ñne© q~Xÿ P ‘Ü¶o {‘iVo.

ñdmÜ¶m¶ 10.7

1. 5 g|.‘r. Am{U 2 g|.‘r. {ÌÁ¶oMr ~mø ñne© H$aUmar XmoZ dVw©io H$mT>m.

2. dVw©i‘Ü¶mVrb A§Va 7 g|.‘r. AgUmar 4.5 g|.‘r. Am{U 2.5 g|.‘r. {ÌÁ¶|Mr XmoZ dVw©io H$mT>m.

3. 4 g|.‘r. Am{U 2.5 g|.‘r. {ÌÁ¶|Mr Am§Vañne© H$aUmar XmoZ dVw©io H$mT>m. Ë¶m§À¶m dVw©i‘Ü¶mVrb A§Va‘moOm Am{U nS>Vmim.

4. Am§Vañne© H$aUmè¶m XmoZ dVw©im§À¶m dVw©i‘Ü¶mVrb A§Va 2 g|.‘r. Amho. Oa EH$m dVw©imMr {ÌÁ¶m 4.8g|.‘r. Agob Va Xþgè¶m dVw©imMr {ÌÁ¶m H$mT>m. Am{U ñne© H$aUmar dVw©io H$mT>m.

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g‘mB©H$ ñn{e©H$m


CXmhaU 1 … gm¶H$brÀ¶m MmH$mH$S>o nhm Or dVw©io Xe©{dVmV, añË¶mdê$Z{’$aUmao MmH$ aofm Xe©{dVo. XmoÝhr MmH$mZm aofm ñne© H$aVo.

CXmhaU 2 … Imbrb AmH¥$VtMo {ZarjU H$am àË¶oH$ KQ>ZoV EH$mM gaiaofobmñne© H$aUmar XmoZ dVw©io AmhoV. gaiaofm hr XmoÝhr dVw©im§Mr ñn{e©H$mAmho. Aem ñn{e©Ho$bm g‘mB©H$ ñn{e©H$m Ago åhUVmV.

g‘mB©H$ ñn{e©H|$À¶m g§X^m©dê$Z dVw©im§À¶m pñWVrMo {ZarjU H$am.Vwåhmbm Ago {XgyZ ¶oVo H$s XmoÝhr dVw©io ñn{e©Ho$À¶m EH$mM ~mOyg AmhoV.

Oa XmoÝhr dVw©io g‘mB©H$ ñn{e©Ho$À¶m EH$mM ~mOyg (A§Jmg) AgVrb Va Ë¶m g‘mB©H$ ñn{e©Ho$bm g‘mB©H$~møñn{e©H$m Ago åhUVmV. (Direct Common Tangent)

CXmhaU 3 … Imbrb AmH¥$VtMo {ZarjU H$am. àË¶oH$ KQ>ZoV gaiaofm hr g‘mB©H$ ñn{e©H$m Amho nU dVw©im§À¶mpñWVrMo {ZarjU H$am. dVw©io hr g‘mB©H$ ñn{e©Ho$À¶m XmoÝhr ~mOyg AmhoV.

Oa XmoÝhr dVw©io g‘mB©H$ ñn{e©Ho$À¶m XmoÝhr ~mOyg (A§Jmg) AgVrb Va Ë¶m g‘mB©H$ ñn{e©Ho$bm g‘mB©H$Am§Vañn{e©H$m Ago åhUVmV. (Tranverse Common Tangent)

H¥$Vr … AmH¥$VtMo {ZarjU H$am àË¶oH$ KQ>Ho$Vrb àH$ma Am{U g‘mB©H$ ñn{e©Ho$Mr g§»¶m AmoiIm.

XmoZ EH$ê$n dVw©imZm g‘mB©H$ ~mø ñn{e©H$m H$mT>Uo.

1. dVw©i‘Ü¶mVrb A§Va 8 g|.‘r. AgUmè¶m 3 g|.‘r. {ÌÁ¶|À¶m XmoZ dVw©imbm g‘mB©H$ ~mø ñn{e©H$m H$mT>m.{Xbobo … r = 3 g|.‘r. = C1 = C2, d = 8 g|.‘r.

A

B

C

D

G HE

F

A B EF

CD

G

H

Sun Earth

Umbra

Penumbra

E

B

A

D

A B

DEC

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nm¶ar 1 … AB = 8 g|.‘r. H$mT>m. A Am{U B ‘Ü¶ AgUmar 3 g|.‘r. {ÌÁ¶|Mr XmoZ dVw©io H$mT>m.

A B

C1 C2

nm¶ar 2 … A Am{U B ‘YyZ XmoZ b§~ ì¶mg H$mT>m. q~XÿZm P, Q, R Am{U S hr Zm§do {bhm.

A B

P Q

R Snm¶ar 3 … PQ Am{U RS OmoS>m. PQ Am{U RS øm Amdí¶H$ (Ano{jV) g‘mB©H$ ~mø ñn{e©H$m AmhoV.

A B

P Q

R S

Q>rn … i) g‘mB©H$ ~møñn{e©H$m H$mT>Ê¶mMr nÕV hr EH$ê$n Agboë¶m§gmR>r g‘mZ AgVo.

a) Z N>oXUmar dVw©io (d>R+r) b) ~mø ñne© H$aUmar dVw©io (d=R+r) Am{U

c) N>oXUmar dVw©io (d<R+r)

ii) XmoZ EH$ê$n dVw©imZm H$mT>boë¶m g‘mB©H$ ~møñn{e©H$m dVw©i‘Ü¶ gm§YUmè¶m aofoer g‘mZAm{U g‘m§Va AgVmV.

XmoZ EH$ê$n Zgboë¶m dVw©im§Zm g‘mB©H$ ~mø ñn{e©H$m H$mT>Uo …

CXmhaU 1 … 5 g|.‘r. Am{U 2 g|.‘r. {ÌÁ¶|À¶m XmoZ dVw©i‘Ü¶mVrb A§Va 10 g|.‘r. Amho Va dVw©im§Zm g‘mB©H$

~mø ñn{e©H$m H$mT>m. g‘mB©H$ ~mø ñn{e©Ho$Mr bm§~r ‘moOm Am{U nS>Vmim.

{Xbobo … d = 10, C1 = R = 5 g|.‘r., C2 = r = 2 g|.‘r., C3 = R – r = 5 – 2 = 3 g|.‘r.

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nm¶ar 1 … AB = 10 g|.‘r. H$mT>m. A Am{U B‘Ü¶ AgUmar AZwH«$‘o 5 g|.‘r. Am{U 2 g|.‘r.{ÌÁ¶|Mr C1 Am{U C2 dVw©io H$mT>m.

AB

C1

C2

nm¶ar 3 … AB bm M ‘Ü¶o N>oXUmam b§~ Xþ mOH$H$mT>m. M ‘Ü¶ AgUmao MA {ÌÁ¶oMo C3 bmP Am{U Q ‘Ü¶o N>oXUmao C4 dVw©i H$mT>m.

AB

C1

C2C3

M

C4

Q

P

nm¶ar 5 … PB ñn{e©H$m H$mT>m.

AB

C1

C2C3

M

C4

Q

P

S

Q>rn … Ë¶mMà‘mUo C1 Am{U C2bm Xþgar g‘mB©H$ ~møñn{e©H$m H$mT>Vm ¶oVo.

nm¶ar 7 … ‘moO‘mnmZo ST = 9.5 g|.‘r.

{heo~mZo DCT (g‘mB©H$ ~møñn{e©H$m)

= 2 2d (R r) = 2 210 (5 2) = 100 9 = 91 = 9.5 g|.‘r.

XmoZ dVw©im§Zm g‘mB©H$ Am§Vañn{e©H$m H$mT>Uo

CXmhaU 1 … 4 g|.‘r. Am{U 2 g|.‘r. {ÌÁ¶oÀ¶m XmoZ dVw©i‘Ü¶mVrb A§Va 10 g|.‘r. Amho. Ë¶m§Zm g‘mB©H$Am§Vañn{e©H$m H$mT>m. g‘mB©H$ Am§Va ñn{e©Ho$Mr bm§~r ‘moOm Am{U nS>Vmim.

{Xbobo , d = 10 g|.‘r., C1 = R = 4 g|.‘r., C2 = r = 2 g|.‘r., C3 = R + r = 4 + 2= 6 g|.‘r.

nm¶ar 2 … A ‘Ü¶ AgUmao 3 g|.‘r. {ÌÁ¶oMoC3 dVw©i H$mT>m.

A B

C1C2

C3

nm¶ar 4 … AP OmoS>m Am{U Vo nwT>o dmT>{dë¶mg C1

bm S ‘Ü¶o N>oXVo.

AB

C1

C2C3

M

C4

Q

P

S

nm¶ar 6 … S ‘Ü¶ gmXmaU BP {ÌÁ¶oMm H§$g H$mT>bmAgVm C2bm T ‘Ü¶oo N>oXVmoo.ST Am{U BT gm§§Ym.

AB

C1

C2C3

M

C4

Q

P

S

T

Q>rn … aMZoMr V§VmoV§VnUm VnmgÊ¶mgmR>rPBTS hm Am¶V Agbm nm{hOo.

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256 UNIT-10

nm¶ar 1 … AB = 10 g|.‘r. A Am{U B ‘Ü¶ AgUmarAZwH«$‘o 4 g|.‘r. Am{U 2 g|.‘r. {ÌÁ¶oMr dVw©io H$mT>m.dVw©imZm C1 Am{U C2 Zm§do Úm.

nm¶ar 3 … AB bm M ‘Ü¶o N>oXUmam b§~ Xþ mOH$ H$mT>m.M ‘Ü¶ AgUmao MA {ÌÁ¶oMo C3 bm P Am{U Q ‘Ü¶oN>oXUmao C4 dVw©i H$mT>m.

nm¶ar 5 … PB ñn{e©H$m H$mT>m.

nm¶ar 7 … ‘moO‘mnmZo ST = 8 g|.‘r.2 2d (R r) = 2 210 (4 2) = 100 36 = 64 = 8 g|.‘r.

A B

C1C2

nm¶ar 2 … A ‘Ü¶ AgUmao 3 g|.‘r. {ÌÁ¶oMo C3

dVw©i H$mT>m.

AB

C1

C2C3

nm¶ar 4 … AP gm§Ybo Agbm C1 bm S ‘Ü¶o N>oXVo.

A B

C3

C2

C1

M

C4P

Q

S A B

C3

C2

C1

M

C4P

Q

A B

C3

C2

C1

M

C4P

Q

S

Q>rn … Ë¶mMà‘mUo C1

Am{U C2 bm Xþgar g‘mB©H$Am§Va ñn{e©H$m H$mT>Vm ¶oVo.

nm¶ar 6 … S ‘Ü¶ ‘mZyZ PB {ÌÁ¶oZo H§$g H$mT>bm AgVmC2 bm T ‘Ü¶o N>oXVmo. ST gm§Ym.

A B

C3

C2

C1

M

C4P

Q

S

T

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Q>rn … i) g‘mB©H$ ~møñn{e©H$m H$mT>Ê¶mMr nÕV hr EH$ê$n Agboë¶m§gmR>r g‘mZ AgVo.

a) Z N>oXUmar dVw©io (d>R+r) (b) ~mø ñne© H$aUmar dVw©io (d=R+r) Am{U (c) N>oXUmar dVw©io

(d<R+r)

ii) dVw©imZm Am§Vañn{e©H$m (TCT) H$mT>Ê¶mMr nÕV hr EH$ê$n Agboë¶m Am{U EH$ê$n Zgboë¶m Z

N>oXUmè¶m dVw©im§gmR>r (d>R+r) g‘mZ AgVo.

ñdmÜ¶m¶ 10.8

I. (A). 1. 3 g|.‘r. {ÌÁ¶|À¶m XmoZ EH$ê$n dVw©i‘Ü¶mVrb A§Va 10 g|.‘r. Amho. g‘mB©H$ ~møñn{e©H$m

H$mT>m.

2. 3.5 g|.‘r. {ÌÁ¶|Mr XmoZ EH$ê$n dVw©io H$mT>m Am{U Ë¶m‘Yrb A§Va 3 g|.‘r. Amho. dVw©imZm XmoZ

g‘mB©H$ ~møñn{e©H$m H$mT>m.

3. 4.5 g|.‘r. {ÌÁ¶|À¶m ~møñne© H$aUmè¶m XmoZ dVw©imZm g‘mB©H$ ~møñn{e©H$m H$mT>m.

4. 2.5 g|.‘r. {ÌÁ¶|À¶m XmoZ dVw©i‘Ü¶mVrb A§Va 4 g|.‘r. Amho. dVw©im§Zm g‘mB©H$ ~mø ñn{e©Ho$Mr

EH$ OmoS>r H$mT>m.

(B). 1. 5 g|.‘r. Am{U 2 g|.‘r. {ÌÁ¶|À¶m XmoZ dVw©i‘Ü¶mVrb A§Va 3 g|.‘r. Amho g‘mB©H$ ~møñn{e©H$m H$mT>m.

2. Am§Vañne© H$aUmè¶m 4.5 g|.‘r. Am{U 2.5 g|.‘r. {ÌÁ¶|À¶m XmoZ dVw©imZm g‘mB©H$ ~møñn{e©H$m H$mT>m.

3. 4 g|.‘r. Am{U 2 g|.‘r. {ÌÁ¶|À¶m XmoZ dVw©i‘Ü¶mVrb A§Va 8 g|.‘r. Amho. dVw©imZm g‘mB©H$

~møñn{e©H$m H$mT>m. ñn{e©Ho$Mr bm§~r ‘moOm Am{U nS>Vmim.

4. 5.5 g|.‘r. Am{U 3.5 g|.‘r. {ÌÁ¶|Mr XmoZ dVw©io EH$‘oH$m§Zm ~mø ñne© H$aVmV. g‘mB©H$

~møñn{e©H$m H$mT>m Am{U Ë¶m§Mr bm§~r ‘moOm.

5. 5 g|.‘r. Am{U 3 g|.‘r. {ÌÁ¶|À¶m XmoZ dVw©i‘Ü¶mVrb A§Va 5 g|.‘r. Amho. dVw©imZm XmoZ

g‘mB©H$ ~møñn{e©H$m H$mT>m.

6. 6 g|.‘r. Am{U 3 g|.‘r. {ÌÁ¶|Mr XmoZ dVw©io 1 g|.‘r. A§Vamda AmhoV. g‘mB©H$ ~møñn{e©H$m

H$mT>m. Ë¶m§Mr bm§~r ‘moOm Am{U nS>Vmim.

II.(A). 1. 6 g|.‘r. Am[¨U 2 g|.‘r. {ÌÁ¶|À¶m XmoZ dVw©i‘Ü¶mVrb A§Va 8 g|.‘r. Amho. dVw©imZm g‘mB©H$

Am§Vañn{e©H$m H$mT>m.

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258 UNIT-10

2. 4.5 g|.‘r. Am[U 2.5 g|.‘r. {ÌÁ¶|Mr XmoZ dVw©io EH$‘oH$m§Zm ~møñne© H$aVmV. dVw©im§Zm g‘mB©H$

Am§Vañn{e©H$m H$mT>m.

3. àË¶oH$s 3 g|.‘r. {ÌÁ¶oÀ¶m XmoZ dVw©i‘Ü¶mVrb A§Va 6 g|.‘r. Amho. dVw©imZm g‘mB©H$ Am§Vañn{e©H$m H$mT>m.

(B). 1. 2.5 g|.‘r. {ÌÁ¶oÀ¶m XmoZ EH$ê$n dVw©i‘Ü¶mVrb A§Va 8 g|.‘r. Amho. g‘mB©H$ Am§Vañn{e©H$m

H$mT>m.

2. 3 g|.‘r. Am{U 2 g|.‘r. {ÌÁ¶|Mr XmoZ dVw©io 8 g|.‘r. A§Vamda AmhoV. g‘mB©H$ Am§Vañn{e©H$m H$mT>m

Am{U {VMr bm§~r ‘moOm.

3. 4 g|.‘r. Am[U 2 g|.‘r. {ÌÁ¶|À¶m XmoZ dVw©imZm 8 g|.‘r. bm§~rÀ¶m Am§Vañn{e©H$m H$mT>m.

4. 4 g|.‘r. Am{U 3 g|.‘r. {ÌÁ¶|À¶m XmoZ dVw©i‘Ü¶mVrb A§Va 10 g|.‘r. Amho. g‘mB©H$ Am§Vañn{e©H$m

H$mT>m. ‘moOm Am{U {heo~mZo nS>Vmim.

5. 2.5 g|.‘r. Am{U 3.5 g|.‘r. {ÌÁ¶|À¶m XmoZ dVw©i‘Ü¶mVrb A§Va 8 g|.‘r. Amho. g‘mB©H$

Am§Vañn{e©H$m H$mT>m. {VMr bm§~r ‘moOm Am{U {heo~mZo nS>Vmim.

* * * *

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CÎmao

ñdmÜ¶m¶ … 10.4

A. 1. 14 g|.‘r. 2. AB = 24g|.‘r. 3. AF = 7g|.‘r. BD = 5 g|.‘r. Am{U CE = 3 g|.‘r.

4. 14 g|.‘r. 5. (i) 056ACB (ii) 0112AOB (iii) 068ADB

ñdmÜ¶m¶ … 10.6

A. 1. 4 g|.‘r., 3 g|.‘r.Am{U 5 g|.‘r. 2. ABC Mr n[a{‘Vr = 16 g|.‘r. 3 .x = 2.4 g|.‘r.

*****

Tangent Properties

Tangent at apoint on a circle

Tangent at the endsof the diameter

Tangents at thenon-centre ends

of the radii

Tangent to a circleif angle between the

tangents is given

Tangent at the endsof the chord

PropertiesConstructions Common tangents totwo circles.

DCT TCT

To twocongruent circles

To two non-congruent circles

* Externally non-touching* Externally touching* Intersecting

Internallytouching

Externallynon-touching

Externallytouching

Tangents from anexternal point to circle Touching circles

Numerical Problems Riders

ñn{e©H$m§Mo JwUY‘©

aMZm

dVw©imdarb q~XÿVyZñn{e©H$m

{ÌÁ¶m§À¶m dVw©i‘Ü¶Zgboë¶m Q>moH$mnmgyZ

ñn{e©H$m

ñn{e©H$m§‘Yrb H$moZ {Xbm AgVmdVw©imbm ñn{e©H$m H$mT>Uo

OrdoÀ¶m Q>moH$mnmgyZñn{e©H$m H$mT>Uo

ì¶mgmÀ¶m Q>moH$mnmgyZñn{e©H$m H$mT>Uo

XmoZ EH$ê$ndVw©imZm

JwUY‘© XmoZ dVw©imZm g‘mB©H$ ñn{e©H$m

XmoZ EH$ê$nZgboë¶m dVw©imZm ~møñne© Z

H$aUmè¶m~møñne©H$aUmè¶m

* ~møñne© Z H$aUmè¶m* ~møñne© H$aUmè¶m* N>oXUmè¶m

Am§Vañne©H$aUmè¶m

ñne© dVw©iodVw©imbm ~møq~XÿVyZH$mT>boë¶m ñn{e©H$m

Cà‘o¶g§»¶mË‘H$ g‘ñ¶m

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260 UNIT-10A

mnU

g‘m

B©H$

ñn{e

©Ho$À¶

m ‘m

{hVr

Mm

gmam

§e n

mhÿ.

{Xb

obr

gmaU

r A

ä¶mg

m.

A.

dVw©i

oA

mH¥$Ë¶

md,

RA

m{U r

g‘m

B©H$ ~

møg

‘mB©H

$ Am§V

ag‘

mB©H$

Z§.‘Y

rb g

§~§Yñn

{e©Ho$

Mr g

§»¶m

ñn{e

©Ho$M

r g§»¶

mñn

{e©Ho$

Mr g

§»¶m

1.Z

N>oXU

mard

> R

+ r

2,2,

4dV

w©io

PQ, R

STU

, VW

2.~m

ø ñ

ne©

d =

R +

r2,

1,3

H$aU

mar d

Vw©io

PQ, R

SXY

3.N>oXU

mar d

Vw©io

d <

R +

r2,

-2

MN

, PQ

4.A

m§Vañ

ne©

d =

R –

r1,

-1

H$aU

mar d

Vw©io

EF

5.g‘

H|$Ðr¶

ZgUmar

d <

R –

r-

--

dVw©io

6.g‘

H|$Ðr¶

dVw©io

d =

0-

--

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¶m KQ>H$m‘wio Vwåhmbm ¶m§Mo gwb^rH$aU hmoVo

* g‘ê$n ~hþ wOmH¥$Vr Am{U g‘ê$n {ÌH$moUm§À¶mAWmªMo ñnï>rH$aU.

* g‘ê$n {ÌH$moU Am{U EH$ê$n {ÌH$moU ¶m‘Yrb’$aH$ AmoiIUo.

* ‘yb yV à‘mUmÀ¶m à‘o¶mMo {dYmZ Am{U {gÕVm.

* ‘y.à. à‘o¶mMm ì¶Ë¶mg Am{U Cn{gÕm§V ¶m§Mo{dYmZ Am{U {gÕVm.

* g‘ê$n {ÌH$moUmgmR>r AA g‘ê$nVoMo {dYmZ Am{U{gÕVm.

* g‘ê$n {ÌH$moUmgmR>r SSS Am{U SAS g‘ê$nVoMo{dYmZ Am{U {gÕVm.

* H$mQ> H$moZ {ÌH$moUmÀ¶m H$mQ>H$moZ {eamoq~XÿVyZ H$Um©daQ>mH$bobm b§~ ¶m g§~§{YV à‘o¶mMr {gÜXVm Am{U{dYmZ

* g‘ê$n {ÌH$moUm§À¶m joÌ’$imdarb à‘o¶m§Mo {dYmZAm{U {gÕVm.

* AZw‘m{ZH$ ¶w³Ë¶m Am{U à‘o¶mda AmYmarVCnà‘o¶m§Mr {gÕVm.

* à‘o¶mda AmYm[aV X¡Z§{XZ OrdZmVrb CXmhaUm§Mon¥W:H$aU Am{U gmoS>{dUo.

Similar Triangles

* g‘ê$n ~hþ^wOmH¥$Vr Am{U g‘ê$n{ÌH$moU

* ‘yb^yV à‘mUmMm à‘o¶

* ‘ y.à.à‘ o¶mMm ì¶Ë¶mg Am{UCn{gÕm§V

* g‘ê$n {ÌH$moUm§À¶m AA Am{U SSSH$gmoQ>r

* g‘ê$n {ÌH$moUm§À¶m AA H$gmoQ>tMr{gÕm§V.

* g‘ê$n {ÌH$moUm§À¶m joÌ’$imdarbà‘o¶

11

{‘boQ>gMm Woëg

(624-546 B.C., J«rg)

Woëg hm n{hbm ‘m{hV Agbobm

VËdkmZr Am{U J{UVVk. ^y{‘Vr‘Yrb

AZw‘m{ZH$ ¶w{º$dmX dmnaÊ¶mMo n{hbo

lo¶ ¶m§Zm {‘iVo. ¶mZo ^y{‘Vr‘Yrb

AZoH$ {gÕm§Vm§Mm emoY bmdbm. ¶m§À¶m

‘VmZwgma B{Oá‘Yrb {na °{‘S>Mr C§Mr

gmdbrMm Cn¶moJ H$ê$Z g‘ê$n

{ÌH$moUmÀ¶m VËdmZo H$mT>Vm ¶ oVo.

The universe cannot be read until wehave learnt the language in which it iswritten. It is written in mathematicallanguage, and the letters are triangles,circles and other geometrical figures,without which it is humanly impossible tocomprehend a single word.

- Galileo Galilei

g‘ê$n {ÌH$moU

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262 UNIT-11

‘mJrb B¶ÎmoV Vwåhr {ÌH$moU Am{U Ë¶mÀ¶m JwUY‘m©{df¶r {eH$bm AmhmV. ¶m JwUY‘mªMm Cn¶moJ {ZË¶

OrdZmVrb CXmhaUo gmoS>{dÊ¶mg hmoVmo. ¶mn¡H$s H$mhtMm ^m¡{VH$emó, agm¶Zgmó BË¶mXr {df¶m§Mm Aä¶mg

H$aÊ¶mg hmoVmo. {ÌH$moU Am{U Ë¶m§À¶m JwUY‘mªMm g‘mdoe Agbobr H$m§hr CXmhaUo Imbr {Xbr AmhoV.

1. ( )( )( )A s s a s b s c

OoWo 2s = a + b + c.

{ÌH$moUm§Mo joÌ’$i H$mT>Ê¶mMo ho gyÌ Amho. ho gyÌ H$go V¶ma

(derived) Pmbo ho Vwåhmbm ‘m{hV Amho H$m?

2. AB = 14 g|.‘r.

Amåhmbm AB 2 : 5. JwUm oÎmamV {d^mJm¶Mo Amho.

àm¶mo{JH$[aË¶m ho H$go H$aVm ¶oBb?

3. 1 1 1u v f

¶mbm Amaí¶mMo gyÌ qH$dm q^JmMo gyÌ åhUVmV, ¶m{df¶r

Vwåhr ^m¡{VH$emóm‘Ü¶o {eH$bm AmhmV. ho gyÌ {‘i{dVm

¶oVo ¶mMm Vwåhr H$Yr {dMma Ho$bm AmhmV H$m?

¶m gd© àým§gmR>r Am{U A{YH$ ‘m{hVrgmR>r ¶m KQ>H$mV Vwåhmbm Cnm¶ {XgyZ ¶oVrb.

Imbrb CXmhaUmMm {dMma H$am.

XmoZ JQ>mVrb {dÚmWu Imbrb àH$ën H$aVmV. EH$m JQ>mVrb

{dÚmWu PmS>mÀ¶m nS>booë¶m gmdbrMr bm§~r ‘moOVmV.

A

B C a

bc

M

N

P

A

BCF

A

B

A B14 cm

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Similar Triangles 263

Ë¶mM doir, PmS>mÀ¶m ~mOybm Agboë¶m VrZ ‘moR>¶m C§MrÀ¶m Cä¶m Im§~mÀ¶m nS>boë¶m gmdbrMr Xþgè¶m

JQ>mVrb {dÚmWu ‘moOVmV. ¶m ‘m{hVrdê$Z Ë¶mZm PmS>mMr C§Mr H$mT>md¶mMr Amho. ho e³¶ Amho H$m?

^y{‘Vr‘Yrb "g‘ê$n {ÌH$ m oU' ¶m ‘hËdmÀ¶m

g§H$ënZoMm Cn¶moJ H$ê$Z PmS>mMr C§Mr H$mT>Uo e³¶ Amho.

g‘ê$n {ÌH$moU g‘OyZ KoÊ¶mgmR>r àW‘ ^m¡{‘{VH$ AmH$ma

Am{U ‘mn ¶m{df¶rÀ¶m ‘hËdmÀ¶m H$ënZm§Mm àW‘ Aä¶mg

H$ê$.

àVbmdarb ^m¡{‘{VH$ AmH¥$Ë¶m§Mm AmH$ma Am{U ‘mn ¶m§À¶m

g§X^m©V VrZ e³¶Vm AmhoV.

e³¶Vm -1 e³¶Vm -2 e³¶Vm-3

AmH¥$Ë¶mZm EH$ Va g‘mZ AmH$ma AmH¥$Ë¶mZm g‘mZ AmH$ma" g‘mZ AmH$ma Agboë¶m na§Vw

qH$dm g‘mZ ‘mn ¶mn¡H$s H$moUVohr Am{U g‘mZ ‘mn AgVo. ‘mn Zgboë¶m AmH¥$Ë¶m.

ZgVo. Mm¡ag Am{U g‘ wO g‘mZ ‘mno AgUmao XmoZ {ÌH$moZ doJdoJir ‘mno Agbobo XmoZ

Mm¡H$moZ g‘mZ g‘H$moZ {ÌH$moU

¶mZm EH$ê$n {ÌH$moU Ago Aem AmH¥$Ë¶mZm g‘ê$n AmH¥$Ë¶m

åhUVmV. Ago åhUVmV.

EH$ê$nVo à‘mUoM y{‘Vr‘Ü¶o g‘ê$nVm g§~§Y XoIrb ‘hËdmMr ^y{‘H$m ~OmdVmo. Amnë¶m X¡Z§{XZ OrdZmV

g‘ê$nVoÀ¶m n[aH$ënZoMm ’$ma {dñVrU© à‘mUmV dmna Ho$bm Amho.

CXmhaUmW©, yJmob ‘Ü¶o dmnabobo Vºo$ g‘ê$nVoÀ¶m n[aH$ënZoda AmYm[aV AmhoV.

‘moR>çm B‘maVtÀ¶m ~m§YH$m‘mnydu ~Z{dboë¶m à{VH¥$Vr g‘ê$n AgVmV.

Amåhr ¶mnydu EH$ê$n {ÌH$moUm{df¶r Aä¶mg Ho$bm Amho. XmoZ {ÌH$moUm§Mm AmH$ma g‘mZ AgyZ Ë¶m§Mo

‘mno g‘mZ Agë¶mg Vo EH$ê$n {ÌH$moU AgVmV. AmVm AmnU Aem AmH¥$Ë¶m§Mm A{YH$ Aä¶mg H$ê$ H$s

Á¶m§Mm AmH$ma g‘mZ Amho nU Ë¶m§Mo ‘mn g‘mZ AgobM Ago Zmhr. Aem AmH¥$Ë¶mZm g‘ê$n AmH¥$Ë¶m

åhUVmV.

OmUyZ ¿¶m!B{VhmgH$mamZr gm§{JVbo Amho H$s, gw‘mao 600BC‘Ü¶o J«rH$Mo à{gÕ J{UV Vk WoëgZr gmdbrÀ¶mgmhmæ¶mZo B{Oá‘Yrb {na°{‘S>Mr C§Mr emoYyZH$mT>br.

A B

CD

P Q

RS

A

B C

D

E F

P

Q R

L

M N

5 06 0

7 0

5 06 0

7 0

3 01 0 0

5 0

3 01 0 0

5 0

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264 UNIT-11

g‘ê$n AmH¥$Ë¶m

Imbrb N>m¶m{MÌm§Mo {ZarjU H$am. Vwåhr Ë¶mM doir gm§Jy eH$mb H$s hr XmoÝhr N>m¶m{MÌo EH$mM

B‘maVrMr ({dYmZ gm¡Y) AmhoV. na§Vw Ë¶m§Mo ‘mn doJdoJio Amho. hr N>m¶m{MÌo g‘ê$n AmhoV H$m? MMm© H$am.

AmVm H$m§hr ^m¡{‘{VH$ g‘ê$n AmH¥$Ë¶m§Mm ZrQ> {dMma H$am.

A B

A

B C

60°

60°60°

D

E F

60°

60° 60°

P Q

RS

R Q

RS

XmoZ dVw©io Zoh‘r XmoZ Mm¡ag Zoh‘r XmoZ g‘H$moZ {ÌH$moU Zoh‘rg‘ê$n AgVmV g‘ê$n AgVmV g‘ê$n AgVmV

darb CXmhaUmdê$Z Amåhr Agm {ZîH$f© H$mT>Vmo H$r

""XmoZ AmH¥$Ë¶m g‘ê$n AgVmV Oa Am{U ’$º$ Oa Ë¶m§Mm AmH$ma g‘mZ AgyZ Ë¶m§Mr ‘mno g‘mZAgVrbM Ago Zm§hr.''

Amåhr ho {MÝh (g‘ê$n Amho Ago dmMm) AmH¥$Ë¶m§Mr g‘ê$nVm Xe©{dÊ¶mgmR>r dmnaVmo.

CXmhaUmW©: darb AmH¥$Ë¶m‘Yrb {Vgè¶m g§MmMm C„oI H$am.

ABC hm DEF bm g‘ê$n Amho.

ABC DEF g‘ê$nVoMr Xþgar g§km III Amho.

‘m{hVr Amho ! g‘ê$n åhUOoM g‘mZ AmH$ma Agbobo.

Oa Amnë¶mH$S>o XmoZ AmH¥$Ë¶m AmhoV Am{U ¶m XmoZ n¡H$s EH$ EH$Va H$‘r H$ê$Z (bhmZ H$ê$Z) qH$dmXþgar AmH$mamV ~Xb Z H$aVm dmT>{dbr (VmUbr) Va Ë¶m XmoZ AmH¥$Ë¶m g‘ê$n AgVmV. {dYmZ gm¡YÀ¶mN>m¶m{MÌm‘Ü¶o AgoM KS>bo Am{U åhUyZ Ë¶m g‘ê$n AmhoV.

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A

B

C

DE

PQ

R

ST

AB

CD

PQ

RS

AmVmn¶ªV MMm© Ho$boë¶m gd© CXmhaUm‘Ü¶o Amåhr Ago J¥hrV Yabo Amho H$r {XgÊ¶mV gd© AmH¥$Ë¶mg‘ê$n AmhoV. na§Vw J{UVmZo AmH¥$Ë¶m§Mr g‘ê$nVm H$er ì¶º$ H$ê$ eH$Vmo? Amåhmbm ¶mgmR>r g‘ê$nVoÀ¶mì¶m»¶oMr JaO Amho. Am{U ì¶m»¶oda AmYmarV H$m§hr {Z¶‘ Am{U AQ>tMm g‘mdoe Ë¶m AmH¥$Ë¶m g‘ê$n AmhoVH$r ZmhrV ho R>a{dÊ¶mg hmoVmo.

AmVm AmnU ~hþ wOmH¥$VtMr g‘ê$nVm Am{U {VÀ¶m ì¶m»¶o {df¶r Aä¶mg H$ê$.

g‘ê$n ~hþ wOmH¥$Vr (Similar polygon)

Imbrb g‘ê$n ^m¡{‘{VH$ AmH¥$Ë¶m§À¶m OmoS>çm§Mm {dMma H$am.

àË¶oH$mÀ¶m ~m~VrV g§JV H$moZ Am{U g§JV ~mOy§À¶m JwUmoÎmamMo {ZarjU H$am.

g‘ê$n ^m¡{‘{VH$ {ZarjUAmH¥$Ë¶m§À¶m OmoS>çm

ABCD Am{U PQRS ‘Ü¶o

1. A P, B Q, C R,D S

Am{U

2.AB BC CD DAPQ QR RS SP

ABCD PQRS

ABCDE Am{U PQRST ‘Ü¶o

1. A P, B Q, C R, D S, E T

AB BC CD DE EAPQ QR RS ST TP

ABCDE PQRST

darb CXmhaUmdê$Z AmnU {ZîH$f© H$mT>Vmo H$s

gmaIrM ~mOy§Mr g§»¶m AgUmè¶m H$moUË¶m ^m¡{‘{VH$ AmH¥$Ë¶m AmhoV, Ë¶m :

(a) g‘ê$n AgVrb qH$dm AgUma Zm§hrV?

(b) Ho$ìhmM g‘ê$n ZgVmV.

AmVm Aem KQ>Zm§Mm {dMma H$ê$.

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266 UNIT-11

Q>rn : g§JV ~mOy§Mo JwUmoÎma ho ~hþ wOmH¥$VrgmR>r à‘mU KQ>H$ ‘mZbo OmVo.

OmUyZ ¿¶m !

{hëgMm {Z¶‘ : ^m¡{‘{VH$ÑîQ>çm g‘ê$n AgUmao àmUr g‘mZ C§MrMr CS>r ‘maVmV.

ñ’${Q>H$ê$nr {g{bH$m°Z hm {hè¶mer g‘mH¥${VH$ (Isomorphous) Amho. åhUOoM Ë¶m§Mr aMZm g‘ê$n Amho.

{ÌH$moUm§Mr g‘ê$nVm

EH$ê$n {ÌH$moUm‘Ü¶| g§JV KQ>H$m§À¶m ghm OmoS>çm g‘mZ AgVmV ¶mMo ñ‘aU H$am. åhUOoM (i.e) g§JVH$moZm§À¶m VrZ OmoS>çm Am{U g§JV ~mOy§À¶m VrZ OmoS>çm. na§Vw H$moUË¶mhr XmoZ {ÌH$moUm‘Yrb EH$ê$nVm {ZpíMVH$aÊ¶mgmR>r Amåhr H$m§hr H$gmoQ>r {‘i{dboë¶m AmhoV H$s Á¶m‘Ü¶o {ÌH$moUmÀ¶m g§JV KQH$m§À¶m ’$º$ VrZOmoS>çm§Mm g‘mdoe Amho. Ë¶m åhUOo ~m-H$mo-~m, ~m-~m-~m Am{U H$mQ>H$moZ-H$U©- wOm H$gmoQ>r. Amåhr AgohrH$mT>bo Amho H$s Ooìhm XmoZ {ÌH$moU g‘H$moZ (AAA H$gmoQ>r) AgVrb Va Vo EH$ê$n AgVrb qH$dm AgUmaZmhrV.


CXmhaU 1:

P

Q R

X

Y Z

P X, Q Y, R Z

PQ = XY, QR = YZ, PR = XZ

CXmhaU 2:P

Q R

X

Y Z

P = X, Q = Y, R = Z; PQ XY, QR YZ, PR XZ

CXmhaU 1 ‘Ü¶o : g§JV H$moZmÀ¶m gd© {VÝhr OmoS>çm Am{U g§JV ~mOyÀ¶m gd© {VÝhr OmoS>çm g‘mZ AmhoV.

PQR hm XYZ er EH$ê$n Amho åhUyZ PQR XYZ

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CXmhaU 2 ‘Ü¶o : g§JV H$moZm§À¶m gd© {VÝhr OmoS>çm g‘mZ AmhoV Am{U g§JV ~mOy§À¶m gd© {VÝhr OmoS>çmAg‘mZ AmhoV.

PQR hm XYZ er EH$ê$n Zm§hr.

Oa XmoZ {ÌH$moU g‘H$moZ AgyZ Vo EH$ê$n ZgVrb Va Ë¶m§À¶m g§JV ~mOy g‘mZ AgVrb qH$dm AgUma

Zm§hrV. Voìhm Ë¶m§À¶m g§JV ~mOy§Mo EH$‘oH$mer Agbobo ZmVo H$go Amho?

Imbrb g‘H$moZ {ÌH$moUm§À¶m OmoS>çm§Mo {ZarjU H$am. V³Ë¶m‘Ü¶o Ë¶m§À¶m g§JV ~mOy§Mr ‘mno {bhm.

4 cm

3 cm

2.5

cm

A

B C

8cm

6 cm

5 cm

P

Q R

A

B

C6 cm

9 cm

R

P Q

2cm

4 cm

3 cm12 cm

g‘H$moZ {ÌH$moU g§JV ~mOy§Mo JwUmoÎma

1. ABC Am{U PQRAB 2.5PQ 5

BCQR =

ACPR

=

2. ABC Am{U PQRABPQ =

BCQR =

ACPR =

darb àË¶oH$ CXmhaUm‘Ü¶o Ë¶m§À¶m g§JV ~mOy§À¶m JwUmoÎmam{df¶r Vwåhr H$m¶ {ZîH$f© H$mT>mb?

Amåhr Ago {ZarjU H$aVmo H$s Ë¶m§Mr JwUmoÎmao g‘mZ AgVmV.

Oa JwUmoÎmao g‘mZ AgVrb Va Ë¶m§À¶m g§JV ~mOy à‘mUmV AmhoV Ago åhUVmV.

Amåhr Agm {ZîH$f© H$mT>Vmo H$s Oa XmoZ {ÌH$moU g‘H$moZ AgVrb Va Ë¶m§À¶m g§JV ~mOy à‘mUmV

AgVmV.

ho {MÝhmZo Ago Xe©{dbo OmVo. ABC Am{U DEF ‘Ü¶o

A

B C

D

E F

2.AB BC CADE EF FD

ABC ~ DEF

1. BAC = EDF ABC = DEF BCA = EFD

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268 UNIT-11

Q>rn: g‘ê$nVoMo ~ ho {MÝh Amho.

{ÌH$moU ho ’$º$ ~hþ wOmH¥$VrMo àH$ma AmhoV H$s OoWo g‘ê$n AmH¥$Ë¶m§À¶m ì¶m»¶o‘Ü¶o {Xboë¶m XmoZ AQ>tMrnyV©Vm hmoV Zm§hr. Oa EImÚm AQ>rMr nyV©Vm hmoV Agob Va Vo {ÌH$moU g‘ê$n AgVmV.

XmoZ {ÌH$moUmZm g‘ê$n åhQ>bo OmVo, Oa

• Ë¶m§Mo g§JV H$moZ g‘mZ AgVmV qH$dm • Ë¶m§À¶m g§JV ~mOy à‘mUmV AgVmV

g‘mZ ~mOy§Mr g§»¶m AgUmè¶m XmoZ ~hþ wOmH¥$Vr g‘ê$n AgVmV.

Oa

AQ> 1 : gd© g§JV H$moZ g‘mZ AgVrb Va

AQ> 2 : gd© g§JV ~mOy gma»¶mM JwUmoÎmamV qH$dm à‘mUmV AgVrb Va

AmVm AmnU g‘ê$n {ÌH$moUm§À¶m g§JV H$moZ Am{U ~mOy ¶m§Mr MMm© H$ê$.

g§JV H$moZ : XmoZ g‘ê$n {ÌH$moUm‘Ü¶o Oo H$moZ g‘mZ AgVmV Ë¶mZm g§JV H$moZ åhUVmV.

Oa ABC ~ DEF

AB BC CADE EF FD

A = D Vo AZwH«$‘o g§JV ~mOy BC Am{U EF À¶m {dê$Õ AmhoV.

¶mMà‘mUo B = E Am{U C = F

g§JV ~mOy : g‘ê$n {ÌH$moUm‘Ü¶o g‘mZ H$moZm§À¶m {déÕ ~mOyg AgUmè¶m ~mOyZm g§JV ~mOy åhUVmV

Am{U Ë¶m à‘mUmV AgVmV.

AB Am{U DE g§JV ~mOy AmhoV H$maU Ë¶m AZwH«$‘o C Am{U F ¶m g‘mZ H$moZm§À¶m {dê$Õ AmhoV.

VgoM,BC Am{U EF g§JV ~mOy AmhoV

CA Am{U FD g§JV ~mOy AmhoV

{ÌH$moUm§Mr EH$ê$nVm Am{U g‘ê$nVm (Congruency and Similarity of Triangles)

EH$ê$nVm hr g‘ê$nVoMr {d{eï> KQ>Zm Amho. XmoÝhr KQ>Zo‘Ü¶o EH$m {ÌH$moUmMo {VÝhr H$moZ Xþgè¶m

{ÌH$moUmÀ¶m VrZ g§JV H$moZm~amo~a AgVmV. EH$ê$n {ÌH$moUm‘Ü¶o g§JV ~mOy g‘mZ AgVmV Va g‘ê$n

{ÌH$moUm‘Ü¶o g§JV ~mOy à‘mUmV AgVmV.

A

B C

D

E F

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3. PQR, ‘Ü¶o PQ da S hm Agm q~Xÿ Amho H$s

ST||QR Am{UPS 3= .SQ 5 Agob.

EH$ê$n {ÌH$moU g‘ê$n {ÌH$moU

A

B C

D

E F

A

B C

D

E F

ABC DEF ABC ~ DEF

A = D, B = E, C = F A = D, B = E, C = F

AB = DE; BC = EF; CA = FD AB DE; BC EF; CA FD

AB BC CA 1DE EF FD

AB BC CA1 < 1

DE EF FD

gmaImM AmH$ma Am{U gmaIoM ‘mn gmaImM AmH$ma na§Vw doJio ‘mn

Amåhr Agmhr {ZîH$f© H$mT>Vmo H$s,

'EH$ê$n {ÌH$moU Zoh‘r g‘ê$n AgVmV.

na§Vw 'g‘ê$n {ÌH$moU ho EH$ê$n AgVrbM Ago Zmhr.


{ÌH$moUmÀ¶m g‘ê$nVoda AmYm[aV g§»¶mË‘H$ CXmhaUo

CXmhaU 1: ~mOyÀ¶m AmH¥$VrV PQR TSR.

g§JV {eamoq~Xÿ, g§JV ~mOy Am{U Ë¶m§Mr JwUmoÎmao AmoiIm.

CH$b :

{Xbobo PQR TSR

g§JV {eamoq~Xÿ g§JV ~mOy

P T Q R R S

Q S P R R T

R R P Q S T

JwUmoÎmao hr

P

Q

R

S

T

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270 UNIT-11

PQ QR PRST RS RT

CXmhaU 2 : Imbrb ‘m{hVrdê$Z ABC DEF g‘ê$n AmhoV qH$dm ZmhrV ho ñnï> H$am Oa

(a) B = 650, C = 820, D = 330 , F = 650

CH$b :

ABC ‘Ü¶o, B = 650, C = 820.

A = 1800 (650 + 820) = 1800 1470 = 330.

DEF ‘Ü¶o, D = 330 , F = 650

E = 1800 – (330 + 650) = 1800 – 980 = 820.

A = D, B = F Am{U C = E.

ABC DEF

ñdmÜ¶m¶ 11.1

1. Imbr {Xboë¶m g‘ê$n {ÌH$moUmÀ¶m OmoS>çm‘Ü¶o, g§JV {eamo[~§Xÿ g§JV ~mOy Am{U Ë¶m§Mr JwUmoÎmao {bhm.

(a)

BA

C

(b)

P Q K

L

M

Z

X

Y

c( )

A

B

D

EC

2. Imbrb AmH¥$Ë¶m§Mm Aä¶mg H$am Am{U àË¶oH$mÀ¶m ~m~VrV {ÌH$moU g‘ê$n hmoVmV H$m¶ Vo gm§Jm. H$maUoÚm

A

D

E

BC

75°

a

F

a

a55°

D

HG

F E

80°

60°

40°

(ii)

M

P

N OQ

2 cm

6 cm6cm

8 cm2 cm

4 cm

X

U V

Y Z

a

b60°

60°3 b

(iii) (iv)

2a

c

2c

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3. Imbrb ‘m{hVrdê$Z ABC hm DEF bm g‘ê$n Amho H$s Zmhr ho ñnï> H$am.

(a) A = 700, B = 800, D = 700, F = 300

(b) AB = 8 g|§.‘r, BC=9g|§.‘r, CA=15 g|§.‘r, DE=4 g|§.‘r, EF=3g|§.‘r, FD=5 g|§.‘r.

4 Imbrb g§»¶m‘Ü¶o g‘ê$n {ÌH$moU hmoÊ¶mgmR>r g§»¶mMo gQ> {ZdS>m.

(i) 3,4,6 (ii) 9, 12, 18 (iii) 8, 6, 12 (iv) 8, 4, 9(v) 2, 4½, 4

à‘mUmdarb ‘yb^yV ^m¡{‘{VH$ n[aUm‘ :

H¥$Vr : Imbrb {ÌH$moUm§Mm H$miOrnyd©H$ Aä¶mg H$am. ABC, ‘Ü¶o BC bm g‘m§Va DE H$mT>bm. aofmI§S> AD,

DB, AE Am{U CE À¶m ‘mnm§Mo {ZarjU H$am. AmVm Ago ‘mZm H$s XmoZ ~mOy§À¶m JwUmoÎmamZm DE Zo ^mJ OmVmo.

¶m JwUmoÎmam{df¶r Vwåhr H$m¶ {ZîH$f© H$mTy eH$mb?

JQ>m‘Ü¶o ¶mMr MMm© H$am.

ABC, ‘Ü¶o DE || BC

AD 2 1DB 6 3

AE 1.5 1EC 4.5 3

AD AE 1DB EC 3

ABC, ‘Ü¶o DE || BC

AD 6 2DB 3

AE 4 2EC 2

AD AE 2DB EC

ABC, ‘Ü¶o DE hr BC ~mOybm g‘m§Va Zm§hr

A

B C

D E

5 cm

6cm

4.5

cm

1.5 cm2 cm

Fig. 1A

B C

D E

4cm

2 cm3 cm

6 cm

5 cm

AmH¥$Vr 2

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272 UNIT-11

AD 2 1DB 4 2

AE 2 1EC 6 3

AD AEDB EC

n{hë¶m XmoZ {ÌH$moUm‘Ü¶o àË¶oH$ JwUmoÎma g‘mZ Amho nU {Vgam {ÌH$moUm‘Ü¶o JwUmoÎma g‘mZ Zm§hrV. åhUyZ

ABC ‘Ü¶o JwUmoÎma g‘mZ hmoÊ¶mgmR>r H$moUVr ‘hËdmMr AQ>/H$gmoQ>r Amho?

{dMma H$am!

ABC, ‘Ü¶o EH$M ‘hËdmMr AQ> Amho. DE hr BC bm g‘m§Va Amho Am{UADDB d

AEEC hr JwUmoÎma g‘mZ

AmhoV.

darb {ZîH$f m©Mo gm‘ mÝ¶rH$aU Ho$b ob o A mho Ë¶mb m ‘ yb ^ yV à‘ mUmMm à‘ o¶ (B.P.T) qH$dm WoëgMm à‘o¶

åhUVmV. ho Ago ñnï> H$aVmV.

"{ÌH$moUmÀ¶m EH$m ~mOyg g‘m§Va AgUmar aofm BVa XmoZ ~mOyZm N>oXV Agob Va Vr aofm Ë¶m ~mOyZm

à‘mUmV {d^mJVo ".

VH©$ewÕ nÕVrZo AmVm WoëgMm à‘o¶ {gÕ H$ê$.

WoëgMm à‘o¶ : (‘yb^yV à‘mUmMm à‘o¶)-1 (Basic propertionality Theorem)

{ÌH$moUmÀ¶m EH$m ~mOyg g‘m§Va AgUmar aofm BVa XmoZ ~mOyZm N>oXV Agob Va Vr aofm Ë¶m ~mOyZm à‘mUmV

{d^mJVo.

J¥hrV : ABC ‘Ü¶o

DE || BC

gmÜ¶ :AD AEDB EC

aMZm : DC Am{U EB OmoS>m EF AB Am{U DN AC H$mT>m.

A

B C

D

E

6cm

5 cm

4cm

2 cm 2 cm

AmH¥$Vr 3

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{gÕVm: {dYmZ H$maU

ADE ADBDE DB

ADE ADBDE DB

ADE AECDE EC

ADE AECDE EC

AD AEDB CE

BDE CDE -1Am{U ‘yb VËd

WoëgMm à‘o¶ {gÕ H$aVmZm XmoZ A{YH$ nÕVrZo AmH¥$Vr H$mT>Vm ¶oVo.

AmH¥$Vr 1 AmH¥$Vr 2

A

B C

DE A

B C

D E

WoëgÀ¶m à‘o¶mMm ì¶Ë¶mg

"Oa EH$ gaiaofm {ÌH$moUmÀ¶m XmoZ ~mOyZm à‘mUmV N>oXV Agob Va Vr gaiaofm {Vgè¶m ~mOybm g‘m§Va

AgVo".

J¥hrV : ABC ‘Ü¶oAD AEDB EC

gmÜ¶ : DE || BC

aMZm: BF || DE H$mT>m

A

B C

DL N

E

( gma»¶mM C§MrÀ¶m {ÌH$moUmMo joÌ’$i

g§JV nm¶mÀ¶m à‘mUV AgVoo.)

12 AD EL12 DB EL

12 AE DN

12 ECX DN

A

B C

D E

F

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274 UNIT-11

A

B C

D E


ABC ‘Ü¶o DE || BF (aMZm)

AD AEDB EF

[WoëgMm à‘o¶]

na§V wAD AEDB EC [J¥hrV]

AE AEEC EF

EC = EF

F q~Xÿ C [~§Xÿer V§VmoV§V OwiVmo.

åhUyZ, DE || BC

AmVm AmnU WoëgÀ¶m à‘o¶mÀ¶m H$m§hr Cn{gÕm§Vm§Mm Aä¶mg H$ê$.

Cn{gÕm§V :1

ABC ‘Ü¶o DE || BC

AD AEDB EC

[ ‘y.à.à‘o¶]

XmoÝhr ~mOy‘Ü¶o 1 {‘idm

AD AE1 1DB EC

AD DB AE ECDB EC

AB ACDB EC ømbm Componendo Ago åhUVmV.

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Cn{gÕm§V 2 :

ABC ‘Ü¶o DE || BC

AD AEDB EC

[ ‘y.à.à‘o¶]

ì¶ñV KoD$Z Amåhmbm {‘iVo

DB ECAD AE

XmoÝhr ~mOy‘Ü¶o 1 {‘idm

DB EC1 1AD AE

DB AD EC AEAD AE

AB ACAD AE

ømbm Componendo Ago åhUVmV.

Cn{gÕm§V 3 :

ABC ‘Ü¶o DE || BC Am{U EF || AB

DEFB g‘m§Va wOMm¡H$moZ [ g‘m§Va^wO Mm¡H$moZmMr ì¶m»¶m]

BF = DE [ g‘m§Va wO Mm¡H$moZm‘Ü¶o {dê$Õ ~mOy g‘mZ

AgVmV.]

ABC ‘Ü¶o DE || BCAD AEAB AC

[ Cn{gÕm§V-1]

ABC ‘Ü¶o EF || ABBF AEBC AC

[ Cn{gÕm§V 2]

DE AEBC AC

[ BF = DE]

AD DE AEDB BC AC

A

B C

D E

F

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276 UNIT-11

{Q>n : gd© {VÝhr Cn{gÕm§VmMm ì¶Ë¶mg AmT>iVmo Am{U Vmo gË¶ Amho.

Cn{gÕm§V 3 Mo {dYmZ Imbrbà‘mUo Amho. Oa {ÌH$moUmÀ¶m EH$m ~mOybm g‘m§Va AgUmar aofm H$mT>br Va

N>oXH$ {ÌH$moUmÀ¶m ~mOy ‘yi {ÌH$moUmÀ¶m ~mOyer à‘mUmV AgVmV.

WoëgÀ¶m à‘o¶mda AmYmarV àm¶m{JH$ CXmhaUo

{Xboë¶m JwUmoÎmamV aofoMo ^mJ H$aUo.

AB = 14 g|.‘r aofmI§S>mM o 2 : 5 JwUmoÎmamV ^mJ H$am.

A 4 cm

N

M

P

BC 10 cm14 cm

2 cm

5 cm

nm¶è¶m : 1. AB = 14 g|.‘r H$mT>m.

2. AB er H$moUË¶mhr ‘mnmMm H$moZ H$aUmam AP H$mT>m.

[g‘Om gmo¶rgmR>r PAB < 600]

3. AP da M Am{U N Aem IwUm H$am H$s AM = 2 g|.‘r, d MN = 5 g|.‘r hmoVrb.

4. N Am{U B OmoS>m.

5. {ÌH$moUr JwÊ¶m (set squares) dmnê$Z MC || NB H$mT>m.

6. AC Am{U CB ‘moOm. AC = ___ g|.‘r CB = ___ g|.‘r

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WoëgMm à‘o¶ dmnê$Z darb n[aUm‘ Imbrb H$m§hr nm¶è¶mdê$Z {gÕ H$aVm ¶oVmo.

J¥hrV : ABN ‘Ü¶oAM 2MN 5 , Am{U MC || NB

gmÜ¶ :AC 2CB 5

{gÕVm : ABN ‘Ü¶o MC || NB

AM AC 2=MN CB 5

[ WoëgMm à‘o¶]


CXmhaU 1 : ABC ‘Ü¶o DEIIBC, AD = 5.7 g|.‘r, BD = 9.5g|.‘r, EC = 6g|.‘r.Va AE H$mT>m.

CH$b :

{Xb ob o : ABC ‘Ü¶o DE || BC

AD AEDB EC ( ‘y.à.à‘o¶)

5.7 AE9.5 6

5.7 6AE9.5

AE = 3.6 g|.‘r

CXmhaU 2 : eoOmarb AmH¥$VrV XY II BC.

AX = p-3, BX =2p-2 Am{UAY 1= .CY 4 Va P H$mT>m.

CH$b :

{Xbobo : ABC ‘Ü¶o XY || BC

AX AYXB YC ( WoëgMm à‘o¶)

A

E

9.5

cm.

D

CB

5.7

cm.

6cm

.

?

A B

C

X

Y

p-3 2p-2

4

1

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278 UNIT-11

3 12 2 4pp

4 (p-3) = 1(2p-2)

4p-12=2p-2

2p = 10

p = 5

CXmhaU 3. PQR, ‘Ü¶o S Am{U T ho XmoZ q~Xÿ AZwH«$‘o PQ Am{U PR da Ago AmhoV H$s PS = 4g|.‘r, SQ

= 3g|.‘r, PT = 6g|.‘r, TR = 4.5g|.‘r Am{U PST = 400. PQR H$mT>m.

CH$b :

PQR ‘Ü¶oPS 4=SQ 3PT 6 4TR 4.5 3

PS PTSQ TR

ST || QR

( WoëgÀ¶m à‘o¶mMm ì¶Ë¶mg)

PST = PQR = 400

CXmhaU 4: ABC ‘Ü¶o g‘ wO Mm¡H$moZ PQRB hm Am§Va{b{IV H$obobm Amho. H$s B hm EH$ Ë¶mMm H$moZ

hmoB©b. P,Q Am{U R ho AZwH«$‘o AB, AC Am{U BC da q~Xÿ AmhoV. Oa AB = 12 g|.‘r. Am{U BC = 6

g|.‘r Va PQRB g‘ wO Mm¡H$moZmÀ¶m ~mOy H$mT>m.

{dYmZ H$maU

CH$b : ABC, PQ || BC g‘ wO Mm¡H$moZm‘Ü¶o g§‘wI ~mOy g‘m§Va AgVmV.

AP PQAB BC

$WoëgMm Cn{gÕm§V

P

T

3cm

S

RQ

6 cm4cm

400

4.5 cm?

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1212 6

x x

12x = 72 – 6x

18x = 72

x = 4 $g|‘r.

$g‘ wO Mm¡H$moZmMr àË¶oH$ ~mOy 4 g|.‘r Amho.

CXmhaU 5. {XdgmÀ¶m EH$m {ZpíMV doir 6 ’y$Q> C§MrÀ¶m ‘mUgmMr gmdbr 8 ’y$Q> bm§~ nS>Vo. Ë¶mM doir 45’y$Q> C§MrÀ¶m B‘maVrMr gmdbr {H$Vr bm§~rMr Agob?

CH$b :

B‘maVrÀ¶m gmdbrMr bm§~rx Amho Ago ‘mZy

Á¶mAWu DE || AC, DBE ABC.

åhUOoM

‘mUgmMr C§Mr ‘mUgmÀ¶m nS>boë¶m gmdbrMr bm§~r

B‘maVrMr C§Mr B‘maVrÀ¶m nS>boë¶m gmdbrMr bm§~r

’y$Q>’y$Q>

’y$Q>6 845 x

’y$Q>8 45 60

6x

B‘maVrÀ¶m nS>boë¶m gmdbrMr bm§~r 60 ’y$Q> Amho.

CXmhaU 6 : PQR ‘Ü¶o 2PM = 3 PN Am{U 2PQ = 3PR

{gÕ H$am MQRN hm g‘b§~ Mm¡H$moZ Amho.

{gÕVm : (i)2PM 3PN2PQ 3PR

[ J¥hrV]PM PNPQ PR

||MN QR [ ‘y.à.à‘o¶mMmì¶Ë¶mg]

A

xP Q

B R

x

x

x

C

A

CB

D6ft

Ex

8ft

45ft

=

M N

P

Q R

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280 UNIT-11

(ii) Á¶m AWu QP Am{U RP ,hr 'P' ‘Ü¶o N>oXVmV, Ë¶m AWu

QP hr RP bm g‘m§Va AgUma Zm§hrV.

MQRN hm g‘b§~ Mm¡H$moZ Amho.

CXmhaU 7: {gÕ H$am H$s g‘b§~ Mm¡H$moZm‘Ü¶o Ag‘m§Va ~mOy§Mo ‘Ü¶ OmoS>Umar

aofm (i) g‘m§Va ~mOyZm g‘m§Va AgVo - Am{U

(ii) g‘m§Va ~mOy§À¶m ~oaOoÀ¶m {Zå‘r AgVo.

J¥hrV : ABCD g‘b§~ Mm¡H$moZm‘Ü¶o(i) AD || BC

(ii) AX = XB(iii) DY = YC

gmÜ¶ : (i) XY II AD

qH$dm XY IIBC

(ii)1

XY (AD BC)2

= +

aMZm : 1. BA Am{U CD Aem dmT>dm H$s Ë¶m Z ‘Ü¶o {‘iVrb

2. A Am{U C OmoS>m XY bm P ‘Ü¶o N>oXVo.

{gÕVm : nm¶ar 1: ZBC ‘Ü¶o

ADII BC [ J¥hrV]

ZA ZDAB DC

= [ ‘y.à.à‘o¶ ]

ZA ZD2AX 2DY

= [ X Am{U Y ho AB Am{U DC Mo

‘Ü¶q~Xÿ AmhoV.]

ZA ZDAX DY

=

XY || AD ---(i) [ ‘y.à.à‘o¶mMm ì¶Ë¶mg]

nm¶ar 2: ABC ‘Ü¶o

1. AX = XB [ J¥hrV]

2. XP || BC [ {gÕ]

AP = PC [ ‘Ü¶-q~Xÿ à‘o¶mMm ì¶Ë¶mg]

XP = ½ BC [ ‘Ü¶-q~Xÿ à‘o¶]

Z

A

B C

DP YX

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||| ly In ADC, PY = ½AD

{‘idyZ AmnUmg {‘iVo XP + PY = ½BC + ½AD

XY = ½ (BC + AD) --- (ii)

hr g‘ñ¶m n¶m©¶r aMZm H$ê$Z XoIrb gmoS>{dVm ¶oVo.

CXmhaUmW© AC OmoS>Ê¶mEodOr DE II AB H$mT>m. Vr XY bm P ‘Ü¶o N>oXVo.

AmVm, JQ> H$ê$Z MMm© H$am Am{U {gÕVm {bhm.

ñdmÜ¶m¶ 11.2

A WoëgÀ¶m à‘o¶mda AmYmarV g§»¶mË‘H$ CXmhaUo.

1.~mOyÀ¶m AmH¥$VrMm Aä¶mg H$am.

‘yb yV à‘mUmÀ¶m à‘o¶mda Am{U

Ë¶mÀ¶m Cn{gÕm§Vmer g§~§{YV JwUmoÎmao

a, b, c Am{U d À¶m ñdê$nmV {bhm.

2.~mOyÀ¶m AmH¥$Vr‘Ü¶ o DE||AB, AD = 7g|.‘r., CD = 5g|.‘r. Am{U BC

= 18g|.‘r. BE Am{U CE H$mT>m.

3. ABC ‘Ü¶o ~mOy AB Am{U AC da AZwH«$‘ o D Am{U E Ago q~Xÿ AmhoVH$s DE||BC hmoB©b.

(i) Oa AD = 6g|.‘r, DB = 9g|.‘r, Am{U AE = 8g|.‘r, Va AC H$mT>m.

(ii) Oa AD = 8g|.‘r, AB = 12g|.‘r, AE = 12g|.‘r, Va CE. H$mT>m.

(iii) Oa AD = 4x-3, BD = 3x-1, AE = 8x-7, CE =5x-3 Va x Mr qH$‘V H$mT>m.

4. PQR ‘Ü¶o PQ Am{U PR ~mOyda AZwH«$‘o E Am{U F q~Xÿ AmhoV. Imbrb àË¶oH$ KQ>ZogmR>r EF||QR

nS>VmiyZ nhm.

A

B CE

DX YP

Z

P Q

R

X

Y

a b

c

d

C

ED

BA

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282 UNIT-11

(i) PE = 3.9g|.‘r, EQ = 3g|.‘r, PF = 3.6g|.‘r, FR = 2.4 g|.‘r.

(ii) PE = 4g|.‘r, QE = 4.5g|.‘r, PF = 8g|.‘r, FR = 9g|.‘r.

(iii) PQ = 1.28g|.‘r, PR = 2.56g|.‘r, PE = 0.18g|.‘r,

PF = 0.36g|.‘r.

5.eoOmarb AmH¥$Vr‘Ü¶ o AC||BD Am{U CE||DF.

Oa OA = 12 g|.‘r, AB = 9 g|.‘r, OC = 8 g|.‘r Am{U EF =

4.5 g|.‘r Va OE H$mT>m.

6.AmH¥$Vr‘Ü¶o PC||QK Am{U BC||HK. Oa

AQ = 6g|.‘r, QH = 4g|.‘r, HP = 5g|.‘r Am{U KC = 18g|.‘r,

Va AK Am{U PB H$mT>m.

7.{XdgmÀ¶m EH$m {ZpíMV doir EH$m PmS>mMr gmdbr 12.5 ’y$Q> bm§~ nS>Vo. Oa

PmS>mMr C§Mr 5 ’y$Q> Agob Va Ë¶mM doir 20 ’y$Q> bm§~ gmdbr

nmS>Umè¶m PmS>mMr C§Mr {H$Vr Agob?

8. Cä¶m q^Vrbm EH$ {eS>r Q>oH$bobr AgyZ q^VrnmgyZ 6 ’y$Q> A§Vamda {eS>rMo Xþgao Q>moH$ O{‘Zrda Amho.

O{‘Zrdarb EH$ ‘mUyg {eS>rda XmoZ V¥{VAm§e ^mJ MT>Vmo Va Ë¶mMo {eS>rnmgyZMo AmVmMo A§Va H$mT>m.

B WoëgÀ¶m à‘o¶mMr àm¶mo{JH$ Cn¶mo{JVm :

1. ABC À¶m AmV x hm H$moUVmhr q~Xÿ Amho XA, XB Am{U XC

OmoS>m. AX d 'E' hm H$moUVmhr q~Xÿ Amho. Oa EF II AB, FG II BC Va

{gÕ H$am H$s ||EG AC . (gyMZm : ‘y.à.à‘o¶ Am{U ‘ybVËd).

O

CA

DB

E

F

A

KQ

CB

H

P

A

B C

E

F G x

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2. ABC ‘Ü¶o, B = C, AB Am{U AC da D Am{U E Ago q~Xÿ

AmhoV H$s BD CE {gÕ H$am DE II BC.

3 ABC, ‘Ü¶o, PQ || BC Am{U BD || DC,

{gÕ H$am PE || EQ

(gyMZm : ABC ‘Ü¶o AD ‘Ü¶Jm BC bm g‘m§Va AgUmè¶m aofobmXþ^mJVo).

4. AmH¥$VrV PR || BC Am{U QR || BD

{gÕ H$am, PQ || CD.

5. ABC ‘Ü¶o DE || BC Am{U CD || EF {gÕ H$am

AD2 = AF × AB

{ÌH$m oUm §À¶m g‘ê$nVogmR>rÀ¶m H$gm oQ >r

{ÌH$moUm§Mr EH$ê$nVm R>a{dÊ¶mgmR>r Amnë¶mH$S>o {ZpíMV H$gmoQ>r AmhoV ho AmR>dm. AemM àH$mao

{ÌH$moUm§Mr g‘ê$nVm Ca{dÊ¶mgmR>r H$moUVr H$gmoQ>r Agob? XmoÝhr {ÌH$moUm§À¶m g§JV KQ>H$m§À¶m ghm OmoS>çm

VnmgÊ¶mMr Amdí¶H$Vm Amho H$m? XmoZ {ÌH$moUm§À¶m g‘ê$nVogmR>rÀ¶m H$gmoQ>rMm Aä¶mg H$ê$. ¶m H$gmoQ>tMm

H¥$Vr H$ê$Z emoY bmdy.

XmoZ {ÌH$moUmgmR>r H$moZ-H$moZ (AA) g‘ê$nVm H$gmoQ>r

Amnë¶mbm ‘m{hV Amho H$s Oa EH$m {ÌH$moUmMo XmoZ H$moZ Xþgè¶m {ÌH$moUmÀ¶m XmoZ g§JV H$moZm~amo~a

AgVrb Va {ÌH$moUmÀ¶m H$moZm§À¶m ~oaOoÀ¶m JwUY‘m©dê$Z Ë¶m§Mm {Vgam H$moZ XoIrb g‘mZ AgVmo. åhUyZ AAA

A

B C

D E

A

B CD

EP Q

A B

C

D

R

P

Q

A

B C

D E

F

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284 UNIT-11

g‘ê$nVm H$gmoQ>r Aem àH$mao gm§JVm ¶oVo. ""Oa EH$m {ÌH$moUmMo XmoZ H$moZ Xþgè¶m {ÌH$moUmÀ¶m XmoZ g§JV

H$moZm~amo~a AgVrb Va XmoZ {ÌH$moU g‘ê$n AgVmV''. ¶mMm C„oI XmoZ {ÌH$moUmMr g‘ê$nVm H$gmoQ>r Agm

H$aVm ¶oB©b.

ho ñnï>nUo Ago gm§JVm ¶oVo. "XmoZ {ÌH$moUm‘Ü¶o Oa g§JV H$moZ g‘mZ AgVrb Va Ë¶m§À¶m g§JV ~mOy

à‘mUmV AgVmV Am{U åhUyZ XmoZ {ÌH$moU g‘ê$n AgVmV".

qH$dm

Oa XmoZ {ÌH$moU g‘H$moZ AgVrb Va Ë¶m§À¶m g§JV ~mOy à‘mUmV AgVmV.

AmVm ho {dYmZ VH©$ewÕ nÕVrZo {gÕ H$ê$.

à‘o¶ (H$moZ-H$moZ g‘ê$nVm H$gmoQ>r)

"Oa XmoZ {ÌH$moU g‘H$moZ AgVrb Va Ë¶m§À¶m g§JV ~mOy à‘mUmV AgVmV".A

B C

G H

D

E F

J¥hrV : ABC Am{U DEF ‘Ü¶o

(i) BAC = EDF

(ii) ABC = DEF

gmÜ¶ : AB BC CADE EF FD

aMZm : AB Am{U AC da 'G' Am{U 'H' q~XÿÀ¶m Aem IwUm H$am H$s

(i) AG = DE Am{U (ii) AH = DF . G Am{U H OmoS>m.


AGH Am{U DEF ‘Ü¶o

AG = DE [ aMZm ]

GAH = EDF [ J¥hrV]

AH = DF [ aMZm]

AGH DEF [ SAS J¥hrVH$ ]

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AGH = DEF [ CPCT]

na§Vw ABC = DEF [ J¥hrV]

AGH = ABC [ ‘ybVËd-1]

GH || BC [ Oa g§JV H$moZ g‘mZ AgVrb Va

aofm g‘m§Va AgVmV.]AB BC CAAG GH HA [ WoëgÀ¶m à‘o¶mMm Cn{gÕm§V]

åhUyZAB BC CADE EF FD

[ AGH DEF]


CXmhaU ABC ‘Ü¶o, AL, BM Am{U CN ho {eamob§~ 'O' ‘Ü¶o {‘iVmV.

{gÕ H$am (i) AMB ~ ANC (2)AN BL CM. . = 1BN CL AM

CH$b :

J¥hrV ({Xbobo) : ABC ‘Ü¶o AL, BM Am{U

CN ho {eamob§~ AmhoV.

gmÜ¶ : (i) AMB ~ ANC

(ii)AN BL CM. . 1BN CL AM

{gÕVm : ‘Ü¶o AMB Am{U ANC

AMB ANC 90 ( J¥hrV)

BAM NAC ( g‘mB©H$ H$moZ)

AMB ~ ANC ( AA H$gmoQ>r)

Ë¶mMà‘mUo BLA ~ BNC

Am{U BMC ~ ALC

O

A

B C

NM

L

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286 UNIT-11

AMB ~ ANCAN ACAM AB

BLA ~ BNCBL ABBN BC

BMC ~ ALCCM BCCL AC

AN BL CM AC AB BCAM BN CL AB BC AC

AN BL CM 1BN LC AM

qH$dm AN × BL × CM = BN × LC × AM

CXmhaU 2. ABCD g‘b§~ Mm¡H$moZm‘Ü¶o (i) AB || DC (ii) AED ~ BEC Va {gÕ H$am H$s AD=BC

J¥hrV : ABCD g‘b§~ Mm¡H$moZm‘Ü¶o

(i) AB||DC(ii) AED ~ BEC

gmÜ¶ : AD=BC

{gÕVm : EDC Am{U EBA Mr VwbZm H$ê$Z

EDC = EBA [ || DC Am{U ì¶wËH«$‘ H$moZ ]

ECD = EAB

EDC ~ EBA g‘H$moZ {ÌH$moU ]

ED ECEB EA

[ H$mo.H$mo.]

ED EBEC EA

nU AED BEC [ J¥hrV ]

ED EA ADEC EB BC

[ H$mo.H$mo.]

EB EAEA EB [ ‘wiVËd]

EA2 = EB2

EA = EB

A B

CD

E

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EA AD 1EB BC

AD = BC

ho OmUyZ ¿¶m

"{ÌH$moUmÀ¶m H$moUË¶mhr XmoZ ~mOy§Mm g‘mdoe AgUma Am¶V hm {ÌH$moUmÀ¶m

{Vgè¶m ~mOydarb {eamob§~ Am{U n[aì¶mgmMm (circum

diameter) g‘mdoe AgUmè¶m Am¶Vm ~amo~a AgVmo." ho {gÕ

H$am.

J¥hrV : ABC ‘Ü¶o AX BC

O hm ABC Mm n[a‘Ü¶ Amho.

AE hm ì¶mg Amho.

gmÜ¶ : AB.AC = AX.AE

aMZm : E Am{U C OmoS>m.

{gÕVm : AXB Am{U AEC ¶m§Mr VwbZm H$am

AXB ACE 90

ABX AEC [ AY© dVw©iI§S>mVrb H$moZ H$mQ>H$moZ AgVmo.]

AXB ||| ACE [ EH$mM dVw©i I§S>mVrb H$moZ]AX ABAE AC

[ g‘H$moZ {ÌH$moU ]

AB.AC = AX.AE [H$mo.H$mo.H$gmoQ>r]

ñdmÜ¶m¶ 11.3

A A A g‘ê$nVm H$gmoQ>rda AmYm[aV Cnà‘o¶

{Xboë¶m AmH¥$Vr‘Ü¶o AE || DB, BC = 7 go.‘r BD=5 go.‘r

DC = 4 go.‘r Oa CE=12 go.‘r Va AE Am{U AC H$mT>m.

O

A

B C

E

X

go§.‘r

go§.‘r

go§.‘r

go§.‘r

12

A BC

D

E

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288 UNIT-11

2. XYZ ‘Ü¶o P hm XY darb H$moUVmhr EH$ q~Xÿ Amho Am{U

PQ XZ Oa XP = 4 g|‘r, XY=16 g|‘r, Am{U

XZ =24 g|‘r, Va XQ H$mT>m.

3. 90 g|.‘r C§Mr AgUmar ‘wbJr 1.2 ‘r/go doJmZo {Xì¶mÀ¶m

Im§ã¶mÀ¶m VimnmgyZ Xÿa Mmbbr Amho. Oa {Xdm O{‘ZrnmgyZ

3.6 ‘r. da Amho, 4 goH§$XmZ§Va gmdbrMr bm§~r H$mT>m.

B A A g‘ê$nVm H$gmoQ>rda AmYm[aV Cnà‘o¶

1. BAC Am{U BDC ¶m XmoZ H$mQ>H$moZ {ÌH$moUm‘Ü¶o g‘mB©H$

H$U© BC Amho. ~mOy AC Am{U BD ho 'P' ‘Ü¶o N>oXVmV.

Va {gÕ H$am H$s AP. PC = DP. PB

2. PQR Am{U TUS ‘Ü¶o

QPR = UTS = 900 Am{U

PR||TS $Va

{gÕ H$am PQR ~ TUS

3. Mm¡H$moZmMo H$U© EH$‘oH$mZm à‘mUmV {d^mJVmV Va Vmo Mm¡H$moZ g‘b§~ Mm¡H$moZ Amho ho {gÕ H$am.

4.||gm ABCD Mm H$U© BD, hm AE bm 'F' ‘Ü¶o

N>oXVmo.

'E' hm BC da H$moUVmhr q~Xÿ Amho.

{gÕ H$am DE.EF = FB. FA

5. eoOmaÀ¶m AmH ¥ $VrV ABC 90 Am{U

AMP 90

{gÕ H$am.

(i) ABC AMP

X

Z

QP

Y

A

B C

D

P

P

Q RS

T

U

A B

CD

EF

A B

C

P

M

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(ii)CA BCPA MP

6. g‘b§~ Mm¡H$moZ ABCD ‘Ü¶o

||AB DC

||EF AB Am{U =DC 2ABBE 3EC 4

{gÕ H$am H$s 7EF 10AB

XmoZ {ÌH$moUm§Mr (SSS) g‘ê$nVm H$gmoQ>r

Vwåhr ‘mJÀ¶m à‘o¶m‘Ü¶o nm{hbo Amho H$s Oa EH$m {ÌH$moUmMo VrZ H$moZ (AAA) Xþgè¶m {ÌH$moUmÀ¶m VrZ

g§JV H$moZm~amo~a (AAA) AgVrb Va Ë¶m§À¶m g§JV ~mOy (SSS) à‘mUmV AgVmV.

AmVm ¶m {dYmZmMm ì¶Ë¶mg ñnï>nUo gm§Jy

"Oa EH$m {ÌH$moUmÀ¶m VrZ ~mOy (SSS) Xþgè¶m {ÌH$moUmÀ¶m VrZ g§JV ~mOy§À¶m (SSS) à‘mUmV

AgVrb, Va Ë¶m§Mo g§JV H$moZ (AAA) g‘mZ AgVmV".

XmoZ {ÌH$moUmgmR>r ~mOy-H$moZ-~mOy (SAS) g‘ê$nVm H$gmoQ>r

"Oa EH$m {ÌH$moUmÀ¶m XmoZ ~mOy Xþgè¶m {ÌH$moUmÀ¶m XmoZ ~mOy§er à‘mUmV AgVrb Am{U XmoZ ~mOy‘Ü¶o

AgUmam H$moZ g‘mZ Agob Va Vo XmoZ {ÌH$moU g‘H$moZ AgVmV Am{U åhUyZ g‘ê$n AgVmV."

ñdmÜ¶m¶ 11.4

1. Imbrb H$moUË¶m KQ>Zm‘Ü¶o {ÌH$moUm§À¶m OmoS>çm g‘ê$n AmhoV ?

àýmMo CÎma XoÊ¶mgmR>r Vwåhr H$moUVr g‘ê$nVm H$gmoQ>r dmnabr Amho Vo {bhm Am{U g‘ê$n {ÌH$moUm§À¶m

OmoS>çm {MÝhm§À¶m ñdê$nmV {bhm.

AmH¥$Vr1

A

B C

D

E F

60°

40° 40°

60°

A

B C

D

E F15 cm5 cm

6.9

cm2.

3c

m

124 cc mm

AmH¥$Vr 2

CD

EF G

A B

CD

EF G

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290 UNIT-11

AmH¥$Vr 3

W X

YZ

77

4 4O

AmH¥$Vr 4

H J5

46

D

E F

3

2

AmH¥$Vr 5

H

M L

A12.5

5

10 4

3

T

AmH¥$Vr 6

X

Y Z

80°2.5

3

T

U V6

5

80°

à‘o¶ : H$mQ>H$moZ {ÌH$moUm‘Ü¶o H$mQ>H$moZ {eamoq~XÿVyZ H$Um©da H$mT>bobm b§~ H$mQ>H$moZ {ÌH$moUmg XmoZ ‘yi

{ÌH$moUmer g‘ê$n AgUmè¶m {ÌH$moUmV {d^mJVmo.

J¥hrV : ABC, ‘Ü¶o (i) ABC = 90°.

(ii) BD AC.

gmÜ¶ : ADB BDC ABC

{dYmZ H$maU

{gÕVm : ADB Am{U ABC

Mr VwbZm H$ê$Z

(i) ADB = ABC = 90° J¥hrV

(ii) BAD = CAB g‘mB©H$ H$moZ

A

B

D

C

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(iii) ABD = ACB {ÌkH$moUmÀ¶m {VÝhr H$moZm§Mr

~oarO 180° AgVo.

ADB ABC (1) g‘H$moZ {ÌH$moU

BDC Am{U ABC er VwbZm H$am.

(i) BDC = ABC = 90° J¥hrV

(ii) BCD = ACB g‘mB©H$ H$moZ

(iii) DBC = BAC {ÌkH$moUmÀ¶m {VÝhr H$moZm§Mr

~oarO 180° AgVo.

BDC ABC (2) g‘H$moZ {ÌH$moZ

AmVm Amåhr ADB BDC ABC

Cn{gÕm§V

à¶ËZ H$am : darb VrZ Cn{gÕm§VmgmR>r Ë¶m§Mo ì¶Ë¶mg ApñVËdmV AgyZ Vo gË¶ AmhoV. àË¶oH$mMr {gÕVm

JQ>m‘Ü¶o MMm© H$am Am{U ñdV§ÌnUo {bhm.

A

B

D

C

Cn{gÕm§V 1

ADB~ BDC [ g‘H$moZ]

AD DB ABAB BC AC

[ H$moH$mo H$gmoQ>r]

AB2 = AC.AD

JwUmoÎma ‘Ü¶ (G.M)Mr ì¶m»¶m AmR>dm.darb g‘rH$aUmdê$Z Amåhr Ago gm§Jy H$sAC Am{U AD Mm JwUmoÎma ‘Ü¶ AB Amho.

Cn{gÕm§V 2

BDC ~ ABC [ g‘H$moZ]

BD DC BCAB BC AC

[ H$moH$mo H$gmoQ>r]

BC2 = AC.DC

JwUmoÎma ‘Ü¶mÀ¶m ì¶m»¶oZwgma Amåhr Agogm§Jy H$s AC$Am{U DC Mm JwUmoÎma ‘Ü¶BC $Amho.

Cn{gÕm§V 3

ADB~ BDC [ g‘H$moZ]

AD DB ABBD DC BC [ H$moH$mo H$gmoQ>r]

BD2 = AD.DC

nwÝhm JwUmoÎma ‘Ü¶ (G.M)À¶m ì¶m»¶odê$ZAmåhr Ago gm§JVmo H$s BD hm AD.DC

Mm JwUmoÎma ‘Ü¶ Amho.

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292 UNIT-11

CXmhaU 1:- ABC ‘Ü¶o, BAC 90° AD hm {eamob§~. Oa

AB = 2 5, BD = 4 . Va BC Am{U AC H$mT>m.

$CH$b : AB2 = BC.BD [ Cn{gÕm§V]

22 5 BC.4

20BC

45 = BC

CD = BC – BD = 5 – 4 = 1

AC2 = BC.CD AC2 = 5 × 1 AC = 5

CXmhaU 2: ABC, ‘Ü¶o ABC 90° , Am{U BM hm {eamob§~ AmhoAm{U

Oa AM = 16MC Va {gÕ H$am AB = 4 BC

CH$b : AB2 = AC.AM [ Cn{gÕm§V]

AB2 = AC. 16 MC AB = 4 AC. MC

BC2 = AC.MC [ Cn{gÕm§V]

BC = AC. MC

na§Vw AB = 4 AC. MC

AB = 4 BC AC. MC BC

ñdmÜ¶m¶ 11.5

1. ABC, ‘Ü¶o ABC = 90° BD AC

(a) BD = 8g|.‘r, AD = 4g|.‘r Va CD H$mT>m.

(b) AB = 5.7g|.‘r, BD = 3.8g|.‘r CD = 5.4g|.‘r Va BC H$mT>m.

(c) AB = 75g|.‘r, BC = 1‘r, AC = 1.25‘r Va BD H$mT>m.

2. ABC ‘Ü¶o BAC = 90° AD BC

BD = 4g|.‘r, DC = 5g|.‘r, Va x Am{U y H$mT>m.

3. PQR ‘Ü¶o PQR =90° QS PR

PQ = a, QR = b, RP = c Am{U QS = p, Va pc = ab ho XmIdm.

4. PQR ‘Ü¶o PQR =90° QD PR

PD = 4DR, Va PQ = 2QR ho {gÕ H$am.

A C

D

B

?

?

4

2 5

CM

B A

AD

B C

y

5 CB D4

x

A

DR

Q P

Q

p

P

bR

Sac

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5. ABC ‘Ü¶ ABC = 90° BM AC

(a) BM = x + 2, AM = x + 7, CM = x Va x H$mT>m.

(b) AM = 8x 2, MC = 2x 2 Va BM Am{U AB H$mT>m.

g‘ê$n {ÌH$moUm§Mo joÌ’$i

AmVmn¶ªV Amåhr g‘ê$n {ÌH$moUm§À¶m H$moZ Am{U ~mOy§À¶m ~m~VrV XmoZ ‘yb yV JwUY‘mªMm Aä¶mg Ho$bm

Amho. AmVm Vwåhr g‘ê$n {ÌH$moUm§À¶m joÌ’$imÀ¶m g§~§Ym{df¶r OmUyZ KoÊ¶mg ’$ma CËgwH$ Agmb.

g§~§Y g‘OyZ KoÊ¶mgmR>r Imbrb H¥$Vr H$am. g‘Om Imbrb à‘mUo ‘mno Agbobo g‘ê$n {ÌH$moU ABC

Am{U DEF AmhoV.

go§.‘rgo§.‘r

go§.‘r

go§.‘r

go§.‘r

go§.‘rgo§.‘r

go§.‘r

A

B C6

3.8

5

3.4

M50°40°

D

E F

1.9

2.5

31.

7

40° 50°

N

ABC Am{U DEF ‘Ü¶o

BAC EDF 90

ABC DEF 40

BCA EFD 50

ABC DEF

AB 5 2DE 2.5

BC 6 2EF 3

CA 3.8 2FD 1.9

AB BC CA 2DE EF FD

AmVm Ë¶m§À¶m joÌ’$im§À¶m JwUmoÎmamMm {dMma H$ê$.

BC

M

A

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294 UNIT-11

g‘Om AM BC Am{U DN EF.

AM = 3.4 g|.‘r., DN = 1.7 g|.‘r.

4noj

noj

12

12

ABC 63

3.41.7

AB BC CADE EF EDDEF

2 2 2

¶mMm AW© Agm H$s ABC Mo joÌ’$i DEF À¶m joÌ’$imÀ¶m Mma nQ> Amho.

XmoZ g‘ê$n H$mQ>H$moZ {ÌH$moUm§gmR>r Am{U XmoZ g‘ê$n {dembH$moZ {ÌH$moUm§gmR>r hr H¥$Vr nwÝhm H$am. Vw‘MoAZw‘mZ H$m¶ AmhoV? JQ>m‘Ü¶o MMm© H$am.

¶mMo Amåhr Imbrbà‘mUo gm‘mÝ¶ {dYmZ H$ê$."g‘ê$n {ÌH$moUm§Mo joÌ’$i Ë¶m§À¶m g§JV ~mOydarb dJmªÀ¶m à‘mUmV AgVo."

à‘o¶ : "g‘ê$n {ÌH$moUm§Mo joÌ’$i Ë¶m§À¶m g§JV ~mOydarb dJmªÀ¶m à‘mUmV AgVo."A

B C

D

E FML

J¥hrV : ABC DEF

AB BC ACDE EF DF

gmÜ¶ :Mo joÌ’$i

Mo joÌ’$i

2

2ABC BCDEF EF

aMZm : AL BC Am{U DM EF H$mT>m.


ALB Am{U DME ¶m§Mr VwbZm H$am.

ABL = DEM [ J¥hrV]

ALB = DME = 900 [ aMZm]

ALB DME [ AA - H$gmoQ>r]

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AL ABDM DE

na§VwBC AB=EF DE [ J¥hrV]

AL BCDM EF

[ g§H«$‘U JwUY‘©]

Mo joÌ’$i

Mo joÌ’$i

2

2

1

1ABC B

EC ALF DMDEF

Mo joÌ’$i

Mo joÌ’$i

12

12

ABC BC AL BC ALDEF EF DM EF DM

AmVmMo joÌ’$i

Mo joÌ’$i( ABC) BC AL( DEF) EF DM

=BC ALEF DM

=BC BCEF EF

AL BCDM EF

=2

2

BCEF

Q>rn

AB BC AC= =DE EF DF ¶m J¥hrVH$mdê$Z

Mo joÌ’$iMo joÌ’$i

2 2 2

2 2 2ABC AB BC AC= = =DEF DE EF DF


1. ABC ‘Ü¶o, XY || BC Am{U XY {ÌH$moUmg g‘mZ joÌ’$i AgUmè¶m XmoZ ^mJmV {d^mJVm§ Va .

BXAB H$mT>m.

J¥hrV : ABC ‘Ü¶o

A

B C

X Y

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296 UNIT-11

(i) XY || BC

(ii) ABC = 2 AXY

gmÜ¶ :BXAB

{gÜXVm : ABC AXY [ XY || BC]

Mo joÌ’$i

Mo joÌ’$i

2

2

( ABC) AB( AXY) AX [ WoëgMm à‘eo¶]

2

2

2 AB1 AX

2 AB1 AX1 AX

AB2

1 AX1 1AB2

2 1 AB AXAB2

2 1 BXAB2

‘hËdmMo n[aUm‘ : 1. AX 1AB 2

2.XB 2 2AB 2

3.AX 1XB 2 1

2. "Oa XmoZ g‘ê$n {ÌH$moUm§Mo joÌ’$i g‘mZ Agob Va Vo EH$ê$n AgVmV." ho {gÕ H$am.

A

B C E

D

FSimilar triangles

g‘ê$n {ÌH$moU

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J¥hrV : ABC ~ DEF

Mo joÌ’$i ( ABC) = Mo joÌ’$i ( DEF)

gmÜ¶ : ABC DEF

{gÕVm :Mo joÌ’$iMo joÌ’$i

2 2 2

2 2 2

( ABC) AB BC CA( DEF) DE EF FD [ à‘o¶]

2 2 2

2 2 2

AB BC CA1DE EF FD

[ J¥hrV dê$Z

joÌ’$i g‘mZ AgVo]

AB2 = DE2 AB = DEBC2 = EF2 BC = EFCA2 = FD2 CA = FD

ABC DEF [ SSS]

3. D, E Am{U F ho ABC À¶m ~mOy§Mo ‘Ü¶ AmhoV. P,Q Am{U R ho DEF À¶m ~mOy§Mo ‘Ü¶ q~Xÿ AmhoV‘Ü¶q~Xÿ§À¶m IwUm H$ê$Z ZdrZ {ÌH$moU ~Z{dÊ¶mMr {H«$¶m AerM Mmby R>odm.V¶ma hmoUmè¶m {ÌH$moUm§À¶m joÌ’$im‘Ü¶o H$m¶ g§~§Y Amho? Vnmg H$am Am{U AZw‘mZ H$mT>m.Amåhr ¶mnyduM {gÕ Ho$bo Amho H$s,

ADF DBE CEF ABCA

B C

D

E

F

Q

P

R


2 2

22

4

ABC BC BC 4DEF DF BC

DEF =14

ABC


2 2

2 2ABC BC BC 16PQR QR BC

PQR =1

16 ABC

åhUyZ joÌ’$io hr1 11, ,4 16 .........

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298 UNIT-11

hr JwUmoÎma H«$‘mV (G.P) AgyZ r =14

OmUwZ ¿¶m ! (Know this)

g‘ê$n {ÌH$moUm§Mr joÌ’$io1. Ë¶m§À¶m g§JV ~mOydarb Mm¡agm§À¶m.2. Ë¶m§À¶m g§JV {eamob§~m§darb Mm¡agm§À¶m.3. Ë¶m§À¶m g§JV ‘Ü¶Jmdarb Mm¡agm§À¶m.4. Ë¶m§À¶m g§JV n[a{ÌÁ¶m§darb Mm¡agm§À¶m.5. Ë¶m§À¶m g§JV H$moZ Xþ mOH$mdarb Mm¡agm§À¶m.6. Ë¶m§À¶m g§JV {ÌÁ¶m§darb Mm¡agm§À¶m

à‘mUmV AgVmV.

gyMZm : darb ghm {dYmZm§Mo ì¶Ë¶mg XoIrb bmJy nS>VmV.

ñdmÜ¶m¶ 11.6

g‘ê$n {ÌH$moUmda AmYm[aV Cnà‘o¶.1. BC ¶m g‘mZ nm¶mda ABC Am{U BDC AmhoV.

{gÕ H$amMo joÌ’$iMo joÌ’$iABC AO

DODBC

2. ABC Am{U BDE ho XmoZ g‘ wO {ÌH$moU AmhoV.

BD = DC, Va ABC Am{U BDE À¶m joÌ’$im‘Yrb JwUmoÎmaH$mT>m.

3. XmoZ g‘{Û wO {ÌH$moUm§Mo {eamoH$moZ (vertical angles) g‘mZ AgyZ Ë¶m§À¶m joÌ’$im§Mo JwUmoÎma 9 : 16

Amho. Ë¶m§À¶m g§JV {eamob§~m§Mo JwUmoÎma H$mT>m.

4. XmoZ g‘ê$n {ÌH$moUm§Mo {eamob§~ AZwH«$‘o 3 g|.‘r Am{U 5 g|.‘r AmhoV. Ë¶m§À¶m joÌ’$im‘Yrb JwUmoÎma

H$mT>m.

5. g‘b§~ Mm¡H$moZ ABCD ‘Ü¶o

A

BC

D

O

A

B CD

E

60

60

6060

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AB || CD, AB = 2CD Am{U

( AOB) Mo joÌ’$i = 84 g|.‘r2, COD Mo joÌ’$i H$mT>m.

6. darb AmH¥$Vr‘Ü¶o Oa AB = 3CD Va AOB Am{U COD ¶m§À¶m joÌ’$im‘Yrb JwUmoÎma H$mT>m.

A B

CD

O

Similar Triangles

Similar Figures

Similar traingles

Criteria for similarityof triangles

A A Criteria Logical proof

Converse of A A criteria-SSS criteria

SAS Criteria

Theorem related toRight angled triangle

Areas of similartriangles Logical proof

Numerical problemsand riders

Corollary 1

Corollary 2

Corollary 3

Basic ProportionalityTheorem (BPT)

Converse of BPT

Numerical problemsand riders

g‘ê$n {ÌH$moU

g‘ê$n AmH¥$Ë¶m

g‘ê$n {ÌH$moU

‘yb yV à‘mUmMm à‘o¶ (‘y.à.à) BPT

‘yb^yV à‘mUmÀ¶m‹ à‘o¶mÀ¶m ì¶Ë¶mg

Cn{gÕm§V 1

Cn{gÕm§V 2

Cn{gÕm§V 3

g§»¶mË‘H$ CXmhaUo Am{UCnà‘o¶H$mQ>H$moZ {ÌH$moUmer g§~§§§{YV

à‘o¶

VH©$ {gÕm§Vg‘ê$n {ÌH$moUm§MrjoÌnio

SAS H$gmoQ>r

AA H$gmoQ>r ì¶Ë¶mgSSS - H$gmoQ>r

AA H$gmoQ>r VH©$ {gÕm§V

{ÌH$moUm§À¶m g‘ê$nVoMrH$gmoQ>r

g§»¶mË‘H$ CXmhaUo Am{UCnà‘o¶

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300 UNIT-11

CÎmao

ñdmÜ¶m¶ 11.2

A 2) BE = 10.5 g|.‘r, CE = 7.5 g|.‘r. 3) (i) 20 g|.‘r. (ii) 6 g|.‘r. (iii) x = 1

5) 6 g|.‘r. 6) AK = 12 g|.‘r, PB = 10 g|.‘r. 7) 8 ’y$Q> 8) 2 ’y$Q>

ñdmÜ¶m¶ 11.3

A 1) 15 g|.‘r. 21g|.‘r. 2) 2.6 g|.‘r. ` 3) 1.6 ‘r.

ñdmÜ¶m¶ 11.5

1) (a) 16 g|.‘r. (b) 6.6 g|.‘r. (c) 0.6 g|.‘r. 2) x =6 g|.‘r. y = 52 g|.‘r. 5) (a) 34

(b) 52 24 4x x

ñdmÜ¶m¶ 11.6

2) 4:1 3) 3:4 4) 9:25 5) 21 g|.‘r2. 6) 9:1

********

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¶m KQ>H$m§‘wio Vw‘Mo ¶m‘Ü¶o gwb^rH$aU hmoVo

* nm¶W°Jmoag n«‘o¶.* nm¶W°JmoagÀ¶m à‘o¶mMm ì¶Ë¶mg* nm¶W°JmoagMr {ÌHo$* nm¶W°JmoagÀ¶m à‘o¶mMr {gÜXVm Am{U Ë¶mMm ì¶Ë¶mg.* nm¶W°Jmoag à‘o¶mda AmYmarV àíZ

Am{U Cnà‘o¶.

12

There is geometry in the humming ofthe strings, there is music in thespacing of spheres.

-Pythagoras

nm¶W°Jmoag à‘o¶Pythagorus Theorem

* nm`W°JmoagÀ`m à‘o¶mMo {dYmZ H$aUo

* VH©$ewÜX nÜXVrZo nm¶W°Jmoag à‘o¶ {gÜX H$aUo

* nm¶W°JmoagÀ¶m à‘o¶mÀ¶m ì¶Ë¶mgmMo ñnï>rH$aU.

* VH©$ewÜX nÜXVrZo nm¶W°Jmoag à‘o¶mMm ì¶Ë¶mg {gÜX

H$aUo.

* nm¶W°JmoagÀ¶m {ÌH$m§À¶m AWm©Mo ñnï>rH$aU.

* AZw‘m{ZH$ ¶w{º$dmX Am{U nm¶W°Jmoag à‘o¶ d Cn

à‘o¶mda AmYm[aV à‘o¶ Am{U Ë¶m§Mo ì¶Ë¶mg m§Mr {gÜXVm.

* nm¶W°Jmoag à‘o¶mda AmYm[aV dmñVd OrdZmdarb

CXmhaUm§Mo n¥W:H$aU H$aUo Am{U gmoS>{dUo.

nm¶W°Jmoag

(569-479 B.C, J«rg)

nm¶W°Jmoag WoëgMm {eî¶ hm H$Xm{MVH$mQ>H$moZ {ÌH$moUmdarb à‘o¶mMr Mm§Jbr‘m{hVr AgUmam, H$s Á¶mbm Ë¶mMo Zm§d{Xbo Amho. nXm§À¶m ‘m{hV Zgboë¶mì¶m»¶m ñnï> ì¶m»¶m, J¥hrVVËdo, VH©$ewÜXAZw‘mZ ¶m §À¶m ‘XVrZo ^ y{‘VrÀ¶mAä¶mgmMr VH©$ewÜX arVrZo àJVr H$aÊ¶mg¶mZo ‘XV Ho$br. J{UV {dkmZ, g§JrV,Y‘© Am{U VËdkmZ, ¶m joÌmVrb Ë¶mMogm‘Ï¶© AmOXoIrb MQ>H$Z bjmV ¶oVo.

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Pythagoras Theorem 302

Amåhr ¶mnydu {eH$bobo H$mQ>H$moZ {ÌH$moUmMo H$m§hr JwUY‘© AmR>dy.1. H$mQ>H$moZmÀ¶m {déÜX AgUmè¶m ~mOybm H$U© åhUVmV.2. H$mQ>H$moZ {ÌH$moUm‘Ü¶o, H$U© hr gdm©V bm§~ ~mOy AgVo.3. g‘{Û wO H$mQ>H$moZ {ÌH$moUm‘Ü¶o àË¶oH$ bKwH$moZ 450 AgVmo.4.H$mQ>H$moZ {ÌH$moUmMo joÌ’$i ho H$mQ>H$moZ H$aUmè¶m ~mOy§À¶m

JwUmH$mamÀ¶ {Zå‘o AgVo.5.H$mQ>H$moZ {eamoq~XÿVyZ H$Um©da b§~ H$mT>ë¶mg Vmo {ÌH$moUmbm XmoZ g‘én

{ÌH$moUm‘Ü¶o {d^mJVmo Oo {Xboë¶m H$mQ>H$moZ {ÌH$moUmbm g‘én AgVmV.AmVm,Amåhr H$mQ>H$moZ{ÌH$moUm§À¶m ~m~VrV AmUIr EH$ A{Ve¶ ‘Zmoa§OH$ JwUY‘m©Mm Aä¶mg H$é.

JwUY‘© g‘OyZ KoÊ¶mgmR>r, Imbrb H¥$Vr H$am.* AB = 4 g|.‘r. BC = 3 g|.‘r., AC = 5

g|.‘r. AgUmam ABC, aMm.* [ZarjU H$am H$s, ABC ~amo~a 90° AgVmo.* {VÝhr ~mOyda Mm¡ag aMm Am{U Ë¶m§Mo àË¶oH$s 1

g|.‘r. ~mOy AgUmè¶m Mm¡agm‘Ü¶o {d^mJ H$am.AmVm {VÝhr ~mOyda AgUmè¶m bhmZ Mm¡agm§Mrg§»¶m ‘moOm. Ë¶m 9,16 Am{U 25 AmhoV. OaAmåhr9 Am{U 16 {‘i{dbo Va Amåhmbm 25{_iVo.{4g|.‘r., 6 g|.‘r. Am{U 10 g|.‘r.}, {5 g|.‘r.,12 g|.‘r, 13 g|.‘r} BË¶mXr, ~mOy AgUmè¶m H$mQ>H$moZ{ÌH$moUm§gmR>r hr H¥$Vr nwÝhm H$am. ¶mdéZAmåhr H$m¶AZw‘mZ H$mT>Vmo? AmåhrAgm AZw‘mZ H$mT>Vmo H$s,""H$mQ>H$m oZ {ÌH$moUm‘Ü¶o, H$Um©darb Mm¡ag hm BVa XmoZ ~mOydarb Mm¡agm§À¶m ~oaOoBVH$mAgVm o''¶m {dYmZm‘wio H$mQ>H$moZ {ÌH$moUmÀ¶m ~mOydarb joÌ’$im‘Ü¶o

AgUmam g§~§Y {‘iVmo Á¶mbm nm¶W°Jmoagà‘o¶ åhUVmV, gw‘mao 500BC À¶m Xaå¶mZ hmoD$Z Joboë¶m J«rH$ J{UVVk nm¶W°JmoagÀ¶m Zm§dmdéZ ¶mbm nm¶W°Jmoag à‘o¶ AgoZm§d nS>bo. ¶m à‘o¶mMm Cn¶moJ J{UVmÀ¶m AZoH$ emIm‘Ü¶o Ho$bm AgyZ Ë¶mZr AZoH$ J{UVVkm§Mo bj AmH${f©V H$éZ KoVbo Am{U AmOAmnë¶mH$S>o ¶mgmR>r eoH$S>mo {d{dY nwamdo AmhoV. nm¶W°Jmoag à‘o¶mMo{dYmZ àm¶mo{JH$[aË¶m Xþgè¶m ‘mJm©Zo {gÜX H$aVm ¶oVo. ho hoÝar no[aJb(1830) ¶m§Zr {Xbo. åhUyZ ¶mbm no[aJb {dÀN>oXZ Ago åhUVmV.

Imbr {Xboë¶m H¥$VrMm Aä¶mg H$am Am{U Vr JQ>m‘Ü¶o H$am.* AmH¥$VrV XmI{dë¶mà‘mUo nwÇ>>çmdaC ‘Ü¶o H$mQ>H$moZ H$aUmam

ABC, aMm.

3 cm

3 cm

3cm

3cm

4cm

4

4 cm

4 cm

cm

5 cm

m

5 cm 5cm

5c

A

B C

BC

P

EF

A

DG

5

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A

B C

D

* AB, BC Am{U CA da Mm¡ag H$mT>m. ¶m Mm¡agm‘Ü¶o, AmH¥$VrVXmI{dë¶mà‘mUo Am{U Imbr {Xboë¶m nm¶è¶m à‘mUo aMZm H$am.

* H$mQ>H$moZ H$aUmè¶m ‘moR>çm BC ~mOydarb Mm¡agm‘Ü¶o P ‘Ü¶q~Xÿ {ZpíMV H$am.(Mm¡agmMo H$U© OmoSy>Zho {‘i{dVm ¶oVo)

* P ‘YyZ DE || AB Am{U FG DE .H$mT>m.* V¶ma hmoUmè¶m Mm¡H$moZm§Zm 1,2,3 Am{U 4, Ago H«$‘m§H$ Úm Am{U AC darb Mm¡agmbm 5 H«$‘m§H$ Úm

( AmH¥$VrMo {ZarjU H$am)* Mm¡H$moZ 1,2,3,4, Am{U 5.H$mnm.* H$U© AB.da H$mT>boë¶m Mm¡agmda H$mnboë¶m AmH¥$Ë¶m Owidm (AmH¥$VrMo {ZarjU H$am)

H$mQ>H$m oZ {ÌH$m oUmÀ¶m ~mOy§À¶m dJm ªÀ¶m joÌ’$im{df¶r Amåhr H$m¶ {ZîH$f© H$mT>Vmo?

nm¶W°Jmoag à‘o¶H$mQ>H$moZ {ÌH$moUm‘Ü¶o, H$Um©darb Mm¡ag hm BVa XmoZ ~mOydarb Mm¡agm§À¶m ~oaOoBVH$m AgVmo.

J¥hrV : ABC ‘Ü¶o, ABC = 90°gmÜ¶ : AB2 + BC2 = CA2

aMZm : BD AC. H$mT>m{gÜXVm : {dYmZ H$m aU

ABC Am{U ADB,Mr VwbZm H$amABC ADB 90 ( J¥hrV Am{U aMZm )

BAD g‘mB©H$

ABC ADB ( g‘H$moZ {ÌH$moU)

ABAD

=ACAB

( A A g‘énVm H$gmoQ>r )

AB2 = AC.AD .....(1)

ABC Am{U BDC, ¶m§Mr VwbZm H$amABC = BDC=900 ( J¥hrV Am{U aMZm )

ACB g‘mB©H$ Amho.

ABC BDC [ g_H$moZ {ÌH$moU]

BC ACDC BC

[ g‘énVm H$gmoQ>r] ( A A g‘énVm H$gmoQ>r )

BC2 = AC.DC .....(2) (1) Am{U (2) {‘idyZ Amåhmbm {‘iVo.

AB2 + BC2 = (AC. AD) + (AC. DC)AB2 + BC2 = AC (AD + DC)AB2 + BC2 = AC. AC = AC2 [ AD + DC = AC) AB2+ BC2 = AC2

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‘m{hV Amho!~m¡ÜXm¶Z ¶m àmMrZ ^maVr¶ J{UVVkmZo 800 BC À¶m Xaå¶mZ V¶maHo$bobr gwwëd gyÌo ho ì¶º$ H$aVmV H$s Am¶VmMm H$U© Ë`mÀ` bm§~r Am{Uê§$XrMr doJir joÌ\$io ~Z{dVmV.

d2 = l2 + b2

¶m H$maUmgmR>r, hm à‘o¶ ""~moYm¶Z à‘o¶'' Agmhr C„oIbm OmVmo.

‘m{hV Amho AmO Amnë¶mH$S>o nm¶W°Jmoag à‘o¶mgmR>r {d{dY nwamdo AmhoV. Ago åhQ>bo OmVo H$s VoWo 300hÿZA{YH$ nwamdo AgyZ Ë¶mn¡H$s H$m§hr {ÌH$moU{‘Vr, gh {ZX}eH$ ^y{‘Vr, gXre BË¶mXrda AmYm[aV AgyZ {gÜXHo$bo AmhoV.

Ë¶mn¡H$s H$m§hr {‘i{dÊ¶mMm à¶ËZ H$a Am{U JQ>m‘Ü¶o MMm© H$am.


nm¶W°Jmoag à‘o¶mda AmYm[aV g§»¶mË‘H$ g‘ñ¶mCXmhaU 1:

H$mQ>H$moZ ABC‘Ü¶o, , AC = 17go.‘r. Am{U AB = 8go.‘r., BC H$mT>m.CH$b : ABC ‘Ü¶o, B= 90° {Xbo Amho

AC2 = AB2 + BC2 nm¶W°Jmoag à‘o¶]BC2 = AC2 - AB2

BC2 = 172 - 82

BC2 = 289 - 64 = 225BC = 15 g|.‘r.

CXmhaU 2 :ABC‘Ü¶o, ABC = 450, AM BC, AM = 4g|.‘r. Am{U BC = 7g|.‘ r. AC Mr bm §~ r

H$mT >m.CH$b : AMB‘Ü¶o,

AMBhm g‘{Û^wO H$mQ>H$moZ {ÌH$moU MA = MB = 4g|.‘r.MC = BC – MB = 7 – 4 = 3go§.‘r MC = 3cm

AMC‘Ü¶o, AC2= AM2 + MC2 [ nm¶W°Jmoag à‘o¶]AC2 = 42 + 32

AC2 = 16 + 9AC2 = 25AC= 5 g|.‘r.

b b

l

l

d

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A

B M C

4cm

7cm

45°

go§.‘rgo§.‘

r

A

B C

178

?

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CXmhaU 3 :

WXYZ ¶m Am¶Vm‘Ü¶o, XY + YZ = 17g|.‘r.Am{UXZ + YW = 26 go§.‘r. Am`VmMr bm §~r Am{U ê§$XrH$T> m.

CH$b : XZ + YW = 26 g|.‘r.

d1 + d2 = 26 g|.‘r.

2d = 26 g|.‘r. [ d1 = d2]

d = 13 g|.‘r.

XZ = YW = 13g|.‘r.

g‘Om bm§~r = XY = x g|.‘r._mZy é§Xr = XW = (17 – x)g|.‘r.

WXY ‘Ü¶o, WX2 + XY2 = WY2 [ nm¶W°Jmoag à‘o¶]

(17 – x)2 + x2 = 132

(289 – 34x + x2) + x2 = 169

(2x2 – 34x + 120 = 0) 2

x2 – 17x + 60 = 0

x2 – 12x – 5x + 60 = 0

x(x – 12) –5(x – 12) = 0

(x – 12)(x – 5) = 0

x – 12 = 0 or x – 5 = 0

x = 12 qH$dm x = 5

bm§~r = 12 g|.‘r., é§Xr = 5 g|.‘r.

CXmhaU 4 :6‘r C§MrÀ¶m Im §~mÀ¶m nm¶mnmgyZ 8‘r. A§Vamda AgUmam EH$ H$sQ>H$ Im§~mÀ¶m {XeoZ o

hihiy gaH$Vm o. H$m §hr A§Va gaH$ë¶mZ§Va Ë¶mMo {Xì¶mÀ¶m Im §~mÀ¶m ‘mÏ¶mnmgyZMo A§Va ho VogaH$boë¶m A§VamBVHo$ AgVo Va Vo H$sQ>H$ {Xì¶mÀ¶m Im§~mÀ¶m nm¶mnmgyZ {H$Vr A§Vamda Amho ?[^mñH$amMm¶m ªÀ¶m brbmdVr]CH$b : H$sQ>H$ Am{U {Xì¶mÀ¶m Im§~mÀ¶m nm¶m‘Yrb A§Va = BD = 8 ‘r.{Xì¶mÀ¶m Im§~mMr C§Mr = AB = 6 ‘r.H$m§§hr A§Va gaH$ë¶mZ§Va, g‘Om H$sQ>H$ C, ¶oWo Amho.

g‘Om AC = CD = x ‘r. BC = (8 - x) ‘r.

X

W Z

Y

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ABC ‘Ü¶o, B 90 AC2 = AB2+BC2 ( nm¶W°Jmoag à‘o¶ )

x2 = 62 + ( 8- x)2

2x = 36 + 64 - 16x + 2x

16x = 100

x =100 6.2516

BC = 8- x

BC = (8-6.25) = 1.75 mH$sQ>H$ {Xì¶mÀ¶m Im§~mÀ¶m nm¶Ï¶mnmgyZ 1.75 ‘r. Xÿa Amho.Q>rn : Amåhr BC = X, ‘mZë¶mg CD = AC = (8 – x)

nm`W°Jmoag à_o`mdarb Cnà_o`.CXmhaU 5: {Xboë¶m AmH¥$Vr‘Ü¶o, AD BC, {gÜX H$am H$s AB2 + CD2 = BD2 + AC2

CH$b : ADC ‘Ü¶o, ADC 90Ð = °

AC2 = AD2 + CD2 (nm¶W°Jmoag à‘o¶mdê$Z ) .......(1)

ADB ‘Ü¶o, ADB 90Ð = ° AB2 = AD2 + BD2 ( nm¶W°Jmoag à‘o¶) .......(2)

( nm¶W°Jmoag à‘o¶)

(2) ‘YyZ (1), dOm H$éZ Amåhmbm {‘iVo,

AB2 – AC2 = AD 2 + BD2 - AD 2 - CD2

AB2 – AC2 = BD2 - CD2

AB2 + CD2 = BD2 + AC2

CXmhaU 6 :ABD‘Ü¶o, BD da C hm Agm q~Xÿ Amho H$s, BC : CD = 1 : 2 Am{U ABC hm g‘^wO

{ÌH$moU Amho Va {gÜX H$am H$s AD2 = 7AC2.CH$b. J¥hrV : ABD ‘Ü¶o, BC : CD = 1 : 2

ABC ‘Ü¶o, AB = BC = CA

gmÜ¶ : AD2 = 7AC2

aMZm: AE BC H$mT>m.

{gÜXVm: ABC ‘Ü¶o

BE = EC =2a

Am{U

A

B

6 =x

=

8-x xC D

A

B

C

D

.

A

B C DE

a a

2a60°60°

32

a

2a

2a

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AE =3

2a

ADE, ‘Ü¶o, AED 90 [ aMZm]

AD2 = AE2 + ED2 [ nm¶W°Jmoag à_o ]

AD2 =2 23 2

2 2a aa

AD2 =223 5

4 2a a

AD2 =2 23 25

4 4a a

AD2 =2 23 25

4a a

AD2 =228

4a

AD2 = 7a2

AD2 = 7 AC2

ñdmÜ¶m¶ 12.1

A nm¶W°Jmoag à‘o¶mdarb g§»¶mË‘H$ CXmhaUo1. H$mQ>H$moZ {ÌH$moUmÀ¶m H$mQ>H$moZ H$aUmè¶m ~mOy 5 g|.‘r. Am{U 12 g|.‘r. AmhoV. {ÌH$moUmMm H$U© H$mT>m.2. 12 g|.‘r. ~mOy Agboë¶m Mm¡agmÀ¶m H$Um©Mr bm§~r H$mT>m.3. Am¶VmH¥$Vr H«$sS>m§JUmÀ¶m H$Um©Mr bm§~r 125 _r.Am{U EH$m ~mOwMr bm§~r 75 _r. Amho Xwgè`m ~mOwMr

bm§~r H$mT>m.

4. LAW‘Ü¶o, LAW 90Ð = ° , LNA 90Ð = °

Am{U LW = 26 g|.‘r.,LN = 6g|.‘r.

Am{U AN = 8 g|.‘r. WA Mr bm§~r H$mT>m.

5. 6 ‘rQ>a é§Xr Agboë¶m XadmOmbm

2 ‘rQ>a C§MrMr H$‘mZ Amho. H$‘mZrMr {ÌÁ¶m H$mT>m.

W A

L

26 cm N8cm

6cm

h = 2m

6m

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6. 9 ’y$Q> C§MrÀ¶m Im§~mdarb ‘moa, Im§~mnmgyZ 27 ’y$Q> A§Vamdarb gmnmbm ~KyZ, H$s Omo Im§~mÀ¶m VimerAgUmè¶m ~rimH$S>o ¶oV Amho Ë¶mbm nH$S>Ê¶mgmR>r CÈ>mU H$aVmo Oa XmoKm§Mm doJ g‘mZ Agë¶mgIm§~mnmgyZ {H$Vr A§Vamda XmoKo EH$Ì ¶oVrb?

B nm¶W°JmoagÀ`m à_o¶mdma AmYmarV Cnà_ o`.

1. MGN ‘Ü¶o, MP GN. Oa MG = a EH$H$, MN = b EH$H$, GP =c

EH$H$ Am{U PN = d EH$H$, Va

{gÜX H$am (a + b)(a – b) = (c + d)(c – d).

2. ABC‘Ü¶o, ABC = 900, BD AC. Oa AB = 'c' EH$H$,BC = 'a' EH$H$, BD = 'p' EH$H$ , CA = 'b' EH$H$.

{gÜX H$am 2 2 2

1 1 1a c p .

3. g‘ wO {ÌH$moUmMr C§Mr Am{U joÌ’$imMo gyÌ aMm.nm¶W°Jm oagMr {ÌHo$

Vwåhr H$mQ>H$moZ {ÌH$moUmÀ¶m VrZ ~mOy§À¶m ‘mnZm‘Ü¶o EH$ {deof nÜXVrMm g§~§Y Amhoho {eH$bm AmhmV.

Oa VrZ Aem, g§»¶m, H$s Á¶m H$mQ>H$moZ {ÌH$moUmÀ¶m {VÝhr ~mOy§Mr ‘mno Agboë¶m YZg§»¶m AgVrb Va Ë¶mZm nm¶W°JmoagMr {ÌHo$ åhUVmV.

Ë¶mn¡H$s H$m§htMr ¶mXr Imbrb V³Ë¶m‘Ü¶o {Xbr Amho Ë¶m§Mm Aä¶mg H$am.A B

3, 4, 5 6, 8, 105, 12, 13 15, 36, 397, 24, 25 14, 48, 5011, 60, 61 44, 240, 244

A ñV§ mVrb àË¶oH$ nm¶W°JmoagÀ¶m {ÌH$mMr VwbZm B ñV§ mVrb g§JV {ÌH$m~amo~a H$am.Amåhr Agm {ZîH$f© H$mT>Vmo H$s, Oa (x, y, z) ho nm¶W°JmoagMo {ÌH$ Agob Va (kx, ky, kz) ho XoIrb nm¶W°JmoagMo{ÌH$ Amho OoWo k N.

OmUyZ ¿¶m!

Imbrb gm‘mÝ¶ gyÌ dmnéZ nm¶W°JmoagMr {ÌHo$ H$mT>Vm ¶oVmV.

Z¡g{J©H$ g§»¶ogmR>r : 2n, (n2 – 1), (n2 + 1) ¶oWo 'n' hr g‘ qH$dm {df‘ AgVo.

{df‘ Z¡g{J©H$ g§»¶m§gmR>r : n, 21 ( 1),2

n 21 ( 1),2

n ¶oWo n {df‘ AgyZ

OoWo n N.

darb gm‘mÝ¶ gyÌmdéZ, 'n' bm qH$‘V XodyZ ‘moR>çm g§»¶oZo nm¶W°JmoagMr {ÌHo$ CËnÞ H$aVm ¶oVmV.

M

G NP

a

d

b

c

a

bpc

A

B C

D

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nm¶W°Jmoag à‘o¶mMm ì¶Ë¶mg (Converse of Pythagorus Theorem)

AmVm AmnU nm¶W°Jmoag à‘o¶mMm ì¶Ë¶mg {eHy$. ¶m gmR>r, XmoZ CXmhaUm§Mm {dMma H$am.

AQ> Aer Amho : ""Oa {ÌH$moUmÀ¶m gdm©V ‘moR>çm ~mOydarb Mm¡ag BVa XmoZ ~mOydarb Mm¡agm§À¶m

~oaOoBVH$m Agob Va Ë¶m Xm oZ ~mOy‘Yrb H$moZ H$mQ>H$moZ AgVmo.

hm nm¶W°Jmoag à‘o¶mMm ì¶Ë¶mg Amho. ¶mMm gd© àW‘ ¶wp³bS>Zo C„oI Ho$bm Am{U {gÜX Ho$bm.

AmVm nm¶W°Jmoag à‘o¶mMm ì¶Ë¶mg VH©$ewÜX nÜXVrZo {gÜX H$é.

CXmhaU 2

P

Q R5 cm

4cm

4.3 cm

• g‘Om PQR hm QR = 5go§.‘r., PQ = 4g|.‘r.Am{U PR = 4.3 g|.‘r.AgUma Xwgam {ÌH$moU Amho

• {VÝhr ~mOydarb Mm¡agm§Mr H$ënZm H$am Am{UË¶m§Mo joÌ’$i H$mT>m.

• ¶oWo gdm©V bm§~ ~mOy QR Amho Am{U Ë¶mMrbm§~r 5 g|.‘r. Amho.QR ¶m gdm©V ‘moR>çm~mOydarb Mm¡agmMo joÌ’$i 25 Mm¡. g|.‘r. Amho.

• BVa XmoZ ~mOy PQ Am{U PR darb (16 + 18.49)

Mm¡.g|.‘r.= 34.49 Mm¡. g|.‘r. Agob Or gdm©Vbm§~ ~mOy QR darb Mm¡agmÀ¶m joÌ’$imBVH$s Zmhr.

• AmVm QPRÐ ‘moOm, Omo gdm©V bm§~ ~mOyQR, bm {déÜX Amho Vwåhmbm Ago {XgyZ ¶oB©bH$s QPR 90° Mm Zmhr.

• åhUyZ, PQR hm H$mQ>H$moZ {ÌH$moU Zmhr.

CXmhaU 1

A

B C5 cm

4 cm

3cm

• g‘Om ABC hm BC = 5 g|.‘r. AB =4go§.‘r Am{U AC =3 go§.‘r AgUmam {ÌH$moU Amho.• {VÝhr ~mOydarb Mm¡agm§Mr H$ënZm H$am Am{U

Ë¶m§Mo joÌ’$i H$mT>m.• ¶oWo gdm©V bm§~ ~mOy BC Am{U Ë¶mMr bm§~r 5

g|.‘r. Amho BC ¶m gdm©V ‘moR>çm ~mOydarbMm¡agmMo joÌ’$i 25 Mm¡. g|.‘r. Amho

• BVa XmoZ ~mOy AB Am{U AC darb Mm¡agm§Mr~oarO (16 + 9) Mm¡.g|.‘r. Agob Vr XoIrb 25Mm¡, go._r.Amho.

• AmVm BACÐ ‘moOm, Omo gdm©V bm§~ ~mOy BC

bm {déÜX Amho. Vwåhmbm ABCÐ = 90°

{‘iob.• åhUyZ, ABC hm A {eamoq~Xyer H$mQ>H$moZ H$aUma

H$mQ>H$moZ {ÌH$moZ Amho• åhUyZ {ÌH$moU H$mQ>H$moZ AgÊ`mgmR>r H$moUVr

_yb yV AQ> Amho darb CXmhaUmVrb XmoZ{ÌH$moUm§À`m ~mOydarb joÌ\$im§Mr VwbZm H$amAm{U Amdí`H$ AQ> g_OyZ KoÊ`mMm à`ËZH$am.

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nm¶W°JmoagÀ¶m à‘o¶mMm ì¶Ë¶mg"Oa {ÌH$moUmÀ¶m gdm©V ‘moR>çm ~mOydarb Mm¡ag hm BVa XmoZ ~mOydarb Mm¡agm§À¶m ~oaOo BVH$m Agob. Va Ë¶mXmoZ ~mOy‘Yrb H$moZ H$mQ>H$moZ AgVmo".

A

B CD

J¥hrV : ABC ‘Ü¶o, AC2 = AB2 + BC2

gmÜ¶ : ABC = 90°Ð

aMZm : B ‘Ü¶o AB b§~ H$mT>m Ë¶mda 'D' q~Xÿ Agm {ZdS>m H$s DB = BC. 'A'Am{U D OmoS>m.

{dYmZ H$m aU

{gÜXVm : ABD‘Ü¶o,

ABD = 90° ( aMZm)

AD2 = AB2 + BD2 (nm¶W°Jmoag à‘o¶)

na§Vw ABC ‘Ü¶o,

AC2 = AB2 + BC2 ( J¥hrV)

AD2 = AC2

AD = AC

ABD Am{U ABC ¶m§Mr VwbZm H$am.

AD = AC ( {gÜX)

BD = BC ( aMZm)

AB hr g‘mB©H$

ABD ABC ( SSS)

ABD = ABC (PCT)

na§Vw ABD + ABC = 180° ( aofr¶ OmoS>r)

ABD = ABC = 90°

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ho OmUyZ ¿¶m : {ÌH$moUmMm {VÝhr ~mOyÀ¶m Mm¡agmdarb g§~§YmMm Aä¶mg H$am.


CXmhaU 1 : Imbrb ‘mno H$mQ>H$moZ {ÌH$moUmÀ¶m ~mOy Xe©{dVmV H$m¶ ho nS>VmiyZ nhm. (a) 6, 8, 10

CH$b. : 6, 8, 10 ¶m ~mOy AmhoV.

Ago ‘mZy H$s ~mOydarb Mm¡agm§À¶m joÌ’$im§Mr ~oarO : 62, 82, 102

i.e., 36, 64, 100

Amåhr Ago {ZarajU H$aVmo H$s, {ÌH$moUmÀ¶m gdm©V ‘moR>çm ~mOydarb Mm¡ag hm BVa XmoZ ~mOydarbMm¡agm§À¶m ~oaOoBVH$m Amho. : 36 + 64 = 100 62 + 82 = 102

nm¶W°Jmoag à‘o¶mÀ¶m ì¶Ë¶mgmdéZ, Ë¶m XmoZ bhmZ ~mOy‘Ü¶o H$mQ>H$moZ AgbmM nm{hOo.

{ZîH$f© : 6, 8 Am{U 10 ~mOy H$mQ>H$moZ {ÌH$moU ~Z{dVmV Á¶m‘Ü¶o H$U© 10 EH$H$ Am{U 6 Am{U 8EH$H$ Agboë¶m ~mOy‘Yrb H$moZ H$mQ>H$moZ AgVmo.

Q>rn: {Xboë¶m ~mOy§À¶m ‘mnmgmR>r àË¶jmV ÌH$moUmMr aMZm Z H$aVm, nm¶W°Jmoag à‘o¶mMm ì¶Ë¶mg dmnéZAmVm Ago gm§JUo e³¶ Amho H$s, Ë¶m ~mOy H$mQ>H$moZ {ÌH$moUmÀ¶m AmhoV ho Xe©{dbo OmVo.

(b)4, 5, 6

~mOy : 4, 5, 6~mOydarb Mm¡agm§Mo joÌ’$i åhUOoM :42, 52, 62

åhUOo, : 16, 25, 36XmoZ bhmZ ~mOydarb Mm¡agm§À¶m joÌ’$im§Mr ~oarO

: 16 + 25 = 41

42 + 52 62

A

B C a

bc

ABC ‘Ü¶o, ABCABC = 90°AC2 = AB2 + BC2

b2 = c2 + a2

Oa b2 = c2 + a2

Va ABC 90°=

A

B C

bc

a*

ABC ‘Ü¶o,ABCABC < 90°

AC2< AB2 + BC2 ì¶Ë¶mgb2 < c2 + a2

Oa b2 < c2 + a2

Va ABC < 90°Ð

*

A

B C a

bc

ABC ‘Ü¶o,ABCABC < 90°AC2 > AB2 + BC2

b2 > c2 + a2

Oa b2 > c2 + a2

Va ABC 90°

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Amåhr {ZarjU H$aVmo H$s {ÌH$moUmÀ¶m gdm©V ‘moR>çm ~mOydarb Mm¡ag hm BVa XmoZ ~mOydarb Mm¡agm§À¶m~oaOoBVH$m Zmhr.

nm¶W°Jmoag à‘o¶mÀ¶m ì¶Ë¶mgmdéZ, ¶m XmoZ ~mOy‘Ü¶o hmoUmam H$moZ H$mQ>H$moZ Zmhr.

åhUyZ, 4, 5, Am{U 6 ¶m H$mQ>H$moZ {ÌH$moUmÀ¶m ~mOy hmoUma ZmhrV..

CXmhaU 2 : Mm¡H$moZ ABCD, ABC 90 Am{U

AD2 = (AB2 + BC2 + CD2). Va {gÜX H$am : H$s ACD 90

A

B C

D

{gÜXVm : ABC‘Ü¶o, ABC < 90°Ð ABC 90= ° [ J¥hrV]

AC2 = AB2 + BC2 [ nm¶W°Jmoag à‘o¶]

na§Vw AD2 = (AB2 + BC2)+CD2 [ J¥hrV]

AD2 = AC2 + CD2 [ J¥hrVmdéZ AB2 + BC2 = AC2]

ACDÐ = 90° [ nm¶W°Jmoag à‘o¶mMm ì¶Ë¶mg]

ñdmÜ¶m¶ 12.2

1. nS>VmiyZ nhm H$s Imbrb ‘mno H$mQ>H$moZ {ÌH$moUmÀ¶m ~mOy Xe©{dVmV.

(i) 1, 2, 3 (ii) 2, 3, 5

(iii) 36 12 6, , (iv)m2 – n2, 2mn, m2 + n2

2. ABC‘Ü¶o, a + b = 18 EH$H$, b + c = 25 EH$H$ Am{U c + a = 17EH$H$ Amho. ABC hm H$moUË¶m àH$maMm {ÌH$moU Amho ? H$maUo Úm.

3. ABC ‘Ü¶o, CD AB, CA = 2AD Am{U BD = 3AD,

{gÜX H$am H$s BCA 90

4. 'A' q~XÿnmgyZ QR Mo H$‘rVH$‘r A§Va 12 g|.‘r.

Amho. 'A' nmgyZ 'Q' Am{U 'R' ho AZwH«$‘o 15 g|.‘r.

Am{U 20 g|.‘r AgyZ AP ~mOyÀ¶m {déÜX AmhoV. {gÜX H$am

QAR 90Ð = °

A

B CA

Q P R

15 c

m

20 cm

12c m

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5. g‘[Û wO ABC ‘Ü¶o,

(i) AB = AC (ii) BC = 18 go§.‘r.

(iii) AD BC (iv) AD = 12 g|.‘r.

(v)BC hr 'E' n¶ªV dmT>{dbr.

(vi) AE = 20 g|.‘r.Va

{gÜX H$am H$s BAE 90Ð = °

6. Mm¡H$moZ ABCD ‘Ü¶o

ADC 90Ð = ° , AB = 9g|.‘r., BC = AD = 6 g|.‘r.

Am{U CD = 3g|.‘r.Va, {gÜX H$am H$s ACB 90Ð =

7. ABCD hm Am¶V Amho. 'P' hm ~mhoaÀ¶m ~mOyg H$moUVmhrq~Xÿ Agm Amho H$s

PA2 + PC2 = BA2 + BD2. Va {gÜX H$am H$s APC 90Ð = °

A

B CD E18 cm

20 cm

12cm

A

B C

D

P

B

CD

9 cm

3 cm

6cm

6 cm

nm¶W°Jmoag à‘o¶

nm¶W°Jmoag à‘o¶ OmVrà{Vkm nm¶W°Jmoag à‘o¶mMm ì¶Ë¶mg nm¶W°JmoagMr{ÌHo$

VH$m©Ë‘H$ {gÕVm {gÕm§V

ñdmÜ¶m¶ 11.11. 13 g§o.‘r 2. 12 2 g§o.‘r 3. 100 ‘r 4. 24 g§o.‘r

5. 3. 25 g§o.‘r 6. 12 ’y$Q>

********

CÎmao

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Vwåhmbm ¶m KQ>H$m‘wio ¶m§Mo gwb^rH$aU hmoVo

* À¶m ~m~VrV (g§X^m©V) H$mQ>H$moZ {ÌH$moUmMr {déÕ,

gb¾ Am{U H$U© AmoiIUo.

* H$mQ>H$moZ {ÌH$moUmÀ¶m ~mOy§er g§~§{YV ghm JwUmoÎmam§Mr

ì¶m»¶m H$aUo.

* {ÌH$moU{‘VrMr ghm JwUmoÎmao AmoiIUo.

* {Xboë¶m H$mQ>H$m oZ {ÌH$moUmÀ¶m, {ÌH$moU{‘VrÀ¶m

JwU§moÎmam§Mr qH$‘V H$mT>Uo.

* 0°, 30°, 45°, 60° Am{U 90° ¶m H$m§hr AmXe©

H$moZm§À¶m, {ÌH$moU{‘VrÀ¶m H$moZm§À¶m qH$‘Vr H$mT>Uo.

* {Xboë¶m {ÌH$moU{‘VrÀ¶m JwUmoÎmamMm H$moQ>rH$moZ H$mT>Uo.

* sin2 + cos2 = 1 gyÌ aMU o

* {Xboë¶m KQ>ZoVrb Cƒ nmVirVrb H$moZ Am{U {Zƒ

nmVirVrb H$moZ AmoiIUo.

* C§Mr Am{U A§Va ¶m§À¶mer g§~§{YV CXmhaUo gmoS>{dUo.

13 Trigonometry

* H$mQ>H$moZ {ÌH$moUmÀ¶m ~mOy

* H$mQ>H$moZ {ÌH$moUmÀ¶m ~mOy §Mr ghmJwUmoÎmao

* {ÌH$moU{‘VrMr AmoiI

* AmXe© H$moZm§À¶m, {ÌH$moU{‘VrÀ¶mJwUmoÎmamÀ¶m qH$‘Vr.

* {ÌH$moU{‘Vr¶ g‘rH$aUo

* C§Mr Am{U A§Vamdarb g‘ñ¶m

{hßn°aH$g

Xþgao eV‘mZ BC ‘Yrb J«rH$ J{UV

Vk {hßn°aH$g ¶mZm {ÌH$moU{‘VrMm

{Z‘m©Vm Ago g‘Obo OmVo.

{ÌH$moU{‘Vr eãXmMm AW© {ÌH$moUmMo

‘mnZ ¶mMo lo¶ ~mgWmobm o‘Z

{n{Q>ñH$g (1561-1613) ¶mZm

{‘iVo.

There is perhaps nothing which so occupies themiddle position of mathematics as trigonometry.

-J.F. Herbart

{ÌH$moU{‘Vr

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Trigonometry 315

Vwåhmbm AZoH$ doim AmíM¶© dmQ>bo Agob H$s J{UVr {hoeo~mZo àM§S> (‘moR>çm) C§Mr Am{U A§Va H$go

H$mT>bo OmVo. Amnë¶m g^modVmbMr H$m§hr {R>H$mUo Imbr {Xbr AmhoV, OoWo Aem {heo~mMr JaO Amho.

Aem gd© g‘ê$n KQ>Zm‘Ü¶o, C§Mr Am{U A§Va J{UVr V§ÌmMm Cn¶moJ H$ê$Z H$mT>Vm ¶oVmV, Á¶m§Mm Aä¶mg

J{UVmÀ¶m Á¶m emIoV Ho$bm OmVmo Ë¶mbm "{ÌH$moU{‘Vr" åhUVmV. {ÌH$moU{‘Vr hm eãX Tri (VrZ) gon(~mOy) Am{U metron (‘mnZ) ¶m J«rH$ eãXm§nmgyZ {‘i{dbm Amho.

åhUyZ {ÌH$moU{‘Vr åhUOo VrZ ~mOy ‘moOUo qH$dm {ÌH$moUmÀ¶m VrZ ~mOy ‘moOUo hm {ÌH$moUmÀ¶m ~mOy

Am{U H$moZm‘Yrb g§~§YmMm Aä¶mg Amho.

åhUyZ {ÌH$moU{‘Vr hr {ÌH$moUm{df¶r gd© H$m§hr Amho.

‘§{XamMr C§Mr Hw$Vw~{‘ZmaMr C§Mr

ZXrÀ¶m nmÌmMr ê$§Xr ‘Zmoam Am{U B‘maV ¶m‘Yrb A§Va

{na°{‘S>Mr C§Mrnd©VmMr C§Mr

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316 UNIT-13

P

Q R75° 35°

{ÌH$moU{‘Vrdarb ¶mnydu ‘m{hV Agboë¶m H$m‘mMr Zm|X B{Oá Am{U ~°{~bm°Z ‘Ü¶o hmoVr. àmMrZH$mimVrb J«rH$ IJmobemók {ÌH$moU{‘VrMm Cn¶moJ M§ÐmMo n¥ÏdrnmgyZMo A§Va H$mT>Ê¶mg H$arV AgV.

AmO XoIrb {ÌH$moU{‘Vrdarb gd© gmYmaU H$ënZm§da AmYmarV Am¡Úmo{JH$ H$bm{dkmZmÀ¶m Ñï>rZo nwT>oAgboë¶m nÕVr ^m¡{VH$ {dkmZ, {dÚwV gma»¶m doJdoJù¶m A{^¶m§{ÌH$s emIm, O{‘ZrMr ‘moOUr, {eënemó,Zm¡H$mZ¶Zemó, IJmobemó Am{U gm‘mÝ¶ {dkmZ ¶m‘Ü¶o AË¶§V Cn¶wº$ AgVmV.

{ÌH$moU{‘Vrdarb gd© g§H$ënZm H$mQ>H$moZ {ÌH$moUmda AmYm[aV AmhoV. H$mQ>H$moZ {ÌH$moUmÀ¶m ~mOy§Mrbm§~r Am{U H$moZm§Mo ‘mn ¶m§À¶mer g§~§{YV AmhoV. H$m§hr {R>H$mUr OoWo {ÌH$moUm§Mo kmZ H$er ‘hËdmMr ^y{‘H$m~OmdVo ho Imbr XmI{dbo Amho.

Á¶m AWu {ÌH$moU{‘Vrbm H$mQ>H$moZ {ÌH$moUmÀ¶m H$moZ Am{U ~mOy~amo~a ~aoM H$mhr H$amd¶mMo Amho, VoìhmàW‘ Amdí¶H$ ‘ybVËdo AmR>dy.

g‘Om H$mQ>H$moZ {ÌH$moU Amho. ~mOy Am{U Ë¶m§À¶m Zmdm§Mo {Z[ajU H$am.

nm¶W°JmoagÀ¶m à‘o¶mV {Xì¶mà‘mUo H$mQ>H$moZ {ÌH$moUmVrb~mOy§À¶m g§~§Ym{df¶r Vwåhr ¶mnydu {eH$bm AmhmV. AmVm AmnUAer H$m§hr CXmhaUo KoD$ H$s OooWo {ÌH$moUmÀ¶m ~mOy qH$dm H$moZm§Mm{hemo~ Ho$bm Amho.

CXmhaU 1 : ABC ‘Ü¶o Oa B = 900, AC = 5 g|.‘r, Am{U AB = 3g|.‘r Va BC Mr bm§~r {H$Vr?AC2 = AB2 + BC2 (nm¶W°JmoagÀ¶m à‘o¶mdê$Z)BC2 = AC2 – AB2

= 52 – 32 = 25 – 9 = 16

BC = 16 = 4 g|.‘r

CXmhaU 2 : PQR ‘Ü¶o P Mo ‘mn {H$Vr?P + Q + R = 180° (H$moZm§À¶m ~oaOoMm JwUY‘©)

P = 180° – ( Q + R)= 180° – (75° + 35°) = 180° – 110° = 70°

P = 700

A

B C

HypotenuseLegs or armsof right angleH$mQ>H$moZ {ÌH$moUmMonm¶ qH$dm ~mhÿ

H$U©

5 cm

3cm

B C

A

g|.‘rg|.‘r

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Trigonometry 317

AmVm Imbrb CXmhaUm§Mo {ZarjU H$am.

CXmhaU 3 : DEF ‘Ü¶o DE H$mT>m. CXmhaU 4 : XYZ ‘Ü¶o X H$mT>m.D

E F9 cm

4 cm

X 10 cm

7 cm

Y

80°

Z

nm¶W°JmoagMm à‘o¶ qH$dm H$moZm§À¶m ~oaOoMm JwUY‘© dmnê$Z Amåhmbm ~mOyMr bm§~r qH$dm H$moZmMo ‘mnH$mT>Vm ¶oB©b H$m?

Vw‘Mo dJ©{‘Ì Am{U {ejH$m§~amo~a dJm©V MMm© H$am.

ho CKS> Amho H$s nm¶W°JmoagMm à‘o¶ qH$dm {ÌH$moUmÀ¶m H$moZm§À¶m ~oaOoMm JwUY‘© dmnê$Z Amåhr darbCXmhaUo gmoS>dy eH$V Zm§hr. Aer CXmhaUo H$er gmoS>{dVmV?

{ÌH$moU{‘Vr dmnê$Z ~mOy Am{U H$moZ H$mT>Uo e³¶ Amho.

{ÌH$moU{‘Vr åhUOo {ÌH$moU gmoS>{dUo Am{U gmoS>{dUo åhUOo {ÌH$moUmÀ¶m ~mOy Am{U H$moZ H$mT>Uo.

{ÌH$moU{‘VrMr ‘yb^yVo (Basics of Trigonometry)AmVm AmnU {ÌH$moU{‘VrÀ¶m ‘yb yVm{df¶r {eHy$. {ÌH$moU{‘VrMr ‘yb yVo

hm ’$º$ H$mQ>H$moZ {ÌH$moUm§er g§~§{YV AmhoV.

EH$ H$moZ 90° AgyZ Ë¶mÀ¶m {dê$Õ AgUmar ~mOy H$U© AgUmè¶mABC Mm {dMma H$am.

Ago g‘Om H$s H$mQ>H$moZ gmoSy>Z BVa H$moZmZm Amåhr IwUm Ho$ë¶m Amåhr EH$Va BAC qH$dm ACBAem IwUm H$aVmo, H$s Oo Zoh‘r bKwH$moZ AgVmV. IyU Ho$bobm H$moZ ' ' Zo Xe©{dbm OmVmo. (J«rH$ eãX, WrQ>mAgm dmMVmV)

Voìhm H$moZmer g§~§{YV AgUmè¶m ~mOyZm H$m¶ åhUVmV?

Imbrb AmH¥$Ë¶m§Mo {ZarjU H$am Am{U ~mOyZm Zm§do H$er {Xbr AmhoV ¶mMm Aä¶mg H$am.A

B C

Hypotenuse

Adja

cent

Opposite

A

B C

Hypotenuse

Opp

osite

Adjacent

À¶m {dê$Õ AgUmè¶m ~mOybm {dê$Õ ~mOy Am{U Xþgè¶m ~mOybm gb¾ ~mOy åhUVmV.

Oa A= Va BC hr {dê$Õ ~mOy Am{U AB hr gb¾ ~mOy Amho.

Oa C= Va AB hr {dê$Õ ~mOy Am{U BC hr gb¾ ~mOy Amho.

' ' H$moZmMo ‘mn "A§e" {H§$dm "H$moZ ‘moOÊ¶mMo EH$H$" ¶mZo Xe©{dbo OmVo.

A

B C

Hypotenuse

g|.‘rg|.‘r

g|.‘r g|.‘r

H$U©

H$U©

gb¾

{dê$Õ

{dê$Õ gb¾

H$U©

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318 UNIT-13

6

4

3

2

ho OmUm:

1 EH$H$ bm§~rÀ¶m H§$gmZo 1 EH$H$ {ÌÁ¶oÀ¶m dVw©imV H|$Ðmer A§mVarV Ho$bobm H$moZ

‘moOÊ¶mMo EH$H$ 1C Zoo Xe©{dVmV 1C = 1800

¶oWo H$m§hr CXmhaUo AmhoV.

A§e H$moZ ‘moOÊ¶mMo EH$H$

30°

45°

60°

90°

180°

360°

àH$aU 10 ‘Ü¶o Amåhr g‘ê$n {ÌH$moUmÀ¶m AAA H$gmoQ>r{df¶r Aä¶mg Ho$bm Amho. ¶mMm Cn¶moJ H$ê$ZAmåhr {ÌH$moU{‘VrMr ‘yb yVo {dH${gV H$ê$.


ABC Am{U DEF ho g‘H$moZ {ÌH$moU AmhoV. åhUyZ Ë¶m§À¶m g§JV ~mOy à‘mUmV AmhoV.

5cm

2cm

4cm

A

B C

2.5cm1cm

2cm

D

E F

ABC ~ DEF, ABDE = BC CA

EF FD=

AmVm EH$m doir H$moUË¶mhr XmoZ JwUmoÎmam§Mm {dMma H$am.

ABDE =

BCEF

hr JwUmoÎmao Aerhr {b{hVm ¶oVmV.

11

1

1800

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Trigonometry 319

ABBC =

DEEF [Oaif than

a c a bb d c d

é ù= =ê úë ûVaif than

a c a bb d c d

é ù= =ê úë û]

{Xboë¶m AmH¥$Ë¶m‘Ü¶o

ABBC = 2cm 1

4cm 2= Am{U

DE 1EF 2

=ABBC = DE 1

EF 2=

¶mMmM AW© ABC, ‘Ü¶o BC hr AB À¶m XþßnQ> Am{U EF hr DE À¶m XþßnQ> Amho.

Amåhr Agm AZw‘mZ H$mTy> H$s ABC À¶m ~mOy AB Am{U BC er AgUmam H$moUVmhr g§~§Y

DEF À¶m g§JV ~mOy DE Am{U EF er gmaImM AgVmo. Oa Vo XmoÝhr {ÌH$moU g‘ê$n AgVrb VaABBC = 2cm 1

4cm 2= Am{U

DE 1EF 2

=ABBC = DE 1

EF 2=

hm {ZîH$f© Xþgè¶m g§JV ~mOy§À¶m OmoS>çm§À¶m ~m~VrV {dñVmarV H$ê$.

åhUyZABBC =

DE 1EF 2

= =BCCA =

EF 1FD 2 =

22.5

CAAB =

FD 2.5DE 1

=

g‘Om, H$mQ>H$moZ {ÌH$moUm§À¶m ~m~VrV ¶m ¶moOZoMm {dñVma H$ê$. H$mQ>H$moZ {ÌH$moUm‘Ü¶o Aem XmoZ {deofKQ>Zm AmhoV.

KQ>Zm (i): 45° – 45°– 90° H$mQ>H$moZ {ÌH$moU qH$dm g‘{Û^wO H$mQ>H$moZ {ÌH$moU

g‘Om XmoZ g‘H$moZ g‘{Û wO H$mQ>H$moZ {ÌH$moU AmhoV.ABC ~ DEF

ABBC = DE 1

EF 1= H$mQ>H$moZ H$aUmè¶m

~mOy g‘mZ AgVmV.

BCCA

= EFFD =

12 H$U© hm ~mOy§À¶m

12 nQ> AgVmo.

ACAB = FD 2

DE 1= H$U© hm ~mOy§À¶m 2 nQ> AgVmo.

KQ>Zm (ii): 30°, 60°, 90° qH$dm H$mQ>H$moZ {ÌH$moU

g‘Om {Xboë¶m ‘mnm§Mo ABC Am{U DEF AmhoV.

A

B C

D

E F2

2 4

1

32 3

A

B C

D

E F

1 2 3 6

3 33

ABC ~ DEF ABC ~ DEF

F2

22 2

45°

45°

A

B C1

1 245°

45°

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320 UNIT-13

ABBC = DE

3EF

=ABBC =

DE 1EF 3

=

AB hr Zoh‘r BC À¶m 3 nQ> AgVo. BC hm Zoh‘r AB À¶m 3 nQ> AgVmo.BCCA =

EF 1FD 2

=BCCA = EF 3

FD 2=

CA (H$U©) Zoh‘r BC À¶m XþßnQ> AgVmo. CA (H$U©) BC À¶m XþßnQ> AgyZ Ë¶mbm 3 Zo ^mJVm oCA FD 2AB DE 3

= = CA FD 2AB DE 1

= =

CA (H$U©) hm Zoh‘r AB À¶m XþßnQ> CA (H$U©) Zoh‘r AB À¶m XþßnQ> AgVmo

AgyZ Ë¶mbm 3 Zo ^mJVmodarb CXmhaUmdê$Z Amåhr Agm {ZîH$f© H$mT>Vmo H$s

"H$mQ>H$moZ {ÌH$moUm‘Ü¶o, {Xboë¶m bKwH$moZmgmR>r H$moUË¶mhr XmoZ ~mOy‘Yrb JwUmoÎma Zoh‘r pñWa AgVo."åhUyZ ¶mnwT>o JwUmoÎma {b{hÊ¶mgmR>r XmoZ g‘ê$n H$mQ>H$moZ {ÌH$moUmEodOr EH$mM H$mQ>H$moZ {ÌH$moZmMm

{dMma H$ê$.{ÌH$moU{‘VrMr JwUmoÎmao : (Trignometric ratios)

g‘Om EH$ H$mQ>H$moZ {ÌH$moU Amho Ago ‘mZy Am{U Ë¶m§À¶m bKwH$moZm§À¶m g§X^m©V Ë¶m§À¶m ~mOy§Mr e³¶{VVH$s JwUmoÎma {bhÿ.

AB AB BC, ,

AC BC ACAC BC AC

, ,AB AB BC

{ÌH$moUmbm VrZ ~mOy AmhoV. JwUmoÎma (fraction) {b{hVmZm A§e VrZ àH$ma Am{U N>oX XmoZ àH$ma{b{hVm ¶oVmo. XmoÝhr {‘iyZ 3 × 2 = 6 àH$mao {b{hVm ¶oVo. åhUyZ {ÌH$moUmÀ¶m ~mOy dmnê$Z Amnë¶mH$S>o ghmdJdoJir JwUmoÎma AmhoV.

{ÌH$moU{‘Vr‘Ü¶o, àË¶oH$ ghm JwUmoÎmamZm Zm§d {Xbo Amho.ghm ‘yb^yV {ÌH$moU{‘VrMr JwUmoÎma o {eH$Ê¶mgmR>r Imbrb V³Ë¶mMm Aä¶mg H$am.

JwUmoÎma JwUmoÎmamMo {ÌH$moU{‘VrVrb Zm§d g§{já ê$nABAC = {dê$Õ

H$U©H$moZmMm gmB©Z gmB©Z (sin

gb½ZH$U©

BCAC = H$moZmMm H$mogmB©Z H$m°g (cos

gb½Z{dê$ÕB

BA

C = H$moZmMm Q>°ZO|Q> Q>°Z (tanH$U©{dê$Õ

ABAC = H$moZmMm H$mogoH§$Q> H$mogoH$ (cosec

BCAC = gb½Z

H$U© H$moZmMm goH§$Q> goH$ (sec

gb½Z{dê$ÕB

BA

C = H$moZmMm H$moQ>°ZO|Q> H$m°Q> (cot

A

B C

Opp

osit

e Hypotenuse

Adjacent

{dê$Õ

gb¾

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Trigonometry 321

¶m ghm {ÌH$moU{‘VrÀ¶m JwUmoÎmamVrb g§~§Y hm "H$mQ>H$moZ {ÌH$moUm‘Ü¶o Oa bKwH$moZ {Xbm Agob VaH$moUË¶mhr XmoZ ~mOy‘Yrb JwUmoÎma H$mT>Vm ¶oVo. CbQ> AWr©, Oa XmoZ ~mOy‘Yrb JwUmoÎma {Xbo Agob Va, H$moZH$mT>Vm ¶oVmo. "AAA g‘ê$nVm H$gmoQ>r‘wio ho gd© e³¶ Amho".

ho ‘m{hV H$ê$Z ¿¶m : gmB©Z (Sine) ¶m H$ënZoMm Cn¶moJ àW‘ Am¶©^Å> (500AD) ¶m§À¶m Am¶© Q>r¶m‘ ¶m H$m‘m‘Ü¶o 500 AD ‘Ü¶o {XgyZ Ambm. na§VwAm¶©^Å>Zo AY©-OrdogmR>r AYu Á¶m eãXmMm Cn¶moJ Ho$bm, Aä¶mg H«$‘mVg§{jáê$nmV Vmo Á¶m qH$dm Ordm Agm {b{hVmV. Am¶© Å>r¶m‘Mo Aao{~H$ ‘Ü¶oê$nm§Va H$aVmZm Ordm eãX VgmM am{hbm. nwT>o Ë¶mMo "gm¶Zg' ‘Ü¶o ê$nm§Va Pmbo,¶mMm b°{Q>Z‘Ü¶o dH«$ Agm AW© hmoVmo. ES>‘§S> Jw§Q>a (1581-1626) ¶mIJmobemómÀ¶m B§J«Or àmÜ¶mnH$mZo gm¶Zg eãX gmB©Z(Sine) ¶m g§{já ê$nm§VaH$ê$Z àW‘ dmnabm. H$mogmB©Z Am{U "Q>°ZO|Q>' eãXm§Mm CJ‘ Iyn Ceram Pmbm.H$moQ>rH$moZmMm gmB©Z ‘moOÊ¶mMr JaO ^mgë¶mda H$mogmB©ZMo H$m¶© gwê$ Pmbo. Am¶© Å>Zo ¶mbm H$moQ>rÁ¶m AgoåhQ>bo. ES>‘§S> Jw§Q>a ~amo~a H$mogm¶ZgMm CJ‘ Pmbm. 1674 ‘Ü¶o ga OmoZg ‘ya ¶m Xþgè¶m B§p½be J{UVVkmZo "H$m°g''cos' ¶m g§{já {MÝhmMm àW‘ Cn¶moJ Ho$bm.

Q>rn :1. H$moZmÀ¶m gmB©ZMo g§{já ê$n gmB©Z (sine Ago Amho, ¶mMm AW© ¶m bKwH$moZmÀ¶m g§X^m©V

{dê$ÜX ~mOy Am{U H$U© ¶m§Mo JwUmoÎma. ¶mbm gmB©Z Am{U ¶m§Mm JwUmH$ma Ago ‘mZVm ¶oV Zm§hr .

BVa {ÌH$moU{‘VrÀ¶m JwUmoÎmam§À¶m ~m~VrV XoIrb AgoM AgVo.

sine Ago Amho H$s-Ë¶mbm {H$§‘V Amho.

sine ho H$mhr Zmhr - ¶mbm H$mhr AW© Zmhr.

gmB©Z Ago H$m§hrVar Amho H$s Á¶mbm qH$‘V Amho. ’$º$ gmB©Zbm H$m§hrM AW© ZgVmo.

2. {ÌH$moU{‘VrMr JwUmÎmao {ÌH$moUmÀ¶m AmH$mamda Adb§~yZ ZgyZ Vr ’$º$ ¶m bKwH$moZmÀ¶m qH$‘Vrda

Adb§~yZ AgVmV.

¶oWo {Xboë¶m {ÌH$moUm§À¶m g§X^m©V

AmnU AmUIr H$m§hr CXmhaUo KoD$Z {ÌH$moU{‘VrMr JwUmoÎmao {bhÿ.

A

B C

A

B C

A

B C30° 30° 30°

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322 UNIT-13

RQ

P

3

4

5


CXmhaU 1: g‘Om H$mQ>H$moZ {ÌH$moU ABC ‘Ü¶o 90ABCÐ = °

CH$b : ¶oWo CAB hm bKwH$moZ Amho. H$moZ A À¶m g§X^m©V ~mOy§À¶m pñWVrMo {Z[ajU H$am.

- BC hr H$moZ A Mr {dê$Õ ~mOy Amho.

- AB hr H$moZ A À¶m g§X^m©V g§b¾ ~mOy Amho.

- AC hm H$mQ>H$moZ {ÌH$moU ABC Mm H$U© Amho.

H$mQ>H$moZ {ÌH$moU ABC Mm H$moZ AÀ¶m {ÌH$moU{‘VrMm JwUmoÎmam§Mr ì¶m»¶m Imbrbà‘mUo Amho.

H$mQ>H$moZ {ÌH$moU ABC Mm H$moZ AÀ¶m {ÌH$moU{‘VrMm JwUmoÎmam§Mr ì¶m»¶m Imbrbà‘mUo Amho.

Sin A =BCAC

=H$moZ Mr {dê$ÜX ~mOyA

H$U© H$U©BCA

AA=

H$moZ Mr g§b½Z ~mOy

Tan A =B

BC

A=

H$moZ Mr {dê$ÜX ~mOyA

H$moZ Mr g§b½Z ~mOyA Cosec A = AH$moZ Mr {dê$ÜX ~mOy C

H$U©B

CA=

Sec A =H$U©

H$moZ Mr g§b½Z ~mOy BC

AA=

A CH$moZ Mr g§b½Z ~mOy

H$moZ Mr {dê$ÜX ~mOyB

BA

C=A

A

AmVm AmnU H$mQ>H$moZ {ÌH$moUm‘Yrb C bKwH$moZmgmR>r {ÌH$moU{‘VrÀ¶m JwUmoÎmamMr ì¶m»¶m ñnï> H$ê$

90ABCÐ = °

{ZarjU H$am H$s H$moZ A À¶m OmJr H$moZ 'C' ‘mZë¶mg ~mOy§Mr pñWVr ~XbVo.

Sin C =ABAC Cos C =

BCAC Tan C =

ABBC

Cosec = ACAB

Sec C =ACBC Cot C =

BCAB

CXmhaU 2: PQR gmR>r {ÌH$moU{‘VrMr JwUmoÎmao {bhm.

CH$b : AmH¥$Vr‘Ü¶o,

H$U© = 5 EH$H$, {dê$Õ = 3 EH$H$, g§b¾ = 4 EH$H$.

sin =H$U©

35

={dê$ÜX cosecH$U©{dê$ÜX

53

=

cos =H$U©

g§b½Z 45

= sec =gb½ZH$U© 5

4=

tan =gb½Z{dê$ÜX 3

4= cot =

gb½Z{dê$ÜX

43

=

A

B C

Hypotenuse

OppositeSide

Adj

acen

t sid

e

{dê$Õ ~mOy

gb¾

~mOy

{dê$Õ ~mOy

H$U©A

B C g§b½Z ~mOy{dê$ÜX

~mOy

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Trigonometry 323

XmhaU 3: XYZ gmR>r {ÌH$moU{‘VrMr JwUmoÎmao {bhm.

CH$b : {Xbobo XY = 15, YZ = 8, XZ = ?

XZ2 = XY2 + YZ2 (>nm¶W°Jmoag à‘o¶mdê$Z)= 152 + 82 = 225 + 64 = 289

XZ = 289 = 17.

sin =1517 cos =

817 tan =

158

cosec = 1715

sec =178 cot = 8

15

CXmhaU 4: sin A = 45 , A hm bKwH$moZ Amho.

2 tan A + 3 sec A + 4 sec A . cosec A Mr qH$‘V H$mT>m.

CH$b : {Xbobo sin A =H$U©

{dê$ÜX 45

=

ABC ‘Ü¶o AC2= AB2 + BC2 (nm¶W°Jmoag à‘o¶mdê$Z) BA2 = AC2 – AB2 = 52 – 42 = 25 – 16 = 9

BA = 9 = 3

åhUyZ tan A= gb½Z{dê$ÜX 4

3= sec A =

gb½ZH$U© 5

3= cosec A =

H$U©{dê$ÜX

54

=

2 tan A + 3 sec A + 4 sec A . cosec A ‘Ü¶o qH$‘Vr KmbyZ

= 3 3 3 44 5 5 52 3 4) ) ) )( ( ( (+ +

= 8 255

3 3+ + = 8 15 25 48

163 3

+ += =

2 tan A + 3 sec A + 4 sec A. cosec A = 16

ì¶ñV g§~§Y (ì¶wËH«$‘ g§~§Y) (Reciprocal Relation)g‘Om PQR ‘Ü¶o PQR = 900

Amåhmbm ‘m{hV Amho sin P =H$U©

{dê$ÜX QRPR

=

ZY

X

15

8

AB

C

4 5

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324 UNIT-13

2 1

1 12 12 2

2 1

1 12 12 2

2 1

1 12 12 2

sin P, Mm ì¶ñV H$mT>m åhUOoM 1 1 PRQRsinP QRPR

= =

na§VwH$U©{dê$ÜX

PRcosec P

QR= =

qH$dm1 1

sinP cosec P =cosec P sinP

=

åhUyZ sin P Am{U cosec P EH$‘oH$mZm ì¶ñV AmhoV.

¶mà‘mUo Amåhr BVa {ÌH$moU{‘VrÀ¶m JwUmoÎmam‘Yrb g§~§Y ì¶º$ H$ê$ eH$Vmo.

cos P =gb½Z

H$U©PQPR

= gb½ZH$U©1 1 PR

secPPQcosP PQPR

= = = =

1 1cosP or sec P

secP cosP\ = =

tan P =gb½Z

{dê$ÜXPQ

QR=

gb½Z

{dê$ÜX

1 1 PQcotP

PQtanP QRPR1 1tanP or cotP

cotP tanP

ì¶ñV g§~§YmMr ¶mXr Imbr {Xbr Amho.

sin A = 1cosec A

cos A = 1sec A

tan A = 1cot A

cosec A = 1sin A

sec A = 1cos A

cot A = 1tan A

AmR>dU H$am.

Imbr XmI{dë¶mà‘mUo eãXm§À¶m AmÚmjamZr ~ZUmè¶m eãXm§Mr AmR>dU H$ê$Z {ÌH$moU{‘VrMoJ wUm oÎma ghOnUo AmR>dVmV.

SOH CAH TOA

S O H Sin {déÕ OppositeH$U© Hypotenuse

C A H Cos g§b¾ Adjacent H$U© Hypotenuse

T O A Tan {dê$Õ Opposite g§b¾ Adjacent

darb VrZ JwUmoÎmam§À¶m ì¶ñVm‘wio AZwH«$‘o cosec , Sec Am{U cot {‘iVo.

RQ

P

Ad j

Opp.

HypH$U©

g§b¾

{déÕ

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Trigonometry 325

{ÌH$moU{‘VrÀ¶m JwUmoÎmam‘Yrb g§~§Y : (Relation between the trigonometric ratios) :

g‘Om ABC ‘Ü¶o B 90

gb½Z{dê$ÜX

BCsinA BCAC tanAABcos A AB

AC

sin Atan Acos A

1 1 cos Acot A = sinAtan A sinA

cos A

cos Acot Asin A

{Xboë¶m JwUmoÎmamÀ¶m nXmV {ÌH$moU{‘VrMr JwUmoÎmao ì¶º$ H$aUo :

AmVm Vwåhr {ÌH$moU{‘VrÀ¶m ghm JwUmoÎmam{df¶r n[a{MV AmhmV. Aem n[apñWVr‘Ü¶o EH$ ‘hËdmMmàý CX^dVmo.

Oa Amåhmbm H$moUVohr EH$ JwUmoÎma ‘m{hV Agob Va Amåhr Xþgar JwUmoÎmaoH$mTy> eH$Vmo?

g‘Om ABC. Sin A = BC K 1AC 3K 3

AmVm BVa {ÌH$moU{‘VrÀ¶m JwUmoÎmam§À¶m qH$‘Vr H$mTy>.

Oa sin A =13

, ¶mMm AW©BC 1AC 3

¶mMm AW© ~mOy BC Am{U AC Mr bm§~r 1 : 3 JwUmoÎmamV Amho.åhUyZ Oa BC ~amo~a K Agob Va AC ~amo~a 3K AgVo, OoWo K hr H$moUVrhr YZ g§»¶m AgVo.

BVa JwUmoÎmao {ZpíMV H$aÊ¶mgmR>r Amåhmbm AB ¶m {Vgè¶m ~mOyÀ¶m bm§~rMr JaO Amho.

ABC ‘Ü¶o ABC = 900

AB2 = AC2 – BC2

= (3K)2 – K2 = 9K2 – K2 = 8K2 = 22 2K

AB = 22 2 2 2K K

cos A = AB 2 2K 2 2AC 3K 3

Am{U tan A =BC K 1AB 2 2K 2 2

cosec A = AC 3K3

BC KAm{U sec A =

AC 3K 3AB 2 2K 2 2

cot A = AB 2 2K2 2

BC K

CB

A

Hypotenuse

Opposite side

Adj

acen

tg§b

¾ ~m

Oy

H$U©

{déÕ ~mOy

A B

C

K3K

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326 UNIT-13

ñnï>rH$aUmËa‘H$ CXmhaUo

CXmhaU 1: Oa sin = 5.

13BVa gd© {ÌH$moU{‘VrÀ¶m JwUmoÎmam §À¶m qH$‘Vr {bhm.

CH$b :sin =Opp 5Hyp 13

ABC ‘Ü¶o, B 90 , C

AC2 = AB2 + BC2 (nm¶W°JmoagÀ¶m à‘o¶mdê$Z)BC2 = AC2 – AB2 = 132 – 52 = 169 – 25 = 144

BC = 144 = 12

cos =gb½Z

H$U©

1213 tan = gb½Z

{dê$ÜX125

cosec =H$U©{dê$ÜX

135

sec =gb½Z

H$U© 1312

cot =gb½Z{dê$ÜX

125

CXmhaU 2: Oa cos = 2425 Va {ÌH$moU{‘VrÀ¶m JwUmoÎmam§À¶m qH$‘Vr H$mT>m.

CH$b : ABC ‘Ü¶o B 90 , C

AB2 = AC2 – BC2 (nm¶W°JmoagÀ¶m à‘o¶mdê$Z)

x2 = 252 – 242 = 625 – 576 = 49

x = 49 = 7 AB = 7

AmVm Amåhr Imbrbà‘mUo {ÌH$moU{‘VrMr JwUmoÎmao {bhÿ eH$Vmo.

sin =H$U©

{dê$ÜX 725

cos =gb½ZH$U©

2425 tan =

gb½Z{dê$ÜX 7

24

cosec =H$U©

{dê$ÜX

257 sec =

gb½ZH$U© 25

24 cot =

gb½Z

{dê$ÜX

247

CXmhaU 3 : ABC ‘Ü¶o, AD BC Am{U BAD =

Oa AC= 20 g|.‘r, CD = 16 g|.‘r, Am{U BD = 5 g|.‘r, Va sin + cos Mr qH$‘V HmT>m.

CH$b : ADC ‘Ü¶o, ADC = 900

AD2 + DC2 = AC2

AD2 +162 = 202

CB

A

5 13

x

XB

A

25

24

x

H$U©

{dê$ÜX

257

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Trigonometry 327

AD2 = 202 162 = 400 256 =144

AD = 144 = 12 g|.‘r

AmVm ADB ‘Ü¶o, ADB = 900

AD2 + BD2 = 122 + 52 = AB2 = 144 + 25 = 169

AB = 169 = 13 g|.‘r

Sin BD 5 AD 12and cos

AB 13 AB 13

Sin 5 12 17cos13 13 13

CXmhaU 4 : {gÕ H$am H$s cos . cosec = cot

CH$b : g‘Om {Xboë¶m g‘rH$aUmMr S>mdr ~mOy (LHS)

cos . cosec = cos .1

sin1cosec

sin

=cossin

= cotcos

cotsin

= RHS cos . cosec = cot

CXmhaU 5: H$mQ>H$moZ {ÌH$moU ABC ‘Ü¶o, B hm H$mQ>H$moZ Amho.

Oa tan A=1, Va 2 sin A cos A =1 ho nS>VmiyZ nmhm.

CH$b : ABC ‘Ü¶o, ABC=900 tan A =BCAB

Á¶mAWu tan A = 1 ({Xbobo) BCAB

= 1 BC = AB

g‘Om AB = BC = k, OoW o k hr g‘g§»¶m Amho.

AmVm AC2 = AB2 +BC2 AC = 2 2AB +BC = 2 2k +k

AC = K 2

Sin A =BC K 1AC K 2 2 , Cos A =

AB K 1AC K 2 2

2 Sin A CosA =1 12 12 2

2 Sin A CosA = 1

A

5 cm 16 cm

20 cm

B CD

CB

A

H$U©

{dê$ÜX

257

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328 UNIT-13

(i) (ii) (iii)

CXmhaU 6: ABCD hm Agm g‘^wO {ÌH$moU Amho H$s Á¶mMm H$U© AC hm ~mOy CD er

H$m oZ H $ aV m o, O oW o cos = 23

. Oa PD=4 g |.‘ r Va g‘^ yO Mm ¡H $ m oZ mMr ~mO y Am{U H $U © H$ mT > m .

CH$b : Amnë¶mbm ‘m{hV Amho H$s g‘ wO Mm¡H$moZmMo H$U© EH$‘oH$m§Zm H$mQ>H$moZmV Xþ mJVmV.

DP = PB Am{U AP = PC

DPC ‘Ü¶o, DPC = 900 Cos =PC 2CD 3

PC = 2k Am{U CD = 3K

na§Vw, CD2 = PD2 +CP2 (3k)2 = 42+ (2k)2

9k2 = 16 + 4k2

5k2 = 16 qH$dm k2 =165

k =45

CD = 3k = 3 × 45 =

125 g|.‘r Am{U PC = 2k = 2 × 4

5 =85 g|.‘r

BD = 2PD = 2×4 = 8 g|.‘r Am{U AC = 2PC = 2 × 85 =

165 g|.‘r

g‘ wO Mm¡H$moZmMr àË¶oH$ ~mOy =12

5 g|.‘r, H$U© BD = 8 g|.‘r Am{U H$U© AC =16

5 g|.‘r.

ñdmÜ¶m¶ 13.1

I. Imbrb CXmhaU sin Am{U cos H$mT>m.

25

7

24 25

20

15 26

24

10

II. Imbrb qH$‘Vr H$mT>m.

1. Oa sin x =35

, cosec x =

2. Oa cos x =2425 , sec x =

3. Oa tan x =724 cot x =

D

A

P

B

C

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Trigonometry 329

4. Oa cosec x = 1525 , sin x =

5. Oa sin A =35

Am{U cos A =45

Va, tan A =

6. Oa cot A =8

15Am{U sin A =

1517

Va, cos A =

III. gmoS>dm :

1. tan A =34

, {Xbo AgVm sin A Am{U cos A Mr qH§$‘V H$mT>m.

2. cot =2021 , {Xbo AgVm cos Am{U cosec emoYm.

3. tan A = 724, {Xbo AgVm H$moZ A À¶m {ÌH$moU{‘VrMr BVa JwUmoÎmao H$mT>m.

4. Oa 2 sin = 3 , Va, cos , tan Am{U cot + cosec H$mT>m.

5. Oa 3 tan =1, Va, sin , cos Am{U cot H$mT>m.

6. Oa sec x = 2, Va sin x, tan x Am{U cot x Am{U cot x + cosec x H$mT>m.

7. Oa 4 sin A - 3 cos A = 0, Va sin A, cos A Am{U sec A, cosec A H$mT>m.

8. Oa 13 sin A = 5 Am{U A hm bKwH$moZ Agob Va5sin A 2cos A

tan A Mr qH$‘V H$mT>m.

9. Oa cos = 513

Am{U hm bKwH$moZ Agob Va5 tan 12cot5 tan 12cot Mr qH$‘V H$mT>m.

10. Oa 13 cos 5 = 0, Vasin cossin cos

H$mT>m.

à‘m{UV H$moZm§À¶m {ÌH$moU{‘VrMr JwUmoÎmao [Trigonometric ratios of Standard angles]

¶m àH$aUm‘Ü¶o AmVmn¶ªV Amåhr H$mQ>H$moZ {ÌH$moUmÀ¶m bKwH$moZm§À¶m {ÌH$moU{‘VrÀ¶m JwUmoÎmam{df¶rMMm© Ho$br. {ÌH$moUm‘Ü¶o 90° Mm H$mQ>H$moZ Am{U 90° nojm bhmZ AgUmam bKwH$moZ AgVmo. bKwH$moZmMo ‘mn90° nojm bhmZ Am{U 0° nojm ‘moR>o Ago H$moUVohr AgVo. na§Vw AZoH$Xm Amåhr 30°, 45° Am{U 60° hoà‘m{UV H$moZ aMZogmR>r dmnaVmo. g‘Om AmVm ¶m H$moZm§À¶m Am{U 0° gmR>r XoIrb {ÌH$moU{‘VrMr JwUmoÎma H$mTy>.

(1) 45° À¶m {ÌH$moU{‘VrMr JwUmoÎmao

g‘Om {ÌH$moU ABC g‘{Û wO H$mQ>H$moZ {ÌH$moU Amho.

OoWo B 90 . Oa Amåhr C 45 , KoVbm Va A XoIrb 45° AgVmo.

AB = BC ( g‘mZ H$moZmg‘moarb {dê$Õ ~mOy)

g‘Om AB = BC = 1 EH$H$ CB

A

45°

2

1

1

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330 UNIT-13

A

B CD

30°

60°

2

22

1 1

90°

3

A

B CD

30°

60°

AC2 = AB2 + BC2 (nm¶W°JmoagÀ¶m à‘o¶mdê$Z)AC2 = 12 + 12 = 1 + 1 = 2

AC = 2 EH$H$

AmVm H$moZ C À¶m g§X^m©V åhUOoM C 45 , {ÌH$moU{‘VrMr JwUmoÎmao

sin 45° =AB 1AC 2 cos 45° =

BC 1AC 2 tan 45° =

AB 11

BC 1

cosec 45° =AC 2

2AB 1

sec 45° =AC 2

2BC 1

cot 45° =BC 1

1AB 1

2) 30° Am{U 60° À¶m {ÌH$moU{‘VrMr JwUmoÎmao

Oar {ÌH$moU{‘VrÀ¶m JwUmoÎmam§Mm Aä¶mg ’$º$ H$mQ>H$moZ {ÌH$moUm§À¶m g§X^m©V(~m~VrV) Ho$bm OmVmo Var H$moUË¶mhr àH$maÀ¶m {ÌH$moUm‘Ü¶o {eamob§~ H$mTy>Z Ë¶mMrCn¶mo{OVm H$aVm ¶oVo. ¶m H$ënZoMm Cn¶moJ 30° Am{U 60° À¶m {ÌH$moU{‘VrMr JwUmoÎmaoH$mT>Ê¶mg hmoVmo.

g‘Om {ÌH$moU ABC g‘ wO Amho.Á¶m AWu gd© H$moZ g‘mZ AgVmV, Amåhmbm {‘iVo A B C 60åhUyZ AB = BC = CAg‘Om A ‘YyZ BC da AD b§~ H$mT>bm

ABD ACD (EH$ê$nVoM RHS J¥hrVH$) BD = DC

BAD = CAD

Á¶m AWu BAC 60 , Amåhmbm {‘iVo, BAD = CAD = 300

AmVm, {ZarjU H$am H$s, ABD hm H$mQ>H$moZ {ÌH$moU Á¶m‘Ü¶o

ADB 90 , BAD 30 Am{U ABD 60

g‘Om nwÝhm EH$Xm ABC g‘ wO {ÌH$moU

{ÌH$moU{‘VrMr JwUmoÎmao H$mT>Ê¶mgmR>r {ÌH$moU ABC À¶m ~mOy§Mr bm§~rAmåhmbm ‘m{hV AgUo JaOoMo Amho.

g‘Om, AB = BC = CA = 2 EH$H$ ‘mZy

ABD ‘Ü¶o, AB = 2 EH$H$, BD = 1 EH$H$

Á¶mAWu ADB 90 , nm¶W°JmoagÀ¶m à‘o¶mdê$Z,AB2 = AD2 + BD2

22 = AD2 + 12

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Trigonometry 331

AD2 = 4 – 1 = 3 AD = 3

i) 300 À¶m {ÌH$m oU{‘VrMr JwUm oÎmao

sin 30° =H$U©

{dê$ÜX BD 1AB 2

cos 30° =gb½Z

H$U©

AD 3AB 2

tan 30° =H$U©

{dê$ÜX BD 1AD 3

cosec 30° =H$U©{dê$ÜX

AB 2BD sec30°=

gb½Z

H$U© AB 2AD 3

cot30°=gb½Z

{dê$ÜXAD 3BD

ii) 600 À¶m {ÌH$moU{‘VrMr JwUmoÎmao

sin 60° =AD 3AB 2

cos 60° =BD 1AB 2 tan 60° =

AD 33

BD 1

cosec 60° =AB 2AD 3 sec 60° =

AB 2 2BD 1

cot 60° =BD 1AB 3

iii) 0° Am{U 90° À¶m {ÌH$moU{‘VrMr JwUmoÎmao

g‘Om H$mQ>H$moZ {ÌH$moU ABC, Á¶m ‘Ü¶o ABC 90 H$ënZm H$am

bKwH$moZ A, Oa eyÝ¶ hmoB©n¶ªV bhmZ Ho$bm Va Ë¶mÀ¶m {ÌH$moU{‘VrÀ¶m JwUmoÎmamMo

H$m¶ hmoB©b.

{ZarjU H$am, AC Am{U AB À¶m bm§~rMo H$m¶ hmoVo?

Og Ogm, A bhmZ Am{U bhmZ hmoVmo Voìhm BC ~mOyMr bm§~r H$‘r hmoVo. Ogm C q~Xÿ B q~XÿÀ¶m

Odi ¶oVmo. AC hr AB bm V§VmoV§V OwiVo.

• Ooìhm A , 0° À¶m AJXr Odi ¶oVmo Voìhm BC Mr bm§~r eyÝ¶mÀ¶m AJXr Odi ¶oVo.

sin A =BCAC

eyÝ¶mÀ¶m AJXr Odi.

• Ooìhm A , 0° hmoVmo Voìhm AC Mr bm§~r Odi-Odi AB BVH$s hmoVo.

cos A =ABAC

1 À¶m AJXr Odi.

g‘Om Ooìhm A 0 Va sin A Am{U cos A Mm AW© ñnï> H$ê$.

A B

C

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332 UNIT-13

Amåhr Ago ñnï> H$aVmo H$s sin 0° = 0 Am{U cos 0° = 1

AmVm {ÌH$moU{‘VrMr BVa JwUmoÎmao g‘OyZ KoD$.

tan 0° =sin0 0

0cos 0 1

cot 0° =1 1

tan0 0¶m qH$‘VrMr ñnï> ì¶m»¶m Zmhr.

(^mJmH$ma ho eyÝ¶mZo ñnï> H$aVm ¶oV Zmhr)sec 0° =

1 11

cos0 1 cosec 0° =

1 1sin0 0

¶m qH$‘VrMr ñnï> ì¶m»¶m Zmhr.

AmVm H$ënZm H$am H$s A dmT>dyZ Am{U 90° n¶ªV dmT>dyZ {ÌH$moU{‘VrÀ¶m qH$‘VrMo H$m¶ hmoVo?

Og Ogm A ‘moR>m ‘moR>m hmoVmo, C bhmZ Am{U bhmZ hmoVmo Voìhm AB ~mOyMr bm§~r H$‘r hmoVo.

Ogm A q~Xÿ B q~XÿÀ¶m Odi ¶oVmo Am{U eodQ>r Ooìhm A , 90° À¶m AJXr Odi ¶oVmo,

C , 0° À¶m AJXr Odi ¶oVmo Voìhm AC Odi-Odi BC BVH$m hmoVmo.

• Ooìhm A , 90° À¶m AJXr Odi AgVmo Voìhm AC, BC bm V§VmoV§V OwiVo.

sin A =BCAC

1 À¶m AJXr Odi.

• Ooìhm A , 90° À¶m AJXr Odi AgVmo AB Mr bm§~r OdiOdi 0 BVH$s H$‘r hmoVo.

cos A =ABAC

0 À¶m AJXr Odi.

Amåhr Ago ñnï> H$aVmo H$s, sin 90° = 1 Am{U cos 90° = 0

AmVm {ÌH$moU{‘VrMr BVa JwUmoÎmao g‘OyZ KoD$.

tan 90° =sin90 1cos 90 0

¶m qH$‘VrMr ñnï> ì¶m»¶m hmoV Zmhr.

cot 90° =cos 90 0sin90 1

0

sec 90° =1 1

cos 90 0¶m qH$‘VrMr ñnï> ì¶m»¶m hmoV Zmhr.

cosec 90° =1 1 1

sin90 1

AmVm AmnU 0°, 30°, 45°, 60° Am{U 90° À¶m {ÌH$moU{‘VrÀ¶m JwUmoÎmam§Mm Vº$m(H$moï>H$) H$ê$.

¶m H$moï>H$mMm Cn¶moJ CXmhaUo gmoS>{dÊ¶mgmR>r V¶ma g§X^© åhUyZ hmoVmo.

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Trigonometry 333

AmR>dm

¶oWo à‘m{UV H$moZm§gmR>r sin À¶m qH$‘VtMo gmono H$moï>H$ ~Z{dÊ¶mMr H¥$Vr {Xbr Amho.

nm¶ar 1 : 0 Vo 4 n¶ªV g§»¶m {bhm 0 1 2 3 4

nm¶ar 2 : àË¶oH$ g§»¶obm 4 Zo ^mJm04

14

24

34

44

nm¶ar 3 : Ë¶m§Mo dJ©‘yi ¿¶m 014

12

34 1

012

12

32

1

¶m qH$‘Vr sin gmR>r AmhoV 00 300 450 600 900

nm¶ar 4 : qH$‘Vr CbQ> H«$‘mV {bhm 132

12

12 0

¶m qH$‘Vr cos gmR>r AmhoV 00 300 450 600 900

nm¶ar 5 : Amnë¶mbm ‘m{hV Amho tancosSin 0

1

1232

12

12

3212

10

013 1 3 ND

nm¶ar 6 : à‘m{UV H$moZm§gmR>r sin cos Am{U tan ¶m§Mo ì¶ñV ¿¶m. Amåhmbm AZwH«$‘ocosec Sec Am{U cot ¶m§À¶m qH$‘Vr {‘iVmV.

0 30 45 60 90

31 1sin 0 12 223 1 1cos 1 02 22

1tan 0 1 33

2cosec 2 2 13

2sec 1 2 23

1cot 3 1 03

ND

ND

ND

ND

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334 UNIT-13


CXmhaU 1 : Imbrb qH$‘Vr H$mT>m.

(i) tan2 60° + 2 tan2 45°

AmnUmbm ‘m{hV Amho tan 600 = 3 Am{U tan 450 = 1

tan2 60° + 2 tan2 45 = 2 23 2.(1) = 3 + 2 = 5

(ii) Cosec 60° – sec 45° + cot 30°

Amnë¶mH$S>o Amh o cosec 60 ° =23 , sec 45° = 2 Am{U cot 30° = 3

g‘Om cosec 60° – sec 45° + cot 30° =2

2 33 =

2 6 3 5 63 3

(iii) sin2 4 + cos2 4 – tan2 3

1sin sin45

4 2,

1cos cos 45

4 2Am{U tan tan60 3

3

=2 21 1 3

2 2

2

=1 1 32 2 = 1 – 3 = –2

(iv)22

cos0.sin .cos .sec2 6 4tan cot3 3

H$moï>H$mdê$Z cos 0 = 1

sin 2 = sin 90° = 1 sec sec 45 24

tan tan60 33

3cos cos 306 2

1cot cot60

3 3

qH$‘Vr dmnê$Z

2231 1 . 2 31 1 22 4

1 3 133 3

=3 3 3 32 4 8

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Trigonometry 335

CXmhaU 2: Oa 2 cos = 1 Am{U hm bKwH$moZ Amho, Voìhm Mr qH$‘V {H$Vr?

CH$b : 2 cos ho J¥hrV cos12

Amnë¶mbm ‘m{hV Amho cos12 cos cos

CXmhaU 3 : Oa tan Am{U hm bKwH$moZ Agob Va sin 3 Am{U cos2 À¶m qH$‘Vr H$mT>m.

CH$b : tan ho J¥hrV tan 13

Amåhmbm ‘m{hV Amho tan 30 0 = 13tan

2 Am{U 3 Sin 3 Sin 900 = 1

Cos 2 = cos 600 =12

CXmhaU 4 : Oa A = 600, B = 300 Va {gÕ H$am.

cos (A + B) = cosA. cosB - SinA, sinBCH$b : LHS = Cos (A+B) = Cos (600 + 300 = cos 900 = 0 RHS = CosA. CosB- SinA. Sin B = Cos 600 , cos 300 - Sin 600 , Sin 300

=1 3 3 12 2 2 2

=3 34 4

= 0

LHS = RHS

ñdmÜ¶m¶ 13.2

I. Imbrb àým§Mr CÎmao Úm :

(1) 00 nmgyZ 900 n¶ªVÀ¶m H$moUË¶m H$moZm§Mr {ÌH$mU{‘VrMr JwUmoÎmao 0 ~amo~a AgVrb?

(2) 00 nmgyZ 900 n¶ªVÀ¶m H$moUË¶m H$moZm§Mr {ÌH$moU{‘VrMr JwUmoÎmao 1 ~amo~a AgVrb?

(3) 00 nmgyZ 900 n¶ªVÀ¶m H$moUË¶m H$moZm§Mr {ÌH$moU{‘VrMr JwUmoÎmao 12 ~amo~a AgVrb?

(4) 00 nmgyZ 900 n¶ªVÀ¶m H$moUË¶m H$moZm§À¶m {ÌH$moU{‘VrÀ¶m JwUmoÎmam§Mr ì¶m»¶m H$aVm ¶oV Zmhr?

(5) 00 nmgyZ 900 n¶ªVÀ¶m H$moUË¶m H$moZm§§Mr {ÌH$moU{‘VrMr JwUmoÎmao g‘mZ AgVmV?

II. H$mT>m, Oa 0 900 Am{U

(i) 2 cos (ii) 3 tan = 1 (iii) 2 sin 3

(iv) 5 sin (v) 2 tan

III. Imbrb qH$‘Vr H$mT>m.

(i) Sin 300 cos 600 - tan2 45 0

(ii) Sin 600 cos 300 + cos 600 sin 300

(iii) cos 600 cos 300 - Sin 600 sin 300

(iv) 2 sin2 300 - 3 cos2 300 + tan 600 + 3 sin2 900

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336 UNIT-13

CB

A

(v) 4 sin2 600 + 3 tan2 300 - 8 sin 450 . cos 450

(vi)0

0 0

cos 45Sec 30 + cosec 30

(vii)2 2

2 2

4 Sin 60° - cos 45°tan 30° + sin 0°

(viii) Sin 300 + tan 450 - Cosec 600 (ix) 5 cos2 600 + 4 sec2 300 - tan2 450

Sec 300 +Cos 600 + cot 450 Sin2 300 + cos2 300

(x) 5 Sin2 300 + cos2 450 - 4 tan2 300

2Sin 300 +Cos 300 + tan 450

IV. Imbrb g‘mZVm {gÕ H$am.(i) Sin 300, Cos 600 + cos 300. Sin600 = Sin 900

(ii) 2cos2 300-1 = 1-2 Sin2 300 = cos 600

(iii) Oa Va 4 cos2 - 3 cos 00 = cos 3 ho {gÕ H$am.

(iv) Oa = 1800 Am{U A =6

{gÕ H$am( 1- cosA) (1+ cosA) 1(1-SinA) (1+SinA) 3

(v) Oa B = 150, {gÕ H$am 4 sin 2B, cos4B. Sin6B= 1

(vi) Oa A= 600 Am{U B = 300, Va {gÕ H$am H$s tan (A - B) =tan A - tan B

1 + tan A. tan B{ÌH$moU{‘VtMr g‘mZVm (Trigonometric identities) :

~rOJ{UVm‘Ü¶o ñ‘aU H$am Vwåhr gyÌo (identities) Aä¶mgbobr AmhoV.

CXmhaUmW©, (a + b)2 = a2 + 2ab + b2 hr EH$ g‘mZVm Amho. a d b ¶m MbnXm§À¶m H$moUË¶mhr

qH$‘VrgmR>r ho g‘rH$aU gË¶ Amho. Ë¶mMà‘mUo H$moZmÀ¶m {ÌH$moU{‘VrÀ¶m JwUmoÎmamMm g‘mdoe AgUmao g‘rH$aU

åhUOoM "{ÌH$moU{‘VrMr g‘mZVm", Oa g‘mdoe AgUmè¶m H$moUË¶mhr H$moZm§À¶m qH$‘VrgmR>r gË¶ Agob Va.

AmVm Amåhr {ÌH$moU{‘VrÀ¶m Ë¶mM g‘mZVm {‘idy.

g‘Om ABC hm H$mQ>H$moZ {ÌH$moU Amho, Á¶m‘Ü¶o B 90

nm¶W°JmoagÀ¶m à‘o¶mdê$Z, Amåhmbm {‘iVo AB2 + BC2 = AC2

g‘rH$aUmÀ¶m àË¶oH$ nXmg AC2 Zo ^mJyZ Amåhmbm {‘iVo.

2 2

2 2

AB BCAC AC

=2

2

ACAC

2 2AB BCAC AC

=2AC

AC

(cos A)2 + (sin A)2 = 1

sin2 A + cos2 A = 1 ............ (i)

0 A 90 ¶m gma»¶m H$moZ A À¶m gd© qH$‘VrgmR>r ho g‘rH$aU gË¶ Amho.

ho g‘rH$aU (1) hr {ÌH$moU{‘VrMr ‘yb yV g‘mZVm Amho.

‘yb yV g‘mZVodê$Z AmVm AmUIr XmoZ {ÌH$moU{‘VtÀ¶m g‘mZVm {‘idy.

sin2 A + cos2 A = 1 ............ (i)

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Trigonometry 337

(i) bm sin2 A Zo ^mJyZ Amåhmbm {‘iVo2 2

2 2 2

sin A cos A 1sin A sin A sin A

2 2sin A cos A 1sin A sin A sinA

1 + (cot A)2 = (cosec A)2

2 21 cot A cosec A ...............(ii)

0° < A 90° ¶m gma»¶m H$moZ A H$moUË¶mhr qH$‘VrgmR>r {ÌH$moU{‘VrMr g‘mZVm (ii) hr gË¶ AgVo.A 0 gmR>r cot A Am{U cosec A Mr ì¶m»¶m hmoV Zmhr.

(i) bm cos2 A Zo ^mJyZ Amåhmbm {‘iVo.2 2

2 2 2

sin A cos A 1cos A cos A cos A

2 2

2 2 2

sin A cos A 1cos A cos A cos A

(tan A)2 + 1 = (sec A)2

2 2tan A 1 sec A ......(iii)

0 A 90 ¶m gma»¶m H$moZ A À¶m H$moUË¶mhr qH$‘VrgmR>r {ÌH$moU{‘VrMr g‘mZVm (iii) hr gË¶

AgVo. A 90 gmR>r, tan A Am{U sec A Mr ì¶m»¶m hmoV ZmhrV.

{ÌH$moU{‘VrÀ¶m g‘mZVm (i), (ii) Am{U (iii) ¶mZm ‘yb yV g§~§Y åhUVmV.(1) sin2 A + cos2A = 1(2)tan2 A + 1 = sec2 A(3) 1 + cot2 A = cosec2 A¶m g‘mZVm nwÝhm Imbrbà‘mUo Aemhr {b{hVm ¶oVrb.

sin2 A + cos2A = 1 sin2 A = 1 – cos2 A; qH$dm cos2 A = 1 – sin2A

tan2 A + 1 = sec2 A tan2 A = sec2 A – 1; qH$dm sec2 A – tan2 A = 1

1 + cot2 A = cosec2 A cot2 A = cosec2 A -1; qH$dm cosec2 A – cot2 A = 1

àË¶oH$ {ÌH$moU{‘VrMo JwUmoÎma Xþgè¶m {ÌH$moU{‘VrÀ¶m JwUmoÎmamÀ¶m g§X^m©V ì¶º$ H$aÊ¶mgmR>r ¶m g‘mZVm§MmAmnU Cn¶moJ H$ê$ eH$Vmo. åhUOoM ¶mn¡H$s EImXo JwUmoÎma ‘m{hV Agob Va BVa {ÌH$moU{‘VrÀ¶m JwUmoÎmamMrqH$‘V Amåhr H$mTy> eH$Vmo.


CXmhaU 1: cos . cosec = cot ho {gÕ H$am

CH$b : LHS cos . cosec

= 1cos .sin

1cosecsin

Amnë¶m gmo¶rgmR>r (sin A)2 hosin2A Ago {b{hVmV Am{U gmB©Zñ¹o$Aa A Ago dmMVmV.

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338 UNIT-13

cossin

= cot = RHS

n¶m©¶r nÕV Amåhmbm ‘m{hV Amho H$s COì¶m ~mOy (RHS) ‘Ü¶o cot =cossin

S>mì¶m ~mOybm Amnë¶mH$S>o cos Amho na§Vw1

cos .sin

Zmhr Ë¶mEodOr cosec Amho.

åhUyZ1cos .

sin ho cosec Ago {bhrVmV.

cos cosec cos 1cos .

sin= cot

CXmhaU 2 : tan A. sin A + cos A = sec A ho XmIdm.

CH$b : g‘Om LHS = tan A . sin A + cos A

= sin A.sin A cos A

cos Asin A

tanAcos A

)

=2sin A

cos Acos A

=2 2sin A cos A

cos A ( sin2A+cos2 A=1)

= 1cos A

= sec A = RHS

CXmhaU 3 : (sin + cos )2 + (sin – cos )2 Mr qH§$‘V H$mT>m.

CH$b : (sin + cos )2 + (sin - cos )2

=sin2 + cos2 + 2sin cos + sin2 + cos2 – 2sin cos

= 2[sin2 + cos2 ] = 2(1) = 2 sin2 + cos2 ]

CXmhaU 4 : cos2 = 2

11 tan

ho {gÕ H$am.

CH$b : LHS = cos2 = 2 2

1 1sec 1 tan

= RHS

Xþgar nÕV RHS= 22 2

1 1 cos1 tan sec

= LHS

CXmhaU 5 : (1 - sin2A) (1 + tan2A) = 1 ho XmIdm.

CH$b : g‘Om LHS = (1 – sin2A) (1 + tan2 A) = cos2 A . sec2 A [ 1 – sin2 A = cos2 A Am{U 1 + tan2 A=sec2 A

= cos2A . 2

1cos A

= 1 = RHS

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Trigonometry 339

ñdmÜ¶m¶ 13.3

I) Imbrb CXmhaUo {gÕ H$am.

1. (1-sin2 ). sec2 = 1 2. (1+tan2 ). cos2 = 1

3. (1+tan2 )(1-sin (1+sin = 1 4. {gÕ H$am H$s1 cos 2 cosec

1 cosSin

Sin

5. 21 sin(sec tan )

1 sin6. cos sin

1 tan 1 cotA A

A A= sin A + cosA

7.2

22

1 tan 1 2sin1 tan

A AA

8. (sin + cos )2 = 1 + 2 sin .cos

9 . sin A. cos A. tan A + cos A. sin A . cot A = 1

10 {gÜX H$am H$s 2

tan sin sec1 cossin

A A AAA

11. tan2 A - sin2 A = tan2A. sin2 A

12. cos2 A - sin2 A = 2 cos2A - 1

H$moQ>rH$moZm§Mr {ÌH$moU{‘VrMr JwUmoÎmao : (Trigonometric Ratios of Complementary angles)

AmnU H$moQ>rH$moZm§~Ôb ¶mnyduM n[a{MV AmhmoV.

AmR>dU H$am, Oa XmoZ H$moZm§Mr ~oarO 90° Agë¶mg Vo H$moQ>rH$moZ AmhoV Ago åhUVmV.

EH$‘oH$m§Mo H$moQ>rH$mooZ AgUmè¶m XmoZ H$moZm§Mm g‘mdoe Agboë¶m {ÌH$moU{‘VrMr JwUmoÎmao Amåhr H$er ì¶º$ H$ê$

eH$Vmo?

g‘Om ABC ‘Ü¶o ABC 90 . ABC ‘Yrb H$moQ>rH$moZm§Mr OmoS>r

AmoiIm. Á¶m AWu B 90 , A C 90

A Am{U C ho H$moQ>rH$moZ AmhoV.

H$moUË¶mhr H$mQ>H$moZ {ÌH$moU‘Ü¶o, H$mQ>H$moZ gmoSy>Z BVa XmoZ bKwH$moZ Zoh‘r H$moQ>rH$moZm§Mr

OmoS>r ~Z{dVmV.

AmVm Aem H$moQ>rH$moZm§gmR>r {ÌH$moU{‘VrÀ¶m JwUmoÎmam§Mr ì¶m»¶m ñnï> H$ê$.

Á¶m AWu A C 90 A 90 C qH$dm C 90 A

[Amnë¶m gmo¶rgmR>r ho H$moQ>rH$moZ (90° – C) qH$dm (90° – A) Ago {b{hVmV.]

A Am{U (90° – A) gmR>rMr {ÌH$moU{‘VrMr JwUmoÎmao Imbrb V³Ë¶m‘Ü¶o {Xbr AmhoV Ë¶m§Mo {ZarjU H$am.

CB

A

A

90-A0

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340 UNIT-13

H$moZ A $gmR>r {ÌH$moU{‘VrMr JwUmoÎmao $H$moZ (90° – A) gmR>r {ÌH$moU{‘Vr JwUmoÎmao

sin A =BCAC

sin (90° – A) =ABAC

cos A =ABAC

cos (90° – A) =BCAC

tan A =BCAB

tan (90° – A) =ABBC

cot A =ABBC

cot (90° – A) =BCAB

sec A =ACAB

sec (90° – A) =ACBC

cosec A =ACBC

cosec (90° – A) =ACAB

$H$moZ A Am{U Ë¶mÀ¶m H$moQ>rH$moZmÀ¶m JwUmoÎmam§Mr VwbZm H$am. Amåhmbm Ago AmT>iVo.

sin (90° – A) = cos A = ABAC

cosec (90° – A) = sec A = ACAB

cos (90° – A) = sin A = BCAC

sec (90° – A) = cosec A = ACBC

tan (90° – A) = cot A = ABBC

cot (90° – A) = tan A = BCAB

Amåhr Agm {ZîH$f© H$mT>Vmo H$s,sin (90° – A) = cos A cosec (90° – A) = sec Acos (90° – A) = sin A sec (90° – A) = cosec Atan (90° – A) = cot A cot (90° – A) = tan A


CXmhaU 1: $qH$‘V H$mT>m0

0

sin65cos 25

CH$b : Amåhmbm ‘m{hV Amho

cos 250 = sin (900-250) = sin 650

0

0

sin65cos 25 =

0

0

cos 25cos 25 =1

$CXmhaU 2: Oa tan 2A = cot (A 180), OoW o 2A hm bKwH$moZ Amho, Va A Mr qH$‘V H$mT>m.

CH$b : Á¶mAWu tan 2 A = cot (900 2A) Amåhmbm {‘iVo cot (900 2A) = cot (A 180)

900 2A = A 180

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Trigonometry 341

3A = 1080 A =0108

3= 360 A = 360

$CXmhaU 3: {gÕ H$am : [cosec (90 – ) – sin (90 – ) [cosec – sin ].[tan + cot ] = 1

CH$b : LHS = [sec – cos ] [cosec – sin ] [tan + cot ]

=1 1 sin coscos sin

cos sin cos sin

=2 21 cos 1 sin 1. .

cos sin cos .sin

=2sin

cos

2cos.

sin1

.cos sin = 1 = RHS

$CXmhaU 4:sin(90 ) cos1 sin 1 cos(90 ) = 2 sec ho {gÕ H$am

CH$b : LHS =sin(90 ) cos1 sin 1 cos(90 )

= cos cos1 sin 1 sin

= 2

cos (1 sin ) cos (1 sin )1 sin

= cos cos sin cos cos sin21 sin

= 2cos 2 2seccos cos

= 2 sec =RHS

ñdmÜ¶m¶ 13.4

1. qH$‘Vr H$mT>m.

(i)0

0

tan65cot 25 (ii)

0

0

sin18cos 72 (iii) cos 480 sin 420

(iv) cosec 310 sec 590 (v) cot 340 tan 560(vi)0 0

0 0

sin36 sin54cos54 cos 36

(vii) sec700 sin200 cos 700 cosec 200 (viii) cos2 130 sin2 770

2. {gÕ H$am

(i) sin 350 sin 550 - cos 350 cos 550 = 0(ii) tan 100 tan 150 tan 750 tan 800 = 1

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342 UNIT-13

(iii) cos 380 cos 520 - sin 380 sin 520 = 03. Oa sin 5 = cos 4 , OoW o 5 Am{U 4 ho bKwH$moZ AmhoV, Va Mr qH$‘V H$mT>m.

4. Oa sec 4A = cosec (A-200), OoWo 4A hm bKwH$moZ Amho Va A Mr qH$‘V H$mT>m.

C§Mr Am{U A§Va (Heights and Distances)

Amåhr ‘moOnÅ>r qH$dm bm§~r ‘moOÊ¶mÀ¶m ’$sVrZo H$m§hr dñVy§Mr C§Mr CXmhaUmW© ‘wbJm, nwVim, PmS>BË¶mXr ‘moOÊ¶mg n[a{MV AmhmoV. Aem àH$maÀ¶m ‘moOÊ¶mbm àË¶j ‘mnZ åhUVmV.

{ZË¶ X¡Z§{XZ OrdZmV Aer H$m§hr {R>H$mUo (n[apñWVr) AgVmV H$s OoWo àË¶j ‘mnZ H$ê$Z C§Mr H$mT>Vm¶oV Zmhr. CXmhaUmW© C§M B‘maVrMr C§Mr, ‘Zmoè¶mMr C§Mr, PmS> Xÿa AgUmam nd©V ZXrMr é§Xr BË¶mXr.

Aer ‘mnZo AàË¶j ‘mnZ nÕVrZo H$ê$Z Ë¶m§Mo ‘moO‘mn H$mT>Vm ¶oVo.

bm§~r qH$dm C§MrMo AàË¶j ‘mnZ H$aÊ¶mÀ¶m EH$m CÎm‘ nÕVr‘Ü¶o JwUmoÎmam§Mm g‘mdoe AgVmo. {ZË¶OrdZmVrb KQ>Zoer g§~§{YV g‘ñ¶m gmoS>{dÊ¶mgmR>r {ÌH$moU{‘Vr‘Ü¶o {dñV¥V Cn¶mo{OVm AmhoV.

C§Mr Am{U A§Va H$mT>Uo hr {ÌH$moU{‘VrMr ‘w»¶ Cn¶mo{OVm Amho. {ÌH$moU{‘VrÀ¶m JwUmoÎmam§Mm Cn¶moJ~m§YH$m‘mMr ¶moOZm, Ajm§e Am{U aoIm§emÀ¶m g§X^m©V ~oQ>mMo ñWmZ {ZpíMV H$aÊ¶mg XoIrb hmoVmo.

AmVm Aem H$m§hr nXm§Mm AW© ñnï> H$ê$ H$s Á¶m§Mm Cn¶moJ C§Mr Am{U A§Va H$mT>Ê¶mg dma§dma hmoVmo.

Ñï>rMr aofm (Line of Sight)Imbrb AmH¥$Ë¶m§Mo {ZarjU H$am.

¶oWo gm‘mdbobo A§Va IynM ‘moR>o Amho Am{U åhUyZ Ñî¶ åhUOoM dñVydarb EH$m q~XÿMo {ZarjU qH$dmdñVy hmM q~Xÿ Amho Ago JhrV Yamdo.

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Trigonometry 343

{ZarjH$mÀ¶m S>moù¶mnmgyZ dñVydarb q~Xÿn¶ªV H$mT>bobr aofm åhUOoM Ñï>rMr aofm

Ñï>r aofm hr O{‘ZrÀ¶m nmVirbm g‘m§Va AgUmar {j{VO g‘m§Va aofm.

na§Vw Zoh‘rM Ñï>r aofm hr {j{VO g‘m§Va aofm AgVoM Ago Zmhr.

Imbrb AmH¥$Ë¶m‘Ü¶o {ZarjUmÀ¶m q~XÿMo {Z[ajU H$am.

AmH¥$Vr (i) ‘Ü¶o EH$ ‘wbJm ‘mZd§XZm XoÊ¶mgmR>r P|S>çmMo {ZarjU H$aVmo. P|S>m {j{VO g‘m§Va nmVirÀ¶m

da Amho Am{U ‘wbmZo dñVy nmhÊ¶mgmR>r S>moHo$ da Ho$bo Amho. ¶m H¥$Vr‘Ü¶o S>moio, Ñï>raofm Am{U {j{VO g‘m§Va

aofm ¶m‘Ü¶o V¶ma Pmboë¶m H$moZm‘Ü¶o hmbVmV. ¶m H$moZmbm Cƒ nmVirVrb H$moZ (angle of elevation)åhUVmV.

AmH¥$Vr (ii) §‘Ü¶o EH$ ‘wbJm EH$m hmoS>rMo {Z[ajU H$aVmo H$s Or {j[VO g‘m§Va nmVirÀ¶m Imbr Amho.

¶mgmR>r ‘wbJm Amnbo S>moHo$ (S>moio) dñVy nhmÊ¶mg Imbr H$aVmo ¶m H¥$Vr‘Ü¶o XoIrb S>moio, Ñï>raofm Am{U g‘m§Va

aofm ¶m‘Ü¶o V¶ma Pmboë¶m H$moZm‘Ü¶o hmbVmV. ¶m H$moZmbm [Zƒ nmVirVrb H$moZ (angle of depres-sion) åhUVmV.

darb CXmhaUmdê$Z Amåhr Agm {ZîH$f© H$mT>Vmo H$s

q~XÿMo {ZarjU H$aVm ¶oUmam Cƒ nmVirVrb H$moZ hm Ñï>raofoZo {j{VO g‘m§Va aofoÀ¶m da AgUmè¶m q~XÿMo

{ZarjU H$aÊ¶mgmR>r Ho$bobm H$moZ Amho. Cƒ nmVirVrb H$moZ {j{VOmnmgyZ daÀ¶m ~mOyg ‘moOVmV.

q~XÿMo {ZarjU H$aVm ¶oUmam [Zƒ nmVirVrb H$moZ hm Ñï>raofoZo {j{VO g‘m§Va aofoÀ¶m Imbr AgUmè¶m

q~XÿMo {Z[ajU H$aÊ¶mgmR>r Ho$bobm H$moZ Amho. Amnë¶m S>moù¶mdarb H$moZ {j{VOmnmgyZ Imbrb ~mOyg

‘moOVmV.

Q>rn :

* Cƒ nmVirVrb H$moZ Am{U {Zƒ nmVirVrb H$moZ ho Zoh‘r {j{VOmer ‘moObo OmVmV.

* {ZarjH$mZo nm{hbobm dñVyMm Cƒ nmVirVrb H$moZ hm dñVynmgyZ {XgUmè¶m {ZarjH$mÀ¶m Zrƒ nmVirVrb

H$moZm BVH$mM AgVmo.* {ZarjH$mMr C§Mr Oa {Xbr Zgob Va {ZarjH$ hm q~Xÿ åhUyZ KoVbobm AgVmo.

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344 UNIT-13

CB

A

50‘ r300

50‘ r

5 0‘ r

RQ

P

ho ‘m{hV Amho : O{‘ZrMr ‘moOUr H$aUmè¶m bmoH$mZr {ÌH$moU{‘VrMm Cn¶moJ AZoH$eVH$mnmgyZ Ho$bm Amho. EH$mo{Ugmì¶m eVH$m‘Yrb {~«Q>re ^maVr¶m§Mm "J«oQ> {Q>¾m°‘o{Q>H$gìh}' hm EH$ Agm ‘moR>çm ‘moOUrMm àH$ën hmoVm H$s Ë¶mgmR>r AmS>do d C o H$moZ‘moOÊ¶mgmR>r AmVmn¶ªV gdm©V ‘moR>r Aer XmoZ ‘moOUr ¶§Ìo ~m§YÊ¶mV Ambr hmoVr.1852 ‘Yrb gd}jUmÀ¶m doir OJmVrb gdm©V C§M nd©VmMm emoY bmJbm hmoVm. 1856‘Ü¶o ¶m {eIambm ga Om°O© EìhaoñQ> Ago Zm§d XoÊ¶mV Ambo H$s Á¶mZo A{V àM§S> Aem‘moOUr ¶§ÌmMm àW‘ Cn¶moJ Ho$bm Am{U gmon{dbobr H$m‘{Jar Ho$br (~mOyÀ¶m AmH¥$Vr ‘Ü¶o nhm) AmS>do Am{UC o H$moZ ‘moOÊ¶mMo ‘moOUrXmam§Mo ¶§Ì AmVm S>ohamSy>Z ‘Yrb gìh} Am°’$ B§{S>¶m ¶m dñVyg§J«hmb¶m‘Ü¶o àX{e©VHo$bo Amho.

ho gd}jUmMo Ago CnH$aU Amho H$s Á¶mMm Cn¶moJ {ZarjH$mMm S>moim Am{U dñVy ¶m‘Yrb H$moZmMo ‘mnZH$aÊ¶mg Ho$bm OmVmo. ho {ÌH$moU{‘VrÀ¶m VËdmda Adb§~yZ Amho Am{U H$moZmMo dmMZ Jmob {’$aUmè¶mXþ{~©Urdarb ^mnZ nÅ>rda Ho$bo OmVo.


CXmhaU 1: O{‘Zrda EH$ ‘Zmoam b§~ê$nmV C^m Amho. ‘Zmoè¶mÀ¶m nm¶mnmgyZ 50 ‘r bm§~ Agboë¶m O{‘ZrdarbEH$m q~XÿnmgyZ ‘Zmoè¶mÀ¶m nm¶mnmgyZ AgUmam Cƒ nmVirVrb H$moZ 30° Amho. ‘Zmoè¶mMr C§Mr H$mT>m.

CH$b : {Xboë¶m ‘m{hVrÀ¶m g§X^m©V AmH¥$Vr H$mT>m. g‘Om AB hr ‘Zmoè¶mMr C§Mr (h)Am{U BC ho ‘Zmoam Am{U {ZarjH$mMm q~Xÿ ¶m‘Yrb A§Va Amho.

ABC ‘Ü¶o, ACB = 300

tan 30° = ABBC = h

501 h

503= h =

503 ‘r.

CXmhaU 2: ‘Zmoè¶mda 50 ‘r C§Mrda EH$ dñVy AgyZ {ZarjH$ Am{U ‘Zmoè¶mMm nm¶m¶m‘Yrb A§Va 50 ‘r Agë¶mg Cƒ nmVirVrb H$moZ H$mT>m.

CH$b : g‘Om PQ hr dñVyMr C§Mr = 50 ‘r.

QR ho {ZarjH$ Am{U ‘Zmoè¶mÀ¶m nm¶m‘Yrb A§Va = 50 ‘r. PQR ‘Ü¶o PQR = 900

tan =PQ 50

1QR 50

tan = 1 = tan 450 ( tan 45° = 1)‘r.tan = tan 450 = 45°

Cƒ nmVirVrb H$moZ 45° Amho.

CXmhaU 3: 50 3 ‘r. C§M Agboë¶m B‘maVrÀ¶m ‘mÏ¶mnmgyZ O{‘Zrdarb H$maMm ZrÀM nmVirVrb H$moZ 60°Agë¶mMo {XgyZ ¶oVo. H$maMo B‘maVrnmgyZ AgUmao A§Va H$mT>m.

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Trigonometry 345

CH$b: g‘Om B‘maVrMr C§Mr PQ Zo Xe©{dbr OmVo, PQ = 50 3 ‘r. QR ho B‘maV Am{U H$ma ¶m‘YrbA§Va Amho.

ZrÀMnmVirVrb H$moZ 600 Amho.

Á¶m AWu, PM || QR, åhUyZ MPR PRQ ( ì¶wËH«$‘ H$moZ)

MPR 60 PRQ 60

PQR ‘Ü¶o, PQR 90 , PRQ 60

tan 60° =PQ 50 3QR QR

3 =50 3QR QR =

50 3

3 = 50

H$ma hr B‘maVrnmgyZ 50 ‘r A§Vamda Amho.

CXmhaU 4 : EH$m eoVO{‘ZrÀ¶m àË¶oH$ ~mOyg EH$ Aem 50 ‘r. Am{U 40 ‘r. C§MrÀ¶m XmoZ ndZ M³³¶m

AmhoV. EH$ ì¶º$s ‘Zmoè¶m‘Yrb EH$m q~XÿnmgyZ ndZ M¸$sÀ¶m ‘mÏ¶m§Mo {ZarjU H$aVoo. XmoÝhr KQ>Zm‘Ü¶o Cƒ

nmVirVrb H$moZ 450 Agë¶mMo {XgyZ ¶oVo. ndZM¸$s‘Ü¶o AgUmao A§Va H$mT>m.

CH$b : CXmhaU ì¶d{ñWV dmMyZ ‘m{hVrMo ê$nm§Va AW©nyU© AmH¥$VrV H$am.

A

B

C

DP

g‘Om ‘Zmoè¶mMr C§Mr AB = 50 ‘r. Amho.

CD hr Xþgè¶m ‘Zmoè¶mMr C§Mr = 40 ‘r.

{ZarjU 'P' ¶oWo Amho Am{U Cƒ nmVirVrb H$moZ APB CPP 45

BD = BP + PD ho BD ‘Yrb A§Va Amho.

ABP ‘Ü¶o, tan =ABBP

tan 45° =50BP

P

Q R

M = 60°

60°?

50 3

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346 UNIT-13

1 =50BP ( tan 45° = 1)

BP = 50 m

CPD ‘Ü¶o, tan =CDPD

tan 45° =40PD ( tan 45° = 1)

1 =40PD PD = 40 ‘r.

BD = BP + PD = 50 + 40 = 90 ‘r.

eoVmÀ¶m XmoÝhr ~mOyg AgUmè¶m ndZM¸$s‘Yrb A§Va 90 ‘r. Amho.

ñdmÜ¶m¶ 13.5

I. Imbrb CXmhaUmVrb 'x' Mr qH$‘V H$mT>m :

(i)x

60 ‘t45°

(ii)

x

90

60°

R

P Q

(iii)x

30°100

K

ML

(iv)

x

100

45°

X

Y Z

(v)75

75

D

F Ex

II. 1. gy¶m©Mr g‘wÐgnmQ>rnmgyZ C§Mr ({j{VOmer Agbobm H$moZ) 300 AgVmZm EH$m C§M ‘Zmoè¶mMr

gmdbr 300 ‘r bm§~ nS>Vo. Va ‘Zmoè¶mMr C§Mr H$mT>m.

2. 50 3 ‘rQ>a C§MrÀ¶m B‘maVrÀ¶m ‘mÏ¶mdê$Z O{‘Zrdarb dñVyÀ¶m {Zƒ nmVirVrb H$moZmMo ‘mn

45° Agë¶mMo AmT>iVo. Va dñVyMo B‘maVrnmgyZMo A§Va H$mT>m.

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Trigonometry 347

3. EH$ PmS> dmè¶mZo VwQ>Vo Am{U O{‘Zrbm H$mQ>H$moZ {ÌH$moU H$aVo Oa ‘moS>bobm ^mJ O{‘Zrer 60°

H$moZ H$aVmo Am{U PmS>mMm ‘mWm AmVm Ë¶mÀ¶m nm¶mnmgyZ 20 ‘r. da Amho. Va PmS> {H$Vr C§MrMo

hmoVo?

4. {j{VOm darb O{‘ZrÀ¶m EH$m q~XÿnmgyZ ÜdOñV§ mÀ¶m (flagpost) ‘mÏ¶mdarb Cƒ nmVirVrb

H$moZ 30° Agë¶mMo {XgyZ ¶oVo. ñV§ mÀ¶m {XeoZo 6 ‘r. MmbV Joë¶mda nmVirVrb H$moZ 15° Zo

dmT>Vmo Va ÜdOñV§ mMr C§Mr H$mT>m.

5.EH$m B‘maVrÀ¶m ‘mWm Am{U Vi ¶mnmgyZ EH$m C§M H$S>çmMo Cƒ nmVirVrb H$moZ AZwH«$‘o 45° Am{U

60° AmhoV. Oa B‘maVrMr C§Mr 24 ‘rQ>a Agob Va H$S>çmMr C§Mr H$mT>m.

6.16 ‘rQ>a C§M B‘maVrÀ¶m ‘mÏ¶mdê$Z EH$m Q>oH$S>rÀ¶m ‘mÏ¶mdarb Cƒ nmVirVrb H$moZ 60° Am{U

Q>oH$S>rÀ¶m nm¶mer AgUmam {ZÀM nmVirVrb H$moZ hm 30° Amho. Va Q>oH$S>rMr C§Mr H$mT>m.

7.Oa 150 g|.‘r C§MrMm {ZarjH$ gmdbrÀ¶m Q>moH$mH$S>o nmhVmo Omo nm¶mnmgyZ 150 3 go.‘r. Amho. Va

{Zƒ nmVirVrb H$moZ H$mT>m.

8.O{‘ZrnmgyZ 50 ‘rQ>a da AgUmè¶m q~XÿnmgyZ T>JmMm Cƒ nmVirVrb H$moZ 30° Am{U Ë¶mÀ¶m

à{Vq~~mMm Zrƒ nmVirVrb H$moZ 60° Amho. O{‘ZrnmgyZ da AgUmè¶m T>JmMr C§Mr H$mT>m.

D

{ÌH$moU{‘Vr

{ÌH$moU{‘VrMr JwUmoÎmao {ÌH$moU{‘VrMr gyÌo C§Mr Am{U A§Va H$mT>Uo

sin cos tan sec cosec cot sin2A+cos2A=1 Cƒ nmVirVrb H$moZ

{ÌH$moU{‘VrÀ¶m JwUmoÎmamMm ì¶ñV 1+ co+2A=cosec2A

{ÌH$moU{‘VrÀ¶m JwUmoÎmamVrb g§~§Y tan2A+ 1= sec2A {Zƒ nmVirVrb H$moZ

{ÌH$moU{‘VrÀ¶m JwUmoÎmamMoà‘m{UV H$moZ 00, 300, 450, 600, 900 H$moQ>r H$moZm§À¶m {ÌH$moU{‘VrMo JwUmoÎma

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348 UNIT-13

CÎmao

ñdmÜ¶m¶ 13.1

I. (i)2425

,725

(ii)35

,45

(iii)5

13,

1213

II. (i)53 (ii)

2524 (iii)

247 (iv) (v)

34 (vi)

817

III. (1)3 4,5 5

(2)20 29,29 21

(3)7 24 7, ,25 25 24

,257

,2524

,247

(4)12 , 3 , 3

(5)1 3

,10 10 , 3 (6) 3 ,

231

,13 , 3 (7)

3 4 5 5, , ,5 5 4 3

(8)1265

(9) (10)177

ñdmÜ¶m¶ 13.2

II. (i) 45° (ii) 30° (iii) 60° (iv) 0º (v) 30°

III.(i)34

(ii) 1 (iii) 0 (iv)5 34

(v) 0 (vi)3

2 2 1 3

(vii)152 (viii)

33

33

44

(ix)6712 (x)

56 4 3

ñdmÜ¶m¶ 13.4

1. (i) 1 (ii) 1 (iii) 0 (iv) 0 (v) 0 (vi) 0 (vii) 0 (viii) 0

3. = 10° 4. A = 22°

ñdmÜ¶m¶ 13.5

I (i) 60 ‘r (ii) 30 3 EH$Ho$ (iii)100

3 EH$Ho$ (iv)100

2 EH$Ho$ (v)4

EH$Ho$

II (1) 100 3 ‘r (2) x = 50 3 ‘r (3) 20 3 2 ‘r (4)6

3 1 ‘r

(5)24243 1+( ) ‘t (6) 64 ‘r (7) 30º (8) 100 ‘r

*****

177

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* gh{ZX}eH$ Ajm§À¶m AWm©Mo ñnï>rH$aU.

* gh{ZX}eH$ Aj H$mT>Uo.

* gh{ZX}eH$ nmVir‘Ü¶o {Xboë¶m q~Xÿ§Mo ñWmZ {ZpíMVH$aUo.

* ax + by = c g‘rH$aUmMm AmboI gaiaofoZo Xe©{dbmOmVmo ho AmoiIUo.

* gaiaofoMm MT> ¶mMm AW© ñnï> H$aUo.

* {Xboë¶m aofoÀ¶m MT>mMm {heo~ H$aUo.

* aofoÀ¶m x- Am§VaN>oX Am{U y - Am§VaN>oX ¶m§À¶m AWm©Moñnï>rH$aU.

* gai aofoÀ¶m g‘rH$aUmMo MT> Am§VaN>oX ñdê$n {‘i{dUo.

* nmVirVrb XmoZ q~Xÿ‘Yrb A§VamMm {heo~ H$aUo.

* aofobm {Xboë¶m JwUmoÎmamV {d^mJUmè¶m gh{ZX}eH$ q~Xÿ§Mm{heo~ H$aUo.

* ‘yb^yV H$ënZm - gh{ZX}eH$ Aj,Ama§ q~Xÿ.

* q~XÿMo gh{ZX}eH$ {b{hUo.

* ax + by = c Mm AmboI(aofr¶ AmboI)

* gai aofoMm MT>.

* MT>-Am§VaN>oX ñdê$nmVrb gaiaofoMrg‘rH$aUo.

* nmVirVrb XmoZ q~Xÿ‘Yrb A§Va.

* nmVirVrb q~XÿMo CJ‘mnmgyZMo A§Va.

* I§S> gyÌ.

14

aoZ. {S>ñH$mQ>©g²[1596 - 1650]

hm ’«|$M V§ËdkmZr, J{UVVk Á¶mÀ¶m

'La geometrie' ¶m H$m‘mMm g‘mdoe

y{‘VrMm ~rOJ{UVm‘Ü¶o Agm dmna

Á¶m‘wio Ambm Amnë¶mH$S>o

"H$mQ>}{g¶Z' y{‘Vr Amho.

Mathematics is the only place where truth andbeauty mean the samething.

- Danica Mckellar

gh{ZX}eH$ ^y{‘VrCo-ordinate Geometry

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350 UNIT-14

Amåhr ghOnUo {ZarjU H$aVmo H$s Q>oH$S>rÀ¶m ‘mÏ¶mn¶ªV nmohmoMUmaoañVo Zoh‘r diUmdiUmMo AgVmV. Q>oH$S>rÀ¶m ‘mÏ¶mn¶ªV gai OmUmao añVoZgVmV. H$m?

darb àýmMo CÎma XoÊ¶mgmR>r Amåhr Aä¶mgboë¶m Cn¶wº$ Am¶VmH¥$Vrgh{ZX}eH$ nÕVrÀ¶m H$ënZoMm {dMma H$ê$.

’«|$M VËdkmZr Am{U J{UVVk (aoZ S>oñH$mQ>©g²) (1546-1950) ¶mZo‘w»¶Ëd J{UVmMr emIm {dH$grV Ho$br åhUyZ Am¶VmH¥$Vr gh{ZX}eH$ nÕVrbmË¶mMmdê$Z ""H$mV}{g¶Z nÕV'' Ago Zm§d nS>bo.

ho J{UVmÀ¶m Aem emIoH$S>o KoD$Z OmVo H$s Á¶mbm ""gh{ZX}eH$ y{‘Vr qH$dm n¥W…H$aUmË‘H$ ^y{‘VråhUVmV.

hr J{UVrMr Aer emIm Amho H$s Or y{‘Vrbm ~rOJ{UVmà‘mUodmJ{dVo.

AmVm g‘Om H$mV}{e¶Z nÕVrÀ¶m g§~§YmV AgUmao eãX AmR>dy.

* XmoZ b§~ aofmZm gh{ZX}eH$ Aj åhUVmV.

* {j{VO g‘m§Va (AmS>dr) aofm X'OX bm x- Aj åhUVmV.

* C^r aofm YOY' bm y- Aj åhUVmV.

* XmoÝhr Ajm§À¶m N>oXZq~Xÿbm Ama§ q~Xÿ 'O' åhUVmV.

gh{ZX}eH$ AjmZo nmVir Mma ^mJm‘Ü¶o {d^mJbr OmVo.

H$mV}{g¶Z (Cartesian) nmVirVrb q~Xÿ {df¶r.

i) Ë¶mÀ¶m y- AjmnmgyZÀ¶m b§~ A§Vambm q~XÿMo x- gh{ZX}eH$(ABSCISSA) åhUVmV.

ii) Ë¶mÀ¶m x- AjmnmgyZÀ¶m b§~ A§Vambm q~XÿMo y- gh{ZX}eH$(ORDINATE) åhUVmV.

i) 'p' q~XÿMo gh{ZX}eH$ p(x, y) Zo Xe©{dVmV.

ii) Ama§ q~XÿMo gh{ZX}eH$ (0, 0)ho AmhoV.

iii) x- Ajmdarb H$moUË¶mhr q~XÿMo gh{ZX}eH$ (x, 0) ¶mñdê$nmV AgVmV.

iv) y- Ajmdarb H$moUË¶mhr q~XÿMo gh{ZX}eH$ (0, y)

ñdê$nmV AgVmV.

‘m{hVr H$ê$Z ¿¶m.ABSCISSA d ORDINATEho b°Q>rZ eãX AmhoV.ABSCISSA åhUOo VwH$S>o Ho$bobo.ORDINATE åhUOo Vo H«$‘mV R>odUo.

OXI X

Y

YI

II-x, + y( )

III

-x, - y( )

MaUMaU

MaU MaU

I+x, + y( )

IV+x, - y( )

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Coordinate Geometry 351

ax + by = c ¶m ñdê$nmMr g‘rH$aUo AmboImZo Xe©{dUo.

aofr¶ g‘rH$aUm§À¶m AmboIm{df¶r Vwåhr n[a{MV AmhmV. Imbrb g‘rH$aUm§À¶m AmboIm§Mo {Z[ajUH$am.

(1)2x + y = 7

y = mx + c À¶m ñdê$nmV ê$nm§Va Ho$ë¶mg Amåhmbm

{‘iVo, y = 7 – 2x

0 1 2 37 5 3 1

xy

(2)3x + 2y = 5

ho g‘rH$aU y =5 3

2- x

Ago nwÝhm {bhm.

0 1 1 32.5 1 4 2

--

xy

darb CXmhaUmdê$Z Ago ñnï> hmoVo H$s ax + by = cñdê$nmVrb AmboI Zoh‘r gai aofm AgVo.

Q>rn … aofm AmboI aMÊ¶mgmR>r XmoZ q~Xþ nwaogo AmhoV. AmUIr EH$ qH$dm XmoZ q~Xÿ VnmgÊ¶mgmR>r qH$dm{ZpíMVrgmR>r KoVbo.

O

87654321

–1–2–3–4–5–6

1 2 3 4 5–1–2–3–4–5

(0, 7)

(1, 5)

(2, 3)

(3, 1)XI X

Y

YI

O

4321

–1–2–3

1 2 3–1–2–3

(3, –2)

(–1, 4)(0, 2.5)

(1, 1)XI X

Y

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352 UNIT-14

gaiaofoMm CVma (Inclination of straight line)

2x + y = 7 Am{U 3x + 2y = 5 ¶m g‘rH$aUm§gmR>r {‘iUmè¶m AmboIm§Mo {ZarjU H$am.

AB Am{U CD ¶m gaiaofm x- Ajmer H$moZ V¶ma H$aVmV. hm H$moZ KS>çmimÀ¶m {XeoZo Am{U KS>çmimÀ¶m{déÕ {XeoZo Aem XmoÝhr {XemZr ‘moOVm ¶oVmo.

x- AjmÀ¶m YZ {Xeoer aofr¶ AmboImZo V¶ma Ho$boë¶m H$moZmbm Ë¶m aofoMr CVaU åhUVmV.hm H$moZ gm‘mÝ¶nUo ‘ ’ Zo Xe©{dbm OmVmo.

gai aofoMr CVaU (inclination) hm Agm H$moZ, H$s Omo x- AjmÀ¶m N>oXZq~Xÿer V¶ma hmoVmo Am{U OmoYZ {XeoZo AgVmo. hm H$moZ x- AjmÀ¶m KS>çmimÀ¶m {déÕ {XeoZo aofoer ‘moObm OmVmo.

Q>rn … Ooìhm AjmnmgyZ aofoer KS>çmimÀ¶m {déÕ {XeoZo H$moZ ‘moObm OmVmo Voìhm aofoMr CVaU hr eyÝ¶mnojm‘moR>r AgVo Aer na§nam Amho.

aofoMm MT> (Slope of a line)


{eS>r q^Vrbm Aer bmdyZ R>odbr Amho H$s {VMm ‘mWm(darb Q>moH$) O{‘ZrnmgyZ EH$m {d{eï> C§Mrda q^Vrbm ñne©H$aVmo.

Q´>H$‘Ü¶o AmoPo ^aVmZm H$m‘Jma bmH$S>r ’$irMm Cn¶moJH$aVmV Vmo CVaË¶m pñWVr‘Ü¶o Agë¶mZo Q>H$‘Ü¶o AmoPo^aUo Ë¶mZm gmono OmVo.

B‘maVrMm {OZm {d{eï> pñWVr‘Ü¶o ~m§Ybobm AgVmo H$sÁ¶m‘wio {OZm MT>Ê¶mMm VmU H$‘r hmoVmo.

B

A

oXI X

Y

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Q

XI X

Y

YI

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darb àË¶oH$ CXmhaUm‘Ü¶o {eS>r/bmH$S>r ’$ir/{OZm Aem pñWVr‘Ü¶o AgVmo H$s MT>Ê¶mMm VmU H$‘rhmoVmo.

q^VrnmgyZ {H$Vr A§Vamda {eS>r R>odmdr H$s Ë¶m‘wio VmU H$‘rV H$‘r hmoB©b?

Imbrb VrZ AmH¥$Ë¶m§Mm Aä¶mg H$am. JQ>m‘Ü¶o MMm© H$am Am{U CH$b J{UVr nÕVrZo H$mT>Ê¶mMm à¶ËZH$am.

(1) (2) (3)C B

AA

BC BC

A

XmoZ bm§~tMo JwUmoÎma KoD$Z ho H$aVm ¶oVo. åhUOo b§~ C§Mr Am{U AmS>do A§Va bjmV R>odm H$s àË¶oH$KQ>Zo‘Ü¶o {eS>rMr bm§~r gmaIrM AgVo.

darb AmH¥$Ë¶m‘Ü¶o, {Z[ajU H$am H$s AB BC Am{U ABC H$mQ>H$moZ {ÌH$moU Amho.

¶oWo, AB bm b§~ A§Va (C§Mr) Am{U CB bm {jVrO (AmS>do) A§Va åhUVmV.

A

BC 4.7 cm

1.6 cm

A

BC 3.5 cm

3.5

cm

A

BC 2.6 cm4.

3cm

AmS>do A§VaC~o A§Va



AB 1.6cm 0.34BC 4.7 cm

AB 3.5cm 1BC 3.5 cm

AB 4.3cm1.65

BC 2.6 cm

àË¶oH$ CXmhaUm‘Ü¶o, {‘iUmè¶m JwUmoÎmambm AC aofoMm MT> åhUVmV.

b§~ A§VamMo {jVrO A§Vamer AgUmè¶m JwUmoÎmambm MT> åhUVmV.


darb CXmhaUm‘Ü¶o,

KQ>Zm (1), AB < BC KQ>Zm (2), AB = BC KQ>Zm (3), AB > BC.

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354 UNIT-14

Amåhr {ZarjU H$aVmo H$s,

* Ooìhm b§~ A§Va ho {jVrO A§Vamnojm H$‘r AgVo Voìhm MT> 1 nojm H$‘r AgVmo.

* Ooìhm b§~ A§Va ho {jVrO A§VamBVHo$ AgVo Voìhm MT> 1 AgVmo.

* Ooìhm b§~ A§Va ho {jVrO A§Vamnojm OmñV AgVo Voìhm MT> 1 nojm A{YH$ AgVmo.

Ooìhm MT> 1 nojm H$‘r AgVmo Voìhm MT>Ê¶mMm VmU H$‘rV H$‘r AgVmo.

aofoMm MT> {ÌH$moU{‘Vr JwUmoÎmamV gwÕm ì¶º$ Ho$bm OmD$ eH$Vmo. Imbrb CXmhaU Aä¶mgm.

aofoMm MT> hm Ë¶mÀ¶m CVaUrÀ¶m H$moZmÀ¶m ñn{e©Ho$ BVa AgVmo. hm gm‘mÝ¶nUo 'm' Zo Xe©{dbm OmVmo.m = tan

gaiaofoÀ¶m MT>mdmMo ‘mZ (Gradient of a straight line)

gaiaofoÀ¶m MT>mdmÀ¶m ‘mZmMr ì¶m»¶m

y ‘Yrb dmT>

x ‘Yrb dmT>aofodarb EH$m q~XÿnmgyZ Xþgè¶m q~XÿH$S>o hmoUmar

hmbMmb Aer H$aVm ¶oVo.

MT>mdmMo ‘mZ (Gradinet) =BCAC

yx

åhUyZ, MT>mdmMo ‘mZ =b§~ C§Mr

AmS>do A§VagaiaofoÀ¶ MT>mdmMo ‘mZ hm Ho$di aofoMm MT> Amho.

Q>rn …

1. x- Aj ñdV…er 0° H$moZ A§mV[aV H$aVmo.åhUyZ x- AjmMm MT> hm m = tan = 0 [ = 0°]

2. y-Aj, x Ajmer 90° Mm H$moZ A§V[aV H$aVmo.åhUyZ, y- AjmMm MT> hm m = tan 90° = ì¶º$ H$aVm ¶oV Zmhr. [ = 90°]

ABC À¶m ~m~VrV

AC Mm MT> =ABBC

BC ~mOyer AC H$moZ ~Z{dVo.

H$moZmÀ¶m ~m~VrV,

AB hr {déÕ ~mOy Am{U BC hr gb¾ ~mOy

MT> ={déÕ ~mOy

gb¾ ~mOy

MT> = tan

C

A

B

AmS>do A§Va

C~oA

§Va

A

BC

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3. H$moZmÀ¶m CVaUrMr qH$‘V MT>mMr qH$‘VValue of slope

= 0° 0

0° < < 90° YZ g§»¶m = 90° ì¶º$ H$aVm ¶oV Zmhr

90° < < 180° F$U g§»¶m

AmVm Vwåhr g‘Oy eH$mb H$s nd©Vmda KoD$Z OmUmao añVo ‘mÏ¶mn¶ªV gai H$m ZgVmV.

Vw‘Mo {‘Ì Am{U {ejH$m§~amo~a MMm© H$am.

{Xboë¶m XmoZ q~XÿVyZ OmUmam gai aofoMm MT>

g‘Om H$mV}{g¶Z àVbmda A(xI, yI) Am{U B(x2, y2)

ho H$moUVohr XmoZ {ZpíMV q~Xÿ AmhoV.

AL x-Aj, BM x-Aj, AN BM H$mT>m.

AmH¥$Vrdê$Z, OL= x1, OM = x2

LM= x2 – x1 = AN

AL= y1 = NM, BM = y2

BN= y2 – y1

ABN ‘Ü¶o, BAN =

À¶m ~m~VrV, tan =BNAN

MT> = tan =2 1

2 1

y yx x

m = 2 1

2 1

y - yx - x

{Xboë¶m XmoZ {ZpíMV q~Xÿ‘YyZ OmUmam gai aofoMm MT>

={Xboë¶m q~Xÿ§Mm y- gh{ZX}eH$m§Mm ’$aH$

Ë¶m§À¶m x- gh{ZX}eH$m§Mm ’$aH$

Vwåhr gaiaofoMm MT> H$mT>Ê¶mMo {eH$bm AmhmV gma»¶mM H$mV}{g¶Z nmVirVrb XmoZ aofm§Mo MT>H$mT>md¶mMo AmhoV, Ago g‘Om.

o

y

( ,,y)x 22

A(,

)x

y 1

1

( – )x2 x1

x2

Lx1 MXI X

Y

YI

y1

(–

)y

y2

1

y 2

B

N ( – )x2 x1

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356 UNIT-14

g‘m§Va aofm (Parallel lines)XmoZ g‘m§Va aofm x-Ajmer gmaImM CVma H$aVmV. åhUyZ

Ë¶mZm gmaImM MT> AgVmo.= [ g§JV H$moZ]

tan = tan

= MT> AB = Mm MT> CD.

g‘m§Va aofmZm g‘mZ MT> AgVmV.CbQ> AWu, Oa XmoZ aofm§Mo MT> g‘mZ AgVrb Va Ë¶m aofm

EH$‘oH$m§Zm g‘m§Va AgVmV.

Q>rn :

* x - Ajmbm MT> 0 Amho. x - Ajmbm g‘m§Va AgUmè¶m H$moUË¶mhr aofoMm MT> XoIrb 0 Amho.

* y- Ajmbm MT> ì¶º$ H$aVm ¶oV Zmhr. y- Ajmbm g‘m§Va AgUmè¶m H$moUË¶mhr aofoMm MT> XoIrb ì¶º$ H$aVm ¶oV Zmhr.

b§~ aofm (Perpendicular lines)Oa AB hr CD bm b§~ Agob Va Ë¶m§Mo CVma AZwH«$‘o Am{U AgVmV.

= 90 + [ ~møH$moZ = Am§Va{déÕ H$moZm§Mr ~oarO]tan = tan (90 + )

tan = – cot [ tan (90 + ) = – cot ]

tan =1

tan tan . tan = – 1

m1 : m2 = –1OoWo m1 AB Mm MT>

m2 CD Mm MT>

m1 =2

1m qH$dm m2 =

1

1m

Oa XmoZ aofm EH$‘oH$m§Zm b§~ AgVrb Va, Ë¶m§À¶m MT>m§MmJwUmH$ma -1 AgVmo.

CbQ> AWu, Oa XmoZ aofm§À¶m MT>m§Mm JwUmH$ma -1 Agob Va Ë¶maofm EH$‘oH$m§Zm b§~ AgVmV.


1. Á¶m aofoMm CVma 60° Amho Va aofoMm MT> H$mT>m.CH$b … = 60°

MT> = m = tanm = tan 60°

m = 3

A

B

C

D

OXI X

Y

YI

A

B

C

DOXI X

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2. Á¶m aofoMm MT> 0 Amho Ë¶m aofoÀ¶m CVmamMm H$moZ H$mT>m.CH$b … MT> = tan

0 = tan

Amåhmbm ‘m{hV Amho, tan 0 = 0

= 0°

3. (3, –2) Am{U (4, 5) q~Xÿ OmoS>Umè¶m aofoMm MT> {ZpíMV H$am.CH$b … (x1, y1) = (3, –2) (x2, y2) = (4, 5)

MT> =2 2

2 1

y yx x =

5 ( 2)4 3 = 5 2

1MT> = 7

4. {gÕ H$am H$s (4, 5) Am{U (0, –2) q~Xÿ OmoS>Umar aofm hr (2, –3) Am{U (–5, 1) q~XÿVyZ OmUmè¶m aofobmb§~ AgVo.

CH$b … aofm 1 gmR>r : (x1, y1) = (4, 5), (x2, y2) = (0, – 2)

MT> =2 1

2 1

y yx x =

2 50 4

MT> =74

= m1

Xþgè¶m aofogmR>r … (x1, y1) = (2, –3), (x2, y2) = (–5, 1)

MT> =2 1

2 1

y yx x =

1 ( 3)5 2

=1 3

7

MT> =47

= m2

m1 × m2=7 44 7

= –1

Ë¶m§À¶m MT>m§Mm JwUmH$ma -1 Amho.

XmoÝhr aofm b§~ AmhoV.

5. P(7, 3) Am{U Q(0, –4) ¶m q~XÿVyZ OmUmam aofoMm MT> H$mT>m. VgoM (i) PQ bm g‘m§Va(ii) PQ bm b§~ AgUmè¶m aofoMm MT> H$mT>m.

CH$b (x1, y1) = (7, 3) (x2, y2) = (0, –4)

MT> =2 1

2 1

y yx x =

4 30 7

=77

MT> = 1

PQ Mm MT> = 1

(i) PQ bm g‘m§Va AgUmè¶m aofoMm MT> = 1 [ g‘m§Va aofm§gmR>r, m1 = m2](ii) PQ bm b§~ AgUmè¶m aofoMm MT> = – 1 [ b§~ aofm§gmR>r, m1 × m2 = –1]

1

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358 UNIT-14

ñdmÜ¶m¶ 14.1

1. aofm§Mm MT> H$mT>m Oa Ë¶m§Mm CVma(i) 90° (ii) 45° (iii) 30° (iv) 0°

2. gai aofm§À¶m CVmamMo H$moZ H$mT>m Á¶m§Mo MT>

(i) 3 (ii) 1 (iii) 13

3. q~Xÿ OmoS>Umè¶m aofm§Mo MT> H$mT>m.(i) (–4, 1) Am{U (–5, 2) (ii) (4, –8) Am{U (5, –2)

(iii) (0, 0) Am{U 3,3 (iv) (–5, 0) Am{U (0, – 7) (v) (2a, 3b)Am{U (a, –b)

4. q~XÿÀ¶m XmoZ OmoS>çm‘YyZ OmUmè¶m aofm H$mT>ë¶mg Ë¶m g‘m§Va AgVmV qH$dm b§~ AgVmV ho H$mT>m.(i) (5, 2), (0, 5) Am{U (0, 0), (–5, 3) (ii) (3, 3), (4, 6) Am{U (4, 1), (6, 7)

(iii) (4, 7), ( 3, 5) Am{U (–1, 7), (1, 6) (iv) (–1, –2), (1, 6) Am{U (–1, 1), (–2, –3)

5. q~Xÿ OmoS>Umè¶m aofmZm b§~ AgUmè¶m aofoMm MT> H$mT>m.(i) (1, 7) Am{U (–4, 3) (ii) (2, –3) Am{U(1, 4)

6. q~Xÿ OmoS>Umè¶m aofm§Zm g‘m§Va AgUmè¶m aofoMm MT> H$mT>m.(i) (–4, 3) Am{U (2, 5) (ii) (1, –5) Am{U (7, 1)

7. (2, 7) Am{U (3, 6) q~XÿVyZ OmUmar aofm (9, a) Am{U (11, 3) ¶m q~XÿZm OmoS>Umè¶m aofobm g‘m§VaAgob Va a H$mT>m.

8. (1, 0) Am{U (4, 3) q~XÿVyZ OmUmar aofm (–2, –1) Am{U (m, 0) ¶m q~XÿZm OmoS>Umè¶m aofobm b§~Agob Va m Mr qH$‘V H$mT>m.

MT> Am§VaN>oX ñdê$nmVrb aofoMo g‘rH$aU (Equation of a line in slope intercept form)

aofr¶ AmboImMo {ZarjU H$am. PQ gaiaofm hr x- Ajmbm P Am{U yAjmbm Q ‘Ü¶o N>oXVo.

OP A§Va åhUOoM, Ama§ q~XÿnmgyZ x- Ajmdarb N>oXZq~Xÿn¶ªV A§Va¶mbm aofoMm x- Am§VaN>oX åhUVmV.

OQ A§Va åhUOoM, Ama§ q~XÿnmgyZ y-Ajmdarb N>oXZ q~Xÿn¶ªVMo A§Va¶mbm aofoMm y -Am§VaN>oX åhUVmV.

darb CXmhaUm‘Ü¶o x-Am§VaN>oX 3 AgyZ y-Am§VaN>oX 4 Amho.Imbrb KQ>Zm‘Ü¶o x Am{U y Am§VaN>oX AmoiIm.

O

4321

–1–2–3

1 2 3–1–2–3P

Q

XI X

Y

YI

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O

4321

–1–2–3–4

1 2–1–2XI X

Y

YI

O

54321

–1–2–3

1 2 3–1–2–3XI X

Y

YI

AmVm Imbrb AmboImMm Aä¶mg H$am.g‘Om aofm 'l ' y-Ajmbm A(0,c) ‘Ü¶o {‘iVo.OA hm l aofoMm y-Am§VaN>oX Amho.g‘Om P hm l aofodarb H$moUVmhr q~Xÿ Amho.PQ OX Am{U AB PQ H$mT>m.

g‘Om H$s 'l ' hm x-Ajmer Q H$moZ H$aVmo.PAB = (g§JV H$moZ)MT> = m = tan

m ={déÕ ~mOyg§b¾ ~mOy

PAB ‘Ü¶o, g§b¾À¶m ~m~VrV

m =PBAB

[PB = PQ – BQ, AB = OQ]

m =cy

xmx = y – c

y = mx + c

'm' MT> Agbobo Am{U Á¶mMm y-Am§VaN>oX hm 'c' Agbobo g‘rH$aU y = mx + c Ago {Xbo Amho.


1. MT> ½ Am{U y-Am§VaN>oX 3 Agbobo aofoMo g‘rH$aU H$mT>m.

CH$b … m= ½, y Am§VaN>oX = c = – 3g‘rH$aU : y = mx + c

y = ½x + (–3) = ½x – 3

y =6

2x

2y = x – 6

qH$dm x – 2y – 6 = 0

O

y

q l

A B

Q xc

x

( –

)y

c

P(, )

xy

(0, c)

XI X

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360 UNIT-14

2. CVaUrMm H$moZ 45° Am{U y-Am§VaN>oX 2 Agbobo aofoMo g‘rH$aU H$mT>m.

CH$b … = 45°, y-Am§VaN>oX = C = 2

MT> = m = tan m = tan 45°

m = 1

g‘rH$aU : y = mx + cy = 1(x) + 2 = x + 2

qH$dm x – y + 2 = 0

3. 3x + 2y + 1 = 0 ho aofoMo g‘rH$aU Amho. Ë¶mMm MT> Am{Uy-Am§VaN>oX H$mT>m.CH$b … g‘rH$aU : 3x + 2y + 1 = 0

y = mx + c ¶m à‘m{UV Z‘wÝ¶mV nwÝhm {bhm.2y = –3x – 1

2 Zo ^mJyZ, Amåhmbm {‘iVo

y =3 12 2

x

y = mx + c er VwbZm H$am

m =32 , c =

12

MT> =32

y Am§VaN>oX =12

ñdmÜ¶m¶ 14.2

1. aofoMo g‘rH$aU H$mT>m, Á¶mMo CVaU H$moZ Am{U y-Am§VaN>oX {Xbo AmhoV.(i) = 60°, y-Am§VaN>oX – 2 Amho. (ii) = 45°, y-Am§VaN>oX 3 Amho.

2. aofoMo g‘rH$aU H$mT>m Á¶mMm MT> Am{U y-Am§VaN>oX {Xbm Amho.(i) MT> = 2, y- Am§VaN>oX = – 4

(ii) MT = 23 , y-Am§VaN>oX = 1

2

(iii) MT = –2, y-Am§VaN>oX = 3

3. aofm§Mo MT> Am{U y-Am§VaN>oX H$mT>m.(i) 2x + 3y = 4 (ii) 3x = y

(iii) x – y + 5 = 0 (iv) 3x – 4y = 5

4. x = 2y hr aofm 2x – 4y + 7 = 0 bm g‘m§Va Amho H$m? (gyMZm … g‘m§Va aofmbm g‘mZ MT> AgVmV.)

5. Ago XmIdm H$s, aofm 3x + 4y + 7 = 0 Am{U 28x – 21y + 50 = 0 EH$‘oH$m§Zm b§~ AmhoV.(gyMZm … b§~ aofm§gmR>r, m1m2 = –1]

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XmoZ q~Xÿ‘Yrb A§Va (Distance between two points)

A Am{U B q~Xÿ§Mo {Z[ajU H$am. Ë¶m‘Yrb A§Va ho, Vo OmoS>Umè¶m aofmI§S>mMr bm§~r.

A BA BAmåhr aofmI§S>mMr bm§~r qH$dm XmoZ q~Xÿ‘Yrb A§Va ‘moOÊ¶mgmR>r ’y$Q>nÅ>r dmnaVmo.

àË¶j Z ‘moOVm XmoZ q~Xÿ‘Yrb A§Va H$mT>Uo e³¶ Amho H$m?

¶mMm Aä¶mg H$aUo ‘Zmoa§OH$ Amho H$s Amåhr Ë¶m‘Yrb A§VaH$mTy> eH$Vmo, Oa Vo H$mV}{g¶Z àVbmda ñWmnZ Ho$bo AgVrb Va.

Oa A Am{U B q~Xÿ X- Ajmda AgVrb, Va Amåhr A Am{U B‘Yrb EH$H$ aofmI§S>m§Mr g§»¶m ‘moOy eH$Vmo.

AB= 4 cm qH$dm

A Mm gh{ZX}eH$ (1, 0) Am{U B Mm (5, 0)

AB= x2 – x1 = 5 – 1

AB = 4 g|.‘r.

gm‘mÝ¶nUo, Oa A Am{U B Mo gh{ZX}eH$ AZwH«$‘o (x1, 0)Am{U(x2, 0) AgVrb Va AB = x2 – x1.

Oa P Am{U Q ho XmoZ q~Xÿ y-Ajmda AgVrb Va Amåhr P Am{UQ ‘Yrb EH$H$ aofmI§S>m§Mr g§»¶m ‘moOy eH$Vmo.

PQ= 6 g|.‘r. qH$dm

P Mm gh{ZX}eH$ (0, 4) Am{U Q Mm (0, –2)

PQ= y2 – y1 qH$dm PQ = y1–y2

= –2 – 4 = –6 = 4 – (–2) = 6PQ = 6 cm = 6

gm‘mÝ¶nUo, Oa P Am{U Q Mo gh{ZX}eH$ AZwH«$‘o (0, y1) Am{U (0,y2) AgVrb Va PQ = y2 – y1

X- Ajmbm g‘m§Va AgUmè¶m aofoMr bm§~rOa AB X- Ajmbm g‘m§Va AgobAA' X- Aj,BB' x - Aj H$mT>m.AA'B'B Zo V¶ma hmoUmar AmH¥$Vr Am¶V Amho.

AB = A'B'A'B' = x2 – x1 = 6 – 1A'B' = 5 AB = 5 g|.‘r.

Q>rn … X- Ajmbm g‘m§Va AgUmè¶m aofodarb gd© q~XÿZm gmaImMy-gh{ZX}eH$ AgVmo.

O

321

–1–2

1 2 3 4 5–1–2XI X

Y

YI

A B

O

54321

–1–2–3

1 2 3–1–2

P

Q

XI X

Y

YI

XI X

YI

–2

–3

Y

O 1 2 3 4 5 6–1–2–1

5

4

3

1

2

AI BI

bjmV R>odm … A§Va ho Zoh‘r YZg§»¶m AgVo.

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362 UNIT-14

y- Ajmbm g‘m§Va AgUmè¶m aofoMr bm§~r

Oa AB, y- Ajmbm g‘m§Va Agob

AA' y-Aj

BB' y-Aj H$mT>m.

AA'B'B Zo V¶ma hmoUmar AmmH¥$Vr Am¶V Amho.AB = A'B'

A'B' = y2 – y1

= –3 – 4 = – 7A'B' = 7 cmAB = 7 cm

Q>rn … y-Ajmbm g‘m§Va AgUmè¶m aofodarb gd© q~XÿZm gmaImMm x-gh{ZX}eH$ AgVmo.

darb gd© CXmhaUm‘Ü¶o, OoWo Amåhmbm XmoZ q~Xÿ‘Yrb A§Va H$mT>VmZm, XmoZ AQ>r bjmV Amë¶m,* q~Xÿ EH$ Va x-Ajmda AgVmV.

* q~Xÿ Aer aofm ~Z{dVmV H$s Or x-Ajmbm qH$dm y-Ajmbm g‘m§Va AgVo.AmVm H$mV}{g¶Z àVbmdarb P Am{U Q q~XÿMo {ZarjU H$am. P Am{U

Q q~Xÿ x-Aj qH$dm y-Ajmda ZgVmV.g‘Om Vo OmoSy>. aofmI§S> PQ hm aofmI§S> x-Ajmbm qH$dm y-Ajmda

g‘m§Va ZgVmV.AmVm, x qH$dm y Ajmda ZgUmè¶m qH$dm Ë¶mZm g‘m§Va ZgUmè¶m XmoZ

q~Xÿ‘Yrb A§Va H$go H$mT>Vm ¶oVo Vo Aä¶mgy.

A§VamMo gyÌ (Distance formula)

Oa A(x1, y1) Am{U B(x2, y2) ho {Xbobo XmoZ q~Xÿ AmhoV, BC || yAj Am{U AC || x-Aj H$mT>m. C Mo gh{ZX}eH$ (x2, y1)

AC= x2 – x1 Am{U BC = y2 – y1

ABC H$mQ>H$moZ {ÌH$moU Amho, nm¶W°JmoagÀ¶m à‘o¶mdê$Z,AB2 = AC2 + BC2

AB2 = (x2 – x1)2 + (y2 – y1)

2

AB = 2 22 1 2 1x x y y

(x1, y1) Am{U (x2, y2) ¶m XmoZ q~Xÿ‘Yrb A§Va

d = 2 22 1 2 1x - x + y - y

d = 2 2difference of absscissa difference of ordinates

Q>rn … gmo¶rgmR>r, n{hë¶m MaUm‘Ü¶o ¶oUmao q~Xÿ gyÌ {‘i{dÊ¶mgmR>r Amåhr KoVbo. na§Vw, H$moUË¶mhrMaUm‘Ü¶o ¶oUmè¶m q~Xÿ§gmR>r gd© gyÌ gË¶ AgVmV.

O

54321

–1–2–3

1 2 3 4–1–2

AI

BI

A(2, 4)

B(2, –3)

–3–4XI X

Y

YI

P

Q

OXI X

Y

YI

OXI X

Y

YI

(x- gh{ZX}eH$mVrb ’$aH$)2 (y- gh{ZX}eH$mVrb ’$aH$)2

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àVbmVrb q~XÿMo Ama§^q~XÿnmgyZ AgUmao A§Va.

A(x, y) Am{U O(0, 0) ho XmoZ q~Xÿ {Xbo AmhoV.

A§VamÀ¶m gyÌmdê$Z,

d = 2 22 1 2 1x x y y

d = 2 20 0x y = 2 2x y


1. (2, 3) Am{U (6, 6) q~Xÿ‘Yrb A§Va H$mT>m.

CH$b … (x1, y1) = (2, 3) Am{U (x2, y2) = (6, 6)

d = 2 22 1 2 1x x y y = 2 26 2 6 3

d = 2 24 3 = 16 9 = 25

d = 5 EH$Ho$

2. Ama§^q~Xÿ Am{U q~Xÿ (12, –5) ¶m‘Yrb A§Va H$mT>m.

CH$b … Ama§^q~Xÿ Am{U (x, y) ‘Yrb A§Va = 2 2x y

¶oW (x, y) = (12, –5)

d = 2 212 ( 5) 144 25 169

d = 13 EH$H$

3. x-Ajmdarb q~Xÿ§Mo gh{ZX}eH$ H$mT>m Oo (6, –3) ¶m q~XÿnmgyZ 5 EH$H$ Xÿa AmhoV.

CH$b … x-Ajmdarb H$moUË¶mhr q~XÿMo gh{ZX}eH$ (x, 0) ñdê$nmV AgVmV.

(x1, y1) = (x, 0) Am{U (x2, y2) = (6, –3) Am{U d = 5

d = 2 22 1 2 1x x y y

5 = 22( 6) 0 ( 3)x = 2 2( 6) 3x

XmoÝhr ~mOy§Mm dJ© H$ê$Z, 52 =2

26 9x

25 = x2 – 12x + 36 + 9

x2 – 12x + 45 – 25 = 0 x2 – 12x + 20 = 0 (x – 10)(x – 2) = 0 x – 10 = 0 or x – 2 = 0 x = 10 or x= 2

x - Ajmdarb Amdí¶H$ q~Xÿ (10, 0) Am{U (2, 0)

O(0, 0)

A( , )x y

XI X

Y

YI

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364 UNIT-14

4. x Mr Aer qH$‘V H$mT>m H$s q~Xy (2, 5) Am{U (x, –7) ¶m‘Yrb A§Va 13 Agob.

CH$b … d= 13, (x1, y1) = (2, 5) Am{U (x2, x2) = (x, –7)

d = 2 22 1 2 1x x y y

13 = 2 22 ( 7 5)x = 2( 2) 144x

XmoÝhr ~mOy§Mo dJ© H$ê$Z, 132= x2 – 4x + 4 + 144169 = x2 – 4x +148

x2 – 4x + 148 – 169 = 0x2 – 4x – 21 = 0

(x – 7) (x + 3) = 0 x = 7 qH$dm x = – 3

5. (8, -4), (9, 5) Am{U (0, 4) {eamoq~Xÿ AgUmam {ÌH$moU hm g‘{Û^wO {ÌH$moU Amho Ago XmIdm.

CH$b … {ÌH$moU hm g‘{Û wO Amho ho XmI{dÊ¶mgmR>r Amnë¶mbm {ÌH$moUmÀ¶m 2 ~mOy§Mr bm§~r g‘mZ AgVo hoXmIdmdoM bmJob.g‘Om A(8, –4), B(9, 5), C(0, 4)

d = 2 22 1 2 1x x y y

AB = 2 29 8 5 ( 4) = 2 21 9 1 81 82

BC = 2 2 2 2(9 0) (4 5) 9 ( 1) 81 1 82

CA = 22 2 2(0 8) 4 ( 4) 8 8 64 64 128

Á¶mAWu AB = BC, {ÌH$moU hm g‘{Û wO Amho.

6. {gÕ H$am H$s q~Xÿ A(–3, –2), B(5, –2), C(9, 3) Am{U D(1, 3) ho g‘m§Va^wO Mm¡H$moZmMo{eam oq~Xÿ Amh oV.

CH$b … Mm¡H$moZmÀ¶m g§‘wI ~mOw g‘mZ AgVrb Va Vmo Mm¡H$moZ g‘m§Va^wO AgVmo.

d = 2 22 1 2 1x x y y

AB = 2 25 ( 3) 2 ( 2) = 2 28 0 64 8

CD = 2 2(9 1) (3 3) = 2 28 0 64 8

BC = 2 2 2 2(5 9) ( 2 3) 4 5 16 25 41(

DA = 2 2 2 21 ( 3) 3 ( 2) 4 5 16 25 41

AB= CD Am{U BC = DA

ABCD hm g‘m§Va wOMm¡H$moZ Amho.

B(9, 5)

C(0, 4) A(8, –4)

A(–3, –2) B(5, –2)

C(9, 3)

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ñdmÜ¶m¶ 14.3

1. Imbrb q~Xÿ§À¶m OmoS>çm‘Yrb A§Va H$mT>m(i) (8, 3) Am{U (8, –7) (ii) (1, –3) Am{U (–4, 7) (iii) (–4, 5) Am{U (–12, 3)

(iv) (6, 5) Am{U (4, 4) (v) (2, 0) Am{U (0, 3) (vi) (2, 8) Am{U (6, 8)(vii) (a, b) Am{U (c, b) (viii) (cos – sin ) Am{U (sin – cos )

2. Ama§ q~Xÿ Am{U q~Xÿ ¶m‘Yrb A§Va H$mT>m.(i) (–6, 8) (ii) (5, 12) (iii) (–8, 15)

3. (i) (3, 1) Am{U (0, x) ¶m q~Xÿ‘Yrb A§Va 5 Amho Va x H$mT>m.

(ii) q~Xÿ P(2, –1) hm (a, 7) Am{U (–3, a) ¶m q~XÿnmgyZ g‘mZ A§Vamda Amho Va 'a' H$mT>m.

(iii) y-Ajmdarb q~Xÿ H$mT>m, Omo (5, 2) Am{U (–4, 3) ¶m q~XÿnmgyZ g‘mZ A§Vamda Amho.

4. {ÌH$moUm§Mr n[a{‘Vr H$mT>m Á¶m§À¶m {eamoq~XÿZm Imbrb gh{ZX}eH$ AmhoV.(i) (–2, 1), (4, 6), (6, –3) (ii) (3, 10), (5, 2), (14, 12)

5. {gÕ H$am H$s q~Xÿ A(1, –3), B(–3, 0) Am{U C(4, 1) ho g‘{Û^wO H$mQ>H$moZ {ÌH$moUmMo {eamoq~Xÿ AmhoV.

6. (-5, 4) dVw©i‘Ü¶ AgUmar Am{U Vr (-7, 1) q~XÿVyZ OmVo Aer dVw©imMr {ÌÁ¶m H$mT>m.

7. {gÕ H$am H$s àË¶oH$ gh{ZX}eH$mMo gQ> ho g‘m§Va wO Mm¡H$moZmMo {eamoq~Xÿ AmhoV.(i) (–5, –3), (1, –11), (7. –6), (1, 2) (ii) (4, 0), (–2, –3), (3, 2), (–3, –7)

8. {ÌH$moUmÀ¶m {eamoq~Xÿ§Mo gh{ZX}eH$ {Xbo AmhoV. {ÌH$moUm§Mo àH$ma AmoiIm.(i) (2, 1) (10, 1) (6, 9)(ii) (1, 6) (3, 2) (10, 8)

(iii) (3, 5) (–1, 1) (6, 2) (iv) (3, –3) (3, 5) (11, –3)

I§S> gyÌ (Section Formula)

g‘Om AB aofm, q~Xÿ A(x1, y1) Am{U B(x2, y2) ¶mZm OmoS>VoAm{U P q~Xÿ AB aofmI§S>mbm m : n JwUmoÎmamV {d^mJVo.

APPB =

mn

AmVm, P q~XÿMo gh{ZX}eH$ {‘i{dÊ¶mMm à¶ËZ H$ê$.g‘Om P Mo gh{ZX}eH$ (x, y) AmhoV.x-Ajmda AL, PM Am{U BN ho b§~ H$mT>m.AR PM Am{U PQ BN

AmH¥$Vrdê$Z, AR= LM = OM – OL = x – x1

PR = PM – RM = PM – AL = y – y1

PQ = MN = ON – OM = x2 – xBQ = BN – QN = BN – PM = y2 – y

{ZarjU H$am.

L M NO

–x x2

–x x1

–y y1

R

PA

( , )b x y2 2

( , )x y ( , )x y1 1

–y

y2

Q

XI XYI

Y

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366 UNIT-14

APR~ PBQ

AR PR AP= =PQ BQ PB [ g‘ê$n {ÌH$moUmÀ¶m g§JV ~mOy à‘mUmV AgVmV.]

g‘OmARPQ

= APPB

KoD$.

1

2

x xx x =

mn

nx + mx = mx2 + nx1

x (m + n) = mx2 + nx1

x(m + n) = mx2 + nx1

x = 2 1mx nxm n

PRBQ

=APPB {dMmamV ¿¶m.

1

2

y yy y =

mn

ny – ny1 = my2 – my

ny + my = my2 + ny1

y(m + n) = my2 + ny1

y = 2 1my nym n )

P Mo gh{ZX}eH$ = (x, y)

P Mo gh{ZX}eH$ = 2 1 2 1mx + nx my + ny,m + n m + n

Oa P hm AB Mm ‘Ü¶q~Xÿ m : n =1:1 Voìhm P Mo gh{ZX}eH$ = 2 1 2 1x + x y + y,2 2

¶mbm ‘Ü¶ q~Xÿ gyÌ (mid point fomula) Agohr åhUVmV.


1. P [~§XÿMo gh{ZX}eH$ H$mT>m, Omo A(4, –5) Am{U B(6, 3) bm 2 : 5 ¶m JwUmoÎmamV OmoSy>Z aofobm Xþ^mJVmo.

CH$b …g‘Om P Mo gh{ZX}eH$$ (x, y) J¥hrV … m : n = 2 : 5

(x1, y1) = (4, –5) Am{U (x2, y2) = (6, 3)

I§S> gyÌmdê$Z.

x = 2 1 2(6) 5(4) 12 20 322 5 7 7

mx nxm n

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y = 2 1 2(3) 5( 5) 6 25 192 (.5) 7 7

my nym n P Mo gh{ZX}eH$ =

32 19,7 7

2. q~Xÿ (5, 4), q~Xÿ (2, 1) Am{U (7, 6) aofm§Zm OmoSy>Z {d^mJV Agob Va Ë¶mMo JwUmoÎma H$mT>m.

CH$b … g‘Om nm{hOo Agbobo JwUmoÎma m : n Amho.

J¥hrV … (x1, y1)= (2, 1) Am{U (x2, y2) = (7, 6), (x, y) = (5, 4)

I§S> gyÌmdê$Z

x = 2 1mx nxm n

qH$dm y = 2 1my nym n

5 =(7) (2)m nm n

4 =(6) (1)m nm n

5 =7 2m nm n

4 =6m nm n

5m + 5n = 7m + 2n 4m + 4n = 6m + n5m – 7m + 5n – 2n = 0 4m – 6m + 4n – n = 0

–2m + 3n = 0 –2m + 3n = 0–2m = –3n –2m = –3mn

mn

=32

mn

=32

m : n = 3 : 2 m : n = 3 : 2

3. q~Xÿ (2, 3) Am{U (4, 7) OmoS>Umè¶m aofmI§S>mÀ¶m ‘Ü¶q~XÿMo gh{ZX}eH$ H$mT>m.

CH$b …(x1, y1) = (2, 3) Am{U (x2, y2) = (4, 7)

‘Ü¶q~Xÿ gyÌmdê$Z, ‘Ü¶q~XÿMo gh{ZX}eH$ = 1 2 1 2,2 2

x x y y

=2 4 3 7,

2 2 =6 10,2 2 = (3, 5)

ñdmÜ¶m¶ 14.4

1) H$moUË¶m JwUmoÎmam‘Ü¶o (-2, 3) q~Xÿ hm (-3, 4) Am{U (4, 9) q~Xÿ OmoS>Umè¶m aofmI§S>mZm {d^mJVmo?2) C(1,1) q~XÿV, A(-2,7) Am{U B OmoS>Umam aofmI§S> 3:2 JwUmoÎmamV {d^mJVmo. Va B Mo gh{ZX}eH$

H$mT>m.3) q~Xÿ (-1, k) hm (-3, 10) Am{U (6, -8) q~XÿZo OmoS>Umè¶m aofmI§S>mbm Xþ mJVmo, Va JwUmoÎma H$mT>m.4) (-3, 10) Am{U (6, -8) q~Xÿ OmoS>Umè¶m aofmI§S>mÀ¶m ‘Ü¶q~XÿMo gh{ZX}eH$ H$mT>m.5) g‘m§Va wO Mm¡H$moZmMo VrZ H«$‘dma {eamoq~Xÿ A(1,2), B(2,3) Am{U C(8,5) AmhoV. Mm¡Wm {eamoq~Xÿ

H$mT>m. (gyMZm … g‘m§Va wO Mm¡H$moZmMo H$U© EH$‘oH$mZm Xþ mJVmV).

* * *

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368 UNIT-14

CÎmao

ñdmÜ¶m¶ 14.1

1. (i) ND (ii) 1 (iii)13 (iv) 0 2. (i) 60º (ii) 45º (iii) 30º

3. (i) –1 (ii) 6 (iii) 3 (iv) 57- (v) 4b

a 4. (i) g‘m§Va (ii) g‘m§Va (iii) b§~ (iv) g‘m§Va

5. (i)54

(ii)17

6. (i)13

(ii) 1 7) 5

8. –3

ñdmÜ¶m¶ 14.2

1. (i)y = 3 x – 2 (ii) y = x + 3 2. (i) y = 2x – 4 (ii) 6y = –4x + 3 (iii) y =- 2x + 3

3. (i) m = 32- , c = 3

4 (ii) m = 3, c = 0 (iii) m = 1, c = 5 (iv) 3 54 4m = ,c = -

ñdmÜ¶m¶ 14.3 1. (i) 10 EH$Ho$ (ii) 55 EH$Ho$ (iii) 172 EH$Ho$ (vi) 5 EH$Ho$ (v) 13 EH$Ho$ (vi) 4 EH$Ho$

(vii) (c–a) EH$Ho$ (viii) 22 (sin cos )EH$Ho$

2. (i) 10 EH$Ho$ (ii) 13 EH$Ho$ (iii) 17 EH$Ho$ 3. (i) 5 qH$dm –3 (ii) 7 (iii) (0, –2)

4. (i)( )8580 +61 + EH$Ho$ (ii) ( )181125 +68 + EH$Ho$ 6) 13 EH$Ho$

ñdmÜ¶m¶ 14.4

1. 1:6 2. (3, –3) 3. 2:7,k = 6 4. 32 1, 5. (7, 4)

*****

Drawing graph ofequation ax+by=c

Coordinate Geometry

Distance betweentwo points

Inclination of astraight line lying on

axisx-lying on

axisy-Length of line

parallel toaxisx-

Length of lineparallel to

axisy-

2 1

2 1

y ym =x x

Distance Formula2 2

2 1 2 1d ( ) ( y )x x y

Section Formula

p ,2 1 2 1mx +nx my +nym+n m+n

Section Formula

p ,2 1 2 1

2 2x + x y + y

Slope of a straightline, m = tan

gh{ZX}eH$ ^y{‘Vr

XmoZ q~XÿVrb A§Vaax+by=c ¶m g‘rH$aUmMmAmboI H$mT>Uo

gai aofoMm CVma y=Ajmdarb

x=Ajmdarb

x=Ajmbm g‘m§VaaofoMr bm§~r

A§VamMo gyÌ

gai aofoMm MTm=tan I§S>mMo gyÌ (‘Ü¶ q~Xþ)

I§S>mMo gyÌ

Am§VaN>oX MT>mÀ¶m aofoÀ¶mg‘rH$aUmMo ñdê$n

y = mx+c

b§~ aofm§MmMT>mMm JwUmH$ma -

1 AgVmo

g‘m§Va aofoMmMT> g‘mZAgVmo

y=Ajmbm g‘m§VaaofoMr bm§~r

XmoZ q~XÿVyZ OmUmè¶mgai aofoMm MT>

2 1 2 1x x y y

2 22 1 2 1x x y ym=

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¶m KQ>H$m‘wio Vwåhmbm ¶m§Mo gwb^rH$aU hmoVo,

* d¥Îm{MÎmr, e§Hy$ Am{U JmobmMr ì¶m»¶m H$aUo.* d¥Îm{MÎmr, e§Hy$ Am{U Jmobm§À¶m JwUY‘m©Mo ñnï>rH$aU

H$aUo.* d¥Îm{MÎmr, e§Hy$ Am{U Jmob ¶m§Mo joÌ’$i Am{U KZ’$i

¶m§Mo gyÌ aMUo.* d¥Îm{MÎmr, e§Hy$ Am{U Jmob ¶m§Mo dH«$n¥ð>’$i Am{U

g§nyU© n¥ð>’$i H$mT>Uo.* d¥Îm{MÎmr, e§Hy$ Am{U Jmob ¶m§Mo KZ’$i H$mT>Uo.* d¥Îm{MÎmr, e§Hy$ Am{U Jmob ¶m§Mo joÌ’$i Am{U KZ’$i

¶mda AmYmarV CXmhaU gmoS>{dUo.* e§Hy$À¶m ’«$ñQ>‘À¶m AWm©Mo ñnï>rH$aU H$aUo.* ’«$ñQ>‘ e§Hy$Mo joÌ’$i Am{U KZ’$i H$mT>Uo.* XmoZ qH$dm Am{YH$ KZnXmWm©nmgyZ ~Zboë¶m dñVy§Zm

AmoiIUo.* EH${ÌV KZ nXmWmªMo joÌ’$i Am{U KZ’$i emoYUo.* A{Z¶{‘V AmH$mamÀ¶m AmH¥$Ë¶m§Mm à‘m{UV AmamIS>m

aMUo.* A{Z¶{‘V AmH$mamÀ¶m AmH¥$Ë¶m§Mo joÌ’$i H$mT>Uo.

‘mnZo (Mensuration)15

Give me a place to stand,and I will move the earth.

- Archimedes of Syracuse

* d¥Îm{MVr, e§Hy$, e§Hy$Mm ’«$ñQ>‘,Jmob Am{U AY©JmobmMm AW©.

* d¥Îm{MVr, e§Hy$, e§Hy$Mm ’«$ñQ>‘,Jm ob Am{U AY©JmobmMo dH «$n¥ð>’$i Am{U KZ’$i.

* EH${ÌV AmH¥$Ë¶m, Ë¶m§Mo joÌ’$iAm{U KZ’$i.

* à‘m{UV AmamIS>m H$mT>Uo.

Am{H©$‘oS>rg Archimedes(287BC-212BC)

gm¶amHw$g Mo Am{H©$‘oS>rg, {ggrbr

ho nwamVZ H$mimVrb EH$ à{gÕ J{UVVk

åhUyZ AmoiIbo OmVmV. ^y{‘Vr‘Yrb

gnmQ > AmH ¥ $Ë¶m §M o j oÌ’ $i Am{U

KZnXmWmªMo joÌ’$i Am{U KZ’$im{df¶r

AË¶§V ‘hËdnyU© ¶moJXmZ {Xbo Amho.

Ë¶m §Zr Jm obmM o KZ’$i h o

AY©JmobmÀ¶m KZ’$imÀ¶m XmoZ V¥VrAm§e

AgVo ho {gÕ Ho$bo Amho.

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370 UNIT-15

nXmWmªÀ¶m ‘mnm§Mm Aä¶mg H$aUo AË¶§V Amdí¶H$ Amho H$maU {ZË¶ OrdZmVrb H$maImZo Am{UA{^¶m§{ÌH$s‘Yrb Ë¶mÀ¶m Cn¶moJm‘wio KZ KZm¶V$, bmobH$ Am{U {MÎmr ¶m§Mo KZnXmWmªÀ¶m n¥ð>^mJmMo joÌ’$iAm{U KZ’$i H$mT>Ê¶mMm Aä¶mg Vwåhr nydu Ho$bobm Amho.

‘mnm§er g§~§{YV Agboë¶m AmUIr H$mhr KZ nXmWmªÀ¶m CXmhaUm§Mm {dMma H$é.

• J«m‘ n§Mm¶VrZo d¥Îm{MÎmrÀ¶m AmH$mamMr C§M nmÊ¶mMr Q>mH$s V¶ma Ho$br Amho. Q>mH$sMmAmH$ma H$gm emoYmb? Ë¶m Q>mH$sbm a§J bmdÊ¶mgmR>r ¶oUmam IM© {H$Vr?

• Q´>H$À¶m d¥Îm{MÎmr AmH$mamÀ¶m B§YZ Q>mH$s‘Ü¶o gmR>{dbobo {S>PobMo à‘mU Vwåhr H$go H$mT>mb?• dmT>{XdgmMr Q>monr ~Z{dÊ¶mgmR>r bmJUmè¶m a§JrV H$mJXmMo à‘mU {H$Vr?• {Xboë¶m ~ogZ bmSy>À¶m dOZmÀ¶m {‘lUmnmgyZ {H$Vr ~ogZ bmSy> V¶ma Ho$bo OmVmV?

darb Z‘yX Ho$boë¶m gd© CXmhaUmMo CH$b ho KZnXmW© Ogo H$s d¥Îm{MÎmr e§Hy$ Am{U Jmob ¶m§À¶m n¥f^mJmMojoÌ’$i Am{U KZ’$i ¶mda Adb§~yZ Amho.

¶m nXmWmªÀ¶m n¥ð>^mJmMo joÌ’$i Am{U KZ’$i H$go H$mT>bo OmVo? ¶mgmR>rH$moUVr gyÌo dmnabr OmVmV?

¶m KQ>H$m‘Ü¶o AmnU d¥Îm{MÎmr e§Hy$ Am{U Jmob ¶m§À¶m n¥ð>^mJmMo joÌ’$i Am{U KZ’$i

¶m§Mm Aä¶mg H$aUma AmhmoV.

d¥Îm{MÎmr (Cylinder)

Vwåhr Aä¶mgbobo Amho H$s, Ooìhm AZoH$ Am¶VmH¥$VrH$mJXmMr n¥ð>o EH$‘oH$mda R>odbr Va Amåhmbm KZm¶V qH$dm Am¶VmH¥$VrVimMm bmobH$ {‘iVmo.

Ooìhm EH$ê$n dVw©imda H$mJXmMr n¥ð>o EH$‘oH$mda R>odbr VaAmnë¶mbm H$m¶ {‘iVo? Imbr {Xboë¶m AmH¥$Ë¶m§Mo {ZarjU H$am.

{‘imboë¶m (V¶ma Pmboë¶m) KZnXmWm©bm d¥Îm{MÎmr Ago åhUVmV.d¥Îm{MÎmr ¶m dVw©imH$ma Vi Agboobo bmobH$ AmhoV.

H¥$Vr … EH$ OmS> H$mJX ¿¶m Am{U ABCD ¶m Am¶Vm‘Ü¶o H$mnm. EH$m ~mOybm EH$ bm§~ H$mR>r {MH$Q>dm. Vmo ABåhUy. Vr Vw‘À¶m XmoÝhm Viì¶mda Yam Am{U OmoamV {’$adm. (Vwåhr EH$ bm§~ Xmoam {MH$Q>dm Am{U Xmoam hmVmVYê$Z XmoÝhr ~mOybm OmoamV {’$adm) hr H¥$Vr JQ>m‘Ü¶o H$am. MMm© H$am Am{U ¶m ¶w³Ë¶m EH$‘oH$mer {d^mJm.

Vwåhr H$moUVo AmH$ma AmoiIy eH$mb?

darb H¥$Vrdê$Z AmnU Agm {ZîH$f© H$mTy> eH$VmoH$s,

Oa Am¶VmMr EH$ ~mOy pñWa R>odyZ Ë¶m ^mdVr{’$a{dë¶mZo V¶ma hmoUmè¶m KZnXmWm©bm b§~dVw©imH$ma d¥Îm{MÎmr Ago åhUVmV.

A

BC

D

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Mensuration 371

V w‘À¶m X ¡Z § {XZ OrdZ m‘Ü¶ o V wåhrd¥Îm{MÎmrÀ¶m AmH$mamÀ¶m AZoH$ dñVy nm{hë¶mAmhoV. Ë¶mn¡H$s H$m§hr Imbrbà‘mUo {Xbobo AmhoV.

d¥Îm{MÎmr ¶m ârd qH$dm nmoH$i Agy eH$VmV.{ZarjU H$am H$s d¥Îm{MÎmrbm XmoZ dVw©imH$ma Vi AgVmV.Vi Am{U ‘mWm Oo EH$‘oH$mbm EH$ê$n AgVmV.

H¥$Vr … Q>o~bmda EH$ d¥Îm{MÎmr R>odm Am{U EH$ n¥ð>mMm VwH$S>m Q>o~bmda g‘m§VaYabm Am{U Vmo d¥Îm{MÎmrÀ¶m darb ^mJmbm ñne© H$aVmo.

Oa Vwåhr nwÇ>m Am{U Q>o~bmMm darb ^mJ (d¥Îm{MÎmrMm Vi) ¶m‘Yrb b§~m§VaH$mT>bo, Va Vr d¥Îm{MÎmrMr C§Mr Amho.

AmVm Imbrb JwUY‘© Agboë¶m d¥Îm{MÎmrMm {dMma H$am. • Ë¶mbm XmoZ g‘m§Va Am{U EH$ê$n dVw©imH$ma Vi AgVmV.

• Ë¶mbm XmoZ Vim§À¶m H$S>m§Zm OmoS>Umao dH«$ n¥ð>^mJ AgVmV. • XmoZ Vim§À¶m ‘Ü¶mbm OmoS>Umam aofmI§S> hm Vimbm b§~ AgVmo. Vr d¥Îm{MÎmrMr C§Mr (h)

AgVo Am{U Ë¶mbm d¥Îm{MÎmrMm Aj Ago gwÕm åhUVmV.Aem d¥Îm{MÎmrbm b§~ dVw©imH$ma d¥Îm{MÎmr Ago åhUVmV. {Vbm Ago åhUVmV H$maUd¥Îm{MÎmrbm dVw©imH$ma Vi Am{U Ë¶mMm Aj hm Vimbm H$mQ>H$moZ AgVmo.¶m KQ>H$m‘Ü¶o Amåhr ’$º$ b§~ dVw©imH$ma d¥Îm{MÎmrer g§~§{YV Amho. VarnU doJù¶m àH$mao

{dMma Ho$bm Var gwÕm d¥Îm{MÎmr ¶m eãXmMm AW© b§~ dVw©imH$ma d¥Îm{MÎmr Amho.

d¥Îm{MÎmrMo dH«$n¥ð>’$i (Surface area of a cylinder)

Amåhmbm ‘m{hV Amho H$s, d¥Îm{MÎmr hr nmoH$i d¥Îm{MÎmr qH$dm ârd d¥Îm{MÎmr ¶mn¡H$s EH$ Amho.’$º$ dH«$ n¥ð>^mJmZr nmoH$i d¥Îm{MÎmr V¶ma hmoVo. nmB©n, nmoH$i Zù¶m (straw) Zù¶m, ~mgar hr nmoH$i

d¥Îm{MÎmrMr CXmhaUo AmhoV.XmoZ dVw©imH$ma gnmQ> n¥ð>^mJ Am{U dH«$ n¥ð>^mJm§Zr ‘¶m©{XV Ho$boë¶m joÌmbm ârd d¥Îm{MÎmr åhUVmV.d¥Îm{MÎmr Im§~, amoS> amobaMr MmHo$, ~mJoVrb amoba, nopÝgb, âbobr d¥Îm{MÎmr Q>mH$s Am{U âbobo H°$Z

(can) hr ârd d¥Îm{MÎmrMr CXmhaUo AmhoV.d¥Îm{MÎmrÀ¶m n¥ð>^mJmMo joÌ’$i (XmoÝhr nmoH$i Am{U ard d¥Îm{MÎmr) ho d¥Îm{MÎmrÀ¶m ~mhoarb joÌ’$imer

g§~§{YV Amho.nmoH$i d¥Îm{MÎmrbm ’$º$ dH«$ n¥ð>’$i AgVo Ë¶mbm nmíd© n¥ð>’$i Agohr

åhUVmV. ¶m n¥ð>^mJmÀ¶m joÌ’$imbm dH«$ n¥ð>^mJmMo joÌ’$i (CSA) qH$dmnmoH$i d¥Îm{MÎmrÀ¶m nmíd© n¥ð>^mJmMo joÌ’$i (LSA) Amho.

ârd d¥Îm{MÎmrbm XmoZ dVw©imH$ma n¥ð>^mJ Am{U dH«$ n¥ð>’$i gwÕm AgVo.åhUyZ ârd d¥Îm{MÎmrÀ¶m dH«$ n¥ð>^mJmMo joÌ’$i ho dH«$ qH$dm nmíd© n¥ð>^mJmMojoÌ’$i Amho.

ârd d¥Îm{MÎmrMo g§nyU© n¥ð>’$i ho XmoZ dVw©imH$ma joÌm§Mo joÌ’$i Am{U dH«$ n¥ð>’$i ¶m§Mr ~oarO Amho.AmVm, d¥Îm{MÎmrMo nmíd© n¥ð>’$i Am{U g§nyU© n¥ð>’$i H$mT>Ê¶mMo gyÌ aMy¶m.

h

Milk can Pipe Gas cylinder Pillars

Top circularbase

Bottom circular

darbdVw©imH$ma

dH«$ qH$dm nmíd©n¥ð>^mJ

ImbrbdVw©imH$ma Vi

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372 UNIT-15

i. d¥Îm{MÎmrMo dH«$ (nmíd©) n¥ð>’$i (Curved surface area of a cylinder)… {Xboë¶m d¥Îm{MÎmr modVr nyU©nUoJw§S>miUmam Am¶VmH¥$Vr H$mJX ¿¶m Am{U Ë¶m§Mr é§Xr hr d¥Îm{MÎmrÀ¶m C§Mrer g‘mZ Amho. nwR²>R>çm§Zr ~Z{dboë¶md¥Îm{MÎmr KodyZ gwÕm hr H¥$Vr Ho$br OmD$ eH$Vo. gai aofoda H$mnyZ nmíd© n¥ð> mJ CKS>bm Va Am¶V {‘iVmo.

b b

l

h

Fold

Cut

2 r

{ZarjU H$am H$s,Am¶VmH¥$Vr H$mJXmMo joÌ’$i = d¥Îm{MÎmrMo dH«$ n¥ð>’$iAm¶VmMr bm§~r = d¥Îm{MÎmrÀ¶m VimMm n[aK l = 2 rAm¶VmMr é§Xr = d¥Îm{MÎmrMr C§Mr b = hAmnë¶mbm ‘m{hV Amho H$s, Am¶VmMo joÌ’$i = l × b = 2 r × h = 2 rh

d¥Îm{MÎmrMo dH«$ (nmíd©) n¥ð>’$i = 2 rh Mm¡. EH$H$

ii. ârd d¥Îm{MÎmrMo g§nyU© n¥ð>’$i (Total surface area of a solid cylinder) …

d¥Îm{MÎmrMo g§nyU© n¥ð>’$i = 2 r (h + r) Mm¡. EH$H$

d¥Îm{MÎmrMo KZ’$i (Volume of a cylinder)Vwåhr nyduM Aä¶mgbobo Amho H$s, bmobH$mMo KZ’$i ho Ë¶m§À¶m VimMo joÌ’$i Am{U C§Mr ¶m§Mm JwUmH$ma Amho.Á¶mAWu d¥Îm{MÎmr hr gwÕm dVw©imH$ma VimMm bmobH$ Amho, {VMo KZ’$i gwÕm gma»¶mM nÕVrZo H$mT>bo OmVo.

(i) ârd d¥Îm{MÎmrMo KZ’$i (Volume of a solid cylinder) :

ârd d¥Îm{MÎmrMo KZ’$i = VimMo joÌ’$i × C§Mr = r2 × h = r2h

d¥Îm{MÎmrMo KZ’$i = r2h KZ EH$H$


CXmhaUo 1 … b§~ dVw©imH$ma d¥Îm{MÎmrMo dH«$ n¥ð>’$i (CSA) H$mT>m Á¶mMr C§Mr Am{U {ÌÁ¶m AZwH«$‘o 30g|.‘r. Am{U 3.5 g|.‘r. Amho.CH$b … d¥Îm{MÎmrMo dH«$ n¥ð>’$i = 2 rh =

222

73.5

0.5cm×30cm CSA = 660 Mm¡.g|.‘r.

h

r

A = r2

d¥Îm{MÎmrMo dH«$n¥ð>’$i

dVw©imH$ma VimMojoÌ’$i

d¥Îm{MÎmrMo g§nyU©n¥ð>’$i = + 2 ×

= 2 rh + 2 × r2

= 2 r (h + r)

KS>r

H$mn

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Mensuration 373

CXmhaUo 2 … EH$m b§~ dVw©imH$ma d¥Îm{MÎmrMo dH«$ n¥ð>’$i 88 Mm¡.g|.‘r. Am{U C§Mr 14 g|.‘r. Amho. Vad¥Îm{MÎmrÀ¶m VimMr {ÌÁ¶m H$mT>m.CH$b … d¥Îm{MÎmrMo dH«$n¥ð>’$i = 2 rh = 88 Mm¡.g|.‘r.=

222 14cm7

r

r =1 2 4 88 2cm 7

2 22 14 2cm = 1 g|.‘r. d¥Îm{MÎmrÀ¶m VimMr {ÌÁ¶m = 1 g|.‘r.

CXmhaU 3 … XmoZ b§~ dVw©imH$ma d¥Îm{MÎmrÀ¶m {ÌÁ¶|Mo JwUmoÎma 2…3 Am{U Ë¶m§À¶m C§MrMo JwUmoÎma 5…4 Amho. VaË¶m§À¶m dH«$ n¥ð>’$im§Mo JwUmoÎma H$mT>m.

CH$b … XmoZ d¥Îm{MÎmtÀ¶m {ÌÁ¶m AZwH«$‘o 2r d 3r Am{U Ë¶m§Mr C§Mr AZwH«$‘o 5h Am{U 4h AmhoV Ago ‘mZy.S1 Am{U S2 ho XmoZ d¥Îm{MÎmtMo dH«$ n¥ð>’$i Amho Ago ‘mZy.S1 {ÌÁ¶m 2r Am{U C§Mr 5h AgUmè¶m d¥Îm{MÎmrMo dH«$ n¥ð>’$i = S1 = 2 × × 2r × 5h = 20 rhS2 {ÌÁ¶m 3r Am{U C§Mr 5h AgUmè¶m d¥Îm{MÎmrMo dH«$ n¥ð>’$i = S2= 2 × × 3r × 4h = 24 rh

1

2

SS =

5 206 24

rh 56rh

S1 : S2 = 5 : 6

CXmhaU 4 … 120 g|.‘r. bm§~ amobaMm ì¶mg 84 g|.‘r. Amho. H«$sS>m§JUmMo gnmQ>rH$aU H$aÊ¶mgmR>r Ë¶mZo 500’o$è¶m nyU© Ho$ë¶m Va 2.50 n¡go Xa Mm¡ag ‘r à‘mUo gnmQ>rH$aU H$aÊ¶mg ¶oUmam IM© H$mT>m.CH$b … h = 120 g|.‘r. r = 42 g|.‘r.

EH$m ’o$ar‘Ü¶o amobaZo AmH«${‘bobo joÌ’$i = amobaMo dH«$ n¥ð>’$i = 2 rh

=22

27

426cm × 120cm

500 Mo ’o$è¶m‘Ü¶o AH«${‘bobo joÌ’$i = (31,680 × 500) Mm¡.g|.‘r. = 15,84,000 Mm¡.g|.‘r.

‘rQ>a‘Ü¶o ì¶mnbobo joÌ’$i = 215840000 m100 100

= 1584 ‘r2

Xa Mm¡ag ‘rQ>a‘Ü¶o {H«$S>m§JUmÀ¶m gnmQ>rH$aUmg ¶oUmam IM© = 2.50 ps

{H«$S>m§JUmÀ¶m gnmQ>rH$aUmg ¶oUmam IM© = 1584 m2 × 2.50 n¡go/‘r2 = < 3936

CXmhaU 5 … YmVwMr nmB©n 77 g|.‘r. bm§~ Amho. Cä¶m N>oXmMm {ÌÁ¶m 4 g|.‘r. Amho. ~mhoarb {ÌÁ¶m 4.4g|.‘r. Amho. Va (a) AmVrb dH«$ n¥ð>’$i (b) ~mhoarb dH«$ n¥ð>’$i (c) g§nyU© n¥ð>’$i H$mT>m.CH$b … ~mhoarb {ÌÁ¶m = R = 2.2 g|.‘r.; AmVrb {ÌÁ¶m = r = 2 g|.‘r.; h = 77 g|.‘r.

(a) AmVrb dH«$n¥ð>’$i (CSA) = S1 = 2 rh =22

27

2cm 7711

cm = 968 Mm¡.g|.‘r.(b) ~mhoarb dH«$n¥ð>’$i (CSA) = S2 = 2 Rh =

222

72.2cm 77

11cmg|.‘r. = 1064.8 Mm¡.g|.‘r.

(c) g§nyU© n¥ð>’$i (TSA) = S1 + S2 + XmoZ VimMo joÌ’$i = S1 + S2 + 2( R2 – r2)

= 968Mm¡.g|.‘r. + 1064.8Mm¡.g|.‘r. +22

27 {(2.2)2 – 22}

=44968 1064.87

0.840.12

Mm¡.g|.‘r.

= (968 + 1064.8 + 5.28) Mm¡.g|.‘r. TSA = 2038.08 Mm¡.g|.‘r.

CXmhaU 6 … Oa VimMr {ÌÁ¶m 7 g|.‘r. Am{U C§Mr 15 g|.‘r. Agob Va b§~ dVw©imH$ma d¥Îm{MÎmrMo KZ’$i

H$mT>m.

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374 UNIT-15

CH$b … r = 7 g|.‘r., h = 15 g|.‘r.

d¥Îm{MÎmrMo KZ’$i = r2h =227

7cm × 7cm × 15cm

KZ’$i = 2310 K.g|.‘r.

CXmhaU 7 … d¥Îm{MÎmrÀ¶m VimMm n[aK 132 g|.‘r. Am{U Ë¶mMr C §Mr 25 g|.‘r. Amho Va d¥Îm{MÎmrMo

KZ’$i H$mT>m.

CH$b …'r' hr d¥Îm{MÎmrMr {ÌÁ¶m Amho Ago g‘Oy.

n[aK = 132 g|.‘r.

2 r = 132 g|.‘r.2227

r = 132 g|.‘r.

r =132

6 3

cm 72 22

r = 21 g|.‘r.

dÎm{MÎmrMo KZ’$i = r2 h = 222 (21cm) 25cm7

=227

213

21 25cmK.g|.‘r.= 34,650 K.g|.‘r.

CXmhaU 8 … 14 g|.‘r. bm§~ YmVyÀ¶m d¥Îm{MÎmr AmH$mamÀ¶m nmB©nÀ¶m ~mhoarb Am{U AmVrb n¥ð>^mJmÀ¶m

joÌ’$imVrb ’$aH$ 44 Mm¡.g|.‘r. Amho. Oa nmB©n 99 K.g|.‘r. À¶m YmVynmgyZ ~Z{dbm Amho. Va nmB©nmMr

~mhoarb Am{U AmVrb {ÌÁ¶m H$mT>m.

CH$b … R hr ~mhoarb {ÌÁ¶m Am{U 'r' hr YmVyÀ¶m nmB©nMr AmVrb {ÌÁ¶m ‘mZy h = 14 g|.‘r.

~mhoarb n¥ð>^mJmMo joÌ’$i – AmVrb n¥ð>^mJmMo joÌ’$i = 44 Mm¡.g|.‘r.

2 Rh – 2 rh = 44 Mm¡.g|.‘r. R – r= 44222 147

= 2 44 72 22 14 2

R – r =12

dmnaboë¶m YmVyMo KZ’$i = 99 K.g|.‘r.~mhoarb KZ’$i – AmVrb KZ’$i = 99 K.g|.‘r.

R2h – r2h = 99 K.g|.‘r.

h(R2 – r2) = 99 K.g|.‘r.227

142(R r)(R r)= 99 K.g|.‘r. [ a2 – b2 = (a + b)(a – b)] Am{U 1R r 2

22 2 1cm(R r)2 = 99 K.g|.‘r. R + r=

999

22 =92

R r 92

R r 12

102R 2

R + r = 92

2R = 5 r + 2.5 = 4.5

g|.‘r. 7

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Mensuration 375

R = 52 = 2.5 g|.‘r. r= 4.5 – 2.5 = 2 g|.‘r.

~mhoarb {ÌÁ¶m = 2.5 g|.‘r.; AmVrb {ÌÁ¶m = 2 g|.‘r.

CXmhaU 9 … d¥Îm{MÎmr AmH$mamÀ¶m ^m§S>çmMm VimMm n[aK 132 g|.‘r. Am{U Ë¶mMr C§Mr 25 g|.‘r. Amho.

Ë¶m‘Ü¶o g‘mdbobo nmUr {H$Vr {bQ>a Amho.

CH$b … n[aK C= 132 g|.‘r. , h = 25 g|.‘r. VimMr {ÌÁ¶m = r ‘mZy

n[aK = 132 g|.‘r. 2 r = 132 g|.‘r.22

2 r7

= 132 g|.‘r.

r=132

2227

=132

6 3

72 22 r = 21 g|.‘r.

^m§S>çmMo KZ’$i = r2h= 2 322 (21) 25cm7

K.g|.‘r.=227

213

21 25 cmK.g|.‘r.= 34650 K.g|.‘r.

KZ’$i {bQ>a‘Ü¶o =346501000

[ 1000 K.g|.‘r. = 1 {bQ>a] = 34.65 {bQ>a

^m§S>çm‘Ü¶o 34.65 {bQ>a BVHo$ nmUr Amho.CXmhaU 10 … Oa b§~ dVw©imH$ma d¥Îm{MÎmrÀ¶m VimÀ¶m {ÌÁ¶oMo XmoZ AY© mJ AmhoV. Ë¶m§Mr C§Mr g‘mZ AmhoZdrZ d¥Îm{MÎmrÀ¶m KZ’$imMo ‘wi d¥Îm{MÎmrÀ¶m KZ’$imer JwUmoÎma {H$Vr?

CH$b … Oa V1 ho ‘wi d¥Îm{MÎmrMo KZ’$i Am{U V2 ho ZdrZ d¥Îm{MÎmrMo KZ’$i Amho Ago ‘mZy,

‘yi d¥Îm{MÎmrMr {ÌÁ¶m = r ZdrZ d¥Îm{MÎmrMr {ÌÁ¶m = 2r

V1= r2h Am{U V2 =2r h

21

2

VV =

2r h2r h4

= 2

1

V4 11 V 4

V2 : V1 = 1 : 4

CXmhaU 11 ^ard d¥Îm{MÎmrMo g§nyU© n¥ð>’$i (TSA) 462 Mm¡.go.‘r. Amho. Ë¶mMo dH«$ n¥ð>’$i (CSA) ho g§nyU©

n¥ð>’$imÀ¶m (TSA) À¶m EH$ V¥Vr¶Am§e Amho. Va d¥Îm{MÎmrMo KZ’$i H$mT>m.

CH$b …d¥Îm{MÎmrMo dH«$ n¥ð>’$i (CSA) =13

d¥Îm{MÎmrMo g§nyU© n¥ð>’$i (TSA)

2 rh =13

× 2 r (h + r) =13

462154

cmMm¡.g|.‘r.

2 rh = 154 Mm¡.g|.‘r.

(TSA) d¥Îm{MÎmrMo g§nyU© n¥ð>’$i = 2 r (h + r)

462 Mm¡.g|.‘r. = 2 rh + 2 r2 = 154 Mm¡.g|.‘r. + 2 r2

462 Mm¡.g|.‘r. – 154 Mm¡.g|.‘r.= 2 r2 308 Mm¡.g|.‘r. = 22227

r 30814 7

72 22

= r2

r2 = 49 r = 7 g|.‘r.

d¥Îm{MÎmrMo dH«$ n¥ð>’$i (CSA) = 154 Mm¡.g|.‘r.

2 rh = 154 Mm¡.g|.‘r.22

27

7cm h = 154 Mm¡.g|.‘r. h =154

7

442 =

72 g|.‘r.

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376 UNIT-15

d¥Îm{MÎmrMo KZ’$i = r2h = 2 322 77 cm7 2 Mm¡.g|.‘r. =

11 227

7492 Mm¡.g|.‘r. = 539 K.g|.‘r.

ñdmÜ¶m¶ 15.1

1. b§~ dVw©imH$ma d¥Îm{MÎmrMr C§Mr 14 g|.‘r. Am{U VimMr {ÌÁ¶m 2 g|.‘r. Amho Va Ë¶mMo (i) CSA(ii) TSA H$mT>m.

2. 20 g|.‘r. bm§~rÀ¶m bmoI§S>r nmB©nMr ~mhoarb {ÌÁ¶m 12.5 g|.‘r. Am{U AmVrb {ÌÁ¶m 11.5 g|.‘r.Amho. Va nmB©nMo g§nyU© n¥ð>’$i (TSA) H$mT>m.

3. XmoZ b§~ dVw©imH$ma d¥Îm{MÎmtÀ¶m {ÌÁ¶oMo JwUmoÎma 2…3 Am{U Ë¶m§À¶m dH«$ n¥ð>^mJm§Mo JwUmoÎma 5…6 Amho.Va Ë¶m§À¶m C§MrMo JwUmoÎma H$mT>m.

4. dVw©imH$ma {d{harMm AmVrb ì¶mg 2.8 ‘r Amho. {VMr Imobr 10 ‘r. Amho Va Ë¶mMo dH«$ n¥ð>’$iH$mT>m. ¶m dH«$ n¥ð>^mJmbm {Jbmdm H$aÊ¶mgmR>r é.42 Xa Mm¡.‘r. à‘mUo ¶oUmam IM© gwÕm H$mT>m.

5. emioVrb H«$mâQ> {ejH$mZo nwR²>R>çmnmgyZ d¥Îm{MÎmrÀ¶m AmH$mamMo noZ R>odm¶Mo gmYZ H$am¶Mo {dÚmÏ¶mªZm{eH${dbo. emioVrb {dÚmÏ¶mªMr g§»¶m 42 Amho, Oa àË¶oH$ 5 g|.‘r. {ÌÁ¶m Am{U 14 g|.‘r. C§MrMonoZ R>odm¶Mo gmYZ V¶ma Ho$bo. Va Ë¶m§Zr dmnabobm nwÇ>m {H$Vr?

6. 1.4 ‘r. ì¶mgmÀ¶m ~mJoVrb amobaMr bm§~r 2 ‘r. Amho. Va 5 ’o$è¶m‘Ü¶o Ë¶mZo AmH«${‘bobo A§Va{H$Vr?

7. {ÌÁ¶m 10.5 g|.‘r. Am{U C§Mr 16 g|.‘r. AgUmè¶m b§~ dVw©imH$ma d¥Îm{MÎmrMo KZ’$i H$mT>m.8. d¥Îm{MÎmr AmH$mamÀ¶m bmH$S>r nmB©nMm AmVrb ì¶mg 24 g|.‘r. Am{U ~mhoarb ì¶mg 28 g|.‘r. Amho.

nmB©nMr bm§~r 35 g|.‘r. Amho. Oa 1 K.g|.‘r.Mo dOZ 0.6 J«°‘ Amho Va nmB©nMo dOZ H$mT>m.9. g‘mZ KZ’$imÀ¶m XmoZ dVw©imH$ma d¥Îm{MÎmtÀ¶m C§MrMo JwUmoÎma 1…2 Amho Va Ë¶m§À¶m {ÌÁ¶oMo JwUmoÎma

H$mT>m.10. 44 g|.‘r. × 20 g|.‘r. AmH$mamMm Am¶VmH¥$Vr H$mJX Ë¶mÀ¶m bm§~r‘Ü¶o Jw§S>miyZ d¥Îm{MÎmr ~Z{dbr.

Va V¶ma Pmboë¶m d¥Îm{MÎmrMo KZ’$i H$mT>m.e§Hy$ (Cone)Imbrb dñVy§Mo {ZarjU H$am.

Ë¶m e§Hy$À¶m AmH$mamÀ¶m AmhoV. Vo nmoH$iqH$dm ^ard Agy eH$VmV.

{ZarjU H$am H$s e§Hy$Mm Vi dVw©imH$maAm{U dH«$ n¥ð >’$i VimnmgyZ {Z‘wiVo hmodyZ{eamoq~XÿV {‘iVmo.

e§Hy§$Mr VwbZm {na°{‘S> (gwMr)er H$am.e§Hy$ ho dVw©imH$ma VimMo {na°{‘S>Mo àH$ma AmhoV.

e§Hy$ ¶m ^ard dñVy AmhoV Ë¶m gnmQ> dVw©imH$ma Vi {Z‘wiVm hmodyZ EH$mq~XÿV {‘iVmo Ë¶mbm {eamo q~Xÿ Ago åhUVmV.

b§~ dVw©imH$ma e§Hy$Mm nm¶m dVw©imH$ma AgyZ Ë¶mMm Aj nm¶mÀ¶m ‘Ü¶mVyZOmVmo Am{U nm¶mbm b§~ AgVmo.

AmB©pñH«$‘MmH$moZ

dmT>{XdgmMrQ>monr

KamMo N>ßna dmiyMr amg

e§Hy$ {na°{‘S>Cone Pyramid

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Mensuration 377

H¥$Vr … EH$ OmS> H$mJX ¿¶m Am{U B = 900 Agbobm H$mQ>H$moZ {ÌH$moU ABC H$mnm. AB À¶m b§~ ~mOybm EH$bm§~ Xmoam {MH$Q>{dbm Ago g‘Oy. {ÌH$moUmÀ¶m XmoÝhr ~mOyMmXmoam hmVmV Yam Am{U {ÌH$moU Xmoè¶m^modVr {’$adm.

V¶ma Pmbobm AmH$ma Vwåhr AmoiIy eH$mb H$m?

V¶ma Pmbobm AmH$ma b§~ dVw©imH$ma e§Hy$Mm Amho.åhUyZ AmnU Agm {ZîH$f© H$mTy> H$s,

Oa H$mQ>H$moZ Agbobr EH$ ~mOy pñWa R>odyZ Ë¶m ^modVr H$mQ>H$moZ {ÌH$moU {’$a{dë¶mZo {Z‘m©U hmoUmè¶mKZmH¥$Vrbm e§Hy$ Ago åhUVmV.e§Hy$À¶m AmH¥$VrMo {ZarjU H$am.

• AB Mr bm§~r qH$dm dVw©imH$ma VimMm ‘Ü¶ q~XÿnmgyZ {eamoq~Xÿ n¶ªVÀ¶m A§Vambme§Hy$Mr C§Mr (AB=h) Ago åhUVmV. Ë¶mbm e§Hy$Mm Aj Agohr åhUVmV.

• BC Mr bm§~r VimMr {ÌÁ¶m Amho. (BC = r)

• AC Mr bm§~r qH$dm {eamoq~XÿnmgyZ dVw©imH$ma Vimdarb H$moUVmhr q~Xþ ¶m‘YrbA§Vambm e§Hy$Mr {VaH$g C§Mr (l) Ago åhUVmV. (AC = AD = l)

e§Hy$Mr {VaH$g C§Mr (l) Am{U C§Mr (h) ¶m‘Ü¶o H$moUVmhr g§~§Y Amho H$m?AmH¥$Vr‘Yrb ABC ¶m H$mQ>H$moZ {ÌH$moUmMm {dMma H$am.

AC2= AB2 + BC2 (nm¶W°JmoagÀ¶m à‘o¶mdê$Z) l2 = h2 + r2

OoWo 'l' hr {VaH$g C§Mr, h hr C§Mr, r hr e§Hy$À¶m VimMr {ÌÁ¶m Amho.

l = 2 2h r , h = 2 2l r , qH$dm r = 2 2l h

AmVm Imbrb JwUY‘© Agboë¶m e§Hy$Mm {dMma H$am.• Ë¶mbm dVw©imH$ma nmVir hm Vi Amho.• e§Hy$Mm Aj Am{U {VaH$g C§Mr N>oXUmè¶m q~Xÿbm {eamoq~Xÿ åhUVmV.• Ë¶mbm dH«$ n¥ð>’$i Amho Oo dVw©imH$ma VimMr H$S>m Am{U {eamoq~Xÿ ¶m§Zm OmoS>Vo.• {eamoq~Xÿ Am{U dVw©imH$ma VimMm ‘Ü¶ ¶m§Zm OmoS>Umar aofm Vimbm b§~ AgVo.

Aem àH$maÀ¶m e§Hy$bm b§~ dVw©imH$ma e§Hy$ Ago åhUVmV. d¥Îm{MÎmrà‘mUo Á¶mAWu AmnU ’$º$ b§~dVw©imH$ma e§Hy$Mm Aä¶mg H$aVmo Voìhm ¶m KQ>H$m‘Ü¶o e§Hy$ åhUOoM b§~ dVw©imH$ma e§Hy$ Agm hmoVmo.

e§Hy$Mo dH«$ n¥ð>’$i Surface area of a cone

e§Hy$À¶m n¥ð>^mJm‘Ü¶o dH«$ n¥ð>^mJ Amho Vmo dVw©imÀ¶mdibobm ^mJ Am{U dVw©imH$ma VimZo ~Zbobm Amho. Omo{d^mJ e§Hy$Mo joÌ ~Z{dVmo Ë¶mbm dH«$ n¥ð>’$i qH$dm e§Hy$Monmíd©n¥ð>’$i Ago åhUVmV.

Imbrb AmH¥$Ë¶m§Mo {ZarjU H$am …

A

BC

A

BC DB

CD

A

A

B CD

h l

{d^mJ … {ÌÁ¶m Am{U H§$g ¶m§Zr ‘¶m©XrVHo$bobm dVw©imH$ma joÌmMm ^mJ.

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378 UNIT-15

hl ll

2 r

Oa nwR²>R>çmMm dVw©imH$ma e§Hy$Mm Vi ~§X Ho$bm Va dH«$n¥ð>’$i Am{U dVw©imH$ma VimMo joÌ’$i ¶m§Mr ~oarOHo$br Va e§Hy$Mo g§nyU© n¥ð>’$i ¶oVo.

i. e§Hy$Mo dH«$ n¥ð>’$i (Curved surface area of cone)

Oa nwR²>R>çmnmgyZ ~Z{dbobm e§Hy$ H$mnbmAm{U ngabm Va Imbr XmI{dë¶mà‘mUo{d^mJ {‘iVmo.

{d^mJmMo joÌ’$i ho e§Hy$Mo dH«$ n¥ð>’$iAmho.

{d^mJmMo joÌ’$i H$go H$mT>bo OmVo?

'r' hr e§Hy$Mr {ÌÁ¶m Amho Ago ‘mZy. 'l ' hr e§Hy$Mr {VaH$g C§Mr Amho.

' ' hm {d^mJmMm H|$Ðr¶ H$moZ Amho.

¶oWo {d^mJmMr {ÌÁ¶m 'r' hr e§Hy$À¶m {VaH$g C§Mr BVH$sM

Amho. r = l

{d^mJmMo joÌ’$i =2

360l .........(1)

{d^mJmÀ¶m H§$gmMr bm§~r 'L' ‘mZy.

Voìhm,2 360lL

.

H§$gmMr bm§~r 'L' =360

2 l ......................(2)

AmH¥$Vr‘Ü¶o,

{d^mJmÀ¶m H§$gmMr bm§~r = e§Hy$À¶m dVw©imH$ma VimMm n[aK

L = 2 r ....................................................(3)

(2) Am{U (3) dê$Z Amåhmbm {‘iob 2 r =3602 l

rl

=360

{d^mJmMo joÌ’$i 'A' ‘mZy.2l

A=

360

A= 2

360l = 2 r

ll

= rl

e§Hy$Mo dH«$ n¥ð>’$i = rl Mm¡. EH$H$

Á¶mAWu rl ho 12

2rl Ago {b{hVmV Am{U 2 r hm e§Hy$À¶m VimMm n[aK Amho, 'l' hr {VaH$g C§Mr

Amho. Amåhmbm {‘iVo,e§Hy$Mo dH«$ n¥ð>’$i = e§Hy$À¶m VimMm n[aK × {VaH$g C§Mr

hl l

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Mensuration 379

e§Hy$Mo dH«$ n¥ð>’$i (CSA) ho e§Hy$À¶m VimMm n[aK (2 r) Am{U {VaH$g C§Mr (l) ¶m§À¶m JwUmH$mamÀ¶m{Zå‘o AgVo.

(ii)^ard b§~ dVw©imH$ma e§Hy$Mo g§nyU© n¥ð>’$i (Surface area of a hollow cone)

e§Hy$Mo g§nyU© n¥ð>’$i = e§Hy$À¶m dH«$ n¥ð>’$i = VimMo joÌ’$i

= rl + r2

= r (l + r)

e§Hy$Mo g§nyU© n¥ð>’$i = r (l + r) Mm¡ag EH$H$

e§Hy$Mo KZ’$i (Volume of cone)e§Hy$Mo KZ’$i H$mT>Ê¶mgmR>r Ho$boë¶m à¶moJmMm {dMma H$am. à¶moJmMm Aä¶mg H$am Am{U bhmZ JQ>mV

nwÝhm-nwÝhm H$am.

EH$ Vi ~§X Agbobr nmoH$i d¥Îm{MÎmr Am{U g‘mZ nm¶md {VaH$g C§MrMm nmoH$i e§Hy$ ¿¶m. Oa Vo nmaXe©H$ dñVynmgyZ~Zbobo AgVrb Va AË¶§V Cn¶wº$ hmoVo. Ë¶m§Mr OmS>r e³¶ {VVH$sH$‘r Agmdr.

ghOnUo {ÌÁ¶m Am{U C§Mr VnmgVm ¶oUmè¶m AmH¥$Ë¶m§Mo{ZarjU H$am.

e§Hy$‘Ü¶o nmUr qH$dm dmiy ^ê$Z Vr d¥Îm{MÎmr‘Ü¶o AmoVm. e§Hy$‘Ü¶o VrZ doim nmUr qH$dm dmiy ^ê$Zd¥Îm{MÎmr‘Ü¶o AmoVbr AgVm Vr nyU©nUo ^aVo.

Amåhmbm ‘mhrV Amho H$s, d¥Îm{MÎmrMo KZ’$i = r2h OoWo 'r' hr Ë¶mÀ¶m dVw©imH$ma VimMr {ÌÁ¶mAm{U 'h' C§Mr Amho.

darb à¶moJmdê$Z AmnU H$gm {ZîH$f© H$mTy> H$s,

e§Hy$Mo KZ’$i =13

× d¥Îm{MÎmrMo KZ’$i

e§Hy$Mo KZ’$i =13 r2 h KZ EH$H$

AmVmn¶ªV Amåhr d¥Îm{MÎmr Am{U e§Hy$ ¶m§Mo joÌ’$i Am{U KZ’$i ¶m§Mm Aä¶mg Ho$bobm Amho.d¥Îm{MÎmr Am{U e§Hy$ ¶m§Mm joÌ’$i Am{U KZ’$i‘Yrb ~Xb ‘Ooera Amho, Oa Ë¶m§Mr {ÌÁ¶m C§Mr XþßnQ>qH$dm AmYu Ho$br AgVm. Imbrb H$moï>H$mMm Aä¶mg H$am.

rl

r2

12

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380 UNIT-15

A. KZnXmW© d¥Îm{MÎmr e§Hy$

Z§. nmíd©n¥ð>’$i g§nyU© n¥ð>’$i KZ’$i KZ’$iLSA TSA V V

1.whenr = h 2 r2 4 r2 r3

13

r3

2.r 2rh h 4 rh 4 r(2r + h) 4 r2h

43

r2h

3.r rh 2h 4 rh 2 r(r + 2h) 2 r2h

23

r2h

4.r2r

h hrh

rr h2

2r h4

2r h12

5. h2

rh

rrh

h2 r r2

2r h2

2r h6

6.r2

h2

rh

rh2

r hr2 2

2r h8

2r h12


CXmhaU 1 … e§Hy$Mm ì¶mg 14 g|.‘r. Am{U {VaH$g C§Mr 10 g|.‘r. Amho. Va Ë¶mMo dH«$ n¥ð>’$i H$mT>m.

CH$b … d= 14 g|.‘r. r =142 = 7 g|.‘r., l = 10 g|.‘r.

e§Hy$Mo dH«$ n¥ð>’$i = rl =227

7´ cm 10cm´ = 220 Mm¡.g|.‘r.

CXmhaU 2 … VimMr {ÌÁ¶m 14 ‘r. Am{U {VaH$g C§Mr 9 ‘r. AgUmè¶m e§Hy$Mo g§nyU© n¥ð>’$i H$mT>m.

CH$b … l = 9 ‘r., r = 14 ‘r.

e§Hy$Mo g§nyU© n¥ð>’$i = r (r + l ) =227

142m (14 9) m‘r. =

227

142

23 mMm¡.‘r.

e§Hy$Mo g§nyU© n¥ð>’$i = 1012 Mm¡.‘r.

CXmhaU 3 … VimMr {ÌÁ¶m 7 ‘r. Am{U C§Mr 24 ‘r. AgUmè¶m e§Hy$À¶m AmH$mamMm V§~y ~Z{dÊ¶mgmR>r 5 ‘r.é§XrMo {H$Vr ‘r. H$mnS> bmJob.

CH$b …r = 7 ‘r, h = 24 ‘r, l = ?

l2 = r2 + h2 = 72 + 242 = 49 + 576 = 625 l = 625 = 25 ‘r.

dmnaboë¶m H$nS>çmMo joÌ’$i = e§Hy$À¶m AmH$mamÀ¶m V§~yMo dH«$ n¥ð>’$i (CSA)

e§Hy$Mo dH«$ n¥ð>’$i = rl =227

7m 25m 550 Mm¡.‘r.

H$nS>çmMo joÌ’$i = 550 Mm¡.‘r.

bm§~r × é§Xr = 550 Mm¡.‘r.

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Mensuration 381

bm§~r =550

ê§$XrMm¡.‘r =

g|.‘r5505

110 ‘r.

dmnaboë¶m H$nS>çmMr bm§~r = 110 ‘r.

CXmhaU 4 … e§Hy$Mo dH«$ n¥ð>’$i CSA 4070 Mm¡.g|.‘r. Am{U ì¶mg 70 g|.‘r. Amho. Va {VaH$g C§Mr {H$Vr?

CH$b … d = 70 g|.‘r. r = 35 g|.‘r.

e§Hy$Mo dH«$ n¥ð>’$i = rl.

4070 Mm¡.g|.‘r. =227

355cmg|.‘r.x l

37 407 011 0 = l l = 37 g|.‘r.

e§Hy$Mo dH«$ n¥ð>’$i = 37 g|.‘r.

CXmhaU 5 … e§Hy$À¶m AmH$mamÀ¶m H$igmMr {VaH$g C§Mr Am{U VimMm ì¶mg AZwH«$‘o 25 ‘r. Am{U 14 ‘r.Amho. Ë¶mÀ¶m dH«$ n¥ð>^mJmbm nm§T>am a§J bmdÊ¶mgmR>r é.210 àË¶oH$ 100 Mm¡.‘r. à‘mUo {H$Vr IM© ¶oVmo?

CH$b … l = 25 ‘r., d= 14 ‘r. r =142 = 7 ‘r. e§Hy$Mo dH«$ n¥ð>’$i = rl =

227

7 25227

7 25 ‘ r‘ r

e§Hy$À¶m AmH$mamÀ¶m H$igmMo dH«$ n¥ð>’$i CSA = 550 Mm¡.‘r.100 Mm¡.‘r.bm nm§T>am a§J bmdÊ¶mgmR>r ¶oUma IM© = < 210

550 Mm¡.‘r.bm nm§T>am a§J bmdÊ¶mgmR>r ¶oUmam IM© =210`

10 0 255 0

m = < 1155

CXmhaU 6 … XmoZ e§Hy$Mm ì¶mg g‘mZ Amho. Oa Ë¶m§À¶m {VaH$g C§MrMo JwUmoÎma 5…4 Amho Va Ë¶m§À¶m dH«$n¥ð>’$imMo (CSA) joÌ’$i H$mT>m.CH$b …n{hë¶m e§Hy$Mo dH«$ n¥ð>’$i S1 Am{U Xþgè¶m e§Hy$Mo dH«$ n¥ð>’$i S2 ‘mZy.

Ë¶m§Mr {VaH$g C§Mr AZwH«$‘o 5l Am{U 4l Amho Ago ‘mZy. e§Hy$Mo dH«$ n¥ð>’$i (CSA) = rl

1

2

SS =

r 5lr 4l S1 : S2 = 5 : 4

CXmhaU 7 …e§Hy$Mr {ÌÁ¶m 10 g|.‘r. Am{U {VaH$g C§Mr 26 g|.‘r. Amho Va KZ’$i H$mT>m.

CH$b … {ÌÁ¶m = 10 g|.‘r., l = 26 g|.‘r.V= ? h=?

C§Mr g |.‘ r22h= l r 56 100 77 242

262

10 66

KZ’$i v = 2 hr13

= 13

227 × 10 × 10 × 24

= 2512 K. g|.‘r.

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382 UNIT-15

CXmhaU 8… 24 g|.‘r. C§Mr AgUmè¶m e§Hy$Mo dH«$ n¥ð>’$i 550 Mm¡.g|.‘r. Amho. Va Ë¶mMo KZ’$i H$mT>m.CH$b … h = 24 g|.‘r. e§Hy$Mo dH«$ n¥ð>’$i = 550 Mm¡.g|.‘r.

Amåhmbm ‘mhrV Amho H$s, l2 = r2 + h2 = r2 + 242 = r2 + 576 l = 2r 576+e§Hy$Mo dH«$ n¥ð>’$i (CSA) = 550 Mm¡.‘r. rl = 550

222r r 576

7 = 550 2r r 576 =

55025

722

2r r 576 = 175

XmoÝhr ~mOyMm dJ© H$ê$Z2

2r r 576 = (175)2 r2 (r2 + 576) = 30625

r4 + 576r2– 30625 = 0 r4 + 625r2– 49r2 + 30625 = 0

r2(r2 + 625) – 49(r2 + 625) = 0 (r2 + 625) (r2 – 49) = 0

If r2 – 49 = 0 r2 = 49

r = 49 r = 7 g|.‘r.

e§Hy$Mo KZ’$i = 21 r h3

=13

227

7 7 248

K.g|.‘r. e§Hy$Mo KZ’$i = 1232 K.g|.‘r.

CXmhaU 9 … e§Hy$À¶m AmH$mamMm H$mMoMm M§~y nmÊ¶mZo ^abobm Amho. M§~yÀ¶m VimMr {ÌÁ¶m 3 g|.‘r. Am{UC§Mr 15 g|.‘r. Amho. Vo nmUr ~§X Vi Agboë¶m g‘mZ {ÌÁ¶m 1.5 g|.‘r. Agboë¶m d¥Îm{MÎmr AmH$mamÀ¶mH$mMoÀ¶m ZirV AmoVbo. Va H$mMoÀ¶m ZirVrb nmÊ¶mMr C§Mr H$mT>m.

CH$b … e §H y $‘Yrb nmÊ¶mM o KZ’$i = d ¥Îm{MÎmr‘Yrb nmÊ¶mM o KZ’$i 2c

1r h

3 c = 2cyr hcy

31 22 3 3 15cm3 7

= cy22 1.5 1.5 h7

hcy=

13

227

32

32

155

227

1.5 1.5hcy= 20 g|.‘r.

H$mMoÀ¶m Zir‘Yrb nmÊ¶mMr C§Mr = 20 g|.‘r.

CXmhaU 10 … 1.8 ‘r. C§M Am{U VimMm ì¶mg 2 ‘r. Agbobr YmVyMr ^ard d¥Îm{MÎmr {dVidyZ 3 ‘r.ì¶mgmÀ¶m b§~ dVw©imH$ma e§Hy$‘Ü¶o nwZ…{Z©‘mU Ho$br. Va e§Hy$Mr C§Mr H$mT>m?

CH$b … d¥Îm{MÎmrMr C§Mr h= 1.8 ‘r., d = 2‘r. r =22 = 1‘r. e§Hy$Mm ì¶mg d = 3‘r., r =

32 = 1.5 ‘r., h = ?

e§Hy$Mo KZ’$i = d¥Îm{MÎmrMo KZ’$i 2cyr hcy = 2

c1

r h3 c

22 1 1 1.8m7

K.‘r. =1 22 1.5 1 m3 7 ch K.‘r.

hc =

227

1 1 1.8

1 223 7

1.5 1.5=

1.86

32

1.5 6 1.5=

125

hC = 2.4,

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Mensuration 383

e§Hy$Mr C§Mr = 2.4 ‘r.

ñdmÜ¶m¶ 15.2

1. {VaH$g C§Mr 60 g|.‘r. Am{U VimMr {ÌÁ¶m 21 g|.‘r. AgUmè¶m e§Hy$Mo dH«$ n¥ð>’$i H$mT>m.

2. e§Hy$Mr {ÌÁ¶m 7 g|.‘r. Am{U dH«$ n¥ð>^mJmMo joÌ’$i 176 Mm¡.g|.‘r. Agob Va Ë¶mMr {VaH$g C§Mr

H$mT>m.

3. Oa e§Hy$Mr {VaH$g C§Mr 8 g|.‘r. Am{U Ë¶mMo dH«$ n¥ð>’$i 60 Mm¡.g|.‘r. Agob Va VimMr {ÌÁ¶m

H$mT>m.

4. e§Hy$Mo dH«$ n¥ð>’$i 308 g|.‘r. Am{U {VaH$g C§Mr 14 g|.‘r. Amho. Va VimMr {ÌÁ¶m Am{U e§Hy$Mo g§nyU©

n¥ð>’$i H$mT>m.

5. b§~ dVw©imH$ma e§Hy$À¶m AmH$mamÀ¶m {dXþfH$mÀ¶m Q>monrÀ¶m VimMr {ÌÁ¶m 7 g|.‘r. Am{U C§Mr 24 g|.‘r.

Amho. Va Aem 10 Q>moß¶m ~Z{dÊ¶mgmR>r bmJUmè¶m nmZmMo (sheet) joÌ’$i H$mT>m.

6. Oa XmoZ e§Hy$À¶m VimMm ì¶mg g‘mZ Am{U {VaH$g C§MrMo JwUmoÎma 4…3 Amho Va Ë¶m§À¶m joÌ’$im§Mo

JwUmoÎma H$mT>m.

7. (i) {ÌÁ¶m 5 g|.‘r.Am{U C§Mr 7 g|.‘r.Amho. Va b§~ dVw©imH$ma e§Hy$Mo KZ’$i H$mT>m.

8. XmoZ e§Hy$À¶m C§MrMo JwUmoÎma 1…3 Am{U Ë¶m§À¶m JwUmoÎma 3…1 Amho. Va Ë¶m§À¶m KZ’$im§Mo JwUmoÎma H$mT>m.

9. H$mQ>H$moZ {ÌH$moUm‘Ü¶o H$mQ>H$moZ H$aUmè¶m ~mOy§Mr bm§~r 6.3 g|.‘r. Am{U 10 g|.‘r. Amho. Ë¶mMr ‘moR>r

~mOy {’$adyZ e§Hy$ ~Z{dbm. Aem àH$mao ~Z{dboë¶m KZmMo KZ’$i H$mT>m.

10. b§~ dVw©imH$ma d¥Îm{MÎmr AmH$mamÀ¶m V§~yMr C§Mr 3 ‘r. n¶ªV Amho Am{U Z§Va O{‘Zrda OmñVrV OmñV

13.5 ‘r. C§MrMm b§~ dVw©imH$ma e§Hy$ ~ZVmo. Oa Ë¶mÀ¶m VimMr {ÌÁ¶m 14 ‘r. Amho Va ê$.2 Xa Mm¡.

‘r. à‘mUo Ë¶m V§~yÀ¶m AmVrb ^mJmZm a§J bmdÊ¶mgmR>r ¶oUmam IM© H$mT>m.

darb Q>moH$ H$mnbobm (’«$ñQ>‘) e§Hy$ (Frustum of a cone)

Amnë¶m X¡Z§{XZ OrdZmV dmnaboë¶m nXmWmªÀ¶m

AmH$mamMm {dMma H$am.

Vo d¥Îm{MÎmr qH$dm e§Hy$ ¶mn¡H$s H$moUVohr Zmhr.

¶m AmH$mam§Zm H$m¶ åhUVmV? Vo H$go {‘i{dVmV?

Tumbler Bucket Cap S>ã~m ~mXbr Q>monr

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‘m{hV Amho !’«$ñQ>‘ hm b°Q>rZ eãX Amho. Ë¶mMm AW© ""H$mnbobm^mJ'' Am{U Ë¶mMo ~hþdMZ ’«$ñQ>m ho Amho.

MMm© H$am …’«$ñQ>‘ e§Hy$Mr VwbZm e§Hy$ Am{U d¥Îm{MÎmrerH$am. Ë¶m‘Ü¶o AgUmam gmaIonUm Am{U ’$aH$H$moUVo?

¶m àým§Mo CÎma XoÊ¶mgmR>r Imbrb H¥$VrMm {dMma H$am.

H¥$Vr … ‘mVr qH$dm bd{MH$ nXmW© KodyZ b§~ dVw©imH$ma e§Hy$ V¶ma H$am. MmHy$Zo Vimbm g‘m§Va ^mJ H$mnm.

Vwåhmbm AmH¥$VrV XmI{dë¶mà‘mUo XmoZ KZnXmW© {‘iVmV.

Oa AmnU darb e§Hy$Mm ^mJ H$mTy>Z Q>mH$bm VaVimer KZnXmW© {e„H$ amhVmo. Ë¶mMm Vi Agboë¶me§Hy$À¶m ^mJmbm e§Hy$Mm ’«$ñQ>‘ (frustum) åhUVmV.

’«$ñQ>‘ e§Hy$bm XmoZ dVw©imH$ma Vi AgVmV. EH$ VimerAm{U Xþgam Ë¶mda AgyZ Ë¶m§Mr {ÌÁ¶m { Þ AgVo.

darb H¥$Vrdê$Z AmnU Agm {ZîH$f© H$mTy> H$s,Oa gnmQ> b§~ dVw©imH$ma e§Hy$Mm Vimer g‘m§Va ^mJ H$mnbm Am{U darb bhmZ e§Hy$ H$mTy>Z Q>mH$bm Va

{e„H$ am{hbobm Vi Agboë¶m e§Hy$À¶m ^mJmbm e§Hy$Mm ’«$ñQ>‘ (Frustum of the cone) Ago åhUVmV.AmVm Vwåhr S>ã~m, ~mXbr Am{U Q>monr À¶m AmH$mamZm

Zm§do Xody eH$Vm.Vwåhr Ë¶mZm Ago H$m åhUVm? JQ> H$ê$Z MMm© H$am.

‘wi e§Hy$ Am{U bhmZ H$mnbobm e§Hy$ ¶m§À¶m ‘mnm‘Ü¶o H$moUVog§~§Y AmhoV H$m? Imbr {Xboë¶m AmH¥$Ë¶m§Mo {ZarjU H$am.

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g‘Om ‘yi e§Hy$À¶m VimMr {ÌÁ¶m, C§Mr Am{U {VaH$g C§Mr R, H Am{U L d bhmZ e§Hy$Mr, r, h Am{Ul Zo Xe©{dbobr Amho.

{ÌH$moU ABO Am{U AED Mo {ZarjU H$am Vo g‘ê$n AmhoV.åhUyZ Ë¶m§À¶m g§JV ~mOy à‘mUmV AmhoV.

i.e.EDBO

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Mensuration 385

Amåhr Agm {ZîH$f© H$mTy> H$s,

Oa e§Hy$À¶m darb Q>moH$mnmgyZ Vimbm g‘m§Va bhmZ e§Hy$ H$mnbm Va XmoZ e§Hy$Mr{VaH$g C§Mr, VimMr {ÌÁ¶m (‘wi e§Hy$ d {‘i{dbobm bhmZ e§Hy$) nañnam§Zm à‘mUmVAgVmV.

e§Hy$À¶m ’«$ñQ>‘Mr {VaH$g C§Mr hr Ë¶mÀ¶m C§Mr Am{U {ÌÁ¶oer H$er g§~§{YV Amho?'l ' hr ’«$ñQ>‘Mr {VaH$g C§Mr, 'h' hr C§Mr, 'R' hr Imbrb VimMr {ÌÁ¶m 'r' hr darb

VimMr {ÌÁ¶m Amho Ago ‘mZy.

g§~§Y l2 = h2 + (R – r)2 qH$dm l = 2 2h (R r) Ago {Xbo OmVo.

e§Hy$À¶m ’«$ñQ>‘Mo n¥ð>’$i Am{U KZ’$i (Surface area and volume of frustum of a cone)

Amåhmbm ‘m{hV Amho H$s, ’«$ñQ>‘ e§Hy$ hm e§Hy$ ABC Mm nm¶mbm g‘m§Va A'B'C hm bhmZ e§Hy$H$mnë¶mZ§Va {‘iVmo.

’«$ñQ>‘Mo = ABC ¶m e§Hy$Mo – A'B'C ¶m e§Hy$ModH«$ n¥ð>’$i nmíd© n¥ð>’$i nmíd© n¥ð>’$i

’«$ñQ>‘Mo g§nyU© n¥ð>’$i = ’«$ñQ>‘Mo LSA + XmoZ dVw©imH$ma VimMo joÌ’$i

’«$ñQ>‘Mo KZ’$i = ABC ¶m ‘yi e§Hy$Mo KZ’$i – A'B'C ¶m bhmZ e§Hy$Mo KZ’$i

Ooìhm ’$º$ ’«$ñQ>‘À¶m XmoZ Vim§Mr {ÌÁ¶m Am{U C§Mr {Xbobr Agob Va Voìhm ’«$ñQ>‘Mo joÌ’$i Am{UKZ’$i H$go H$mT>bo OmVo?

Oa h C§Mr l {VaH$g C§Mr r1Am{U r2 hr XmoZ ’«$ñQ>Z e§Hy$À¶m VimMr {ÌÁ¶m (r1> r2)Amho Ago ‘mZy.

Imbr {Xbobr gyÌo dmnê$Z àË¶jnUo e§Hy$À¶m ’«$ñQ>‘Mo joÌ’$i Am{U KZ’$iH$mT>bo OmD$ eH$Vo.

* e§Hy$À¶m ’«$ñQ>‘Mo nmíd© n¥ð>’$i

= (r1+r2)l OoWo l = 2 21 2h +(r r )

* e§Hy$À¶m ’«$ñQ>‘Mo g§nyU© n¥ð>’$i

= (r1+r2)l + r12+ r2

2 = {(r1+r2)l + r12+r2

2} OoWo l = 2 21 2h +(r r )

* e§Hy$À¶m ’«$ñQ>‘Mo KZ’$i =13

h (r12+r2

2+r1r2)

[ Q>rn … {ÌH$moUmMr g‘ê$nVm dmnê$Z darb gyÌo H$mT>br OmD$ eH$VmV. na§Vw ¶oWo AmnU Ë¶m§Mo gyÌ V¶ma H$aVZmhr. AmnU gyÌo dmnê$Z CXmhaUo gmoS>{dVmo.


CXmhaU 1 … e§Hy$À¶m ’«$ñQ>‘Mr {VaH$g C§Mr 4 g|.‘r. Am{U Ë¶mÀ¶m dVw©imH$ma VimMr n[a{‘Vr AZwH«$‘o 18g|.‘r. d 6 g|.‘r. Amho. Va ’«$ñQ>‘Mo dH«$ n¥ð>’$i Am{U g§nyU© n¥ð>’$i H$mT>m.

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CH$b … {Xbobo 2 r1 = 18 g|.‘r., 2 r2 = 6 g|.‘r., l = 4 g|.‘r.

r1 =9

g|.‘r. r2 =3

g|.‘r.

’«$ñQ>‘Mo dH«$ n¥ð>’$i = (r1+r2)l

=9 3

4=12 4

’«$ñQ>‘Mo CSA = 48 Mm¡.g|.‘r.’«$ñQ>‘Mo g§nyU© n¥ð>’$i = (r1+r2)l + r1

2+ r22

= 48 9 9 3 348 +

81 +

9

= 48 +81 7 9 7

22 22 = 48 + 25.8 + 2.9

’«$ñQ>‘Mo TSA = 76.7 Mm¡.g|.‘r.

CXmhaU 2 … ’«$ñQ>‘À¶m AmH$mamVrb H$Mam noQ>rÀ¶m XmoZ dVw©imH$ma Q>moH$m§Mr {ÌÁ¶m 15 g|.‘r. Am{U 8 g|.‘r.Amho. Oa Ë¶mMr Imobr 6.3 g|.‘r. Amho. Va H$Mam noQ>rMo KZ’$i H$mT>m.

CH$b … {Xbobo r1=15 g|.‘r., r2 = 8 g|.‘r., h = 63 g|.‘r.

H$Mam noQ>rMo KZ’$i (’«$ñQ>‘) =13

h (r12+r2

2+r1r2)

=1 22 633 7

(152+82+15×8)

=1

13

2271

639

3

(225+64+120)

= 66 × 409 = 26,994H$Mam noQ>rMo KZ’$i = 26,994 K.g|.‘r.

CXmhaU 3 … e§Hy$À¶m VimMr {ÌÁ¶m 12 g|.‘r. Am{U 20 g|.‘r. C§Mr Agboë¶me§Hy$À¶m darb Q>moH$mnmgyZ 3 g|.‘r. VimMr {ÌÁ¶m Agbobm e§Hy$ H$mnbm. Va{eamoq~XþnmgyZ Vmo {H$Vr A§Vamda H$mnbm nm{hOo? Aem [aVrZo {‘i{dboë¶m ’«$ñQ>‘MoKZ’$i H$mT>m.

CH$b … {Xbobo r1 = 12 g|.‘r., h1= 20 g|.‘r., r2 = 3 g|.‘r. h2 =?

Amåhmbm ‘m{hV Amho H$s, 1 1

2 2

r hr h 2

12 203 h h2 =

3 20 5 cm12 g|.‘r

{eamoq~XþnmgyZ 5 g|.‘r. A§Vamdê$Z Vmo H$mnbm nm{hOo.

’«$ñQ>‘Mo KZ’$i =13

h (r12+r2

2+r1r2)

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=1 22 153 7

(122+32+12×3) =110

7× (144+9+36)

=110

7× 189 = 2,970 K.g|.‘r.

’«$ñQ>‘Mo KZ’$i = 2,970 K.g|.‘r

CXmhaU 4 … e§Hy$À¶m AmH$mamÀ¶m JwimÀ¶m T>onrMm VimMm ì¶mg 10 g|.‘r. Am{U C§Mr10 g|.‘r. Amho. {eamoq~XÿnmgyZ 4 g|.‘r. Vimbm g‘m§Va bhmZ e§Hy$ H$mnbm. AemàH$mao {‘i{dboë¶m ’«$ñQ>‘Mo jooÌ’$i Am{U KZ’$i H$mT>m.

CH$b …ABO ¶m e§Hy$Mm ì¶mg = 10 g|.‘r. {ÌÁ¶m r1 =102 = 5 g|.‘r.

ABO ¶m e§Hy$Mr C§Mr = h1 = 10 g|.‘r.

AI BI O ¶m e§Hy$Mr C§Mr = h2 = 4 g|.‘r.

àW‘ BVa hdr Agbobr ‘m{hVr H$mTy>.

AI BI O e§Hy$Mr {ÌÁ¶m = r2

l1 Am{U l2 ¶m AZwH«$‘o ABO Am{U AI BI O ¶m e§Hy$Mr {VaH$g C§Mr Amho.

1 1

2 2

r hr h 2

5 10r 4 r2 =

5 410

= 2 g|.‘r.

l12 = h1

2+ r12 = 102+ 52 = 100 + 25 = 125 l1 = 125 5 5 cm g|.‘r.

l22 = h2

2+ r22 = 42 + 22 = 16 + 4 = 20 l2 = 20 2 5 cmg|.‘r.

(i) ABO ¶m e§Hy$Mo nmíd© n¥ð>’$i = r1l1 = 5 5 5 25 5

ABO ¶m e§Hy$Mo LSA = 25 5 Mm¡. g| ‘r.

e§Hy$ AI BI O nmíd© (dH«$) n¥ð>’$i = r2l2 = 2 2 5 4 5

AI BI O ¶m e§Hy$Mo LSA = 4 5 Mm¡.g|.‘r.

’«$ñQ>‘ e§Hy$Mo nmíd© n¥ð>’$i = ABO e§Hy$Mo LSA - AI BI O e§Hy$Mo LSA

= 25 5 4 5 21 53 21 225

7 1 = 66 5 Mm¡.g| ‘r.

’«$ñQ>‘ e§Hy$Mo nmíd© n¥ð>’$i = 66 5 Mm¡.g|.‘r. qH$dm. 147.84 Mm¡.g|.‘r.

(ii) ABO e§Hy$Mo g§nyU© joÌ’$i = r1 (r1+l1)

5 5 5

ABO e§Hy$Mo TSA = 25 5 Mm¡.g| ‘r.

AI BI O e§Hy$Mo g§nyU© joÌ’$i = r2 (r2+l2)

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2 5 5

AI BI O e§Hy$Mo TSA = 4 5 Mm¡.g| ‘r.

’«$ñQ>‘Mo g§nyU© joÌ’$i = ABO e§Hy$Mo TSA - AI BI O e§Hy$Mo TSA

5 - 4 5

5 - 4) = 21 53 21 221 5

7 1

’«$ñQ>‘Mo g§nyU© joÌ’$i= 66 5 Mm¡.g|.‘r. qH$dm 213.84 Mm¡.g|.‘r.

(iii) ABO e§Hy$Mo KZ’$i =13

r12 h1 =

13

5 5 10

ABO e§Hy$Mo KZ’$i2503

K.g|.‘r.

AI BI O e§Hy$Mo KZ’$i =13

r22 h2=

13

2 2 4

AI BI O e§Hy$Mo KZ’$i =163

K.g|.‘r.

’«$ñQ>‘ e§Hy$Mo KZ’$i = ABO e§Hy$Mo KZ’$i AI BI O e§Hy$Mo KZ’$i

=2503

–163

=2343

’«$ñQ>‘ e§Hy$Mo KZ’$i = 78 K.g|.‘r. qH$dm 245.14 K.g|.‘r.

AmnU àË¶jnUo dmnê$Z gwÕm e§Hy$À¶m ’«$ñQ>‘Mo nmíd© n¥ð>’$i, g§nyU© n¥ð>’$i Am{U KZ’$i H$mTy> eH$Vmo.

ñdmÜ¶m¶ 15.3

1. (âbm°da nm°Q>) ’y$bXmUr hr ’«$ñQ>‘ e§Hy$À¶m ñdê$nmV Amho. Q>moH$m§Mr n[a{‘Vr 44 g|.‘r. Am{U 8.4g|.‘r. Amho. Oa {VMr Imobr 14 g|.‘r. Amho Va Ë¶m‘Ü¶o {H$Vr nmUr ‘mdVo?

2. ~mXbr hr ’«$ñQ>‘ ñdê$nmVrb AgyZ {VÀ¶m darb Am{U Imbrb dVw©im§Mr {ÌÁ¶m 15 g|.‘r. Am{U 10g|.‘r. AmhoV. {VMr Imobr 12 g|.‘r. Amho. Va {VMo dH«$ n¥ð>’$i Am{U g§nyU© n¥ð>’$i H$mT>m. (CÎma À¶mñdê$nmV ì¶º$ H$am)

3. e§Hy$À¶m VimMr {ÌÁ¶m 24 g|.‘r. Am{U C§Mr 45 g|.‘r. d {VaH$g C§Mr 45 g|.‘r. Agboë¶m e§Hy$À¶mdarb Q>moH$mnmgyZ 17 g|.‘r. C§Mrda H$mnbr. {e„H$ am{hboë¶m ’«$ñQ>‘ e§Hy$Mo KZ’$i H$mT>m. (CÎma À¶mñdê$nmV ì¶º$ H$am)

4. ^m§S>o ’«$ñQ>‘ e§Hy$À¶m AmH$mamMo Amho. Ë¶mMr EH$m Q>moH$mdarb {ÌÁ¶m 8 g|.‘r.Am{U C§Mr 14 g|.‘r. Amho. Ë¶mMo KZ’$i

56763

K.g|.‘r. Amho. VaË¶mÀ¶m Xþgè¶m Q>moH$mMr {ÌÁ¶m H$mT>m.

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Jmob (Sphere)

‘mJrb ^mJmV Ho$boë¶m H¥$Vrà‘mUoM H¥$Vr H$am.

EH$ dVw©imH$ma MH$Vr ¿¶m Am{U Ë¶mÀ¶m ì¶mgmda Xmoar {MH$Q>dm. Vr{’$adm Am{U ZdrZ V¶ma hmoUmè¶m KZmH¥$VrMo {ZarjU H$am.

Vo H$emgmaIo {XgVmV? Ë¶mbm Jmob Ago åhUVmV.

Oa dVw©imH$ma MH$Vr Ë¶mÀ¶m ì¶mgm^modVr {’$a{dbr Va V¶ma hmoUmè¶mKZmH¥$Vrbm Jmob Ago åhUVmV.

dVw©imMm ‘Ü¶ Ooìhm Vmo {’$a{dbm AgVm Jmob ~ZVmo Voìhm Vmo JmobmMm ‘Ü¶~ZVmo. V¶ma Pmboë¶m Jmobm‘Ü¶o ‘Ü¶ q~Xÿ Ë¶mÀ¶m AdH$memVrb gd© q~Xÿ nmgyZ g‘mZA§Vamda AgVmV.

åhUyZ Jmob hr {Ì‘rVr d AmH¥$Vr qH$dm KZmH¥$Vr AgVo. Or AdH$memVrb gd©q~XÿZr ~Zbobr AgVo Oo pñWa A§Vamda AgVmV Ë¶m§Zm {ÌÁ¶m åhUVmV Am{U pñWaq~Xÿbm JmobmMm ‘Ü¶ Ago åhUVmV.

^ard Jmob hm eãX KZmH¥$VrgmR>r dmnaVmV {OMm n¥ð>^mJ Jmob AgVmo {Vbm joÌ’$i Am{U KZ’$i XmoÝhrAgVmV.

Amnë¶m g^modVmbr JmobmÀ¶m AmH$mamÀ¶m AZoH$ dñVy AmhoV.

Imbr {Xboë¶m CXmhaUm§Mm {dMma H$am.

JmobmÀ¶m AmH$mamÀ¶m nXmWm©‘Ü¶o Vwåhr {H$Vr n¥ð>^mJ nmhmb?

VoWo ’$º$ EH$M AgVmo Omo dH«$mH$ma AgVmo.

AmVm M|Sy> gmaIm Jmob Amho Ago g‘Om Ë¶mÀ¶m ‘Ymo‘Y ~amo~a ‘Ü¶mVyZ OmUmao gnmQ> VwH$S>o H$am.

JmobmMo XmoZ g‘mZ ^mJ hmoVmV. JmobmÀ¶m àË¶oH$ ^mJmbm

AY©Jmob Ago åhUVmV.

Jmob Am{U AY©JmobmÀ¶m n¥ð>^mJmMo joÌ’$i (Surface area ofsphere and hemisphere)

‘Ooera H¥$VrÀ¶m ghmæ¶mZo JmobmÀ¶m n¥ð>^mJmMo joÌ’$iH$mT>Ê¶mMo gyÌ {‘i{dbo OmVo. Vwåhmbm ^moJè¶mÀ¶m g^modVmbr Xmoar Jw§S>miyZ IoiÊ¶mÀ¶m Ioim{df¶r ‘m{hVrAmho. Imbrb H¥$Vr Á¶m‘Ü¶o JmobmH$ma dñVw modVr Xmoam Jw§S>miyZ JmobmMo n¥ð>’$i H$mT>Ê¶mMo gyÌ aMÊ¶mg ‘XVH$aVo. H¥$VrMm Aä¶mg H$am Am{U JQ> H$ê$Z H$aÊ¶mMm à¶ËZ H$am.

• ßb°ñQ>rH$Mm M|Sy> ¿¶m.

• M|Sy>À¶m darb Q>moH$mda Q>mMUr Q>moMm.

• M|Sy>da g‘mZ[aË¶m g§nyU© n¥ð>^mJmda Xmoam Jw§S>mië¶mZo dH«$ n¥ð>’$i ¶oVo. Xmoam pñWa R>odÊ¶mgmR>r Ë¶mdaQ>mMUr Q>moMm.

M|Sy> JmoQ>r

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r

• Xmoè¶mÀ¶m gwadmVrÀ¶m Am{U eodQ>À¶m q~Xÿda IwU H$am.

• gmdH$menUo JmobmÀ¶m n¥ð>^mJmda Jw§S>mibobm Xmoam gmoS>m Am{U Ë¶mMr bm§~r ‘moOm.

• Xmoè¶mMo g‘mZ Mma ^mJ H$am.

• M|Sy>Mm ì¶mg Am{U {ÌÁ¶m ‘moOm.

• EH$ H$mJXmMo nmZ ¿¶m Am{U M|Sy>À¶m {ÌÁ¶oMr g‘mZ {ÌÁ¶oMr Mma dVw©io H$mT>m.

• AmH¥$VrV XmI{dë¶mà‘mUo EH$m VwH$S>çmZo àË¶oH$ dVw©i âm.

¶m H¥$Vrdê$Z Vwåhr H$m¶ nmhmb Am{U H$moUVm AZw‘mZ H$mT>mb?

Xmoè¶mZo JmobmMo (M|Sy>Mr) nyU© joÌ’$i ì¶mnbobo hmoVo Ë¶mZo nyU©nUo Mma dVw©im§Mo joÌ âVo. Ë¶m gdmªMr{ÌÁ¶m Jmobmer g‘mZ Amho.

ho gwM{dVo H$s JmobmMo dH«$ n¥ð>’$i ho dVw©imÀ¶m n¥ð>’$imÀ¶m Mma nQ> Amho. OoWo XmoÝhr Jmob Am{U dVw©i¶m§Mr {ÌÁ¶m g‘mZ Amho.

JmobmMo dH«$ n¥ð>’$i = 4 × dVw©imMo joÌ’$i = 4 × r2 = 4 r2

JmobmMo dH«$ n¥ð>’$i = 4 r2 Mm¡.EH$H$ qH$dm JmobmMo n¥ð>’$i 4 r2 Mm¡.EH$H$

¶mH$S> o bj XoUo ‘Oo{ea Amho H$s, Jm obmMo n ¥ð>’$i ho Ë¶m‘Ü¶o Agboë¶m d¥Îm{MÎmrÀ¶m dH«$n ¥ð >’$imBVHo $ AgVo.

AmH¥$VrMo {ZarjU H$am.'r' {ÌÁ¶oÀ¶m Jmob d¥Îm{MÎmrÀ¶m AmV R>odbobm Amho. Ë¶mMr {ÌÁ¶m gwÕm 'r'

Am{U C§Mr '2r' Amho.JmobmMo n¥ð>’$i = 4 r2

d¥Îm{MÎmrMo dH«$ n¥ð>’$i = 2 rh = 2 r × 2r = 4 r2

JmobmMo n¥ð>’$i = d¥Îm{MÎmrMo nmíd© n¥ð>’$iOoWo Jmob Am{U d¥Îm{MÎmrÀ¶m {ÌÁ¶m g‘mZ AmhoV Am{U d¥Îm{MÎmrMr C§Mr hm JmobmÀ¶m ì¶mgmer g‘mZ

Amho.AmVm, AmnU AY©JmobmMo dH«$ n¥ð>’$i H$mTy>. {ZarjU H$am H$s Amnë¶mOdi ârd AY©Jmob Am{U

nmoH$i AY©Jmob AmhoV.Amnë¶mbm ‘m{hV Amho H$s AY©Jmob hm AJXr ~amo~a AYm© Jmob Amho. hr ¶wº$s dmnê$Z ârd Am{U

nmoH$i AY©JmobmMo n¥ð>’$i H$mTy>.

(i) ârd AY©JmobmMo dH«$ n¥ð>’$i (Curved surface area of solid hemisphere)

ârd AY©JmobmMo dH«$ n¥ð>’$i (C.S.A) = ard JmobmMo (C.S.A)=

24 r2

= 2 r2

2

ârd AY©JmobmMo dH«$ n¥ð>’$i = 2 r2 Mm¡.EH$H$

(ii) nmoH$i AY©JmobmMo g§nyU© n¥ð>’$i (Total surface area of solidhemisphere)

ârd AY©Jmob

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Mensuration 391

ârd AY©JmobmMo n¥ð>’$i (TSA) = AY©JmobmMo n¥ð>’$i (CSA) + dVw©imH$ma VimMo joÌ’$i= 2 r2 + r2 = 3 r2

ârd AY©JmobmMo g§nyU© n¥ð>’$i (T.S.A) = 3 r2 Mm¡. EH$H$

(Vwåhr nm{hbo Agob H$s, dmT>{XdgmÀ¶mdoir {H«$g‘g PmS>mda JmobmH$ma gOmdQ>rÀ¶m dmnaboë¶m dñVy B.

hm Jmob AZoH$ e§Hy$nmgyZ ~Zbobm Amho.

JmobmMr H$ënZm H$am, Or AZoH$ bhmZ e§Hy$ dmnê$Z ~Z{dbobm Amho, Á¶mMr {ÌÁ¶m 'R' Am{U C§Mr 'h' Amho.Agm {dMma H$ê$ eH$Vm H$s, Jmob bhmZ-bhmZ e§Hy$nmgyZ ~Zbobm Amho. Á¶m§Mr C§Mr 'h' hr JmobmMr {ÌÁ¶m'r' er g‘mZ Am{U dVw©imH$ma nm¶m AgVmo.

àË¶oH$ e§Hy$Mo KZ’$i =13

× VimMo joÌ’$i × C§Mr.

n{hë¶m e§Hy$Mo KZ’$i =13

× B1 × r, Xþgè¶m e§Hy$Mo KZ’$i =13

× B2 × r

{Vgè¶m e§Hy$Mo KZ’$i =13

× B3 × r, n ì¶m e§Hy$Mo KZ’$i =13

× Bn × r

JmobmMo KZ’$i = gd© e§Hy$À¶m KZ’$im§Mr ~oarO

=13

B1r +13

B2r +13

B3r+........+13

Bnr =13

r(B1 + B2 + B3 + .......Bn)

Amåhr nmhVmo H$s, Jmob ~Z{dUmè¶m gd© e§Hy$À¶m Vim§À¶m joÌ’$im§Mr ~oarO hr JmobmÀ¶m n¥ð>’$imerg‘mZ AgVo. Á¶mAeu JmobmMo n¥ð>’$i 4 r2 Amho.

Amåhmbm {‘iob,

JmobmMo KZ’$i =13

r(4 r2) = 34r

3

JmobmMo KZ’$i = 34 r3

KZ EH$H$

{ÌÁ¶m (r) hr g‘mZ Agboë¶m Jmob Am{U e§Hy$À¶m KZ’$imÀ¶m g§~§Ym{df¶r AË¶§V Hw$VwhbmMr Jmoï>Amho.

g§~§YmMm Aä¶mg H$am.

JmobmMo KZ’$i ho e§Hy$À¶m KZ’$imÀ¶m Mma nQ> Amho.

JmobmMo KZ’$i = 4 × e§Hy$Mo KZ’$i

= 4 ×13

r2h = 4 ×13

r2 × r = 4 ×13

r2 =43

r3

JmobmMo KZ’$i =43

r3

ârd AY©JmobmMo KZ’$i (Volume of a solid hemisphere)

ârd AY©JmobmMo KZ’$i =12 × ârd JmobmMo KZ’$i = 3 31 4 2r r

2 3 3

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392 UNIT-15


CXmhaUo 1 … 21 g|.‘r. {ÌÁ¶oÀ¶m JmobmMo n¥ð>’$i H$mT>m. r = 21 g|.‘r.

CH$b … JmobmMo n¥ð>’$i = 4 r2 =22

47

213

21cmMm¡.g|.‘r.

JmobmMo n¥ð>’$i = 5544 Mm¡.g|.‘r.

CXmhaU 2 … 14 g|.‘r. {ÌÁ¶oÀ¶m AY©JmobmMo dH©$ n¥ð>’$i (CSA) Am{U g§nyU© n¥ð>’$i (TSA) H$mT>m.

CH$b … AY©JmobmMo dH«$ n¥ð>’$i (CSA) = 2 r2 =22

27

142

14cmMm¡.g|.‘r.

AY©JmobmMo dH«$ n¥ð>’$i (CSA) = 1232 Mm¡.g|.‘r.

AY©JmobmMo g§nyU© n¥ð>’$i (TSA) = 3 r2 =22

37

142

14cmMm¡.g|.‘r.

AY©JmobmMo g§nyU© n¥ð>’$i (TSA) = 1848 Mm¡.g|.‘r.

CXmhaU 3 … 3 g|.‘r. {ÌÁ¶oÀ¶m JmobmMo KZ’$i H$mT>m.

CH$b … r = 3 g|.‘r. JmobmMo KZ’$i = 34r

3 =

43

22 37

3 3cmK.g|.‘r. = 113.14 K.g|.‘r.

CXmhaU 4 … AY© JmobmH$ma dmQ>rMm AmVrb ì¶mg 9 g|.‘r.Amho Va bm ‘Ü¶o gm‘mdUmè¶m XþYmMo KZ’$iH$mT>m.

CH$b … d = 9 g|.‘r. r =92 g|.‘r.

AY©JmobmMo KZ’$i = 32r

3=

23

22 97

3

29 92 2

K.g|.‘r.=5346 190.92cm28

K.g|.‘r.

AY©JmobmH$ma dmQ>rMo KZ’$i = 190.92 [1 K.g|.‘r.=1 {‘.br. ]Ë¶m‘Ü¶o gm‘mdboë¶m XþYmMo KZ’$i = 190.92 {‘.br. [1 K.g|.‘r.=1 {‘.br. ]

CXmhaU 5 … AY©JmobmH$ma dmQ>rMr AmVrb {ÌÁ¶m 18 g|.‘r. AgyZ Vr ’$imÀ¶m agmZo âbobr Amho. hm ag 3g|.‘r. {ÌÁ¶m Am{U 9 g|.‘r. C§Mr Agboë¶m d¥Îm{MÎmr AmH$mamÀ¶m ~mQ>br‘Ü¶o hm ag âbm Va dmQ>r nyU©nUoImbr H$aÊ¶mgmR>r {H$Vr ~mQ>ë¶m âmì¶m bmJVrb?

CH$b … AY©Jmob … r = 18 g|.‘r. V = ? AY©JmobmMo KZ’$i = 32 r3

AY©JmobmH$ma dmQ>rMo KZ’$i =2 22

18 18 18cm3 7

g|.‘r.

d¥Îm{MÎmr … r = 3, h = 9

d¥Îm{MÎmrMo KZ’$i = r2h EH$m ~mQ>brMo KZ’$i =22 3 3 9cm7

K.g|.‘r.

~mQ>btMr g§»¶m = AY©JmobmH$ma dmQ>r‘Yrb ’$imÀ¶m agmMo KZ’$i

EH$m d¥Îm{MÎmr AmH$mamÀ¶m ~mQ>brVrb ’$imÀ¶m agmMo KZ’$i

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Mensuration 393

=

23

227

186

2

186

182

227

3 3 9 = 48

Amdí¶H$ Agboë¶m ~mQ>ë¶m§Mr g§»¶m = 48

CXmhaU 6 … YmVyÀ¶m Jmoù¶mMm ì¶mg 4.2 g|.‘r. Amho. Ë¶mbm {dVidyZ 8.4 g|.‘r. C§MrMm b§~ dVw©imH$ma

e§Hy$ ~Z{dbm Va e§Hy$À¶m VimMm ì¶mg H$mT>m..

CH$b … d = 4.2 g|.‘r. r =4.22 = 2.1 g|.‘r. e§Hy$ : h = 8.4 g|.‘r.

JmobmMo KZ’$i = e§Hy$Mo KZ’$i

3s

4r

3= 2

c c1

r h3

2Cr =

43

3Sr

13 Ch =

4 2.1 2.1 2.18.4 2.1

= 2.1 × 2.1 rC = 2.1 2.1 = 2.1 cm

e§Hy$À¶m VimMr {ÌÁ¶m = 2.1 g|.‘r.

CXmhaU 7 … nmoH$i AY©JmoimH$mam AmH$mamÀ¶m e§ImMm AmVrb Am{U ~mhoarb ì¶mg AZwH«$‘o 6 g|.‘r. Am{U10 g|.‘r. AmhoV. Vr {dVidyZ 14 g|.‘r. ì¶mgmMm ârd e§Hy$ ~Z{dÊ¶mV Ambm. Aem[aVrZo ~Z{dboë¶me§Hy$Mr C§Mr H$mT>m.

CH$b … nmoH$i AY©Jmob … ~mhoarb {ÌÁ¶m R =102 = 5g|.‘r. AmVrb {ÌÁ¶m r =

62 = 3 g|.‘r.

e§Hy$ … r =142 = 7 g|.‘r. h = ?

nmoH$i AY©JmobmMo KZ’$i = e§Hy$Mo KZ’$i3 32

(R r )3

= 21r h

3

h=

2 23

3 3(R r )

13

3 3

22

2(R r )r

r=

3 3

2

2(5 3 ) 2(125 27)497

=98

22

49 = 4 g|.‘r.

V¶ma Pmboë¶m e§Hy$Mr C§Mr = 4 g|.‘r.

CXmhaU 8 … XmoZ ârd YmVyÀ¶m Jmoù¶mMr {ÌÁ¶m r1 Am{U r2 Amho. Jmob {dVidyZ EH${ÌVnUo (r1 + r2) C§MrMm

ârd e§Hy$ ~Z{dbm. Va e§Hy$Mr {ÌÁ¶m 2 21 2 1 22 r r r r ho XmIdm.

CH$b … (Jmob 1 + Jmob 2) Mo KZ’$i = e§Hy$Mo KZ’$i

3 31 2

4 4r r

3 3=

2C 1 2

1r (r r )

3

3 31 2

4 r r3

= 2C 1 2

1 r (r r )3

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394 UNIT-15

rC=

43

3 31 2r r

13 1 2r r =

1 24 r r 2 21 1 2 2

1 2

r r r r

r r

2Cr = 2 2

1 1 2 24 r r r r

rC=2 2

1 1 2 24 r r r r = 2 21 1 2 22 r r r r

e§Hy$À¶m VimMr {ÌÁ¶m = 2 21 1 2 22 r r r r

CXmhaU 9 … 1 g|.‘r. {ÌÁ¶oMm ârd Jmob {dVidyZ 100 g|.‘r. bm§~rMr Vma ~Z{dÊ¶mV Ambr. VmaoMr{ÌÁ¶m H$mT>m.

CH$b … Jmob r = 1 g|.‘r.

Vma hr d¥Îm{MÎmr Amho.

d¥Îm{MÎmr h = 100 g|.‘r. r = ?

JmobmMo KZ’$i = Vma (d¥Îm{MÎmr)Mo KZ’$i

3S

4r

3 = 2

cy cyr h 2Cyr =

43

3sr

cyh=

3s

cy

4r3 4 =

4 31 1 1cm3 100 25cm =

175 K.g|.‘r.

rCy=1 175 5 3

g|.‘r.= 0.11 g|.‘r.

VmaoMr {ÌÁ¶m = 0.11 g|.‘r.

ñdmÜ¶m¶ 15.4

1. JmobmMo n¥ð>’$i H$mT>m Oa Ë¶mMr {ÌÁ¶m (i) 14 g|.‘r.

2. 5 g|.‘r. {ÌÁ¶oÀ¶m AY©JmobmMo g§nyU© (TSA) n¥ð>’$i H$mT>m.

3. bmH$S>mÀ¶m AY©JmobmH$ma dmQ>rMm AmVrb ì¶mg 10.5 g|.‘r. Amho Va 100 Mm¡.g|.‘r. bm < $.12 à‘mUoAmVrb ^mJmbm a§J bmdÊ¶mgmR>r ¶oUmam IM© H$mT>m.

4. 15 g|.‘r. ~mOyÀ¶m KZmnmgyZ H$mnboë¶m ‘moR>çmV ‘moR>çm JmobmMo joÌ’$i (dH«$ n¥ð>’$i) H$mT>m.

5. JmobmMo KZ’$i H$mT>m. Oa Ë¶mMr {ÌÁ¶m 7 g|.‘r.Amho.

6. JmobmMo dH«$ n¥ð>’$i 154 Mm¡.g|.‘r. Amho. Va Ë¶mMo KZ’$i H$mT>m.

7. ârd AY©JmobmMo KZ’$i 1152 K.g|.‘r. Amho. Va Ë¶mMo dH«$ n¥ð>’$i H$mT>m.

8. Am¡fYmÀ¶m JmoirMm AmH$ma JmobmH$ma AgyZ Ë¶mMm ì¶mg 3.5 {‘.‘r. Amho. hr H°$mgwbMr JmoirâÊ¶mgmR>r {H$Vr Am¡fY bmJob?

9. b§~ dVw©imH$ma YmVyÀ¶m e§Hy$Mr C§Mr 20 g|.‘r. Am{U VimMr {ÌÁ¶m 5 g|.‘r. Amho. Vmo {dVidyZ Jmob~Z{dbm Va JmobmMr {ÌÁ¶m H$mT>m

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Mensuration 395

10.EH$m YmVyÀ¶m Jmoù¶mMm ì¶mg 18 g|.‘r. Amho. Ë¶mbm {dVidyZ 0.4 g|.‘r. ì¶mgmMr Vma ~Z{dÊ¶mV Ambr.

Va Ë¶m VmaoMr bm§~r H$mT>m.

KZ nXmWmªMm g§¶moJ (Combination of solids)

KZnXmW© Ogo H$s, KZ, KZm¶V, bmobH$, {na°{‘S>, d¥Îm{MÎmr, e§Hy$ Am{U Jmob {df¶r Vwåhmbm ‘m{hVr

Amho. Vwåhmbm Ë¶m§Mo joÌ’$i Am{U KZ’$i H$go H$mT>bo OmVo Vo ‘m{hV Amho. Ago KZ AmH$ma Amnë¶m

g^modVmbÀ¶m nXmWm©‘YyZ AmoiIUo gd© gmYmaU Amho.

AmVm Imbrb nXmW© Am{U Ë¶m§Mo AmH$ma ¶m§Mo {ZarjU H$am. JQ>m‘Ü¶o MMm© H$am Am{U àË¶oH$mMm AmH$ma

AmoiIm

àË¶oH$ KQ>ZoVrb nXmW© ho XmoZ qH$dm A{YH$ KZnXmWm©nmgyZ ~Zbobo AmhoV. åhUyZ Ë¶mn¡H$s àË¶oH$ hmKZnXmWmªMm g§¶moJ Amho.

IoiUo V obmMr Q>mH$s

Cone + Hemisphere

harV J¥ham±Ho$Q>a

ewH«$+KZm¶VAY© d¥V{dVr +KZm¶V

Aem KZnXmWmªMo KZ’$i Am{U joÌ’$i H$go H$mT>bo OmVo? KZnXmWmªÀ¶m g§¶moJmMo joÌ’$i Am{U KZ’$iH$mT>Ê¶mMm Aä¶mg H$ê$.

Hemisphere + Cylinder+ Hemisphere AY©Jmob + d¥Îm{MÎmr

e§Hw$ + AY©Jmob

IoiUo VobmMm Q>±H$a Ap½Z~mU Z‘wZm harV J¥h PmonS>r

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396 UNIT-15

KZnXmWmªÀ¶m g§¶moJmMo joÌ’$i … (Surface area of combination of solids)

IoiÊ¶mÀ¶m CXmhaUmMm {dMma H$am. Amåhmbm Ago {XgyZ ¶oVo H$s, IoiÊ¶mMo joÌ’$i hoAY©JmobmMo dH«$n¥ð>’$i Am{U e§Hy$Mo n¥ð>’$i ¶m§Mr ~oarO Amho..

IoiÊ¶mMo g§nyU© joÌ’$i (TSA) = AY©JmobmMo n¥ð>’$i (CSA)

IoiÊ¶mMo dH«$ n¥ð>’$i (CSA) = AY©JmobmMo (CSA) + e§Hy$Mo CSA

AemM àH$mao BVa CXmhaUmVrb AmnU g§nyU© n¥ð>’$i Am{U dH«$ n¥ð>’$i Imbrbà‘mUo H$mTy> eH$Vmo

VobmÀ¶m Q>mH$sMo TSA = AY©JmobmMo CSA + d¥Îm{MÎmrMo CSA + AY©JmobmMo CSA

VobmÀ¶m Q>mH$sMo CSA = AY©JmobmMo CSA + d¥Îm{MÎmrMo CSA + AY©JmobmMo CSA

A{¾~mU à{VH¥$VrMo TSA = d¥Îm{MÎmrMo CSA + VimMo joÌ’$i + e§Hy$Mo CSA

A{¾~mU à{VH¥$VrMo CSA = d¥Îm{MÎmrMo CSA + e§Hy$Mo CSA

PmonS>rMo TSA = KZm¶VmMo CSA + VimMo joÌ’$i + d¥Îm{MÎmrMo AY} CSA

PmonS>rMo CSA = KZm¶VmMo CSA + d¥Îm{MÎmrMo AY} CSA

darb CXmhaUmdê$Z AmnU Agm {ZîH$f© H$mTy> H$s KZnXmWmªÀ¶m g§¶moJmMog§nyU© n¥ð>’$i ho EH${ÌV Ho$boë¶m gd© KZnXmWmªMr ~oarO AgVo qH$dm ZgVo.

AmnU Imbrb Zm|Xr bjmV R>ody.

• KZnXmWmªÀ¶m g§¶moJm§Mo joÌ’$i ho ’$º$ EH${ÌV Ho$boë¶m KZnXmWmªÀ¶m g§¶moJmMo

joÌ’$i ZgVo H$maU OmoS>V AgVmZm g§nyU© joÌ’$imMm H$mhr ^mJ Zmhrgm

hmoVmo.

• EH${ÌV Ho$boë¶m KZnXmWmªÀ¶m dH«$ n¥ð>’$im§Mr ~oarO gwÕm ZgVo H$maU dH«$

n¥ð>^mJ gmoSy>Z BVa ^mJ Ë¶m§À¶m g§nyU© joÌ’$imV {‘gibm OmVmo.

• KZnXmWmªÀ¶m g§¶moJmMo dH«$ n¥ð>’$i ho EH${ÌV Ho$boë¶m KZnXmWmªÀ¶m g§¶moJmÀ¶m dH«$ n¥ð>’$im§Mr ~oarO

Agy eH$Vo.

• KZnXmWmªÀ¶m g§¶moJmMo dH«$ n¥ð>’$i ho EH${ÌV Ho$boë¶m KZnXmWmªÀ¶m g§¶moJmÀ¶m dH«$ n¥ð>’$im§Mr ~oarO

Agy eH$V Zmhr. H$maU dH«$ n¥ð>^mJ gmoSy>Z BVa ^mJ gwÕm {‘i{dbo OmVmV.

AmVm KZ nXmWmªÀ¶m g§¶moJmÀ¶m dH«$ n¥ð>’$imda AmYmarV H$mhr CXmhaUo gmoS>dy¶m.

IoiUo

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Mensuration 397


CXmhaU 1 … d¥Îm{MÎmr AmH$mamÀ¶m ^m§S>çm§Mr {ÌÁ¶m 6 g|.‘r. Am{U C§Mr 15 g|.‘r. AgyZ Vo AmB©gH«$s‘Zo

^abobo Amho. darb Q>moH$ AY©dVw©imH$ma Agboë¶m e§Hy$Zo 10 {dÚmÏ¶mª‘Ü¶o g‘mZ[aË¶m g§nyU© AmB©gH«$s‘

dmQ>bm. e§Hy$Mr C§Mr VimÀ¶m {ÌÁ¶oÀ¶m 4 nQ> Amho. Va AmB©gH«$s‘À¶m e§Hy$Mr (H$moZmMr) {ÌÁ¶m H$mT>m.

4r r r

6 cm

15 c

m

CH$b … darb Q>moH$ AY©dVw©imH$ma AgUmè¶m e§Hy$Mo KZ’$i

= 2 31 2r h r3 3

= 2 31 2r 4r r3 3

= 3 34 2r r3 3

=6

2

33r = 2 r3

AY©JmobmH$ma darb Q>moH$mÀ¶m 10 e§Hy$Mo KZ’$i = 10 × 2 r3 = 20 r3

d¥Îm{MÎmr AmH$mamÀ¶m ^m§S>çmVrb AmB©gH«$s‘ KZ’$i = r2h × 6cm × 6cm × 15 cm = 540 cm3

AY©JmobmH$ma darb Q>moH$mÀ¶m 10 e§Hy$Mo KZ’$i= d¥Îm{MÎmr AmH$mamÀ¶m ^m§S>çmVrb AmB©gH«$s‘ Mo

KZ’$i

20 r3 = 540 r3 =54

270

2 0 = 27 r = 3 27 r = 3 g|.‘r.

CXmhaU 2 … b§~ dVw©imH$ma d¥Îm{MÎmrda e§Hy$À¶m AmH$mamMm KZnXmW© Amho. XmoÝhtÀ¶m VimMr {ÌÁ¶m g‘mZ

Amho. d¥Îm{MÎmrÀ¶m Vimda e§Hy$Mm Vi ~g{dbobm Amho. Oa e§Hy$À¶m VimMr {ÌÁ¶m Am{U C§Mr AZwH«$‘o 7

g|.‘r. Am{U 10 g|.‘r. Amho. Am{U EHy$U KZnXmWm©Mr C§Mr 30 g|.‘r. Amho Va KZnXmWm©Mo KZ’$i H$mT>m.

CH$b … d¥Îm{MÎmr ^mJ … r = 7 g|.‘r., h = 30 – 10 = 20 g|.‘r.

e§Hy$Mm ^mJ … r = 7 g|.‘r., h = 10 g|.‘r.

KZnXmWm©Mo KZ’$i = d¥Îm{MÎmrMo KZ’$i + e§Hy$Mo KZ’$i

= r2h + 21r h

3=

227

7 7 20cmK.g|.‘r. +1 223 7

7 7 10

= 3080 K.g|.‘r.+1540

3K.g|.‘r.

= 3080 + 513.33 K.g|.‘r.

KZnXmWm©Mo KZ’$i = 3593.33 K.g|.‘r.

CXmhaU 3 … EH$ IoiUo g‘mZ {ÌÁ¶oÀ¶m AY©Jmobmda e§Hy$ ñdê$nmV ~g{dbobo Amho. e§Hy$À¶m AmH$mamMmì¶mg 6 g|.‘r. Am{U C§Mr 4 g|.‘r. Amho. Va KZnXmWm©Mo dH«$ n¥ð>’$i Am{U KZ’$i H$mT>m.

g|.‘r.

g|.‘r.

g|.‘r.g|.‘r.

g|.‘r

.

g|.‘

r.A

B

D

30

10

20

E7

7

7

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398 UNIT-15

126 m

21m

16m

30 m 30 mO5m

16 ‘

r.

30 ‘r. 30 ‘r.

‘r.

‘r.

‘r.

CH$b … e§Hy$Mm ^mJ JmobmH$ma mJr = 3 g|.‘r., h = 4 g|.‘r. r = 3 g|.‘r.

l2 = r2 + h2 = 32 + 42 = 9 + 16l2 = 25

l = 25 = 5 g|.‘r.

IoiÊ¶mMo dH«$ n¥ð>’$i= AY©JmobmMo CSA + e§Hy$Mo CSA

= 2 r2 + rl = 2 222 222 3 3cm 3 5cm

7 7Mm¡.g|.‘r.

= 2 2396 330cm cm

7 7Mm¡.g|.‘r. = 7

726 Mm¡.g|.‘r.

IoiÊ¶mMo dH«$ n¥ð>’$i = 103.71 cm2

IoiÊ¶mMo KZ’$i = AY©JmobmMo KZ’$i + e§Hy$Mo KZ’$i

= 3 22 1r r h3 3

=23

22 37

3 13 3cm3

22 37

3 4cmK.g|.‘r.

= 3 3396 264 660cm cm

7 7 7K.g|.‘r.

IoiÊ¶mMo KZ’$i = 94.28 K.g|.‘r.

CXmhaU 4 … VmS>nÌrnmgyZ gH©$gMm V§~y ~Z{dbm Amho Am{U Vmo Imbrb ~mOybm b§~ dVw©imH$ma d¥Îm{MÎmrAm{U darb ~mOybm b§~ dVw©imH$ma e§Hy$À¶m ñdê$nmV Amho. d¥Îm{MÎmrÀ¶m ^mJmÀ¶m V§~yMm ì¶mg Am{U C§MrAZwH«$‘o 126 ‘r. Am{U 5 ‘r. Amho. V§~yMr EHy$U C§Mr 21 ‘r. Amho. Oa Xa ‘r2.Mm VmS>nÌrMm Xa < .15à‘mUo V§~y ~Z{dÊ¶mgmR>r dmnaboë¶m VmS>nÌrMm EHy$U IM© H$mT>m.

CH$b … d¥Îm{MÎmr ^mJ … r =126

2 = 63 ‘r., h = 21 – 5 = 16 ‘r.

l = 2 2r h = 2 263 16

= 3969 256 = 4225

l = 65 ‘r.

EHy$U dmnabobr VmS>nÌr = 14850 Mm¡.‘r.

< .16 Xa Mm¡.‘r. à‘mUo VmS>nÌrMm ¶oUmam IM© = 14,850 × 15 = 2,22,750

VmS>nÌrMm IM© = < 2,22,750

CXmhaU 5 … d¥Îm{MÎmr AmH$mamÀ¶m KZnXmWm©Mr Q>moHo$ AY©dVw©imH$ma AmhoV. Oa KZnXmWm©Mr EHy$U C§Mr 104g|.‘r. Am{U àË¶oH$ AY©JmobmH$ma Q>moH$mMr {ÌÁ¶m 7 g|.‘r. Amho. 100 Mm¡.‘r.bm < 4 XamZo Ë¶mMm n¥ð>^mJCOmim XoÊ¶mgmR>r ¶oUmam IM© {H$Vr?

A B

C

4 cm

3 cm3 cm

g|.‘r.g|.‘r.

4 g|.

‘r.

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Mensuration 399

CH$b … AY©JmobmH$ma ^mJ … r = 7 g|.‘r.

d¥Îm{MÎmrMo ^mJ … r = 7 g|.‘r., h = 104 – (7 + 7) = 90 g|.‘r.

KZnXmWm©Mo EHy$U joÌ’$i = d¥Îm{MÎmr ^mJmMo CSA + 2 × AY©JmobmH$ma ^mJmMo CSA

= 2 rh + 2 × 2 r2 = 2 r (h + 2r)

=22

27

7 90 2(7) cmMm¡.g|.‘r.

= 44 × 104 Mm¡.g|.‘r. = 4576 Mm¡.g|.‘r.

KZnXmWm© TSA = 4.576 Mm¡.g|.‘r.100 Mm¡.Mo ‘r. bm ê$.4 XamZo COmim XoÊ¶mgmR>r ¶oUmam IM©

= 4576 ×4

100`

= < 183.04

KZnXmWm©bm COmim XoÊ¶mgmR>r ¶oUmam IM© = `< 183.04

ñdmÜ¶m¶ 15.5

1. noQ´>mob Q>mH$sÀ¶m XmoÝhr Q>moH$m§Zm g‘mZ {ÌÁ¶oMo AY©Jmob ~g{dbobo AmhoV. Q>mH$sMr EHy$U bm§~r 6 ‘r.Am{U {ÌÁ¶m 1 ‘r. Amho. Va Q>mH$sMr j‘Vm {bQ>a‘Ü¶o {H$Vr?

2. d¥Îm{MÎmr AmH$mamÀ¶m A{¾~mUmÀ¶m EH$m Q>moH$mbm e§Hy$ Am{U Xþgè¶m Q>moH$mbm AY©Jmob OmoS>bobm Amho.gdmªMr {ÌÁ¶m g‘mZ AgyZ Vr 1.5 ‘r. Amho. A{¾~mUmMr EHy$U bm§~r 7 ‘r. Am{U e§Hy$Mr C§Mr 2 ‘r.Amho. Va A{¾~mUmMo KZ’$i {H$Vr?

3. AY©JmobmH$ma H$n d¥Îm{MÎmrÀ¶m ñdê$nmV Amho. d¥Îm{MÎmr ^mJmMr C§Mr 8 g|.‘r. Am{U H$nmMr EHy$U C§Mr11.5. g|.‘r. Amho. Va H$nmMo EHy$U n¥ð>’$i {H$Vr?

4. dVw©imH$ma d¥Îm{MÎmrÀ¶m AmH$mamÀ¶m gmR>{dÊ¶mÀ¶m Q>mH$sÀ¶m XmoÝhr Q>moH$m§Zm AY©JmobmH$ma ^mJ ~g{dboAmhoV. d¥Îm{MÎmrMm ~mhoarb ì¶mg 1.4 g|.‘r. Am{U bm§~r 8 ‘r. Amho. ~mhoarb ~mOyZo Xa Mm¡.‘r. bm

10 < .à‘mUo a§J bmdÊ¶mgmR>r ¶oUmam IM© H$mT>m.

5. AY©JmobmH$ma e§Hy$ ~g{dboë¶m AmH$mamMo bmH$S>mMo IoiUo Amho. e§Hy$À¶m VimMm ì¶mg 16 g|.‘r.

Am{U C§Mr 15 g|.‘r. Amho. 100 Mm¡.‘r.bm < .7 à‘mUo a§J bmdÊ¶mgmR>r ¶oUmam IM© {H$Vr?

6. gH©$gMm V§~y 3 ‘r. C§Mrn¶ªV d¥Îm{MÎmr AgyZ Ë¶mdarb e§Hy$À¶m AmH$mamMm Amho. Oa VimMm ì¶mg 105‘r. Am{U e§Hy$À¶m ^mJmMr {VaH$g C§Mr 53 ‘r. Amho. Xa Mm¡.‘r. bm 100 à‘mUo V§~y ~Z{dÊ¶mgmR>rdmnaboë¶m VmS>nÌrMm EHy$U IM© {H$Vr?

eoVmMm AmamIS>m (Scale drawing) :

gnmQ> ^m¡{‘VrH$ AmH¥$Ë¶m§Mo joÌ’$i H$mT>Ê¶mgmR>r dmnaboë¶m gyÌm{df¶r Amåhmbm ‘m{hVr Amho. Imbr {Xboë¶m

104

cm

90 cm7 cm

7 cm

7 cm

7 cm

g|.‘r

g|.‘r

g|.‘r

g|.‘r

g|.‘r

g|.‘r

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400 UNIT-15

AmH¥$Ë¶m§Mo {ZarjU H$am Am{U Ë¶m§Mo joÌ’$i H$mT>Ê¶mÀ¶m gyÌm§Mo ñ‘aU H$am.

.

D b C

h

A

B

S

P

R

b

Ql

K

N

L

h

Mb(a) {ÌH$moU (b) Am¶V (c) g‘m§Va yO Mm¡H$moZ

(A =12 × nm¶m × C§Mr) (A = bm§~r × é§Xr) (A = nm¶m × C§Mr)

A = 1/2 b×h A = l×b A = b×h

A B

EdF

CD

h1

h2

G

D E

Fb

a

h

(d) Mm¡H$moZ (e) g‘b§~ Mm¡H$mZA = 1/2 × H$U© × ~mOy§Mr ~oarO A = 1/2 × C§Mr × g‘m§Va ~mOy§Mr ~oarOA = 1/2 × d (h1+ h2) A = 1/2 h (a + b)

Imbrb AmH¥$Ë¶m§Mo {ZarjU H$am.

B

CA

E D

E D

C

BA

F G

A BC

DE

F

¶m AmH¥$Ë¶m§Mo joÌ’$i H$go {‘iVo?

Amåhr Amnë¶m X¡Z§{XZ OrdZm‘Ü¶o A{Z¶{‘V AmH$mamZo ~Zboë¶m AmH¥$Ë¶m§À¶m KQ>Zoer OmoS>bobmo AmhmoV.Imbr {Xboë¶m AmH¥$Ë¶m nmhm.

Aem KQ>Zm‘Ü¶o OoWo AmH¥$Ë¶m‘Ü¶o {ÌH$moU qH$dm Mm¡H$moZmMo AmH$ma Am{U AmH¥$Ë¶m§Mo A{Z¶{‘V AmH$maAmhoV. gmYo V§Ì åhUOo {Xbobr AmH¥$Vr {ÌH$moU Am{U Mm¡H$moZm‘Ü¶o {d^mJyZ Z§Va Ë¶m§Mo joÌ’$i H$mT>Uo.

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Mensuration 401

Aem AmH¥$Ë¶m ‘m{hV Agboë¶m AmH$mam‘Ü¶o Imbrbà‘mUo {d^mJë¶m OmD$ eH$VmV. Ë¶m§Mm Aä¶mgH$am.

{ZarjU H$am H$s àË¶oH$ AmH¥$Vr {ÌH$moU Am{U g‘b§~ Mm¡H$moZm‘Ü¶o {d^mJbobr Amho.

Vwåhr H$gm {dMma H$amb H$s, V¶ma Ho$boë¶m doJdoJù¶m {ÌH$moUmMo Am{U g‘b§~ Mm¡H$moZmMo joÌ’$i

dmnê$Z àË¶oH$ AmH¥$VrMo EHy$U joÌ’$i H$mT>bo OmVo.

JQ> H$ê$Z dJm©‘Ü¶o MMm© H$am.

H¥$Vr 1 …

AmboI H$mJX ¿¶m

A C B E D hr n§M ~hþ wOmH¥$Vr aMm.

AB gm§Ym, hr nm¶mMr aofm Amho.

ABbm C, D Am{U E ho b§~ aMm.

F, G, H ¶m q~XþÀ¶m IwUm H$am.

V¶ma Pmbobo {ÌH$moU Am{U g‘b§~ Mm¡H$moZ AmoiIm.

Ë¶m§À¶m ~mOw§Mr bm§~r ‘moOm.

ADF, ACB Am{U EHB ¶m àË¶oH$ {ÌH$moUmMo

Am{U g‘b§~ Mm¡H$moZ FHED Mo joÌ’$i H$mT>m.

¶m§À¶m joÌ’$im§Mr ~oarO H$mT>m.

Vwåhr n§M ~hþ wOmH¥$VrMo joÌ’$i H$mT>bo Amho. nydu MMm© Ho$boë¶m BVa AmH¥$Ë¶mgmR>r hr H¥$Vr nwÝhm H$ê$Z

nmhm.

AmVm eoVmÀ¶m VwH$S>çmÀ¶m CXmhaUmMm {dMma H$am. eoVH$ar hm Ë¶mMm ‘mbH$ Am{U O{‘Z {ZarjH$mbm

Ë¶mMo joÌ’$i H$mT>m¶Mo Amho Ago g‘Oy. Vo H$go Ho$bo OmVo?

O{‘ZrÀ¶m A{Z¶{‘V AmH$mamMo ‘moO‘mn Am{U joÌ’$i H$mT>Ê¶mgmR>r {ZarjH$mZo dmnaboë¶m nm¶è¶m

Imbrbà‘mUo {Xbo AmhoV.

A{Z¶{‘V AmH$mamMr O‘rZ ‘m{hV Agboë¶m ^m¡{‘VrH$ {d^mJm‘Ü¶o {d^mJUo.

Jw§Q>g© gmIir (Guntor’s chain) dmnê$Z ~mOy§Mr ‘mno H$mT>Uo.

{ZarjH$mÀ¶m O{‘ZrÀ¶m nwñVH$m‘Ü¶o ¶m ‘mnm§Mr Zm|X H$aUo.

O 1–1–2–3–4–5X

Y

A

D

E

B

G

F

H

C

2 3 4 5

1

2

3

4

5

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402 UNIT-15

O X

Y

A

D

E

B

G

F

H

C

I1

2

3

4

5

1 2 3 4 5

A{Z¶{‘V AmH$mamMr O‘rZ à‘mUm‘Ü¶o {MÌ qH$dm

AmH¥$VrV Xe©{dUo.

àË¶oH$ I§S>mMo joÌ’$i H$mT>Uo.

àË¶oH$ I§S>mÀ¶m joÌ’$imMr ~oarO H$ê$Z O{‘ZrMo EHy$U

joÌ’$i H$mT>bo OmVo.

O{‘ZrÀ¶m Zm|X dhr‘Ü¶o hr ‘mno H$er KoVmV Am{U Zm|X H$aVmV?

eoVmMm AmamIS>m H$gm H$mT>bm OmVmo? ho g‘OyZ KoÊ¶mgmR>r

Imbrb H¥$Vr H$ê$Z nhm.

H¥$Vr 2 …

J«m’$ (AmboI) H$mJX ¿¶m.

AmH¥$VrV XmI{dë¶mà‘mUo A{Z¶{‘V AmH$ma H$mT>m.

e³¶ VoWo eodQ>À¶m q~XþMr IyU H$am.

àW‘ darb Am{U Imbrb q~XÿMr IyU H$am Ë¶m‘wio Vo OmoSy>Z nm¶m aofm {‘iVo AB hr nm¶m aofm Amho.

nm¶m aofoÀ¶m XmoÝhr ~mOybm BVa A§Ë¶q~XÿMr IyU H$am. Vo q~Xÿ C,D, E Am{U F AmhoV.

nm¶m aofoda C,D,E Am{U F ho b§~ H$mT>m.

CG, DJ, EK Am{U FG ¶m b§~ aofm AmhoV Ago g‘Oy.

A Am{U C, C Am{U G, D Am{U J, B Am{U F, E Am{U K, D Am{U A OmoS>m.

AmVm Vwåhr ACDBEF hm {Z¶{‘V AmH$ma {‘i{dbobm Amho.

aofm§Mr bm§~r ‘moOm Am{U Imbr {Xboë¶m H$moï>H$m‘Ü¶o Zm|X H$am.

H$moï>H$m‘Ü¶o {Xboë¶m qH$‘Vr dmnê$Z H¥$Vr 1 ‘Ü¶o MMm© Ho$ë¶mà‘mUo joÌ’$i H$mT>m.

O‘rZ gd}jH$mMr O‘rZ ‘moOyZ Zm|X H$aÊ¶mMr nÕV darb à‘mUo MMm© Ho$ë¶mà‘mUoM AgVo. na§Vy ‘mno

gm‘mÝ¶nUo ‘rQ>a‘Ü¶o AgVmV. Ë¶m‘wioM H$ƒo {MÌ H$mT>VmZm ¶mo½¶ à‘mU KoVmV Am{U gwadmVrMr (‘wi) ‘mno

Ë¶mà‘mUmZwgma ê$nm§VarV H$aVmV.

AmVm Imbrb CXmhaUm§Mm Aä¶mg H$am.

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Mensuration 403


CXmhaU 1 … Imbr {Xboë¶m ‘m{hVrdê$Z ßb°Z (AmamIS>m) H$mT>m Am{U joÌ’$i H$mT>m.

CH$b … ¶mo½¶ à‘mU ¿¶m. Vo 20 ‘r. = 1 g|.‘r. g‘Oy.

{Xboë¶m ‘mnm§Mo ê$nm§Va H$am.

200 ‘r. =120 × 200 = 10 g|.‘r.

140 ‘r. =120 ×140 = 7 g|.‘r.

120 ‘r. =120 × 120 = 6 g|.‘r.

40 ‘r. =120 × 40 = 2 g|.‘r.

60 ‘r. =12 × 60 = 3 g|.‘r.

50 ‘r. =12 × 50 = 2.5 g|.‘r.

30 ‘r. =12 × 30 = 1.5 g|.‘r.

C^r aofm AD = 10g|.‘r. (200 ‘r.) H$mT>m.

AD da P, Q, R ho q~Xÿ ¿¶m Ago H$s AP = 2 g|.‘r. (40‘r.), hmoB©b.

AQ = 6 g|.‘r. (120 ‘r.) qH$dm PQ = 4g|.‘r. ( 6-2 KoVbo OmVo)

AR = 7 g|.‘r. (140g|.‘r.) qH$dm QR = 1g|.‘r. ( 7-6KoVbo OmVo)

{Xboë¶m ‘mnmZwgma P,Q, R nmgyZ b§~ H$mT>m.

PB = 1.5 g|.‘r. (30‘r.) QE = 3 g|.‘r. (60‘r.)

RC = 2.5 g|.‘r. (50‘r.)

gd© q~Xÿ OmoS>ë¶mg AB C DE hr AmH¥$Vr {‘iVo

‘mnm§Mr ZmoX H$am.

ABCDE Mo joÌ’$i H$mT>m

ABCDE Mo joÌ’$i = ABP Mo joÌ’$i + g‘b§~ Mm¡H$moZ PBCR Mo joÌ’$i +

CRD Mo joÌ’$i + DEQ Mo joÌ’$i + EQA Mo joÌ’$i.

D n¶ªV ‘r.‘Ü¶o

200

140 C H$S>o 50

120

40 B H$S>o 30

E H$S>o 60

A nmgyZ

D

C

BP

A

E

R

Q

60

502060

80

30

40

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404 UNIT-15

=12 × AP × PB +

12 ×PR(PB+RC) +

12 × RC× RD+

12 × EQ × QD

12 × EQ × QA

=12 × 40×30 +

12 ×100(30 + 50) +

12 × 50 × 60 +

12 ×60×80+

12 × 60 × 120

= 600 + 4000 + 1500 + 2400 + 3600

ABCD Mo joÌ’$i = 12,100 Mm¡.‘r.

CXmhaU 2 … Zm|X dhrVrb Zm|XrZwgma Imbrb ‘m{hVrMm AmamIS>m H$mT>m Am{U joÌ’$i H$mT>m.


150100 C H$S>o 70

E H$S>o 80 80

30 B H$S>o 40

A nmgyZ

CH$b … à‘mU 20 ‘r. = 1 g|.‘r.

{ZarjU H$am

AM = 30 ‘r.

AN = 80 ‘r.

MP = (100 – 30) = 70 ‘r.

ND = (150 – 80) = 70 ‘r.

PD = (150 – 100) = 50 ‘r.

(1) ABM Mo joÌ’$i =1 1bh 30 40 600 sq.m.2 2 Mm¡.‘r.

(2) MBCP ¶m g‘b§~ Mm¡H$moZmMo joÌ’$i=1 1h a b 70 70 40 3850 sq.m.2 2 Mm¡.‘r.

(3) DPC Mo joÌ’$i =1 50 70 1750 sq.m.2 Mm¡.‘r.

(4) DEN Mo joÌ’$i =1 80 70 2800 sq.m.2 Mm¡.‘r.

(5) NEA Mo joÌ’$i =1 80 80 3200 sq.m.2 Mm¡.‘r.

ABCDE Mo joÌ’$i=600+3850+1750+2800+3200=12,200 Mm¡.‘r.

E80

P

20

7050

5040

30

A

M B

D

C

N

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Mensuration 405

ñdmÜ¶m¶ 15.6

1. Imbrb ‘m{hVrdê$Z eoVmMm AmamIS>m H$mT>m Am{U eoVmMo joÌ’$i H$mT>m.

C n¶ªV ‘r.‘Ü¶o

220

D H$S>o 120 210

120 B H$S>o 200

E H$S>o 180 80A nmgyZ

2. O‘rZ gdo©jH$mÀ¶m Zm|X dhrV {Xboë¶m ‘m{hVrdê$Z eoVmMm AmamIS>m H$mT>m. Am{U Ë¶mMo joÌ’$iH$mT>m.

E n¶ªV ‘r.‘Ü¶o350

D H$S>o 100 300 F H$S>o 150C H$S>o 75 250

150 G H$S>o 100B H$S>o 50 50

A nmgyZ

3. Imbrb ‘m{hVrdê$Z ABCDEFG Mm H$ƒm AmamIS>m H$mT>m Am{U O{‘ZrMo joÌ’$i H$mT>m.


225E H$S>o 90 175

125 C H$S>o 20F H$S>o 60 100G H$S>o 15 80

60 B H$S>o 70A nmgyZ

4. Imbr {Xboë¶m AmH¥$Vrdê$Z O{‘ZrMo joÌ’$i H$mT>m. (‘mno ‘rQ>a‘Ü¶o AmhoV.)

D

40 30

C 35F

E

25

G

30

70

B

A

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406 UNIT-15

ñdmÜ¶m¶ 15.1

1. (i) 176 Mm¡.g|.‘r (ii) 201.14 Mm¡.g|.‘r 2. 3168 Mm¡.g|.‘r 3. 5:4

4. 88Mm¡.‘r.< 3696 5.21,780 Mm¡.g|.‘r. 6. 44 Mm¡.‘r. 7. 5,544 K.g|.‘r.

8. 3432 gm 9. 2 :1 10.3080 K.g|.‘r.

ñdmÜ¶m¶ 15.2

1. 3960 Mm¡.g|.‘r 2. 8 g|.‘r. 3. 7.5 g|.‘r. 4. 7 g|.‘r Am{U 462 Mm¡.g|.‘r 5. 5,500 Mm¡.g|.‘r 6. 4:3 7. 183.33 K.g|.‘r. 8. 3:1 9. 415.8 K.g|.‘r.

10. < 2.068

ñdmÜ¶m¶ 15.3

1. 1408.58 K.g|.‘r. 2. CSA = 1021.42 Mm¡.g|.‘r, TSA = 1335.71 Mm¡.g|.‘r 3. 26148.57 K.g|.‘r. 4. 5 g|.‘r.ñdmÜ¶m¶ 15.4

1. 2,464 Mm¡.g|.‘r 2. 235.71 Mm¡.g|.‘r 3. 20.79 4. 707.14 Mm¡.g|.‘r

5. 1437.33 K.g|.‘r. 6.179.66 K.g|.‘r. 7. 905.14 Mm¡.g|.‘r 8. 22.45 K.‘r.‘r.

9. 5 g|.‘r. 10. 243 ‘r.

CÎmao

Cylinder

Statistics

Cone Frustum ofa cone

Sphere Hemisphere

Combinationof solids

Scaledrawing

2 r(h+r)

r h2

2 rh rl

r (l+r)

r h213

(r +r )1 2

{(r +r )l+1 2

r +r1 22 2 }

2 2

h(r +r +r +r )1 2 1 2

13

4 r2

r343

2 r2

3 r2

r323

LSA

TSA

Mensuration‘mnZo

d¥Îm{MÎmr e§Hy$ ’«$ñQ>‘ e§Hy$ Jmob AY©Jmob

KZ’$i

KZnXmWmªMmg§¶moJ

eoVmMmAmamIS>m

4 r2

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Mensuration 407

When you gain something,

let it be a,

When you make use of it for ntimes,

then it will be na,

When you share it with n persons,

ñdmÜ¶m¶ 15.5

1. 16,761.9 l. 2. 36.53 K.‘r. 3. 253 Mm¡.g|.‘r 4. ` 413.6 5. 58.08 6. < 97.350000

ñdmÜ¶m¶ 15.6

1. 49,300 Mm¡.‘r 2,50625 Mm¡.‘r 3.15,250 Mm¡.‘r 4. 46,00 Mm¡.‘r

*****

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408

n¶m©¶r àíZ ‹( Optional Questions)1.doZ AmH¥$Vr dmnéZ, Imbrb g§~§Y gË¶ Amho H$m Vo Vnmgm.

(i) A B = B A (ii) A \ B B \ A

(iii) (A B)I = AI BI (iv)A \ (B C) = (A \ B) (A \ C)

(v) A (B C) = (A B) (A C) (vi)A (B C) = (A B) (A C)

2. doZ AmH$¥$Vr dmnê$Z (A B) (A \ B ) = A ho XmIdm.

3. {Xboë¶m doZ AmH$¥$Vrdê$Z Imbrb qH$‘Vr H$mT>m.

1 (i)n(A B C) (ii) n(A B)

(iii) n(B C) (iv) n(A C)

(v) n(A B C) (vi) n

(vii) n [A

2 Imbrb g§~§Y nS>VmiyZ nmhm.

a. n(A B C) =n (A) +n (B) + n(C)-[n(A B) +n(A C) + n(AC) ]+n(A B C)

b. n (A = n(A B)-n(A B C)

c. n(A B C)1 n n(A B C)

4. \$ŠV doZ AmH$Vr H$mT>y>Z gmoS>dm.

(i) EH m$ dmVm©ha g§KQ>Zobm Ë`mÀ`m 170 J«mhH$mn¡H$s 115 OU XyaXe©Z dmnaVmV, 110 OU ao{S>`mo Am{U130 OU _m{gHo$ dmnaVmV Ago AmT>iyZ Ambo, 85 OU XwaXe©Z Am{U _m{gHo$, 75 OU XwaXe©Z Am{Uao{S>Amo, 95 OU ao{S>Amo Am{U _m{gHo$ Am{U 70 OU {VÝhr dmnaVmV Ago gwÜXm AmT>ibo Va H$mT>m (i)\$ŠV XwaXe©Z {H$Vr OU dmnaVmV? (ii) \$ŠV ao{S>Amo {H$Vr OU dmnaVmV (iii) {H$Vr OU XwaXe©Z Am{U_m{gHo$ dmnaVmV na§Vw ao{S>Amo dmnaV ZmhrV.

CÎma (i) 25 (ii)10 (iii)15

(ii)EH$ eham_Ü o 85% gm`H$b, 40% _moQ>ma JmS`m Am{U 20% H$ma JmS>`m dmnaVmV 32%

gm`H$b Am{U _moQ>ma Jmô`m 13% gm`H$b Am{U H$maJmS`m VgoM 10% _moQ>maJmS`m Am{U

H$ma JmS`m dmnaVmV Va gd© {VÝhr àH$maMr dmhZo dmnaUmè`m bmoH$m§Mr Q>ŠHo$dmar H$mT>m

CÎma 10 %

5. EH$m A. P Mo n{hbo nX 'a' Xþgao nX 'b' Am{U eodQ>Mo nX 'c' Amho Va A. P Mr ~oarO

(a+b) (b+c-2a)2(b-a) ho XmIdm.

576

81

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24

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6 .EH$m ‘wbmbm gdm©V darb AmoirV 1 {dQ> Xþgè¶m AmoirV 2, {Vgè¶m AmoirV 3 Am{U BË¶mXr ¶mà‘mUo

IoiÊ¶mVrb {dQ>m§Mr {ÌH$moUmH¥$Vr Wßnr aMmd¶mMr Amho. Oa Ë¶mÀ¶mOdi 100 {dQ>m AmhoV. Va Ë¶mZo

nyU© Ho$boë¶m AmoitMr g§»¶m Am{U Ë¶mZo {eëbH$ R>odboë¶m {dQ>m§Mr g§»¶m {H$Vr?

CÎma : 13 Amoir, 9 {dQ>m>

7. EH$m Ioim‘Ü¶o g‘mZ 6 ’y$Q> A§Vamda Q>monbr Am{U 16 ~Q>mQ>o EH$m gai AmoirV R>odbobo AmhoV. Ë¶mZo

Q>monbrnmgyZ gamgar 12 ’y$Q> Xa goH§$X doJmZo gwédmV Ho$br. Va EH$m Z§Va EH$ ¶m à‘mUo ~Q>mQ>o Q>monbr‘Ü¶o

R>odÊ¶mgmR>r IoimSy>§Zm {H$Vr doi bmJob?

CÎma : 136 goH§$X

8. Oa, a,b, c, d ho JwUmoÎma H«$‘mV AmhoV Va {gÜX H$am.

i) (b-c)2 + (c-a)2 + (d-b)2 = (a-d)2 ii) (a-b+c) (b+c+d) = ab+bc+cd

iii) (a+b), (b+c), (c+d) ho gwÜXm G.P. ‘Ü¶o AmhoV.

9. EH$ CgiUmam M|Sy> ( Bouncing ball) àË oH$doir _mJrb C§MrÀ`m {Zåå¶m C§Mrda nwÝhm CgiVmo

(Rebounds) Oa Ë`mbm 10 _r C§Mrdê$Z Q>mH$bo Va

a) Ë`mdoir O{_Zrbm AmT>iV AgVmZm AmH«${_bobo EHy$U A§Va H$mT>m.

b) pñWa pñWVrbm oÊ`mnydu AmH«${_bobo EHy$U A§Va H$mT>m.

CÎma a) 29.96 $‘r. b) 30 ‘r.

10. XmoZ g§»¶m§Mm A§H$J{UVr ‘Ü¶ Ë¶m§À¶m JwUmoÎma ‘Ü¶mnojm 2 Zo OmñV Amho. Am{U Ë¶m§Mm JwUmH$maì¶ñV‘Ü¶ hm ‘moR>çm g§»¶oÀ¶m 1/5 Amho Va Ë¶m XmoZ g§»¶m H$mT>m. CÎma 9 Am{U 1

11. XmoZ g§»¶m§Mm A§H$J{UVr ‘Ü¶ Am{U JwUmoÎma ‘Ü¶ ¶m‘Yrb g§~§Y AM:GM=5:4. Amho. Oa JwUmoÎma ‘Ü¶

Am{U JwUmH$ma ì¶ñV ‘Ü¶ ¶m‘Yrb ’$aH$45- , Amho Va, Ë¶m g§»¶m H$mT>m. CÎma -2 Am{U-8

12 .Oa 'a' hm b Am{U c, Mm A§H$J{UVr ‘Ü¶ Am{U 'b' hm a d c, ‘Yrb JwUmoÎma ‘Ü¶ Amho Va 'c' hm

a Am{Ub ‘Yrb JwUmH$ma ì¶ñV ‘Ü¶ Amho ho {gÜX H$am.

13. H$moÊ¶mË¶mh>r n YZ nyUmªH$mgmR>r n3-n bm 6 Zo ^mJ OmVmo ho {gÜX H$am.

14. H$moUVmhr YZ, g‘ nyUmªH$ hm 4q +1 qH$dm 4q + 3 ñdénmV Amho ho XmIdm, OoWo q hr nyU© g§»¶m Amho.

15. VrZ K§Q>m 4.7 Am{U 17 {‘{ZQ>mZr dmOVmV. Oa gd© VrZ K§Q>m gH$mir 6 dmOVm hmoVAgVrb Va Ë¶m EH${ÌV

nUo Ho$ìhm dmOVrb ? CÎma 6 dmOyZ 28 {‘ZrQ>.

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16. Oa1 19! 10! 11!

nVa n Mr qH$‘V H$mT>m CÎma 121

17 . gai ê$n Úm:( 1)!

( 2)! (n–1)n

n ! CÎma n (n+1)

18. Oa 16 nP3 = 13. n+1P3 Va 'n' H$mT>m. CÎma 15

19. Oa n–1P3 : n+1P3 = 5 : 12 Va 'n' H$mT>m. CÎma 8

20 .Oa 10Pr + 1 : 11Pr = 30 : 11, Va 'r' H$mT>m. CÎma 5

21.n+1Pr+1 = (n + 1) nPr ho {gÜX H$am.

22. Oa nCr-1: nCr: nCr +1 = 3 : 4 : 5, Va r H$mT>m. CÎma 2723.àíZn{ÌHo$‘Ü¶o 5 àíZ AmhoV. Va EImXm ‘wbJm EH$ qH$dm OmñV àíZ {H$Vr àH$mao gmoS>dy

eH$Vmo? CÎma 3124.EH$m {ZdS>UwH$s‘Ü¶o 7 C‘oXdma AmhoV Ë¶mn¡H$s 3 OUm§Mr {ZdS> Ho$br OmD$ eH$Vo. ‘VXmam§Zm 3 nojm OmñV C‘oXdmam§Zm ‘VXmZ H$aVm ¶oUma Zmhr. Va Vo {H$Vr àH$mao ‘VXmZ H$é eH$VmV? CÎma 6325.dVw©imdarb 20 q~XÿVyZ {H$Vr Ordm H$mT>Vm ¶oVmV? CÎma 19026.3 nmoï> noQ>çm (nÌnoQ>çm) ‘Ü¶o 5 nÌo Q>mH$m¶Mr AmhoV Va {H$Vr àH$mao Q>mH$Vm ¶oVmV, Oa gd©{VÝhr nÌ noQ>çm‘Ü¶o {H$Vrhr àH$mao nÌo Q>mH$br OmD$ eH$VmV? CÎma 24327.26 B§J«Or ‘wimjao Agbobo H$mS>© (nwÝhm Z KoVm) ì¶dpñWV {ngyZ EH$m nmH$sQ>‘Ü¶o R>odbr Oa H$mS>© ghOnUo{ZdS>bo Va Imbrb Ajao ¶oÊ¶mMr g§ mì¶Vm {H$Vr?

(i) ñda (ii) ì¶§OZ

(iii)p Z§Va ¶oUmao Aja (iv) m nydu ¶oUmao Aja

(iv)g Am{U 't' ‘Ü¶o ¶oUmar Ajao

CÎmao (i) (ii) iii) (iv) (iv)

28. aº$XmZ Ho$boë¶m 60 {dÚmÏ¶mªMo aº$ JQ> Imbrb H$moï>H$mV XmI{dbo AmhoV.

aº$ JQ> A B AB O

{dÚmÏ¶mªMr g§»¶m 16 12 9 23

Oa ghOnUo EH$m {dÚmÏ¶m©Mr {ZdS> Ho$br. Imbrb aº$JQ> ¶oÊ¶mMr g§ mì¶Vm {H$Vr?

(i) A (ii) B (iii) AB (iv) O

(v) B Zmhr (vi) AB Zmhr (vii) A qH$dm AB (viii) B qH$dm O

CÎmao (i) (ii) (iii) (iv) ( v) (vi) (vii) (viii)¥

526

2126

1226

10 26

1226

356 0

166 0

126 0

96 0

236 0

486 0

516 0

256 0

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29. A Am{U B ho XmoZ XmoñV AmhoV. Oa XmoKm§Mmhr dmT>{Xdg ({bn df© Z KoVm)

(i) EH$mM {Xder (ii) doJdoJù¶m {Xder AgÊ¶mMr g§ mì¶Vm {H$Vr?

CÎma (i) (ii)

30. EH$m noQ>rV 12 M|Sy> AmhoV ¶mn¡H$s ‘x’ ho bmb M|Sy> AmhoV. Va

(i) Oa EH$ M|Sy> H$mT>bm Va bmb M|Sy> ¶oÊ¶mMr g§ mì¶Vm {H$Vr?

(ii) Oa noQ>r‘Ü¶o AmUIr ghm M|Sy> KmVbo Va bmb M|Sy> CMbÊ¶mMr g§ mì¶Vm nyduÀ¶m XþßnQ> AmhoV.Va ‘x’ H$mT>m. CÎma (i) (ii) x = 3

31.Ioim‘Ü¶o XmoZ ’$mgo AmhoV. Oa ~oarO 2, 3, 4, 5, 10, 11 qH$dm 12 Agob Va Vmo IoimSy> A qOH$Vmo. OaVo 6, 7, 8 qH$dm 9 Agob Va IoimSy> B qOH$Vmo. Va Vwåhr A qH$dm B ho IoimSy> AmhmV? dU©Z H$am.

CÎma IoimSy> B

32. VrZ añVo (‘mJ©) EH$m {R>H$mUr {‘iVmV. EH$m añË¶m§Mo ~m§YH$m‘ Mmby Amho. añË¶m§Mr {ZdS> ghOnUoHo$br. ~m§YH$m‘ Mmby Agboë¶m añË¶m§Mr Am{U ~m§YH$m‘ Mmby Zgboë¶m añË¶m§Mr {ZdS> H$aÊ¶mMrg§ mì¶Vm hr añË¶m§Mo ~m§YH$m‘ Mmby Agboë¶m qH$dm añË¶mMo ~m§YH$m‘ Mmby Zgboë¶mer g‘mZ AgVoho {gÕ H$am.

33 .XmoZ ’$mgo EH${ÌVnUo ’o$H$bo n¥ð>^mJmdarb g§»¶m§À¶m JwUmH$mambm 5 qH$dm 7 Zo ^mJ OmVmo Va Ë¶m§Mrg§ mì¶Vm H$mT>m. CÎma

34. ghO KS>Umè¶m à¶moJmMm {ZH$mbmMr CËnÎmr ¶eñdr qH$dm A¶eñdr ¶mn¡H$s EH$ Amho. Oa ¶emMrg§ mì¶Vm hr An¶emÀ¶m g§ mì¶VoÀ¶m {VßnQ> Amho. Va ¶emMr g§ mì¶Vm {H$Vr? CÎma

35. ~wÕr~i nwð>m(~moS>©)Vrb VrZ Mm¡ag ghOnUo {ZdS>bo. Va XmoZ Mm¡ag EH$m a§JmMo Am{U EH$ Mm¡ag doJù¶ma§JmMm ¶oÊ¶mMr g§ mì¶Vm {H$Vr? CÎma

36. W§S> hdoÀ¶m (hill station) {R>H$mUr EH$m AmR>dS>çm‘Ü¶o Zm|X Ho$bobo H$‘rVH$‘r({ZƒÎm‘) Vmn‘mZ

Imbrb à‘mUo {Xbobo Amho.

gmo‘dma ‘§Jidma ~wYdma Jwê$dma ewH«$dma e{Zdma a{ddma4.1 3.2 2.1 1.8 1.6 2.2 1.4

ì¶Ë¶mg Am{U à‘m{UV {dMbZ H$mT>m Am{U {ZH$mbmMo ñnï>rH$aU H$am. CÎma v =0.805, =0.897

37. Imbrb dma§dmaVm {dVaU gmaUrdê$Z (i) A§H$J{UVr ‘Ü¶ (ii) à‘m{UV {dMbZ H$mT>m.

C.I 20-25 25-30 30-35 35-40 40-45 45-50f 8 3 15 12 8 4

CÎma ‘Ü¶ = 34.6, =7.286

364365

x12

1365

34

1136

1621

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38. Imbrb {ÌÁ¶m§Va I§S>mboImMm Aä¶mg H$am Am{U Ë¶mImbrb àíZm§Mr CÎmao Úm.

‘mZdr earamVrb KQ>H$m§Mr (g¶y§JmMr) Q>¸$odmar

(i)EHy$U ‘mZdr earamÀ¶m dOZmer g‘mZ AgUmè¶m ‘mZdr

earamÀ¶m ËdMoVrb à{WZm§Mr Q>¸o$dmar {H$Vr?

(ii) 50 {H$.J«°. dOZmÀ¶m EH$m ì¶º$sÀ¶m earamVrb nmÊ¶mMo à‘mU {H$Vr?

(iii)ñZm¶y‘Yrb à{WZmÀ¶m {d^mJUrMo hmS>m§‘Yrb à{WZm§À¶m {d^mJUrer JwUmoÎma {H$Vr?

(iv) ‘mZdr earamVrb H$moUVm ^mJ hmS>o Am{U ËdMm ¶mnmgyZ ~Zbobm Zmhr.

(v)‘mZdr earam‘Ü¶o à{WZm§Mr {d^mJUr Am{U BVa dmibobo KQ>H$ Xe©{dÊ¶mMm H|$ÐñW H$moZ {H$Vr?

CÎma i) 1.6 % (ii) 35 {H$.J«° (iii) 2:1 (iv) 415 (v) 1080

39. nwñVH$m§À¶m àXe©Zm‘Ü¶o ¶oUmè¶m IMm©À¶m {dVaUmMr Q>¸$odmar Xe©{dUmam {ÌÁ¶m§Va I§S>mboI ~mOybm

{Xbobm Amho. {ÌÁ¶m§Va I§S>mboImMm Aä¶mg H$am Am{U àíZm§Mr

CÎmao Úm.

(i) Oa EHy$U ¶oUmam IM© <3,60,000 Amho. Va àË¶oH$ dñVybm

¶oUmam IM© {H$Vr?

(ii) Oa H$m§hr nwñVH$mÀ¶m ‘wÐUmb¶mMm IM© < 30,600 àH$meH$mZo

{Xbm nm{hOo Va Ë¶m nwñVH$m§Zm {Xbobo ‘mZYZ {H$Vr?

(iii) nwñVH$m§Mo ‘mZYZ ho H$mJXmÀ¶m Xamnojm {H$Vr Q> o$ H$‘r Amho.

(iv) nwñVH$ ~m§YUrMm IM© hm XiUdiU (dmhVyH$) IMm©À¶m {H$Vr

Q> o$ OmñV Amho.

(v) Oa dmhVyH$ IM© <36,600 Amho Va nwñVH$ ~m§YUr IM© {H$Vr?

(vi) Oa ‘wÐUmb¶mbm ¶oUmam IM© <52,600 Amho Va nwñVH$ ~m§YUr IM© hm ‘wÐUmb¶ IM©mnojm {H$VrZo H$‘r

Amho.

Binding20%

Transport10%

Paper cost25%

Royalty15%

Printingcost

20%

Adverti-

semen

t10% ‘wÐU IM©

Om[hamVr

‘mZYZ

OmoS>H$m‘ H$mJXmMm IM©

XiU diU

Water70%

Proteins

Other dryelements 14%

16%Bones

Skin

MusclesHormonesandEnzymes

110

25

61

31

{dH$ao Am{Ug§àoaHo$

ñZm¶y

ËdMmhmS>o

nmUrà{WZo

BVa H$moaS>oKQ>H$

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(vii) Oa <5500 nwñVH$m§Mo àH$meZ H$aÊ¶mV Ambo Am{U Ë¶mgmR>r dmhVyH$ IM© < 82,500 Pmbm Va

àH$meH$mbm 25% Z’$m hmoÊ¶mgmR>r nwñVH$m§Mr {dH«$sMr qH$‘V {H$Vr?

(viii) nwñVH$mMr qH$‘V IaoXr qH$‘VrÀ¶m 20% Zo dmT>{dbr. Oa nwñVH$mMr N>m{nb qH$‘V < 180 Amho Va

EH$m nwñVH$mgmR>r dmnaboë¶m H$mJXmMr qH$‘V {H$Vr?

CÎma o XiUdiU = <36,000 ; H$mJX IM© = < 90,000; OmoS>H$m‘ = < 72,000;‘mZYZ = <54,000; Om{hamV = < 36,000, N>nmh© = < 72,000, (ii) ‘mZYZ = <22, 950(iii) 10% (iv) 10% (v) OmoS>H$m‘ = < 73,200 (vi) 0 (vii) {d.{H$ = <187.50 (viii) <37.5

40. Oa y =+ - -+ + -

x a x ax a x a

Va1 2

+ =x

yy a Ago XmIdm.

41.eof à‘o¶ dmnê$Z ~mH$s H$mT>m. p(x) = 3x4 + 2x3 –2

3x

–9x

+227

g(x) = x +23

CÎma r (x) = 0

42. Oa p(x) = x4 – 2x3 + 3x2 – ax + b hr Aer ~hþnXr Amho H$s {Vbm (x + 1) Am{U (x – 1), Zo ^mJboAgVm ~mH$s AZwH«$‘o 19 Am{U 5 amhVo. p(x) bm (x–2) Zo ^mJbo AgVm ~mH$s H$mT>m.

CÎma 10

43. Oa , (x + 2) hm (4x4 + 2x3 – 3x2 + 8x + 5a) Mm A d¶d Agob Va a Mr qH$‘V H$mT>m. CÎma a = - 4

44. (x4 + ax3 + 2x2 – 3x + b) bm (x2 – 1) Zo ^mJ OmVmo Va a Am{U b Mr qH$‘V H$mT>m. CÎma a = 3,b=- 3

45. x3 – 2x2 –12x – 9 ‘Ü¶o H$m¶ {‘i{dbo AgVm x2 + x – 6 Zo nyU©nUo ^mJ OmB©b ?CÎma 3x +27

46. 2x3 + 3x2 – 22x + 12 ‘Ü¶o H$m¶ {‘i{dbo AgVm n[aUm‘r ({‘iUmè¶m) ~hþnXrbm (2x2 + 5x – 14)

Zo nyU©nUo ^mJ OmB©b ? CÎma 3x + 247. x3 – 6x2 – 5x + 80 ‘YyZ H$m¶ dOm Ho$bo AgVm n[aUm‘r ({‘iUmè¶m) ~hþnXrbm x2 + x – 12 Zo nyU©nUo^mJ OmB©b? CÎma 14 x - 448. Oax + a hm Imbrb àË¶oH$s XmoÝhr ~hþnXtMm Ad¶d Amho Va a Mr qH$‘V H$mT>m.

(i) x3 + ax2 – 2x + a + 4 (ii) x4 – a2x2 + 3x – a CÎma (i) a= - 4 3 (ii) a=0

49. Imbrb dJ© g‘rH$aUo nyU© dJ© H$ê$Z gmoS>dm.

(i)3x (x – 5) = 2x(x + 7) (ii)5 7

1xx

= 3x + 2

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(iii) a2x2 – 3abx + 2b2 = 0 (iv) 4x2 + 4bx – (a2 – b2) = 0

CÎmao (i) 29, 0 (ii) 3, –1 (iii)2b ba a

, (iv)_b a

2

50. gyÌmMm Cn¶moJ H$ê$Z Imbrb dJ© g‘rH$aUo gmoS>dm.

i.x2 – 7x + 12 = 0 ii.p = 5 – 2p2 iii. 4x2 – 4ax + (a2 – b2) = 0

iv. 2 9 13x x v . 36x2 – 12ax + (a2 – b2) = 0 vi.1 1 1 0

2 3 4x x x

vii.3 2 8

5 4 2b b b

CÎma o i) 4, 3 ii)41-1

4 iii)

b

2

aiv) 20, 8 v)

b

6

a vi)

39

3vii )

58

13, 2

51. ‘moR>çm Mm¡agmÀ¶m joÌ’$imVyZ bhmZ Mm¡agmÀ¶m joÌ’$imMr XþßnQ> dOm Ho$ë¶mg CÎma 14 Mm¡.g|.‘r. ¶oVo.VWm{n Oa ‘moR>çm Mm¡agmÀ¶m joÌ’$imÀ¶m XþßnQ>rV bhmZ Mm¡agmÀ¶m joÌ’$imMr {VßnQ> {‘i{dë¶mg CÎma203 Mm¡.g|.‘r. ¶oVo. XmoÝhr Mm¡agm§À¶m ~mOy AmoiIm. CÎma 5 g|.‘r., 8 g|.‘r.

52. . EH$ H$U O{‘Z nmVirnmgyZ àjo{nV Ho$bobm Agm Amho H$s O{‘ZrnmgyZ Ë¶mMr C§Mr t goH§$XmZ§Va

(20t – 5t2) ‘r. Agob. {H$Vr goH§$XmZ§Va O{‘ZrnmgyZ 15 ‘r. C§Mrda H$U Agob? VoWo XmoZ CÎmam§Mre³¶Vm Amho? ¶m~Ôb Vwåhr WmoS>³¶mV dU©Z H$am. CÎma 3 qH$dm 1

53. C1 Am{U C2 dVw©io EH$‘oH$m§Zm A ‘Ü¶o Am§Vañne© H$aVmV Am{U C1 dVw©imMr AB Ordm C2

bm P ‘Ü¶o N>oXVo. Va {gÕ H$am H$s AP = PB.

54 . ABC À¶m BC ~mOybm EH$ dVw©i P ‘Ü¶o ñne© H$aVo. AB Am{U AC dmT>{dë¶m AgVm Ë¶mdVw©imbm AZwH«$‘o Q Am{U S ‘Ü¶o ñne© H$aVmV. {gÕ H$am H$s AQ=1/2 ( ABC Mr n[a{‘Vr)

55 .Imbrb àË¶oH$ AmH¥$Ë¶m‘Yrb ‘m{hV Zgboë¶m qH$‘Vr H$mT>m. gd© bm§~r g|Q>r‘rQ>a‘Ü¶o {Xë¶m AmhoV. (‘mno

à‘mUmV Zm§hrV)

CÎma (i) x = 4, y = 9 (ii) x = 2.64 , y = 2 0.96 (iii) a = 9, b = 3

OOI

A

P

B

C2

C1

A G

FCB

D E

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y

8 6

24

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M

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3

5

4

6

D

G

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H

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b

b

a

5 7

(ii) (iii)

8

12

62i

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E F

56. "'‘Ü¶ q~XÿÀ¶m à‘o¶mMr"" OmVrà{Vkm gm§Jm.WoëgMm ì¶Ë¶mg à‘o¶ dmnê$Z {gÜX H$am.

57.ABCD Agm Mm¡H$moZ Amho Á¶m ‘Ü¶o W, X, Y Am{U Z AZwH«$‘o AB,

BC, CA Am{U DA ~mOy§Mo {Ì^mOH$ q~Xÿ AmhoV. {gÕ H$am WXYZ

g‘m§Va wO Mm¡H$moZ Amho.

(gyMZm : A, C OmoS>m)

58.ABCD Agm g‘b§~ Mm¡H$moZ Amho H$s Á¶m‘Ü¶o AB || DC H$U© 'O' ‘Ü¶o

N>oXVmV. Ago XmIdm H$sAO COBO OD (‘y.à.à‘o¶ dmnê$Z {gÕ H$am)

(gyMZm: 'O' ‘YyZ EF || AB || CD H$mT>m.)

59. ABCD hm g‘ê$n Mm¡H$moZ Amho. BC da 'P' hm H$moUVmhr

q~Xÿ Amho. AP OmoS>br Am{U Aer dmT>{dbr H$s

DC dmT>dyZ Vr Q ‘Ü¶o {‘iVo.

{gÕ H$am1 1 1

BC PC CQ= -

60. Oa {ÌH$moUmÀ¶m VrZ ~mOy§Mo ‘Ü¶q~Xÿ H«$‘mZo OmoS>bo Va [gÕ H$am H$s V¶ma hmoUmao Mma {ÌH$moU

EH$‘oH$m§er g‘ê$n Am{U ‘yi {ÌH$moUmerhr g‘ê$n AgVmV.

61.{gÕ H$am H$s XmoZ g‘ê$n {ÌH$moUm§À¶m joÌ’$im§Mo JwUmoÎma ho Ë¶m§À¶m g§JV ‘Ü¶Jm§À¶m JwUmoÎmamÀ¶m

dJm©BVHo$ AgVo.

62. A A H$gmoQ>r dmnê$Z q^Jo qH$dm AmaemMo gyÌ aMm.

63.H$mQ>H$moZ {ÌH$moUmÀ¶m ~mOyda g‘ wO {ÌH$moU H$mT>bo AmhoV.

Ago XmIdm H$s H$Um©darb {ÌH$moUmMo joÌ’$i ho BVa XmoZ ~mOydarb

{ÌH$moUm§À¶m joÌ’$im§À¶m ~oaOoBVHo$ AgVo.

64. g‘b§~ Mm¡H$moZmMo joÌ’$i dmnéZ nm¶W°Jmoag à‘o¶ {gÜX H$am.

A B

CD

P

Q

M

D

P

N

F C A

B

A

B

2 234

+x yA

B C

R

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Q

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y

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234

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2

+

A

E

B DC

a

cb

a c

A B

CD

1

1

1

1

2

2

2

x

y

z

w 2

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65. ABC ‘Ü¶o, (i) ACBACB = 60° (ii) AD BC. VaAC Am{U BC. Mo AB gmR>r ì¶º$ H$éZ {dYmZ {‘idm

66. ABC ‘Ü¶o, H$U© hm XmoZ ~mOy§n¡H$s EH$m ~mOynojm 2 EH$H$ Zo ‘moR>m Am{U Xþgè¶m ~mOyÀ¶m XþßnQ>rnojm1 EH$H$ Zo A{YH$ Amho. ~mOyMr ‘mno H$mT>m.

CÎmao AB =15 EH$H$ BC = 8 EH$H$ AC = 17 EH$H

67. H$mQ>H$moZ {ÌH$moUmÀ¶m ~mOy A§H$J{UVr loUr‘Ü¶o AmhoV.H$s {ÌH$moUmÀ¶m ~mOy 3 : 4 : 5.JwUmoÎmamV AmhoVAgo XmIdm .

68. LAB‘Ü¶o, ALB 90Ð = ° Am{U LM AB

{gÜX H$am2

2

LA AMLB AB

=

69. Oa 2 sin2 + 5 cos = 4. Va cos =12 Ago XmIdm

70.sin cos

1cosec sec

ho {gÕ H$am.

71. Oa x = r sin A cos B, y = r sin A sin B, z = r cos A, Va x2+y2+z2=r2. ho {gÜXH$am.

72.XmoZ {ZarjH$ EH$‘oH$mnmgyZ 1 {H$‘r Xÿa AgyZ EH$mM b§~àVbm‘Ü¶o ’w$Jm åhUyZ AmhoV. na§Vw Ë¶mÀ¶m

{dê$Õ ~mOyg 60° Am{U 45° Ago Cƒ nmVirVrb H$moZ {XgyZ ¶oVmV. ’w$Jm O{‘ZrnmgyZ {H$Vr

C§Mrda Amho? CÎmao3

1 3{H$bmo ‘rQ>a

73.XrnJ¥hmnmgyZ XrnJ¥hmÀ¶m {dê$Õ ~mOyg AgUmè¶m XmoZ OhmOm§Mo {ZÀM nmVirVrb H$moZ 450 Am{U

600 Agë¶mMo {XgyZ ¶oVo. Oa XrnJ¥hmMr C§Mr 120 ‘rQ>a Am{U XmoZ OhmOmZm OmoS>Umar aofm

XrnJ¥hmÀ¶m nm¶mVyZ OmVoo, XmoZ OhmOm‘Yrb A§Va H$mT>m. CÎmao 120

L

MA B

A

B CD

E

60°

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113 ‘rQ> a

74. ‘r. Imob H$mT>boë¶m {dhrarMm AmVrb ì¶mg 10 ‘r. Amho. {dhrarVyZ H$mT>boë¶m

‘mVrMm 5 ‘r. é§XrMm âmd Q>mH$bm. Va Ë¶m âmdmMr C§Mr H$mT>m.

CÎma h = 4.66 ‘r.

75.PmS>mMr ’$m§Xr d¥Îm{MÎmr AmH$mamMr Amho. Ë¶mMm n[aK 176 g|.‘r. Amho. Oa ’$m§XrMr bm§~r 3 ‘r. Amho.Va Ë¶m ’$m§XrnmgyZ {‘i{dboë¶m bmH$S>mMo KZ’$i H$mT>m.

CÎma 7,39,200, K.g |.‘r.76.14 ‘r. ì¶mg Am{U 8 ‘r. Imob {dhra ImoXbr. ¶m‘YyZ H$mT>boë¶m ‘mVrMm g^modVmbr 21 ‘r OmS>rMm

âmd Q>mH$bm Va âmdmMr C§Mr H$mT>m. CÎma 0.53 ‘r.77.EH$m JmdmVrb OÌo‘Ü¶o XþH$mZXmamZo d¥Îm{MÎmr AmH$mamÀ¶m VimMr {ÌÁ¶m 15 g|.‘r. Agboë¶m ^m§S>çm‘Ü¶o

32 g|.‘r. C§Mrn¶ªV g§Í¶mMm ag ^ê$Z R>odbm. Ë¶mZo Vmo 3 g|.‘r. {ÌÁ¶m Am{U 8 g|.‘r. C§M Agboë¶m

d¥Îm{MÎmr AmH$mamÀ¶m ½bmg‘YyZ àË¶oH$s <.3 ¶mà‘mUo {dH$bm. Vmo ag nyU©nUo {dHy$Z Ë¶m XþH$mZXmamZo

{H$Vr ê$n¶o {‘i{dbo? CÎma < 30078. ârd e§Hy$Mm ì¶mg 14 g|.‘r. Am{U b§~ C§Mr 51 g|.‘r. Amho. Á¶m nXmWm©nmgyZ Vo ~Zbobo Amho Ë¶mMo

dOZ 10 J«°‘/K.g|.‘r. Va Ë¶mMo dOZ H$mT>m. CÎma 26.18 {H$.J °.79. b§~ dVw©imH$ma e§Hy$Mr C§Mr 3.6 g|.‘r. Am{U Ë¶mÀ¶m VimMr {ÌÁ¶m 1.6 g|.‘r. Amho. Vmo {dViyZ 1.2

g|.‘r. {ÌÁ¶oMm b§~ dVw©imH$ma e§Hy$ ~Z{dÊ¶mV Ambm. Va Aem [aVrZo V¶ma Pmboë¶m e§Hy$Mr C§Mr H$mT>m. CÎm a 6.4 g |.‘r .

80.e§Hy$ AmH$mamMm M§~y nmÊ¶mZo nyU©nUo âbobm Amho. M§~yÀ¶m VimMr {ÌÁ¶m 'a' Am{U C§Mr '2a' Amho. ho nmUr23a

VimÀ¶m d¥Îm{MÎmr AmH$mamÀ¶m M§MwnmÌm‘Ü¶o AmoVbo. d¥Îm{MÎmr AmH$mamÀ¶m M§MwnmÌmVrb nmÊ¶mMr

C§Mr H$mT>m. CÎma 3 2a

81. YmVyÀ¶m nÍ¶mnmgyZ ~Z{dbobo ^m§S>o darb Q>moH$mda CKS>o AgyZ ’«$ñQ>‘ e§Hy$À¶m AmH$mamMo Amho. Ë¶mMr C§Mr16 g|.‘r. Am{U darb Am{U Imbrb Q>moH$m§À¶m {ÌÁ¶m 20 g|.‘r. d 8 g|.‘r. AmhoV. ^m§S>o XþYmZo nyU©nUoâÊ¶mgmR>r ê$.40 Xa {bQ>aà‘mUo ¶oUmam IM© H$mT>m. 100 Mm¡.‘r.bm ê$.8 à‘mUo YmVy nÍ¶mnmgyZ ^m§S>o

~Z{dÊ¶mgmR>r ¶oUmam IM© H$mT>m. ( = 3.14 ¿¶m) CÎma < 418, < 156.7582. YmVyÀ¶m M|Sy>Mm ì¶mg 3.5 g|.‘r. Amho. Oa YmVyMr KZVm 8.9 g/Mm¡.g|.‘r. Amho. Va M|Sy>Mo dOZ H$mT>m.

(gyMZm … dñVw‘mZ = KZ’$i × KZVm) CÎma 199.8 J°.83.ârd AY©JmobmH$ma ‘oUmMr {ÌÁ¶m 12 g|.‘r. Amho. Vmo {dVidyZ 6 g|.‘r. VimMr {ÌÁ¶m Agbobm e§Hy$

~Z{dbm. Va e§Hy$Mr C§Mr H$mT>m. CÎma 96 g|.‘r.84.YmVyÀ¶m JmobmH$ma e§ImMr AmVrb Am{U ~mhoarb {ÌÁ¶m AZwH«$‘o 4 g|.‘r. d 8 g|.‘r. Amho. Ë¶mbm

{dVidyZ b§~ dVw©imH$ma d¥Îm{MÎmr ~Z{dÊ¶mV Ambr {VMr C§Mr1

93

g|.‘r. Amho. Va d¥Îm{MÎmrÀ¶mVimMm ì¶mg H$mT>m. CÎma 16 g|.‘r.

85. 28 g|.‘r. ì¶mgmMm ârd Jmob {dVi{dbm Am{U ì¶mg243

g|.‘r. d C§Mr 3 g|.‘r. Agbobo AZoH$

5m5m5m

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nwadUr (Appendix)

J{UVr Z‘yZo (Mathematical Modelling)

Amnë¶m {ZË¶ OrdZmVrb KQ>Zm Am{U dmñV{dH$ OJm‘Ü¶o Amnë¶mbm AZoH$ g‘ñ¶m§Zm Vm|S> Úmdo bmJVAmho, Ë¶mda J{UV Vk, d¡km{ZH$, AW©emñÌk, ‘mZgmonMma Vk Am[U BVam§Zr Cnm¶ emoYbo AmhoV.

Imbrb CXmhaUm§Mm Aä¶mg H$am.* nmohMy Z eH$Umè¶m {eIam§Mr C§Mr H$mT>Uo.* g‘wÐ Am{U ZÚm ¶m§Mr Imobr emoYUo.* 10 dfm©Z§Va eha qH$dm amîQ´>m§À¶m bmoH$g§»¶oMr ^{dî¶dmUr H$aUo.* gy¶m©À¶m n¥îR>^mJmdarb Vn‘mZmMm A§XmO H$aUo.* ‘§Ji J«hmda CnJ«h gmoS>Uo.* CnJ«hmÀ¶m ghmæ¶mZo g§nH©$Omb ( ) V¶ma H$aUo.

Aem gd© dmñV{dH$ OJmVrb g‘ñ¶m gmoS>{dÊ¶mgmR>r Or g§H$ënZm dmnabr OmVo {Vbm "J{UVr Z‘yZo 'åhUVmV.J{UVr Z‘yZo åhUOo H$m¶?

J{UVr Z‘yZo ho X¡Z§{XZ OrdZmVrb H$mhr KQ>Zm§Mo J{UVr dU©Z H$aVo. J{UVr Z‘yZo V¶ma H$aÊ¶mMr hr à{H«$¶mAmho Am{U {VMm Cn¶moJ H$éZ g‘ñ¶m§Mo n¥W:H$aU H$éZ Ë¶m gmoS>rdë¶m OmVmV.

J{UVr Z‘yZo qH$dm à{VH¥$Vr hr J{UVr g§H$ënZm Am{U ^mfm dmnéZ dU©Z H$aÊ¶mMr nÜXV Amho.J{UVr Z‘yÝ¶m§À¶m {dH$mgmÀ¶m à{H«$¶obm J{UVr Z‘yZo åhQ>bo OmVo. J{UVm‘Ü¶o J{UVr Z‘yÝ¶m§Mm dmna ¶mgmR>r

Ho$bm OmVmo.* dmñV{dH$ OJmVrb JwUY‘mªMo dU©Z H$aÊ¶mgmR>r,* dmñV{dH$ OJmVrb KQ>Zm§{df¶r ‘hËdmMo àûZ VnmgÊ¶mgmR>r,* J{UVr ñdénmVrb dmñV{dH$ OJmVrb JwUY‘mªÀ¶m ñnîQ>rH$aUmgmR>r,* J{UVr nÜXVrZo ¶w³Ë¶m§Mr MmMUr H$aÊ¶mgmR>r,* dmñV{dH$ OJmVrb ^{dî¶dmUr H$aÊ¶mgmR>r,

dmñV{dH$ OJmVrb Xe©{dbobo J{UVr Z‘yZo {Xboë¶m àdmhrV V³Ë¶m‘Yrb joÌm§er g§~§{YV AmhoV.

J{UVr Z‘yZo ho da gm§{JVboë¶m gd© joÌ, Ë¶m§Mo CnjoÌ Am{U kmZmÀ¶m gd© emIm‘Ü¶o dmnaVmV.

dmñV{dH$ OJmMm g§X^©

ewÜX{dkmZ

g‘mO{dkmZ

eoVr ì¶dgm¶ Ioig§~§{YV{dkmZ

^m¡{VH$emñÌ agm¶ZemñÌ OrdemñÌ g‘mOemñÌ AW©emñÌ n[ag§ñWm

Am¡fYA{^¶m§{ÌH$s hdm‘mZemñÌO¡{dH$ agm¶ZemñÌ

OÝ¶y A{^¶m§{ÌH$s O¡{dH$ V§ÌkmZ

gyú‘OrdemñÌ

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Imbr {Xboë¶m AmH¥$Vr‘Ü¶o J{UVr Z‘wÝ¶m§À¶m {H«$¶m Xe©{dboë¶m AmhoV.

J{UVr Z‘wÝ¶m§Mr à{H«$¶m hr Imbrb n[aUm‘H$maH$ Am{U H«$‘dma nm¶è¶m‘Ü¶o Ho$br OmVo.J{UVr Z‘wÝ¶m§À¶m nm¶è¶m :nm¶ar 1 : g‘ñ¶m§Mo AmH$bZ

¶m nm¶ar‘Ü¶o dmñV{dH$ OJmVrb g‘ñ¶m àË¶jnUo g‘OyZ KoD$Z H$mht KQ>H$ Am{U J¥hrVH$mÛmao Ë¶m§Mr ì¶m»¶mAm{U ñnîQ>rH$aU H$aUo.CXmhaUmW© : A^¶maÊ¶mVrb dZñnVtÀ¶m g§»¶oMm A§XmO H$aUo, Oo àË¶jmV ‘moOUo e³¶ Zmhr.gm‘mÝ¶nUo O§JbmÀ¶m joÌmMm Z‘wZm {dMmamV KoVbm Va Am{U O§JbmVrb EHy$U PmS>m§À¶m g§»¶oMm A§XmO Ho$bm OmVmo.nm¶ar 2 : J{UVr Z‘wÝ¶mMo gyÌrH$aU

J{UVr nXo, KQ>H$ Am{U J¥hrVm§Mr {ZdS> H$éZ dmñV{dH$ OJmVrb g‘ñ¶m§er g§~§Y àñWm{nV H$éZ Xe©{dUo.J{UVr nXo Am{U ^mfo‘Ü¶o J{UVr Z‘wZo {dH${gV H$aUo Am{U Ë¶m§Mo dU©Z H$aUo. J{UVr Z‘wÝ¶m§Mo dU©Z H$aÊ¶m§Mo H$mhràH$ma :* Mbm§Mr ì¶m»¶m H$aUo.* gm‘wJ«r g§J«{hV H$éZ H$moîQH$mV ‘m§S>Uo.* g‘rH$aUo Am{U gyÌo aMUo.* AmboI H$mT>Uo.CXmhaUmW© : O§JbmVrb EHy$U PmS>m§À¶m g§»¶oMm A§XmO H$aÊ¶mgmR>r J{UVr Z‘wZo {dH${gV H$aUo.O§JbmMo joÌ’$i 100 Mm¡. {H$. ‘r. Amho Ago g‘Oy. Vo VrZ àH$maÀ¶m joÌmV {d^mJbo OmVo.A KZXmQ> PmS>m§Zr ì¶mnbbo joÌB >{dai PmS>m§Zr ì¶mnbbo joÌC KZXmQ> qH$dm {dai ¶mn¡H$s H$moUË¶mhr PmS>mZr Z ì¶mnbobo joÌ

g§nyU© O§Jb àË¶oH$r AÜ¶m© {H$.‘r. à‘mUo 200 ^mJm‘Ü¶o {d^mJbo Ago ‘mZy. ho à‘m{UV AmamIS¶m‘Ü¶o ZH$memXe©dyZ H$aVm ¶oVo. 200 ^mJmn¡H$s àË¶oH$ ^mJMr g§µ»¶m emoYbr OmVo Am{U A:B:C=3:1:2 ho JwUmoÎma {‘iVo.

Ago g‘Om H$s 12 ^mJ ¶mÑpÀN>H$ à¶moJmZo {ZdS>bo, Á¶m‘Ü¶o A àH$maMo 6, B àH$maMo 2 Am{U C àH$maMo 4AmhoV. ¶m 12 ^mJmVrb Pm§S>mZm g§»¶m XodyZ ‘moObo OmVo. hr gm‘J«r n¥W:H$aUmgmR>r KoVbr OmVo.nm¶ar 3: J{UVr g‘ñ¶m§Mo {ZdmaU

doJdoJir J{UVr V§Ìo dmnê$Z nm¶ar 2 ‘Yrb J{UVr Z‘wÝ¶mZm gaiê$n XodyZ gmoS>{dUo.CXmhaUmW©, {ZdS>boë¶m àË¶oH$ 12 ^mJmVrb PmS>m§Mr g§»¶m ‘moOyZ Pmë¶mZ§Va PmS>m§À¶m g§»¶oMr gamgar H$mT>bo OmVo.

g‘Om H$s. 12 ^mJmVrb PmS>m§Mr gamgar g§»¶m 3556 Amho.Vo§ìhm, EH$m ^mJmVrb PmS>m§Mr gamgar g§»¶m 3556/12= 296 Amho.

nm¶ar 4: Cnm¶ qH$dm CH$bmMo ñnîQ>rH$aU:nm¶ar 3 ‘Ü¶o {‘i{dboë¶m CH$bm§Mo AmVm dmñV{dH$ OJmVrb KQ>Zm qH$dm g‘ñ¶m§Mo ñnîQ>rH$aU Ho$bo OmVo.

åhUOoM, dU©Z [Xbr OmVmV Am{U g‘ñ¶m§À¶m {ZdmaUm{df¶r ^{dî¶dmUr H$obr OmVo.CXmhaUmW©, nm¶ar 1 ‘Yrb g‘ñ¶m§Mo nwZ:n[ajU H$aUo Am{U CH$b dmnaUo. åhUOoM 100 Mm¡. {H$.‘r. ‘Yrb O§JbmVrbEHy$U PmS>m§Mr g§»¶m hr A§XmOo 59,200 Amho.

dmñV{dH$ OJmVrb g‘ñ¶m§Mo {ZdmaU J{UVr Cnm¶

gyÌrH$aU J{UVr g‘ñ¶mdmñV{dH$ OJmVrb g‘ñ¶m

ñnîQ>rH$aUMmM

Ur

n¥W:H

$aU

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nm¶ar 5 : Z‘wÝ¶m§Mo à‘mUrH$aUnm¶ar 3 ‘Ü¶o {‘i{dbobo {ZH$mb Am{U nm¶ar 4 ‘Yrb ñnîQ>rH$aUmMr VnmgUr H$ê$Z Ë¶mbm AW© bmJVmo H$m Vo

nmhVmV. Omo n¶ªV J¥hrVo ~XbVmV qH$dm ZdrZ ‘m{hVr {‘iVo qH$dm Z‘wZo V¶ma hmoVmV Vmo n¶ªV Z‘wZo dmnaVmV.n¶m©¶r Z‘wZm :

g‘Om H$s O§JbmMo hdoVyZ N>m¶m{MÌ KoVbo Am{U {MÌm‘Yrb àË¶oH$ PmS> EH$m {Q>§~mUo Xe©{dbo.PmS>m§Mr g§»¶m {‘i{dÊ¶mgmR>r Z‘yZm {dH${gV H$am.J{UVr Z‘yZo {‘i{dÊ¶mgmR>r ì¶dhmar J{UVm‘YyZ AmUIr EH$m CXmhaUmMm {dMma H$ê$.

CXmhaU. amOoebm gm¶H$b IaoXr H$am¶Mr Amho.Vmo XwH$mZmV OmVmo Am{U < 1800 qH$‘VrMr gm¶H$b {ZdS>Vmo Ë¶mÀ¶mOdi< 600 AmhoV. XþH$mZXmamZo Ë¶mÀ¶m nwT>o hßVo~§XrZo IaoXr H$aÊ¶mMr ¶moOZm Imbrbà‘mUo {Xbr.

amOoe < 600 hr a³H$‘ IaoXrÀ¶mdoir XoVmo Am{U am{hbobr a³H$‘ àË¶oH$s <610 Mo XmoZ ‘m{gH$ hßVo XodyZ IaoXrH$ê$ eH$Vmo. amOoeH$S>o XmoZ àñVmd AmhoV, EH$ Va Vmo hßVo~§Xr ¶moOZonm’©$V IaoXr qH$dm 10 % dm{f©H$ gai ì¶O XamZo~±Ho$H$Sy>Z H$O© KodyZ g§nyU© a³H$‘ ^ê$ eH$Vm. H$moUVm n¶m©¶ A{W©H$ ÑîQ¶m Ë¶mÀ¶mgmR>r Cn¶w³V Amho ?CH$bnm¶ar 1: (g‘ñ¶m g‘OyZ KoUo)

amOoeH$S>o XmoZ àñVmd AmhoV. Ë¶mbm XþH$mZXmamZo {Xbobm àñVmd pñdH$mamdm H$s Zmhr ¶mMm {ZíM¶ H$aÊ¶mMr JaOAmho. ¶mH$[aVm Ë¶mbm XmoÝhr ì¶mOmMo Xa g‘OyZ KoVbo nm{hOoV, EH$ hßVo~§Xr ¶moOZo‘Ü¶o XþH$mZXmamZo ~Xbbobo Am{U Xþgao~±H$oZo {Xbobo Xa. XmoÝhr§Mr VwbZm H$ê$Z Am{W©H$ÑîQ¶m ’$m¶ÚmMr {ZpíMVr H$aUo.nm¶ar 2: (J{UVr Z‘wÝ¶m§Mo gyÌrH$aU)

nmhm H$s hmßVo~§Xr ¶moOZo‘Ü¶o g§nyU© a³H$‘ EH$ dfm©V âmdr bmJob Am{U Ë¶m‘wio Ë¶mbm gai ì¶mO AmH$maboOmB©b.

gm¶H$bMr IaoXrMr qH$‘V = < 1800

hßVo~§Xr ¶moOZo‘Ü¶o gwê$dmVrbm âmdr bmJUmar a³H$‘ = <600hßVo~§Xr ¶moOZoV âmdr bmJUmar am{hbobr a³H$‘ =<(1800-600) = < 1200XþH$mZXmamZo AmH$maboë¶m ì¶mOmMm Xa r % X.gm.X.eo. Amho Ago g‘Oy.àË¶oH$ hßË¶mMr a³H$‘ = <610hßVo~§Xr‘Ü¶o âbobr EHy$U a³H$‘ =<610+ <610 = <1200

hßVo~§Xr ¶moOZo‘ Ü¶o ^ ab ob o ì ¶mO =<1220 - <1200 = <20............(i)

o n{hë¶m ‘{hÝ¶mMr ‘wÔb = <1200Xþgè¶m ‘{hÝ¶mMr ‘wÔb =<‹(1200-610) =< 590

Xþgam hßVm = Xþgar ‘wÔb +AmH$mabobo ì¶mO= < 590 + <20 = < 610

EH$m ‘{hÝ¶mMr EHy$U ‘wÔb < 1200 + <590 = <1790

Amåhmbm ‘m{hV Amho H$s, ì¶mO= P T R100

r1790100

=I ...............(ii)

nm¶ar 3: (g‘ñ¶m gmoS>{dUo)(i) Am{U (ii) dê$Z Amåhmbm {‘iob.

r1790100 = 20

20 12001790

= = 13.14 %r

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nm¶ar 4: (CH$bm§Mo ñnîQ>rH$aU>)hßVo~§Xr ¶moOZo‘Ü¶o AmH$maboë¶m ì¶mOmMm Xa =13.14 %~±Ho$Zo AmH$maboë¶m ì¶mOmMm Xa =10 %åhUyZ amOoeZo gm¶H$b IaoXr H$aÊ¶mgmR>r ~±Ho$H$Sy>Z ì¶mOmZo n¡go KoVbo nm{hOo. Oo Am{W©H$ ÑîQ¶m ’$m¶ÚmMo Amho. nm¶ar 5: (Z‘wÝ¶m§Mo à‘m{UH$aU>)

¶m CXmhaUm‘Ü¶o, Z‘wÝ¶mÀ¶m à‘mUrH$aUmbm {VVHo$go ‘hËd {Xbo OmV Zmhr H$maU {H$§‘Vr pñWa ({ZpíMV) AmhoV.VarnU, H$O© KÊ¶mMr à{H«$¶m doJir Amh qH$dm ~°±H$ MH«$$dmT> ì¶mO AmH$mê$ eH$Vo. Ë¶m‘wio ì¶mO§mÀ¶m XamMr

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