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9MATHEMATICS

amîQ´>r` e¡j{UH$ g§emoYZ Am{U à{ejU n[afX²NATIONAL COUNCIL OF EDUCATIONAL RESEARCH AND TRAINING

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ForewordThe National Curriculum Framework (NCF), 2005, recommends that children’s life at school must be linked to their life outside the school. This principle marks a departure from the legacy of bookish learning which continues to shape our system and causes a gap between the school, home and community. The syllabi and textbooks developed on the basis of NCF signify an attempt to implement this basic idea. They also attempt to discourage rote learning and the maintenance of sharp boundaries between different subject areas. We hope these measures will take us significantly further in the direction of a child-centred system of education outlined in the national Policy on Education (1986).

The success of this effort depends on the steps that school principals and teachers will take to encourage children to reflect on their own learning and to pursue imaginative activities and questions. We must recognize that, given space, time and freedom, children generate new knowledge by engaging with the information passed on to them by adults. Treating the prescribed textbook as the sole basis of examination is one of the key reasons why other resources and sites of learning are ignored. Inculcating creativity and initiative is possible if we perceive and treat children as participants in learning, not as receivers of a fixed body of knowledge.

This aims imply considerable change is school routines and mode of functioning. Flexibility in the daily time-table is as necessary as rigour in implementing the annual calendar so that the required number of teaching days are actually devoted to teaching. The methods used for teaching and evaluation will also determine how effective this textbook proves for making children’s life at school a happy experience, rather then a source of stress or boredom. Syllabus designers have tried to address the problem of curricular burden by restructuring and reorienting knowledge at different stages with greater consideration for child psychology and the time available for teaching. The textbook attempts to enhance this endeavour by giving higher priority and space to opportunities for contemplation and wondering, discussion in small groups, and activities requiring hands-on experience.

The National Council of Educational Research and Training (NCERT) appreciates the hard work done by the textbook development committee responsible for this book. We wish to thank the Chairperson of the advisory group in science and mathematics, Professor J.V. Narlikar and the Chief Advisor for this book, Professor P. Sinclair of IGNOU, New Delhi for guiding the work of this committee. Several teachers contributed to the development of this textbook; we are grateful to their principals for making this possible. We are indebted to the institutions and organizations which have generously permitted us to draw upon their resources, material and personnel. We are especially grateful to the members of the National Monitoring Committee, appointed by the Department of Secondary and Higher Education, Ministry of Human Resource Development under the Chairpersonship of Professor Mrinal Miri and Professor G.P. Deshpande, for their valuable time and contribution. As an organisation committed to systemic reform and continuous improvement in the quality of its products, NCERT welcomes comments and suggestions which will enable us to undertake further revision and refinement.

DirectorNew Delhi National Council of Educational20 December 2005 Research and Training

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TexTbook DevelopmenT CommiTTee

Chairperson, aDvisory Group in sCienCe anD maThemaTiCs

J.V. Narlikar, Emeritus Professor, Chairman, Advisory Committee, Inter University Centre for Astronomy & Astrophysics (IUCAA), Ganeshkhind, Pune University, Pune

Chief aDvisor

P. Sinclair, Director, NCERT and Professor of Mathematics , IGNOU, New Delhi

Chief CoorDinaTor

Hukum Singh, Professor (Retd.), DESM, NCERT

members

A.K. Wazalwar, Professor and Head, DESM, NCERTAnjali Lal, PGT, DAV Public School, Sector-14, GurgaonAnju Nirula, PGT, DAV Public School, Pushpanjali Enclave, Pitampura, DelhiG.P. Dikshit, Professor (Retd.), Department of Mathematics & Astronomy, Lucknow University, LucknowK.A.S.S.V. Kameswara Rao, Associate Professor, Regional Institute of Education, BhubaneswarMahendra R. Gajare, TGT, Atul Vidyalya, Atul, Dist. ValsadMahendra Shanker, Lecturer (S.G.) (Retd.), NCERTRama Balaji, TGT, K.V., MEG & Centre, ST. John’s Road, BangaloreSanjay Mudgal, Lecturer, CIET, NCERTShashidhar Jagadeeshan, Teacher and Member, Governing Council, Centre for Learning, BangaloreS. Venkataraman, Lecturer, School of Sciences, IGNOU, New DelhiUaday Singh, Lecturer, DESM, NCERTVed Dudeja, Vice-Principal (Retd.), Govt. Girls Sec. School, Sainik Vihar, Delhi

member-CoorDinaTor

Ram Avtar, Professor (Retd.), DESM, NCERT (till December 2005)R.P. Maurya, Professor, DESM, NCERT (Since January 2006)

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aCknowleDGemenTs

The Council gratefully acknowledges the valuable contributions of the following participants of the Textbook Review Workshop: A.K. Saxena, Professor (Retd.), Lucknow University, Lucknow; Sunil Bajaj, HOD, SCERT, Gurgaon; K.L. Arya, Professor (Retd.), DESM, NCERT; Vandita Kalra, Lecturer, Sarvodaya Kanya Vidyalya, Vikas Puri, District Centre, New Delhi; Jagdish Singh, PGT, Sainik School, Kapurthala; P.K. Bagga, TGT, S.B.V. Subhash Nagar, New Delhi; R.C. Mahana, TGT, Kendriya Vidyalya, Sambalpur; D.R. Khandave, TGT, JNV, Dudhnoi, Goalpara; S.S. Chattopadhyay, Assistant Master, Bidhan Nagar Government High School, Kolkata; V.A. Sujatha, TGT, K.V. Vasco No. 1, Goa; Akila Sahadevan, TGT, K.V., Meenambakkam, Chennai; S.C. Rauto, TGT, Central School for Tibetans, Mussoorie; Sunil P. Xavier, TGT, JNV, Neriyamangalam, Ernakulam; Amit Bajaj, TGT, CRPF Public School, Rohini, Delhi; R.K. Pande, TGT, D.M. School, RIE, Bhopal; V. Madhavi, TGT, Sanskriti School, Chanakyapuri, New Delhi; G. Sri Hari Babu, TGT, JNV, Sirpur Kagaznagar, Adilabad; and R.K. Mishra, TGT, A.E.C. School, Narora.

Special thanks are due to M. Chandra, Professor and Head (Retd.), DESM, NCERT for her support during the development of this book.

The Council acknowledges the efforts of Computer Incharge, Deepak Kapoor; D.T.P. Operator, Naresh Kumar; Copy Editor, Pragati Bhardwaj; and Proof Reader, Yogita Sharma.

Contribution of APC–Office, administration of DESM, Publication Department and Secretariat of NCERT is also duly acknowledged.

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7.2 Mm¡H$moZmÀ¶m gd© H$moZm§À¶m ~oaOoMm Jw‘Y‘© 124

7.3 Mm¡H$moZmMo àH$ma 125

7.4 g‘m§Va^wO Mm¡H$moZmMo JwUY‘© 126

7.5 Mm¡H$moZ hm g‘m§Va^yO Mm¡H$moZ hmoÊ¶mgmR>r BVa AQ>r : 133

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x

7.6 ‘Ü¶q~XyMm à‘o¶ 136

7.7 gmam§e 139

AZw~§Y - 1 141 - 162 A 1.1 àñVmdZm 141

A 1.2. J{UVmZo pñdH$maÊ`m`mo½` {dYmZo 142

A 1.3. AZw_m{ZH$ {dMmagaUr (`wpŠVdmX) 146

A 1.4. à_o`, AZw_mZ Am{U _ybVËdo 149

A 1.5 J{UVr {gÜXVm åhUOo H$m`? 155

A 1.6 gmam§e 162

CÎma gyMr 163 - 174

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g§»`m nÜXVr 1

àH$aU 1

g§»`m nÜXVr (Number systems)

1.1 àñVmdZm

Vwåhr _mJrb dJm©V g§»`maofm Am{U g§»`maofoda doJdoJù`m àH$maÀ`m g§»`m H$em XmI{dVmV `m§Mm Aä`mg Ho$bm AmhmV (AmH¥$Vr 1.1 nhm.)

AmH¥$Vr 1.1 g§»`maofm

H$ënZm H$am H$s Vwåhr 0 (eyÝ`) nmgyZ gwédmV H$ê$Z g§»`maofoda YZ {XeoZo MmbV OmV AmhmV. Vw_À`m S>moù`m§Zm \$ŠV g§»`m, g§»`m Am{U g§»`mM {XgVmV

-

AmH¥$Vr 1.2

AmVm g_Om Vwåhr g§»`maofoda MmbV OmV AmhmV d g§»`m O_m H$aV AmhmV. g§»`m O_m H$aÊ`mgmR>r EH$ {nedr V`ma R>odm.

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2 J{UV

Vwåhr Z¡g{J©H$ g§»`m CXm :1,2,3 .. B. AZoH$ g§»`m CMbm`bm gwédmV Ho$br Va Vwåhmbm _mhrV Amho H$s `m g§»`m dmT>VM OmUma (ho gË` Amho H$m?) åhUyZ Vw_À`m {nedrV A_`m©X Z¡g{J©H$ g§»`m AgVrb. AmR>dU H$am `m gd© g§»`mÀ`m g_whmbm N {MÝhmZo Xe©{dVmV.

N = (Natural numbers)

AmVm nmR>r_mJo {\$am d MmbV Om. eyÝ` CMbm Am{U {nedrV R>odm. Vw_À`mH$S>o AmVm nyU© g§»`m§Mm g_yh Amho. Ë`m§Zm " W ' `m AjamZo Xe©{dVmV.

W = (Whole numbers)

AmVm Vwåhr g_moaÀ`m G$U g§»`m KoD$Z Ë`m {nedrV ^am. Vw_Mm ZdrZ g§J«h H$m` Amho? AmR>dm, ho nwUm©H$m§Mo EH$ÌrH$aU Amho Am{U ho " Z' `m {MÝhmZo Xe©{dVmV.?

H$mhr g§»`m AOyZ g§»`maofoda AmhoV H$m? ZŠH$sM 12

3

4, qH$dm −

2005

2006 `mgma»`m g§»`m

CaVmV. Oa `mgma»`m g§»`m Vwåhr {nedrV R>odë`m Va Vmo g§J«h n[a_o` g§»`m§Mm hmoB©b.

n[a_o` g§»`m§À`m EH$ÌrH$aUmbm Q `m AjamZo Xe©{dVmV. n[a_o` (Rational) hm eãX ‘Ratio’ `m eãXmnmgyZ Am{U Q hm eãX Quotient `m eãXmnmgyZ ~Zbm Amho.

Vwåhr n[a_o` g§»`oMr ì`m»`m AmR>dm.

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g§»`m nÜXVr 3

Or g§»`m pq `m ñdê$nmV {b{hVm òVo OoWo p Am{U q ho nyUmªH$ Am{U q ≠ 0 Agob Va

Ë`m g§»òbm n[a_o` g§»`m (Rational numbers) Ago åhUVmV. (q ≠ 0 da OmñV â {Xbm Amho. _mhrV Amho H$m?)

bjmV R>odm {ne{d_Üò Agboë`m g§»`m pq `m ñdê$nmV {bhÿ eH$Vmo OoWo p Am{U q ho

nyUmªH$ Am{U q ≠ 0.

CXmhaU : – 25 ho −25

1 Ago {bhrVmV, ho p = – 25 Am{U q = 1. åhUyZM Z¡g{J©H$ g§»`m,

nyU© g§»`m d nyUmªH$ ho gwÜXm n[a_o` g§»òV òVmV.

Vwåhm§bm _mhrV AmhoV {H$ n[a_o` g§»`m§Mo ñdê$n pq ho H$mhr EH$_od Zmhr. òWo

p Am{U q nyUmªH$ AmhoV Am{U q ≠ 0.

CXm : 1

2

2

4

10

20

25

50

47

94= = = = BË`mXr.

n[a_o` g§»`m AnyUmªH AmhoV. VWmnr Ooìhm Amåhr pq hr n[a_o` g§»`m åhUVmo, qH$dm

Ooìhm Amåhr g§»`m aofoda pq KoVmo, Voìhm Amåhr g_OVmo {H$ q ≠ 0 Am{U p d q _Üò H$moUVmhr

gm_mB©H$ Ad`d 1 {edm` (p Am{U q _wig§»`m). ZgVmo. åhUyZ g§»àofoda 1

2AnyUmªH$m gmaIo

AZoH$ AnyUmªH$ AgVmV. gdm©gmR>r AmnU 1

2 KoD$> eH$Vmo. AmVm, _mJrb dJm©V Aä`mg

Ho$boë`m H$mhr doJdoJù`m g§»`m§Mr CXmhaUo gmoS>dy.

CXmhaU 1: Imbrb dmŠò MwH$ H$s ~amo~a AmhoV Vo gm§Jm. Vw_À`m CËVamMo H$maU Úm.

(i) àËòH$ nyU© g§»`m hr Z¡g{J©H$ g§»`m Amho.

(ii) àËòH$ nyUmªH$ hm n[a_o` g§»`m Amho.

(iii) àËòH$ n[a_o` g§»`m hr nyUmªH$ Amho.

CH$b :

(i) MwH$, H$maU eyÝ` (0) nyU© g§»`m Amho na§Vw Z¡g{J©H$ g§»`m Zmhr.

(ii) ~amo~a , H$maU àËòH$ nyUmªH$ m hm m1 `m ñdê$nmV _m§S>Vm òVmo åhUyZ hr n[a_o` g§»`m

Amho.

(iii) MwH$, H$maU 35

hm nyUmªH$ Zmhr.

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4 J{UV

CXmhaU 2: 1 Am{U 2 _Yrb nm§M n[a_o` g§»`m emoYm.

Amåhr ho CXmhaU 2 nÜXVrZo gmoS>dy eH$Vmo.

CH$b 1: AmR>dm, y Am{U s _Yrb nmM n[a_o` g§»`m. Vwåhr r d s Mr ~oarO H$ê$Z 2 Zo ^mJm

r s+2

. y d s À`m _Üò òVo. åhUyZ 3

2 hr g§»`m 1 Am{U 2 À`m _Üò òVo.

Vwåhr `m à_mUo nw‹T>o Jobm Va AZoH$ 1 d 2 _Yrb n[a_o` g§»`m {_iVmV Ë`m g§»`m

AmhoV 54

11

8

13

8, , Am{U

7

4.

CH$b 2: Xþgè`m nÜXVrV EH$mM nmàrV AmnU 5 n[a_o` g§»`m emoYy eH$Vmo. Amåhmbm 5 n[a_o` g§»`m nmhrOo. 1 d 2 `m n[a_o` g§»`m KoD$Z N>oX 5 + 1 KoD$.

Ë`m åhUOo 16

6= Am{U 2

12

6= `mà_mUo 7

6

8

6

9

6

10

6, , , Am{U

11

6 `m

1 d 2 _Yrb 5 n[a_o` g§»`m AmhoV. Ë`m nm§M g§»`m 76

4

3

3

2

5

3, , , Am{U

11

6AmhoV.

Q>rn : CXmhaU 2 Mo {ZarjU Ho$ë`mg, Vwåhmbm 1 d 2 _Yrb 5 n[a_o` g§»`m emoYm`À`m hmoË`m nU 1 d 2 _Üò AZoH$ n[a_o` g§»`m AmhoV. gd©gm_mÝ`nUo {Xboë`m H$moUË`mhr XmoZ n[a_o` g§»ò_Üò AZoH$ n[a_o` g§»`m AgVmV.

AmVm nwÝhm g§»`maofm nhm. Ë`mdarb gd© g§»`m KoVë`m AmhoV H$m? Zmhr. gË` ho Amho H$s g§»àofoda AOwZ AZoH$ g§»`m {eëbH$ AmhoV. VoWo Vwåhr KoVboë`m g§»`m§À`m [aH$må`m OmJm AmhoV Ë`m AZoH$ AmhoV. AmíM`m©Mr JmoîQ> hr Amho H$s [aH$må`m Agboë`m 2 g§»`mÀ`m _Üò gwÜXm AZoH$ g§»`m AmhoV.

åhUyZ Imbrb àíZ Amnë`mbm nS>VmV.

1. g§»`maofoda {eëbH$ am{hboë`m g§»`mZm H$m` åhUVmV?

2. Ë`m§Zm Vwåhr H$go AmoiImb? Ë`m§Zm n[a_o` g§»`mnmgyZ H$go doJio H$amb? `m àíZm§Mr CËVao nwT>rb ^mJmV {_iVrb.

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g§»`m nÜXVr 5

ñdmÜ`m` 1.1

1. eyÝ` (0) n[a_o` g§»`m Amho H$m? 1 bm pq mdê$nmV {bhÿ eH$Vm OoWo p Am{U q nyUmªH$

Am{U q ≠ 0.

2. 3Am{U 4 _Yrb 6 n[a_o` g§»`m emoYm..

3. 3

5 Am{U 4

5_Yrb 5 n[a_o` g§»`m emoYm.

4. Imbrb dmŠò MyH$ {H$ ~amo~a AmhoV Vo gm§Jm, Vw_À`m CËVamMo H$maU gm§Jm.

(i) àËòH$ Z¡g{J©H$ g§»`m hr nyU© g§»`m Amho.

(ii) àËòH$ nyUmªH$ g§»`m hr nyU© g§»`m Amho.

(iii) àËòH$ n[a_o` g§»`m hr nyU© g§»`m Amho.

1.2 An[a_o` g§»`m (Irrational Numbers)

_mJrb ^mJmV AmnU nmhrbo H$s, g§»`maofmdarb gd© g§»`m `m n[a_o` g§»`m ZmhrV.

`m ^mJmV AmnU Ë`m g§»`m emoYm`À`m AmhoV. åhUyZM pq ñdê$nmVrb g§»`m OoWo p Am{U q ho

nyUmªH$ Am{U q ≠ 0 `m g§»`m Amboë`m AmhoV Am{U Vwåhr {dMmamb pq `m ñdê$nmV Zgboë`m

g§»`m VoWo AmhoV H$m? Aem g§»`m IamoIaM {VWo AmhoV.

400 BC _Üò J«rH$Mm à{gÜX J{UVk Am{U VËdk nm`WmJmoag d Ë`mMo ghH$mar `m§Zr `m g§»`mMm emoY bmdbm Or n[a_o` Zmhr Ë`m g§»`m§Zm An[a_o` g§»`m åhUVmV. H$maU Ë`m nwUm©H$m§À`m JwUmoÎmamV {b{hVm òV ZmhrV nm`WmJmoagMm EH$ ghH$mar H«$moQ>moZ {hnm°Šgg² Ûmam emoYboë`m An[a_o` g§»`mg§~§YmV AZoH$ X§VH$Wm AmhoV. {hnm°ŠggMm eodQ> AË`§V XþX¡dr Pmbm. `mMo H$maU `m JmoîQ>rMm emoY Agob H$s 2 hr An[a_o` g§»`m Amho qH$dm `m emoYm g§~§YrVrb _m{hVr ~mø-OJmV àH$Q> Ho$br Agob. nm`WmJmoag

(569 BC- 479 BC)

AmH¥$Vr 1.3AmVm `m g»`m§Mr ì`m»`m nmhþ`m.

g§»`m "s' bm An[a_o` åhUVmV Oa Vr pq `m ñdê$nmV {b{hVm òV Zmhr, OoWo p Am{U

q ho nyUmªH$ Am{U q ≠ 0 AgVo.Vwåhmbm AJmoXa _mhrV AmhoV H$s n[a_o` g§»`m Ag§»` AmhoV Ë`mM à_mUo AZoH$ An[a_o`

g§»`m AmhoV. `mdê$Z hoM g_OVo H$s VoWo A_`m©{XV AZoH$ An[a_o` g§»`m gwÜXm AmhoV. H$mhr CXmhaUo,

2 3 15 0 10110111011110, , , , . .......π

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6 J{UV

{Q>n : AmR>dm, Ooìhm Amåhr " ' ho {MÝh dmnaVmo Voìhm AmnU _mZVmo H$s ho g§»òMo YZ^marV dJ©_yi Amho åhUyZ, 4 2= VWm{n +2 Am{U –2 hr 4 Mr dJ©_yio Amho.

H$mhr darb An[a_o` g§»`m Vw_À`m n[aM`mÀ`m AmhoV. CXmhaU da {Xbobr AZoH$ dJ©_wio d g§»`m p, AZoH$ doim Ambobr AmhoV.

nm`WmJmoag `m§Zr {gÜX Ho$bo Amho H$s 2 hr An[a_o` g§»`m Amho. Z§Va gw_mao 425 B. C. _Üò {WAmoS>ag Am{U gËàoZ `m§Zr 3 5 6 7 10 11 12 13 14 15, , , , , , , , , ©Am{U

17 `m gwÜXm An[a_o` g§»`m AmhoV ho XmI{dbo. 2 3 5, , ........ BË`mXr An[a_o` g§»`m§Mr {gÜXVm AmnU B`Îmm Xhmdr _Üò {eHy$. p hr g§»`mhOmamo dfm©nmgyZ {d{dY g§ñH¥$Vrer nar{MV Amho. 700 À`m eVH$mÀ`m CËVamYm©V bm°å~Q>© Am{U brOoS>} `m§Zr p hr EH$ An[a_o` g§»`m Amho Agm emoY bmdbm hmoVm. nwT>rb n[aÀN>oXmV AmnU m JmoîQ>rda MMm© H$ê$ H$s 0.10110111011110 Am{U p An[a_o` H$m AmhoV ? _mJrb ^mJmÀ`m eodQ>r hr Omo àíZ hmoVm Ë`m H$S>o OmD$`m, bjmV R>odm {nedrV n[a_o` g§»`m AmhoV. Oa Amåhr gd© n[a_o` g§»`m, Ë`m {nedrV âë`m Va _J g§»`maofoda H$moUVr g§»`m {eëbH$ amhrb?

CËVa Amho, Zmhr. `mdê$Z ho g_OVo H$s, n[a_o` g§»`m d An[a_o` g§»`m `m§À`m g§Mmbm AmnU dmñVd g§»`m åhUVmo. Ë`m R . `m AjamZo Xe©{dVmV. dmñVd g§»`m EH$ Va n[a_o` AgVo ZmhrVa An[a_o` AgVo. Amåhr Agohr åhUy eH$Vmo àËòH$ dmñVd g§»`m hr g§»`maofodarb EH$_od q~Xÿ Amho. g§»àofodarb àËòH$ q~Xÿ hm EH$_od dmñVd g§»`m Amho. åhUyZ g§»`maofobm AmnU dmñVd g§»`m aofm Ago hr åhUy eH$Vmo.

Or H¡$ÝQ>a

(1845-1918)

AmH¥$Vr 1.4

1870 _Üò XmoZ O_©Z J{UVk H¡$ÝQ>a Am{U Ama. S>oS>oqH$S> `m§Zr ho XmIdyZ {Xbo {H$ àËòH$ dmñVd g§»òH$[aVm g§»`m aofoda EH$ q~Xÿ {ZJS>rV AgVmo Am{U àËòH$ q~Xÿer EH$ dmñVd g§»`m {ZJS>rV AgVo.

Ama.S>oS>oqH$S> (1831-1916)

AmH¥$Vr 1.5

AmVm AmnU g§»`maofoda H$mhr An[a_o` g§»`m H$em XmI{dVmV Vo nmhÿ.

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g§»`m nÜXVr 7

CXmhaU 3: g§»`maofoda 2 Xe©dm.

CH$b : J«rH$m§Zr 2 Mm emoY H$gm bmdbm ho nmhÊ`mg gmono Amho. 1 EH$H$ bm§~r Agbobm Mm¡ag OABC ¿`m. (AmH¥$Vr 1.6 nhm). Z§Va Vwåhr nm`WmJmoag à_o`mZo nhm

OB = 1 1 22 2+ = . g§»`maofoda 2 H$er XmI{dVmV? ho gmono Amho. AmH¥$Vr

1.6 hr g§»`maofoda Aer ñWbm§Va H$am H$s AmH¥$VrMm O {eamoq~Xÿ eyÝ`mer Owiob (AmH¥$Vr 1.7 nhm)

AmH¥${V 1.7

AmVm AmnU nm{hbo Amho H$s, OB = 2 Amho. AmVm H§$nmgÀ`m ghmæ`mZo O `m q~Xÿbm H|$Ð KoD$Z OB `m {ÌÁ`oZo H§$g H$mT>m. Vmo g§»`maofobm P q~XÿV N>oXVmo. Voìhm g§»`maofodarb P q~Xÿ åhUOoM 2 g§»`m hmo`.

CXmhaU 4: g§»`maofoda 3 Xe©dm.

CH$b : AmnU AmH¥$Vr 1.7 bm nwÝhm KoD$ `m.

AmH¥${V 1.8

. OB bm b§~ BD H$mT>m (AmH¥$Vr 1.8 nhm) Z§Va Vwåhr nm`WmJmoag à_o`mZwgma

OD = 2 1 32( ) + =

2 Amho.

AmVm H§$nmgÀ`m gmhmæ`mZo O bm H|$Ðq~Xÿ Am{U OD bm {ÌÁ`m Q KoD$Z EH$ H§$g H$mT>m. Vmo g§»`maofobm Q q~XÿV N>oXVmo. Voìhm g§»`maofodarb Q q~Xÿ åhUOoM 3 hmo`.

`mMà_mUo KZ nwUmªH$ n gmR>r n Mo ñWmZ Xe©dm. Z§Va n −1 Mo ñWmZ Xe©dm.

AmH¥${V 1.6

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8 J{UV

ñdmÜ`m` 1.2

1. Imbrb {dYmZo MyH$ H$s ~amo~a Vo gm§Jm. CÎma nS>VmiyZ nhm.

(i) àËòH$ An[a_o` g§»`m hr EH$ dmñVd g§»`m Amho.

(ii) g§»`maofodarb àËòH$ q~Xÿ m ñdê$nmV Amho. OoWo m hr Z¡g{J©H$ g§»`m Amho.

(iii) àËòH$ dmñVd g§»`m hr An[a_o` g§»`m AgVo.

2. gd© YZ nyUmªH$ g§»`m§Mo dJ©_yi An[a_o` g§»`m AgVo H$m? Oa Zmhr, Va g§»`m§Mo dJ©_yi ho n[a_o` g§»`m Amho `mMo EH$ CXmhaU Úm.

3. g§»`maofoda 5 Mo ñWmZ Xe©dm.

4. dJm©Vrb H¥$Vr (MH«$mH$ma dJ©_yi aMZm) :

EH$ _moR>m H$mS>©erQ> nona KoD$Z Imbrb à_mUo MH«$mH$ma dJ©_wimMr aMZm H$am (square root spiral). O q~XÿnmgyZ gwédmV H$ê$Z 1 EH$H$ KoD$Z aofmI§S> OP1 H$mT>m. p1p2 aofmI§S> H$mTy>Z OP1 bm b§~ p1p2 1 EH$H$ ¿`m (AmH¥$Vr 1.9 nhm). AmVm p2p3 aofmI§S> H$mTy>Z OP2 bm b§~ p2p3 H$mT>m. Z§Va OP3 da b§~ aofmI§S> p3p4 H$mT>m.

`màH$mao AmnU q~Xÿ OPn–1.da EH$H$ bm§~rMm aoImI§S> H$mTy>Z aofmI§S> pn–1 pn {_idy eH$Vmo. O, P1, P2, P3, .... Pn......... H$mTy> eH$Vmo. Ë`m§Zm OmoSy>Z 2 3 4, , ..... Xe©{dUmar EH$ gw§Xa MH«$mH$ma dJ©_yi aMZoMr AmH¥$Vr {_iob.

1.3 dmñVd g§»`m Am{U Ë`m§Mm Xem§e {dñVma :

`m ^mJmV AmnU n[a_o` d An[a_o` g§»òMm doJù`m ÑîQ>rH$moZmVyZ Aä`mg H$é`m. `mgmR>r AmnU dmñVd g§»òÀ`m Xem§e {dñVmamMm {dMma n[a_o` g§»`m d An[a_o` g§»`m `m§À`mV \$aH$ H$aÊ`mgmR>r H$aVm òVmo H$m? Vo nmhÿ Ë`m§À`m Xem§e {dñVmamMm Cn`moJ H$ê$Z Amåhr g§»`maofoda dmñVd g§»`m§Mo {dñVmarV ê$n XmI{dVm òVo Vo nmhÿ`m. n[a_o` g§»`m Am_À`m n[aM`mÀ`m AmhoV

Ë`mnmgyZ gwédmV H$ê$. 10

3

7

8

1

7, , hr VrZ CXmhaUo KoD$.

~mH$sH$S>o {deof bj Úm Am{U `mV H$mhr {deofVm AmT>iVo H$m Vo nhm.

P41 1

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g§»`m nÜXVr 9

CXmhaU 5: 103

7

8, Am{U

1

7 `m§Mo Xem§e {dñVma ê$n H$mT>m..

CH$b:

~mH$s- 1, 1, 1, 1, 1 ~mH$s : 6, 4, 0 ~mH$s- 3, 2, 6, 4, 5, 1

^mOH$ : 3 ^mOH$ : 8 ^mOH$ : 7

Vwåhr òWo H$emMr Zm|X KoVbr? Vwåhr H$_rVH$_r VrZ ~m~tMr Zm|X KoVbrAgob.

(i) H$mhr nm`è`mZ§Va ~mH$s eyÝ` òVo qH$dm ^mJmH$ma nwÝhm Ë`mMH«$_mZo nwÝhm gwê$ hmoVmo.

(ii) ~mH$s nwÝhm-nwÝhm Amboë`m g§» oÀ`m {R>H$mUr ~mH$s hr ^mOH$nojm bhmZ AgVo. ( 1

3

`mg§»òV 1 hm A§H$ nwÝhm nwÝhm òVmo Am{U ^mOH$ Amho 3, 17

`m g§»òV 3, 2, 6, 4, 5, 1 ho A§H$ nwÝhm òVmV Am{U ^mOH$ Amho 7)

(iii) Oa ~mH$s Mr nwZamd¥ËVr hmoV Agob Va Amnë`mbm ^mJmH$mamÀ`m A§H$m_Üò nwZamd¥ËVrMm

EH$ I§S> {_iVmo (1

3 _Üò ^mJmH$mamV 3 Mr nwZamd¥Îmr hmoVo Am{U 1

7 _Üò ^mJmH$mam_Üò

nwZamd¥ËVr I§S> 142857 _Üò {_iVmo).

Var Vwåhr g_Obm AgUmaM H$s, darb CXmhaUmV bmJy Ho$bobm Z_wZm hm gd© n[a_o`

g§»`m§Zm bmJy nS>Vmo Á`m pq (q ≠ 0) ñdê$nmV AgVmV. p bm q Zo ^mJbo AgVm _w»`

XmoZ JmoîQ>r{XgyZ òVmV - ~mH$s eyÝ` òVo qH$dm ~mH$s H$YrM eyÝ` òV Zmhr. Aemdoiog Amnë`mbm ~mH$sMr EH$ gmIir {_iVo. AmVm AmnU àËòH$ àH$ma doJdoJim nmhþ-

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10 J{UV

àH$ma (i): ~mH$s eyÝ` AgVo Voìhm

7

8`m CXmhaUmV AmnU nm{hbo Amho H$s, H$mhr nm`è`mZ§Va ~mH$s eyÝ` òVo Am{U

7

8 Mm

Xem§e {dñVma 0.875 Amho. Xþgar H$mhr CXmhaUo : 1

20 5

639

2502 556= =. ; . Amho. `m gd© CXmhaUmV

_`m©XrV nm`è`mZ§Va Xem§e {dñVmamMm eodQ> hmoVmo. Aem g§»òÀ`m {dñVmambm gm§V Xem§e {dñVma (terminating) åhUVmV.

àH$ma (ii): ~mH$s Ho$ìhmhr eyÝ` ZgVo Voìhm

1

3 Am{U 1

7 `m CXmhaUmV AmnU nmhrbo Amho H$s H$mhr nm`mè`mZ§Va ~mH$sMr nwZamd¥ËVr

hmoVo d ^mJmH$mamMr {H«$`m nwT>o Mmbw amhVo. Xþgè`m eãXmV ^mJmH$mamÀ`m A§H$mV nwZamd¥ËVr AI§S>nUo Mmby amhVo. Ë`mbm AI§S> AmdVu Xem§e {dñVma (Non-termination recurring) åhUVmV.

CXm. 1

3 = 0.3333....... Am{U

1

7 = 0.142857142857142857...... Amho.

BWo 1

3 À`m ^mJmH$mamV 3 Mr nwZamd¥ËVr hmoVo Am{U `mbm 0 3. ê$nmV {bhVmo. `mMàH$mao

1

7

À`m AnyUm©H$m§À`m Xem§e ê$nmV 142857 `m A§H$m§Mm JQ> Ë`mM H«$_mZo ^mJmH$mamV gVV òV amh-Vmo. n[a_o` g§»`m Am{U AI§S> AmdVu Xem§e {dñVma EoH$_oH$mg g§b½Z AmhoV. Wmo‹S>Š`mV Xem§e _m§S>UrV Oa EH$ A§H$ AWdm H«$_dma A§H$ JQ> H$Yrhr Z g§nUmè`m AmdVu nÜXVrZo òV AgVrb Va Aem _m§S>Urg AI§S> AmdVu Xem§e _m§S>Ur åhUVmV. àËòH$ n[a_o` g§»`m Xem§e nÜXVrV AI§S> AmdVu nÜXVrZo _m§S>Vm òVo. AI§S> AmdVu Xem§e {dñVmamV AmdVu A§H$ qH$dm A§H$ JQ> nwÝhm {b{hÊ`m EodOr Ë`m A§H$mda qH$dm A§H$ JQ>mda AmS>dr aofm H$mT>Ê`mMr qH$dm Ë`mVrb àW_

Am{U A§Ë` A§H$mda qQ>~ XoÊ`mMr nÜXV Amho. CXm. 1

7 ho 0142857. Ago {b{hVmV. `m A§H$mMm

JQ> Ë`mM H«$_mZo gVV òV amhVmo. VgoM 3.57272 ho 3 572. `m àH$mao {b{hVmV. hr gd© CXmhaUo AI§S> AmdVu Xem§e {dñVma Xe©{dVmV. `mMà_mUo, n[a_o` g§»`m§Mm Xem§e {dñVma XmoZ àH$mao H$aVm òVmo. gmV Xem§e {dñVma qH$dm AI§S> AmdVu Xem§e {dñVma.

g_Om g§»`maofoda Mmbë`mda AmnUmbm 3.142678 gma§»`m g§»`m {_iVmV. Á`m§Mm Xem§e {dñVma gm§V Xem§e {dñVma Amho qH$dm. 1.272727.. AWm©V² 1 27. gma»`m g§»`m Á`m AI§S> AmdVu Xem§e {dñVmamV _m§S>VmV mdê$Z Vwåhr m n[a_o` g§»`m AmhoV Agm AZw_mZ H$mT>mb? CËVa hmo` Amho. na§Vw Ë`mMr {gÜXVm AmnU Aä`mgUma Zmhr. nU H$mhr CXmhaUo KoD$Z Vo AmnU XmIdy eH$Vmo. gm§V Xem§e _m§S>Ur AJXr gmonr Amho.

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g§»`m nÜXVr 11

CXmhaU 6: 3.142678 hr n[a_o` g§»`m Amho ho XmIdm. Xþgè`m eãXmV 3.14268 hr pq

ñdê$nmV {bhm. OoWo p Am{U q ho nwUmªH$ Am{U q ≠ 0

CH$b : 3.142678 = 3142678

1000000 Am{U åhUyZM n[a_o` g§»`m Amho. AmVm AmnU Ooìhm

Xem§e{dñVma hm AI§S> AmdVu {dñVma `m nÕVrMm {dMma H$ê$

CXmhaU 7: 0 3333 0 3. .= ho pq `m ñdê$nmV XmIdm OoWo p Am{U q nyUmªH$, q ≠ 0.

CH$b :0 3. H$m` AmhoV Amnë`mbm _mhrV Zmhr. åhUyZ `mbm x _mZy Am{U åhUyZ.

x = 0.3333.........

AmVm `oWo nhm

10x = 10 × (0.3333.......) = 3.333.........

AmVm 3.333 = 3 + x

Am{U x = 0.3333........

åhUyZ 10x = 3 + x

x Mr qH$_Vr Amåhmbm {_iob

9x = 3

åhUyZ . x = 13

CXmhaU 8: 1.272727...... = 1 27. ho pq

`m ñdê$nmV XmIdm OoWo p Am{U q ho nyUmªH$ AmhoV Am{U q ≠ 0.

CH$b : x = 1.272727...... `oWo XmoZ A§H$ nwÝhm Ambobo AmhoV Amåhr x bm 100 Zo JwUy. 100 x = 127.2727......

Am{U 100 x = 126 + 1.272727........= 126 + x

åhUyZ 100 x – x = 126

99 x =126

åhUOo x =12699

14

11=

Vwåhr Vnmgm 1411

= 1 27.

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12 J{UV

CXmhaU 9 : 0.2353535 = 0 235. ho pq

ñdê$nmV XmIdm OoWo

p Am{U q ho nyUmªH$ AmhoV Am{U q ≠ 0

CH$b x = 0 235. bjmV R>odm òWo 2 A§H$ nwÝhm Ambm Zmhr nU 35 hm g§M nwÝhm nwÝhm Ambm Amho. åhUOoM 2 A§H$ nwÝhm Ambo AmhoV. Amåhr x bm 100 Zo JwUy.

100 x = 23.53535.....

100 x = 23.3 + 0.2353535..........= 23.3 + x

åhUyZ 99 x = 23.3

\99 x= 233/10→x=233

990

Vwåhr Vnmgm 233

990 = 0 235.

åhUyZ, àËòH$ g§»òMm AI§S> AmdVu Xem§e {dñVma hm Zoh_r pq ( q ≠ 0) m ñdê$nmV {b{hVm oVmo

OoWo p & q ho nyUmªH$ AmhoV. Am_À`m CÎmamMm gmam§e Aem ñdê$nmV Amåhr _m§Sy> eH$Vmo.

n[a_o` g§»òMm Xem§e {dñVma hm gm§V qH$dm AI§S >AmdVu Xem§e Agob Va Vr n[a_o` g§»`m AgVo.

åhUyZ AmVm n[a_o` g§»òMm Xem§e {dñVma H$m` AgVmo Vwåhmbm _mhrV Amho. An[a_o` g§»òMm Xem§e {dñVma H$m` Amho? darb nwamì`mdê$Z Amåhr AZw_mZ H$mT>bo Amho H$s Ë`m§Mm Xem§e {dñVma AZ§V AZmdVu AgVmo.

åhUyZ An[a_o` g§»`m§Mo JwUY_© ho n[a_o` g§»`m§Mm JwUY_m©à_mUo bmJy nS>VmV.

Á`m n[a_o` g§»`m§Mm Xem§e {dñVma hm AZ§V AZmdVu AgVmo Ë`m g§»`m An[a_o` g§»`m AgVmV.

_mJrb ^mJmV AmnU EH$ An[a_o` g§»`m 0.10110111011110 Mr MMm© Ho$br hmoVr. s = 0.10110111011110 ... hr g§»`m AZ§V AZmdVu Amho. Ë`m_wio darb JwUY_m©Zwgma Vr An[a_o` g§»`m Amho. gmo~V ho gwÜXm bjmV ¿`m {H$ AmnU s `m AjamZo AZoH$ An[a_o` g§»`m XmIdy eH$Vmo.

2 d p`m An[a_o` g§» o~m~V AmnU H$m` OmUVmo? `m An[a_o` g§»`m§Mo AZ§V AZmdVu Xem§e {dñVma H$mhr nm`è`mn`ªV {Xbm Amho.

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g§»`m nÜXVr 13

2 = 1.4142135623730950488016887242096.........

p= 3.14159265358979323846264338327950.............

(bjmV R>odm p Mr OdiMr qH$_V 22

7 KoVmo na§Vw π ≠ 22

7 )

{H$VrVar dfm©nmgyZ J{UVkmZr An[a_o` g§»`m§Mm Xem§e {dñVma A{YH$m{YH$ A§H$ {_i{dÊ`mMo V§ÌkmZ {dH${gV Ho$bo Amho. CXmhaUmW© mJmH$ma nÜXVrZo (division method) 2 qH$_V H$mT>Vm§Zm Xem§e ê$nmV A§H$m§Zm _m{hV H$aÊ`mMr nÜXV Vwåhr Aä`mgbr Amho. hr EH$ _Zmoa§OH$ ~m~ Amho H$s, gwb^m gwÌm_Ü`o Omo EH$ d¡{XH$ H$mimVrb (800BC - 500BC) J{UVmdarb J«§W Amho. Ë`mZwgma

2 Mr AXm§Oo qH$_V Imbrbà_mUo àmßV Pmbr Amho.

2 11

3

1

4

1

3

1

34

1

4

1

31 4142156= + + ×

− × ×

= .

bjmV ¿`m H$s, ho VoM Xem§e ê$n Amho Oo da nm§M A§H$mn`ªV Xem§e {dñVmamV {Xbobo Amho. p À`m Xem§e {dñVmamV A{YH$m{YH$ A§H$ {_i{dÊ`mMm B{Vhmg A{YH$ _Zmoa§OH$ Amho.

J«rH$Mm à{gÜX VËddoVm Am{H©${_S>rO hr n{hbr ì`ŠVr hmoVr Á`m§Zr p M Xem§e {dñVma _m§S>bm. Ë`m§Zr XmI{dbo {H$ 3.140845 < p< 3.142857 Amho. Am`©^Å> (B. gZ476 – 550 ) Oo EH$ _hmZ ^maVr` J{UVk Am{U IJmobemñÌk hmoVo. Ë`m§Zr 4 Xem§e ñWmZmn`ªV pMr qH$_V ( 3.1416 ) AMwH$ H$mT>br hmoVr. AmVm gwna H$m±åß`wQ>a Am{U AmYw{ZH$ AëJmo[aW_ dmnê$Z 1.24 {Q´>{b`Z nojm OmñV Xem§e ñWmZmn`ªV p Mo _wë` _mhrV Ho$bo Amho. Am{H©${_{S>µO

(287BCE - 212 BCE)

AmH¥$Vr1.10

AmVm AmnU nmhÿ H$s$ H$emàH$mao An[a_o` g§»`m {_i{dVmV..

CXmhaU 10: 17

Am{U 2

7 _Yrb An[a_o` g§»`m emoYm.

CH$b Amåhr nm{hbo Amho 1

70142857= . Am{U 2

70 285714= . ghO H$mTy> eH$mb.

1

7 Am{U 2

7

_Yrb An[a_o` g§»`m H$mT>>Ê`mgmR>r Amåhr Aer g§»`m emoYy {H$ Ë`m

XmoÝhr_Yrb

AZ§V AZmdVu Xem§e Agob. `m àH$maÀ`m AZoH$ g§»`m AmnU _mhrV H$ê$ eH$Vmo.`m àH$maÀ`m EH$m g§»`oMo CXmhaU, 0.150150015000150000......... Amho

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14 J{UV

ñdmÜ`m` 1.3

1. Imbrb AnwUmªH$m§Zm Xem§e ê$nmV {bhm Am{U H$moUË`m àH$maMo Xem§e ê$n Amho Vo {bhm.

(i) 36100

(ii) 111

(iii) 41

8

(iv) 313

(v) 211

(vi) 329

400

2. Vwåhmbm _mhrV Amho {H$ 1

70142857= . . XrK© ^mJmH$ma Z H$aVm

2

7

3

7

4

7

5

7

6

7, , , , `m§Mm

Xem§e {dñVma AmoiImb H$m? Oa hmo`, Va H$go?

({Q>n 1

7 Mr qH$_V H$mT>VmZm ~mH$sMm H$miOrnyd©H$ Aä`mg H$am.)

3. Imbrb AmdVu Xem§em§Zm pq `m ñdê$nmV _m§S>m. OoWo p Am{U q ho nyUmªH$ Am{U q ≠ 0.

(i) 0.6 (ii) 0.47 (iii) 0.001

4. 0.99499 ho pq `m ñdê$nmV _m§S>m. Vwåhr H$mT>boë`m CÎmamZo AmíM`©M{H$V Pmbm H$m?

Amnë`m {ejH$m§~amo~a Am{U dJ©{_Ìm~amo~a CÎmamÀ`m gË`Vo~Ôb MMm© H$am.

5. 117

À`m Xem§e {dñVmamV A§H$m§À`m nwZamdVu JQ>mV A§H$m§Mr OmñVrV OmñV g§»`m {H$Vr

Agob? Amnbo CÎma VnmgyZ nmhÊ`mgmR>r ^mJmH$ma {H«$`m H$am.

6. pq

(q ≠ 0) À`m ñdê$nmVrb n[a_o` g§»òMr AZoH$ CXmhaUo ¿`m. òWo p Am{U q nwUmªH$

AmhoV. `m§Mm 1 ì`{V[aŠV H$moUVmhr gm_mB©H$ Ad`d Zmhr Am{U Á`m§Mo gm§V Xem§e ê$n Amho. q bm H$moUVo ~§YZ nmiUo Amdí`H$ Amho?

7. Aem VrZ g§»`m {bhm Á`m§Mm Xem§e {dñVma AZ§V AZmdVu Amho.

8. n[a_o` g§»`m 57

Am{U 9

11 `m§À`m Xaå`mZÀ`m VrZ doJdoJù`m An[a_o` g§»`m AmoiIm.

9. Imbrb g§»`m§Mo n[a_o` g§»`m Am{U An[a_o` g§»`m_Üò dJuH$aU H$am.

(i) 23 (ii) 225 (iii) 0.3796

(iv) 7.478478..... (v) 1.101001000100001....

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g§»`m nÜXVr 15

1.4 g§»`maofoda dmñVd g§»`m XmI{dUo.

_mJrb ^mJmV Vwåhr nmhrbm AmhmV {H$ dmñVd g§»òbm Xem§e {dñVma AgVmo. `mÀ`m ghmæ`mZo AmnU Ë`mg§»`m§Zm g§»`maofoda XmIdy eH$Vmo. H$go Vo nhm.

g_Om, Amnë`mbm g§»`maofoda 2.665 Mo ñWmZ {ZpíMV H$am`Mo Amho. Amåhm§bm _mhrV Amho H$s Vr 2 Am{U 3 À`m _Üò òVo. AmnU g§»`maofoda 2 d 3 `m§À`m Xaå`mZMm ^mJ nmhÿ. g_Om, AmnU Ë`mMo 10 g_mZ ^mJ H$ê$Z àËòH$ ^mJmbm AmH¥$Vr 1.11 (i) à_mUo àËòH$ q~Xÿbm A§{H$V H$ê$.

Voìhm 2 À`m COì`m ~mOyH$S>rb n{hbo {MÝh 2.1 Xe©{dVo Xþgao {MÝh 2.2 hr g§»`m Xe©{dVo. Amnë`mbm AmH¥${V 1.11(i) _Üò 2 Am{U 3 _Yrb {dÔZ q~XÿZm nhmÊ`mgmR>r H$mhr AS>MU {Z_m©U hmoV Agob Va Ë`m§Zm ñnîQ> ê$nmV nmhÊ`mgmR>r Vwåhr ~{h©dH$ {^JmMm Cn`moJ H$am. ho AmH¥$Vr 1.11 (ii) à_mUo {Xgob. AmVm 2.6 Am{U 2.7 `m§À`m Xaå`mZ 2.665 hr g§»`m Amho. åhUyZ `m AmnU 2.6 Am{U 2.7 `m_Yrb ^mJmda bj H|${ÐV H$ê$. `m ^mJmV nwÝhm 10 ^mJ {d^mOZ H$ê$. `mVrb n{hbm ^mJ 2.61Xe©dob, Xþgam ^mJ 2.62 Xe©dob 3 `mbm ñnîQ> ê$nmV nmhÊ`mgmR>r AmH¥$Vr. 1.12 (ii) _Üò XmI{dbo Amho.

AmH¥$Vr 1.12

AmH¥$Vr 1.11

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16 J{UV

AmVm `m AmH¥$VrV, 2.665 nwÝhm 2.66 d 2.67 À`m Xaå`mZ Amho. `mH$[aVm `m g§»`maofodarb `m ^mJmda bj H|${ÐV H$ê$. (AmH¥$Vr nhm 1.13 (i) ) Am{U. H$ënZm H$am H$s hm ^mJ g_mZ ^mJmV dmQ>bm Jobm Amho. `mbm A{YH$ ñnîQ> nUo nmhÊ`mgmR>r Ë`mbm {dembZ H$ê$ Ogo {H$ AmH¥$Vr. 1.13 (ii) _Üò XmI{dbo Amho. òWo n{hbo {MÝh 2.661 Xe©{dVo. nwT>o {MÝh 2.662 bm Xe©{dVo B. eodQ>r 2.665 hr g§»`m 5 À`m ^mJmH$S>o Amho.

AmH¥$Vr 1.13

EH$m {dembH$mZo g§»`maofodarb g§»`m§Mo {dembZ H$ê$Z XmI{dÊ`mÀ`m nÜXVrbm H«$_mZo òUmao {dembZ åhUVmV.

AemàH$mao `m nÜXVrZo gm§V Xem§e ê$nVrb dmñVd g§»`m§Mr g§»`maofodarb pñW{V AmnU ñnîQ>nUo nmhÿ eH$Vmo.

AmVm AmnU g§»`maofodarb AZ§V AZmdVu Xem§e {dñVmamVrb dmñVd g§»`m§Mr pñW{V nmhÊ`mÀ`m à`ËZ H$ê$. EH$m {dembH$mMm Cn`moJ H$ê$Z g§»`maofodarb g§»`m§Mo ñWmZ nmhþ eH$Vmo.

CXmhaU 11 : 5 37. Mo 5.37777 n`ªV g§»`maofoda 5 Xem§e ñWimn`ªV {dembZ XmIdm. CH$b : nwÝhm EH$Xm AmnU H«$_mZo òUmao {dembZ dmnê$ Á`mV 5 37. Mm g_mdoe Agboë`m g§»`maofoMr bm§~r H$_r H$aV Om. àË`j nmhþ {H$ 5 37. ho 5 d 6 À`m Xaå`mZ AmhoV. nwT>rb nmàrbm Amåhr 5 37. Mo ñWmZ 5.3 Am{U 5.4 À`m Xaå`mZ {ZpíMV H$am. A{YH$ n[anyU© ñdê$nmV ñWmZ nmhÊ`mgmR>r Amåhr g§»`maofoMr 5.37 Vo 5.38 Xaå`mZ 10 g_mZ ^mJmV {d^mJUr H$am. naV A{YH$ n[anyU© ñdê$nmV nmhÊ`mgmR>r 5.377 Am{U 5.378 `m Xaå`mZ nwÝhm 10 g_mZ ^mJmV {d^mJUr H$am.

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g§»`m nÜXVr 17

Am{U 5 37. Mo ñWmZ nmhÿ ho AmH¥$Vr 1.14 (iv) _Üò XmI{dbo Amho. bjmV ¿`m {H$, ho 5.3777 nojm 5.3778 A{YH$ Odi AmhoV (AmH¥$Vr 1.14 (iv) nhm).

AmH¥$Vr 1.14

gwMZm : EH$m {dembZ q^JmÀ`m gmhmæ`mZo g§»`maofoda Á`m {R>H$mUr 5 37. XmI{dbo Amho Ë`m g§»`maofoÀ`m ^mJmMr bm§~r _`m©{XV KoD$Z . åhUOoM H$_r H$ê$Z d {Za§Va {dembZ H$ê$Z Ë`m g§»òÀ`m àíZm~Ôb AMwH$Vm {_io n`ªV hr H¥$Vr Mmby R>odyZ nmhÿ eH$Vmo.

Vwåhm§bm AmVm g_Obo AgUma H$s g§»`maofoda dmñVd g§»`m§Mo dY©Z (_moR>o) AI§S>rV nwÝhm nwÝhm òUmÀ`m Xem§e {dñVmamà_mUo g_mZ {H«$`m AgVo.

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18 J{UV

darb MMm© Am{U dY©Z H$ê$Z Amåhr Ago åhUy eH$Vmo H$mo àËòH$ dmñVd g§»òbm g§»`maofoda EH$ A{ÛVr` q~XÿZo Xe©{dVm oWo. gmo~V g§»`maofodarb àËòH$ q~Xÿ EH$ Am{U EH$M dmñVd g§»òer {ZJS>rV AgVmo.

ñdmÜ`m` 1.4

1. A{YH$m{YH$ dY©Z H$ê$Z g§»`maofoda 3.765 XmIdm.

2. dY©Z H$ê$Z 4 26. bm 4 Xem§e ñWimn`ªV XmIdm.

1.5 dmñVd g§»`mdarb {H«$`m (Operations on Real Number)

_mJrb B`ËVoV AmnU {eH$bm AmhmV H$s n[a_o` g§»`m H«$_{ZanojVm, gmhM`© Am{U {dVaU JwUY_© ~m~VrV g§d¥ËV AgVmV Am{U AmnU ho gwÜXm {eH$bmo {H$ Oa XmoZ n[a_o` g§»`m§Mr AmnU ~oarO, dOm~mH$s, JwUmH$ma (eyÝ` gmoSy>Z), ^mJmH$ma Ho$bm Va Amåhmbm n[a_o` g§»`mM {_iVo. AmnU ho gwÜXm nmhVmo H$s An[a_o` g§»`m gwÜXm H«$_{ZanojVm, gmhM`© Am{U {dVaU {Z`_m§Mo g_mYmZ H$aVmV VWm{n An[a_o` g§»`m§Mr ~oarµO, ^mJmH$ma Am{U JwUmH$ma hm Zoh_r An[a_o` g§»`m AgVo. CXm.

6 6 2 2 3 3( ) + −( ) ( ) − ( ) ( ) × ( ), , Am{U 17

17`m n[a_o` g§»`m AmhoV.

AmVm AmnU nmhÿ H$s Oa EH$m n[a_o` g§»òV An[a_o` g§»`m {_i{dbr Am{U EH$m n[a_o` g§»òbm EH$m An[a_o` g§»òZo JwUbo Va H$m` hmoVo? CXm: 3 hr An[a_o` Amho. 2 3+ Am{U 2 3 ~m~VrV H$m`? 3 hr EH$ An[a_o` g§»`m Amho (AZ§V AZmdVu Xem§e) åhUyZM 2 3+ Am{U

2 3 `m H$arVm gwÜXm eŠ` Amho. 2 3+ Am{U 2 3 `m gwÜXm An[a_o` g§»`m AmhoV.

CXmhaU 12: 7 57

52 21 2, , ,+ −π `m An[a_o` g§»`m AmhoV H$s ZmhrV ho Vnmgm.

CH$b 5 2 236= . ....

2 1 4142= . ....

π = 3 1415. ....

Z§Va

7 5 = 15.652.............

7

5 = 7 5

5 5

7 5

53 1304= = . ...........

2 21+ = 22.4142............

π − 2 = 1.1415........

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g§»`m nÜXVr 19

ho gd© AZ§V AZmdVu Xem§e AnyUmªH$ AmhoV. åhUyZ `m gd© An[a_o` g§»`m AmhoV.

CXmhaU 13: 2 2 5 3+( ) Am{U 2 3 3− `m§Mr ~oarO H$am.

CH$b 2 2 5 3 2 3 3+( ) + −( ) = 2 2 2 5 3 3 3+( ) + −( ) = 2 1 2 5 3 3+( ) + −( ) = 3 2 2 3+

CXmhaU 14: 6 5 bm 2 5 Zo JwUm.

CH$b 6 5 2 5× = 6 2 5 5× × . = 12 × 5 = 60

CXmhaU 15: 8 15 bm 2 3 Zo ^mJm.

CH$b 8 15 2 3÷

= 8 3 5

2 3

×

= 4 5

`m CXmhaUmdê$Z AmnU Imbrb ~m~r gË` AmhoV. Aer Amem H$aVmo.

1. EH$ An[a_o` d n[a_o` g§»òMr ~oarO qH$dm dOm~mH$s An[a_o` g§»`m AgVo.

2. EH$ An[a_o` g§»ò gmo~V EH$ eyÝòËVa (non-zero) n[a_o` g§»`m§Mm JwUmH$ma qH$dm ^mJmH$ma An[a_o` g§»`m AgVo.

3. Oa AmnU XmoZ n[a_o` g§»`m§Mr ~oarµO, dOm~mH$s JwUmH$ma qH$dm ^mJmH$ma H$aVmo Voìhm CÎma n[a_o` qH$dm An[a_o` g§»`m Agy eH$Vo.

AmVm Amåhr dmñVd g§» oÀ`m dJ©_yi {H«$òH$‹S>o diy`m.

Oa a hr Z¡g{J©H$ g§»`m Agob Va a b= åhUOoM b2 = a, åhUOoM b > 0 Am{U hrM ì`m»`m YZ dmñVd g§»`|Zm gwÜXm bmJyy Ho$br OmD$ eH$Vo.

Oa, a > 0 Va dmñVd g§»`m Agob Va a b= åhUZoM b2 = a Am{U b > 0.

1.2 ^mJmV Amåhr nmhrbo Amho H$s g§»`maofoda n bm n YZ nwUmªH$ Amho ho H$go XmI{dbo OmVo Vo nmhrbo Amho.

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20 J{UV

AmVm AmnU ho XmIdy {H$ H$emàH$mao x bm. x hr {Xbobr YZmË_H$ dmñVd g§»`m Amho ho ^m¡{_VrH$ ê$nmZo _mhrV Ho$bo OmVo CXmhaU åhUyZ x = 3.5 H$aVm {_idy. AWm©V² Amåhr 3 5. ^m¡{_VrH$ ê$nmZo (geometrically) {_idy.

AmH¥$Vr 1.15

{Xboë`m aofoda EH$ pñWa q~Xÿ A nmgyZ 3.5 EH$H$ A§Vamda B q~Xÿ {_iVmo. AB = 3.5 EH$H$ Amho. (AmH¥${V 1.15) B q~XÿnmgyZ 1 EH$H$ A§Vamda EH$ ZdrZ q~Xÿ ¿`m. Ë`mbm C _mZm. AC Mm _Ü`q~Xþ AmoiIyZ Ë`mbm O Zmd Úm. OC {ÌÁòZo AY©dVw©i H$mT>m. B _YyZ OmUmar AC b§~ aofm H$mT>m Am{U Vr AY©dVw©imbm D _Üò N>oXVo Voìhm BD = 3 5. Amho.

A{YH$ ì`mnH$ ñdê$nmV x Mr qH$_V H$mT>Ê`mgmR>r, x hr YZdmñVd g§»`m Amho. B q~Xÿ Agm KoD$ {H$ AB = x EH$H$ Amho (ho AmH¥$Vr 1.16 XmI{dbo Amho). C q~Xÿ Agm ¿`m {H$ BC = 1 EH$H$ Agob. Z§Va x = 3.5 {H«$ò à_mUo BD = x {_iob.

Amåhr ho CÎma nm`WmJmoagÀ`m à_o`mZo {gÜX H$ê$ eH$Vmo.

bjmV R>odm, AmH¥$Vr 1.16 _Üò DOBD hm H$mQ>H$moZ {ÌH$moU AgyZ dV©wimMr {ÌÁ`m x +12

EH$H$ Amho.

åhUyZ OC = OD = OA = x +12

EH$H$

OB = x x− +

1

2=

x −12

åhUyZ nm`WmJmoag à_o`mZwgma Amnë`mbm {_iob.

AmH¥$Vr 1.16

BD2 =OD2 – OB2 = x x x x+

− −

= =1

2

1

2

4

4

2 2

`mdê$Z {gÜX hmoVo BD = x `m aMZodê$Z ho XmI{dÊ`mMo Ñí`r` d ^m¡{_VrH$ àH$ma àmßV Pmë`mZo gd© dmñV{dH$ g§»`m

x > 0 H$[aVm x bm ApñVËdmV AmhoV. Oa Vwåhr g§»`maofoda x Mo ñWmZ _m{hV H$ê$ eH$mb.

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g§»`m nÜXVr 21

Oa AmnU aofm BC bm g§»`maofm _mZyZ B bm eyÝ` d C bm 1 EH$H$ _mZbo BË`mXr. B H|$Ð _mZyZ. BD {ÌÁòZo EH$ Mmn H$mT>m. Omo g§»`maofobm E _Üò N>oXVmo (AmH¥$Vr 1.17 nhm) Voìhm B ho x Xe©{dVo.

AmH¥$Vr 1.17

AmVm AmnU dJ©_wimÀ`m g§H$ënZobm KZ_yi, MVwW©_yi Am{U gm_mÝ`nUo n do _yi. òWo YZnyUmªH$ Amho. Vwåhr AmR>dU H$am H$s _mJrb B`ÎmoV Vwåhr dJ©_yi Am{U KZ_yimMm Aä`mg Ho$bm Amho.

83 ho H$m` Amho? AmnUmg _mhrV Amho {H$ hr EH$ YZ g§»`m AgyZ {OMm KZ 8 Amho. Am{U Vwåhr AZw_mZ Ho$bo Agob {H$ 83 = 2 Amho. AmVm 2435 Mr qH$_V H$mT>Ê`mMm à`ËZ H$ê$. Vwåhm§bm _mhrV Amho 'b' EH$ g§»`m Amho. b Mr {H$_V H$mT>m. b5 = 243 Mo CÎma Amho 3. åhUyZ 243 35 = Amho.

`m CXmhaUmdê$Z Vwåhr an Mr ì`m»`m H$ê$ eH$Vm Á`mV a > 0 EH$ dmñVd g§»`m Amho mm{U 'n' hm YZ nyUmªH$ Amho.

g_Om a > 0 EH$ dmñVd g§»`m Amho Am{U 'n' hm YZnwUmªH$ Amho Va an = b Oa b2 = a Am{U b > 0 . bjmV R>odm 2 83, , an _Üò Cn`moJ Ho$bobo {MÝh " ' bm H$aUr {MÝh (radical sign) åhUVmV.

Amåhr AmVm dJ©_wimer g§~§YrV H$mhr {ZË` g_rH$aUmMr `mXr {Xbr Amho. Or {d{dY {H«$ò_Üò Cn`moJr Amho. _mJrb dJm©V `mVrb H$mhr gwÌmer AmnU n[a{MV AmhmV. am{hbobr gwÌo dmñVd g§»`mÀ`m ~oaOoda d JwUmH$mamÀ`m {dVaU {Z`_mZo Am{U {ZË` g_rH$aU (x + y)(x – y) = x2 – y2 Zo Á`mV x Am{U y `m dmñVd g§»`m AmhoV g_Om {H$ a Am{U b KZ dmñVd g§»`m AmhoV Voìhm. (i) ab a b=

(ii) ab

ab

=

(iii) a b a b a b+( ) −( ) = −

(iv) a b a b a b+( ) −( ) = −2

(v) a b c d ac ad bc bd+( ) +( ) = + + +

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22 J{UV

(vi) a b a ab b+( ) = + +2

2

`m {ZË`g_rH$aUm§Mr H$mhr CXmhaUo nmhÿ.

CXmhaU 16: Imbrb ametZm gai ê$n Úm.

(i) 5 7 2 5+( ) +( ) (ii) 5 5 5 5+( ) −( ) (iii) 3 7

2

+( ) (iv) 11 7 11 7−( ) +( )CH$b : (i) 5 7 2 5+( ) +( ) = 10 5 5 2 7 35+ + +

(ii) 5 5 5 5+( ) −( ) = 5 52

2

− ( ) =25 – 5 = 20

(iii) 3 72

+( ) = 3 2 3 7 72 2( ) + + ( ). =3 2 21 7+ + =10 2 21+

(iv) 11 7 11 7−( ) +( ) = 11 72 2( ) − ( ) = 11 – 7 = 4

{Q>n : darb CXmhaUmV gai ê$n Úm `mMm AW© åhUOo n[a_o` AWdm An[a_o` g§»òMr ~oarµO Agm {b{hVmV.

1

2 g§»`maofoda H$moR>o Xe©{dVmV `m g_ñòda {dMma H$ê$Z `m ^mJmbm òWo g§ndy. Vwåhmbm

_mhrV Amho H$s hr EH$ An[a_o` g§»`m Amho. Oa N>oX EH$ n[a_o` g§»`m Agob Va `mbm ghO gmoS>{dVm òVo. AmnU nmhÿ {H$ `mÀ`m N>oXmMo n[a_o`rH$aU (n[a_o` H$aUo) H$aVm òVo H$m? åhUOoM N>oXmbm n[a_o` g§»òV ê$nm§VarV H$aVm òVo H$m. `m H$arVm Amnë`mbm dJ©_yi g§~§YrV gwÌm§Mr Amdí`H$Vm Amho. AmVm AmnU nmhÿ {H$ ho H$go H$aVmV.

CXmhaU 17: 1

2 `m N>oXmMo n[a_o`H$aU H$am.

CH$b : AmnU 1

2 bm g_Vwë` amerV {bhm {H$ Á`mV EH$ n[a_o` g§»`m Agob. Amnë`mbm

_mhrV Amho {H$ 2 {h n[a_o` g§»`m Amho. AmnUm§bm ho gwÜXm _mhrV Amho.

1

2 bm

22

Zo JwUë`mZ§Va Amnë`mbm EH$ g_Vwë` amer àmßV hmoVo H$maU 2

21= Amho `m

XmoZ ametZm EH$Ì KoVë`mZ§Va AmnUmg {_iVo.

1

2

1

2

2

2

2

2= × =

`mñdê$nmV 1

2 Mo g§»`maofoda ñWmZ {ZpíMV H$aUo gmono hmoVo. hr g§»`m 0 Am{U 2 `m§À`m_Üò

pñWV Amho.

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g§»`m nÜXVr 23

CXmhaU 18: 1

2 3+ À`m N>oXmMo n[a_o`H$aU H$am.

CÎma : da {Xbobo {ZË`g_rH$aU (iv) dmnê$Z 1

2 3+ bm 2 3− Zo JwUyZ d ^mJmH$ma H$ê$Z

Amåhmbm {_iVo.

1

2 3

2 3

2 3+× −

− = 2 3

4 32 3

−−

= −

CXmhaU 19: 5

3 5− À`m N>oXmMo n[a_o`H$aU H$am.

CH$b : da {Xbobo {ZË`g_rH$aU (iii) dmnê$.

AmVm, 5

3 5− = 5

3 5

3 5

3 5−× +

+

= 5 3 5

3 5

+( )−

= −

+( )5

23 5

CXmhaU 20: 1

7 3 2+ À`m N>oXmMo n[a_o`H$aU H$am..

CH$b: : 1

7 3 2+ =

1

7 3 2

7 3 2

7 3 2+× −

−

= 7 3 2

49 18

−−

= 7 3 2

31

−

`mdê$Z Ooìhm EImÚm amerÀ`m N>oXmV dJ©_yi Agbobo nX AgVo Voìhm {Vbm EH$ Aem g__wë` amerV Á`mMm N>oX EH$ n[a_o` g§»`m Agob Ago ê$nm§V[aV H$aÊ`mÀ`m {H«$`obm N>oXmMo n[a_o`H$aU (rationalising the denominator) åhUVmV.

ñdmÜ`m` 1.5

1. Imbrb g§»`m§Mo n[a_o` d An[a_o` g§»`oV dJuH$aU H$am.

(i) 2 5− (ii) 3 23 23+( ) −

(iii) 2 7

7 7 (iv)

1

2 (v) 2p

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24 J{UV

2. Imbrb àËòH$ ametZm gai ê$n Úm.

(i) 3 3 2 2+( ) +( ) (ii) 3 3 3 3+( ) −( ) (iii) 5 2

2

+( ) (iv) 5 2 5 2−( ) +( ) 3. p Mr ì`m»`m dVw©imMm n[aK ( C) Am{U Ë`mMm ì`mg ( d) `m§À`m JwUmoËVamV ì`ŠV Ho$bo

OmVo. AWm©V π =cd , òWo {damoYm^mg {XgyZ òVmo H$maU p hr An[a_o` g§»`m Amho. hm

{damoYm^mg H$gm gmoS>dmb?

4. g§»`maofoda 9 3. XmIdm.

5. Imbrb ametÀ`m N>oXmMo n[a_o`H$aU H$am.

(i) 1

7 (ii)

1

7 6−

(iii) 1

5 2+ (iv) 1

7 2−1.6 dmñVd g§»`mH$[aVm KmVm§H$mMo {Z`_

Vwåhm§bm AmR>dVo H$m `m§Zm gai ê$n H$go XoVmV?

(i) 172 . 175 = (ii) (52)7 =

(iii) 23

23

10

7 = (iv) 73. 93 =

Amnë`mbm `m§Mr CÎmao {_iVmV H$m? Vr Imbrbà_mUo AmhoV.?

(i) 172. 175 = 177 (ii) (52)7 = 514

(iii) 2323

23

10

7

3= (iv) 73.93 = 633

darb CÎmao {_i{dÊ`mgmR>r Vwåhr Imbrb KmVm§H$m§Mm {Z`_m§Mm Cn`moJ Ho$bm Agob. Oo Vwåhr _mJrb dJm©V {eH$bm AmhmV. (a, n Am{U m `m Z¡g{J©H$ g§»`m AmhoV. . a hm AmYmam§H$, m d n ho KmVm§H$).

(i) am . an = am + n (ii) (am)n = amn

(iii) aa

a m nm

nm n= >( )− (iv) ambn = (ab)m

(a)0 åhUOo H$m`? `mMr qH$_V 1 Amho åhUyZ {Z`_ (iii) dmnê$Z 1a

ann= − ho {_idy

eH$Vmo. Amåhr AmVm G$U KmVm§H$ gwÜXm KoD$ eH$Vmo.

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g§»`m nÜXVr 25

CXmhaUo :

(i) 172.17–5 = 17–3 = 1

173 (ii) (52)–7 = 5–14

(iii) 23

2323

10

7

17

−−= (iv) (7)–3.(9)–3 = (63)–3

Amåhm§bm Imbrb qH$_Vr {_idm`À`m AmhoV.

(i) 2 2

2

3

1

3⋅ (ii) 3

1

5

4( ) (iii) 7

7

1

5

1

3

(iv) 13 17

1

5

1

5⋅

hr CXmhaUo Amåhr H$er gmoS>dy eH$Vmo ? `m AJmoXa KmVm§H$mÀ`m {Z`_mMm Aä`mg Ho$bm Amho. Vo `oWo bmJy hmoVmV. VargwÕm Ooìhm AmYmam§H$ hm YZ dmñVd g§»`m Amho Am{U KmVm§H$ n[a_o` g§»`m Agob (Z§Va Vwåhr KmVm§H$ dmñVd g§»`m AmhoV `mMm Aä`mg H$amb). na§Vy ho {Z`_ gm§JÊ`mnydu Am{U `m {Z`_m§Zm bmJy H$aÊ`mnydu ho g_OyZ KoUo Amdí`H$ Amho {H$ CXmhaU

4

3

2 Mm AW© H$m`

Amho.

^mJ 1.4 _Ü`o Amåhr an Mr ì`m»`m Ho$br Amho OoWo dmñVd g§»`m a > 0 Imbrbà_mUo Amho.

g_Om a > 0 hr YZ dmñVd g§»`m Amho Am{U 'n' hm YZ nwUmªH$ Amho Ooìhm an = b AgVmo Oa bn = a Am{U b > 0 Agob KmVrH$m§À`m ^mfoV Amåhr a an n=

1 ì`m»`m H$aVmo. åhUyZ {ZpíMnUo 2 23

13= AmVm AmnU 4

32 H$S>o XmoZ nÜXVrZr nmhÿ eH$Vmo.

4 4 2 8

3

2

1

2

3

3= ( ) = =

4 4 64 8

3

2 3

1

2

1

2= ( ) = ( ) =

`mdê$Z Amåhr Imbrb ì`m»`m H$é eH$Vmo.

g_Om a > 0 ho KZ dmñVd g§»`m Amho. m Am{U n ho nyUmªH$ AgyZ m d n Mo 1 {edm` Xþgao gm_mB©H$ Ad`d ZgVrb Am{U n > 0 Va.

a a amn n

m mn= ( ) =

Amåhmbm Imbrb KmVm§H$mMo {Z`_ {dñV¥V ê$nmV {_iVmV.

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26 J{UV

g_Om a > 0 hr dmñVd g§»`m AgyZ p Am{U q `m n[a_o` g§»`m AmhoV Voìhm Amåhm§bm -

(i) ap. aq = qp + q (ii) (ap)q = apq )

(iii) aa

ap

qp q= − (iv) apbp = (ab)p

Vwåhr AmVm m AJmoXa {dMmaboë`m àíZm§Mo CÎma emoYÊ`mgmR>r m {Z`_m§Mm Cn`moJ H$ê$ eH$Vm.

CXmhaU 21: gai ê$n Úm.

(i) 2 2

2

3

1

3⋅ (ii) 31

5

4( ) (iii) 7

7

1

5

1

3

(iv) 13 17

1

5

1

5⋅

CH$b :

(i) 2 2

2

3

1

3⋅ = 22

3

1

3+

= 23

3

= 21

= 2

(ii) 3

1

5

4( ) = 34

5

(iii) 7

7

1

5

1

3

= 71

5

1

3−

= 73 5

15

−

= 72

15

−

(iv) 13 17

1

5

1

5⋅ = 13 171

5×( ) = 2211

5( )

ñdmÜ`m` 1.6

1. emoYm

(i) 641

2 (ii) 321

5 (iii) 1251

3

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g§»`m nÜXVr 27

2. emoYm

(i) 93

2 (ii) 322

5 (iii) 163

4 (iv) 1251

3

−

3. gai ê$n Úm:

(i) 2 2

2

3

1

5⋅ (ii) 1

33

7

(iii)

11

11

1

2

1

4

(iv) 7 8

1

2

1

2⋅

1.7 gmam§e (Summary)

`m àH$aUmV Vwåhr Imbrb ~m~tMm Aä`mg Ho$bm.

1. g§»`m r bm n[a_o` g§»`m åhUVmV Oa Vr pq

`m ñdê$nmV {bhrVm oV Agob. oWo p Am{U q nyUmªH$ AgyZ q ≠ 0

2. g§»`m s bm An[a_o` g§»`m åhUVmV Oa Vr pq `m ñdê$nmV {b{hVm òV Zgob. òWo p

Am{U q nyUmªH$ AgyZ q ≠ 0.

3. EH$m n[a_o` g§»òMm Xem§e {dñVma EH$Va gm§V (I§S>rV) qH$dm AZ§V AmdVu AgVo. VWm{n Á`m g§»òMm Xem§e {dñVma gm§V qH$dm AZ§V AmdVu AgVmo Vr n[a_o` g§»`m AgVo.

4. EH$m An[a_o` g§»òMm Xem§e {dñVma AZ§V AZmdVu AgVmo na§Vy, Á`m g§»òMm Xem§e {dñVma AZ§V AZmdVu AgVmo Vr An[a_o` g§»`m AgVo.

5. gd© n[a_o` Am{U An[a_o` g§»`m§À`m EH$ÌrH$aUmbm dmñVd g§»`m åhUVmV.

6. g§»`m aofoda àËòH$ q~Xÿer {ZJS>rV EH$_od dmñVd g§»`m AgVo. g§»`maofoda dmñVd g§»òer {ZJS>rV EH$_od q~Xÿ AgVmo.

7. Oa r n[a_o` Am{U s An[a_o` g§»`m AmhoV Va r + s Am{U r – s An[a_o` g§»`m AgVmV

Am{U s Am{U r s `m gwÜXm An[a_o` g§»`m AgVmV òWo r ≠ 0.

8. YZ dmñVd g§»`m a Am{U b gmR>r Imbrb {ZË`g_rH$aUo bmJy hmoVmV.

(i) ab a b= (ii) ab

ab

=

(iii) a b a b a b+( ) −( ) = − (iv) a b a b a b+( ) −( ) = −2

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28 J{UV

(v) a b a ab b+( ) = + +2

2

9. 1a b+ À`m N>oXmMo n[a_o`H$aU H$aÊ`mgmR>r AmnU

a ba b

−−

Zo JwUVmo oWo, a Am{U b nyUmªH$ AmhoV.

10. g_Om a > 0 EH$ dmñVd g§»`m Agob Am{U p Am{U q n[a_o` g§»`m AgVrb Va.

(i) ap . aq = ap+q (ii) (ap)q = apq

(iii) aa

ap

qp q= − (iv) apbp = (ab)p

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`wŠbrS>À`m ^y{_VrMm n[aM` 27

àH$aU - 2

`wŠbrS>À`m ^y{_VrMm n[aM` (Introduction to Euclid's Geometry)

2.1 àñVmdZm

^y{_Vr (Geometry) m eãXmMm CJ_ J«rH$ mfoVrb eãX {O`mo (Geo) åhUOoM y (O_rZ) Am{U _oQ´>r (Metry) åhUOoM "_mnZ'`mnmgyZ Pmbm Amho. `mdê$Z AmnUmbm bjmV òVoH$s, ^y_rVrMr H$ënZm hr n¥ÏdrÀ`m qH$dm O{_ZrÀ`m _moO_mnZmgmR>r Amdí`H$Vm Agë`m_wio CJ_ Pmbm Amho. J{UVmÀ`m `m emIoMm Aä`mg doJdoJù`m àmMrZ ì`ŠVrH$Sy>Z doJdoJù`m ê$nmV Ho$bo Jobo. BWoo B{OßV Agmo, MrZ, ^maV, J«rH$ qH$dm BZH$mg B. ì`dhmam_Üò AZoH$ àH$maÀ`m g_ñ`m Amë`m. Ë`mMo {ZdmaU H$aÊ`mgmR>r Ë`mZm ^y{_VrÀ`m doJdoJù`m JwUY_m©Mm {dH$mg H$aUMoo `mo½` dmQ>bo.

CXmhaUmW©, Ooìhm ZmB©b (Nile) ZXrbm nya òV Ago Voìhm Á`m§À`m O{_Zr (eoVr) ZXrÀ`m Odi AgV Ë`m eoVrMo ~m§Y nwamVyZ dmhÿZ OmV AgV. Aem dmhÿZ Joboë`m ~m§Ym§Zm naV ~m§Ybo OmV Ago. `m H$m_mgmR>r B{OßVdmgr`m§Zr joÌ\$i H$mT>Ê`mgmR>r AZoH$ ^y{_VrÀ`m {Z`_mMm Cn`moJ Ho$bm. Aem ^y{_VrMm Cn`moJ AÞgmR>m H$aÊ`mgmR>r, Am`VmH¥$Vr ^m§S>çm§Mr {Z{_©Vr qH$dm {na°{_S> V`ma H$aÊ`mgmR>r hmoV Ago. Vo EH$ H$mnboë`m {na°_{S>Mo (truncated pyramid) KZ\$i H$mT>Ê`mMo gyÌ nU OmUV AgV. (AmH¥$Vr 2.1 nhm) AmnUmg _mhrV Amho H$s, {na°{_S> hr EH$ Aem àH$maMr KZmH¥$Vr Amho {OMm nm`m {ÌH$moUmH¥$Vr qH$dm Mm¡agmH¥$Vr qH$dm ~hþ^wOmH¥$Vr AgVmo Am{U {VMo darb gd©> {ÌH$moUr n¥îR^mJ EH$mM q~Xÿ_Üò da {_iVmV.

AmH¥$Vr 2.1

H$mnbobm {na°{_S>>

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^maVmÀ`m CnI§S>m_Yrb hS>ßnm Am{U _mohZOmoXmamo B. {R>H$mUr Ooìhm ImoXH$m_ Ho$bo Voìhm AmnUmg Ago AmT>iyZ Ambo H$s, qgYy Imoè`mVrb bmoH$m§Zr (B.g. nyd© 3000) WmoS>çm à_mUmV ^y{_VrMm dmna Ho$bm hmoVm VoWrb. ehao AË`mYw{ZH$ àH$mao {dH$grV hmoVr Am{U Vr `moOZm~ÜX nÜXVrZo {Z_m©U H$aÊ`mV Ambr hmoVr. añVo nañnam§Zm g_m§Va hmoVo Am{U ^y{_JV JQ>matMr hmoVr. VoWrb a{hdmer A§H$J{UVm_Üò nyU©nUo {ZnwU hmoVo. Ë`mZm joÌ_mnZmMo kmZ hmoVo. ~m§YH$m_mgmR>r bmJUmè`m drQ>m ^Å>rV ^mOë`m Om`À`m Am{U Ë`m§Mr bm§~r : é§Xr : C§Mr hr AZwH«$_o 4 : 2 : 1 Agm`Mr.

àmMrZ ^maVmV ""gwb^mgwÌ'' (8000 BC Vo 500 BC) hm ^y{_Vr aMZoMm _hËdnyU© J«§W hmoVm. d¡{XH$ H$mimV ^y{_VrMm Cn`moJ hm d¡{XH$ nyOogmR>r bmJUmao doJdoJù`m àH$maMo A½ZrHw§$S> {Z_m©U H$aÊ`mgmR>r H$aÊ`mV Ambm. n{dÌ Ap½Z A{YH$ à^mdembr hmoÊ`mgmR>r Ë`m§Mo ñWmZ, Ë`m§Mm AmH$ma Am{U joÌ\$im§~m~VrV ñnîQ>nUo V`ma Ho$boë`m {Z`_mZwgma Agm`Mo. KaJwVr Ym{_©H$ {H«$`m§gmR>r Mm¡H$moZr qH$dm JmobmH$ma AmH¥$VtMm dmna H$aV AgV. na§Vw gmd©O{ZH$ nyOoÀ`m {R>H$mUr Zoh_r Am`VmH¥$Vr qH$dm {ÌH$moUmH¥$Vr qH$dm g_b§~mH¥$Vr AmH$mamMr A½ZrHw§$S>o ~Zbobr Agm`Mr. {Xboë`m AWd©doXmV "lr`§Ì' _Üò ZD$ g_{X^yO {ÌH$moU "EH$_oH$m§Zm' OmoSy>Z V`ma Ho$bobo `§Ì Amho. ho `§Ì AemàH$mao OmoS>bobo AgVmV H$s, Á`m_wio 43 bhmZ {ÌH$moU {Z_m©U hmoVmV. AmH¥$VrMr aMZm H$aÊ`mgmR>r ^y{_VrÀ`m doJdoJù`m à`moJmMm Cn`moJ hmoV Ago na§Vy `m g§~§YrV {gÜXVoMr H$moUË`mM àH$mao MMm© Ho$br Zmhr.

darb CXmhaUo ho Xe©{dVmV H$s, ^y{_VrMm {dH$mg Am{U à`moJ OJmVrb àËòH$ ñWimda Ho$bm Amho nU `mMm Cn`moJ ì`dpñWV hmoV Zgo. àmMrZ H$mimV `m ^y_rVrÀ`m {dH$mgmMr EH$ amo_m§MH$mar JmoîQ> åhUOoM `m ^y{_VrMo kmZ EH$m {nT>rnmgyZ Xþgè`m {nT>rn`ªV nmohM{dÊ`mgmR>r EH$ Va Vm|S>r qH$dm VmS>mÀ`m d¥jmÀ`m nmZm§da Ago g§Xoe {bhÿZ doJdoJù`m _mJm©Zo g§H«${_V Ho$bo OmV hmoVo Ë`mMà_mUo AmnU nmhÿ eH$Vmo H$s, àmMrZ _mZd åhUOo ~o~rbmo{Z`m_Üò ^y{_Vr hr \$ŠV ì`dhm[aH$ ÑîQ>rH$moZmn`ªV _`m©{XV hmoVr. Ë`mMà_mUo ^maV Am{U amo__Üò ^y{_VrMm {dMma `mM ÑîQ>rH$moZmVyZ Pmbm.

B{OßV bmoH$m_Üò {dH$grV Ho$boë`m ^y{_VrMo n[aUm_r {dYmZoM CnbãY hmoVr. `m_Üò H¥$Vr ~ÔbMo H$moUVoM ì`mnH$ {Z`_ {Xbobo ZìhVo. dñVwV… ~o~rbmoZ Am{U BñÌmB©b bmoH$m§Zr ^y{_VrMm Cn`moJ EH$ nÜXVera {dkmZmÀ`m ê$nmV {dH${gV H$aÊ`mgmR>r Iyn H$_r à_mUmV H$m_ Ho$bo. na§Vy J«rH$ gma»`m bmoH$m§Zr H$maUmda (VH$m©da) OmñV â {Xbm, Aem {dYmZm_Üò J«rH$m§Zr Amnbr A{^éMr {Z_m©U Ho$br. {Z`_~ÜX H$maUmMm (deductive reasoning) dmna H$ê$Z Ë`m§Zr Ë`mMm nS>Vmim H$ê$Z g§emoYZ Ho$bo.

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àW_ {gÜXVm _m§S>Umao J«rH$ J{UVk Woëg (Thales) Vwåhmbm AmR>dV AgVrb. Ë`m§Zr EH$m dVw©imMm ì`mg dVw©imbm Xþ^mJVmo `mMr {gÜXVm _m§S>br. WoëgMm EH$_od {eî` åhUOoM nm`WmJmoag (Pythagoras) `m§Mo Zmd AmnU EoH$boM Amho. nm`WmJmoag Am{U Ë`m§À`m H$mhr ghH$mè`m§Zr AZoH$ ^y{_Vr JwUY_mªMm emoY bmdbm Am{U ^y{_VrÀ`m {gÜXm§Vm§Mm {dH$mg Ho$bm. ho H$m`© 300 BC n`ªV MmbyM amhrbo. `m doiobm B{OßV_Yrb AboŠP±{S´>`m `m J{UVmÀ`m {ejH$mZo Am{U `wwŠbrS>Zo J{UVmMr gd© {dYmZo EH${ÌV Ho$br. Ë`mbm "X Bboo_|Q²>g' (The Elements) Ago Zmd {Xbo. `m Bbo_|Q²>gbm 13 AÜ`m``m {d^mJUr Ho$br. àËòH$mbm "nwpñVH$m' Ago Zmd XoÊ`mV Ambo. `m nwpñVHo$VyZ nwT>rb {nT>rbm ^y{_VrMr _m{hVr nmohM{dÊ`mMm à`ËZ Ho$bm.

`m àH$aUmV AmnU `wŠbrS>À`m ^y{_Vr{df`r AgUmè`m ÑîQ>rH$moZm§Mr MMm© H$aUma AmhmoV Am{U ^y{_VrMr hr g§H$ënZm AmnU dV©_mZmV XoIrb dmnaÊ`mMm à`ËZ H$ê$

Woëg (B©.ny 640 - B©.ny. 546)> AmH¥$Vr 2.2

`yŠbrS> (B©.ny 352 - B©.ny. 265)> AmH¥$Vr 2.3

2.2 `wŠbrS>À`m ì`m»`m, _wbVËdo Am{U J¥hrVËdo.

`wŠbrS>À`m H$mimVrb H$mhr J«rH$ J{UVkm§Zr `wŠbrS>bm ^y{_VrMm Z_yZm (model) _mZbo. q~Xÿ (point), aofm (line), àVb (plane) BË`mXr g§H$ënZm Aem dñVydê$Z _m§S>Ê`mV Amë`m H$s. Ë`m dñVy Ë`m§À`m Odinmg ApñVËdmV hmoË`m. AmH$me(space) Am{U Ë`m§À`m OdinmgÀ`m KZ dñVy§À`m {gÜXm§VmMr g§H$ënZm {dH${gV H$aÊ`mV Ambr. EH$m KZmbm AmH$ma (size), ê$n (shape), ñWmZ (position) VgoM EH$m OmJodê$Z Xþgè`m OmJoda KoVmZm ñWmZ~Xb H$aV AgVo. Ë`mÀ`m H$S>m§Zr {d{eîQ> Pmboë`m mJmbm n¥îR>^mJ (surface) Ago åhUVmV. hr n¥îR>o KZmÀ`m EH$m ^mJmnmgyZ Xþgè`m ^mJmbm OmoS>VmV nU Ë`m§Zm {d{eîQ> Om‹S>r ZgVo. hr n¥îR>o gaiaofoV qH$dm dH«$aofoV AgVmV. `m aofm q~XÿZr ~Zboë`m AgVmV.

KZmnmgyZ q~Xÿn`ªV n¥îR>^mJ-H$S>m-{eamoq~Xÿ `m VrZ MaUmMm {dMma H$ê$. `m àËòH$ MaUm_Üò EH$ {dñVma H$_r hmoVmo. Ë`mbm _mno (dimension) åhUVmV. `mH$[aVm AmnU Ago åhUy eH$Vmo H$s. EH$m KZmMr VrZ _mno. EH$m n¥îR>mMr XmoZ _mno, EH$m aofoMr EH$ Am{U EH$m q~Xÿbm _mno ZgVmV. `yŠbrS>Zo m MaUmbm ì`m»òÀ`m ê$nmV ì`ŠV Ho$bo Amho. Ë`m§Zr m ì`m»`mgmR>r Amnë`m "Bbo_|Q²>g' À`m nwpñVH$m- 1 _Üò 23 ì`m»`m {Xboë`m AmhoV. Ë`mVrb H$mhr ì`m»`m Imbrb à_mUo.

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30 J{UV

1. EH$m q~Xÿ Mo nwÝhm ^mJ hmoV ZmhrV.

2. EH$ aofobm é§Xr ZgVo nU bm§~r AgVo.

3. àËòH$ aofobm A§Ë`q~Xÿ AgVmV.

4. EH$ gaiaofm Aer AgVo H$s {OÀ`mda q~Xÿ§Mo ñWmZ gnmQ> ê$nmV pñWa AgVo.

5. EH$m n¥îR>^mJmbm bm§~r Am{U é§Xr AgVo.

6. n¥îR>^mJmÀ`m H$S>m åhUOo aofm AgVmV.

7. EH$ gnmQ> n¥îR>^mJ Agm AgVmo H$s, Omo ñdV…da gai aofm§Zr gnmQ> ê$nmV pñWa AgVmo.

Oa AmnU bjnyd©H$ `m ì`m»`m§Mm Aä`mg Ho$bm Va AmnUmg bjmV òV H$s, H$mhr ^mJ bm§~r, é§Xr, gnmQ> n¥îR>^mJ B. JmoîQ>r§Mo ñnîQ> kmZ AmnUmbm {_iVo. CXmhaUmW©, `yŠbrS>Zo q~XÿMr {Xbobr ì`m»`m AmnU nmhÿ. `m ì`m»òV EH$ ^mJ `m JmoîQ>rMm CbJ‹S>m hmoUo Amdí`H$ Amho. AmnU Aer H$ënZm H$ê$ H$s, ì`m»ò_Üò Z_wX Ho$bobm EH$ ^mJ åhUOo joÌ Omo q~XÿZr _`m©XrV AgVmo. na§Vy naV AmnU XmoZ `m eãXmMr ì`m»`m nmhÿ eH$Vmo. åhUOoM EH$m dñVyMr ì`m»`m XoÊ`mgmR>r AmnUmg AZoH$ dñVy§Mr ì`m»`m XoÊ`mMr Amdí`H$Vm Amho Am{U Aem àH$mao AmnUmg H$YrM Z g§nUmar ì`m»`mMr EH$ l¥§Ibm {_iVo.

`m JmoîQ>rMm {dMma H$ê$Z J{UVkmZr Amnbo Ago _V _m§S>bo H$s, H$mhr ^m¡{_VrH$ ^mJmMr (nXmMr) ì`m»`m H$aVm òV Zmhr. AemàH$mao AmnU EH$m q~XÿÀ`m ^y_rVr` g§H$ënZoda {Xboë`m ì`m»òÀ`m VwbZoV EH$ `mo½` Am§VakmZmË_H$ Aä`mg àmßV H$ê$ eH$Vmo. Ë`mgmR>r EH$m q~Xÿbm EH$ gyú_ q~XÿÀ`m H$mhr Zm H$mhr A§V AgVmo qH$dm Ë`mbm H$mhr {g_m AmhoV.

AemàH$maMr g_ñ`m da {Xboë`m ì`m»`m 2 _Üò Amho. `m_Üò bm§~r Am{U é§XrMm g§X^© Ë`mbm Amho. Am{U `m_Üòhr H$moUË`mhr nXmMr n{hë`m§Xm ì`m»`m Ho$bobr Zmhr. `mMgmR>r H$mhr {df`m§À`m Aä`mgmgmR>r H$mhr nXm§À`m ì`m»`m H$aUo Amdí`H$ Zmhr. Ago g_OyZ Ë`m Aì`m»`rV R>odë`m AmhoV, `mgmR>r ^y{_Vr_Üò AmnU q~Xÿ, aofm Am{U àVb (`yŠbrS>À`m ^mfoV gnmQ> n¥îR>^mJ) Aì`m»`rV KQ>H$ AmhoV. nU Ë`mM~amo~a hr JmoîQ> gwÜXm Amdí`H$ Amho H$s, `m nXm§Zm AmnU A§VakmZmË_H$ ê$nmV {ZX}erV H$ê$ eH$Vmo AWdm ^m¡{VH$ à{VH¥$Vr (dñVy) À`m ghmæ`mZo Xe©dy eH$Vmo.

`m ì`m»`m§Mr gwédmV H$ê$Z `wŠbrS>Zo H$mhr {dYmZmZm {gÜX Z H$aVmM gË` _mZbr AmhoV. `m H$ënZm dmñVdm_Üò ñnîQ> hmoË`m. Ë`m§Zr Ë`m§Mr XmoZ {d^mJmV {d^mJUr Ho$br Vo åhUOo _ybVËdo (axioms) Am{U J¥hrVËdo (postulates). Ë`mZr J¥hrVËdo Or _w»`H$ê$Z ^y{_VrgmR>rM dmnabr OmV hmoVr Va _wbVËdo hr ^y{_Vr_Üò H$mhr {dYmZo V`ma H$aÊ`mgmR>r Á`m§Mr gË`Vm H$moUË`mhr nwamì`m {edm` _mÝ` H$amdr bmJV AgV. `wŠbrS>Mr hr H$mhr _wbVËdo Imbrb à_mUo AmhoV.

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1. Oo KQ>H$ EH$mM KQ>H$mer g_mZ AgVmV Vo nañnamer g_mZ AgVmV.

2. g_mZ KQ>H$mV g_mZ KQ>H$ {_i{dë`mg CÎma g_mZ òVo.

3. g_mZ KQ>H$mVyZ g_mZ KQ>H$ dOm Ho$ë`mg CÎma g_mZ òVo.

4. Á`m dñVy EH$_oH$mer g_mZ AgVmV Ë`m nañnamer g_mZ AgVmV.

5. nyU© hm Ë`mÀ`m I§S>mnojm _moR>m AgVmo.

6. g_mZ KQ>H$m§Mr XþßnQ> hr g_mZ AgVo.

7. g_mZ KQ>H$m§Mr {Zå_nQ> hr g_mZ AgVo.

hr gm_mÝ` _wbVËdo n[aUm_[aË`m gm§{JVbr AmhoV. n{hë`m gm_mÝ` _wbVËdmMm EH$ àVbr` AmH¥$VrgmR>r à`moJ Ho$bm OmD$ eH$Vmo. CXmhaUmW©, Oa EH$m {ÌH$moUmMo joÌ\$i ho EH$m Am`VmÀ`m joÌ\$imer g_mZ Agob Am{U `m Am`VmMo joÌ\$i ho EH$m Mm¡agmÀ`m joÌ\$imer g_mZ AgVo.

EH$mM àH$maÀ`m KQ>H$m§Mr VwbZm Ho$br OmD$ eH$Vo qH$dm {_i{dbr OmD$ eH$Vo. na§Vy {^ÝZ àH$maÀ`m KQ>H$m§Mr VwbZm H$aVm òV Zmhr. CXmhaUmW© EH$m aofobm EH$m Am`V~amo~a {_i{dbo OmD$ eH$V Zmhr. qH$dm EH$m H$moZmbmhr EH$m n§M^yOmH¥$Vrer VwbZm Ho$br OmD$ eH$V Zmhr.

darb {Xboë`m Mm¡Ï`m _wbVËdoMm AmYmao AmnUmg ho g_OVo H$s, Oa XmoZ dñVy g_mZ AgVrb Va Ë`m g_mZ AgVmV. åhUOoM H$moUVrhr dñVy hr ñdV…er g_mZM AgVo. _wbVËd - 5 "Á`m nojm _moR>m' Mr ì`m»`m XoVmo. CXmhaUmW© : Oa EImXm KQ>H$ B. H$moUË`mVar doJù`m KQ>H$mMm A Mm EH$ ^mJ Amho. Va . A `m KQ>H$mbm B Am{U BVa KQ>H$ C À`m g_mZVoÀ`m ê$nmV {b{hVm òD$ eH$Vo. gm§Ho$Vrb ê$nmV A>B Mm Amho H$s C KQ>H$ A = B + C Aer Amho.

AmVm `wŠbrS>À`m 5 ì`m J¥hrVËdm§Mr MMm© H$ê$.

J¥hrVËd 1 : EH$m q~XÿnmgyZ Xþgè`m q~Xÿn`ªV EH$ gai aofm H$mT>Vm òVo.

H$mhr _wbVËdo AmnUmg _mhrV Amho H$s, XmoZ {^ÝZ q~Xÿ _YyZ OmUmar aofm AmnU H$mTy> eH$Vmo na§Vw `mdê$Z AmnUmg _mhrV hmoV Zmhr H$s, AemàH$maÀ`m EH$m nojm A{YH$ gaiaofm AmnU H$mTy> eH$V Zmhr. na§Vw Amnë`m gd© Aä`mgmV `yŠbrS>Zo H$mhrhr Z gm§JVm, XmoZ {^ÝZ q~Xÿ_YyZ OmUmar EH$ Am{U EH$M aofm AmnU H$mTy> eH$Vmo `m g§H$ënZoda â {Xbm Amho. `m {ZH$mbmbm AmnU EH$m _wbVËdmbm Imbrb ê$nmV _m§Sy> eH$Vmo.

_wbVËd 2.1 : {Xboë`m XmoZ q~XÿVyZ EH$ Am{U EH$M aofm H$mT>Vm òVo.

q~Xÿ P _YyZ OmUmè`m {H$Vr Ag§»` aofm H$mT>Vm òVmV. Ë`m Q q~XÿVyZM OmVmV? (AmH¥$Vr 2.4 nhm) Ho$db EH$ hr aof PQ Amho. Q q~XÿVyZ OmUmè`m Aem {H$Vr aofm AmhoV Á`m P q~XÿVyZ

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32 J{UV

OmVmV? Ho$di EH$ PQ aofm Aem àH$mao darb {dYmZ ho EH$ ñd`§{gÜX gË` Amho. Am{U åhUyZM Ë`mbm EH$m _wbVËdmÀ`m ê$nmV _m§S>Ê`mV Ambo.

AmH¥$Vr 2.4

J¥hrVËd 2 : EH$ aofmI§S> XmoÝhr ~mOybm A_`m©{XV dmT>{dVm òVmo.

bjmV ¿`m Á`mbm AmnU aofmI§S> (line segment) åhUVmo Ë`mbm `wŠbrS>Zo gm§V aofm åhQ>bo Amho. åhUyZM AmVmÀ`m ^mfo_Ü o, Xþgao J¥hrVËd ho gm§JVo H$s, EH$m aofmI§S>mbm XmoÝhr ~mOyZo A_`m©{XVnUo dmT>{dVm òVo (AmH¥$Vr 2.5 nhm)

AmH¥$Vr 2.5

J¥hrVËd 3 : EImXm gai aofmI§S> {Xbm AgVm, aofmI§S> hr {ÌÁ`m Am{U Ë`mMm EH$ AË`q~Xÿ _Ü` KoD$Z dVw©i H$mT>Vm òVo.

J¥hrVËd 4 : gd© H$mQ>H$moZ EH$ê$n AgVmV.

J¥hrVËd 5 : Oa EH$ gaiaofm Xþgè`m XmoZ gaiaofm§Zm AemàH$mao N>oXVo H$s. EH$mM ~mOybm AgUmao XmoZ A§Va H$moZ EH$ÌnUo XmoZ H$mQ>H$moZmnojm bhmZ AgVmV. Oa Xþgar gaiaofm Ë`m ~mOybm dmT>{dbr Va N>oXë`m_wio Oo H$moZ hmoVmV Vo XmoZ H$mQ>H$moZmnojm bhmZ AgVmV.

CXmhaUmW©, AmH¥$Vr 2.6 _Üò aofm PQ, aofm AB Am{U CD bm AemàH$mao N>oXVo H$s Am§VaH$moZ 1 Am{U 2 Mr ~oarO PQ À`m COì`m A§Jmg Amho. Am{U 180° nojm H$_r Amho. åhUOoM AB Am{U CD eodQ>r PQ À`m S>mì`m ~mOyg EH$_oH$m§Zm N>oXVmV

AmH¥$Vr 2.6

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darb gd© J¥hrVËdmdê$Z AmnUmg bjmV òVo H$s nmMdo J¥hrVËd A{YH$ Jw§VmJw§VrMo Amho. XþgarH$S>o nmhVm 1 Vo 4 J¥hrVËdo gai Am{U ñnîQ> AmhoV. Á`m ñd`§{gÜX gË` _mZë`m OmVmV. na§Vy Ë`m§Zm {gÜX H$aVm òV Zmhr åhUyZ `m {dYmZm§Zm {gÜXoVo{dZmM pñdH$maVm òVo. H$maU nmMì`m J¥hrVËdmÀ`m Aä`mgmgmR>r nwT>rb ^mJmH$S>o A{YH$ bj {Xbo Amho.

AmOH$mb _wbVËdo Am{U J¥hrVËdo `m XmoÝhr KQ>H$m§Zm EH$ Xþgè`mgmR>r à`moJ Ho$bm Jobm Amho. dmñV{dH$Vm _wbVËdo åhUOoM {H«$`m Amho. Ooìhm AmnU Ago åhUVmo H$s, "`m AmnU EH$ _wb VËd' H$ê$. åhUOoM "AmnU OJmV KS>Umè`m KQ>Zm§À`m AmYmao {dYmZo H$ê$'' `m gË`VoMo {ZarjU Z§Va Ho$bo Jobo. Oa Vo gË` hmoV Agob Va Va Vo _wbVËdmÀ`m ê$nmV pñdH$mamdo bmJbo.

H$mhr J¥hrVËdmZm EH$ pñWa nÜXV _mZÊ`mV Ambr Amho. Oa `m J¥hrVËdmnmgyZ Ago {dYmZ H$aUo AgË` Amho Oo AÝ` J¥hrVËd qH$dm {gÜX Ho$boë`m {dYmZmÀ`m {damoYmV Amho. åhUOoM Oa J¥hrVËdmMr H$mhr nÜXV {Xbobr Agob Va hr {ZpíMV H$aUo Amdí`H$ Amho H$s `m nÜXVr pñWa AmhoV.

`yŠbrS>Zo AÝ` n[aUm_ {gÜX H$aÊ`mgmR>r Amnbr _wbVËdo Am{U J¥hrVËdo dmnabr Am-hoV. `mMm Cn`moJ H$ê$Z H$mhr {Z`_mZË_H$ KQ>H$m§Zm {gÜX Ho$bo Amho. Á`m {dYmZm§Zm Ë`mZr {gÜX Ho$bo Ë`mbm "à_o`' (theorem) Ago åhQ>bo OmVo. `wŠbrS>Zo Amnë`m J¥hrVËdm§Mm, _wbVËdm§À`m, ì`m»`m§À`m Am{U n{hë`m {gÜX Ho$boë`m à_o`mMm Cn`moJ H$ê$Z EH$ VH©$ewÜX l¥§Ibo_Üò 465 KQ>H$m§Mo AZw_mZ H$mT>bo. y{_VrÀ`m nwT>rb H$mhr àH$aUm_Üò AmnU m à_o`m§Mm Cn`moJ H$ê$Z H$mhr {dYmZo {gÜX H$aUma AmhmoV.

AmVm AmnU nwT>rb H$mhr CXmhaUm_Üò ho nmhUma AmhmoV H$s `wŠbrS>Zo H$mhr n[aUm_mÀ`m {gÜXVogmR>r Ë`mÀ`m _wbVËdm§Mm Am{U J¥hrVËdm§Mm Cn`moJ Ho$bm Amho.

CXmhaU 1 : Oa A, B Am{U C ho EH$mM aofoda pñWa Agbobo q~Xÿ AgyZ q~Xÿ B hm A

d C À`m _Üò pñWa Amho (AmH¥$Vr 2.7 nhm) Va {gÜX H$am H$s AB + BC = AC

AmH¥$Vr 2.7

CH$b :darb AmH¥$VrV AB + BC hm AC Mm g§nmVr Amho. Ë`mM~amo~a, `wŠbrS>Mo _wbVËd (4) dê$Z Á`m dñVy g_mZ AgVmV Ë`m nañnamer g_mZ AgVmV. åhUyZ ho {gÜX hmoD$ eH$Vo H$s

AB + BC = AC

bjmV ¿`m, `m CH$bm_Üò Ago _mZbo Jobo Amho H$s, XmoZ q~XÿVyZ OmUmar EH$ Am{U EH$M aofm H$mT>Vm òVo.

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CXmhaU 2 : {gÜX H$am H$s, {Xboë`m EH$m aofmI§S>mda EH$m g_^wO {ÌH$moUmMr aMZm H$aVm òVo.

CH$b: darb {dYmZmdê$Z AB hr {Xboë`m bm§~rMr aofmI§S> Amho (AmH¥$Vr 2.8(i) nhm).

AmH¥$Vr 2.8

`wŠbrS>À`m _wbVËd (3) Mm Cn`moJ H$ê$. H$mhr aMZm H$aUo Amdí`H$ Amho. AmnU q~Xÿ A bm H|$Ð Am{U AB bm {ÌÁ`m _mZyZ EH$ H§$g H$mTy> (AmH¥$Vr 2.8(ii) nhm). `mMàH$mao B bm H|$Ð Am{U BA bm {ÌÁ`m _mZyZ AmUIrZ EH$ H§$g H$mTy>. `m H§$gmMm N>oXZ q~Xÿ C _mZy. AmVm aofmI§S> AC d AB gm§Yy Am{U hm DABC hmoB©b (AmH¥$Vr 2.8(iii) nhm)

`mgmR>r AmnUmg {gÜX H$mamd`mMo Amho H$s. hm {ÌH$moU g_^wO {ÌH$moU Amho. åhUOoM AB = AC = BC hmoB©b.

AmVm AB = AC Amho, H$maU `m EH$mM dVw©imÀ`m {ÌÁ`m AmhoV. (EH$mM dVw©imÀ`m gd© {ÌÁ`m g_mZ AgVmV.)

Ë`mMà_mUo AB = BC.... (EH$mM dVw©imÀ`m {ÌÁ`m)

darb XmoÝhr {dYmZmdê$Z Am{U _wbVËd (1) dê$Z AmnU Agm

{ZîH$f© H$ê$ eH$Vmo H$s AB = BC = AC Amho.

åhUyZ ∆ABC hm EH$ g_^wO {ÌH$moU Amho.

bjmV R>odm H$s, òWo `wŠbrS>Zo H$mhrhr Z gm§JVm ho _mÝ` Ho$bo Amho H$s H|$Ð A d B q~XÿVyZ H$mT>bobo. H§$g EH$_oH$m§Zm EH$m q~XÿV N>oXVmV.

AmVm AmnU EH$ à_o` {gÜX H$ê$ H$s, Oo {d{dY {ZH$mbm_Üò AZoH$ doim Cn`moJmV AmUbo OmVo.

à_o` 2.1 : XmoZ {^ÝZ aofm_Üò EH$nojm A{YH$ gm_mB©H$ q~Xÿ$ Agy eH$V ZmhrV.

{gÜXVm : òWo AmnUmg XmoZ aofm l d m {Xboë`m AmhoV. AmnUmg ho {gÜX H$amd`mMo Amho H$s, l d m `m§Zm Ho$di EH$M gm_mB©H$ q~Xÿ Amho.

H$mhr doimgmR>r AmnU Ago _mZy H$s XmoZ aofm XmoZ {^Þ q~Xÿ P d Q _Üò nañnam§Zm N>oXVmV. `mM~amo~a XmoZ {^ÝZ P d Q {~Xÿ_YyZ OmUmè`m Amnë`mH$S>o l d m `m XmoZ aofm AmhoV. na§Vy ho

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{dYmZ à_o` 2.1 À`m {déÜX Amho. Ë`mÀ`mZwgma XmoZ {^ÝZ q~Xÿ_YyZ OmUmar EH$ Am{U EH$M aofm AgVo. åhUyZ AmnU hr H$ënZm _mZVmo H$s, XmoZ aofm {^ÝZ q~XÿVyZ OmVo ho MwH$sMo Amho.

`mdê$Z AmnU H$moUVm {ZîH$f© bmdy eH$Vmo? AmnU {ZîH$f© bmdy eH$Vmo H$s, XmoZ aofm_Üò EH$mnojm A{YH$ q~Xÿ g_mB©H$ ZgVmV.

ñdmÜ`m` 2.1

1. Imbrb {dYmZmn¡H$s H$moUVr {dYmZo gË` qH$dm AgË` AmhoV? Vw_À`m CÎmambm H$maUo Úm.

(i) EH$m q~XÿVyZ OmUmar Ho$di EH$ Am{U EH$M aofm H$mT>Vm òVo.

(ii) XmoZ {^ÝZ q~XþVyZ OmUmè`m Ag§»` aofm AgVmV.

(iii) EH$ gm§V aofm ( aofmI§S>) XmoÝhr ~mOy§Zr A_`m©{XVnUo dmT>{dbr OmD$ eH$Vo.

(iv) Oa XmoZ dVw©io g_mZ AgVrb Va Ë`m§À`m {ÌÁ`m hr g_mZ AgVmV.

(v) AmH¥$Vr 2.9 _Üò Oa AB = PQ Am{U PQ = XY Amho, Va AB = XY AgVo.

AmH¥$Vr 2.9

2. Imbrb àËòH$ KQ>H$m§Mr ì`m»`m {bhm. `m_Üò Aer H$mhr nXo AmhoV H$s Á`m§Mr ì`m»`m H$aUo Amdí`H$ Amho? Vr H$moUVr AmhoV? Am{U Ë`mMr ì`m»`m AmnU H$er H$amb.

(i) g_m§Va aofm (ii) b§~ aofm

(iii) aofmI§S> (iv) dVw©imÀ`m {ÌÁ`m

(v) Mm¡ag

3. Imbrb {Xboë`m XmoZ _wbVËdmMm {dMma H$am.

(i) XmoZ {^ÝZ q~Xÿ A d B {Xbobo AgVmZm. EH$ {Vgam q~Xÿ C Agm ñWmnZ H$am H$s Omo A d B À`m _Üò Agob.

(ii) òWo AemàH$mao {VZ q~Xÿ ñWmnZ H$am H$s, Oo EH$m aofoda ñWm{nV ZmhrV.

`m _wbVËdm_Üò H$moUVr ì`m»`m Z H$aUmam Agm eãX Amho H$m? Va m A{damoYr AmhoV H$m`? `m `wŠbrS>À`m _wbVËdmdê$Z àmßV hmoVmo H$m? ñnîQ> H$am.

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4. Oa A d B `m XmoZ q~XÿÀ`m_Üò C hm q~Xÿ Agm Amho H$s, AC = BC Va {gÜX H$am H$s

AC AB=12

. AmH¥$Vr H$mTy>Z ñnîQ> H$am.

5. CXmhaU 4 _Üò C q~Xÿ aofmI§S> AB Mm _Ü` Amho. Va {gÜX H$am H$s, EH$m aofmI§S>mMm EH$ Am{U EH$M _Ü` q~Xÿ AgVmo.

6. AmH¥$Vr 2.10 _Üò Oa AC = BD Amho Va, {gÜX H$am AB = CD

AmH¥$Vr 2.10

7. `wŠbrS>À`m J¥hrVËdmÀ`m gy{M_Üò {Xboë`m J¥hrVËd 5 bm EH$ gd©ì`mnr gË` H$m _mZÊ`mV Ambo (bjmV R>odm hm àíZ nmMì`m J¥hrVVËdmer g§~§YrV Amho.)

2.3. `wŠbrS>À`m nmMì`m J¥hrVVËdmMo g_mZ ê$nm§Va

J{UVmÀ`m B{VhmgmV `wŠbrS>À`m nmMì`m J¥hrVËdmMo A{YH$ _hËd Amho. KQ>H$ 2.1 _Yrb `m _wbVËdmbm naV S>moù`mg_moa AmUm. `m _wbVËdmMr {ZînVrZwgma Oa XmoZ aofm§Zm N>oXUmar EH$ aofm Ë`m_wio V`ma hmoUmè`m Ë`m aofoÀ`m EH$m A§JmÀ`m Am§VaH$moZm§Mr ~oarO 180° AgVo. Va Ë`m aofm H$YrM EH$_oH$m§Zm N>oXV ZmhrV. `m J¥hrVËdmÀ`m g_mZ ê$nm§Vao AZoH$ AmhoV. `m_Üò EH$ ßbo\o$AaMo J¥hrVH$ (Playfair's axiom) ho EH$ Amho. (ßbo\o$Aa ñH$m°Q>b±S>À`m J{UVkmZo 1729 _Üò V`ma Ho$bo) Vo Imbrb à_mUo Amho.

àËòH$ aofm l Am{U Ë`mdarb pñWV Agbobo àËòH$ q~Xÿ P gmR>r, EH$ A{ÛVr` aofm m Aer Amho H$s Or q~Xÿ P _YyZ OmVo Am{U l bm g_m§Va AgVo.

AmH¥$Vr 2.11 _Üò AmnU nmhÿ eH$Vmo H$s, q~Xÿ P _YyZ OmUmMm gd© aofmn¡H$s Ho$di m hr aofm l bm g_m§Va Amho.

AmH¥$Vr 2.11

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`m {ZînVrbm Imbrb àH$mao ì`ŠV Ho$bo OmD$ eH$Vo. XmoZ {^ÝZ N>oXUmè`m aofm EH$mM aofobm

g_m§Va Agy eH$V ZmhrV.

`wŠbrS>bm Amnbo n{hbo 28 à_o` {gÜX H$aÊ`mgmR>r 5 ì`m J¥hrVËdmMr Amdí`H$Vm ^mgbr Zmhr.`wŠbrS>gh ~è`mM J{UVkmZm Ago dmQ>V hmoVo H$s 5 do J¥hrVËd ho EH$ à_o` Amho. d Ë`mMr {gÜXVm n{hë`m Mma _wbVËd Am{U J¥hrVVËdmMm AmYmao XoVm òB©b. na§Vw `mH$arVm Ho$bobo gd© à`ËZ Anèñdr à`ËZmVyZ Z{dZ ^y{_H$m ñnîQ> Pmbr hr ^y{_H$m `wŠbrS>À`m ^y{_Vrnojm doJir Amho. åhUyZ `m ^y{_Vrbm `wŠbr{S>`Z ^y{_Vr g§~moYÊ`mV Ambo. `m Zì`m ^y{_Vrbm B{VhmgmVrb EH$ Zdo nd© Ago _mZÊ`mV Ambo. H$maU Vmon`ªV AmH¥$Vr 2.12

g§nyU© OJmV `wŠbrS>Mr ^y{_H$m hrM EH$ ^y{_Vr Amho Agm g_O hmoVm. AmnU øm {dídmV amhVmo Ë`m {dídmÀ`m ^y{_Vrg `wŠbrS>r`Z ^y{_Vr Ago g§~moYÊ`mV Ambo. àË`jmV hr Jmobr` ^y{_Vr Amho. Jmobr` ^y{_Vr_Üò aofm `m gaiaofm ZmhrV, `m aofm àË`jmV dVw©imMm ^mJ Amho. (àË`jmV hr dVw©io åhUOo Jmob d Ë`m_YyZ OmUmar àVbo `m§À`m N>oXm_wio V`mar Pmbr AmhoV).

AmH¥$Vr 2.12 _Üò aofm AN d BN (`m aofm åhUOo JmobmÀ`m dVw©imMm H$mhr ^mJ) `m aofm AB bm b§~ AmhoV. `m EH$_oH$mg N>oXë`m Var Ë`m aofoÀ`m EH$mM A§Jmg V`ma hmoUmè`m XmoZ H$moZm§À`m _mnm§Mr ~oarO XmoZ H$mQ>H$moZmnojm H$_r Zmhr (åhUOo 90° + 90° = 180°). bjmV ¿`m H$s NAB `m {ÌH$moUmÀ`m H$moZmÀ`m _mnmMr ~oarO hr 180° nojm OmñV Amho. Ogo ∠A + ∠B = 180°. åhUyZM `yŠbrS>Mr ^y{_H$m hr \$ŠV àVbr` AmH¥$Ë`mZmM bmJy hmoVo. na§Vy dH«$mH$ma qH$dm Jmobr` AmH¥$Ë`m§Zm bmJy hmoV Zmhr.

CXmhaU 3 : nwT>rb {dYmZmda {dMma H$am. gaiaofm§Mr EH$ Omo‹S>r Aer Amho H$s, EH$

Xþgè`mnmgyZ àËòH$ q~Xÿ nmgyZ g_mZ A§Vamda AgVo H$m` ? ho {dYmZ `wŠbrS>À`m nmMì`m

J¥hrVËdmMm àË`j {ZîH$f© Amho H$m? ñnîQ> H$am.

CH$b : EH$ aofm l ¿`m Am{U EH$ q~Xÿ Agm ¿`m H$s Omo aofm l da pñWa Zmhr. Voìhm

ßbo\o$Aa J¥hrVVËdmZygma Oo `wŠbrS>À`m nmMì`m J¥hrVVËdmer g§~§YrV Amho. AmnUmg _mhrV Amho

H$s, P _YyZ OmUmar EH$ aofm m Amho. Am{U Or l bm g_m§Va Amho.

AmVm q~XÿMo EH$m q~XÿnmgyZMo A§Va ho Ë`m q~XÿnmgyZ Ë`m aofoda Q>mH$boë`m b§~mÀ`m bm§~r

BVHo$ Amho. m da pñWa AgUmè`m q~XÿMo l aofon`©VMo A§Va Am{U l da pñWa EH$m q~XÿMo m aofm

`m àËòH$ ñWmZmda EH$ Xþgè`mnmgyZ g_mZ Amho. åhUOoM `m XmoZ aofm l d m àËòH$ ñWmZmda EH$

Xþgè`mnmgyZ g_mZ A§Vamda Amho.

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{Q>n : nwT>rb H$mhr àH$aUm_Üò AmnU Oo {eH$Uma AmhmoV Vr `wŠbrS>Mr ^y{_H$m Agob na§Vy

Ë`m_Üò Am_À`mÛmao à`moJ Ho$bo Jobobr J¥hrVËdo Am{U à_o` yŠbrS>À`m J¥hrVËdo Am{U à_o`mnmgyZ

{^ÝZ Agy eH$Vrb.

ñdmÜ`m` 2.2

1. AmnU `wŠbrS>À`m nmMì`m J¥hrVVËdmbm H$em àH$mao _m§Sy> eH$Vm H$s, Á`m_wio AmnUmg Vr

gai Am{U gmoß`m ^mfoV g_OyZ KoVm òB©b.

2. `wŠbrS>À`m nmMì`m J¥hrVVËdm_wio g_m§Va aofm§Mo ApñVËd {ZYm©[aV hmoVo H$m? ñnîQ> H$am.

2.4 gmam§e

`m àH$maUm_Üò AmnU Imbrb JmoîQ>r§Mm Aä`mg Ho$bm Amho:

1. Oar wŠbrS>Zo q~Xÿ, aofm Am{U àVbmÀ`m ì`m»`m Ho$ë`m AmhoV na§Vy J{UVkm§Zr m ì`m»`m§Mm

pñdH$ma Ho$bobm Zmhr. åhUyZ ^y{_Vr_Üò AmVm Ë`m§Zm An[a^m{fH$ nXm§À`m ê$nmV pñdH$mabo

Amho.

2. J¥hrVËdo Am{U _wbVËdo `m Aem H$ënZm Amho H$s Ë`m ñfîQ>nUo gd©ì`mnr gË` AmhoV. `mZm

{gÜX Ho$bo OmD$ eH$V Zmhr.

3. à_o` hr Aer {dYmZo AmhoV Á`m§Zm ì`m»`m J¥hrVHo$ Am{U n{hbr {gÜX Ho$bobr {dYmZo Am{U

VH©$ewÜX nÜXVrZo {gÜX Ho$bo Amho.

4. `yŠbrS>Mr H$mhr _wbVËdo Aer AmhoV.

(i) Oo KQ>H$ EH$mM KQ>H$mer g_mZ AgVmV Vo nañnamer g_mZ AgVmV.

(ii) g_mZ KQ>H$mV g_mZ KQ>H$ {_i{dÊ`mg CÎma g_mZ òVo.

(iii) g_mZ KQ>H$mVyZ g_mZ KQ>H$ dOm Ho$bo CÎma g_mZ òVo.

(iv) dñVy Á`m EH$_oH$mer g_mZ AgVmV Ë`m nañnamer g_mZ AgVmV.

(v) nyU© hm Ë`mÀ`m I§S>mnojm _moR>m AgVmo.

(vi) g_mZ KQ>H$m§Mr XþßnQ> hr g_mZ AgVo.

(vii) g_mZ KQ>H$m§Mr {Zå_nQ> hr g_mZ AgVo.

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5. `ypŠbS>Mo J¥hrVËdo Imbrb à_mUo AmhoV.

J¥hrVËd 1 : EH$m q~XÿnmgyZ Xþgè`m q~Xÿn`ªV OmUmar EH$ gaiaofm H$mT>Vm òVo.

J¥hrVËd 2 : EH$m aofmI§S>mbm XmoÝhr ~mOybm A_`m©{XVnUo dmT>{dVm òVo.

J¥hrVËd 3 : EImXm gai aofmI§S> {Xbm AgVm, aofmI§S> hr {ÌÁ`m Am{U Ë`mMm

EH$ A§Ë`q~Xÿ _Ü` KoD$Z dVw©i H$mT>Vm òVo.

J¥hrVËd 4 : gd© H$mQ>H$moZ EH$ê$n AgVmV.

J¥hrVËd 5 : Oa EH$ gai aofm Xþgè`m XmoZ gaiaofm§Zm N>oXVo Aem àH$mao H$s EH$m

~mOybm AgUmao XmoZ Am§VaH$moZ EH$ÌnUo XmoZ H$mQ>H$moZmnojm bhmZ

AgVmV. Oa Xþgar gabaofm Ë`m ~mOybm dmT>{Vbr Va N>oXbobo H$moZ

ho XmoZ H$mQ>H$moZmnojm bhmZ AgVmV.

6. `yŠbrS>À`m nmMì`m J¥hrVËdoMr Imbrb XmoZ g_mZ ê$nm§Vao AmhoV.

(i) àËòH$ aofm Am{U Ë`mda pñWV Zgbobm àËòH$ q~Xÿ P gmR>r EH$ EH$_od aofm m Aer

Amho H$s P _YyZ OmdyZ {Vbm g_m§Va AgVo.

(ii) XmoZ {^ÝZ nañnam§Zm N>oXÊmmè`m aofm EH$mM aofobm g_m§Va AgV ZmhrV.

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40 J{UV

àH$aU - 3

aofm Am{U H$moZ (Lines and Angles)

3.1 àñVmdZm

àH$aU 2 _Üò AmnU {eH$bmo H$s, EH$ aofm H$mT>Ê`mgmR>r H$_rV H$_r XmoZ q~XÿMr Amdí`H$Vm AgVo. AmnU H$mhr J¥hrVH$m§Mm gwÜXm Aä`mg Ho$bm Amho, Am{U Ë`m§Mm Cn`moJ H$ê$Z H$mhr à_o` gwÜXm {gÜX Ho$bo AmhoV. `m àH$aUm_Üò AmnUmbm H$moZ Am{U Ë`m§Mo JwUY_© `m§Mm Aä`mg H$aVm òB©b. Ooìhm XmoZ aofm nañnam§Zm N>oXVmV Am{U XmoZ g_m§Va aofm§Zm EH$ N>oXrH$m N>oXVo Aem JwUY_m©Mm Aä`mg H$ê$. Ë`mM~amo~a JwUY_m©Zm {gÜX H$aÊ`mgmR>r VH©$ewÜX nÜXVrMm dmna H$aÊ`mV òVmo ho AmnU _mJrb B`ÎmoV {gÜXVmÛmao `m JwUY_m©Mr nS>VmiUr H$ê$Z nmhrbr Amho.

AmnU Amnë`m X¡Z§{XZ OrdZmV gnmQ> àVbmÀ`m H$S>m_Üò ~Zboë`m AZoH$ àH$maMo H$moZ nm{hbo AgVrb. gnmQ> àVbmMm Cn`moJ H$ê$Z EH$mM àH$maMo _m°S>ob ~Z{dÊ`mgmgmR>r AmnUmg H$moZm§Mr {dñVmarV _mhrVr AgUo Amdí`H$ Amho. CXmhaUmW© AmnU Amnë`m emboÀ`m {dkmZ àXe©ZmgmR>r ~m§~yÀ`m H$mS>`m§Mm Cn`moJ H$ê$Z EH$ PmonS>rMr à{VH¥$Vr ~Zdy BpÀN>V Agmb Va, {dMma H$am AmnU Vo H$go ~ZdUma. H$mhr ~m§~yÀ`m H$mS>çm§Zm AmnU nañna g_m§Va R>odUma Am{U H$mhtZm {VaŠ`m Ooìhm EH$ A{^`§Vm (Architect) EH$ ~hþ_Obr B_maVrMo EH$ aoIm{MÌ H$mT>Vmo, Voìhm Ë`mbm {^ÝZ H$moZm§da nañna N>oXUmè`m qH$dm g_m§Va aofm H$mT>mì`m bmJVmV. AmnU Agm {dMma H$ê$ H$s, Vr aofm Am{U H$moZm§À`m _mhrVr {edm` `m B_maVrMo aoIm{MÌ (Plan) H$mTy> eH$Vmo H$m?

{dkmZm_Üò, AmnU àH$memÀ`m JwUY_m©À`m {H$aUm§Mo AmboI(Ray diagrams) H$mTy>Z Ë`mMm Aä`mg H$aVmo. CXmhaUmW© àH$memMo AndV©Z (refraction) `m JwUY_m©Mm Aä`mg H$aÊ`mgmR>r Ooìhm àH$me {H$aU EH$m _mÜ`_mVyZ Xþgè`m _mÜ`_mV àdoe H$aVmV Voìhm AmnU nañnam§Zm N>oXUmè`m Am{U g_m§Va aofm§Mm Cn`moJ H$aVmo. Ooìhm EImÚm dñVyda XmoZ AWdm A{YH$ ~b H$m`© H$aVmV Voìhm AmnU `m ~bm§Mm Ë`m dñVyda Pmbobm n[aUm_r ~b _mhrVr H$ê$Z KoÊ`mgmR>r AmboI H$mT>Vmo. H$s Á`m_Üò ~bmMr _m§S>Ur aofmI§S>mÛmao Ho$bobr AgVo.

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aofm Am{U H$moZ 41

`mdoir AmnUmg Aem H$moZm_Yrb g§~§Y _m{hVr AgUo Amdí`H$ Amho H$s Á`m§Mr {H$aUo nañnam§Zm g_m§Va qH$dm N>oXUmar AgVrb. EH$m _Zmoè`mMr C§Mr qH$dm EImÚm OhmOmdarb àH$me, Kam§Mo A§Va _m{hVr H$ê$Z KoÊ`mgmR>r AmnUmg {jVrO Am{U ÑîQ>r aofm (line of sight) `m Xaå`mZ V`ma Pmboë`m H$moZm§Mr _m{hVr AgUo Amdí`H$ Amho.

Ooìhm H$moZ Am{U aofm {Xbo AgVm AemàH$maÀ`m Xþgè`m CXmhaUmV dmnê$ eH$Vmo. ^y{_VrÀ`m `m CnKQ>H$m_Ü`o Vwåhr H$moZ Am{U aofm `m§Mo JwUY_© AZoH$ àH$mao Cn`moJmV AmUy eH$Vm.

àW_ _mJrb B`ÎmoV H$moZ Am{U aofm `m§Mr ì`m»`m Am{U _w»` _wÔo {eH$bm AmhmV Vo AmR>dm.

3.2 H$mhr _wÔoo Am{U ì`m»`m (Basic terms and definition)

AmR>dm H$s, XmoZ A§Ë` q~Xÿ AgUmè`m EH$m aofoÀ`m ^mJmbm aofmI§S> åhUVmV Am{U EH$ A§Ë` q~Xÿ AgUmè`m EH$m aofoÀ`m ^mJmbm {H$aU åhUVmV. AB aofmI§S> AB `m {MÝhmZo Xe©{dVmV.

Ë`mMà_mUo {H$aU AB hm AB → `m {MÝhmZo Xe©{dVmV$. H$mhr doiobm aofmI§S> d {H$aU ho l,m,n `m

gma»`m bhmZ Ajam_Ü`o gwÜXm Xe©{dVmV.

EH$mM aofoda VrZ qH$dm VrZnojm A{YH$ q~Xÿ AgVrb Va Ë`m§Zm EH$aofr` q~Xÿ (collinear points) åhUVmV.

Ooìhm XmoZ {H$aUm§Zm EH$M A§Ë` q~Xÿ Agob Voìhm VoWo H$moZ {Z_m©U hmoVmo. H$moZ {Z_m©U H$aUmè`m {H$aUmbm H$moZmMr ~mOy åhUVmV Am{U A§Ë`q~Xÿbm H$moZmMm {eamoq~Xÿ Ago åhUVmV. Vwåhr doJdoJù`m H$moZm§Mm Aä`mg H$ê$ eH$Vm, Vo åhUOoM, bKwH$moZ, H$mQ>H$moZ, {dembH$moZ, gaiH$moZ Am{U à{dembH$moZ (AmH¥$Vr 3.1 nhm).

AmH¥$Vr : 3.1 H$moZm§Mo àH$ma

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42 J{UV

bKwH$moZm§Mo _mn 0° Vo 90° À`m_Üò d H$mQ>H$moZ 90° er g_mZ AgVmo. 90° nojm _moR>m Am{U 180° nojm bhmZ AgUmè`m H$moZmbm {demb H$moZ Ago åhUVmV. VgoM AmR>dm H$s, gaiH$moZ 180° AgVmo. 180° nojm _moR>m Am{U 360° nojm bhmZ AgUmè`m H$moZmbm à{demb H$moZ åhUVmV. Ë`mMà_mUo Oa XmoZ H$moZm§Mr ~oarµO 90° Agob Va Ë`m H$moZm§Zm H$mo{Q>H$moZ åhUVmV Am{U Oa XmoZ H$moZm§Mr ~oarO 180° Agob Ë`m H$moZm§Zm nyaH$ H$moZ åhUVmV.

_mJrb B`ÎmoV g§b½Z H$moZm{df`r {eH$bm AmhmV. (AmH¥$Vr 3.2 nhm) XmoZ H$moZ gb½Z hmoVmV, Ooìhm XmoZ H$moZm§Zm EH$M gm_mB©H$ ~mOy AgVo. AmH¥$Vr 3.2 _Üò

ABC∠ d ∠DBC ho g§b½Z H$moZ AmhoV. BD {H$aU ho gm_mB©H$ ~mOy Am{U B ho gm_mB©H$ {eamoq~Xÿ Amho. {H$aU BA d BC ho `m Agmg_mB©H$ ~mOy AmhoV. Ooìhm XmoZ H$moZ gb½Z AgVrb Voìhm Agm_mB©H$ ~mOyer A§VarV Ho$boë`m H$moZmMr ~oarO Ë`mÀ`mer g_mZ AgVo. ho AmnU Ago {bhÿ eH$Vmo.

∠ABC = ∠ABD + ∠DBC

AmH¥$Vr : 3.2 g§b½Z H$moZ

bjmV R>odm H$s, ∠ABC d ∠ABD ho gb½Z H$moZ ZmhrV. H$m? H$maU BA `m gm_mB©H$ ~mOyer BD d BC `mo Agm_ Agm_mB©H$ ~mOy AmhoV.

AmH¥$Vr 3.2 _Yrb Oa BA d BC ho Agm_mB©H$ ~mOy AgyZ Ë`m AmH¥$Vr 3.3 _Üò Xe©{dë`m AmhoV. `m_Üò ∠ABD d ∠DBC `m H$moZmZm nyaH$ H$moZ åhUVmV.

AmH¥$Vr : 3.3 nyaH$ H$moZ

Vwåhr nwÝhm nmhÿ eH$Vm H$s, 0 q~XÿV N>oXUmè`m AB d CD `m XmoZ aofm§_Yrb {eamo{déÜX H$moZ A§VarV AmhoV (AmH¥$Vr 3.4 nhm) {eamo{déÜX H$moZmÀ`m òWo XmoZ OmoS>`m AmhoV.

∠AOD d ∠BOC hr EH$ OmoS>r Amho Va Xþgar Omo‹S>r Vwåhr emoYy eH$Vm H$m? AmH¥$Vr : 3.4 {eamo{déÜX H$moZ

3.3 nañnam§Zm N>oXUmè`m Am{U Z N>oXUmè`m aofm

EH$m H$mJXmda PQ d RS `m XmoZ {^ÝZ aofm H$mT>m. AmH¥$Vr 3.5(i) d AmH¥$Vr 3.5 (ii) _Üò Xe©{dboë`m à_mUo XmoZ {^ÝZ aofm§Mr aMZm H$am.

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aofm Am{U H$moZ 43

(i) N>oXUmè`m aofm (ii) Z N>oXUmè`m (g_m§Va) aofm

AmH¥$Vr 3.5: XmoZ {^ÝZ àH$maÀ`m aofm§Mr aMZm.

XmoZ aofm ~mø~mOy XmoÝhr ~mOyZr dmT>rdbr AgVm aofoMm JwUY_© AmR>dm. AmH¥$Vr 3.5(i) _Üò PQ d RS `m aofm N>oXboë`m aofm AmhoV Am{U AmH¥$Vr 3.5(ii) _Üò g_m§Va aofm AmhoV. bjmV R>odm H$s, {^ÝZ q~XÿVrb gm_mB©H$ b§~mVam§Mr bm§~r g_m§Vaaofofr g_mZ AgVo. `m g_mZ bm§~rbm g_m§Va aofo_Yrb A§Va Ago åhUVmV.

3.4. H$moZm§À`m OmoS>çm

KQ>H$ 3.2 _Üò nyaH$ H$moZmMr ì`m»`m H$mhr AgUmè`m gb½Z H$moZ, gai H$moZ, H$moQ>r H$moZ, nyaH$ H$moZ B `m~Ôb {eH$bm Am-hmV. `m H$moZm§À`m g§~§YrV Vwåhr {dMma H$ê$ eH$Vm H$m? AmVm, EH$m aofoda C^m AgUmam {H$aU `m g§~§YrV nmhÿ `m, AmH¥$Vr 3.6 _Üò Xe©{dë`mà_mUo EH$m aofoda EH$ {H$aU C^m Amho Aer aMZm H$am. Ë`m aofobm AB d {H$aUmbm OC Ago Zmd Úm. 0 `m q~XÿVyZ H$moUVm H$moZ V`ma hmoVmo? Vo åhUOo ∠AOC, ∠BOC Am{U

AmH¥$Vr : 3.6 nyaH$ H$moZ

∠AOB + ∠BOC = ∠AOB .....(1)

Ago {bhÿ eH$Vmo H$m? hm. (H$m? `mgmR>r KQ>H$ 3.2 _Yrb g§b½Z H$moZ nhm) ∠AOB `m H$moZmMo _mnZ H$m`? Vo 180 Amho, (H$m?) .....(2)

(1) d (2) dê$Z ∠AOC + ∠BOC = 180 Ago gm§Jy eH$Vm H$m? darb MMm© hr AmnU Imbrb _wbVËdmZr gmoS>dy eH$Vmo.

_wbVËd 3.1: Oa EH$m aofoda EH$ {H$aU C^m Agob Va ~Zboë`m XmoZ g§b½Z H$moZm§Mr ~oarO 180o AgVo.

AmR>dm, Oa XmoZ g§b½Z H$moZm§Mr ~oarO 180o Agob Va Ë`m H$moZm§Zm nyaH$ H$moZ Ago åhUVmV.

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_wbVËd 3.1 _Üò "aofoda EH$ {H$aU C^m Amho' Ago {Xbobo Amho. `mdê$Z AmnU AZw_mZ H$ê$ eH$Vmo H$s XmoZ g§b½Z H$moZm§Mr ~oarO 180o AgVo. _wbVËd 3.1 Xþgè`m nÜXVrZo {bhÿ eH$Vmo H$m? åhUOoM _wbVËd 3.1 _Üò AZw_mZ KoD$ Am{U AZw_mZ ho {Xbobo Amho.

Ë`mdê$Z :

(A) Oa EH$m aofoda EH$ {H$aU C^m Agob Va XmoZ g§b½Z H$moZm§Mr ~oarO 180o AgVo. (Va hr aofodr Ag_mZ ~mOy Amho).

AmVm. Vwåhr nmhÿ eH$Vm H$s {dYmZ (A) Am{U _wbVËd 3.1 ho EH$_oH$m§Mm ì`Ë`mg Amho. {dYmZ (A) ho ~amo~a Amho H$s Zmhr ho AmnUmbm _mhrV Zmhr. AmH¥$Vr 3.7 _Üò XmI{dë`mà_mUo doJdoJù`m nÜXVrZo gb½Z H$moZmMr aMZm H$am. Ag_mZ AgUmao H$moZ EH$mM ~mOyg Xe©dm.

AmH¥$Vr 3.7. BVa _mnmMo g§b½Z H$moZ

AmH¥$Vr 3.7 (iii) _YrbM emoYy. _moOnÅ>rÀ`m XmoÝhr Ag_mZ AgUmè`m ~mOyg A, O Am{U B `m q~Xÿda OC hm {H$aU C^m Amho. VgoM ∠AOC + ∠COB = 125o + 55o = 180o nmhÿ eH$Vmo. `mdê$Z AZw_mZ H$mTy> eH$Vmo H$s. {dYmZ (A) ~amo~arV Amho. åhUyZ AemM àH$mao AmnU nwT>rb nÜXVrZo _wbVËd gmÜ` H$ê$ eH$Vmo.

_wbVËd 3.2 : Oa XmoZ g§b½Z H$moZm§Mr ~oarO 180o Agob Va Ag_mZ ~mOyg AgUmao H$moZ EH$ aofm V`ma H$aVmV.

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aofm Am{U H$moZ 45

`m H$maUmgmR>r, darb XmoZ _wbVËdm§Zm nyaH$ _wbVËdo åhUVmV. AmVm XmoZ aofm EH$_oH$m§Zm N>oXVmV ho CXmhaUmÛmao nmhÿ. Ooìhm XmoZ aofm EH$_oH$m§Zm N>oXVmV Voìhm {eamo{déÜX H$moZ g_mZ AgVmV ho _mJrb B`ÎmoV {eH$bm AmhmV. {dYmZ-1 _Üò {gÜX Ho$bobr nmàr Imbrb {Xboë`m gmÜ`gmR>r bjmV R>odm.

à_o` 3.1 : Oa XmoZ aofm nañnam§Zm N>oXV AgVrb Va {eamo{déÜX H$moZ g_mZ AgVmV.

{gÜXVm : XmoZ aofm nañnam§Zm N>oXVmV ho darb {dYmZmV {Xbo Amho. AmH¥$Vr 3.8 _Üò AB d CD `m XmoZ aofm EH$_oH$m§Zm 0 q~XÿV N>oXVmV. òWo XmoZ {eamoq~XÿÀ`m Omo‹S>`m {Xë`m AmhoV.

(i) ∠AOC d ∠BOD (ii) ∠AOD d ∠BOC. ∠AOC = ∠BOD d ∠AOD =∠BOC

ho AmnUmbm {gÜX H$am`Mo Amho.

AmVm CD aofoda OA {H$aU C^m Amho. AmH¥$Vr 3.8 {eamo{déÜX H$moZ

\ ∠AOC + ∠AOD = 180o (nyaH$ H$moZ ) .............. (1)

∠ AOD + ∠ BOD =180o Ago {bhÿ eH$Vmo H$m? .............. (2)

(1) d (2) dê$Z

∠ AOC + ∠ AOD = ∠ AOD + ∠ BOD\ \∠AOC = ∠ BOD (¨KQ>H$ 2.2 _Yrb _wbVËd 3)

Ë`mMà_mUo : ∠ AOD =∠ BOC

AmVm nwaH$ _wbVËd Am{U à_o` 3.1 dê$Z H$mhr CXmhaUo gmoS>dy `m.

CXmhaU 1 : AmH¥$VrV 3.9 _Üò PQ d RS `m aofm EH$_oH$m§Zm O q~XÿV N>oXVmV Oa

∠ POR : ∠ ROQ = 5:7 Va gd© H$moZ AmoiIm.

AmH¥$Vr 3.9

CH$b : ∠ POR + ∠ ROQ = 180o .......... (nyaH$ H$moZ) nU ∠ POR : ∠ ROQ = 5:7

Va, ∠ POR = 5

12 × 180o = 75o

VgoM ∠ ROQ = 7

12 × 180o = 105o

AmVm ∠ POS = ∠ ROQ = 105o ..........({eamo{déÜX H$moZ)

Am{U ∠ SOQ = ∠ POR = 75o ..........({eamo{déÜX H$moZ)

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CXmhaU 2 : AmH¥$Vr 3.10 _Ü`o, POQ `m aofoda OS hm {H$aU C^m Amho. ∠ POS Am{U ∠ SOQ `m H$moZmMo H$moZXþ^mOH$ AZwH«$_o OR d OT {H$aU AmhoV. Oa. ∠ POS = x Va ∠ ROT H$mT>m.

AmH¥$Vr 3.10

CH$b: POQ `m aofoda OS hm {H$aU C^m.

\ ∠ POS + ∠ SOQ = 180o

nU ∠ POS = x

Va x + ∠ SOQ = 180o

åhUyZ, ∠ SOQ = 180o – x

AmVm {H$aU OR hm ∠ POS Mm H$moZXþ^mOH$ Amho.

∠ ROS = 12 ∠ POS

=12 × x

= 2x

Ë`mMà_mUo, ∠ SOT = 12 × ∠ SOQ

=12 × (180o – x)

= 90o– 2x

AmVm ∠ ROT = ∠ ROS + ∠ SOT

= 2x + 90o – 2

x

= 90o

CXmhaU 3 : AmH¥$Vr 3.11 _Ü`o OP, OQ, OR, Am{U ‹OS hr Mma {H$aUo AmhoV. Va .

∠ POQ + ∠ QOR + ∠ SOR + ∠ POS = 360° {gÜX H$am.

AmH¥$Vr 3.11

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aofm Am{U H$moZ 47

CH$b : AmH¥$Vr 3.11 _Üò _mJrb q~Xÿ Xe©{dUmao OP,

OQ, OR qH$dm OS {H$aU V`ma hmoVmV. Va OT `m {H$aU_YyZ T `m q~Xÿn`ªV TOQ aofm dmT>{dbr. AmVm TOQ `m aofoda OP {H$aU C^m.

AmH¥$Vr 3.12

\ ∠ TOP + ∠ POQ = 180o ...... (1) (nyaH$ H$moZ)

Ë`mMà_mUo TOQ aofoda OS {H$aU C^m.

\ ∠ TOS + ∠ SOQ = 180º ..... (2)

nU ∠ SOQ = ∠ SOR + ∠ QOR

åhUyZ (2) dê$Z

∠ TOS + ∠ SOR + ∠ QOR = 180o ..... (3)

AmVm (1) d (3) Mr ~oarO H$ê$ ∠ TOP + ∠ POQ + ∠ TOS + ∠ SOR + ∠ QOR = 360° (4)

nU ∠ TOP + ∠ TOS = ∠ POQ

(4) dê$Z

∠ POQ + ∠ QOR +∠ SOR + ∠ POS = 360o

ñdmÜ`m` 3.1

(1) AmH¥$Vr 3.13 _Üò AB d CD aofm O `m q~XÿV N>oXVmV. Oa ∠ AOC + ∠ BOE = 70o Am{U ∠BOD = 40o Va∠ BOE Am{U ∠ COE H$mT>m

AmH¥$Vr 3.13

(2) AmH¥$Vr 3.14 _Üò XY d MN aofm O q~XÿV N>oXVmV. Oa ∠POY = 90º Am{U a : b = 2 : 3 Va c Mr qH$_V H$mT>m.

AmH¥$Vr 3.14

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48 J{UV

(3) AmH¥$Vr 3.15 _Üò ∠ PQR =∠ PRQ Va ∠ PQS = ∠ PRT {gÜX H$am.

AmH¥$Vr 3.15

(4) AmH¥$Vr 3.16 _Üò Oa x + y = w + z Va AOB hr EH$ aofm Amho Ago {gÜX H$am.

AmH¥$Vr 3.16

(5) AmH¥$Vr 3.17 _Üò POQ hr aofm Amho. PQ aofoda OR {H$aU b§~ Amho. OP Am{U OR `m {H$aUmÀ`m _Ymo_Y OS hm AmUIrZ EH$ {H$aU Amho. Va {gÜX H$am.

∠ ROS = 12 (∠ QOS – ∠ POS)

AmH¥$Vr 3.17

(6) ∠XYZ = 64o Am{U P hm q~Xÿ XY _YyZ Amho. `m _m{hVrdê$Z AmH¥$Vr H$mT>m. Oa ∠ZYP d H$moZXþ^mOH$ YQ Agob Va ∠XYQ Am{U ∠QYP Mr qH$_V H$mT>m.

3.5 g_m§Va aofm Am{U N>o{XH$m

AmR>dm, XmoZ qH$dm XmoZmnojm A{YH$ aofm§Zm EH$m aofoZo EH$m q~XÿV N>oXboë`m aofog N>o{XH$m åhUVmV. (AmH¥$Vr 3.18 nhm). m d n `m aofm§Zm l hr N>o{XH$m P d Q q~XÿV N>oXVo. åhUyZ m d n `m aofrMr l hr N>o{XH$m Amho. {ZarjU H$am H$s P d Q `m q~XÿVyZ àËòH$ Mma H$moZ {XgyZ òVmV.

AmH¥$Vr 3.18

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aofm Am{U H$moZ 49

Va ∠ 1, ∠ 2, ...... ∠ 8 Ago Ë`m H$moZm§Zm Zm§do XoD$ AmH¥$Vrb 3.18 _Ü`o nhm.

∠ 1, ∠ 2, ∠ 7 d ∠ 8 `mZm ~møH$moZ åhUVmV Va ∠ 3, ∠ 4, ∠ 5 d ∠ 6 `m§Zm Am§VaH$moZ åhUVmV.

AmR>dm H$s, XmoZ aoofm§Zm EH$m N>oXrHo$Zo N>oXbo AgVm V`ma hmoUmè`m H$moZm§Mm Aä`mg _mJrb B`ÎmoV Vwåhr Ho$bm AmhmV Vo Imbrb à_mUo AmhoV.

a) g§JV H$moZ (correponding angles) :

(i) ∠ 1 d ∠ 5 (ii) ∠ 2 d ∠ 6

(iii) ∠ 4 d ∠ 8 (iv) ∠ 3 d ∠ 7

b) Am§Va ì`wËH«$_ H$moZ (alternate interior angles):

(i) ∠ 4 d ∠ 6 (ii) ∠ 3 d ∠ 5

c) ~mø ì`wËH«$_ H$moZ (alternate exterior angles) :

(i) ∠ 1 d ∠ 7 (ii) ∠ 2 d ∠ 8

d) N>o{XHo$À`m EH$mM A§Jmg AgUmao Am§VaH$moZ

(i) ∠ 4 d ∠ 5 (ii) ∠ 3 d ∠ 6

N>o{XHo$À`m EH$mM A§Jmg AgUmè`m H$moZm§Zm H«$_mJV Am§VaH$moZ qH$dm g§JV Am§VaH$moZ (consecutive interior angles qH$dm allied angles) åhUVmV.

VgoM gmÜ`m eãXmV AmnU ì`wËH«$_ H$moZ Ago gwÜXm åhUy eH$Vmo.

AmVm, m hr aofm n bm g_m§Va AgVmZm H$moZm§À`m OmoS>çm g§~§YrV AgUmao {dYmZ nmhÿ `m Vwåhmbm _mhrV Amho, Vw_À`m dhr_Yrb aofm g_m§Va AgVmV. åhUyZ nopÝgb d _moOnÅ>r KoD$Z XmoZ g_m§Va aofm H$mTy>Z Ë`m§Zm EH$m N>o{XHo$Zo N>oXbobr aofm H$mT>m. (AmH¥$Vr 3.19 _Ü`o Xe©{dbo Amho).

AmVm,g§JV H$moZm§À`m OmoS>çm H$mT>m Am{U Ë`m g§~§YrV {dYmZ _m§S>m. AmH¥$Vr 3.19

Vwåhr H$mTy> eH$Vm H$s ∠ 1 =∠ 5, ∠ 2 =∠ 6, ∠ 4 =∠ 8 Am{U ∠ 3 =∠ 7 `mgmR>r Imbrb _wbVËdmMm dmna H$ê$.

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50 J{UV

_wbVËd 3.3 : XmoZ g_m§Va aofm§Zm EH$m N>o{XHo$Zo N>oXbo AgVm hmoUmao g§JVH$moZ g_mZ AgVmV.

_wbVËd 3.3 bm gwÜXm g§JV H$moZmMo _wbVËdo åhUy eH$Vmo. AmVm, `m _wbVËdmMo ì`ñV Imbrb à_mUo MMm© H$ê$ eH$Vmo.

Oa XmoZ aofm§Zm EH$m N>o{XHo$Zo N>oXbo AgVm g§JV H$moZ g_mZ AgVrb Va Ë`m aofm g_m§Va AgVmV.

ho {dYmZ ~amo~a Amho H$m? ho Imbrb à_mUo {gÜX H$ê$ eH$Vmo. AD aofm H$mTy>Z Ë`mda B d C

q~Xÿ ¿`m. AmH¥$Vr 3.20 (i) _Ü`o Xe©{dë`mà_mUo B d C Mr aMZm Ho$br AgVm ∠ ABQ d ∠ BCS EH$_oH$m§er g_mZ hmoVrb

AmH¥$Vr 3.20

AD À`m Xþgè`m ~mOyg QB d SC dmT>{dbo AgVm PQ d RS `m XmoZ aofm {_iVmV. (AmH¥$Vr 3.20 (ii)) nhm). XmoZ aofm EH$_oH$m§Zm N>oXV Zmhr ho Vwåhr {ZarjU H$ê$ eH$Vm. Vwåhr Aer aMZm H$am H$s PQ d RS `m XmoZ gm_mB©H$ b§~ aofm AmhoV Am{U Ë`mMr bm§~r _moOm. Vo g_mZ AmhoV Ago gm§Jy eH$Vm. åhUyZ Ë`m aofm g_m§Va AmhoV Ago Xe©dy eH$Vm. åhUyZM g§JV H$moZmMo ì`ñV _wbVËd gË` Amho åhUyZ Imbrb _wbVËd AmnUmbm {_iVo.

_wbVËd 3.4 : Oa XmoZ aofm§Zm EH$m N>o{XHo$Zo N>oXbo AgVm hmoUmè`m g§JV H$moZmMr OmoS>r g_mZ Agob Va Ë`m XmoZ aofm nañnamg g_m§Va AgVmV..

Ooìhm XmoZ g_m§Va aofm§Zm N>o{XHo$Zo N>oXbo AgVm ì`wËH«$_ H$moZmÀ`m g§~§YrV AgUmao g§JV H$moZmMo _wbVËd H$mTy> eH$Vmo H$m? AmH¥$Vr 3.21 _Ü`o AB d CD `m XmoZ g_m§Va aofm§Zm PS hr N>o{XH$m AZwH«$_o Q d R q~XÿV N>oXVo.

AmH¥$Vr 3.21

∠ BQR =∠ QRC d ∠ AQR =∠ QRD Amho H$m?

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aofm Am{U H$moZ 51

Vwåhmbm _mhrV Amho H$s, ∠ PQA = ∠ QRC (1) (g§JV H$moZmMo _wbVËd)

∠ PQA = ∠ BQR (2) (ì`wËH«$_ H$moZ)

(1) d (2) dê$Z

∠ BQR =∠ QRC

\ ∠ AQR =∠ QRD

Imbrb {Xboë`m ào_`mgmR>r hm {ZîH$f© g§~§YrV Amho.

à_o` 3.2 : XmoZ g_m§Va aofm§Zm EH$m N>o{XHo$Zo N>oXbo AgVm ì`wËH«$_ H$moZm§À`m OmoS>`m g_mZ AgVmV.

AmVm, g§JV H$moZmMm ì`ñV _wbVËd dmnê$Z AmnU Oa ì`wËH«$_ H$moZmÀ`m OmoS>`m g_mZ AgVmV Va XmoZ aofm g_m§Va AmhoV Ago Xe©dy eH$Vmo H$m? AmH¥$Vr 3.22 _Ü`o AB d CD `m aofm§Zm PS hr N>o{XH$m AZwH«$_o Q d R q~XÿV N>oXVo. Va

∠ BQR =∠ QRC

AB || CD Amho H$m?

∠ BQR =∠ PQA (H$m ?) (1)

nU ∠ BQR =∠ QRC ([Xbobo) (2)

AmH¥$Vr 3.22

(1) d (2) dê$Z

∠ PQA =∠ QRC

nU Vo g§JV H$moZ AmhoV.

åhUyZ AB || CD (g§JV H$moZmMo ì`ñV _wbVËd)

Imbrb {Xboë`m à_o`mgmR>r hm {ZîH$f©.

à_o` 3.3 : Oa XmoZ aofm§Zm EH$m N>o{XHo$Zo N>oXbo AgVm ì`wËH«$_ H$moZm§À`m OmoS>`m g_mZ AgVrb Va Ë`m XmoZ aofm g_m§Va AgVmV. Ë`mMà_mUo, N>o{XHo$bm EH$mM A§Jmg AgUmao Am§VaH$moZmer g§~§YrV Imbrb XmoZ à_o` Xe©{dbo AmhoV.

à_o` 3.4 : Oa XmoZ g_m§Va aofm§Zm N>o{XHo$Zo N>oXbo AgVm EH$mM A§Jmg AgUmè`m Am§VaH$moZmÀ`m Omo‹S>`m nyaH$ AgVmV.

à_o` 3.5 : Oa XmoZ aofm§Zm N>o{XHo$Zo N>oXbo AgVm EH$mM A§Jmg AgUmè`m Am§VaH$moZmÀ`m OmoS>`m nyaH$ Agob Va Ë`m XmoZ aofm g_m§Va AgVmV.

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52 J{UV

Vwåhr AmR>dU H$am H$s, darb gd© à_o` Am{U _wbVËdo _mJrb dJm©V {gÜX Ho$bobr AmhoV. Va ho gd© naV EH$Xm Aä`mgm.

3.6 EH$mM aofoer AgUmao g_m§Va aofm

Oa g_mZ aofm XmoZ g_m§Vaaofm AgVrb Va Vo EH$_oH$m§er g_m§Va AgVmV H$m? ho nSV>miy`m. AmH¥$Vr 3.23 _Üò m || l Am{U n || l

l, m Am{U n `m aofm§Zm t hr N>o{XH$m H$mT>br Amho.

m || l Am{U n || l {Xbo Amho.

Voìhm ∠1 = ∠2 Am{U ∠1 = ∠3

(g§JV H$moZmMo _wbVËd)

∠2 = ∠3 (H$m?) AmH¥$Vr 3.23

nU ∠2 Am{U ∠3 g§JV H$moZ AgyZ Vo g_mZ AmhoV.

åhUyZM aofm m || aofm n .

(g§JV H$moZmMo ì`ñV _wbVËd)

Imbrb {Xboë`m à_o`m_Üò hm {ZH$mb Xe©{dbobm Amho.

à_o` 3.6 : EH$mM aofobm g_m§Va AgUmè`m aofm nañnam§er g_m§Va AgVmV.

{Q>n XmoZ aofm§gmR>r gwÜXm darb JwUY_© gmÜ` H$ê$ eH$Vmo. AmVm g_m§Va aofo g§~§YrV H$mhr CXmhaU gmoS>dy`m.

CXmhaU 4 : AmH¥$Vr 3.24 _Üò Oa PQ || RS, ∠ MXQ = 135o Am{U ∠ MYR = 40o

Va∠ XMY H$mT>m.

AmH¥$Vr 3.24 AmH¥$Vr 3.25

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aofm Am{U H$moZ 53

CH$b : òWo M q~XÿVyZ OmUmar AB hr aofm PQ bm g_m§Va H$mT>br ho AmH¥$Vr 3.25 _Üò H$mT>bo Amho. AmVm AB || PQ Am{U PQ || RS

Ë`mMà_mUo AB || RS (H$m ?)

AmVm ∠ QXM + ∠ XMB =180o

(AB || PQ , XM N>o{XHo$À`m EH$mM A§Jmg AgUmao Am§VaH$moZ

nU ∠ QXM = 135o

åhUyZ 135o+ ∠ XMB = 180o

\ ∠ XMB = 45o (1)

AmVm ∠ BMY = ∠ MYR (AB || RS ì`wËH«$_ H$moZ)

∠ BMY = 40o (2)

(1) d (2) Mr ~oarO H$ê$

∠ XMB + ∠ BMY = 45o + 40o

∠ XMY = 85o

CXmhaU 5: Oa XmoZ aofm§Zm EH$m N>o{XHo$Zo N>oXbo AgVm Am§VaH$moZ H$moZm§Mr Omo‹S>r g_m§Va aofoV AgVo Va {gÜX H$am H$s, Ë`m XmoZ aofm EH$_oH$m§Zm g_m§Va AgVmV.

CH$b : AmH¥$Vr 3.26 _Üò PQ d RS XmoZ aofm§Zm AD hr N>o{XH$m AZwH«$_o B d C _Üò N>oXVo. ∠ ABQ _YyZ BE ho {H$aU H$mT>bo Am{U ∠ BCS _YyZ CG {H$aU H$mT>bo. Am{U BE || CG. AmH¥$Vr 3.26

AmnUmg PQ || RS {gÜX H$amd`mMo Amho.

∠ ABQ _YyZ BE {H$aU H$mT>bo Amho

\ ∠ ABE =12 ∠ ABQ ............ (1)

VgoM ∠BCS _YyZ CG {H$aU H$mT>bo.

\ ∠BCG = 12 ∠BCS ........... (2)

nU BE||CG Am{U AD hr N>o{XH$m.

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54 J{UV

\ ∠ ABE =∠ BCG ........ (3) (g§JV H$moZ)

(1) d(2) ho (3) _Üò _m§Sy>

12 ∠ABQ =

12 ∠BCS

\ ∠ABQ = ∠ BCS

nU PQ Am{U RS bm AD `m N>o{XHo$er Am§VarV AgUmao g§JVH$moZ g_mZ AmhoV.

\ PQ || RS (g§JVH$moZmMo {dYmZ)

CXmhaU 6 : AmH¥$Vr 3.27 _Üò AB || CD Am{U CD || EF VgoM EA ⊥ AB Oa.

∠BEF = 55o Agob Va x, y Am{U z À`m qH$_Vr H$mT>m.

CH$b : y + 55o = 180o

(ED `m N>o{XHo$À`m EH$mg A§Jmg AgUmao Am§VaH$moZ)

\ y = 180o – 55o = 125o

x =y

(AB || CD g§JVH$moZmMo {dYmZ)

x = 125o AmH¥$Vr 3.27

AmVm AB || CD Am{U CD || EF VgoM AB || EF

\ ∠EAB +∠FEA =180o (EA `m N>o{XHo$À`m EH$mM A§Jmg AgUmao Am§VaH$moZ)

\ 90o + Z + 55o = 180o

Z = 35o

ñdmÜ`m` 3.2

1) AmH¥$Vr 3.28 _Üò, x Am{U y À`m qH$_Vr H$mTy>Z AB || CD {gÜX H$am.

AmH¥$Vr 3.28

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aofm Am{U H$moZ 55

2) AmH¥$Vr 3.29 _Üò, Oa AB || CD, CD || EF Am{U y : z =3:7 Agob Va x ¨H$mT>m.

AmH¥$Vr 3.29

3) AmH¥$Vr 3.30 _Üò, Oa AB || CD, EF ⊥ CD Am{U∠GED =126o Va ∠AGE, ∠GEF Am{U ∠FGE H$mTm>..

AmH¥$Vr 3.30

4) AmH¥$Vr 3.31 _Üò, Oa PQ || ST, ∠PQR =

110o Am{U ∠RST = 130o Va ∠QRS H$mT>m. ({Q>n : R q~XÿVyZ ST bm g_m§Va AgUmar aofm H$mT>m.)

AmH¥$Vr 3.31

5) AmH¥$Vr 3.32 _Ü o, Oa AB || CD, ∠APQ = 50o Am{U ∠PRD = 127o Va x Am{U y H$mT>m.

AmH¥$Vr 3.32

6) AmH¥$Vr 3.33 _Üò PQ Am{U RS ho XmoZ Amaeo

EH$_oH$m§Zm g_m§Va_Üò R>odbo. PQ `m Amaemda B q~Xÿ

AB hm nVZ {H$aU Amho. RS `m Amaemda C q~XÿVyZ

òUmam namd{V©V {H$aU BC Amho Am{U naV Vmo CD bm

namd{V©V H$aVr Va {gÜX H$am AB || CD. AmH¥$Vr 3.33

3.7 {ÌH$moUmÀ`m H$moZm§À`m ~oaOoMm JwUY_©

{ÌH$moUmÀ`m gd© H$moZm§Mr ~oarµO 180o AgVo ho _mJrb dJm©V Vwåhr Aä`mg Ho$bm AmhmV. ho {dYmZ g_m§Va aofoÀ`m g§~§YrV à_o` Am{U _wbVËdo KoD$Z Amåhr {gÜX H$ê$ eH$Vmo.

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56 J{UV

à_o` 3.7: {ÌH$moUmÀ`m {VÝhr H$moZm§Mr ~oarµO 180o AgVo.

{gÕVm : darb {Xboë`m {dYmZmdê$Z AmnUmbm _mhrV Amho AmnUmbm H$m` {gÜX H$amd`mMo Amho.

PQR hm {ÌH$moU {Xbobm AgyZ DPQR _Üò ∠1, ∠2 Am{U ∠3 ho H$moZ AmhoV. (AmH¥$Vr 3.34 nmhm).

AmH¥$Vr 3.34

∠1 + ∠2 + ∠3 = 180o ho AmnUmg {gÜX H$am`Mo Amho.

AmH¥$Vr 3.35 _Üò XmI{dë`mà_mUo P `m q~XÿVyZ XPY hr QR bm g_m§Va AgUmar aofm H$mT>m. Ë`m_wio g_m§Va aofoMo JwUY_© AmnU Cn`moJ H$ê$ eH$Vmo.

AmVm, XPY hr aofm Amho Or QR bm g_m§Va H$mT>br Amho. AmH¥$Vr 3.35

\ ∠4 + ∠1 + ∠5 = 180o. .............................(1)

nU XPY || QR d PQ, QR `m N>o{XH$m

åhUyZ ∠4 =∠2 Am{U ∠5 =∠3 (ì`wËH«$_ H$moZmÀ`m OmoS>`m)

∠4 Am{U ∠5 (1) _Üò _m§Sy>

\ ∠2 + ∠1 + ∠3 = 180o

\∠1 + ∠2 + ∠3 = 180o

AmR>dm {ÌH$moUmÀ`m ~møH$moZmMm Aä`mg _mJrb dfu B`ÎmoV Ho$bm Amho (AmH¥$Vr 3.36 nhm) QR hr ~mOy S n`ªV dmT>{dbr ∠PRS hm DPQR Mm ~møH$moZ Amho.

AmH¥$Vr 3.36

∠3 + ∠4 = 180o (H$m?) .... (1)

AmVm ∠1 + ∠2 + ∠3 = 180o (H$m?) .... (2)

(1) d (2) H$ê$Z

∠4 = ∠1 + ∠2

hm {ZH$mb EH$m à_o`mà_mUo AgyZ Vmo Imbrb {Xbm Amho.

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aofm Am{U H$moZ 57

à_o` 3.8: {ÌH$moUmMr H$moUVrhr EH$ ~mOy dmT>{dbr AgVm hmoUmam ~møH$moZ hm Am§Va{déÜX H$moZmÀ`m ~oaOoBVH$m AgVmo.

darb à_o`mdê$Z Ago R>aVo H$s, {ÌH$moUmMm ~møH$moZ hm Am§Va{déÜX EH$m H$moZmnojm _moR>m AgVmo.

AmVm darb à_o`mdê$Z H$mhr CXmhaU gmoS>dy.

CXmhaU : 7: AmH¥$Vr 3.37 _Ü o, Oa QT ⊥ PR, ∠TQR = 40o Am{U ∠SPR = 30o Va x d y H$mT>m.

AmH¥$Vr 3.37

CH$b : DTQR _Ü`o 90o + 40o + x =180o ({ÌH$moUmÀ`m H$moZmÀ`m ~oaOoMo JwUY_©)

x = 50o

AmVm y =∠SPR + x ...... (à_o` 3.8)

y = 30o + 50o

y = 80o

CXmhaU 8 : AmH¥$Vr 6.38 _Ü`, DABC _Ü` AB d AC ~mOy AZwH«$_o E d D q~Xÿn`ªV dmT>{dbr. Oa ∠CBE d ∠BCD AZwH«$_o BO d CO N>o{XH$m O q~Xÿn`ªV {_iVo. Va

∠BOC = 90o – 12

∠BAC {gÜX H$am.

AmH¥$Vr 3.38

CH$b: ∠CBE hm {H$aU BO Mr N>o{XH$m Amho.

∠CBO =12 ∠CBE

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58 J{UV

=12(180º – y)

= 90º – 2y ...... (1)

Ë`mMà_mUo ∠BCD hm {H$aU CO Mr N>o{XH$m.

\ ∠BCO =12

∠BCD

=12 (180º – z)

=90 – 2z ...... (2)

DBOC _Üò ∠BOC + ∠BCO + ∠CBO = 180º (3)

(1) d (2) ho (3) _Üò _m§Sy>

∠ BOC + 90º – 2z+ 90º –

2y

=180º

∠BOC =2z +

2y

qH$dm ∠BOC = 12

(y + z) ...... (4)

nU x + y + z = 180o ({ÌH$moUmÀ`m H$moZmMo JwUY_©)

y + z = 180º – x

(4) dê$Z ∠BOC =12

(180º – x)

= 90º – 2x

= 90º – 12 ∠BAC

ñdmÜ`m` 3.3

1) AmH¥$Vr 3.39 _Üò ∆PQR _Üò QP d RQ `m ~mOy AZwH«$_o S d T n`ªV dmT>{dë`m. Oa ∠SPR =135o Am{U ∠PQT = 110o Va ∠PRQ H$mT>m.

2) AmH¥$Vr 3.40 _Üò, ∠X = 62o, ∠XYZ =54o Oa DXYZ _Üò ∠XYZ d∠XZY Mo AZwH«$_o YO d ZO N>o{XH$m AgVrb Va ∠OZY d ∠YOZ H$mT>m.

3) AmH¥$Vr 3.41 _Üò Oa AB || DE, ∠BAC =35o Am{U ∠CDE = 53o Va ∠DCE H$mT>m.

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aofm Am{U H$moZ 59

AmH¥$Vr 3.39 AmH¥$Vr 3.40 AmH¥$Vr 3.41

4) AmH¥$Vr 3.42 _Üò PQ d RS `m aofm T `m q~XÿV N>oXV AgVrb, ∠PRT = 40º , ∠RPT = 95º Am{U ∠TSQ = 75º Va ∠SQT H$mT>m.

5) AmH¥$Vr 3.43 _Üò Oa PQ ⊥ PS, PQ || SR, ∠SQR = 28º Am{U ∠QRT = 65º Agob Va x Am{U y Mr qH$_V H$mT>m.

AmH¥$Vr 3.42 AmH¥$Vr 3.43

6) AmH¥$Vr 3.44 _Üò ∆PQR Mr ~mOy QR hr S q~Xÿn`ªV dmT>{dbr. Oa ∠PQRd∠PRS

N>o{XHo$Vrb H$moZ T q~XÿV {_iVmV Va {gÜX H$am ∠QTR=12 ∠QPR

AmH¥$Vr 3.44

3.8 gmam§e

`m àH$aUmV Vwåhr Imbrb JmoîQ>tMm Aä`mg Ho$bmV.

1. Oa EH$m aofoda EH$ {H$aU C^m Agob Va XmoZ g§b½Z H$moZm§Mr ~oarO 180o AgVo `m JwUY_m©bmM gai aofodarb _wbVËdo åhUVmV.

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60 J{UV

2. Oa XmoZ aofm EH$_oH$m§Zm N>oXV AgVrb Va {eamo{déÜX H$moZ g_mZ AgVmV.

3. Oa XmoZ g_m§Va aofm§Zm EH$m N>oXrHo$Zo N>oXbo AgVm,

(i) àËòH$ g§JV H$moZm§À`m OmoS>`m g_mZ AgVmV.

(ii) àËòH$ ì`wËH«$_ H$moZm§Mr Omo‹S>r g_mZ AgVo.

(iii) N>o{XHo$À`m EH$mM A§Jmg AgUmè`m Am§VH$moZm§Mr àËòH$ Omo‹S>r nyaH$ AgVo.

4. Oa XmoZ g_m§Va aofm§Zm EH$m N>o{XHo$Zo N>oXbo AgVm,

(i) g§JV H$moZm§Mr H$moUVrhr EH$ Omo‹S>r g_mZ AgVo qH$dm,

(ii) ì`wËH«$_ H$moZm§Mr H$moUVrhr EH$ Omo‹S>r g_mZ AgVo qH$dm,

(iii) N>o{XHo$À`m EH$mM A§Jmg AgUmè`m Am§VaH$moZm§Mr H$moUVrhr EH$ Omo‹S>r nyaH$ Agob Va Vr aofm g_m§Va AgVo.

5. {Xbobr aofm Xþgè`m aofoer g_m§Va Agob Va Ë`m aofm nañnamer g_m§Va AgVmV.

6. {ÌH$moUmÀ`m {VÝhr H$moZm§Mr ~oarO 180º AgVo.

7. Oa {ÌH$moUmMr H$moUVrhr EH$ ~mOy dmT>{dbr AgVm hmoUmam ~møH$moZ hm Am§Va{déÜX H$moZm§À`m

~oaOo BVH$m AgVmo.

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~hþnXr 61

àH$aU - 4

~hþnXr (Polynomials)

4.1 àñVmdZm

Vwåhr _mJrb dJm©V ~¡{OH$ amerMr ~oarO, dOm~mH$s, JwUmH$ma Am{U mJmH$ma {eH$bm AmhmV. ~¡{OH$ ametMo Ad`d H$go nmS>VmV ho gwÜXm Vwåhr {eH$bm AmhmV. Vwåhr ~¡{OH$ {ZË`g_rH$aUo AmR>dyZ Ad`drH$aUmMo Cn`moJ gwÜXm AmR>dm. (x + y)2 = x2 + 2xy + y2

(x – y)2 = x2 – 2xy + y2

Am{U x2 – y2 = (x + y) (x – y)

`m àH$aUm_Üò Amåhr _hËdmÀ`m ~¡{OH$ ametÀ`m àH$mamZo gwédmV H$aUma Amho Ë`m§Zm ~hþnXr åhUVmV Am{U Ë`m§Mr n[a^mfm (eãXmdbr) ~hþnXrer g§~§YrV Amho. Amåhr eof {gÜXm§V Am{U Ad`d {gÜXm§V VgoM Ë`m§Mo ~hþnXrÀ`m Ad`dmVrb Cn`moJ gwÜXm {eH$Uma AmhmoV. m{edm` Amåhr AZoH$ ~¡OrH$ g_rH$aUo d Ë`m§Mo Ad`drH$aUmVrb Cn`moJ m§Mm Aä`mg H$aUma AmhmoV Am{U H$mhr {Xboë`m amerÀ`m qH$_Vr H$mT>Uma AmhmoV..

4.2 EH$ MbnX ~hþnXr (Polynomials in one variable)

AmR>dU H$am H$s MbnX ho {MÝhmZo Xe©{dVmV. Vr åhUOo H$moUVrhr dmñVd qH$_V. Amåhr x, y, z,

B. Ajao dmnaVmo. Ë`m§Zm MbnXo åhUVmV. 2 3 12

x x x x, , ,− − `m ~¡OrH$ amer AmhoV. gd© ~¡OrH$

amer ( x m Z_wÝ`mV AmhoV. AmVm g_Om amer (pñWanX) × (MbnX) m Z_wÝ`mV {bhm`Mr Agob Va pñWanX H$moUVo Amho _mhrV Zmhr. Aem {H«$ò_Üò pñWanX åhUyZ a, b, c B. {b{hVmo. åhUyZ ax hr amer Amho Ago åhUVmo.

VWm{n, pñWanXmMo Aja d MbnXmMo Aja _mZÊ`mV \$aH$ Amho. {Xboë`m àg§JmV pñWanX-mMr qH$_V eodQ> n`ªV pñWa AgVo. åhUOoM {Xboë`m CXmhaUmV pñWanXmMr qH$_V ~XbV Zmhr na§Vw MbnXmMr qH$_V ~XbV AgVo.

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62 J{UV

AmVm, g_Om 3 EH$H$ ~mOy Agbobm Mm¡ag Amho. (AmH¥$Vr 4.1 nhm) `mMr n[a{_Vr {H$Vr Amho? Vwåhm§bm _mhrV Amho Mm¡agmÀ`m Mma ~mOy§À`m bm§~rMr ~oarµO åhUOoM n[a{_{V hmo`. òWo àËòH$ ~mOy 3 EH$H$ Amho. åhUyZ `mMr n[a{_{V 4 × 3 Amho, åhUOoM 12 EH$H$. 10 EH$H$ ~mOy Agboë`m Mm¡agmMr n[a{_{V {H$Vr Agob? n[a{_Vr 4 × 10 Amho. åhUOoM 40 EH$H$. àËòH$ ~mOyMr bm§~r x EH$H$. (AmH¥$Vr 4.2 nhm). n[a{_{V 4x EH$H$ {Xbr Amho. åhUOoM Oa ~mOyMr bm§~r dmT>br Va n[a{_Vr gwÜXm dmT>Vo.

AmH¥$Vr 4.1

AmH¥$Vr 4.2

Mm¡ag PQRS Mo joÌ\$i H$mTy> eH$mb H$m? ho x × x = x2 Mm¡ag EH$H$ Amho.

x2 hr ~¡OrH$amer Amho. Vwåhm§bm. 2x, x2 + 2x, x3 – x2 + 4x + 7. `m ~¡OrH$ amer _mhrV AgVrbM. bjmV R>odm, KoVboë`m gd© ~¡OrH$ amerÀ`m MbnXm§Mm KmVm§H$ nyUmªH$ g§»`m Amho. `m amerÀ`m Z_wÝ`mbm EH$ MbnX ~hþnXr åhUVmV. darb CXmhaUmV MbnX x Amho. CXmhaUmW© x3 – x2 + 4x + 7. hr x MbnX ~hþnXr Amho. Ë`mM ~amo~a 3y2 + 5y hr y MbnX ~hþnXr d t2 + 4 hr t MbnX ~hþnXr Amho.

x2 + 2x `m ~hþnXrV x2 Am{U 2x `m§Zm ~hþnXrMr nXo åhUVmV. Ë`m~amo~a 3y2 + 5y + 7 `m ~hþnXrV VrZ nXo AmhoV. Ë`m§Mr Zmdo 3y2 , 5y Am{U 7. Vwåhr – x3 + 4x2 + 7x – 2 m ~hþnXrMr nXo {bhÿ eH$mb H$m? `m ~hþnXrbm – x3, 4x2, 7x Am{U – 2 Aer 4 nXo AmhoV.

~hþnXrVrb àËòH$ nXmbm ghJwUH$ AgVmo. åhUyZ – x3 + 4x2 + 7x – 2 _Yrb x3 Mm ghJwUH$ –1, x2 Mm ghJwUH$ + 4, x Mm ghJwUH$ 7 Am{U – 2 hm x0 Mm ghJwUH$ (AmR>dm x0 = 1) Amho. x2 – x + 7 _Yrb. x Mm ghJwUH$ _mhrV Amho H$m? Vmo – 1 Amho.

2 hr gwÜXm ~hþnXr Amho. 2, – 5, 7 B.hr pñWa ~hþnXrMr CXmhaUo Amho. 0 pñWa ~hþnXtbm eyÝ` ~hþnXr åhUVmV. hm ~hþnXtÀ`m gd© àH$mam_Üò _hËdmMr ^y{_H$m ~OmdVo ho darb dJm©V nhmb.

AmVm ~¡OrH$ amer x xx

x+ +1 3, Am{U y y3 2+ nhm. Vwåhm§bm _mhrV Amho x

x+

1 =

x + x–1 Aer {b{hVmo. òWo Xþgè`m nXmMm åhUOoM x–1 Mm KmVm§H$ – 1 Amho. åhUOoM nyUmªH$ g§»`m Zmhr. åhUyZ hr ~¡OrH$ amer ~hþnXr Zmhr.

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~hþnXr 63

nwÝhm x + 3 ho x12 3+ Ago {b{hVmV. òWo x Mm KmVm§H$

12 . Amho. Vr nwUmªH$ g§»`m Zmhr.

åhUyZ x + 3 hr ~hþnXr Amho? Zmhr. y y3 2+ m~Ôb H$m`? hr gwÜXm ~hþnXr Zmhr. H$m Zmhr?

Oa ~hþnXrVrb MbnX x Agob Va Ë`m ~hþnXrbm AmnU p(x) qH$dm q(x) qH$dm r(x) BË`mXr {bhÿ eH$Vmo. CXmhaU åhUyZ Aer {bhÿ eH$Vmo.

p(x) = 2x2 + 5x – 3

q(x) = x3 – 1

r(y) = y3 + y + 1

s(u) = 2 – u – u2 + 6u5

~hþnXr hr H$moUË`mhr g§»òÀ`m (_`m©XrV) nXm§Mr AgVo. CXmhaUmW© .

x150 + x140 + ..... + x2 + x + 1 hr ~hþnXr 151 nXm§Mr Amho.

2x, 2, 5x3, – 5x2, y Am{U u4 `m ~hþnXr nhm. Vwåhr nmhrbm H$m àËòH$ ~hþnXr hr \$ŠV EH$ nXmMr Amho. \$ŠV EH$ nX Agboë`m ~hþnXrbm EH$nXr (monomials) åhUVmV. (mono åhUOo EH$)

AmVm Imbrb àËòH$ ~hþnXrMo {ZarjU H$am :

P(x) = x + 1, q(x) = x2 – x, r(y) = y30 + 1, t(u) = u43 – u àËòH$m_Üò {H$Vr nXo AmhoV? àËòH$ ~hþnXr_Üò XmoZ nXo Amho. XmoZ nXo AgUmè`m ~hþnXrbm {ÛnXr åhUVmV (binomials). (bi åhUOo XmoZ).

Ë`mMà_mUo \$ŠV VrZ nXo AgUmè`m ~hþnXrbm {ÌnXr (trinomials) åhUVmV. (tri åhUOo VrZ). {ÌnXrMr H$mhr CXmhaUo -

p(x) = x + x2 + π q(x) = 2 2+ −x x

r(u) = 4 + 42 – 2 t(y) = y4 + y + 5

AmVm p(x) = 3x7 – 4x6 + x + 9 hr ~hþnXr nhm. x Mm _moR>m KmVm§H$ Agbobo H$moUVo nX Amho? Vo 3x7 Amho. `m nXmV xMm KmVm§H$ 7 Amho. Ë`mMà_mUo q(y) = 5y6 – 4y2 – 6 `m ~hþnXrV y Mm _moR>m KmVm§H$ Agbobo nX 5y6. Amho. `mV y `m nXm§Mm KmVm§H$ 6 Amho. ~hþnXrVrb MbnXmÀ`m _moR>çm KmVm§H$mbm Amåhr ~hþnXrMr H$moQ>r Ago åhUVmo. åhUyZ 3x7 – 4x6 + x + 9 `m ~hþnXrMr H$moQ>r 7 Amho Am{U 5y6 – 4y2 – 6 `m ~hþnXm§Mr H$moQ>r 6 Amho. eyÝ` ahrV pñWanX ~hþnXtMr H$moQ>r eyÝ` AgVo.

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CXmhaU 1: Imbr {Xboë`m àËòH$ ~hþnXtMr H$moQ>r emoYm..

(i) x5 – x4 + 3 (ii) 2 – y2 – y3 + 2y8 (iii) 2

CH$b: (i) MbnXmMm _moR>m KmVm§H$ 5 Amho. åhUyZ ~hþnXrMr H$moQ>r 5 Amho.

(ii) MbnXmMm _moR>m KmVm§H$ 8 Amho åhUyZ. ~hþnXtMr H$moQ>r 8 Amho.

(iii) 2 ho EH$M nX Amho Vo 2x0 Agohr {b{hVmV. åhUyZ x Mm KmVm§H$ 0 Amho.åhUyZM ~hþnXrMr H$moQ>r 0 Amho.

AmVm, ~hþnXr p(x) = 4x + 5 , q(y) =2y, r (t) = t + 2 Am{U s(u)= 3 – u m§Mo {ZarjU H$am. `m_Üò H$mhr gmå` nm{hbm H$m? àËòH$ ~hþnXrMr H$moQ>r 1 Amho. `mbm aofr` ~hþnXr (linear polynomial) åhUVmV. 1 MbnX Agboë`m H$mhr aofr` ~hþnXr 2 1 2 1 2x y u− + −, , `m AmhoV.

AmVm x MbnXmMr 3 nXo AgUmè`m aofr` ~hþnXr à`ËZ H$am Am{U emoYm? Vwåhr Vo emoYUma Zmhr H$maU x À`m aofr` ~hþnXr _Üò OmñVrV OmñV 2 nXo AgVmV. åhUyZ H$moUË`mhr x À`m aofr` ~hþnXrMm gm_mÝ` Z_wZm ax + b AgVmo. òWo a Am{U b ho pñWanX Am{U a ≠ 0 (H$m?) Ë`mMà_mUo y _Yrb aofr` ~hþnXr ay + b Amho.

AmVm `m ~hþnXr nhm :

2x2 + 5, 5x2 + 3x + π, x2 Am{U x x2 25

+

Vwåhm§bm _mÝ` Amho H$m `m§Mm H$moQ>r 2 Amho? ~hþnXrMm H$moQ>r 2 AgVmo Voìhm Ë`mbm dJ© ~hþnXr (quadratic polynomial) åhUVmV.

5 – y2, 4y + 5y2 Am{U 6 – y – y2 hr H$mhr dJ© ~hþnXrMr CXmhaUo AmhoV. EH$ MbnX Amåhr doJdoJir Mma nXo AgUmè`m dJ© ~hþnXr {bhÿ eH$mb H$m? EH$ MbnXmÀ`m OmñVrV OmñV doJdoJir 3 nXo AgUmè`m dJ© ~hþnXr emoYm. Oa Vwåhr AmUIr H$mhr dJ© ~hþnXrMr `mXr Ho$bm Va Vwåhm§bm g_OyZ òB©b H$s x _Yrb dJ©~hþnXrMm gm_mÝ` Z_wZm ax2 + bx + c Amho. òWo a ≠ 0 Am{U a, b, c hr pñWanXo AmhoV. Ë`mMà_mUo y- _Yrb dJ© ~hþnXrMm gm_mÝ` Z_wZm ay2 + by + c OoWo a ≠ 0 Am{U a, b, c hr pñWanXo AmhoV.

Amåhr VrZ H$moQ>r Agboë`m ~hþnXrbm KZ ~hþnXr (cubic polynomial) åhUVmo. x _Yrb 4x3, 2x3 + 1, 5x3 + x2, 6x3 – x, 6 – x3, 2x3 + 4x2 + 6x + 7 hr H$mhr KZ ~hþnXrMr CXmhaUo Am-hoV. EH$ MbnXmÀ`m KZ ~hþnXr_Üò {H$Vr nXo AgVmV {dMma H$am? `m_Üò OmñVrV OmñV 4 nXo AgVmV. hr ax3 + bx3 + cx + d `m gm_mÝ` Z_wÝ`mV {b{hVmV. OoWo a ≠ 0 Am{U a, b, c Am{U d hr pñWanXo AmhoV.

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~hþnXr 65

AmVm 1 H$moQ>r, 2 H$moQ>r Am{U 3 H$moQ>r ~hþnXr H$em AmhoV Vo Vwåhr nm{hbm AmhmV. H$moUVrhr Z¡g{J©H$ g§»`m n _Yrb EH$ MbnXmMr n H$moQ>r Agbobr ~hþnXr {bhÿ eH$mb H$m? EH$ MbnX x, H$moQ>r n Agboë`m ~hþnXrMm gm_mÝ` Z_wZm.

anxn + an–1x

n–1 + .... + a1x + a0

òWo a0 , a1 , a2 , a3 , an ho pñWanX Am{U a ≠ 0 VWm{n Oa a0 = a1 = a2 = a3 .... = an = 0 (gd© pñWanXo eyÝ` AmhoV) Amåhm§bm {_iVo. eyÝ` ~hþnXr {Vbm 0 (eyÝ`) Ago hr åhUVmV. eyÝ` ~hþnXrMr H$moQ>r {H$Vr? eyÝ` ~hþnXr H$moQ>rMr ì`m»`m H$aVm òV Zmhr.

åhUyZM Amåhr EH$ MbnXmMr ~hþnXr dmnaVmo. EH$mnojm OmñV MbnXm§Mm ~hþnXr gwÜXm AmhoV. CXm. x2 + y2 + xyz (MbnXo x, y Am{U z AmhoV ). hr VrZ nXm§Mr ~hþnXr Amho. Ë`mMà_mUo p2 + q10 + r (òWo p, q d r MbnXo AmhoV). u2 + v2 (òWo u d v hr MbnXo AmhoV). `m ~hþnXr VrZ d XmoZ MbnXmÀ`m AmhoV. Aem ~hþnXtMm gImob Aä`mg Vwåhr Z§Va H$amb.

ñdmÜ`m` 4.1

1. Imbrbn¡H$s H$moUË`m amer EH$ MbnX Agboë`m Am{U EH$ MbnX Zgboë`m ~hþnXr Am-hoV? CÎmamMo H$maU gm§Jm.

(i) 4x2 – 3x + 7 (ii) y2 2+ (iii) 3 2t t+

(iv) yy

+2

(v) x10 + y3 + t50

2. Imbrb àËòH$ x2 Mm ghJwUH$ {bhm.

(i) 2 + x2 + x (ii) 2 – x2 + 4x3 (iii) π2

2x x+ (iv) 2 1x −

3. {ÛnXrMr H$moQ>r 35 Am{U EH$nXrMr H$moQ>r 100 Agbobo àËòH$s EH$ CXmhaU Úm.

4. Imbrb àËòH$ ~hþnXrMr H$moQ>r {bhm..

(i) 5x3 + 4x2 + 7x (ii) 4 – y2 (iii) 5 7t − (iv) 3

5. Imbrb ~hþnXtMo aofr`, dJ© d KZ ~hþnXr_Üò dJuH$aU H$am.

(i) x2 + x (ii) x – x3 (iii) y + y2 + 4 (iv) 1 + x

(v) 3t (vi) r2 (vii) 7x3

4.3 ~hþnXrMr eyÝò (Zeroes of Polynomials)

p(x) = 5x3 – 2x2 + 3x – 2 hr ~hþnXr nhm.

Oa Amåhr p(x) _Üò gd© {R>H$mUr x Mr qH$_V KoVbr Va Amåhmbm {_iVo.

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66 J{UV

p(1) = 5(1)3 – 2(1)2 + 3(1) – 2

= 5 – 2 + 3 – 2

= 4

åhUyZ p(x) _Üò x = 1 KoD$Z qH$_V 4 òVo

Ë`mMà_mUo p(0) = 5(0)3 – 2(0)2 + 3(0) – 2 = – 2

Vwåhr p(–1) Mr qH$_V H$mT>mb H$m?

CXmhaU 2: MbnXm§À`m {Xboë`m qH$_VrZwgma Imbrb àËòH$ ~hþnXtMr qH$_V H$mT>m.

(i) p(x) = 5x2 – 3x + 7 Oa x = 1

(ii) q(y) = 3 4 113y y− + Oa y = 2

(iii) p(t) = 4t4 + 5t3 – t2 + a Oa t = a

CH$b : (i) p(x) = 5x2 – 3x + 7

p(x) `m ~hþnXrV x = 1 qH$_V KoD$Z.

p(1) = 5(1)2 – 3(1) + 7 p(1) = 5 – 3 + 7 p(1) = 12 – 3 p(1) = 9

(ii) q(y) = 3 4 113y y− +

q(y) `m ~hþnXrV y = 2 qH$_V KoD$Z.

q(2) = 3 2 4 2 113( ) − ( ) + q(2) = 3 8 8 11( ) − +

= 24 8 11− +

= 16 11+

(iii) p(t) = 4t4 + 5t3 – t2 + 6

p(t) `m ~hþnXrV t = a qH$_V KoD$Z. p(a) = 4a2 + 5a3 – a2 + 6

AmVm p(x) = x – 1 hr ~hþnXr KoD$Z. p(1) åhUOo H$m`?

bjmV R>odm : p(1) = 1 – 1 = 0

p(1) = 0 åhUOoM 1 hm ~hþnXr p(x) Mm eyÝ` Amho.

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~hþnXr 67

Ë`mMà_mUo Vwåhr 2 hm q(x) Mm eyÝ` Amho. Vo Vnmgm Ooìhm q(x) = x – 2.

gm_mÝ`nUo, Oa c hr g§»`m ~hþnXr p(x) Mr eyÝ` Agob Va Amåhr åhUy eH$Vmo {H$ p(c) = 0.

Vwåhr {ZarjU Ho$bm AgUmaM H$s, ~hþnXr x – 1 Mm ewÝ` hm 0 ~amo~a AgVmo. åhUyZ x – 1 = 0 Vo x = 1 òVo. Amåhr p(x) = 0 bm ~hþnXr g_rH$aU åhUVmo Am{U, p(x) = 0 `m ~hþnXrÀ`m g_rH$aUmMo dJ©_yi (root of polynomial equation) 1 Amho. åhUyZ 1 hm ~hþnXr x – 1 Mo eyÝ` Amho Ago åhUVmo. qH$dm x – 1 `m ~hþnXrÀ`m g_rH$aUmMo dJ©_yi Amho.

AmVm 5 hr pñWa ~hþnXr nhm. `mMm eyÝ` H$m` Amho gm§Jy eH$mb? `mbm eyÝ` Zmhr H$maU 5x0 _Üò x À`m EodOr H$moUVrhr g§»`m KoVbr Va 5 {_iVmV.

IaoVa eyÝàhrV pñWa ~hþnXrbm eyÝò ZgVmV. eyÝ` ~hþnXrMr eyÝò H$moUVr AmhoV? {Z`_mZwgma, àËòH$ dmñVd g§»`m hr eyÝ` ~hþnXrMr eyÝò AgVmV.

CXmhaU 3: ~hþnXr x + 2 Mo eyÝò – 2 Am{U 2 AmhoV H$m Vnmgm.

CH$b : Oa p(x) = x + 2

p(2) = 2 + 2 = 4

p (–2) = – 2 + 2 = 0

åhUyZ ~hþnXr x + 2 Mo eyÝ` – 2 Amho. nU 2 Zmhr. åhUyZ ~hþnXr x + 2 Mm eyÝ` Zmhr.

CXmhaU 4: ~hþnXr p(x) =2x + 1 Mr eyÝò emoYm.

CH$b : p(x) Mr eyÝ` emoYÊ`mgmR>r p(x) = 0 `mgmaIo g_rH$aU gmoS>dm.

AmVm, x = −12 KoVë`mg 2x + 1 = 0 òVo

åhUyZ −12 ho, ~hþnXr 2x + 1 Mo eyÝ` Amho.

AmVm, Oa p(x) = ax + b, a ≠ 0 hr aofr` ~hþnXr Amho. p(x) Mo eyÝ` emoYÊ`mgmR>r ~hþnXr g_rH$aU

p(x) = 0 hmo qH$_V KoD$Z Vr gmoCdm.

AmVm p(x) = 0 åhUOoM ax + b, a ≠ 0

åhUyZ, ax = – b

x = −ba

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68 J{UV

åhUyZ x = −ba

ho \$ŠV p(x) Mo eyÝ` Amho i.e. aofr` ~hþnXrbm \$ŠV EH$ Am{U EH$M eyÝ`

AgVo.

AmVm AmnU åhUyZ eH$Vmo H$s x – 1 Mo eyÝ` 1 Amho. Am{U x + 2 Mo eyÝ` – 2 Amho.

CXmhaU 5: x2 – 2x `m ~hþnXrMr eyÝò 2 Am{U 0 AmhoV nS>Vmim.

CH$b : Oa p(x) = x2 – 2x

p(2) = 22 – 2(2) = 4 – 4 = 0

Am{U p(0) = 0 – 0 = 0

x2 – 2x `m ~hþnXrMr eyÝò 2 Am{U 0 XmoÝhr AmhoV.

AmVm AmnU Am_À`m {ZarjUm§Mr `mXr V`ma H$ê$.

(i) eyÝ` ~hþnXrMo eyÝ` (0) nmhrOoM Ago Zmhr.

(ii) eyÝ` (0) ho ~hþnXrMo eyÝ` Agy eH$Vo.

(iii) àËòH$ aofr` ~hþnXrMm \$ŠV EH$ Am{U EH$M eyÝ` AgVmo.

(iv) ~hþnXrbm EH$mnojm OmñV ewÝò Agy eH$VmV.

ñdmÜ`m` 4.2

1. ~hþnXr 5x – 4x2 + 3 Mr qH$_V emoYm Ooìhm -

(i) x = 0 (ii) x = – 1 (iii) x = 2

2. Imbrb àËòH$ ~hþnXrMo p(0), p(1) Am{U p(2) emoYm.

(i) p(y) = y2 – y + 1 (ii) p(t) = 2 + t + 2t2 – t3

(iii) p(x) = x3 (iv) p(x) = (x – 1) (x + 1)

3. g_moa {Xbobr eyÝò Ë`m ~hþnXrMr AmhoV H$m nS>Vmim.

(i) p(x) = 3x + 1 ; x = − 13

(ii) p(x) = 5x – π ; x =45

(iii) p(x) = x2 – 1 ; x = 1, – 1

(iv) p(x) = (x + 1) (x – 2) ; x = – 1, 2

(v) p(x) = x2 ; x = 0

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~hþnXr 69

(vi) p(x) = lx + m ; xml

= −

(vii) p(x) = 3x2 – 1 ; x = − 13

23

,

(viii) p(x) = 2x + 1 ; x =12

4. Imbrb àËòH$ ~hþnXtMr eyÝò emoYm.

(i) p(x) = x + 5 (ii) p(x) = x – 5

(iii) p(x) = 2x + 5 (iv) p(x) = 3x – 2

(v) p(x) = 3x (vi) p(x) = ax, a ≠ 0

(vii) p(x) = cx + d, c ≠ 0. c, d `m dmñVd g§»`m.

4.4 eof {gÜXm§V (Remainder Theorem)

15 d 6 `m g§»`m Vwåhm§bm _mhrV AmhoV. 15 bm 6 Vo ^mJbo Va AmnUmbm ^mJmH$ma 2 d ~mH$s 3 {_iVo. Vwåhm§bm AmR>dVo H$m ho H$go {b{hVmV? Amåhr 15 bm Ago {b{hVmo.

15 = (6 × 2) + 3

Amåhr {ZarjUmZo gm§Jy eH$Vmo H$s ~mH$s 3 hm ^mOH$ 6 nojm bhmZ Amho. Ë`mMà_mUo Oa Amåhr 12 bm 6 Zo ^mJbo Va AmnUmg {_iVo -

12 = (6 × 2) + 0

òWo ~mH$s {H$Vr Amho? òWo ~mH$s 0 Amho, Am{U Amåhr Ago åhUy eH$Vmo H$s 12Mm Ad`d 6 Amho qH$dm 12 ho 6 À`m nmQ>rV AmhoV.

AmVm, àíZ Agm Amho H$s Amåhr EH$m ~hþnXrbm Xþgè`m ~hþnXrZo ^mJ XoD$ eH$Vmo H$m? à`ËZ H$ê$`m H$s ^mOH$ EH$nXr Agob. åhUyZ ~hþnXr 2x3 + x2 + x bm EH$nXr x Zo ^mJy`m.

(2x3 + x2 + x) ÷ x = 2 3 2xx

xx

xx

+ +

= 2x2 + x + 1

Vw_À`m bjmV Ambo AgUma H$s (2x3 + x2 + x) àËòH$ nXmV x gm_mB©H$ Amho. åhUyZ 2x3 + x2 + x ho x (2x2 + x + 1) Ago {bhÿ eH$Vmo. Amåhr Ago åhUyZ eH$Vmo H$s, 2x3 + x2 + x Mo Ad`d x Am{U 2x2 + x + 1 AmhoV. Am{U 2x3 + x2 + x ho x Am{U 2x2 + x + 1 À`m nQ>rV AmhoV.

AmUIr 3x2 + x + 1 d x hr EH$ ~hþnXrMr OmoS>r KoD$.

òWo (3x2 + x + 1) ÷ x = (3x2 ÷ x) + (x ÷ x) + (1 ÷ x)

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~hþnXrMo nX {_i{dÊ`mgmR>r Amåhr 1 bm x Zo ^mJy eH$V Zmhr. åhUyZ òWo AmnU Wm§~bo nmhrOo Am{U bjmV R>odm 1 hr ~mH$s Amho. åhUyZ Amåhr

3x2 + x + 1 =[ x × (3x + 1)] + 1

`m KQ>Zo_Üò 3x + 1 hm ^mJmH$ma Am{U 1 hr ~mH$s Amho. Vwåhr {dMma H$ê$ eH$Vm x hm 3x2 + x + 1 Mm Ad`d Amho H$m? ~mH$s 0 Zmhr åhUyZ Vmo Ad`d Zmhr. AmVm ewÝàhrV ~hþnXrbm

^mJ H$gm XoVmV, EH$ CXmhaU nmhÿ. .

CXmhaU 6: p(x) bm g(x) Zo ^mJm òWo p(x) = x + 3x2 – 1 Am{U g(x) = 1 + x

CH$b : Amåhr Imbrb nm`è`m§Mr ^mJmH$mamMr {H«$`m H$ê$..

nmàr 1: Amåhr ^mÁ` x + 3x2 – 1 d ^mOH$ 1 + x gm_mÝ` Z_wÝ`mV {bhÿ. Vr åhUOo {Xbobr

nXo Ë`m§À`m H$moQ>rÀ`m CVaË`m H«$_mV _m§Sy> åhUyZ ^mÁ` (standard form) 3x2 + x – 1 d ^mOH$

x + 1 hmoB©b.

nmàr 2 : Amåhr ^mÁ`mÀ`m n{hë`m nXmbm ^mOH$mÀ`m n{hë`m nXmZo ^mJ XoD$ åhUOo Amåhr 3x2

bm x Zo ^mJy. Amnë`mbm 3x. {_iVmV. hm n{hë`m nXmMm ^mJmH$ma {_iVmo.32

32x x=

nmàr 3 : Amåhr n{hë`m nXmÀ`m ^mJmH$mamZo ^mOH$mbm JwUyZ

^mÁ`mVyZ dOm~mH$s H$ê$ åhUOo Amåhr 3x Zo x + 1 bm JwUyZ ^mÁ`

3x2 + x – 1 _YyZ 3x2 + 3x dOm H$ê$. ~mH$s – 2x – 1 {_iVo.

nmàr 4: ~mH$s – 2x – 1 hm ZdrZ ^mÁ` Pmbm. ^mOH$ VmoM Amho. Amåhr nmàr 2 naV dmnê$

Amnë`mbm mJmH$maMo nwT>Mo nX {_iVo Vo åhUOo Amåhr ZdrZ mÁ`mMo n{hbo nX – 2x bm mOH$mMo

n{hbo nX x Zo ^mJy {_iVmV -2 åhUOoM -2 ^mJmH$maMo 2 ao nX Amho.

− = −2 2xx

^mJmH$mamMo Xþgao nX ZdrZ ^mÁ` = 3x – 2.

nmàr 5: Amåhr ^mJmH$mamÀ`m Xþgè`m nXmZo ^mOH$mbm JwUyZ ^mÁ`m_YyZ dOm~mH$s H$ê$. åhUyZ Amåhr –2x Zo (x + 1) bm JwUyZ –2x – 2 ho ^mÁ`m_YyZ dOm H$ê$. Amnë`mbm 1 ~mH$s {_iVo.

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~hþnXr 71

hr {H«$`m ~mH$s eyÝ` (0) {_ion`ªV Mmby R>odm. qH$dm Z{dZ ^mÁ`mMr H$moQ>r hr ^mOH$mÀ`m

H$moQ>rnojm bhmZ Ambr n{hOo. `mdoir ZdrZ ^mÁ` hm ~mH$s ~ZVmo Am{U ^mJmH$mamÀ`m gd© nXm§Mr

~oarO ^mJmH$ma ~ZVo.

nmàr 6: åhUyZ nyU© ^mJmH$ma 3x – 2 d ~mH$s 1 Amho.

AmVm nhm nyU© {H«$`m H$er Ho$br Vo -

bjmV R>odm 3x2 + x – 1 =[ (x + 1) (3x – 2)] + 1..

^mÁ`= (^mOH$ × ^mJmH$ma) +~mH$s

gd©gm_mÝ`nUo Oa p(x) Am{U g(x) `m XmoZ ~hþnXr AgyZ p(x) Mr H$moQ>r ≥ g(x) Mr H$moQ>r

Am{U g(x) ≠ 0 Va ~hþnXr q(x) Am{U r(x) Aer emoYy eH$Vmo H$s.p(x) = g(x) q(x) + r(x)

òWo r(x) = 0 qH$dm r(x) Mr H$moQ>r < g(xMr H$moQ>r. òWo Amåhr Ago åhUy eH$Vmo p(x) bm g(x)

Zo ^mJ {Xbm AgVm q(x) hm ^mJmH$ma, r(x) hr ~mH$s {_iVo.

darb CXmhaUmV, ^mOH$ hm aofr` ~hþnXr Amho, Aem doir Oa ~mH$s Am{U ^mÁ`mÀ`m H$mhr

qH$_Vr_Üò H$mhr g§~§Y Amho H$m nmhmdm bmJob.

p(x) = 3x2 + x – 1 Oa x Mr qH$_V – 1 KoD$Z.

p(.–1) = 3(–1)2 + (–1) – 1= 1

åhUyZ, p(x) = 3x2 + x – 1. bm x + 1..Zo mJbo AgVm ~mH$s {_iVo Vr p(x) ~hþnXrMr qH$_V

d x + 1 `m ~hþnXtMr eyÝò `m§Mr qH$_V g_mZ AgVo åhUOoM –1.

AmUIr H$mhr CXmhaUo nmhÿ.

CXmhaU 7: ~hþnXr 3x4 – 4x3 – 3x – 1 bm x – 1 Zo ^mJm.

CH$b : XrK© ^mJmH$ma nÜXVrZo

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72 J{UV

`oWo ~mH$s – 5 Amho. AmVm x – 1 Mo eyÝ` 1 Amho. åhUyZ p(x) _Ü`o x = 1 KoD$Z Amåhr nmhÿ eH$Vmo.

p(x) = 3(1)4 – 4(1)3 – 3(1) – 1

= 3 – 4 – 3 – 1 = – 5 hr ~mH$s Amho.

CXmhaU 8: ~hþnXr p(x) = x3 + 1 bm x + 1 Zo ^mJyZ ~mH$s H$mT>m.

CH$b : XrK© ^mJmH$ma nÜXVrZo -

åhUyZ ~mH$s 0 Amho ho emoYbo.

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~hþnXr 73

òWo p(x) = x3 + 1 Am{U x + 1 = 0.Mo ~rO x = – 1 Amho Amåhr nmhÿ eH$Vmo p(–1) = (–1)3 + 1 = – 1 + 1

= 0

XrK© ^mJmH$ma nÜXVrZo H$mT>boë`m ~mH$s BVH$sM hr nU ~mH$s Amho.~hþnXrbm aofr` ~hþnXrZo ^mJyZ ~mH$s {_i{dÊ`mMr nÜXV hr gmonr Amho H$m? eof {gÜXm§VmÀ`m gm_mÝ` Z_wÝ`mZwgma nwamdm XoD$Z hm {gÜXm§V Iam Amho ho gwÜXm Amåhr Vwåhm§bm XmIdy eH$Vmo.

eof {gÜXm§V(Remainder Theorem) : Oa p(x) hr H$moUVrhr ~hþnXr AgyZ {VMm H$moQ>r 1 qH$dm 1 nojm _moR>m Agob Am{U a hr dmñVd g§»`m Amho. Va p(x) bm aofr` ~hþnXr (x – a) Zo ^mJbo Va ~mH$s p(a) BVH$s AgVo.

{gÜXVm : Oa p(x) hr ~hþnXr AgyZ {VMr H$moQ>r 1. qH$dm 1 nojm _moR>r Amho. g_Om p(x) bm (x – a) Zo ^mJbo Va ^mJmH$ma q(x) Amåhr ~mH$s r(x) AgVo.

p(x) = (x – a) q(x) + r(x)

(x – a) Mr H$moQ>r 1 Am{U r(x) Mr H$moQ>r hr (x – a) À`m H$moQ>rnojm bhmZ r(x) = 0. åhUOoM r(x) ho pñWa Amho. Ë`mbm r _mZy.

åhUyZ x àËòH$ qH$_V r(x) = r

p(x) = (x – a) q(x) + r

VWm{n Oa x = a KoD$Z g_rH$aU Ago {_iVo.

p(a)= (a – a) q(a) + r

= r

\ {gÜXm§V {gÜX Pmbm.

Xþgè`m CXmhaUmbm ho CÎma bmJy H$ê$.

CXmhaU 9 : x4 + x3 – 2x2 + x + 1 bm x – 1 Zo ^mJ {Xbm Va ~mH$s emoYm.

CH$b: òWo p(x) = x4 + x3 – 2x2 + x + 1 Am{U x – 1 Mo eyÝ` 1 Amho.

p(1) = (1)4 + (1)3 – 2(1)2 + 1 + 1

= 2

åhUyZ Ooìhm x4 + x3 – 2x2 + x + 1 bm (x – 1) Zo ^mJbo Va eof {gÜXm§VmZo ~mH$s 2 òVo.

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74 J{UV

CXmhaU 10: ~hþnXr q(t) = 4t3 + 4t2 – t – 1 ho (2t + 1) À`m nQ>rV Amho H$m Vo Vnmgm.

CH$b: Vwåhm§bm _mhrV AmhoM q(t) ho (2t + 1) À`m nQ>rV AmhoV.

Oa q(t) bm (2t + 1) Zo ^mJbo Va ~mH$s eyÝ` amhVo. AmVm (2t + 1) = 0 \ t = − 12

KoD$Z

q −

12

= 4 12

4 12

12

13 2

−

+ −

− −

−

= − + + − =12

1 12

1 0

åhUyZ q(t) bm (2t + 1) Zo ^mJë`mg ~mH$s 0 {_iVo.

åhUyZ {Xbobr ~hþnXr q(t) Mm 2t + 1 hm Ad`d Amho. åhUOoM q(t) hm (2t + 1) À`m nQ>rV Amho.

ñdmÜ`m` 4.3

1. x3 + 3x2 + 3x + 1 bm Imbrb ~hþnXrZo ^mJbo AgVm ~mH$s H$mT>m.

(i) x + 1 (ii) x −12

(iii) x

(iv) x + π (v) 5 + 2x

2. Ooìhm x3 – ax2 + 6x – a bm x – a Zo ^mJbo AgVm ~mH$s H$mT>m.

3. 3x3 + 7x Mm 7 + 3x hm Ad`d Amho H$m ? Vo Vnmgm.

4.5 ~hþnXtMo Ad`d : (Factorisation of Polynomials)

CXmhaU 10 Mo H$miOrnyd©H$ {ZarjU H$am. q t−

= +( )1

20 2 1, hr ~mH$s Amho.

(2t + 1) hm q(t) Mm Ad`d Amho åhUyZ g(t) ~hþnXrÀ`m ~m~VrV q(t) = (2t + 1) g(t). Imbrb {gÜXm§VmMo ho EH$ CXmhaU Amho.

Ad`d à_o` (Factor Theorem)

Oa ~hþnXtMr H$moQ>r n ≥ 1 Am{U a hr H$moUVrhr dmñVd g§»`m Agob Va, (i) (x – a) hm p(x) Mm Ad`d Amho Ooìhm p(a) = 0 Am{U (ii) p(a) = 0 Ooìhm (x – a) hm p(x) Mm Ad`d Amho.

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~hþnXr 75

{gÜXVm :

eof {gÜXm§VmZwgma p(x) = (x – a) q(x) + p(a)

(i) Oa p(a) = 0 Va p(x) = (x – a) q(x), (x – a) hm p(x) Mm Ad`d Xe©{dVmo.

(ii) p(x) hm (x – a) Mm Ad`d Amho Ooìhm p(x) = (x – a) g(x) ~hþnXr g(x) _Ü`o p(a – a) g(a) = 0 darb gmÜ` eof{gÕm§Vmdê$Z bJoM àmßV H$aVm `oVo.

nwT>rb CXmhaUo nhm.

CXmhaU 11: x3 + 3x2 + 5x + 6 Am{U (2 x + 4 ) Mm x + 2 hm Ad`d Amho H$m Vo Vnmgm.

CH$b : (x + 2) Mo eyÝ` –2 Amho Oa p(x) = x3 + 3x2 + 5x + 6 Am{U s(x)= 2x + 4

Va p(2) = (–2)3 + 3(–2)2 + 5(–2) + 6

= –8 + 12 – 10 + 6 = 0

åhUyZ eof {gÜXm§VmZwgma (x + 2) hm x3 + 3x2 + 5x + 6 Mm Ad`d Amho.

nwÝhm s(–2)= 2(–2) + 4 = 0

åhUyZ (x + 2) hm (2x + 4) `m Ad`d AmhoV. Vwåhr Ad`d à_o` bmJy H$am`À`m AJmoXa nU Vnmgy eH$Vm 2x + 4 = 2 (x + 2)

CXmhaU 12: 4x3 + 3x2 – 4x + k Mm (x – 1) hm Ad`d Agob Va..Mr qH$_V H$mT>m.

CH$b : (x – 1) hm p(x) = 4x3 + 3x2 – 4x + k M Ad`d Amho p(1) = 0

AmVm p(1) = 4(1)3 + 3(1)2 – 4(1) + k

åhUyZ 4 + 3 – 4 + k = 0

åhUyZ k = –3.

H$mhr ~hþnXtMr H$moQ>r 2 d 3 Agboë`m ~hþnXrMo Ad`drH$aU H$aÊ`mgmR>r Amåhr Ad`d à_o` dmnê$. Vwåhm§bm _mhrV AmhoM H$s, x2 + lx + m. m gma»`m dJ© ~hþnXrMo Ad`drH$aU H$go H$aVmV. _Ybo nX lm Mo ax + bx m ñdê$nmV Ago Ad`d nmS>m H$s ab = m, x2 + lx + m = (x + a) (x + b) hmoB©b. AmVm Amåhr ax2 + bx + c `m ñdê$nmÀ`m dJ© ~hþnXrMo Ad`d H$mT>Ê`mMm à`ËZ H$ê$. OoWo a ≠ 0 Am{U a,b, c hr pñWanXo AmhoV.

~hþnXr ax2 + bx + c Mo Ad`drH$aU H$aVmZm _Yë`m nXmMr \$moS> Imbrb à_mUo H$aVmV.

Oa Ad`d (px + q) Am{U (rx + s) AgVrb Va.

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ax2 + bx + c = (px + q) (rx + s)

= prx2 + (ps + qr) x + qs

x2 À`m ghJwUH$mMr VwbZm Ho$brVa a = pr {_iVmo Ë`mMà_mUo

x À`m ghJwUH$mMr VwbZm Ho$br Va b = ps + qr {_iVmo. Am{U pñWanXmMr VwbZm c = qs {_iVo.... ps Am{U pr `m nXm§Mr ~oarO b Xe©{dVo Am{U JwUmH$ma (ps) (qr) = (pr) (qs) = ac `oVmo.

ax2 + bx + c À`m Ad`drH$aUmgmR>r XmoZ g§»`mMr ~oarO åhUOo b d Ë`m§Mm JwUmH$ma ac KoVbm nmhrOo ho CXmhaU 13 dê$Z Amnë`mbm g_Oob.

CXmhaU 13: _Yë`m nXmMr \$mo‹S> H$ê$Z 6x2 + 17x + 5 Mo Ad`drH$aU Ad`d à_o`mZo H$am.

CH$b 1 _Ü` nXmMr \$mo‹S> H$aÊ`mMr nÜXV (By splitting method)

p d q `m XmoZ g§»`m Aem ¿`m H$s p + q = 17 Am{U pq = 6 × 5 = 30 oVrb. Z§Va Ë`mMo Ad`d KoD$.

AmVm 30À`m Ad`d OmoS> m nhm. 1 Am{U 30, 2 Am{U 15, 3 Am{U 10, 5 Am{U 6.

`mn¡H$s p + q = 17 ... åhUyZ 2 + 15 KoVm oB©b.

6x2 + 17x + 5 = 6x2 + (2 + 15)x + 5

= 6x2 + 2x + 15x + 5

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

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

CH$b 2 (Ad`d à_o` dmnê$Z)

6x2 + 17x + 5 = 6 176

56

2x x+ +

= 6p(x) Oa a Am{U b ho p(x) Mo eyÝ o Va

6x2 + 17x + 5 = 6(x – a) (x – b), ab =56

a Am{U b gmR>r

H$mhr qH$_V ± ± ± ± ±12

13

53

52

1, , , , AmVm

p 12

14

176

12

56

0

= +

+ ≠

na§Vy p −

=

13

0 åhUyZ x +

13

hm p(x) Mm Ad`d Amho

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~hþnXr 77

Ë`mMà_mUo x +

52 hm p(x) Mm Ad`d gwÜXm Ad`d Amho.

åhUyZ 6x2 + 17x + 5 = 6 13

52

x x+

+

= 6 3 13

2 52

x x+

+

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

darb {d^mJUr nÜXVrMm dmna `mo½` dmQ>Vmo. AmUIr H$mhr CXmhaUo nmhþ.

CXmhaU 14 : y2 – 5y + 6 Mo Ad`drH$aU Ad`d à_o` dmnê$Z H$am.

CH$b : Oa p(x) = y2 – 5y + 6, AmVm p(y) = (y – a) (y – b) Vwåhm§bm _mhrV AmhoM

H$s ab ho. pñWanX Amho åhUyZ ab = 6, p(y) Mo Ad`d H$mT>m`Mo åhUOoM 6 Mo Ad`d H$mT>Uo

hmo`.

∴ 6 Mo Ad`d 1, 2 d 3 AmhoV.

AmVm p(2) = 22 – 5(2) + 6 = 0

(y – 2) hm p(y) Mm Ad`d Amho.

Ë`mMà_mUo p(3) = 32 – 5(3) + 6 = 0

åhUyZM y2 – 5y + 6 Mm (y – 3) gwÜXm Ad`d Amho.

åhUyZM y2 – 5y + 6 = (y – 2) (y – 3)

bjmV R>odm, _Ybo nX – 5y Mr \$moS>> H$ê$Z gwÜXm y2 – 5y + 6 Mo Ad`d H$mT>Vm `oVmV.

AmVm KZ ~hþnXtMo Ad`drH$aU nmhÿ. `oWo _Yrb nX \$moS> H$aÊ`mMr nÜXV `mo½` Zmhr.

Amnë`mbm H$_rV H$_r EH$ Ad`d H$mT>m`bm nmhrOo ho Vwåhr Imbrb CXmhaUmV nmhÿ eH$mb.

CXmhaU 15: x3 – 23x2 + 142x – 120 Mo Ad`d nm‹S>m.

CH$b : Oa p(x) = x3 – 23x2 + 142 – 120.

àW_ –120 Mo Ad`d nmhÿ. ± 1, ± 12, ± 15, ± 20, ± 24, ± 30, ± 60 ho H$mhr Ad`d

AmhoV.

à`ËZmZo p(1) = 0 {_iVo åhUyZ x – 1 hm p(x) Mm Ad`d Amho.

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78 J{UV

AmVm Amåhr nmhÿ eH$Vmo

x3 – 23x2 + 142x – 120

= x3 – x2 – 22x2 + 22x + 120x – 120

= x2 (x – 1) – 22 (x – 1) + 120 (x – 1) ............(H$m?)

= (x – 1) (x2 – 22x + 120) ∵[(x – 1) gm_B©H$ KoD$Z]

p(x) bm (x – 1) Zo ^mJyZ gwÜXm ho {_iVo.

AmVm x2 – 22x + 120 Mo Ad`drH$aU EH$Va _Yë`m nXmMr \$mo‹S> H$ê$Z$ qH$dm Ad`d à_o` dmnê$Z H$aVm òVo.

_Yë`m nXmMr \$moS> H$ê$Z Amåhm§bm {_iVo :

x2 – 22x + 120 = x2 – 12x – 10x + 120

= x(x – 12) – 10(x – 12)

= (x – 12) (x – 10)

åhUyZ x3 – 23x2 + 142x – 120 = (x – 1) (x – 10) (x – 12)

ñdmÜ`m` 4.4

1. Imbrb H$moUË`m ~hþnXrMm Ad`d (x + 1) {gÜX H$am.

(i) x3 + x2 + x + 1 (ii) x4 + x3 + x2 + x + 1

(iii) x4 + 3x3 + 3x2 + x + 1 (iv) x x x3 2 2 2 2− − +( ) +

2. Ad`d à_o` dmnê$Z Imbrb àËòH$m_Üò g(x) hm p(x) Mm Ad`d Amho. Vo R>adm.

(i) p(x) = 2x3 + x2 – 2x – 1, g(x) = x + 1

(ii) p(x) = x3 + 3x2 + 3x + 1, g(x) = x + 2

(iii) p(x) = x3 – 4x2 + x + 6, g(x) = x – 2

3. Oa p(x) Mm (x – 1) hm Ad`d Agob Va Imbrb àËòH$m_Yrb k Mr qH$_V H$mT>m.

(i) p(x) = x2 + x + k (ii) p x x kx( ) = + +2 22

(iii) p x kx x( ) = − +2 2 1 (iv) p(x) = kx2 – 3x + k

4. Ad`d nmS>m

(i) 12x2 – 7x + 1 (ii) 2x2 + 7x + 3

(iii) 6x2 + 5x – 6 (iv) 3x2 – x – 4

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~hþnXr 79

5. Ad`d nmS>m

(i) x3 – 2x2 – x + 2 (ii) x3 – 3x2 – 9x – 5

(iii) x3 + 13x2 + 32x + 20 (iv) 2y3 + y2 – 2y – 1

4.6 ~¡OrH$ {ZË`g_rH$aUo (Algebraic Identities)

_mJrb dJm©V Vwåhr ~¡OrH$ g_rH$aUmVrb ~¡OrH$ {ZË` g_rH$aUo AmR>dm Or MbnXmÀ`m H$moUË`mhr qH$_Vrbm gË` AgVmV. _mJrb dJm©V Imbrb ~¡OrH$ {ZË`g_rH$aUm§Mm Aä`mg Ho$bm AmhmV.

{ZË`g_rH$aU (I) : (x + y)2 = x2 + 2xy + y2

{ZË`g_rH$aU (II) : (x – y)2 = x2 – 2xy + y2

{ZË`g_rH$aU (III) : x2 – y2 = (x + y) (x – y)

{ZË`g_rH$aU (IV) : (x + a) (x + b) = x2 + (a + b)x + ab

Vwåhr ~¡OrH$ ametMo Ad`drH$aU H$aÊ`mgmR>r darb ~¡OrH$ {ZË`g_rH$aUo dmnabm AmhmV. Vwåhr `m§Mm Cn`moJ g§JUH$m_Ü`o gwÜXm nmhmb.

CXmhaU 16: Imbrb JwUmH$ma `mo½` {ZË`g_rH$aUo dmnê$Z H$am.

(i) (x + 3) (x + 3) (ii) (x – 3) (x + 5)

CH$b : (i) `oWo {ZË`g_rH$aU I dmnam : (x + y)2 = x2 + 2xy + y2

y = 3 hr {H$_V KoD$Z Amåhm§bm {_iVo

(x + 3) (x + 3) = (x + 3)2 = x2 + 2x(3) + (3)2

= x2 + 6x + 9

(ii) darb {ZË`g_rH$aU IV dmnam (x + a) (x + b) = x2 + (a + b)x + ab

Amnë`mbm {_iob.

(x – 3) (x + 5) = x2 + (–3 + 5)x + (–3) (+5)

= x2 + 2x – 15

CXmhaU 17: gai JwUmH$ma Z H$aVm 105 × 106 Mr qH$_V H$mT>m.

CH$b: (105) × (106) = (100 + 5) × (100 + 6)

= (100)2 + (5 + 6)100 + (5 × 6)

= 10000 + 1100 + 30

= 11,130

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80 J{UV

Vwåhr nm{hbm AmhmV ~¡OrH$ ametMm JwUmH$ma H$mT>>Ê`mgmR>r H$mhr da {Xboë`m {ZË`g_rH$aUmMm Cn`moJ Ho$bm. `m {ZË`g_rH$aUmMm Cn`moJ ~¡OrH$ amerMo Ad`drH$aU H$aÊ`mgmR>r gwÜXm H$aVmV ho Vwåhr Imbrb CXmhaUmV nhmb.

CXmhaU 18 : Ad`d nmhm.

(i) 49a2 + 70ab + 25b2 (ii) 254 9

22

x y−

CH$b: (i) `oWo Vwåhr nhm

49a2 = (7a)2

25b2 = (5b)2

70ab = 2(7a) (5b)

x2 + 2xy + y2 ~amo~a {Xboë`m amerMr VwbZm H$ê$.

x = 7a Am{U y = 5b {ZarjUmZo {ZË`g_rH$aU I dmnê$Z Amåhm§bm {_iob.

49a2 + 70ab + 25b2 = (7a + 5b)2

= (7a + 5b) (7a + 5b)

(ii) 254 9

22

x y− =

52 3

2 2

x y

−

AmVm {ZË`g_rH$aU III Mr VwbZm H$ê$Z Amåhm§bm {_iob.

254 9

22

x y− =

52 3

2 2

x y

−

= 52 3

52 3

x y x y+

−

åhUyZ ~hÿnXrMm JwUmH$ma H$aVmZm {ZË`g_rH$aUmMm g_mdoe AgVmo.

AmVm {ZË`g_rH$aU I {ÌnXr (x + y + z) n`ªV dmT>dy.

Amåhr (x + y + z)2 {ZË`g_rH$aU I dmnê$Z H$ê$.

Oa x + y = t _mZy

∴ (x + y + z)2 = (t + z)2

= t2 + 2tz + t2 ({ZË`g_rH$aU I dmnê$Z)

= (x + y)2 + 2(x + y)z + z2 ( t Mt qH$_V KmbyZ)

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~hþnXr 81

( ∴ t = x + y)

= x2 + 2xy + y2 + 2xz+ 2yz + z2 ({ZË`g_rH$aU I dmnê$Z)

= x2 + y2 + z2 + 2xy + 2yz + 2zx (nXm§Mr AXbm~Xb H$ê$Z )

åhUyZ Amåhm§bm Imbrb {ZË`g_rH$aU {_iVo.

{ZË`g_rH$aU (V) : (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx

{Q>n : S>mì`m ~mOyMr amer åhUOoMo COì`m ~mOyMm {dñVma hmo`.

bjmV R>odm (x + y + z)2 `m_Ü`o VrZ dJm©Mr nXo Am{U VrZ JwUmH$ma nXo AmhoV.

CXmhaU 19: (3a + 4b + 5c)2 Mo {dñVmarV ê$n {bhm.

CH$b : (x + y + z)2 ~amo~a amerMr VwbZm H$ê$Z Amåhmbm {_iob x = 3a, y = 4b Am{U z = 5c

{ZË`g_rH$aU V dmnê$Z Amnë`mbm {_iob.

(3a + 4b + 5c)2 = (3a)2 + (4b)2 + (5c)2 + 2(3a)(4b) + 2(4b)(5c) + 2(5c)(3a)

= 9a2 + 16b2 + 25c2 + 24ab + 40bc + 30ca

CXmhaU 20: (4a – 2b – 3c)2 {dñVma H$am.

CH$b: {ZË`g_rH$aU V dmnê$Z Amåhm§bm {_iob.

(4a – 2b – 3c)2 = [4a +(– 2b )+(– 3c) ]2

= (4a)2 + (–2b)2 + (–3c)2 + 2(4a)(–2b) + 2(–2b)(–3c) + 2(–3c)(4a)

= 16a2 + 4b2 + 9c2 – 16ab + 12bc – 24ca

CXmhaU 21: 4x2 + y2 + z2 – 4xy – 2yz + 4zx Mo Ad`d nmhm.

CH$b: 4x2 + y2 + z2 – 4xy – 2yz + 4zx

= (2x)2 + ( – y)2 +( z)2 + 2(2x)(–y) + 2(–y)(z) + 2(z)(2x)

({ZË`g_rH$aU V dmnê$Z) = (2x – y + z)2

= (2x – y + z) (2x – y + z)

Amåhr XmoZ H$moQ>r Agboë`m nXm§~amo~a {ZË`g_rH$aUo H$er dmnaVmV Vo nmhrbo.

AmVm {ZË`g_rH$aU I ho

(x + y)3 qH$_V H$mT>Ê`mgmR>r dmTdy.

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(x + y)3 = (x + y) (x + y)2

= (x + y) (x2 + 2xy + y2)

= x(x2 + 2xy + y2) + y(x2 + 2xy + y2)

= x3 + 2x2y + xy2 + x2y + 2xy2 + y3

(x + y)3 = x3 + 3x2y + 3xy2 + y3

(x + y)3 = x3 + y3 + 3xy(x + y)

åhUyZ AmnUmbm Imbrb {ZË`g_rH$aU {_iVo.

{ZË`g_rH$aU (VI) : (x + y)3 = x3 + y3 + 3xy(x + y)

{ZË`g_rH$aU VI _Ü`o y Mr qH$_V –y KoD$Z AmnUmbm {_iob.

(x – y)3 = x3 – y3 – 3xy(x – y)

{ZË`g_rH$aU (VII) : (x – y)3 = x3 – y3 – 3xy(x – y)

CXmhaU 22: Imbrb KZm§Mo {dñVmarV ê$n {bhm.

(i) (3a + 4b)3 (ii) (5p – 3q)3

CH$b

(i) (x + y)3 Mr {Xboë`m amer~amo~a VwbZm H$ê$Z Amnë`mbm {_iob

x = 3a Am{U y = 4b.

åhUyZ {ZË`g_rH$aU VI dmnê$Z Amnë`mbm {_iob.

(3a + 4b)3 = (3a)3 + (4b)3 + 3(3a) (4b) (3a + 4b)

= 27a3 + 64b3 + 108a2b + 144ab2.

(ii) (x – y)3 Mr {Xboë`m amer~amo~a VwbZm H$ê$Z Amnë`mbm {_iob.

x = 5p, y = 3q

åhUyZ {ZË`g_rH$aU VII dê$Z Amnë`mbm {_iob.

(5p – 3q)3 = (5p)3 – (3q)3 – 3(5p) (3q) (5p – 3q)

= 125p3 – 27q3 – 225p2 q + 135pq2

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CXmhaU 23: `mo½` {ZË`g_rH$aU dmnê$Z Imbrb qH$_Vr H$mT>m

(i) (104)3 (ii) (999)3

CH$b : (i) Amåhmbm {_iob

(104)3 = (100 + 4)3

= (100)3 + (4)3 + 3(100)(4) (100 + 4) ({ZË`g_rH$aU VI dmnê$Z)

= 1000000 + 64 + 124800

= 1124864

(ii) Amåhmbm {_iob 9993 = (1000 – 1)3

= (1000)3 – (1)3 – 3(1000)(1)(1000 – 1) ({ZË`g_rH$aU VII dmnê$Z)

= 1000000000 – 1 – 2997000

= 997002999.

CXmhaU 24: 8x3 + 27y3 + 36x2y + 54xy2 Mo Ad`d H$mT>m.

CH$b : {Xbobr amer Aer {b{hVm `oVo

(2x)3 + (3y)3 + 3(4x2)(3y) + 3(2x)(9y2)

= (2x)3 + (3y)3 + 3(2x)2 (3y) + 3(2x)(3y)2

= (2x + 3y)3 ({ZË`g_rH$aU VI dmnê$Z)

= (2x + 3y) (2x + 3y) (2x + 3y)

AmVm (x + y + z) (x2 + y2 + z2 – xy – yz – zx)

AmnUmbm {dñVma {_iVmo.

x (x2 + y2 + z2 – xy – yz – zx) + y (x2 + y2 + z2 – xy – yz – zx) + z (x2 + y2 + z2 – xy – yz – zx)

= x3 + xy2 + xz2 – x2y – xyz – zx2 + x2y + y3 + yz2 – xy2 – y2z – xyz + x2z + y2z + z3 – xyz – yz2 – xz2

= x3 + y3 + z3 – 3xyz (gai ê$n XoD$Z)

åhUyZ, Amnë`mbm Imbrb {ZË`g_rH$aU {_iVo.

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84 J{UV

{ZË`g_rH$aU VIII : (x3 + y3 + z3 – 3xyz) = (x + y + z) (x2 + y2 + z2 – xy – yz – zx)

CXmhaU 25: 8x3 + y3 + 27z3 – 18xyz Mo Ad`d nmS>m.

CH$b: `oWo AmnUmbm {_iVo.

= (2x)3 + y3 + (3z)3 – 3(2x)(y)(3z)

= (2x + y + 3z)[(2x)2 + y2 + (3z)2 – (2x)(y) – (y)(3z) – (2x)(3z)]

= (2x + y + 3z)(4x2 + y2 + 9z2 – 2xy – 3yz – 6xz)

ñdmÜ`m` 4.5

1. {ZË`g_rH$aU dmnê$Z JwUmH$ma H$am.

(i) (x + 4) (x + 10) (ii) (x + 8) (x – 10)

(iii) (3x + 4) (3x – 5) (iv) y y2 232

32

+

−

(v) (3 – 2x) (3 + 2x)

2. gai JwUmH$ma Z H$aVm {ZË`g_rH$aU dmnê$Z Imbrb qH$_Vr H$mT>m.

(i) 103 × 107 (ii) 95 × 96 (iii) 104 × 96

3. `mo½` {ZË`g_rH$aU dmnê$Z Ad`d nmS>m.

(i) 9x2 + 6xy + y2 (ii) 4y2 – 4y + 1 (iii) x2– y2

100

4. `mo½` {ZË`g_rH$aU dmnê$Z Imbrb àË`oH$mMm {dñVma H$am.

(i) (x + 2y + 4z)2 (ii) (2x – y + z)2 (iii) (–2x + 3y + 2z)2

(iv) (3a – 7b – c)2 (v) (–2x + 5y – 3z)2 (vi) 14

12

12

a b− +

5. Ad`d H$mT>m.

(i) 4x2 + 9y2 + 16z2 + 12xy – 24yz – 16xz

(ii) 2 8 2 2 4 2 82 2 2x y z xy yz xz+ + − + −

6. Imbrb KZm§Mo {dñVmarV ê$n {bhm.

(i) (2x + 1)3 (ii) (2a – 3b)3

(iii) 32

13

x +

(iv) x y−

23

3

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~hþnXr 85

7. `mo½` {ZË`g_rH$aU dmnê$Z Imbrb nXm§À`m qH$_Vr H$mT>m.

(i) (99)3 (ii) (102)3 (iii) (998)3

8. Imbrb ~hþnXtMo Ad`d H$mT>m.

(i) 8a3 + b3 + 12a2b + 6ab2 (ii) 8a3 – b3 – 12a2b + 6ab2

(iii) 27 – 125a3 – 135a + 225a2 (iv) 64a3 – 27b3 – 144a2b + 108ab2

(v) 27 1216

92

14

3 2p p p− − +

9. {gÜX H$am

(i) x3 + y3 = (x + y)(x2 – xy + y2) (ii) x3 – y3 = (x – y)(x2 + xy + y2)

10. Ad`d H$mT>m.

(i) 27y3 + 125z3 (ii) 64m3 – 343n3

({Q>n : àíZ 9 nhm)

11. Ad`d nmS>m : 27x3 + y3 + z3 – 9xyz

12. {gÜX H$am x y z xyz3 3 3 3 12

+ + − = (x + y + z)[(x – y)2 + (y – z)2 + (z – x)2]

13. Oa x + y + z = 0 Va {gÕ H$am x3 + y3 + z3 = 3xyz.

14. gai KZ Z H$aVm Imbrb àË`oH$mMr qH$_V H$mT>m.

(i) (–12)3 + (7)3 + (5)3 (ii) (28)3 + (–15)3 + (–13)3

15. Imbr Am`VmMo joÌ\$i {Xbobo Amho. Ë`mÀ`m eŠ` bm§~r d é§XrMo g_rH$aU {bhm.

joÌ\$i : 27a2 – 35a + 12 joÌ\$i : 35y2+13y – 12(i) (ii)

16. Imbr KZm`VmMo KZ\$i {Xbo Amho. Ë`m§À`m _mnmMo eŠ` g_rH$aU {bhm.

KZ\$i : 3x2–12x KZ\$i : 12ky2 + 8ky – 20k(i) (ii)

4.7 gmam§e (Summary)

`m àH$aUmV Vwåhr Imbrb JmoîQ>tMm Aä`mg Ho$bm AmhmV.

1. x MbnX amerÀ`m ~¡OrH$ amer_Yrb ~hþnXr P(x) Mo ñdê$n -

p(x) = anxn + an –1x

n –1+ ---- +a2x2 + a1x + a0 `oWo a0, a1, a2 ------ an ho pñWanX

Am{U an ≠ 0.

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86 J{UV

AZwH«$_ a0, a1, a2 ------ an ho x0, x1, x2 ----xn Mo ghJwUm§H$ Am{U n bm ~hþnXrMr H$moQ>r

Ago åhUVmV.

anxn, an–1x

n–1,--- a0 (an ≠ 0) bm ~hþnXr p(x) Mr nXo åhUVmV.

2. EH$ nX Agboë`m ~hþnXrbm EH$nXr åhUVmV.

3. XmoZ nXo Agboë`m ~hþnXrbm {ÛnXr åhUVmV.

4. VrZ nXo Agboë`m ~hþnXrbm {ÌnXr åhUVmV.

5. EH$ H$moQ>r Agboë`m ~hþnXrbm aofr` ~hþnXr åhUVmV.

6. XmoZ H$moQ>r Agboë`m ~hþnXrbm dJ© ~hþnXr åhUVmV.

7. VrZ H$moQ>r Agboë`m ~hþnXrbm KZ ~hþnXr åhUVmV.

8. Oa a hr dmñVd g§»`m AgyZ a hm ~hþnXr p(x) Mo eyÝ` Agob Va p(a) = 0 `m àH$mamV, a bm g_rH$aU p(x) Mo dJ©_yi gwÜXm åhUVmV åhUyZ p(x) = 0

9. àËòH$ EH$aofr`h ~hþnXrMo EH$ MbnX eyÝ` AgVo, eyÝòÎma pñWa ~hþnXrMm eyÝ` ZgVmo Am{U àËòH$ dmñVd g§»`m eyÝ` ~hþnXrMm eyÝ` AgVmo. .

10. eof {gÕm§V: Oa p(x) hr H$moUVrhr ~hþnXr AgyZ {VMm H$moQ>r 1 qH$dm 1 nojm _moR>m Agob Am{U p(x) bm aofr` ~hþnXr (x – a) Zo ^mJbo Va ~mH$s p(a) òVo.

11. Ad`d à_o` : Oa p(a) = 0 d (x – a) hm ~hþnXr p(x) Mm Ad`d AgVmo VgoM Oa (x – a) hm p(x) M Ad`d Agob Va p(a) = 0

12. (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx

13. (x + y)3 = x3 + y3 + 3xy (x + y)

14. (x – y)3 = x3 – y3 – 3xy (x – y)

15. x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2 – xy – yz – zx)

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{ÌH$moU 87

àH$aU - 5

{ÌH$moU (Triangles)

5.1 àñVmdZm

_mJrb B`ÎmoV Vwåhr {ÌH$moU Am{U Ë`mÀ`m JwUY_m© {df`r Aä`mg Ho$bm AmhmV. Vwåhmbm _mhrV Amho H$s EH$m àVb-mVrb VrZ aofm nañnamg A§Ë`q~XÿV gm§YyZ ~§{YñV hmoUmè`m. g_Vb AmH¥$Vrbm {ÌH$moU åhUVmV. ({Ì åhUZoM VrZ) qH$dm (Tri- VrZ). {ÌH$moUmbm VrZ H$moZ, VrZ ~mOy Am{U VrZ {eamoq~Xÿ AgVmV. CXmhaUmV {ÌH$moU ABC hm ∆ABC Agm {b{hVmV (AmH¥$Vr 5.1 nhm) AB, BC, CA `m VrZ ~mOy, ∠A, ∠B, ∠C ho VrZ H$moZ Am{U A, B, C. ho VrZ {eamoq~Xÿ AmhoV. AmH¥$Vr 5.1

àH$aU 6 _Ü`o, {ÌH$moUmÀ`m H$mhr JwUY_mªMm Aä`mg Vwåhr Ho$bm AmhmV. `m àH$maUmV Vwåhr {ÌH$moUm§Mr EH$ê$nVm, EH$ê$nVoMo {Z`_, H$mhr {ÌH$moUm§Mo JwUY_© Am{U {ÌH$moUm§Mr Ag_mZVm `m§Mm {dñVmarV Aä`mg H$amb.

5.2 {ÌH$moUm§Mr EH$ê$nVm (Congruence of Triangles)

Vwåhr nhm H$s, EH$mM AmH$ma AgUmar Vw_Mr XmoZ N>m`m§{MÌo gmaIrM (g_mZ) AgVmV. Ë`mMà_mUo XmoZ g_mZ AmH$mamÀ`m ~m§JS>`m, EH$mM ~±H$_YyZ KoVbobr XmoZ ATM H$mS>} AmH$mamUo gmaIrM AgVmV. EH$mM dfm©_Ü`o V`ma Ho$bobr EH$ ê$n`mMr XmoZ qH$dm A{YH$ ZmUr EH$_oH$mda R>odë`mg Vr nyU©nUo EH$_oH$m§Zm PmHy$Z KoVmV.

Vwåhr AmR>dy eH$Vm H$m, Aem AmH¥$Vr§Zm H$m` åhUVmV? Va Ë`m§Zm EH$ê$n AmH¥$Ë`m åhUVmV. (EH$ê$nVm åhUOoM Ë`m AmH¥$VrMm AmH$ma Am{U _mn XmoÝhr g_mZ AgVo).

g_mZ {ÌÁ`m AgUmar XmoZ dVw©io EH$_oH$m§da R>odm, H$m` {ZarjU H$ê$ eH$mb? Vr EH$_oH$mer OwiVmV åhUyZ Amåhr Ë`m§Zm EH$ê$n dVw©io åhUy eH$Vmo.

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88 J{UV

nwÝhm EH$ H¥$Vr H$ê$. g_mZ _mnmMo XmoZ Mm¡ag ~mOy ~mOybm R>odë`mg (AmH¥$Vr 5.2 nhm) qH$dm Xþgè`m Mm¡agmdaÀ`m OmoJoda R>ody qH$dm XmoZ g_mZ AmH$mamMo XmoZ {ÌH$moU (Equilateral) EH$_oH$mda R>odë`mg Vo AmnUmg EH$ ê$n AmhoV Aer Om{Ud hmoVo. AmH¥$Vr 5.2

Amåhr EH$ê$nVoMm Aä`mg H$m H$é`m? {eVnoQ>r_Yë`m (Refrigerator) ~\$m©À`m gmMmMo (ice tray) {ZarjU H$am. åhUOoM gmMm_Yrb Agbobo gd© ~\©$ EH$ê$nVoMo bjU Xe©{dVmV. gmMm_Yrb Agboë`m ~\$m©Mr OmJm gwÜXm EH$ê$nVm Xe©{dVo. (ho gd© Am`VmH¥$Vr qH$dm {ÌH$moUmH¥$Vr qH$dm dVw©imH¥$Vr Agy eH$VmV). åhUyZ Ooìhm gma»`mM dñVy ~Zdm`À`m AgVmV Voìhm Ago gmMo ~Z{dÊ`mgmR>r EH$ê$nVoÀ`m g§H$ënZoMm dmna H$aVmV.

H$mhr doiobm Vwåhmbm Vw_À`m noZMr [a\$sb (Refill) ~XbÊ`mg gwÜXm AS>MU òD$ eH$Vo. Oa ZdrZ [a\$sb Vw_À`m noZÀ`m AmH$mamMr Zgob Va Vr noZ_Üò ì`dpñVV ~gV Zmhr. Ooìhm OwZr [a\$sb Am{U ZdrZ [a\$sb EH$ê$n AgVo VoìhmM Vr noZ_Üò ~gVo. AemàH$mao Amnë`m X¡Z§{XZ OrdZm_Üò Aer AZoH$ CXmhaUo AmhoV H$s Á`mV EH$ê$nVoMo JwUY_© AmT>iVmV.

Vwåhr EH$ê$nVoMr Aer AmUIrZ CXmhaUo gm§Jy eH$mb H$m?

AmH¥$Vr 5.3 _Üò, H$moUVo Mm¡ag EH$_oH$m§er EH$ê$n ZmhrV?

AmH¥$Vr 5.3

AmH¥$Vr 5.3 (ii) Am{U (iii) ho AmH¥$Vr 5.3 (i) er EH$ê$n ZmhrV. nU AmH¥$Vr 5.3 (iv) _Yrb Mm¡ag AmH¥$Vr 5.3 (i) _Yrb Mm¡agmer AmH¥$Vr EH$ê$n Amho.

AmVm XmoZ {ÌH$moUmÀ`m EH$ê$nVo~Ôb MMm© H$ê$.

Vwåhmbm _mhrV Amho H$s, Oa EH$m {ÌH$moUmÀ`m VrZ ~mOy Am{U VrZ H$moZ ho Xþgè`m {ÌH$moUmÀ`m g§JV ~mOy Am{U g§JV H$moZmer g_mZ AgVrb Va Vo XmoZ {ÌH$moU EH$ê$n AgVmV.

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{ÌH$moU 89

AmVm AmH¥$Vr 5.4 (i) _Üò {Xboë`m ∆ABC er H$moUVm {ÌH$moU EH$ê$n Amho?

AmH¥$Vr 5.4

AmH¥$Vr 5.4 (ii) Vo (v) _YyZ àËòH$ {ÌH$moU ~mOybm H$am Am{U {\$adm Am{U ∆ABC bm g_mZ hmoVrb Ago à`ËZ H$am. AmH¥$Vr 5.4 (ii), (iii) Am{U (iv) _Yrb {ÌH$moU ho ∆ABC er EH$ê$n AmhoV VgoM AmH¥$Vr 5.4 (v) _Yrb ∆ TSU, ho ∆ ABC EH$ê$n ZmhrV.

Oa ∆PQR hm ∆ABC er EH$ê$n Amho Va ∆PQR ≅ ∆ABC Ago {bhÿ eH$Vmo.

Ooìhm ∆PQR ≅ ∆ABC AmhoV Voìhm ∆ABC Mo g§JV ~mOy ∆PQR À`m g§JV ~mOyer g_mZ Amho Am{U H$moZmÀ`m g§X^m©V AgoM hmoVo.

åhUyZM PQ bm AB, QR bm BC, RP bm CA OwbVo, ∠P bm ∠A, ∠Q bm ∠B Am{U ∠R bm ∠C OwiVo. VgoM Ë`m§Mo AZwH«$_o {eamoq~Xÿ gwÜXm OwiVmV. åhUyZ P Mm g§JV {eamoq~Xÿ A, Q Mm g§JV {eamoq~Xÿ B, R Mm g§JV {eamoq~Xÿ C . ho AmnU Imbrb à_mUo {bhÿ eH$Vmo.

P ↔ A, Q ↔ B, R ↔ C

`m g§JV KQ>H$mdê$Z ∆PQR ≅ ∆ABC hmoVmV. nU ∆QRP ≅ ∆ABC Ago {b{hVm òV Zmhr.

Ë`mMà_mUo 5.4 (iii) dê$Z

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90 J{UV

FD ↔ AB, DE ↔ BC Am{U EF ↔ CA

Am{U F ↔ A, D ↔ B, Am{U E ↔ C

\ ∆FDE ≅ ∆ABC. nU ∆DEF ≅ ∆ABC ho ~amo~a Zmhr.

AmH¥$Vr 5.4 (iv) Am{U ∆ABC ho EH$ê$n {ÌH$moU hmoVmV.

g§Ho$VmV {ÌH$moUmMr EH$ê$nVm {b{hVmZm Ë`m§Mo g§JV {eamoq~Xÿ g_mZ Agbo nm{hOoV.

bjmV R>odm H$s, EH$ê$n {ÌH$moUm§Mo g§JV KQ>H$ g_mZ AgVmV Am{U ho Ago {b{hVmV E.{Ì. g§.K (EH$ê$n {ÌH$moUmMo g§JV KQ>H$) qH$dm CPCT (corresponding parts of congruent triangles)..

5.3 EH$ê$n {ÌH$moUmÀ`m AQ>r

_mJrb B`ÎmoV EH$ê$n {ÌH$moUmÀ`m EHy$U 4 AQ>r {eH$bm AmhmV. Ë`m AmVm AmR>dy`m.

EH$ ~mOy 3cm AgUmao H$moUVohr XmoZ {ÌH$moU aMm. Vo EH$ê$n {ÌH$moU AmhoV H$m? {ZarjU H$am H$s, Vo EH$ê$n ZmhrV. (AmH¥$Vr 5.5 nhm).

AmH¥$Vr 5.5

AmVm EH$ ~mOy 4cm Am{U EH$ H$moZ 50° AgUmao XmoZ {ÌH$moU aMm (AmH¥$Vr 5.6 nhm) Vo EH$ê$n hmoVmV H$m?

AmH¥$Vr 5.6

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{ÌH$moU 91

nhm H$s, ho XmoZ {ÌH$moU EH$nê$n ZmhrV.

{ÌH$moUmÀ`m OmoS>`mMr hr H¥$Vr nwÝhm H$ê$.

åhUOo, {ÌH$moUmÀ`m EH$ê$nVogmR>r {Xbobo g_mZVm AgUmao EH$ ~mOy§Mr OmoS>r qH$dm EH$ H$moZm§Mr Omo‹S>r BVHo$M. `mo½` R>aV Zmhr.

Oa Xþgar ~mOy§Mr OmoS>r hr H$moZmer g_mZ Am{U g_mZM AgobVa ?

AmH¥$Vr 5.7 _Üò BC = QR, ∠B =∠Q Am{U VgoM AB = PQ AmVm, ∆ABC Am{U ∆PQR `m EH$ê$nVobm Vwåhr H$m` åhUy eH$mb?

AmH¥$Vr 5.7 _Üò ∆ABC d ∆PQR nS>Vmim H$s, {Xbobo XmoZ {ÌH$moU EH$ê$n AmhoV Ago `m§d g§~§YrV _mJrb B`ÎmoV {eH$bm AmhmV Vo AmR>dm.

Xþgè`m {ÌH$moUmÀ`m OmoS>çm KoD$Z hr H¥$Vr naV H$am. EH$ê$n {ÌH$moUmgmR>r XmoZ ~mOy Am{U Ë`mg§~§YrV H$moZ g_mZVm Amho Ago {ZarjU H$ê$ eH$mb H$m?

AmH¥$Vr 5.7

{ÌH$moUmÀ`m EH$ê$nVogmR>r hr n{hbr AQ> Amho.

_wbVËd 5.1 : EH$ê$nVoMo ~m. H$mo. ~m. J¥{hVH$ :

Oa EH$m {ÌH$moUmÀ`m XmoZ ~mOy Am{U g_m{dîQ> H$moZ Xþgè`m {ÌH$moUmÀ`m XmoZ g§JV ~mOy Am{U g_m{dîQ> H$moZmer g_mZ Agob Va Vo {ÌH$moU EH$ê$n AgVmV.

`m J¥hrVH$mbm `m nydu _mhrV Agboë`m J¥hrVH$mÀ`m ghmæ`mZo {gÜX H$aVm òV Zmhr Am{U åhUyZ `mbm EH$ J¥hrVÀ`m ê$nmZo gË` _mZbo Jobo Amho.

AmVm `mgmR>r EH$ CXmhaU KoD$.

CXmhaU 1 : AmH¥${V 5.8 _Üò OA = OB Am{U OD = OC. Va {gÜX H$am H$s,

(i) ∆AOD ≅ ∆BOC Am{U (ii) AD || BC .

CH$b : (i) ∆AOD Am{U ∆BOC Mo {ZarjU H$am.

OA OBOD OC

= = ({Xbobo KQ>H$) AmH¥$Vr 5.8

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92 J{UV

VgoM Oa ∠AOD Am{U ∠BOC `m {eamo{déÜX H$moZm§À`m Omo‹S>`m AmhoV, Va

∠AOD = ∠BOC

åhUyZ ∆AOD ≅ ∆BOC (¨~m.H$mo.~m. J¥hrVH$)

(ii) ∆AOD d ∆BOC `m EH$ê$n {ÌH$moUmV BVa g§JV KQ>H$ gwÜXm g_mZ AmhoV.

åhUyZ ∠OAD = ∠OBC. Am{U AD d BC aofoer g§~§YrV AgUmao ho {déÜX H$moZ AmhoV.

\AD || BC

CXmhaU 2 : AB hr aofm AgyZ `m aofoda b§~ AgUmar l hr N>oXrH$m Amho. Oa l `m aofoda P q~Xÿ Agob Va {gÜX H$am H$s. A d B bm P hm g_mZ A§Vamda Amho.

CH$b: aofm l ⊥ AB Am{U AB _YyZ N>oXÿZ OmUmam C hm _Ü`q~Xÿ Amho (AmH¥$Vr 5.9 nhm). PA = PB Ago AmnU {gÜX H$ê$`m.

∆ PCA d ∆ PCB _Ü`o AmH¥$Vr 5.9

AC = BC (C hm AB `m _Ü`q~Xÿ)

∠PCA = ∠PCB = 90o ({Xbobo)

PC = PC (gm_mB©H$)

åhUyZ, ∆PCA ≅ ∆PCB (¨~m.H$mo.~m J¥hrVH$)

Am{U PA = PB, ho EH$ê$n {ÌH$moUmÀ`m g§JV ~mOy AmhoV.

AmVm, {ÌH$moUmÀ`m g§JV ~mOyer Z AgUmam H$moZ Am{U 4cm d 5cm ~mOy Am{U H$moUVmhr EH$ H$moZ 50° AgUmaoè`m XmoZ {ÌH$moUmMr aMZm H$am (AmH¥$Vr 5.10 nhm).

ho XmoZ {ÌH$moU EH$ê$n AmhoV H$m?

AmH¥$Vr 5.10

ho XmoZ EH$ê$n {ÌH$moU ZmhrV Ago _mZy.

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{ÌH$moU 93

{ÌH$moUmÀ`m H$mhr Omo‹S>`m KoD$Z hr H¥$Vr nwÝhm H$ê$. ho {ÌH$moU EH$ê$n AmhoV Ago AmnU {ZarjU Ho$bo Amho. `mgmR>r g_mZ ~mOy§Mr Omo‹S>r AgUmao g§~§YrV g_mZ H$moZ _hËdmMo AgVmV.

åhUyZ, ~m.H$mo.~m (SAS) ho EH$ê$nVoMo J¥hrVH$ ({Z`_) Amho nU d H$mo.~m.~m (ASS) qH$dm ~m.~m. H$mo (SSA) ho hmoV Zmhr.

Z§Va, XmoZ {ÌH$moUmMr Aer aMZm H$am H$s, Ë`m XmoZ {ÌH$moUmMo g§JV H$moZ 60º Am{U 45º AgyZ `m H$moZm g§~§YrV 4cm Mr g§JV ~mOy hmoB©b. (AmH¥$Vr 5.11 nhm).

AmH¥$Vr 5.11

ho XmoZ {ÌH$moU {d^ŠV H$ê$Z Vo EH$_oH$mda R>odm `mgmdê$Z H$m` {ZarjU H$amb? EH$ {ÌH$moU Xþgè`m {ÌH$moZmer nyU©nUo PmHy$Z OmB©b. {ÌH$moUmÀ`m H$mhr OmoS>`m KoD$Z hr H¥$Vr nwÝhm H$ê$. `mdê$Z AmnU {ZarjU H$amb H$s, {ÌH$moUmÀ`m EH$ê$nVogmR>r XmoZ g§JV H$moZmer g_m{dîQ> AgUmar gJV EH$ ~mOy nwaoer Amho.

EH$ê$nVoÀ`m `m {ZH$mbmbm H$moZ-~mOy-H$moZ J¥hrVH åhUVmV Am{U ho H$mo.~m.H$mo (ASA) `m ñdê$nmV {b{hVmV. ho J¥hrVH$ _mJrb B`ÎmoV nS>Vmibm Amho AmVm ho J¥hrVH$ {gÜX H$ê$.

~m.H$mo.~m ho EH$ê$nVoMo J¥hrVH$ dmnê$Z AmnU hm {ZîH$f© {gÜX H$ê$ Am{U `mbm à_o` Ago åhUy..

à_o` 5.1: (H$mo.~m.H$mo J¥{hVH$)

Oa EH$m {ÌH$moUmMo XmoZ H$moZ Am{U EH$ ~mOy Xþgè`m {ÌH$moUmÀ`m g§JV XmoZ H$moZmer Am{U EH$ g§JV ~mOyer g_mZ Agob Va Vo {ÌH$moU EH$ê$n AgVmV.

CH$b: ∆ ABC d ∆ DEF ho XmoZ {ÌH$moU {Xbobo AmhoV.

∠ B =∠ E, ∠C =∠ F

Am{U BC = EF

AmnU {gÜX H$ê$ H$s, ∆ ABC ≅ ∆ DEF .

ho XmoZ EH$ê$n {ÌH$moU {gÜX H$aÊ`mgmR>r Imbrb VrZ nm`è`m {Xë`m AmhoV.

n`mar (i) AmVm AB = DE (AmH¥$Vr 5.12 nhm)

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94 J{UV

Vwåhr H$m` {ZarjU Ho$bmV? Vwåhr {ZarjU Ho$bm H$s.

AB = DE (_mZy)

∠ B = ∠ E ({Xbobo)

BC = EF ({Xbobo)

åhUyZ ∆ ABC ≅ ∆ DEF (~m.H$mo.~m. J¥hrVH$)

AmH¥$Vr 5.12

nmàr (ii): Oa AB > DE Agob Va, AB `m ~mOyda P q~Xÿ KoD$Z

PB = DE hmoVo. AmVm ∆ PBC d ∆ DEF _Üò ( AmH¥$Vr 5.13 nhm)

AmH¥$Vr 5.13

{ZarjU H$am ∆ PBC d ∆ DEF _Üò

PB = DE (aMZodê$Z)

∠ B = ∠ E ({Xbobo)

BC = EF ({Xbobo)

`mdê$Z, ∆ PBC ≅ ∆ DEF (H$mo.~m.H$mo. J¥hrVH$)

Oa g§JV KQ>H$ g_mZ AgVrb Va {ÌH$moU EH$ê$n AgVmV.

åhUyZ, ∠ PCB = ∠ DEF

nU ∠ ACB = ∠ DEF {Xbobo Amho.

\ ∠ ACB = ∠ DEF

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{ÌH$moU 95

ho eŠ` Amho H$m?

Oa A hm P er g§JV Agob Va {gÜX hmoVo.

qH$dm BA = ED

∆ ABC ≅ ∆ DEF...... (~m.H$mo.~m.J¥hrVH$)

nmàr (iii): Oa AB < DE Voìhm DE ~mOyda M hm q~Xÿ KoVbm AgVm nmàr (ii) À`m {gÜXVoZwgma ME = AB hmoVo. Va Amåhr AB = DE KoD$ eH$Vmo Am{U åhUyZ ∆ABC ≅ ∆DEF hmoVo.

g_Om Oa, XmoZ {ÌH$moUm_Üò XmoZ H$moZm§À`m OmoS>çm Am{U g§JV ~mOy g_mZ AgVmV nU `m ~mOy Ë`m g§JV H$moZm§À`m OmoS>rer A§VJ©V (g§~§YrV) ZgVmV ho XmoZ {ÌH$moU AmVm gwÜXm EH$ê$n AgVrb H$m?

Vwåhr {ZarjU H$am H$s {ÌH$moU EH$ê$n AmhoV. `mMo H$maU gm§Jy eH$mb H$m?

Vwåhmbm _{gV Amho H$s. {ÌH$moUmÀ`m {VÝhr H$moZm§Mr ~oarO 180o AgVo. åhUyZ {ÌH$moUmÀ`m XmoZ H$moZm§À`m OmoS>`m g_mZ AgVrb Va {Vgè`m H$moZm§Mr OmoS>r gwÜXm g_mZ AgVo. ({ÌH$moUmÀ`m VrZ H$moZm§Mr ~oarO 180o)

åhUyZ, XmoZ {ÌH$moUm_Üò EH$m {ÌH$moUmMo Oa XmoZ H$moZ d EH$ ~mOy ho Xþgè`m {ÌH$moUmMo XmoZ g§JV H$moZ d EH$ g§JV ~mOy `m§À`mer g_mZ AgVrb Va Vo XmoZ {ÌH$moU EH$ê$n AgVmV. åhUyZ `mbm H$mo.H$mo.~m.(AAS) J¥hrVH$ åhUVmV.

AmVm, Imbrb H¥$Vr H$ê$.

40o, 50o Am{U 90o H$moZ AgUmao {ÌH$moU aMm. Ago Vwåhr {H$Vr {ÌH$moUm§Mr aMZm H$ê$ eH$mb?

~mOy§Mr bm§~r H$_r OmñV KoD$Z AmnU eŠ` hmoB©b {VVHo$ {ÌH$moU H$mTy>. (AmH¥$Vr 5.14 nhm)

AmH¥$Vr 5.14

{ZarjU H$am H$s, {ÌH$moU nañnamer EH$ê$n AmhoV qH$dm ZmhrV.

åhUyZ, {ÌH$moUmMr g_mZVm {ÌH$moUmÀ`m EH$ê$nVogmR>r nyaoer Zmhr. åhUyZ, {ÌH$moUmÀ`m EH$ê$nVogmR>r VrZ KQ>H$mn¡H$s EH$ ~mOy Agbr nm{hOo.

AmVm `mgmR>r AmUIr H$mhr CXmhaUo KoD$.

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96 J{UV

CXmhaU 3 : aofmI§S> AB ho aofmI§S> CD bm g_m§Va AgyZ. O hm AB Mm _Ü`q~Xÿ Amho. (AmH¥$Vr 5.15 nhm) Va {gÜX H$am.

(i) ∆ AOB ≅ ∆ DOC (ii) O hm gwÜXm BC Mm _Ü`q~Xÿ Amho.

CH$b:

(i) ∆ AOB d ∆ DOC _Üò ∠ ABO = ∠ DCO

(ì wËH«$_ H$moZ AB || CD d BC hr N>o{XH$m)

∠ AOB = ∠ DOC ({eamo{déÜX H$moZ)

OA = OD ({Xbobo)

\ ∆ AOB ≅ ∆ DOC (H$mo.~m.H$mo)

(ii) OB = OC

åhUyZ 'O' hm BC Mm _Ü`q~Xÿ Amho.

ñdmÜ`m` 5.1

(1) Mm¡H$moZ ACBD _Üò AC = AD Am{U ∠A M AB hm Xþ^mOH$ (AmH¥$Vr 5.16 nhm) {gÜX H$am ∆ABC ≅ ∆ABD. BC d BD `m~Ôb H$m` gm§Jy eH$mb?

AmH¥$Vr 5.16

(2) ABCD `m Mm¡H$moZm_Üò AD = BC d ∠DBA=∠CBA Agob Va {gÜX H$am.

(1) ∆ ABD ≅ ∆ BAC

(2) BD = AC

(3) ∠ABD = ∠BAC

AmH¥$Vr 5.17

AmH¥$Vr 5.15

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{ÌH$moU 97

(3) AB hr aofm g_mZ AgUmè`m BC d AD bm b§~ Amho (AmH¥$Vr 5.18 nhm) Va {gÜX H$am. AB

hr CD bm Xþ^mOH$ Amho.

AmH¥$Vr 5.18

(4) p d q `m g_m§Va AgUmè`m aofm l d m g_m§Va AgUmè`m aofobm N>oXVmV ( AmH¥$Vr 5.19 nhm) Va {gÜX H$am ∆ABC ≅ ∆CDA.

AmH¥$Vr 5.19

(5) ∠A bm l hm Xþ^mOH$ AgyZ. B hm l da H$moUVmhr EH$ q~Xÿ Amho. ∠A À`m ~mOyda AgUmè`m B {~Xÿda BP Am{U BQ b§~ Q>mH$bobm Amho. (AmH¥$Vr 5.20 nhm).

{gÜX H$am

(i) ∆ APB ≅ ∆ AQB

(ii) BP = BQ qH$dm ∠A H$moZmÀ`m ~mOynmgyZ B hm g_mZ A§Vamda Amho.

AmH¥$Vr 5.20

(6) AmH¥$Vr 5.21 _Ü`o AC = AE, AB = AD Am{U ‹∠BAD = ∠EAC Va {gÜX H$am BC = DE.

AmH¥$Vr 5.21

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98 J{UV

(7) AB hm aofmI§S> AgyZ P Ë`mdarb _Ü`q~Xÿ Amho. AB À`m EH$mM ~mOyg AgUmao D d E ho q~Xÿ AgyZ ∠BAD = ∠ABE d ∠EPA = ∠DPB Amho. (AmH¥$Vr 5.22 nhm) Va {gÜX H$am.

(i) ∆DAP ≅ ∆EBP

(ii) AD = BE AmH¥$Vr 5.22

(8) H$mQ>H$moZ {ÌH$moU ABC _Ü`o. C hm H$mQ>H$moZ Amho. AB `m H$Um©da M hm _Ü`q~Xÿ Amho. C ho M n`ªV gm§Ybo Am{U D n`ªV Ago dmT>{dbo H$s DM=CM hmoB©b VgoM D d B gm§Kbo (AmH¥$Vr 5.23 nhm) Va {gÜX H$am H$s.

AmH¥$Vr 5.23

(i) ∆AMC ≅ ∆BMD

(ii) ∠DBC hm H$mQ>H$moZ Amho.

(iii) ∆DBC ≅ ∆ACB

(iv) CM = 1 2 AB

5.4 {ÌH$moUmMo H$mhr JwUY_©

_mJrb ^mJmV AmnU {ÌH$moUmÀ`m EH$ê$nVoÀ`m XmoZ J¥hrVH$m§Mm Aä`mg Ho$bm AmhmV. XmoZ ~mOy g_mZ AgUmè`m {ÌH$moUmÀ`m JwUY_m©Mm Aä`mg H$aÊ`mgmR>r `m {ZîH$fm©Mm dmna H$ê$.

Imbrb {Xbobr H¥$Vr H$am.

3.5cm g_mZ AgUmè`m XmoZ ~mOy Am{U 5cm {Vgar ~mOy AgUmar {ÌH$moUmMr aMZm H$am (AmH¥$Vr 5.24 nhm). Aer aMZm _mJrb B`ÎmoV Ho$br AmhmV.

Aem {ÌH$moUmbm H$m` åhUVmV ho Vwåhmbm AmR>dVo H$m?

AmH¥$Vr 5.24

Oa EH$m {ÌH$moUmÀ`m XmoZ ~mOy g_mZ AgVri Va Ë`m {ÌH$moUmbm g_{Û^yO {ÌH$moU åhUVmV. åhUyZ AmH¥$Vr 5.24 _Ü`o ∆ABC hm g_{X^yO {ÌH$moU AB = AC Amho.

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{ÌH$moU 99

AmVm ∠B d ∠C _moOm, H$m` {ZarjU H$amb?

{^ÝZ ~mOy§Mm Cn`moJ H$ê$Z g_{Û^yO {ÌH$moUmMr hr H¥$Vr nwZ… H$am.

AmnU nmhVm H$s, Aem àËòH$ {ÌH$moUmV g_mZ ~mOy g_moarb H$moZ g_mZ AgVmV.

hm EH$ g_{Û^yO {ÌH$moUmgmR>r _hËdmMm JwUY_© gË` R>aVmo. `m JwUY_m©bm Imbrb nÜXVrZo {gÜX H$ê$.

à_o` 5.2 : EH$m g_{Û^yO {ÌH$moUm_Üò g_mZ ~mOyg_oarb H$moZ g_mZ AgVmV.

{gÜXVm : ∆ABC `m g_{Û^yO {ÌH$moUm_Üò AB = AC

Amho Va AmnUmbm

∠B = ∠C {gÜX H$amd`mMo Amho.

AmVm, ∠A Mm H$moZXþ^mOH$ H$mTy>Z D hm BC bm {_iVmo (AmH¥$Vr 5.25 nhm).

AmH¥$Vr 5.25

∆BAD d ∆CAD _Üò

AB = AC ({Xbobo)

∠BAD = ∠CAD (aMZm)

AD= AD (gm_mB©H$)

åhUyZ ∆BAD ≅ ∆CAD (¨~m.H$mo.~m)

åhUyZ ∠ABD = ∠ACD (EH$ê$n {ÌH$moUmMo g§JV H$moZ)

\ ∠B = ∠C

`mMm ì`Ë`mg gwÜXm gË` hmoVmo H$m? åhUOoM Oa EH$mÚm {ÌH$moUmMo XmoZ H$moZ g_mZ AgVrb Va Ë`m§À`m gå_wI (g_moarb) ~mOy g_mZ AgVmV. Ago R>ady eH$Vmo H$m?

Oa EImYm (g_oarb) ~mOy g_mZ AgVmV. Ago {h R>ady eH$Vmo H$m?

Imbrb {Xbobr H¥$Vr H$am.

∆ ABC Mr Aer aMZm H$am H$s, BC hr H$moUVrhr bm§~r AgyZ ∠B = ∠C = 50o Amho. ∠A Mm Xþ^mOH$ D hm BC bm {_iVmo (AmH¥$Vr 5.26 nhm).

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100 J{UV

{ÌH$moUmÀ`m AmH$mamMm EH$ nona H$mnm (Vmo ∆ ABC, Amho) Am{U {eamoq~Xÿ C òB©b Agm {ÌH$moU ABC, AD dê$Z Xþ_S>m.

AC Am{U AB `m ~mOy~Ôb AmnbU H$m` gm§Jy eH$mbm?

{ZarjU H$am H$s, AC hm AB bm nyU©nUo PmHy$Z OmB©b. AmH¥$Vr 5.26

åhUyZ AC = ABAZoH$ H$mhr {ÌH$moU KoD$Z hr H¥$Vr nwZ… H$am.

àËòH$ doiog AmnUmg {XgyZ òB©b H$s. g_mZ H$moZmg_moarb ~mOy g_mZ AgVmV. ho Imbrb à_mUo {gÜX H$ê$ eH$Vmo.

à_o` 5.3: EH$m {ÌH$moUmÀ`m g_mZ H$moZmg_moarb ~mOy g_mZ AgVmV. hm à_o` 5.2 Mm Ë`Ë`mg Amho.

hm à_o` Vwåhr H$mo.~m.H$mo (ASA) `m EH$ê$nVoÀ`m J¥hrVH$mdê$Z {gÜX H$ê$ eH$Vm.

`m {ZîH$fm©gmR>r Imbrb CXmhaUo gmoS>dy ?

CXmhaU 4: ∆ABC _Üò BC da b§~ AgUmam AD hm ∠A Mm H$moZXþ^mOH$ Amho. (AmH¥$Vr 5.27 nhm) Va {gÜX H$am AB = AC Am{U ∆ABC hm g_{Û^yO Amho.

CH$b : ∆ ABD d ∆ ACD _Üò

∠BAD = ∠CAD ({Xbobo)

AD = AD (gm_mB©H$)

∠ADB = ∠ADC = 90o ({Xbobo)

∆ABD ≅ ∆ACD (H$mo.~m.H$mo)

AmH¥$Vr 5.27

åhUyZ AB = AC (EH$ê$n ∆ Mo g§JV KQ>H$.)

qH$dm ∆ABC hm g_{Û^yO {ÌH$moU Amho.

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{ÌH$moU 101

CXmhaU 5: ∆ABC Mo AB d AC g_mZ ~mOy AgyZ AZwH«$_o E d F _Ü`q~Xÿ Amho (AmH¥$Vr 5.28 nhm). Va {gÜX H$am BF = CE.

CH$b : ∆ ABF d ∆ ACE _Üò

AB = AC ({Xbobo)

∠A = ∠A (gm_mB©H$ H$moZ)

AF = AE (g_mZ ~mOy§Mr {Zå_nQ>)

∆ABF ≅ ∆ACE (¨~m.H$mo.~m.)

åhUyZ BF = CE (EH$ê$n ∆ Mo g§JV KQ>H$)

AmH¥$Vr 5.28

CXmhaU 6 : g_{Û^yO ∆ABC _Üò AB = AC Amho. BC darb D

d E q~Xÿ AgyZ BE = CD (AmH¥$Vr 5.29 nhm) Va {gÜX H$am AD = AE.

CH$b : ∆ ABD d ∆ ACE _Üò

AB = AC ({Xbobo) ... (1)

∠B = ∠C (g_mZ ~mOyg_moarb H$moZ) ... (2) AmH¥$Vr 5.29 VgoM BE = CD

\ BE – DE = CD – DE

BD = CE ... (3)åhUyZ, ∆ABD ≅ ∆ACE

((1), (2) d (3) d ~m.H$mo.~m {Z`_mZwgma) AD = AE (EH$ê$n ∆ Mo g§JV KQ>H$.)

ñdmÜ`m` 5.2

(1) g_{Û^yO ∆ABC _Üò AB = AC Amho. ∠B Am{U ∠C Mo H$moZmXþ^mOH$ nañnam§Zm O q~XÿV N>oXVmV. A d O

gm§Ym. Va {gÜX H$am

AmH¥$Vr 5.30

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102 J{UV

(i) OB = OC (ii) ∠A Mm AO hm Xþ^mOH$ Amho.

(2) ∆ABC _Üò AD ⊥ BC (AmH¥$Vr 5.30 nhm) Va {gÜX H$am ∆ABC hm g_{Û^yO {ÌH$moU AgyZ AB = AC Amho.

AmH¥$Vr 5.31

(3) g_{Û^yO ∆ABC _Üò, CF ⊥ AB d BE ⊥ AC Amho (AmH¥$Vr 5.31) Va {gÜX H$am CF = BE.

AmH¥$Vr 5.32

(4) ∆ABC _Üò AB d AC da H$mT>bobm {eamob§~ AZwH«$_o CF d BE g_mZ Amho. (AmH¥$Vr 5.32 nhm) Va {gÜX H$am.

(i) ∆ABE ≅ ∆ACF

(ii) AB = AC åhUOoM ∆ABC hm g_{Û^yO Amho.

AmH¥$Vr 5.33

(5) BC `m EH$mM nm`mda Cô AgUmao ABC d DBC ho XmoZ g_{Û^yO {ÌH$moU AmhoV (AmH¥$Vr 5.33 nhm) Va {gÜX H$am ∠ABD =∠ACD

AmH¥$Vr 5.34

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{ÌH$moU 103

(6) ∆ ABC `m g_{Û^yO {ÌH$moUm_Ü`o AB = AC Amho. BA ~mOy D n`ªV Aer dmT>{dbr H$s, AD = AB hmoB©b (AmH¥$Vr 5.34 nhm) {gÜX H$am ∠BCD hm H$mQ>H$moZ Amho.

(7) ABC hm H$mQ>H$moZ {ÌH$moU AgyZ ∠A = 90o Am{U AB = AC. Agob Va ∠B d ∠C H$mT>m.

(8) {gÜX H$am H$s, EH$m g_^yO {ÌH$moUmMm àË`oH$ H$moZ 60o AgVmo.

5.5 {ÌH$moUmÀ`m EH$ê$nVoMr H$mhr J¥hrVHo$

AmnU `m nmR>mV nm{hbo Amho H$s XmoZ {ÌH$moUmÀ`m EH$ê$nVogmR>r EH$m {ÌH$moUmMo VrZ H$moZ Xþgè`m {ÌH$moUmÀ`m VrZ H$moZmer g_mZ Agbo Var Vo EH$ê$nVogmR>r Amdí`H$ Zmhr. nU `mdê$Z Amnb {dMma H$ê$ eH$Vmo H$s, XmoZ {ÌH$moUmÀ`m EH$ê$nVogmR>r EH$m {ÌH$moUmÀ`m VrZ ~mOy Xþgè`m {ÌH$moUmÀ`m VrZ ~mOyer g_mZ AgVrb Va Vo EH$ê$nVogmR>r Amdí`H$ hmoVo. _mJrb B`ÎmoV AmnU {eH$bm AmhmV H$s hr pñW{V `mo½` R>aVo.

ho {ZpíMV H$aÊ`mgmR>r 4cm, 3.5cm Am{U 4.5cm ~mOy AgUmao XmoZ {ÌH$moUmMr aMZm H$am. (AmH¥$Vr 5.35 nhm) `m {ÌH$moUm§Zm H$mnm Am{U Vo EH$_oH$m§da R>odm. `mdê$Z Vwåhr H$m` {ZarjU H$amb? Oa g_mZ ~mOy EH$_oH$m§da g_mZ AgVrb Va Vo XmoZ {ÌH$moU nañnamer g_m{dîQ> hmoVrb. åhUyZ XmoÝhr {ÌH$moU EH$ê$n hmoVrb.

AmH¥$Vr 5.35

AZoH$ àH$maMo {ÌH$moU KoD$Z hr H¥$Vr H$am H$s, AmnU EH$ê$nVoMo EH$ J¥hrVH$ nmhÿ eH$Vmo.

à_o` 5.4 : (~m.~m.~m. J¥hrVH$) : Oa EH$m {ÌH$moUmÀ`m VrZ ~mOy XþgÝ`m {ÌH$moUmÀ`m g§JV VrZ ~mOyer g_mZ AgVrb Va Vo XmoZ {ÌH$moU EH$ê$n AgVmV.

EH$ Cn`wŠV aMZm H$ê$Z hm à_o` {gÜX H$ê$.

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104 J{UV

`m AJmoXa AmnU ~m.H$mo.~m(SAS) J¥hrVH$m_Ü`o nmhrbmV H$s, XmoZ {ÌH$moUmÀ`m XmoZ ~mOy§À`m OmoS>çm Am{U EH$ H$moZm§Mr OmoS>r g_mZ Zgob Va Vo XmoZ {ÌH$moU EH$ê$n ZgVmV.

hr H¥$Vr H$am.

XmoZ H$mQ>H$moZ {ÌH$moU Ago aMm H$s, Ë`m§Mm H$U© 5 cm Am{U EH$ ~mOy 4 cm hmoB©b (AmH¥$Vr 5.36 nhm). :

AmH¥$Vr 5.36

`m XmoZ {ÌH$moUm§Zm nañnamda EH$_oH$mda Ago R>odm H$s, EH$m {ÌH$moUmMr g_mZ ~mOy Xþgè`m {ÌH$moUmÀ`m g_mZ ~mOyda Agob. Am{U Vo {\$adm. `mdê$Z AmnU H$m` {ZarjU H$amb?

XmoZ {ÌH$moU EH$_oH$m§er g§nyU©nUo EH$mVM g_m{dîQ> Pmbo AmhoV. Am{U Vo EH$ê$n AmhoV. H$mQ>H$moZ {ÌH$moUmÀ`m BVa Omo‹S>`m KoD$Z hr H¥$Vr nwÝhm H$am. `mdê$Z H$moUVo {ZarjU H$amb?

Vwåhr nmhÿ eH$Vm H$s, {ÌH$moUmMr EH$ ~mOy§Mr Omo‹S>r Am{U H$U© g_mZ AgVrb Va Vo H$mQ>H$moZ {ÌH$moU EH$ê$n AmhoV. Ago Vwåhr _mJrb B`ÎmoV n‹S>VmiUr Ho$br Amho.

`mdê$Z AmnU Imbrb EH$ê$nVoMo J¥hrV nmhÿ eH$Vmo..

à_o` 5.5 : (EH$ê$nVoMo H$U© yOm J¥hrVH$) Oa XmoZ H$mQ>H$moZ {ÌH$moUm_Ü`o EH$m {ÌH$moUmMr EH$ ~mOy Am{U H$U© Xþgè`m {ÌH$moUmÀ`m g§JV ~mOyer Am{U g§JV H$Um©er g_mZ Agob Va Vo XmoZ H$mQ>H$moZ {ÌH$moU EH$ê$n AgVmV.

bjmV R>odm H$s, RHS åhUOoM R (Right Angle) H$mQ>H$moZ, H- Hypotenuse H$U© S (side) ~mOy.

AmVm H$mhr CXmhaUo nmhÿ`m.

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{ÌH$moU 105

CXmhaU 7: AB hr aofmI§S> Amho, P Am{U Q, AB `m aofoda Ago q~Xÿ AmhoV H$s Vo A d B q~XÿnmgyZ g_mZ A§Vamda AmhoV (AmH¥$Vr 5.37 nhm) Va {gÜX H$am PQ hm AB Mm b§~Xþ^mOH$ Amho.

CH$b : PA = PB Am{U QA = QB {Xbo AgyZ

PQ ⊥ AB d PQ hm AB M b§~Xþ^mOH$ Amho Ago

{gÜX H$ê$ PQ hr aofm

AB bm C q~§XÿV N>oXVo.

`m AmH¥$Vr_Üò Vwåhmbm XmoZ EH$ê$n

{ÌH$moU {XgyZ òVmV H$m?

AmH¥$Vr 5.37

∆ PAQ d ∆ PBQ _Üò

AP = BP ({Xbobo)

AQ = BQ ({Xbobo)

PQ = PQ (gm_mB©H$)

åhUyZ, ∆ PAQ ≅ ∆ PBQ (~m.~m.~m)

∠APQ =∠BPQ (EH$ê$n {ÌH$moUmMo g§JV KQ>H$)

AmVm ∆ PAC d ∆ PBC _Üò

AP = BP ({Xbobo)

∠APC = ∠BPC (∠APQ = ∠BPQ {gÜX Ho$bo)

PC = PC (gm_mB©H$)

åhUyZ ∆ PAC ≅ ∆ PBC (¨~.H$mo.~m)

AC = BC (EH$ê$nVoMo g§JV KQ>H$) ... (1)

Am{U ∠ACP = ∠BCP (EH$ê$nVoMo g§JV KQ>H$)

VgoM ∠ACP + ∠BCP = 180o (nyaH$ H$moZ)

åhUyZ 2∠ACP = 180o

qH$dm ∠ACP = 90o ... (2)

(1) d (2) dê$Z Vwåhr {ZîH$f© H$mTy> eH$Vm H$s, PQ hm AB Mm b§~Xþ^mOH$ Amho.

(Q>rn : Oa AP = BP {Xbo AgVmZm ∆ PAQ d ∆ PBQ EH$ê$n Xe©{dë`m {edm` Vwåhr ∆PAC ≅ ∆PBC {gÜX H$ê$ eH$V Zmhr. VgoM

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106 J{UV

PC = PC (gm_mB©H$)

Am{U ∠PAC = ∠PBC (∆APB _Üò g_mZ ~mOyMo {déÜX H$moZ)

`m {ZîH$fm©Mo _w»` H$maU åhUOoM EH$ê$nVogmR>r ~m.~m.H$mo (SSA) J¥hrVH$ `mo½` Zmhr VgoM H$moZ hm Ë`mÀ`m g§JV AgUmè`m ~mOyÀ`m OmoS>rer Zmhr.)

`mgmR>r H$mhr A{YH$ CXmhaUo KoD$.

CXmhaU 8 : l d m `m XmoZ aofm A `m q~XÿV N>oXVmV Am{U Ë`m aofm P q~XÿnmgyZ g_mZ A§Vamda Amho. (AmH¥$Vr 5.38 nhm) Va {gÜX H$am H$s, XmoÝhr aofoÀ`m _Yrb H$moZm§bm Xþ^mJUmar aofm AP Amho.

CH$b : {Xbobo Amho H$s, l d m `m aofm EH$_oH$m§Zm A `m q~XÿV N>oXVmV

PB ⊥ l, PC ⊥ m {Xbo AgVm

PB = PC hmoVo.

∠PAB = ∠PAC Ago AmnUmbm Xe©dm`Mo Amho..

∆ PAB d ∆ PAC _Üò

PB = PC ({Xbobo)

∠PBA = ∠PCA = 90o ({Xbobo)

PA = PA (gm_mB©H$)

åhUyZ ∆ PAB ≅ ∆ PAC (H$U© ^wOm J¥hrV$)

∠PAB = ∠PAC (EH$ê$nVoMo g§JV KQ>H$)

bjmV ¿`m H$s. darb {ZH$mb hm ñdmÜ`m` 5.1 _Yrb CXm. 5 Mm ì`Ë`mg Amho.

ñdmÜ`m` 5.3

1. ∆ ABC d ∆ DBC ho XmoZ g_{Û^yO {ÌH$moU BC `m EH$mM nm`mda Cô AmhoV Am{U BC

darb. A d D _YyZ OmUmar aofm Amho.(AmH¥$Vr 5.39 nhm). Oa AD dmT>{dbr AgVm BC da P q~XÿV N>oXVo Va

(i) ∆ABD ≅ ∆ACD

(ii) ∆ABP ≅ ∆ACP

(iii) ∠A Am{U ∠D Mm AP hm Xþ^mOH$ Amho.

(iv) AP hm BC Mm b§~Xþ^mOH$ Amho Ago {gÜX H$am.

AmH¥$Vr 5.39

AmH¥$Vr 5.38

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{ÌH$moU 107

2. g_{Û^yO ∆ABC _Üò AD

hm {eamob§~ AgyZ AB=AC Amho Va {gÜX H$am.

(i) AD hm BC bm Xþ^mJVmo (ii) ∠A bm AD bm Xþ^mJVmo.

3. ∆ ABC _Üò AB d BC `m XmoZ ~mOyMr _Ü`Jm PQ d QR ~mOyÀ`m PN _Ü`Joer g_mZ Amho (AmH¥$Vr 5.40 nhm). Va {gÜX H$am.

(i) ∆ ABM ≅ ∆ PQN

(ii) ∆ ABC ≅ ∆ PQRAmH¥$Vr 5.40

4. ∆ABC _Üò BE d CF ho XmoZ g_mZ {eamob§~ AmhoV. H$U© ^wOm J«hrVH$mMm dmna H$ê$Z ∆ABC hm g_{Û^yO {ÌH$moU Amho Ago {gÜX H$am.

5. ∆ABC hm g_{Û^yO {ÌH$moU AgyZ AB = AC Amho, VgoM AP ⊥ BC Agob Va {gÜX H$am ∠B = ∠C .

5.6 {ÌH$moUm_Yrb Ag_mZVm

AmVmn`ªV AmnU {ÌH$moUmÀ`m ~mOy Am{U H$moZmÀ`m g_mZVoMm Aä`mg Ho$bm AmhmV. H$mhrdoiobm Amnë`m g_moa Ag_mZVm gwÜXm {XgyZ òVo Am{U Ë`mMr VwbZm AmnUmbm H$amdr bmJVo. CXmhaUmW© AmH¥$Vr 5.41 (i) _Üò AB aofmI§S> hr CD aofmI§S> nojm _moR>r Amho Am{U AmH¥${V 5.41 (ii) _Üò ∠A hm ∠B nojm _moR>m Amho.

AmH¥$Vr 5.41

AmVm AmnU n[ajU H$ê$ H$s, EImÚm {ÌH$moUmV Ag_mZ H$moZ d Ag_mZ ~mOy `m§À`mV g§~§Y Amho H$m? `mgmR>r Imbrb H¥$Vr H$ê$.

H¥$Vr: EH$ S´>m°B§J ~moS>©da B d C `m XmoZ Q>mMÊ`m Q>moMm Am{U Ë`m§Zm EH$m Xmoè`mZo ~m§YyZ BC {ÌH$moUmMr ~mOy {_idm.

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108 J{UV

EH$m Xmoè`mMo EH$ Q>moH$ C q~Xÿda Am{U Xþgè`m Q>moH$mbm noÝgrb ~m§Ym. noÝgrbZo A q~Xÿ Xe©dyZ ∆ABC aMm (AmH¥$Vr 5.42 nhm) AmVm noÝgrbZo CA ~mOydê$Z A' hm q~Xÿ aMm.

åhUyZ, A'C > AC (bm§~rMr VwbZm Ho$br AgVm).

A' d B gm§Ym H$s ∆A'BC V`ma hmoVo ∠A'BC d ∠ABC

`m~Ôb H$m` gm§Jy eH$mb?

VwbZm H$am H$s, `mdê$Z H$m` {ZarjU hmoB©b?

ñnîQ>nUo, ∠A'BC >∠ABCAmH¥$Vr 5.42

nwÝhm CA ~mOy hmoD$Z AmUIrZ EH$ A'' q~Xÿ aMm Am{U Ë`mdê$Z {ÌH$moUmMr aMZm H$am.

Vwåhr {ZarjU H$amb H$s, Oer AC ~mOyMr bm§~r dmT>V OmB©b VgVgo {déÜX AgUmam ∠B dmT>V OmVmo.

AmVm nwÝhm EH$ Xþgar H¥$Vr H$ê$.

H¥$Vr : EH$ {df_^yO {ÌH$moU H$mT>m (`m {ÌH$moUmV {VÝhr ~mOy§Mr bm§~r Ag_mZ AgVo) `m ~mOy§Mr bm§~r _moOm.

AmVm, H$moZ _moOm. `mdê$Z H$moUVo {ZîH$f© H$mT>mb?

∆ ABC _Üò (AmH¥$Vr 5.43 nhm) BC hr gdm©V bm§~ ~mOy Am{U AC hr bhmZ ~mOy Amho.

VgoM ∠A hm _moR>m Am{U ∠B hm bhmZ Amho.

XþgÝ`m {ÌH$moUmdê$Z hr H¥$Vr naV H$am. AmH¥$Vr 5.43

`mdê$Z AmnU {ÌH$moUmÀ`m Ag_mZVoÀ`m JwUY_©H$‹S>o OmVmo. `mgmR>r Imbrb à_o` {gÜX Ho$bo Amho.

à_o` 5.6 : Oa {ÌH$moUmÀ`m XmoZ ~mOy Ag_mZ AgVrb Va, _moR>çm ~mOyg_oarb H$moZ hm _moR>m AgVmo.

AmH¥$Vr 5.43 _Üò, BC da P q~Xÿ KoD$Z CA = CP à_o` H$aVm òVmo.

AmVm AmUIr EH$ H¥$Vr H$ê$.

H¥$Vr : AB aofmI§S> H$mT>m, A bm H|$Ð _mZyZ P, Q, R, S,

T Ago {^ÝZ àH$maÀ`m {ÌÁ`m aMm. gd© q~Xÿbm A Am{UAmH¥$Vr 5.44

B bm gm§Ym (AmH¥$Vr 5.44 nhm) OgOgo AmnU P H$Sy>Z T H$‹S>o OmVmo VgVgo ∠A hm dmT>V OmVmo. `mÀ`m {déÜX ~mOyMr bm§~r H$m` hmoVo? {ZarjU H$am H$s ~mOy§Mr bm§~r dmT>VM OmVo. åhUyZ ∠TAB>∠SAB> ∠RAB >∠QAB >∠PAB Am{U TB > SB > RB > QB > PB.

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{ÌH$moU 109

AmVm, Ag_mZ H$moZ AgUmè`m {ÌH$moUmMr aMZm H$am. Am{U Ë`mÀ`m ~mOy§Mr bm§~r _moOm (AmH¥$Vr 5.45 nhm). {ZarjU H$am H$s, _moR>çm H$moZmg_oarb ~mOy _moR>r AgVo. AmH¥$Vr 5.45 _Üò ∠B hm _moR>m H$moZ Va AC hr _moR>r ~mOy Amho.

AZoH$ {ÌH$moU KoD$Z `màH$mamMr H¥$Vr nwÝhm H$ê$ Am{U hmoM à_o` 5.6 Mm ì`Ë`mg {gÜX R>aVmo. `mà_mUo AmnUmbm Imbrb à_o` {_iVmo. AmH¥$Vr 5.45

à_o` 5.7 : H$moUË`mhr {ÌH$moUmV _moRçm H$moZmg_moarb ~mOw _moR>r AgVo.

`m à_o`mbm {damoYm^mg (Method of contradiction) nÜXVrZo {gÜX H$aVm òB©b.

Vwåhr nmhmb H$s,AB + BC > AC

BC + AC > AB Am{U AC + AB > BC

`màH$mao BVa {ÌH$moUmÀ`m ~m~VrV hrM H¥$Vr H$am d Ë`mMàH$mao Vwåhmbm nwT>rb à_o` {_iob.

à_o` 5.8 : {ÌH$moUmÀ`m H$moUË`mhr XmoZ ~mOy§Mr ~oarO hr {Vgè`m ~mOynojm A{YH$ AgVo.

AmH¥$Vr 5.46 _Üò ∆ABC _Üò BA hr ~mOy D n`ªV Aer dmT>Vm H$s, AD=AC hmoB©b. ∠BCD >∠BDC Am{U BA + AC > BC Ago {gÜX H$ê$ eH$Vm H$m?

Vwåhr darb à_o`m§À`m {gÜXVon`ªV nmohMbmV H$m?

AmVm `m {ZH$mbmgmR>r H$mhr CXmhaU gmoS>dy. AmH¥$Vr 5.46

CXmhaU 9 : ∆ ABC _Üò BC `m ~mOyda D hm Agm q~Xÿ Amho H$s AD = AC (AmH¥$Vr 5.47 nhm) Va {gÜX H$am AB > AD.

CH$b : ∆ DAC _Üò

AD = AC ({Xbobo)

åhUyZ ∠ADC = ∠ACD

(g_mZ ~mOyg_moarb H$moZ)AmH¥$Vr 5.47

AmVm ∠ADC hm ∆ ABD Mm ~møH$moZ Amho.

\ ∠ADC > ∠ABD

qH$dm ∠ACD> ∠ABD

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110 J{UV

qH$dm ∠ACB > ∠ABC

åhUyZ AB > AC ∆ ABC _Yrb _moRçm H$moZmMr {déÜX ~mOy)

qH$dm AB > AD......(AD = AC)

ñdmÜ`m` 5.4

(1) {gÜX H$am H$s, H$mQ>H$moZ {ÌH$moUm_Üò H$U© hr gdm©V _moR>r ~mOy Amho.

(2) AmH¥$Vr 5.48 _Üò, ∆ ABC _Üò AB d AC

`m ~mOy AZwH«$_o P d Q q~Xÿn`ªV dmT{dbo VgoM ∠PBC < ∠QCB Va, {gÜX H$am AC > AB.

AmH¥$Vr 5.48

(3) AmH¥$Vr 5.49 _Üò ∠B < ∠A Am{U ∠C < ∠D Va {gÜX H$am AD < BC.

AmH¥$Vr 5.49

(4) ABCD `m Mm¡H$moZm_Üò AB d CD hr ~mOy AZwH«$_o bhmZ d _moR>r Amho. (AmH¥$Vr 5.50 nhm). {gÜX H$am ∠A > ∠C Am{U∠B >∠D

AmH¥$Vr 5.50

(5) AmH¥$Vr 5.51 _Üò, PR > PQ Am{U ∠QPR Mm PS hm H$moZ Xþ^mOH$ Amho. Va {gÜX H$am ∠PSR >

∠PSQ.

AmH¥$Vr 5.51

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{ÌH$moU 111

(6) {gÜX H$am H$s, {Xboë`m aofo~mhoarb q~XÿVyZ Ë`m aofon`ªV H$mT>boë`m aofmI§S>mn¡H$s b§~ aofmI§S> gdm©V bhmZ AgVo.

ñdmÜ`m` 5.5 (Optional EopÀN>H$)

(1) ABC hm {ÌH$moU Amho. {ÌH$moUmÀ`m A§VJ©V Agm q~Xÿ emoYm H$s Omo {ÌH$moUmÀ`m {eamoq~XÿnmgyZ {ÌH$moUmbm g_mZ A§Vamda AgVmo.

(2) H$moUË`mhr {ÌH$moUmÀ`m A§VJ©V EH$ Agm q~Xÿ emoYm H$s, Omo {ÌH$moUmÀ`m gd© ~mOynmgyZ g_mZ A§Vamda AgVmo.

(3) EH$m _moR>`m CÚmZmV, VrZ q~Xÿda H$mhr bmoH$ O_bo AmhoV

(AmH¥${V 5.52 nhm)

A : OoWo _wbm§H$arVm KgaJw§S>r Am{U PmoHo$ AmhoV.

B : Á`m§À`m nmer EH$ _mZd {Z{_©V Vio Amho.

C : Oo EH$m _moR>çm nmH$sªJ ñWimOdi Am{U ~mhoa OmÊ`mÀ`m añË`mOdi AmhoV.

EH$ AmB©ñH«$s_Mm ñQ>m°b H$moR>o bmdmdm OoWo OoUoH$ê$Z A{YH$mYrH$ bmoH$ nmohmoMy eH$Vrb.

(gwMZm : ñQ>m°b A, B Am{U C nmgyZ g_mZ A§Vamda Agbo nmhrOo.)

AmH¥$Vr 5.52

(4) 5 cm µ~mOy Agboë`m g_^wO {ÌH$moUmMr aMZm fîQ>^yOmH¥$Vr Am{U Vmè`mÀ`m AmH$mamÀ`m am§JmoirV H$am (AmH¥$Vr 5.53 (i) d (ii) nhm) Va àË`oH$ AmH¥$Vrdê$Z {ÌH$moUmMr g§»`m H$mT>m. ? H$moUË`m AmH¥$VrV A{YH$ {ÌH$moU AmhoV?

AmH¥$Vr 5.53

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112 J{UV

5.7 gmam§e :

`m àH$aUm_Ü`o AmnU Imbrb JmoîQ>tMm Aä`mg Ho$bm AmhmV

(1) Oa {ÌH$moUmMm AmH$ma Am{U {ÌH$moUmMr _mno g_mZ AgVrb Va XmoZ {ÌH$moUmÀ`m AmH¥$Ë`m

EH$ê$n AgVmV.

(2) g_mZ {ÌÁ`m§Mr XmoZ dVw©io EH$ê$n AgVmV..

(3) g_mZ ~mOy Agbobo XmoZ Mm¡ag EH$ê$n AgVmV.

(4) ABC d PQR `m XmoZ EH$ê$n {ÌH$moUmMo g§JV KQ>H$ A ↔ P, B ↔ Q Am{U C ↔ R

Agob Va ∆ ABC ≅ ∆ PQR `m {MÝhmZo Xe©{dVmV.

(5) Oa EH$m {ÌH$moUmÀ`m XmoZ ~mOy Am{U EH$ H$moZ Xþgè`m {ÌH$moUmÀ`m g§JV XmoZ ~mOy Am{U

g§JV EH$ H$moZmer g_mZ Agob Va Vo XmoZ {ÌH$moU EH$ê$n hmoVmV. (~m.~mo.~m J¥hrVH$)

(6) Oa EH$m {ÌH$moUmMo XmoZ H$moZ Am{U Ë`mZr g_m{dîQ> Ho$bobr EH$ ~mOy Xþgè`m {ÌH$moUmMo XmoZ

H$moZ Am{U Ë`mZr g_m{dîQ> Ho$boë`m EH$ ~mOyer g_mZ Agob Va Vo XmoZ {ÌH$moU EH$ê$n

AgVmV (H$mo.~m.H$mo. J¥hrVH$)

(7) Oa EH$ {ÌH$moUmMo XmoZ H$moZ Am{U EH$ ~mOy Xþgè`m {ÌH$moUmMo XmoZ H$moZ Am{U g§JV EH$

~mOyer g_mZ Agob Va Vo XmoZ {ÌH$moU EH$ê$n AgVmV (H$mo.H$mo. ~m J¥hrVH$)

(8) {ÌH$moUmÀ`m g_mZ ~mOyg_oarb H$moZ g_mZ AgVmV..

(9) {ÌH$moUmÀ`m g_mZ H$moZmg_oarb ~mOy g_mZ AgVmV.

(10) H$moUË`mhr g_^wO {ÌH$moUmMm àË`oH$ H$moZ 60O AgVmo.

(11) Oa EH$m {ÌH$moUmÀ`m VrZ ~mOy Xþgè`m {ÌH$moUmÀ`m VrZ ~mOyer g_mZ AgVrb Va Vo XmoZ

{ÌH$moU EH$ê$n AgVmV (~m.~m. J¥hrVH$).

(12) Oa XmoZ H$mQ>H$moZ {ÌH$moUmV EH$m {ÌH$moUmMr EH$ ~mOy Am{U H$U© Xþgè`m {ÌH$moUmÀ`m EH$

~mOy Am{U H$Um©er g_mZ Agbrb Va Vo XmoZ {ÌH$moU EH$ê$n AgVmV. (H$U© ^yOm J¥hrVH$)

(13) H$moUË`mhr {ÌH$moUmV _moR>`m ~mOyg_moarb H$moZ _moR>m AgVmo.

(14) H$moUË`mhr {ÌH$moUmV _moR>`m H$moZmg_moarb ~mOy _moR>r AgVo.

(15) H$moUË`mhr {ÌH$moUmV XmoZ ~mOy§Mr ~oarO {Vgè`m ~mOynojm A{YH$ AgVo.

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aMZm 113

àH$aU- 6

aMZm (Construction)

6.1 àñVmdZm

_mJrb àH$aUmV à_o` {gÜX H$aÊ`mgmR>r qH$dm H$mhr CXmhaUo gmoS>{dÊ`mgmR>r H$mhr AmH¥$Ë`m H$mT>>mì`m bmJVmV. Ë`m Aem H$mT>ë`m OmVmV H$s, Ë`m_wio Amnë`mbm Ë`m àg§JmMr g_O Am{U H$maU{__m§gmMo àmW{_H$ CnMma g_OyZ òVmV. H$m§hr doim AmnUm§g _mnmMr AmH¥$Vr bmJVo. CXm. Ka~m§YUrMm ZH$mem, CnH$aUm§À`m à{VH¥$Vr ~Z{dUo Am{U H$m§hr `§ÌmMo ^mJ ~Z{dUo Am{U añË`m§Mo ZH$meo V`ma H$aUo B0. darb AmH¥$Ë`m H$mT>Ê`mgmR>r H$mhr _wb^yV ^m¡{_VrH$ gmYZm§Mr JaO AgVo. Vw_À`mH$S>rb H§$nmg noQ>r_Üò Imbrb dñVy AgVrb.

(i) _moOnÅ>r (Scale) : Ë`mÀ`m EH$m ~mOybm g|._r d {_.{_ À`m IwUm AgyZ Xþgè`m ~mOyg B§M d Ë`mMo ^mJ`m§À`m IwUm Ho$boë`m AgVmV.

(ii) XmoZ {ÌH$moUr JwÊ`m (set-squares) : EH$ 90°, 60° d 30° Am{U Xþgar 90°, 45° Am{U 45°.

(iii) H$H©$Q>H$ ({d^mOH$):

(iv) H§$nmg : (H$_r OmñV H$aVm òUmao.) Ë`m_Üò EH$m Q>moH$mbm noÝgrb ~gdy eH$VmV.

(v) H$moZ _mnmH$ (Protractor)

AmH¥$Ë`m H$mT>Ê`mgmR>r gm_mÝ`nUo `m gmYZm§Mm Cn`moJ Ho$bm OmVmo. CXm. {ÌH$moU, dVw©i, Mm¡H$moZ d ~hþ^wOmH¥$Vr. na§Vw ^m¡{_VrH$ AmH¥$Vr {Xboë`m _mnmdê$Z H$mS>Ê`mg ^m¡{_VrH$ aMZm H$aÊ`mMr nÜXV Aer Amho H$s, \$ŠV XmoZ gmYZo (1) _moOnÅ>r Am{U (2) H§$nmg `m§Mm dmna H$aVmV. aMZo_Ü o O|ìhm _moO_mn bmJVo V|ìhm _moOnÅ>r d H$moZ_mnH$ dmnê$ eH$Vm. `m àH$aUm_Üò H$m§hr _wb^yV aMZm§Mm Ho$bm Amho. `m§À`m aMZm§Mm H$m§hr {ÌH$moUm§À`m aMZm H$aÊ`mg Cn`moJ Ho$bm Amho.

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114 J{UV

6.2 _yb^wV aMZm

B`Îmm VI dr _Üò Vwåhr dVw©i H$mT>Uo, aofm§Mo b§~Xþ^mOH$ H$mT>Uo, 30°, 45°, 60°, 90° Am{U 120° H$moZ _mnmMo aMUo, Am{U H$moZm§Mo Xþ^mOH$ H$mT>Uo Aem aMZm H$mhr H$maU{__m§gm Z XoVm Ho$bm AmhmV. AmVm `m emIoV Vwåhr aMUo _mJrb H$maU XoD$Z aMZm H$em gË` AmhoV ho XmIdyZ H$mhr aMZm H$amb.

aMZm 6.1 : {Xboë`m H$moZmMm H$moZXþ^mOH$ H$mT>Uo.

CXm ∠ ABC hm {Xbobm H$moZ Amho dm H$moZmMm H$moZXþ^mOH$ H$mT>Uo.

aMZoÀ`m nm`è`m

1. B hm _Ü` KoD$Z gmo`rñH$a {ÌOoZo EH$ H§$g H$mT>m. Vmo AB d BC bm AZwH«$_o E Am{U D

N>oXVmo. (AmH¥$Vr. 6.1 (i) nhm.)

2. Z§Va D d E _Ü` KoD$Z gmo`rñH$a {ÌOoZo XmoZ EH$_oH$m§g N>oXUmao H§$g H$mT>m N>oXZq~Xÿg F Zmd Úm.

3. BF {H$aU H$mT>m (AmH¥$Vr.6.1 (i) nhm) {H$aU BF hm ∠ ABC Mm BpÀN>V H$moZXþ^mOH$

Amho.

AmH¥$Vr 6.1

AmVm hr nÜXV H$er H$moZXþ^mOH$ XoVo Vo nmhÿ.

DF Am{U EF OmoS>m

∆BEF Am{U ∆ BDF `m_Üò

BE = BD (EH$mM H§$gmÀ`m {ÌÁ`m)

EF = DF (EH$mM {ÌÁòMo H§$g)

BF = BF (g_mB©H$)

\ ∆ BEF ≅ ∆ BDF (SSS ~mOy ~mOy ~mOy {Z`_)

`m_wio , ∠EBF = ∠DBF (CPCT)

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aMZm 115

aMZm 6.2 : {Xboë`m aofmI§S>mMm b§~Xþ^mOH$ H$mT>Uo.

AB hm aofmI§S> {Xbobm Amho. `mMm b§~Xþ^mOH$ H$mT>md`mMm Amho.

aMZoÀ`m nm`è`m

1. A Am{U B _Ü` KoD$Z d AB À`m {Zåå`mnojm OmñV {ÌÁòZo AB À`m XmoÝhr A§Jmg EH$_oH$m§g N>oXUmao H§$g H$mT>m.

2. H§$g EH$_oH$m§g P d Q _Üò N>oXÿ Úm. PQ OmoS>m (AmH¥$Vr. 6.2 nhm)

3. PQ b§~Xþ^mOH$ AB bm M q~XÿV N>oXÿ Úm.

V|ìhm PMQ hr aofm AB Mr BpÀN>V b§~Xþ^mOH$ hmoB©b.

AmVm AB Mm b§~Xþ^mOH$ m nÜXVrZo H$gm {_iVmo Vo nmhÿ. A d B XmoÝhr P d Q bm OmoS>m. AP, AQ, BP d BQ V`ma hmoVrb.

AmH¥$Vr 6.2

∆ PAQ d ∆ PBQ `m_Üò,

AP = BP (EH$mM {ÌOoMo H§$g)

AQ = BQ (EH$mM {ÌOoMo H§$g)

PQ = PQ (g_mB©H$)

\ ∆ PAQ ≅ ∆ PBQ

(~m.~m.~m.{Z`_)

åhUyZ ∠APM = ∠BPM (CPCT)

AmVm ∆ PMA d PMB `m_Üò (~m.~m.~m.{Z`_)

AP = BP (EH$mM {ÌOoMo H§$g)

PM = PM (g_mB©H$)

∠APM = ∠BPM ({gÜX)

∆ PMA ≅ ∆ PMB (~m.~m.~m.{Z`_)

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116 J{UV

Ë`m_wio AM = BM Am{U ∠PMA = ∠PMB (CPCT)

na§Vw ∠PMA + ∠PMB = 180° (aofr` OmoS>r J¥{hVH$)

\ ∠PMA = ∠PMB = 90°

PM hrM PMQ hr AB Mm b§~Xþ^mOH$ Amho.

aMZm 6.3: {Xboë`m {H$aUmÀ`m Ama§^q~XÿMr 60° Mm H$moZ aMUo.

AB hm {Xbobm {H$aU Amho. Ë`mMm Ama§^q~Xÿ A Amho (AmH¥$Vr.6.3 (i) nhm).

AmnUm§g AC hm {H$aU Agm H$mT>bm nm{hOo H$s ∠CAB = 60° hmoB©b..

aMZoÀ`m nm`è`m

1. A _Ü` KoD$Z gmo`rñH$a {ÌÁ`oZo AB bm D q~XÿV N>oXUmam H§$g H$mT>m.

2. D _Ü` KoD$Z Ë`mM {ÌOoZo n{hë`m H§$gmg E _Ü`o N>oXUmam H§$g H$mT>m.

3. E _YyZ AC {H$aU H$mT>m. (Am.6.3 (ii) nhm)

∠CAB hm 60° M H$moZ V`ma hmoVmo..

AmVm `m nÜXVrZo AmnUmg hdm AgUmam 60° Mm H$moZ H$gm {_iVmo Vo nmhÿ DE OmoS>m.

AmH¥$Vr 6.3

Voìhm AE = AD = DE (aMZonmgyZ)

\ ∆ EAD hm g_^wO {ÌH$moU Amho Am{U ∠EAD åhUOoM ∠CAB =60°.

ñdmÜ`m` 6.1 1. 90° Mm H$moZ aMm d Ë`mMr {gÜXVm gm§Jm.

2. {Xboë`m {H$aUmÀ`m Ama§^mq~Xÿer 45° Mm H$moZ aMyZ aMZoMr {gÜXVm gm§Jm.

3. Imbrb _mnmMo H$moZ aMm

(i) 30° (ii) 1222

° (iii) 15°

4. Imbrb H$moZ aMm d H$moZ_mnH$mZo _moOyZ nS>Vmim.

(i) 75° (ii) 105° (iii) 135°

5. H$moUË`mhr EH$ ~mOy KoD$Z g_^wO {ÌH$moU aMm d {gÜXVm nhm.

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aMZm 117

6.3 H$mhr {ÌH$moUm§À`m aMZm

H$m§hr _wb^yV aMZmMm {dMma Ho$bm. nwT>o H$mhr {ÌH$moUm§À`m aMZm§Mm _mJrb dJm©Vrb aMZm§Mm Cn`moJ H$ê$Z Ho$bm OmB©b. àH$aU 5 _Yrb ~mOy H$moZ ~mOy, ~mOy ~mOy ~mOy, H$moZ ~mOy H$moZ d H$mQ>H$moZ H$U© d EH$ ~mOy RHS {Z`_ dmnê$Z XmoZ {ÌH$moU EH$ê$n XmI{dVmV. åhUyZ {ÌH$moU hm EH$_od Amho Oa

(i) XmoZ ~mOy d g_{dîQ>H$moZ {Xbm AgVm.

(ii) VrZ ~mOy {Xë`m AgVm

(iii) XmoZ H$moZ d EH$ ~mOy {Xbo AgVm, "

(iv) H$mQ>H$moZ {ÌH$moUmV, H$U© d EH$ ~mOy {Xë`mg Vwåhr 7 ì`m B`Îmo_Üò Aem àH$maÀ`m {ÌH$moUm§À`m aMZm H$em Ho$ë`m OmVmV ho {eH$bm AmhmV. AmVm {ÌH$moUmÀ`m AmUIr H$mhr aMZm nmhÿ. Vwåhmbm g_Obo AgboM H$s H$_rV H$_r VrZ JmoîQ>r {Xë`mg {ÌH$moUmMr aMZm H$aVm òVo. na§Vy H$mhr aMZm VrZ JmoîQ>tÀ`m g_whmZo hmoD$ eH$V ZmhrV.

CXmhaUmW© : Oa {ÌH$moUmÀ`m XmoZ ~mOy d H$moZ (g_m{dîQ> Zgbobm) d BVa XmoZ ~mOy§Mr ~oarO {Xë`mg {ÌH$moUmMr aMZm H$aUo eŠ` Zmhr.

aMZm 6.4 : ∆ Mr nm`mMr ~mOy, nm`mdarb EH$ H$moZ d BVa XmoZ ~mOy§Mr ~oarO {Xë`mg {ÌH$moUmMr aMZm H$aUo.

∆ ABC _Üò BC nm`m, ∠B Am{U AB + AC {Xbo AmhoV.

AmnUm§g ∆ ABC aMmd`mMm Amho.

aMZoÀ`m nm`mè`m :

1. BC nm`m H$mT>m B q~Xÿer H$moZ ∠XBC aMm H$s Vmo {Xboë`m H$moZm EdT>m hmoB©b.

2. AB + AC EdT>r BX aofm D q~XÿV N>oXm.

3. DC OmoS>m d ∠DCY =∠BDC hmoD$ Úm.

4. CY aofm BX bm A q~XÿV N>oXVo.

∆ ABC hm BpÀN>V {ÌH$moU hmoB©b..AmH¥$Vr 6.4

Vwåhmbm Oê$ar AgUmam {ÌH$moU H$gm {_imbm Vo nmhÿ.

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118 J{UV

BC nm`m Am{U ∠B hm nm`mdarb H$moZ ho {Xbo AmhoV.

∆ ACD _Ü`o, ∠ACD = ∠ADC (aMZm)

åhUyZ AC = AD

AB = BD – AD = BD – AC

∴ AB + AC = BD.

n`m©`r nÜXV :

darb XmoZ nm`è`m AmhoV VemM dmnam . CD Mm PQ

b§~Xþ^mOH$ Agm H$mT>m H$s BD bm A q~XÿV N>oXob.

AC Omo‹S>m ∆ ABC V`ma hmoB©b. BWo A hm CD aofoÀ`m b§~Xþ^mOH$mdarb q~Xÿ åhUyZ AD = AC Oa AB + AC ≤ BC {ÌH$moUmMr aMZm eŠ` Zmhr.

AmH¥$Vr 6.5

aMZm 6.5 : {ÌH$moUmMm nm`m, EH$ H$moZ Am{U BVa XmoZ ~mOy_Yrb \$aH$ {Xbm AgVm {ÌH$moUmMr aMZm H$aUo.

∆ ABC _Ü`o BC nm`m d ∠B nm`mdarb H$moZ Am{U AB – A qH$dm AC – AB {Xbo Amho

∆ ABC Mr aMZm H$amo Amho.

g_Om AB > AC åhUOoM AB – AC {Xbo Amho.

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aMZm 119

aMZm :

1. BC nm`m ¿`m d B q~Xÿer {Xboë`m ∠B EdT>m∠XBC

aMm.

2. BX darb AB – AC EdT>r BD H$mT>m.

3. DC OmoS>m d DC Mm b§~Xþ^mOH$ PQ H$mT>m.

4. b§~Xþ^mOH$ BX bm A q~XÿV N>oXÿ Úm. (AmH¥$Vr. 6.6 nhm)

AmVm ∆ ABC V`ma hmoB©b.

Vwåhmbm hdm AgUmam {ÌH$moU H$gm {_imbm Vo nmhÿ.

nm`m BC d H$moZ ∠B {Xbobo AmhoV. A hm q~Xÿ DC

À`m b§~Xþ^mOH$mda Amho.

\AD= AC

AmH¥$Vr 6.6

∴ BD = AB – AD = AB – AC

KQ>Zm (ii) : g_Om AB < AC åhUOoM AC – AB {Xbo Amho.

aMZoÀ`m nm`mè`m :

1. darb (i) à_mUoM

2. BX dmT>dyZ AC – AB EdT>r BD H$mT>m.

3. DC OmoS>m Am{U DC Mm b§~Xþ^mOH$ PQ H$mT>m.

4. PQ b§~Xþ^mOH$ bm A _Ü`o N>oXÿ Úm.(Am. 6.7 nhm)

AmVm {ÌH$moU ABC V`ma hmoB©b.

KQ>Zm (i) à_mUo aMZoMr gË`Vm nS>Vmim.

AmH¥$Vr 6.7

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120 J{UV

aMZm 6.6 : {ÌH$moUmÀ`m nm`mdarb XmoZ H$moZ d n[a{_Vr {Xbr AgVm {ÌH$moUmMr aMZm H$aUo.

∠B Am{U ∠C nm`mdarb H$moZ Am{U AB + BC + CA `m JmoîQ>r {Xboë`m AmhoV. {ÌH$moU ABC Mr aMZm H$amd`mMr Amho.

aMZoÀ`m nm`mè`m :

1. AB + BC + CA `m n[a{_VrEdT>r XY aofmI§S> H$mT>m.

2. ∠B EdT>m ∠LXY Am{U ∠C EdT>m ∠MYX H$mT>m.

3. ∠LXY Am{U ∠MYX Mo H$moZXþ^mOH$ H$mT>m. ho H$moZXþ^mOH$ EH$_oH$m§g A q~XÿV N>oXÿ Úm. (Am. 6.8 (i) nhm).

AmH¥$Vr 6.8 (i)

4. AX Mm b§~Xþ^mOH$ PQ Am{U AY Mm b§~Xþ^mOH$ RS H$mT>m.

5. PQ b§~Xþ^mOH$ XY bm B _Ü`o Am{U RS b§~Xþ^mOH$ XY bm C _Ü`o N>oXÿ Úm.

AB d AC OmoS>m (Am. 6.8 (ii) nhm).

AmH¥$Vr 6.8 (ii)

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aMZm 121

{ÌH$moU ABC hm BpÀN>V {ÌH$moU Amho.

aMZoÀ`m gË`Vo{df`r, Vwåhmbm Ago {XgyZ òB©b H$s, B q~Xÿ AX Mm b§~Xþ^mOH$ PQ da Amho. VgoM

XB = AB Am{U Ë`mMà_mUo CY = AC

nwÝhm AB + BC + CA = BC + XB + CY = XY

∠BAX = ∠AXB Ogo [∆ AXB _Üò AB = XB]

∠ABC = ∠BAX + ∠AXB = 2 ∠AXB = ∠LXY

Ë`mMà_mUo ∠ACB = ∠MYX Amdí`H$VoZwgma

CXmhaU 1 : ∠B = 60°, ∠C = 45° Am{U AB + BC + CA = 11 cm {Xbo AgVm {ÌH$moU ABC aMm.

aMZoÀ`m nm`mè`m :

1. PQ = 11 cm aofmI§S> H$mT>m. (AB + BC + CA= PQ)

2. P q~Xÿer 60° Mm d Q q~Xÿer 45° Mm H$moZ aMm.

AmH¥$Vr 6.9

3. ho XmoÝhr H$moZ Xþ^mJm XmoZ Xþ^mOH$ A q~XÿV N>oXÿ Úm.

4. AP Mm b§~Xþ^mOH$ DE aofm PQ bm C _Üò N>oXÿ Úm.

5. AB Am{U AC OmoS>m (Am. 6.9 nhm)

{ÌH$moU ABC BpÀN>V {ÌH$moU Amho.

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122 J{UV

ñdmÜ`m` 6.2

1. {ÌH$moU ABC Agm aMm H$s, BC = 7 cm, ∠B = 75° Am{U AB + AC = 13 cm g|._r hmoB©b.

2. {ÌH$moU ABC Agm aMm H$s, BC = 8 cm, ∠B = 45° Am{U AB – AC = 3.5 cm hmoB©b.

3. {ÌH$moU PQR Agm aMm H$s, QR = 6 cm, ∠Q = 60° Am{U PR – PQ = 2 cm hmoB©b.

4. {ÌH$moU XYZ Agm aMm H$s, ∠Y = 30°, ∠Z = 90° Am{U XY + YZ + ZX = 11 cm hmoB©b.

5. H$mQ>H$moZ {ÌH$moU Agm aMm H$s, Ë`mMm nm`m 12 cm Am{U Ë`mMm H$U© d BVa ~mOy `m§Mr

~oarO 18 cm hmoB©b.

6.4 gmam§e

`m àH$aUmV Vwåhr _moOnÅ>r d H§$nmg dmnê$Z Imbrb aMZm H$aÊ`mg {eH$bmV.

1. {Xbobm H$moZ Xþ^mJUo.

2. {Xboë`m aofmI§S>mMm b§~Xþ^mOH$ H$mT>Uo.

3. 60° Mm H$moZ aMUo.

4. {ÌH$moUmMm nm`m, nm`mdarb H$moZ d BVa XmoZ ~mOw§Mr ~oarO, dOm~mH$s (\$aH$) {Xbm AgVm {ÌH$moU aMUo.

5. {ÌH$moUmMm nm`m, nm`mdarb H$moZ Am{U BVa XmoZ ~mOwMr dOm~mH$s (\$aH$) {Xbm AgVm {ÌH$moU aMUo.

6. nm`mdarb XmoZ H$moZ d n[a{_Vr (~mOw§Mr) {Xbr AgVm {ÌH$moU aMUo.

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Mm¡H$moZ 123

àH$aU - 7

Mm¡H$moZ (Quadrilaterals)

7.1 àñVmdZm

àH$aU 3 d 5 ‘Ü¶o {ÌH$moUmÀ¶m AZoH$ JwUY‘m©{df¶r Aä¶mg Ho$bm AmhmV Am{U Ë¶mdê$Z Vwåhmbm g‘OyZ Ambo AgobM H$s, VrZ Z¡H$aofr¶ q~Xþ H«$‘mZo OmoS>ë¶mg ~ZUmar AmH¥$Vr {ÌH$moU Amho. AmVm Mma Z¡H$aofr¶ q~Xþ ¿¶m d XmoZ XmoZ q~Xÿ H«$‘mZo OmoS>m nhm H$moUVr AmH¥$Vr ~ZVo.

AmH¥$Vr 7.1

Oa gd© q~Xÿ EH$m aofoV AgVrb Va AmnUm§g EH$ aofm {‘iVo (AmH¥$Vr 7.1 (i) nhm.) Oa Mman¡H$s VrZ q~Xÿ EH$aofr¶ AgVrb Va AmnUm§g {ÌH$moU AmH¥$Vr {‘iVo(AmH¥$Vr 7.1 (ii) nhm.) Am{U Oa Mman¡H$s VrZ q~Xÿ EH$aofr¶ ZgVrb Va {‘iUmar ‘¶m©{XV Mma ~mOy§§Mr AmH¥$Vr {‘iVo (AmH¥$Vr 7.1 (iii) d (iv) nhm ).

Mma Z¡H$aofr¶ q~Xÿ H«$‘mZo OmoSy>Z ~Zboë¶m AmH¥$Vrbm Mm¡H$moZ Agoo åhUVmV. ¶m àH$aUmV AmnU AmH¥$Vr 7.1 (iii) àH$maÀ¶m Mm¡H$moZmMm {dMma H$ê$¶m na§Vw , AmH¥$Vr 7.1 (iv) àH$maÀ¶m Mm¡H$moZmMm Zmhr.

Mm¡H$moZmbm Mma ~mOy, Mma H$moZ d Mma {eamoq~Xÿ d XmoZH$U© AgVmV (AmH¥$Vr 7.2 (i) d (ii) nhm )

AmH¥$Vr 7.2

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124 J{UV

Mm¡H$moZ ABCD ‘Ü¶o AB, BC, CD Am{U DA ¶m Mma ~mOy A,B,C d D ho Mma {eamoq~Xÿ Am{U ∠ A, ∠ B, ∠ C d ∠D ho Mma H$moZ {eamoq~XÿÀ¶m {R>H$mUr V¶ma Pmbo AmhoV AmVm A d C Am{U B d D ho {dê$ÜX {eamoq~Xÿ OmoS>m (AmH¥$Vr 7.2 (ii) nhm). AC Am{U BD ho Mm¡H$moZ ABCD Mo H$U© AmhoV.

¶m àH$aUm‘Ü¶o AmnU Mm¡H$moZmMo àH$ma d Ë¶m§Mo JwUY‘© {deofV: g‘m§Va^wO Mm¡H$moZmMm Aä¶mg H$ê$. Vwåhmbm àûZ nS>ob H$s, AmnU Mm¡H$moZmMm Aä¶mg H$ê$¶m. Vw‘À¶m AdVr ^dVr nhm Vwåhmbm Mm¡H$moZr AmH$mamÀ¶m AZoH$ dñVy AmT>iVrb. KamÀ¶m {^§Vr, O‘rZ, N>V, {IS>³¶m, ’$im, S>ñQ>aMr àË¶oH$ ~mOy, nwñVH$mMo àË¶oH$ nmZ, Aä¶mgmÀ¶m Q>o~bmMm n¥îR>^mJ BË¶mXr. Imbr H$mhr AmH¥$Ë¶m {Xë¶m AmhoV.

AmH¥$Vr 7.3

Amnë¶mbm ~è¶mMem dñVy Am¶VmH¥$Vr {XgVmV. AmnU g‘m§Va^wO Mm¡H$moZmMm Aä¶mg H$ê$.Am¶V hm gwÜXm g‘m§Va^wO Mm¡H$moZ Amho d g‘m§Va^wO Mm¡H$moZmMo JwUY‘© Am¶Vmbm bmJy hmoVmV.

7.2 Mm¡H$moZmÀ`m gd© H$moZm§À`m ~oaOoMm JwUY_©

Mm¡H$moZmÀ¶m gd© H$moZm§Mr ~oarO 360° AgVo. hm JwUY‘© Mm¡H$moZmMm EH$ H$U© OmoSy>Z Ë¶mMo XmoZ {ÌH$moUmV {d^mJyZ nS>VmiyZ nmhVm ¶oVmo. AmH¥$Vr 7.4 ‘Ü¶o ABCD hm Mm¡H$moZ Amho. AC hm H$U© Amho.

∆ ADC À¶m gd© H$moZm§Mr ~oarO {H$Vr ? AmH¥$Vr 7.4

Vwåhmbm ‘m{hV Amho ∠ DAC + ∠ ACD + ∠ D = 1800 ...(i)

Ë¶mMà‘mUo {ÌH$moU ∆ ABC ‘Ü¶o ∠ CAB + ∠ ACB + ∠ B = 1800 ...(ii)

(i) d ii) ¶m§Mr ~oarO Ho$ë¶mg ∠ DAC +∠ ACD + ∠ D + ∠ CAB + ∠ ACB + ∠ B = 1800 + 1800 = 3600

VgoM ∠ DAC +∠ CAB = ∠ A Am{U ∠ ACD + ∠ ACB = ∠ C \ ∠ A + ∠ D + ∠ B + ∠ C = 3600

åhUOoM Mm¡H$moZmÀ¶m gd© H$moZm§Mr ~oarO 3600 Amho.

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Mm¡H$moZ 125

7.3 Mm¡H$moZmMo àH$ma

Imbr {Xbobo gd© Mm¡H$moZ nhm.

AmH¥$Vr 7.5

{ZarjUo :

• AmH¥$Vr 7.5 (i) ‘Ü¶o ABCD hr g§‘wI ~mOyMr OmoS>r g‘m§Va Amho. ¶m Mm¡H$moZmg g‘b§~ Mm¡H$moZ åhUVmV.

• AmH¥$Vr 7.5 (ii), (iii), (iv) Am{U (v) ‘Ü¶o g§‘wI ~mOyÀ¶m XmoÝhr OmoS>çm g‘m§Va AmhoV. Aem Mm¡H$moZmg g‘m§Va^wO Mm¡H$moZ åhUVmV.

• AmH¥$Vr 7.5 (iv) ‘Yrb Mm¡H$moZ PQRS hm g‘m§Va^wO Mm¡H$moZ Amho. VgoM AmH¥$Vr 7.5 (iii), (iv) Am{U (v) ho Mm¡H$moZ g‘m§Va^wO Mm¡H$moZ AmhoV.

• g‘m§Va^wO Mm¡H$moZ MNRS AmH¥$Vr 7.5 (iii) ‘Ü¶o ∠M hm H$mQ>H$moZ Amho. Va ¶m Mm¡H$moZmbm H$m¶ åhUVmV ? ¶mbm Am¶V åhUVmV.

• g‘m§Va^wO Mm¡H$moZ DEFG ‘Ü¶o (AmH¥$Vr 7.5 (iv) ) gd© ~mOy g‘mZ AmhoV. åhUyZ Ë¶mbm g‘^wO Mm¡H$moZ (Rhombus) åhUVmV.

• g‘m§Va^wO Mm¡H$moZ ABCD (AmH¥$Vr 7.5 (v) ) ‘Ü¶o ∠A= 90° Am{U gd© ~mOy g‘mZ AmhoV. åhUyZ Ë¶mbm Mm¡ag (Square) åhUVmV.

• g‘m§Va^wO Mm¡H$moZ ABCD (AmH¥$Vr 7.5 (vi) ) ‘Ü¶o AD = CD Am{U AB = CB åhUOoM g§b½Z ~mOwÀ¶m XmoZ OmoS>çm g‘mZ AmhoV. ¶mbm g‘m§Va^wO Mm¡H$moZ åhUV ZmhrV. ¶mbm nV§J åhUVmV. bjmV R>odm Mm¡ag, Am¶V Am{U g‘^wO Mm¡H$moZ ho gd© g‘m§Va^wO Mm¡H$moZ AmhoV.

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126 J{UV

• Mm¡ag hm Am¶V Am{U g‘^wO Mm¡H$moZ Amho

• Am¶V hm g‘m§Va^wO Am{U g‘^wO Mm¡H$moZ Amho.

• nV§J hm g‘m§Va^wO Mm¡H$moZ Zmhr.

• g‘b§~ Mm¡H$moZ hm g‘m§Va^wO Mm¡H$moZ Zmhr. (H$maU ’$³V EH$M g‘w§I ~mOw§Mr OmoS>r g‘m§Va AgVo Xþgar Zmhr.) g‘m§Va^wO Mm¡H$moZm§V XmoÝhr g§‘wI OmoS>çmg‘m§Va Agë¶m nm{hOoV.

• Am¶V d g‘^wO Mm¡H$moZ ho Mm¡ag ZmhrV. AmH¥$Vr 8.6 nhm. Am¶V d g‘m§Va^wO Mm¡H$moZ g‘mZ n[a{‘VrMo 14 g|. ‘r AmhoV.

AmH¥$Vr 7.6

¶m {R>H$mUr g‘m§Va^wO Mm¡H$moZ ABCD Mo joÌ’$i DP × AB Am{U Am¶VmMo joÌ’$i AB × AD

nojm H$‘r Amho. DP < AD gd©gmYmaUnUo {‘R>mB©dmbm ~’$s© g‘m§Va^wO Mm¡H$moZmÀ¶m AmH$mamV H$mnVmo.

H$maU Ë¶mbm Q´>o ‘Ü¶o OmñVrV OmñV ZJ aVm ¶oVrb. Vwåhr ~’$s© ImÊ¶mAJmoXa Ë¶mMm AmH$ma nhm.

g‘m§Va^wO Mm¡H$moZmÀ¶m H$mhr JwUY‘mªMr COiOr H$ê.$

7.4. g‘m§Va^wO Mm¡H$moZmMo JwUY‘©

H¥$Vr: H$mJXr nwÇ>¶m‘YyZ EH$ g‘m§Va^wO Mm¡H$moZ H$mnyZ ¿¶m

d Vmo H$Um©À¶m {XeoZo H$mnm. Vwåhmbm XmoZ {ÌH$moU {‘iVrb.

¶m {ÌH$moUm{df¶r Vwåhr H$m¶ gm§Jy eH$mb?

EH$ {ÌH$moU Xþgè¶m {ÌH$moUmda R>odm, Vwåhmbm H$m¶ {XgyZ

¶oB©b? XmoÝhr {ÌH$moU EH$ê$n {ÌH$moU AmhoV. Ago AZoH$

g‘m§Va^yO Mm¡H$moZ KoD$Z darb à‘mUo H¥$Vr H$am. àË¶oH$ doir

Vwåhmbm XmoZ EH$ê$n {ÌH$moU {‘iVmV. ¶mdê$Z AmH¥$Vr 7.7

g‘m§Va^wO Mm¡H$moZmMm àË¶oH$ H$U© Ë¶mbm XmoZ EH$ê$n {ÌH$moUm§V {d^mJVmo. hm {ZîH$f© {gÜX

H$ê$¶m.

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Mm¡H$moZ 127

à‘o¶ 7.1 : g‘m§Va^wO Mm¡H$moZmMm H$U© Mm¡H$moZmog XmoZ EH$ê$n {ÌH$moUm§V {d^mJVmo.

{gÜXVm : ABCD hm g‘m§Va^wO Mm¡H$moZ Amho AC hm Ë¶mMm H$U© (AmH¥$Vr 7.8) AC g‘m§Va^wO Mm¡H$moZ ABCD bm XmoZ EH$ê$n {ÌH$moUm§V {d^mJVmo

∆ ABC Am{U ∆CDA ho XmoZ {ÌH$moU EH$ê$n AmhoV Ago {gÜX H$amd¶mMo Amho.

∆ ABC d ∆ CDA ¶m‘Ü¶o BC || AD d AC hr N>o{XH$m Amho.

∠ BCA = ∠ DAC (ì¶wËH«$‘ H$moZ)

AB || DC d AC N>o{XH$m

∠ BAC = ∠ DCA (ì¶wËH«$‘ H$moZ)

Am{U AC = CA (g‘mB©H$ ~mOy)

∆ ABC ≅ ∆ CDA ( ASA {Z¶‘)AmH¥$Vr 7.8

qH$dm AC H$U© g‘m§Va^wO Mm¡H$moZmg XmoZ EH$ê$n ‘Ü¶o {d^mJVmo

AmVm g‘m§Va^wO Mm¡H$moZmÀ¶m {dê$ÜX ~mOy ‘moOm. AB= DC d AB= BC AmhoV. ¶mdê$Z g‘m§Va Mm¡H$moZmMm AmUIr EH$ JwUY‘© Imbrb à‘mUo

à‘o¶ 7.2 : g‘m§Va^wO Mm¡H$moZmV g§‘wI ~mOy g‘mZ AgVmV. ¶mnydu Vwåhr {gÜX Ho$bo Amho H$s, H$U© g‘m§Va^wO Mm¡H$moZmg XmoZ EH$ê$n {ÌH$moUm§V {d^mJVmo. g§‘wI ~mOw{df¶r d g§‘wI H$moZm{df¶r H$m¶ gm§Jmb?

Vo g‘mZ AmhoV.

åhUyZ AB = DC Am{U AD = BC

AmVm ¶m {ZîH$fm©Mm ì¶Ë¶mg H$m¶? Vwåhmbm ‘mhrV Agboë¶m à‘o¶mZwgma ì¶Ë¶mg gwÜXm à‘o¶mà‘mUoM {gÜX H$amdo bmJVmV à‘o¶ 7.2 Imbrb à‘mUo gm§{JVbm OmVmo.

Oa g‘m§Va^wO MmoH$moZ Agob Va Ë¶mMm g§‘wI ~mOwÀ¶m OmoS>çm g‘mZ AgVmV.¶mMm ì¶Ë¶mg Agm Amho.

à‘o¶ 7.3 : Oa Mm¡H$moZmÀ¶m g§‘wI ~mOwÀ¶m OmoS>çm g‘mZ AgVrb Va Vmo g‘m§Va^wO Mm¡H$moZ AgVmo.

¶mMr H$maU{‘‘m§gm Vwåhr H$amb H$m?

Mm¡H$moZ ABCD ‘Ü¶ AB = CD d AD = BC.

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128 J{UV

(AmH¥$Vr 7.9 : nhm ) H$U© AC H$mT>m:

∆ ABC ≅ ∆ CDA (H$m ?)

åhUyZ ∠ BAC = ∠ DCA

∠ BCA = ∠ DAC (H$m ?)

AmVm ABCD hm g‘m§Va^wO Mm¡H$moZ Amho gm§Jy eH$mb

H$m?

AmH¥$Vr 7.9

Vwåhr AmVm nm{hbo AmhoM H$s g‘m§Va^wO Mm¡H$moZmÀ¶m g§‘wI ~mOy g‘mZ AgVmV. ì¶Ë¶mgmZo Oa g‘w§I ~mO§wMr OmoS>r Mm¡H$moZmV g‘mZ Agob Va Vmo Mm¡H$moZ g‘m§Va^wO Mm¡H$moZ Amho.

AgmM {ZîH$f© AmnU g‘w§I H$moZm§À¶m OmoS>r~Ôb H$mTy> eH$Vmo.

g‘m§Va^wO Mm¡H$moZ H$mT>m d Ë¶mMo H$moZ ‘moOm, Vwåhmbm H$m¶ {XgyZ ¶oVo.

g‘w§I H$moZm§Mr àË¶oH$ OmoS>r g‘mZ Amho.

à‘o¶ 7.4 ‘Ü¶o g‘m§V^wO Mm¡H$moZmMr g‘wI H$moZmMr OmoS>r g‘mZ AgVo ho nm{hbo. AmVm ¶m {ZîH$fm©Mm ì¶Ë¶mg gË¶ Amho H$m? hmo¶. H$moZm§À¶m ~oaOoÀ¶m JwUY‘m©Mm Cn¶moJ H$ê$Z Am{U g‘m§Va ~mOy N>o{XHo$Zo N>oXë¶mZo AmnU Ago åhUy eH$Vmo H$s, ì¶Ë¶mg Amho gË¶ AgwZ AmUIr EH$ à‘o¶ Imbrb à‘mUo V¶ma hmoVmo.

à‘o¶ 7.5 : EImÚm Mm¡H$moZmV g§‘wI H$moZmMr àË¶oH$ OmoS>r g‘mZ Agob Va Vmo g‘m§Va^wO Mm¡H$moZ AgVmo.

BWo g‘m§Va^wO Mm¡H$moZmMm Xþgam JwUY‘© Amho. Ë¶mMm Aä¶mg H$ê$¶m ABCD g‘m§Va^wO H$mT>m. Mm¡H$moZmMo H$U© EH$‘oH$mg O ‘Ü¶o N>oXÿ Úm (AmH¥$Vr 7.10 nhm).

OA, OB, OC d OD ‘moOm. Vwåhmbm H$m¶ AmT>iVo?

OA = OC Am{U OB = OD

'O' hm XmoÝhr H$UmªMm ‘Ü¶ q~Xÿ Amho. hrM H¥$Vr AZoH$

g‘m§Va^wO Mm¡H$moZ KoD$Z H$am. àË¶oH$ doir AmnUm§g Ago {XgyZ

¶oVo H$s O hm XmoÝhr H$UmªMm ‘Ü¶q~Xÿ Amho. åhUyZ à‘o¶ Imbrb à‘mUo

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Mm¡H$moZ 129

à‘o¶ 7.6 : g‘m§Va^wO Mm¡H$moZmMo H$U© EH$‘oH$m§g

Xþ^mJVmV. Oa Mm¡H$moZmMo H$U© EH$‘oH$mg Xþ^mJV AgVrb

Va H$m¶ KS>ob ? Vmo Mm¡H$moZ g‘m§Va^wO Mm¡H$moZ Agob

H$m¶? hmo¶ ho Iao Amho. hm {ZîH$f© à‘o¶ 7.6 À¶m

{ZîH$fm©Mm ì¶Ë¶mg Amho. Vmo Imbrb à‘mUo.

AmH¥$Vr 7.10

à‘o¶ 7.7 : Oa Mm¡H$moZmMo H$U© EH$‘oH$m§g Xþ^mJV

AgVrb Va Vmo g‘m§Va^wO Mm¡H$moZ AgVmo.

¶mMo H$maU Vwåhr emoYy eH$Vm ?

AmH¥$Vr 7.11 nhm

OA = OC Am{U OB = OD {Xbo Amho.

åhUyZ ∆ AOB ≅ ∆ COD (H$m ?) AmH¥$Vr 7.11

∠ABO = ∠CDO (H$m ?)

¶mnmgyZ AB || CD

VgoM BC || AD

ABCD hm g‘m§Va^yO Mm¡H$moZ AgVmo.

CXmhaU 1 : Am¶VmMm àË¶oH$ H$moZ H$mQ>H$moZ AgVmo ho {gÜX H$am.

Am¶V åhUOo H$m¶?

Am¶V hm g‘m§Va^wO Mm¡H$moZM Amho Á¶mMm àË¶oH$ H$moZ H$mQ>H$moZ AgVmo.

CH$b : ABCD hm Am¶V Amho d ∠A= 900 Amho.

AmnUmg {gÜX H$am¶Mo Amho Amho. ∠B = ∠C = ∠D = 900

AD || BC Am{U AB N>o{XH$m$ Amho.

∠A + ∠B = 1800 No{XHo$À¶m EH$mM A§Jmg AgUmao H$moZ)

na§Vw ∠A = 900

åhUyZ ∠B = 1800 – ∠A = 1800 - 900 = 900

AmVm ∠C = ∠A Am{U ∠D =∠B AmH¥$Vr 7.12

[g‘m§Va^wO Mm¡H$moZmMo g‘wI H$moZ ] åhUyZ ∠C = 900 d ∠D = 900

åhUyZ Am¶VmMm àË¶oH$ H$moZ H$mQ>H$moZ AgVmo.

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130 J{UV

CXmhaU 2 : g‘^wO Mm¡H$moZmMo H$U© EH$‘oH$m§g b§~ AgVmV. ho XmIdm.

CH$b : ABCD hm g‘^wO Mm¡H$moZ Amho (Am. 7.13 nhm).

Vwåhmbm ‘mhrV Amho AB = BC = CD = DA (H$m?)

AmVm ∆ AOD d ∆ COD ¶m‘Ü¶o

OA = OC (g‘m§Va^wO Mm¡H$moZmMo H$U© EH$‘oH$m§g

Xþ^mJVmV.)

OD = OD (g‘mB©H$)

AD = CD (J¥hrV)

åhUyZ ∆ AOD ≅ ∆ COD (~m.~m.~m. {Z¶‘) AmH¥$Vr 7.13

åhUyZ ∠AOD = ∠COD (g§b½Z H$moZ)

na§Vw ∠AOD + ∠COD = 1800 (g§b½Z H$moZ)

2 ∠AOD = 1800

∠AOD = 900

åhUyZ g‘^wO Mm¡H$moZmMo H$U© EH$‘oH$m§g b§~ AgVmV.

CXmhaU 3 : ∆ ABC hm g‘pìX^wO {ÌH$moU Amho. Á¶m‘Ü¶o AB = AC , ~mô¶ H$moZ ∠PAC hm AD

Zo Xþ^mJVmo Am{U CD || AB (AmH¥$Vr 7.14 nhm ).

(i) ∠ DAC = ∠ BCA Am{U (ii) ABCD hm g‘m§Va^wO Mm¡H$moZo Amho.

CH$b : ∆ ABC hm g‘pìX^wO {ÌH$moU Amho

Á¶m‘Ü¶o AB = AC (J¥{hV)

∠ ABC = ∠ ACB (g‘mZ ~mOwg‘moarb H$moZ)

VgoM ∠ PAC = ∠ ABC + ∠ ACB (~mô¶ H$moZ)

qH$dm ∠ PAC = 2 ∠ ACB … (1)

AmVm AD hm ∠PAC bm Xþ^mJVmo

åhUyZ ∠ PAC = 2 ∠ DAC … (2) AmH¥$Vr 7.14

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Mm¡H$moZ 131

((1) Am{U (2)dê$Z)

2 ∠DAC = 2 ∠ACB åhUyZ ∠DAC =∠ACB

(ii) AmVm ho g‘mZ H$moZ ì¶wËH«$‘ H$moZm§Mr OmooS>r ~ZVo Ooìhm BC d AD ¶m§Zm AC hr N>oXrH$m

Amho.

BC || AD

VgoM BA || CD (J¥{hV)

AmV ABCD g§‘wI ~mOw§À¶m OmoS>çm g‘m§Va AmhoV.

ABCD hm g‘m§Va^wO Mm¡H$moZ Amho.

CXmhaU 4: XmoZ l d m g‘m§Vaaofm§Zm p ¶m No{XHo$Zo N>oXbo Amho (AmH¥$Vr 7.15 nhm ).

Va Am§VaH$moZm§À¶m Xþ^mOH$mZr ~Zbobm Mm¡H$moZ Am¶V Amho ho XmIdm.

BWo : PS || QR Xrbo Amho N>o{XH$m PQ Zoo Ë¶mZm A d C ¶m q~XÿV N>oXbo Amho.

AmnUmbm ABCD hm Am¶V Amho ho XmIdmdo bmJUma Amho.

AmVm ∠ PAC = ∠ ACR (l || m) Am{U aofm PQ Zo V¶ma Pmbobo d ì¶wËH«$‘ H$moZ

åhUyZ 12

∠ PAC = 12 ∠ ACR

∠ BAC = ∠ ACD

hr H$moZm§Mr OmoS> ì¶wËHw$‘ H$moZm§Mr OmoS>r ~ZVo

AB Am{U DC aofm AC Zo N>oXë¶m AmhoV d Vo g‘mZ

H$moZ AmhoV.

AB || DCAmH¥$Vr 7.15

Ë¶mMà‘mUo BC || AD (∠ ACB Am{U ∠ CAD ¶m§Mm g‘mZ H$moZ AmhoV)

ABCD hm Mm¡H$moZ g‘m§Va^wO Mm¡H$moZ Amho.

VgoM ∠ PAC + ∠ CAS = 1800 (EH$aofr¶ OmoS>r )

åhUyZ 12

∠ PAC + 12 ∠ CAS =

12 ×1800 = 900

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132 J{UV

qH$dm ∠ BAC +∠ CAD = 900

åhUyZ ∠ BAD = 900

åhUyZ ABCD hm g‘m§Va^wO Mm¡H$moZ Á¶m‘Ü¶o EH$ H$moZ 900 Amho.

åhUyZ ABCD hm Am¶V Amho.

CXmhaU 5 : g‘m§Va^wO Mm¡H$moZmÀ¶m àË¶oH$ H$moZm§À¶m Xþ^mOH$m‘wio Am¶V V¶ma hmoVmo.

CH$b : g‘Om ABCD g‘Om g‘m§Va^wO Mm¡H$moZmÀ¶m ∠A d ∠ B , ∠ B d ∠ C, ∠ C d

∠ D, ∠ D Am{U ∠A ¶m§À¶m H$moZ Xþ^mOH$m§Mo N>oXZ{~Xþ AZwH«$‘o P, Q, R, d S AmhoV .

∆ASD ‘Ü¶o Vwåhr H$m¶ {ZarjU H$amb?

∠D aof DS bm Xþ^mJVo d ∠ A bm AS Xþ^mJVo.

∠ DAS + ∠ ADS = 12 ∠A +

12

∠D

= 12

(∠ A + ∠D) AmH¥$Vr 7.15

= 12 ×1800 (∠A d ∠ D ho N>oXrHo$À¶m EH$mM A§Jmg AgUmao H$moZ AmhoV)

= 900

VgoM ∠DAS + ∠ADS + ∠DSA =1800 ( {ÌH$moUm§À¶m H$moZm§Mr ~oarO)

qH$dm 900 +∠DSA = 1800

qH$dm ∠DSA = 900

VgoM ∠PSR = 900 (∠DSA Mm N>oXZ {dê$ÜX H$moZ)

Ë¶mMà‘mUo ∠APB = 900 qH$dm ∠SPQ = 900 XmI{dVm ¶oB©b

Mm¡H$moZ PQRS ‘Ü¶o àË¶oH$ H$moZ H$mQ>H$moZ Amho. AmVm AmnU {ZîH$f© H$mTy> eH$Vmo H$s hm

Am¶V Amho? XmI{dbo Amho. H$s ∠SPQ = ∠SRQ = 900 Am{U ∠SPQ = ∠SRQ = 900

Ë¶m‘wio g§‘wI ~mOwMo XmoÝhr H$moZ g‘mZ hmoVmV åhUyZ Mm¡H$moZ PQRS hm g‘m§Va^wO Mm¡H$moZ

AgyZ Ë¶mMm àË¶oH$ H$moZ 900 Amho Am{U Ë¶m‘wioM PQRS hm Am¶V Amho.

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Mm¡H$moZ 133

7.5: Mm¡H$moZ hm g‘m§Va^yO Mm¡H$moZ hmoÊ¶mgmR>r BVa AQ>r :

¶m àH$aUm§V Vwåhr g‘m§Va^wO Mm¡H$moZmoÀ¶m JwUY‘mªMm AÜ¶mg Ho$bm Amho Am{U Oa

EImÚm Mm¡H$moZmV H$moUVmhr EH$ JwUY‘© bmJy hmoV Agob Va hmo Mm¡H$moZ g‘m§V^wO Mm¡H$moZ hmoVmo&

AmVm AmnU Xþgè¶m AQ>r{df¶r AÜ¶mg H$ê$ Mm¡H$moZ hm g‘m§Va^wO Mm¡H$moZ hmoÊ¶mgmR>r

H$‘rV H$‘r H$moUVr AQ> Amdí¶H$ Amho.

Vr Imbrb à‘o¶mZwgma gm§{JVbr Amho.

à‘o¶ 7.8 : Mm¡H$moZmÀ¶m g‘w§I ~mOy g‘mZ d g‘m§Va AgVrb Va Vmo g‘m§Va^wO Mm¡H$moZ

hmoVmo..

AmH¥$Vr 7.17 nhm ¶m‘Ü¶o AB = CD Am{U AB || CD

H$U© AC H$mT>m.

∆ ABC ≅ ∆ CDA (~m.~m.~m. {Z¶‘)

BC || AD (H$m?)

¶mgmR>r Imbrb CXmhaU KoD$Z g‘m§Va^wO Mm¡H$moZmMm

JwUY‘© bmJy H$ê$.AmH¥$Vr 7.17

CXmhaU 6 : ABCD g‘m§Va^wO Mm¡H$moZm§V P d Q ho AZwH«$‘o AB d CD ~mOwMo ‘Ü¶q~Xþ

AmhoV.( Am 7.18 nhm)$Oa AQ aof DP $bm S ‘Ü¶o Am{U BQ aof CP bm R ‘Ü¶o N>oXV Agob Va

XmIdm

(i) APCQ hm g‘m§Va^wO Mm¡H$moZ Amho.

(ii) DPBQ hm g‘m§Va^wO Mm¡H$moZ Amho.

(iii) PSQR hm g‘m§Va^wO Mm¡H$moZ Amho.

CH$b : (i) Mm¡H$moZ APCQ ‘Ü¶o AP || QC ( H$maU AB ||

CD åhUyZ) ...(1)AmH¥$Vr 7.18

AP= AB, CQ= CD12

12

(J¥{hV )

VgoM AB = CD (H$m?)

Ë¶m‘wio AP = QC ...(2)

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134 J{UV

(i) d 2 dê$Z APCQ hm g‘m§Va^wO Mm¡H$moZ Amho.

(ii) Ë¶mMà‘mUo DPBQ Mm¡H$moZ g‘m§Va^yO Mm¡H$moZ Amho. H$maU

DQ || PB Am{U DQ = PB

(iii) PSQR Mm¡H$moZm§V SP || QR (DP hr SP Mm ^mJ d QB hm QR Mm ^mJ)

Ë¶mMà‘mUo SQ || PR

åhUyZ PSQR hm g‘m§Va^wO Mm¡H$moZ Amho.

ñdmÜ¶m¶ 7.1

1. EH$m Mm¡H$moZmMo H$moZ 3 : 5 : 9 : 13 ¶m JwUmoÎmamV AmhoV Va H$moZm§Mr ‘mno H$mT>m.

2. Oa Mm¡H$moZmMo H$U© g‘mZ AgVrb Va Vmo Am¶V AgVmoo ho XmIdm.

3. Oa Mm¡H$moZmMo H$U© EH$‘oH$mg H$mQ>H$moZmV Xþ^mJV AgVrb Va Vmo g‘^wO Mm¡H$moZ Amho ho

XmIdm.

4. Mm¡agmMo H$U© g‘mZ AgyZ EH$‘oH$mg H$mQ>H$moZmZ Xþ^mJVmV ho {gÜX H$am.

5. Oa Mm¡H$moZmMo H$U© g‘mZ AgyZ EH$‘oH$m§g H$mQ>H$moZmV Xþ^mJV AgVrb Va Vmo Mm¡ag AgVmo

Ago XmIdm.

6. g‘m§Va^wO Mm¡H$moZ ABCD Mm AC hm H$U© ∠A bm Xþ^mJVmo

(AmH¥$Vr 7.19 nhm).

Va (i) AC hm H$U© ∠ C bm gwÜXm Xþ^JVmo

(ii) ABCD hm g‘^wO Mm¡H$moZ Ago XmIdm

AmH¥$Vr 7.19

7. ABCD hm XmIdm g‘^wO Mm¡H$moZ Amho Va Ago XmIdm H$s AC aofm ∠A d ∠C bm Xþ^mJVo

Am{U BD aofm ∠B d ∠D bm Xþ^mJVo$.

8. ABCD hm Am¶V Amho. AC H$U© ∠A d. ∠C bm Xþ^mJVmo Va

$ (i) ABCD hm Mm¡ag Amho.

(ii) BD H$U© ∠B d ∠D bm Xþ^mJVmo

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Mm¡H$moZ 135

9. ABCD g‘m§Va^wO Mm¡H$moZm‘Ü¶o DP = BQ hmoB©b

Ago H$U© BD da P d Q ho XmoZ q~Xþ KoVbo Am-

hoV (AmH¥$Vr 7.20 nhm.) Va XmIdm H$s, (i) ∆ APD ≅ ∆ CQB (ii) AP = CQ (iii) ∆ AQB ≅ ∆ CPD (iv) AQ = CP

(v) APCQ hm g‘m§Va^yO Mm¡H$moZ Amho.

AmH¥$Vr 7.20

10. ABCD g‘m§Va^yO Mm¡H$moZ Amho. AP d CQ

¶m b§~aofm AZwH«$‘o {eamoq~Xÿ A d C nmgyZ BD

H$m‘©dm H$mT>boë¶m AmhoV (AmH¥$Vr 7.21) Va

Ago XmIdm H$s

(i) ∆ APB ≅ ∆ CQD (ii) AP = CQ AmH¥$Vr 7.21

11. ∆ ABC d ∆ DEF ‘Ü¶o AB = DE, AB ||DE,

BC = EF Am{U BC || EF {eamo~tXÿ AB d C

{eamo~tXÿ D, E d F ¶m§Zm AZwH«$‘ OmoS>bo AmhoV

(AmH¥$Vr 7.22 nhm ). Va XmIdm.

(i) Mm¡H$moZ ABED g‘m§Va^yO Mm¡H$moZ Amho AmH¥$Vr 7.22

(ii) Mm¡H$moZ BEFC g‘m§Va^yO Mm¡H$moZ Amho

(iii) AD || CF Am{U AD = CF

(iv) Mm¡H$moZ ACFD g‘m§Va^wO Mm¡H$moZ Amho. (v) AC = DF (vi) ∆ ABC ≅ ∆ DEF

12. ABCD hm g‘b§~ Mm¡H$moZ Amho. Á¶m‘Ü¶o

AB||CD Am{U AD = BC (Am. 7.23 nhm)Va

Ago XmIdm. (i) ∠A = ∠B (ii) ∠C = ∠D (iii) ∆ ABC ≅ ∆ BAD

(iv) H$U© AC = H$U© BD.AmH¥$Vr 7.23

[ AB dmT>dm Am{U C ‘YyZ aofm Aer H$mT>m H$s DA bm g‘m§Va hmoB©b d AB dmT>dyZ E ‘Ü¶o {‘iob]

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136 J{UV

7.6 ‘Ü¶q~XyMm à‘o¶ (Midpoint Theorem)

Vwåhr {ÌH$moU d Mm¡H$moZmÀ¶m JwUY‘mªMm Aä¶mg Ho$bm Amho AmVm AmUIrZ EH$m JwUY‘m©Mm

Aä¶mg H$ê$ Omo {ÌH$moUmÀ¶m ~mOw§À¶m ‘Ü¶q~Xÿer g§~§YrV Amho.

EH$ {ÌH$moU aMm Am{U Ë¶mÀ¶m XmoZ ~mOy§Mo ‘Ü¶ q~Xÿ

E d F {ZûMV H$am. E Am{U F OmoS>m (AmH¥$Vr 7.24 nhm ).

EF Am{U BC ‘moOm ∠ AEF d∠ABC ‘moOm Vwåhmbm H$m¶

AmT>iVo? Vwåhmbm Ago {XgyZ ¶oB©b H$s

EF = 12 BC Am{U ∠AEF = ∠ABC

Ë¶m‘wio EF || BC {‘iVo AmH¥$Vr 7.24

hrM H¥$Vr doJdoJio {ÌH$moU KoD$Z nS>VmiyZ nhm. Vwåhr Imbrb à‘o¶mda ¶oD$Z nmohmoMmb.

à‘o¶ 7.9 : ∆ À¶m H$moUË¶mhr XmoZ ~mOw§Mo ‘Ü¶{~Xÿ gm§YUmar aofm hr {Vgè¶m ~mOyg g‘m§Va

AgVo. Vwåhr hm à‘o¶ Imbrb nÜXVrZo gmoS>dy eH$Vm.

Am. 7.25 nhm Á¶m‘Ü¶o E Am{U F ho AZwH«$‘ AB

Am{U AC Mo ‘Ü¶q~Xÿ AmhoV d CD || BA.

∆ AEF ≅ ∆ CDF (H$mo. ~m.H$mo. {Z¶‘ )

åhUyZ EF = DF Am{U BE = AE = DC (H$m?)

BCDE hm g‘m§Va^wO Mm¡H$moZ Amho. AmH¥$Vr 7.25

¶mnmgyZ EF || BC

EF = 12

ED = 12

BC. .

à‘o¶ 7.9 Mm ì¶Ë¶mg Vwåhr gm§Jy eH$mb ? ì¶Ë¶mg gË¶ Amho H$m? ¶m à‘o¶mMm ì¶Ë¶mg

gË¶ Agob Va Vmo Vwåhmbm Imbrb à‘o¶mV gm{JVë¶m à‘mUo {XgyZ ¶oB©b

à‘o¶ 7.10 : {ÌH$moUmÀ¶m EH$m ~mOyÀ¶m ‘Ü¶q~XÿVyZ OmUmar gai aofm Oa Xþgè¶m ~mOyg

g‘m§Va Agob Va Vr aofm {Vgè¶m ~mOyg Xþ^mJVo.

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Mm¡H$moZ 137

AmH¥$Vr 7.26 : ‘Ü¶o Ago {XgyZ ¶oVo H$s AB Mm

‘Ü¶{~§Xÿ E AgyZ hr l aofm E ‘YyZ OmVo Am{U BC bm g‘m§Va

Amho. Am{U CM || BA .

∆ AEF Am{U ∆CD F ¶m§Mm EH$ê$nVoMm Cn¶moJ H$ê$Z

AF = CF ho {gÜX H$am.

AmH¥$Vr 7.26

CXmhaU 7 : ∆ ABC ‘Ü¶o AB, BC Am{U CA $ ¶m§Mo ‘Ü¶o q~Xÿ AZwH«$‘o D, E Am{U F $AmhoVm

($AmH¥$Vr 7.27) Va ∆ ABC$Mo Mma EµH$ê$n {ÌH$moUmV {d^mOZ hmoVo Amho XmIdm.

CH$b : ∆ ABC$‘Ü¶o AB Am{U BC $¶m§Mo D Am{U E $ho ‘Ü¶o q~Xÿ AmhoV. à‘o¶ 7.9$Zwgma DE || AC

DF || BC $Am{U EF || AB.

ADEF, BDFE Am{U DFCE ho g‘m§Va^yO

Mm¡H$moZ AmhoV BDFE Mm H$U© DE hm Amho.

∆ BDE ≅ ∆ FED

Ë¶mM à‘mUo ∆ DAF ≅ ∆ FED AmH¥$Vr 7.27

Am{U ∆ EFC ≅ ∆ FED

Mma hr {ÌH$moU EH$ê$n AmhoV.$

CXhaU 8 : l, m Am{U n ¶m g‘m§Va VrZ g‘m§Va aofm§Zm p d q ¶m N>o{XH$mZr Ago N>oXbo Amho

H$s > p N>oXrHo$Mo AB Am{U BC g‘mZ ^mJ hmoVmV Va Ago XmIdm H$s l, m , n ho q N>o{XHo$Mo XmoZ

^mJ H$aVmV. DE Am{U EF .

CH$b : AB = BC . J¥[hV

DE = EF ho {gÜX Ho$bo nmhrOo. AF OmoS>m Vr m aofog G

q~Xÿ N>oXVo. g~§b~ Mm¡H$moZ ACFD ¶m XmoZ {ÌH$moUmV

{d^mJVmo. ∆ ACF Am{U ∆ AFD ∆ ACF ‘Ü¶o B hm AC Mm

‘Ü¶q~Xÿ Amho (AB = BC)

Am{U BG || CF (m || n ) AmH¥$Vr 7.28

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138 J{UV

AF Mm ‘Ü¶q~Xy G Amho (à‘o¶ 7.10 Zwgma )

AmVm ∆ AFD ‘Ü¶o AF Mm ‘Ü¶q~Xÿ G Amho GE||AD Am{U Ë¶m‘wio à‘o¶ 7.10 dê$Z E hm

DF Mm ‘Ü¶ q~Xÿ Amho

DE = EF

Xþgè¶m eãXmV g§mJm¶Mo Pmë¶mg l, m Am{U n d q da g‘mZ ^mJ N>oXVmV.

ñdmÜ¶m¶ 7.2

1. ABCD ¶m Mm¡H$moZmV P, Q, R, d S ho AZwH«$‘o AB, BC, CD

Am{U DA Mo ‘Ü¶ q~Xÿ AmhoV. AC hm H$U© Amho Va Ago

XmIdm H$s,

(i) SR || AC Am{U SR =12 AC

(ii) PQ = SR

(iii) PQRS hm g‘m§Va^wO Mm¡H$moZ Amho.AmH¥$Vr 7.29

2. ABCD hm g‘^wO Mm¡H$moZ Amho Am{U P, Q, R Am{U S ho AZwH«$‘o AB, BC, CD d DA Mo

‘Ü¶{~Xÿ AmhoV. Va PQRS hm Am¶V Amho ho {gÜX H$am.

3. ABCD hm Am¶V Amho. P, Q, R Am{U S ho AZwH«$‘o AB, BC, CD d DA Mo ‘Ü¶{~§Xþ AmhoV

Va PQRS hm g‘^wO Mm¡H$moZ Amho XmIdm.

4. ABCD hm g‘b§~ Mm¡H$moZ Amho Á¶m‘Ü¶o AB || DC, BD H$U© AD Mm ‘Ü¶{~§Xÿ E Amho.E

‘YyZ H$mT>bobr aofm BC bm F ‘Ü¶o N>oXVo (AmH¥$Vr 7.30). Va F hm BC Mm ‘Ü¶ q~Xÿ Amho ho {gÜX H$am.

AmH¥$Vr 7.30

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Mm¡H$moZ 139

5. ABCD ¶m g‘m§Va^wO Mm¡H$moZm§V E d F ho AZwH«$‘o AB d CD Mo ‘Ü¶q~Xÿ AmhoV. (AmH¥$Vr

7.31nhm). Ago XmIdm H$s, aofmI§S> AF d EC ‘wio H$U© BD $bm VrZ g‘mZ ^mJmV {d^mJVmV.

AmH¥$Vr 7.31

6. Mm¡H$moZmÀ¶m {dê$ÜX ~mOw§Mo ‘Ü¶q~Xy gm§YUmao aofmI§S> EH$‘oH$m§g Xþ^mJmVmV ho XmIdm

7. ABC hm ∠C {eamo{~§Xÿer H$mQ>H$moZ {ÌH$moU Amho.H$U© AB Mm ‘Ü¶q~Xÿ M ‘YyZ H$mT>bobm

AgyZ d BC bm g‘m§Va AgUmar aofm AC bm D ‘Ü¶o N>oXVo

Va (i) D hm AC Mm ‘Ü¶{~Xÿ Amho.

(ii) MD ⊥ AC

(iii) CM = MA = 12AB Ago XmIdm

7.7. gmam§e

¶m àH$aUm‘Ü¶o Imbr JmoîQ>tMm Vwåhr Aä¶mg Ho$bm Amho.

1. Mm¡H$moZmÀ¶m gd© H$moZm§Mr ~oarO 3600 AgVo.

2. g‘m§Va^wO Mm¡H$moZmMm H$U© Ë¶mbm XmoZ EH$ê$n {ÌH$moUmV {d^mJVmo.

3. g‘m§Va^yO Mm¡H$moZm§V

(i) {dê$ÜX ~mOy g‘mZ AgVmV.

(ii) {dê$ÜX H$moZ g‘mZ AgVmV.

(iii) H$U© EH$‘oH$m§g Xþ^mJVmV.

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140 J{UV

4. Mm¡H$moZ hm g‘m§Va^wO Mm¡H$moZ hmoVmo Oa,

(i) {dê$ÜX ~mOy g‘mZ AgVrb.

(ii) {dê$ÜX H$moZ g‘mZ AgVrb.

(iii) H$U© EH$‘oH$m§g Xþ^mJV AgVrb.

(iv) {dê$ÜX ~mOw§Mr OmoS>r g‘mZ d g‘m§Va Agob Va.

5. Am¶VmMo H$U© EH$‘oH$m§g Xþ^mJVmV d H$U© g‘mZ AgVmV Ë¶mMm ì¶Ë¶mg gË¶ AgVmo.

6 . g‘^wO Mm¡H$moZmMo H$U© EH$‘oH$m§g H$mQ>H$moZmV Xþ^mJVmV Am{U Ë¶mMm ì¶Ë¶mg gË¶ AgVmo.

7. Mm¡agmMo H$U© EH$‘oH$m§g H$mQ>H$moZmV Xþ^mJVmV d g‘mZ AgVmV Am{U CbQ> {H«$¶m.

8. ∆ À¶m H$moUmË¶mhr XmoZ ~mOyMo ‘Ü¶q~Xÿ gm§YUmar aofm {Vgè¶m ~mOwÀ¶m {Zå‘r AgyZ {Vbm

g‘m§Va AgVo.

9. ∆ À¶m EH$m ~mOyÀ¶m ‘Ü¶mVyZ OmUmar aofm Xþgè¶m ~mOyg g‘m§Va Agob Va Vr {Vgè¶m ~mOyg

Xþ^mJVo.

10. Mm¡H$moZmÀ¶m ~mOy§Mo ‘Ü¶q~Xÿ gm§YyZ V¶ma hmoUmam Mm¡H$moZ g‘m§Va^wO Mm¡H$moZ AgVmo.

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J{UVm_Yrb {gÜXm§V (nwamdo) 141

AZw~§Y - 1

J{UVm_Yrb {gÜXm§V (nwamdo)(Proofs in Mathematics)

A 1.1 n[aM` (Introduction)

g_Om Vw_À`m Hw$Qw>§~mMm ñdV…Mm EH$ ßbm°Q>(^yI§S>)Amho. Ë`mbm g^modVr Hw§$nU KmVbobo Zmhr. Vw_À`m eoOmaÀ`m ßbm°Q> _mbH$mZo Vwåhmb Z H$idVm Amnë`m ßbm°Q> g^modVr Hw§$nU KmVbo. Ë`mMo Hw§$nU KmbyZ Pmë`mZ§Va Vw_À`m bjmV Ambo H$s Vw_À`m ßbm°Q>Mm H$mhr ^mJ Ë`m§À`m Hw§$nU KmVboë`m ßbm°Q>_Üò Jobm Amho. AmVm Vwåhr Vw_À`m ßbm°Q>da A{VH«$_U Pmbo Amho ho H$go {gÜX H$amb? Vw_Mr n{hbr nmàr Aer Agob H$s, JmdmVrb d`ñH$a bmoH$m§Zm ~mobdyZ Ë`m§À`m _XVrZo ßbm°Q>Mr hÔ ~amo~a H$ê$Z KoD$ eH$mb. na§Vw g_Om d`ñH$am§À`m_Üò {ZU©` XoÊ`m§V _VôX Pmbo Va, H$mhr OU åhUVrb Vw_Mo ~amo~a Amho Va H$mhr OU åhUVrb Vw_À`m eoOmaÀ`m§Mo ~amo~a Amho. Ago Pmë`mg Vwåhr H$m` H$amb? Vw_À`mH$‹S>o AmVm EH$M n`m©` Agob H$s Vw_À`m ßbm°Q>Mr hÔ `mo½` hmoB©b Am{U Vo gdmªZm _mÝ` H$amdo bmJob. Vmo n`m©` åhUOo Vw_À`m JmdMm gaH$mar _mÝ`Vm Agbob gìh}jUmMm ZH$mem. Amdí`H$Vm ^mgë`mg H$moQ>m©À`m {Z`_mZwgma hÔr `mo½` hmoVrb Am{U Vw_Mo ~amo~a d eoOmaÀ`m§Mo MwH$ ho {gÜX hmoB©b.

AmVm AmnU Xþgar KQ>Zm nmhÿ. g_Om Vw_À`m AmB©Zo Am°JñQ> 2016 _{hÝ`mMo KamMo {dÚwV ~rb âbobo Amho. na§Vw gßQ>|~a 2016 `m _{hÝ`mMo {dÚwV ~rb Ambo Ë`m_Üò Am°JñQ> _{hÝ`mMo âbobo Zmhr Agm CëboI hmoVm. AmVm Vwåhr {dÚwV _§S>imZo Ho$bobm CëboI MwH$sMm Amho Ago H$go {gÜX H$amb? Vwåhr Am°JñQ> _{hÝ`mMr {~bmMr âbobr nmdVr XmIdyZ Am°JñQ>Mo {~b âbobo Amho Ago {gÜX H$ê$ eH$mb.

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142 J{UV

X¡Z§{XZ OrdZmV Ago AZoH$ àg§J òVmV Ë`mn¡H$s XmoZ CXmhaUo AmnU nm{hbr. H$mhr doim AZoH$ {dYmZo qH$dm Xmdo {gÜX H$ê$Z ~amo~a qH$dm MyH$ {gÜX H$amdr bJVmV.

H$mhr doim ~arMer CXmhaUo H$moUVrhr {gÜXVm Z XoVm Amåhr pñdH$maVmo. na§Vy J{UVm_Üò VH©$emñÌmZwgma {gÜX hmoUmarM {dYmZo MyH$ qH$dm ~amo~a pñdH$maVmo (H$mhr _ybVËdo gmoSy>Z).

J{UVm_Yrb AZoH$ {gÜXVm hOmamo dfm©nmgyZ kmV AmhoV Am{U Ë`m J{UVmÀ`m àËòH$ emIoMo H|$Ð AmhoV. _m{hV Agbobr n{hbr {dídmg{Z` hr {gÜXVm J«rH$ VËddoVm Am{U J{UVk Woëg `m§Zr {Xbr. _ogmonmoQ>°{_`m, B{OßV>, {MZ Am{U ^maV `m gma»`m àmMrZ g§ñH¥$Vt_Üò _Ü` (H|$Ð)J{UV hmoVo. na§Vw Vo AmVmÀ`m gmaIo {gÜXm§V dmnaV hmoVo H$s Zmhr `mMo ñnîQ> nwamdo ZmhrV.

`m àH$aUmV AmnU {dYmZ åhUOo H$m`, J{UVmV `wpŠVdmXmMo H$moUVo àH$ma g_m{dîQ> hmoVmV Am{U J{UVr {gÜXVo_Üò H$moUË`m JmoîQ>r AgVmV ho nmhÿ.

A 1.2. J{UVmZo pñdH$maÊ`m`mo½` {dYmZo (Mathematically Acceptable Statements)

`m {d^mJmV AmnU J{UVmZo pñdH$maÊ`m`mo½` {dYmZm§À`m AWm©Mo ñfîQ>rH$aU H$aÊ`mMm à`ËZ H$ê$. {dYmZ ho dmŠ` AmXoedOm qH$dm AmûM`©H$maH$ ZgVo Am{U Ë`m_wio {dYmZ hm àý ZgVmo¡ !

CXmhaUmW©, ""VwÂ`m Ho$gmMm a§J H$moUVm Amho?'' ho {dYmZ Zmhr, hm àý Amho.

(a) ""H¥$n`m Om Am{U Wmo‹S>o nmUr KoD$Z ò'' hr {dZ§Vr qH$dm AmXoe Amho, {dYmZ Zmhr.

(b) ""ìdm, H$m` gw§Xa gw`m©ñV Amho ! ho EH$ CX²JmadmMH$ dmŠ` Amho. ho {dYmZ Zmhr.

gm_mÝ`nUo, {dYmZ ho Imbrb n¡H$s EH$ Agy eH$Vo."

• Zoh_r gË` (always true)

• Zoh_r AgË` (always false)

• g§{X½Y (_moK_) (ambiguous)

òWo ""g§{X½Y (_moK_)'' eãXmgmR>r Wmo‹S>o ñnîQ>rH$aU H$aÊ`mMr JaO Amho. g§{X½Y {dYmZ H$aÊ`mg XmoZ pñW{V AgVmV. n{hbr pñW{V åhUOo Ooìhm {dYmZ Zoh_r Iao qH$dm Zoh_r MyH$ (AgË`) `mMm {ZU©` KoD$ eH$V Zmhr. CXmhaUmW©, ""CÚm Jwédma Amho'' ho g§{X½Y Amho, {dYmZ ~amo~a {H$ MyH$ `mMm {ZU©` KoÊ`mg _mJMm nwT>Mm g§X^© {Xbobm Zmhr.

g§{X½YVm R>a{dÊ`mg Xþgar pñW{V åhUOo Ooìhm {dYmZ ì`pŠV{ZîR> AgVo. åhUOo H$mhr bmoH$m§Zm {dYmZ Iao Amho Va H$mhrZm ImoQ>o Amho Ago dmQ>Vo CXmhaUmW© ""Hw$Ìo ~wpÜX_mZ AgVmV' ho g§{X½Y Amho. H$maU H$mhr bmoH$$ `mda {dídmg R>odVmV Va H$mhr OU Zmhr.

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CXmhaU 1 : Imbrb {dYmZo Zoh_r Iar (gË`), Zoh_r (MyH$) AgË` qH$dm g§{X½Y (_moK_) AmhoV H$m Vo gm§Jm. Vw_Mr CÎmao nS>Vmim.

(i) EH$m AmR>dS>`mV 8 {Xdg AgVmV.

(ii) òWo nmD$g nS>V Amho.

(iii) npûM_ob gy`m©ñV hmoVmo.

(iv) Jm¡ar EH$ ào_i _wbJr Amho.

(v) XmoZ {df_ nyUmªH$mMm JwUmH$ma g_ AgVmo..

(vi) XmoZ Z¡g{J©H$ g_ g§»`m§Mm JwUmH$ma g_g§»`m òVo.

CH$b :

(i) ho {dYmZ Zoh_r AgË` H$maU EH$m AmR>dS>`m_Üò 7 {Xdg AgVmV.

(ii) ho {dYmZ g§{X½Y Amho, H$maU "òWo' åhUOo H$moR>o ho Zo_Ho$ ñnîQ> hmoV Zmhr.

(iii) ho {dYmZ Zoh_r gË` Amho, H$maU gw`m©ñV Zoh_r npûM_obmM hmoVmo AmnU H$moR>ohr ahmV Agbmo Var.

(iv) ho {dYmZ g§{Y½Y Amho. H$maU ho ì`pŠV{ZîR> {dYmZ Amho. Jm¡ar H$mhr OUm§À`m ~m~VrV à_oi Agob na§Vw H$mhr OUm§Mr ~m~VrV Zgob.

(v) ho {dYmZ Zoh_r AgË` Amho. H$maU XmoZ {df_ g§»`m§Mm JwUmH$ma Zoh_r {df_ g§»`m òVo.

(vi) ho {dYmZ Zoh_r gË` Amho. `mgmR>r Amåhr H$mhr Var H¥$Vr H$aÊ`mMr Amdí`H$Vm Amho. {d^mJ A 1.4 _Üò {gÜX Ho$bo Amho.

_mJo gm§{JVë`mà_mUo, Amnë`m X¡Z§{XZ OrdZmV AmnU {dYmZm§À`m ~§YZH$maVo {df`r \$ma H$miOr H$aV Zmhr. CXmhaUmW©, g_Om Vw_À`m {_Ìm§Zo gm§{JVbo H$s Owb¡ _{hÝ`mV Ho$ai_Yrb _Z§VdmS>r òWo XaamoO nmD$g AgVmo. Vwåhr g§^mì`Vm Yê$Z Ë`mda {dídmg R>odVm. EH$ XmoZ {Xdg nmD$g Zgob Var hr Vwåhr {_Ìmer `mda MMm© H$aV Zmhr.

Xþgao CXmhaU AmnU {H$ËòH$Xm AmnmngmV ~mobVmo Ogo ""AmO \$maM Ja_r Amho?'' Aer {dYmZo AmnU bJoM pñdH$maVmo ho Oar _moK_ dmŠ` Agbo Varhr H$maU Amåhmbm _mJMm-nwT>Mm g§X^© _m{hV AgVmo.

""AmO \$maM Ja_r Amho?'' `mMm doJdoJim AW© Agy eH$Vmo. H$maU Hw$_m§ZÀ`m bmoH$m§Zm OmñV Ja_r dmQ>ob, na§Vy

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144 J{UV

MoÞB©À`m bmoH$m§Zm OmñV Ja_r dmQ>Uma Zmhr.

J{UVr dmŠò g§{X½Y (_moK_) ZgVmV. J{UVr {dYmZo gË` qH$dm AgË` AgVmV Vr \$ŠV pñdH$maÊ`m`moJ qH$dm d¡Y AgVmV.

CXmhaUmW© : 5 + 2 = 7 Zoh_r ~amo~a, åhUyZ 5 + 2 = 7 ho gË` {dYmZ d 5 + 3 = 7 ho AgË` {dYmZ.

CXmhaU 2: Imbrb {dYmZo gË` H$s AgË` Vo gm§Jm.

(i) {ÌH$moUmÀ`m {VÝhr Am§VaH$moZm§Mr ~oarµO 180° AgVo.

(ii) 1 nojm _moR>r AgUmar àËòH$ {df_ g§»`m hr _yi (A{d^mÁ`) g§»`m AgVo.

(iii) x `m H$moUË`mhr dmñVd g§»ògmR>r 4x + x = 5x.

(iv ) x `m àËòH$ dmñVd g§»ògmR>r 2x > x.

(v) x `m àËòH$ dmñVd g§»ògmR>r x2 > x.

(vi) Oa Mm¡H$moZmÀ`m gd© ~mOy g_mZ AgVrb Va, Vmo Mm¡ag AgVmo.

CH$b :

(i) ho {dYmZ gË` Amho. àH$aU 6 _Üò Amåhr {gÜX Ho$bobo Amho..

(ii) ho {dYmZ AgË` Amho. CXmhaUmW© 9 hr _yi g§»`m Zmhr.

(iii) ho {dYmZ gË` Amho.

(iv) ho {dYmZ AgË` Amho, CXmhaUmW© x = -1 Am{U 2 × (-1) = -2 Am{U -2 > -1.

(v) ho {dYmZ AgË` Amho, CXmhaUmW© 12

x = Am{U 21 1

2 4 =

Am{U 14

hm> 12

. nojm _moR>m Zmhr.

(vi) ho {dYmZ AgË` Amho. CXmhaUmW©, g_^wO Mm¡H$moZmÀ`m gd© ~mOy g_mZ AgVmV na§Vw Vmo Mm¡ag ZgVmo.

Vw_À`m bjmV Ambo Agob H$s gm§{JVbobr {dYmZo gË` H$s AgË` R>>a{dÊ`mg J{UVmZo nS>VmiyZ nmhmdr bJVmV åhUyZ

CXm (ii) _Üò 9 hr _yi g§»`m Zmhr. {dYmZ Ago H$s 1 nojm _moR>r AgUmar {df_ g§»`m hr_yi g§»`m AgVo. na§Vw ho gË` Zmhr. Aer {dYmZo AgUmè`m CXmhaUm§Zm {déÕ CXmhaU (counter example) åhUVmV. Amåhr `m~Ôb A 1.5 ^mJmV A{YH$ MMm© H$ê$.

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Vw_À`m ho gwÜXm bjmV Ambo Agob H$s, {dYmZo (iv) ,(v) d (vi) hr {dYmZo AgË` AmhoV. Ë`m§À`mda H$mhr à{V~§Y bmXÿZ AmnU Ë`m§Zm gË` R>ady eH$Vmo.

CXmhaU 5: Imbrb MwH$sMr {dYmZo ~amo~a H$ê$Z nwÝhm gm§Jm

(i) x `m dmñVd g§»ògmR>r 2x > x.

(ii) x `m àËòH$ g§»ògmR>r x2 > x.

(iii) EH$m g§»òbm Ë`mM g§» oZo ^mJë`mg Zoh_r CÎma 1 {_iVo.

(iv) dVw©imV OrdHo$Sy>Z dVw©i n[aKmda hmoUmam H$moZ 90° Mm AgVmo.

(v) Oa Mm¡H$moZmÀ`m gd© ~mOy g_mZ AgVrb Va Vmo Mm¡ag AgVmo.

CH$b :

(i) Oa x > 0 Va 2x > x.

(ii) Oa x < 0 qH$dm x > 1 Va x2 > x.

(iii) H$moUË`mhr g§»òbm 0 (ewÝ`) {edm` Ë`mM g§»òZo ^mJë`mg CÎma 1 {_iVo.

(iv) dVw©imV ì`mgmH$Sy>Z dVw©i n[aKmda hmoUmam H$moZ 90° Mm AgVmo.

(v) Oa Mm¡H$moZmÀ`m gd© ~mOy d gd© Am§VaH$moZ g_mZ AgVrb Va Vmo Mm¡H$moZ Mm¡ag AgVmo.

ñdmÜ`m` A 1.1

1. Imbrb {dYmZo Zoh_r gË`, Zoh_r AgË` qH$dm g§{X½Y AmhoV Vo gm§Jm Vw_Mr CÎmao nS>VmiyZ nhm.

(i) EH$m dfm©_Üò 13 _{hZo AgVmV.

(ii) {Xdmir JmdmMo òUma Amho.

(iii) _mJXrVrb Vmn_mZ 26°C Amho.

(iv) ¨n¥Ïdrbm EH$ M§Ð Amho.

(v) Hw$Ìm CSy> eH$Vmo.

(vi) \o$~«ydmar _{hÝ`mV 28 {Xdg AgVmV.

2. Imbrb {dYmZo gË` qH$dm AgË` Vo gm§Jm. Vw_À`m CÎmam§Zm H$maUo Úm.

(i) Mm¡H$moZmÀ`m gd© Am§VaH$moZm§Mr ~oarO 350° AgVo.

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146 J{UV

(ii) H$moUË`mhr dmñVd g§»`m x gmR>r x2 > 0

(iii) g_^wO Mm¡H$moZ hm g_m§Va^wO Mm¡H$moZ AgVmo.

(iv) XmoZ g_g§»`m§Mr ~oarµO g_g§»`m AgVo.

(v) XmoZ {df_g§»`m§Mr ~oarO {df_g§»`m AgVmV.

3. `mo½` à{V~§Y bmXÿ>Z Imbrb {dYmZo gË` H$ê$Z nwÝhm gm§Jm.

(i) gd© A{d^mÁ` g§»`m {df_g§»`m AgVmV..

(ii) EH$m dmñVd g§»òMr XþßnQ> Zoh_r g_g§»`m AgVo.

(iii) x `m H$moUË`mhr g§»ògmR>r 3x + 1 > 4.

(iv) x `m H$moUË`mhr g§»ògmR>r, x3 > 0.

(v) àËòH$ {ÌH$moUmVrb _Ü`Jm hr H$moZ Xþ^mOH$ gwÜXm AgVo.

A 1.3. AZw_m{ZH$ {dMmagaUr (`wpŠVdmX) (Deductive Reasoning)

EH$m Ag§{X½Y (_moK_) ZgUmè`m {dYmZmMo gË` emoYÊ`mg dmnabobr VH©$ewÜX nÜXV åhUOo AmZw_m{ZH$ {dMmagaUr hmo`. AmZw_m{ZH$ {dMmagaUr H$m` Amho ho g_OyZ KoÊ`mg, Vw_À`mgmR>r EH$ H$mo‹S>o gmoS>{dÊ`mg KoD$.

Vw_À`mH$S>o Mma H$mS>} {Xbobr AmhoV. àËòH$ H$mS>m©da EH$m ~mOybm EH$ g§»`m d Xþgè`m ~mOybm B§J«Or Aja N>mnbobo Amho.

g_Om Vwåhmbm H$mS>m©g§~§Yr Imbrb {Z`_ gm§{JVbo :

""H$mS>m©À`m EH$m ~mOybm Oa g_g§»`m Agob, Va Xþgè`m ~mOybm ñda (vowel) Agob' {Z`_ gË` Amho H$m? ho VnmgÊ`mg H$mS>m©da gdm©V bhmZg§»`m H$moUVr? AWm©VM gd© H$mS>} naVyZ VnmgÊ`mg Vwåhmbm n`m©` Amho. na§Vw R>am{dH$ H$_r H$mS>mªÀ`m g§»òZo ho Vwåhr H$ê$ eH$mb?

bjmV ¿`m, Vwåhmb gm§{JVboM Amho H$s Á`m H$mS>m©À`m EH$m ~mOybm g_ g§»`m Agob Ë`mÀ`m Xþgè`m ~mOwbm ñda Agob. Ago gm§{JVbobo Zmhr H$s H$mS>m©À`m EH$m ~mOybm ñda Agob Va Xþgè`m ~mOybm g_g§»`m AgobM. Vr Agob qH$dm AgUma gwÜXm Zmhr. {Z`_ Agm gwÜXm gm§{JVbobm Zmhr H$s, H$mS>©À`m EH$m ~m§Owbm{df_ g§»`m Agob Va Xþgè`m ~mOwbm ì`§OZ Agob. Vgo Agob gwÜXm qH$dm AgUma nU Zmhr.

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åhUyZ, A Agbobr ~mOw naVÊ`mMr JaO Amho H$m? Zmhr. VoWo Xþgè`m ~mOwbm g_g§»`m qH$dm {df_g§»`m Agob {Z`_ Oar hr bmJy AgVmo.

5 Mm g§»ò~Ôb H$m`? nwÝhm Amåhmbm H$mS>© CbQ>{dÊ`mMr JaO Zmhr H$maU VoWo Xþgè`m ~mOybm ñda qH$dm ì`§OZ Agob {Z`_ VgmM amhrb.

na§Vw Vwåhmbm V qH$dm 6 CbQ>{dÊ`mMr JaO Amho, Oa V À`m Xþgè`m ~mOybm g_g§»`m Agob Va {Z`_ ^§J Ho$ë`mgmaIo hmoB©b VgoM 6 À`m Xþgè`m ~mOybm ì`§OZ Agob Var gwÜXm {Z`_ ^§J Ho$ë`mgmaIo hmoB©b.

ho H$moS>o gmoS>{dÊ`mg Amåhr Á`m V§ÌmMm à`moJ Ho$bm Ë`mbm AZw_m{ZH$ {dMmagaUr åhUVmV. `mb AZw_m{ZH$ åhUVmV H$maU Amåhr {ZH$mbmer (AZw_mZ H$mT>Ê`mn`ªV) qH$dm VH©$emñÌmMm Cn`moJ H$ê$Z nydu àñWmnrV Ho$ë`m Joboë`m {dYmZmnmgyZ EH$ {dYmZ àmßV H$ê$ eH$Vmo. CXmhaUmW© darb H$moS>`mV, VH©$emñÌmÀ`m g§dmXmÀ`m H«$_mZo Amåhr AZw_mZ H$mT>bo H$s Amåhmbm \$ŠV V Am{U 6 Agbobr H$mS>} CbQ>dÊ`mMr Amdí`H$Vm Amho.

AZw_m{ZH$ {dMmagaUr R>am{dH$ {dYmZ gË` H$s AgË` R>am{dÊ`mg _XV H$aVo H$maU gm_mÝ` {dYmZ Iao Amho ho OmUyZ KoÊ`mMr hr {deof ~m~ Amho. CXmhaUmW©, EH$Xm Amåhr XmoZ {df_ g§»`mMm JwUmH$ma {df_ g§»`m òVo ho {gÜX Ho$bo H$s Amåhr bJoM gm§JVmo H$s 70001 × 134563 hm JwUmH$ma {df_ g§»`m òVo. H$maU `m {df_ g§»`m AmhoV.

AZw_m{ZH$ {dMmagaUr AZoH$ eVH$m§nmgyZ _mZdr {dMmam§Mm EH$ A§e AgwZ Am_À`m X¡Z§{XZ OrdZmV Ë`mMm Cn`moJ àËòH$ doir hmoV Ambm Amho. CXmhaUmW© : g_Om {dYmZ H$s gmobm[ag \w$b VoìhmM C_iVo Ooìhm AmXë`m {XderMo Vmn_mZ 28°C nojm OmñV AgVo Am{U H$mën{ZH$ Xar_Üò 15 gßQ>|~a 2005 amoOr \y$b \w$bVo Aer gË` {dYmZo AmhoV. Z§Va AZw_m{ZH$ {dMmagaUrMm Cn`moJ H$ê$Z Agm {ZîH$f© H$mT>Vmo H$s 14 gßQ>|~a 2005 amoOr H$mën{ZH$ Xar_Üò Vmn_mZ 28°C nojm OmñV hmoVo.

XþX£dmZo Amåhr Am_À`m X¡Z§{XZ OrdZmV Zoh_r ~amo~a wpŠVdmX dmnaV Zmhr. AZoH$Xm MwH$sÀ`m `wpŠVdmXm_wio AZoH$ {ZîH$fm©da òVmo. CXmhaUmW©, Oa Vw_Mm {_Ì EH$ {Xdg ôQ>ë`mZ§Va hgbm Zmhr Va Vwåhr g_OVm Vmo _mÂ`m da amJmdbobm Amho. "" Oa Vmo _mÂ`mda amJmdbob Agob åhUyZ Vmo _mÂ`mer hgbm Zmhr.'' ho gwÜXm gË` Agob H$s Ë`mMo S>moHo$ XþIV AgUma åhUyZ Vmo hgbm Zgob.X¡Z§{XZ KQ>Zm_Üò Vwåhr H$mhr {ZîH$fm©da nS>VmiUr H$ê$ eH$V Zmhr H$m? ho {ZîH$f© VH$m©Yma AmYm[aV AmhoV H$s ZmhrV `mMr ImÌr H$am. MwH$sÀ`m VH$m©da AmYm[aV AmhoV H$s ~amo~a VH$m©da AmYm[aV AmhoV `mMr ImÌr H$am.

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ñdmÜ`m` A 1.2 1. Imbrb àíZm§Mr CÎmao XoÊ`mg AmZw_m{ZH$ {dMmagUrMm Cn`moJ H$am.

(i) _mZd hm gñVZ àmUr Amho. gd© gñVZr n¥îR>d§er` AgVmV. m XmoZ {dYmZmda AmYm[aV Vwåhr _mZdm~Ôb H$m` {ZîH$f© H$mTy> eH$mb?

(ii) A°ÝWmoZr hm EH$ Ýhmdr Amho. {XZoeZo Ë`mMo Ho$g H$mnbo. Va A°ÝWmoZrZo {XZoeMo Ho$g H$mnbo Agm {ZîH$f© Vwåhr H$mTy> eH$mb?

(iii) _§Ji J«hmdarb a{hdmem§Mr Or^ bmb Amho. JwbJ hm _§Jimdarb Amho. `m XmoZ {dYmZm§À`m AmYmao JwbJ ~Ôb Vwåhr H$m` {ZîH$f© H$mT>mb?

(iv) EH$m R>am{dH$ {Xder Mma Vmgmnojm OmñV nmD$g nS>bm H$s Xþgè`m {Xder JQ>mar ñdÀN> H$amì`m bmJVmV. AmO 6 Vmg nmD$g n‹S>bm Va JQ>matÀ`m CÚmÀ`m pñWVr~Ôb Vwåhr H$m` {ZîH$f© H$amb?

(v) Imbrb ì`§J{MÌmV JmB©À`m `wŠVrdmXmV H$moUVm {damoYm^mg Amho?

2. nwÝhm EH$Xm Vwåhmbm Mma H$mS>} {Xbr, àË`oH$ H$mS>m©À`m EH$m ~mOybm EH$ g§»`m N>mnbobr Amho Am{U Xþgè`m ~mOybm B§J«Or _wimja N>mnbo Amho. Ë`mn¡H$s \$ŠV XmoZ H$mS>} naVyZ nm{hë`mg Imbrb {Z`_ eŠ` AmhoV H$m`?

""Oa EImÚm H$mS>m©À`m EH$m ~mOybm ì`§OZ Agob Va XþgÀ`m ~mOybm {df_ g§»`m nm{hOo.''

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A 1.4. à_o`, AZw_mZ Am{U _ybVËdo

AmVmn`ªV AmnU H$mhr Am{U Ë`m§Mr {dYrJ«møVm VnmgUo `m~Ôb MMm© Ho$bobr Amho. `m ^mJm_Üò Vwåhr VrZ doJdoJù`m àH$maÀ`m dmŠ`m_Yrb oX H$m` Amho `m~Ôb Aä`mg H$amb. J{UV ho Imbrb VrZ {dYmZmnmgyZ ~Zbobo Amho Vr åhUOo. à_o`, AZw_mZ Am{U _wbVËdo.

AmVmn`ªV Vwåhr H$mhr à_o`m§Mr _m{hVr KoVbobrM Amho. Va gm§Jm à_o` åhUOo H$m`?

CXmhaUmW© Imbrb gd© {dYmZo hr à_o` AmhoV Vo H$go ho ^mJ A 1.5 _Üò nhmb.

à_o` A 1.1 {ÌH$moUmÀ`m {VÝhr Am§VaH$moZm§Mr ~oarO 180°CAgVo.

à_o` A 1.2 XmoZ g_ Z¡g{J©H$ g§»`mMm JwUmH$ma g_ AgVmo.

à_o` A 1.3 H$moUË`mhr VrZ H«$_dma Z¡g{J©H$ g_ g§»`m§À`m JwUmH$mambm 16 Zo {Z…eof ^mJ

OmVmo.

AZw_mZ ho Ago {dYmZ Amho Oo J{UVmMo nyd©kmZ Am{U AZw^d `mdê$ZM _m§S>bo OmVo. åhUOoM

Amnë`m J{UVmVrb A§Vkm©Z ( VmËH$mbrZ kmZmdê$Z) AZw_mZ ho gË` Agob qH$dm AgË` Agob.

Oa AmnU Vo {gÜX Ho$bo Va Ë`mbm à_o` åhUVmV. J{UVk EImXo AZw_mZ KodyZ Ë`mMm Z_wZm nmhÿZ

~wÜXrÀ`mMmVw`m©Zo Ë`mMm VH©$ d AZw_mZmMo à`moJ H$aVmV. AmVm AmnU H$mhr Z_wZo KodyZ Ë`mda

~wÜXrÀ`mMmVw`m©Zo VH©$ H$go H$aVm òVo Vo nmhÿ.

CXmhaU 4: H$moUË`mhr VrZ H«$_mda g_g§»`m ¿`m Am{U Ë`m§Mr ~oarO H$am.

2 + 4 + 6 = 12

4 + 6 + 8 = 18

6 + 8 + 10 = 24

8 + 10 + 12 = 30

20 + 22 + 24 = 66

`m ~oaOodê$Z Vwåhr H$mhr Z_wZo V`ma H$ê$ eH$mb H$m? Ë`m~Ôb Vwåhr H$m` AZw_mZ XoD$

eH$mb?

CH$b : AZw_mZ Ago Agob

(i) VrZ H«$_dma g_g§»`mMr ~oarO hr g_ AgVo.

(ii) VrZ H«$_dma g_g§»`m§À`m ~oaOobm 6 Zo ^mJ OmVmo.

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150 J{UV

CXmhaU 5: g§»`mMm Imbrb Z_wZm Kodw`m Á`mbm nmñH$bMm {ÌH$moU Ago åhUVmV.

7 ì`m Am{U 8 ì`m AmoirVrb A§H$m§À`m ~oaOo~Ôb Vwåhr H$moUVm AZw_mZ H$mT>mb? 21 ì`m AmoirVrb A§H$m§Mr ~oarO {H$Vr Agob? Vwåhr hm Z_wZm nm{hbmV? VH©$ H$ê$Z n ì`m AmoirVrb A§H$m§Mr ~oarO H$mT>Ê`mMo gyÌ V`ma H$am.

CH$b:

7 ì`m AmoirVrb g§»`m§Mr ~oarO = 2 × 32 = 64 = 26

8 ì`m AmoirVrb g§»`m§Mr ~oarO = 2 × 64 = 128 = 27

21 ì`m AmoirVrb g§»`m§Mr ~oarO = 220

n ì`m AmoirVrb g§»`m§Mr ~oarO = 2n - 1

CXmhaU 6: Imbrb Z_wÝ`mH$S>o nhm. g_Om Tn hr EH$ {ÌH$moUr g§»`m Amho..

AmH¥$Vr A 1.1

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AmH¥$VrV XmI{dë`mà_mUo qQ>~m§Mr aMZm {ÌH$moUmÀ`m AmH$mamV Ho$br Amho.

òWo T1 = 1, T2 = 3, T3 = 6, T4 = 10 ............ Am{U BË`m{X Va T5 À`m qH$_VrMm A§XmO H$am? T6 {H$Vr Agob? VgoM Tn {H$Vr Agob?

Tn ~Ôb AZw_mZ H$am.

Vwåhr Imbrb à_mUo AmH¥$Ë`m H$mTy>Z KoVë`mg _XV hmoB©b.

AmH¥$Vr A 1.2

CH$b :

T5 = 1 + 2 + 3 + 4 + 5 = 15 = 5 × 6

2

T6 = 1 + 2 + 3 + 4 + 5 + 6 = 21 = 6 × 7

2

T =× +1

2n

n n( )

AZw_mZ ~ÔbMo EH$ _hËdmMo CXmhaU Amnë`m g_moa òV Amho (Oo AmVmn`ªV gË` {H$ AgË` ho {gÜX Ho$bobo Zmhr) Vo åhUOo JmoëS>~°MMo AZw_mZ. Oo EH$ _hmZ J{UVk {¼pñV`Z JmoëS>~°MÀ`m (1690-1764) Zmdmdê$Z Ambo Amho.

ho AZw_mZ Ago gm§JVo H$s ? 4 nojm _moR>m g_ nyUmªH$ hm XmoZ {df_ _wig§»`mÀ`m ~oaOo_Üò ì`ŠV H$aVm òVmo.

AmnU gwÜXm ho gË` H$s AgË` ho nS>VmiyZ nmhÿ eH$Vm.

Vwåhr H$Xm{MV AmíM`©M{H$V Pmbm Agmb.

H$Xm{MV² J{UVmV Amnë`m g_mao òUmè`m gd© JmoîQ>rMr {gÜXVm XoD$ eH$Vmo. Am{U Oa Zmhr Va> H$m Zmhr?

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152 J{UV

ho Iao Amho {H$ J{UVmVrb àËòH$ joÌ ho H$mhr {dYmZmda Adb§~yZ AgVo Or H$moUË`mhr {gÜXVo {edm` J«mø _mZbr OmVmV. Or ñd`§ {gÜX gË` AmhoV `m {dYmZmZm _ybVËdo åhUVmV. àH$aU - 5 _Üò AmnU `wpŠbS>>Mr _ybVËdo Am{U J¥hrVVËdo `m~Ôb Aä`mg Ho$bm AmhmV. (A{bH$S>o Amåhr _wbVËdo d J¥{hVVËdo `m _Üò ôX H$aV Zmhr).

CXmhaUmW© : `wpŠbS>>À`m n{hë`m J¥{hVVËdmZwgma H$moUVohr XmoZ q~Xÿ Omo‹Sz>Z EH$ gai aofm H$mT>Vm òVo Am{U {Vgè`m J¥{hVËdmZwgma EImXm gai aofmI§S> {Xbm AgVm aofmI§S> {h {ÌÁ`m Am{U Ë`mMm A§Ë`q~Xÿ _Ü` KodyZ dVw©i H$mT>Vm òVo.

darb {dYmZo nyU©nUo gË` AmhoV Am{U wpŠbS>Zo Vr gË` AmhoV Ago J«mø _mZbo. H$m? H$maU gd© H$mhr Amåhr {gÜX H$ê$ eH$V Zmhr. Am{U AmnU H$moRy>Z Var gwédmV {h Ho$brM nm{hOo. H$mhr {dYmZo Iar AmhoV Ago Amåhr KoVbo nm{hOo Am{U Amnbo kmZ ho VH©$emñÌmÀ`m {Z`_mZwgma _yb-VËdmMm dmna H$ê$Z _O~yV Ho$bo nm{hOo.

AmnU AmíM`©M{H$V Pmbm Agmb H$s {h {dYmZo ñd`§{gÜX AgyZ gwÜXm AmnU Ë`mM gd© {dYmZm§Zm Iar åhUyZ H$m pñdH$mé eH$V Zmhr. `mbm AZoH$ H$maUo AmhoV. AZoH$Xm Amnbo VH©$kmZ MwH$sMo Agy eH$Vo. {MÌo qH$dm Z_wZo \$gdy eH$VmV Am{U ImÌrZo EH$M n`m©` AgVmo H$s Vo {gÜX H$aÊ`mgmR>r H$mhrVar gË` AgVo. CXmhaUmW©, Amåhr gd©OU ho _mZVmo H$s EH$m A§H$mZo Xÿgè`m A§H$mbm JwUbo AgVm oUmam JwUmH$ma hm Ë`m XmoÝhr A§H$mnojm _moR>m AgVmo nU Amåhmbm _mhrV Amho H$s ho Zoh_rM Iao ZgVo CXmhaUmW©, 5 × 0.2 = 1 òWo 1 hm 5 nojm bhmZ Amho.

AmH¥$Vr A 1.3 nhm. AB qH$dm CD `mn¡H$s

H$moUVm aofmI§S> bm§~ Amho?

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AmH¥$Vr A 1.3

òWo XmoÝhr aofmI§S> ho g_mZ bm§~rMo AmhoV, VarnU AB hm bhmZ {XgVmo.

_wbVËdm§À`m gË`nUm~Ôb Vwåhm§bm AmíM`© dmQ>V Agob na§Vy _wbVËdo hr Amnë`m Am§Vkm©ZmMm AmYma KoD$Z d Ë`m§À`m ñd`§{gÜX _m{hVrdê$Z {ZdS>br OmVmV. Ë`m_wio AmnU _wbVËdm§Zm gË` åhUyZ pñdH$maVmo. Ago AgwZgwÜXm AmnUmg Ago AmT>iyZ Ambo H$s EImXo _wbVËd MwH$sMo Agy eH$Vo. `meŠ`VoÀ`m {déÜX ~Mmd H$m`? Amåhr nwT>rb H«$_ KoVmo.

(i) _wbVËdm_Üò A{V Imobda Z OmVm gÜ` n[apñWVrV \$ŠV _ybVËdo Am{U `wpŠbS>Mr nmM J¥hrVVËdo KodyZ Amhr eoH$S>mo à_o` {gÜX H$ê$ eH$Vmo.

(ii) ho {ZpíMV H$am H$s _ybVËdo {h gwg§JV AmhoV.

Amåhr Ago g_Oy {H$ gd© _ybVËdo {h gwg§JV ZmhrV, Oa Amåhr EH$ _wbVËd ho dmnê$Z Xÿgao _ybVËd MyH$sMo Amho ho Xe©{dbo Va. CXmhaUmW© : Imbrb XmoZ {dYmZo bjmV KodyZ Amåhr _ybVËdo {h gwg§JV ZmhrV ho XmIdy eH$Vmo.

{dYmZ 1 : H$moUVrhr nyU© g§»`m hr {VÀ`m nwT>rb g§»ònojm _moR>r ZgVo.

{dYmZ 2 : EH$m nyU© g§»òbm eyÝ`mZo ^mJbo AgVm EH$ nyU© g§»`mM {_iVo.

(bjmV R>odm, eyÝ`mZo ^mJbo AgVm ( ∞ ) ì`m»`m Z H$aUmao CVa òVo. nU H$m§hr jUmnwaVo AmnU ho gË` _mZyZ H$m` hmoVo Vo nmhÿ).

{dYmZ 2 MwH$sMo _ybVËd ho AmVm qH$dm Z§Va ZŠH$sM CÎmambm {damoY Xe©{dVo. Amåhr Ago _mZy H$s ZŠH$sM òWo {damoY Amho, Ooìhm EH$ {dYmZ Am{U Ë`mMoM ZH$mamË_H$ {dYmZ {h XmoÝhr {dYmZo gË` AmT>iVmV.

CXmhaUmW© : darb {dYmZ 1 Am{U {dYmZ 2 nwÝhm ¿`m.

{dYmZ 1 dê$Z Amåhr 2 ≠ 1 ho {gÜX H$ê$ eH$Vmo.

AmVm x2 -x2 Mo Ad`d Imbrb XmoZ nÜXVrZo Kody.

(i) x2 -x2 = x (x - x) Am{U

(ii) (x2 -x2) = (x + x) (x - x) | (a + b) (a - b) = a2 - b2 Zwgma

åhUyZ x(x -x) = (x + x) (x - x)

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154 J{UV

{dYmZ 2 dê$Z XmoÝhr ~mOy_Yrb (x – x) Zmhrgo H$ê$Z

x = 2x ho {_iVo

`mdê$Z ho {gÜX hmoVo {H$ 2 = 1.

åhUOoM Amnë`mH$‹S>o 2 ≠ 1 Am{U Ë`mMo ZH$mamË_H$ 2 = 1

{h XmoÝhr {dYmZo gË` AmhoV. hmM {damoYm^mg Amho. MwH$sMo _ybVËd KoVb`m_wio hm {damoYm^mg CX²^dbm. Oo nyU© g§»òbm eyÝ`mZo ^mJbo AgVm nyU© g§»`m§M {_iVo.

åhUyZ Amåhr Or {dYmZo _ybVËdo åhUyZ KoVmo Ë`m§À`m ~m~VrV ânya {dMma {d{Z_` Am{U gwú_ÑpîQ> nm{hOo. AmnU ho ZŠH$s Ho$bo nm{hOo H$s {h {dYmZo AgmVË` qH$dm VH©$ewÕ {damoYm^mg Xe©{dUma ZmhrV. {edm` _ybVËdmMr {ZdS> {h H$mhr doim Amåhmbm Z{dZ emoYmH$S>o KodyZ OmVmV. àH$aU 5 _Üò AmnUmbm 5 ì`m J¥{hVVËdm~Ôb VgoM `wŠbrS>À`m ^y{_Vr ~Ôb _m{hVr AmhoM. AmnU nm{hbmM AmhmV H$s J{UVkm§À`m _Vo nmMdo J¥{hVVËd ho J¥{hVVËd ZgyZ Vmo EH$ à_o` Amho Omo Mm¡Ï`m J¥hrVVËdmÀ`m ghmæ`mZo {gÜX H$aVm òVmo. AX²^yV[aË`m `m gd© à`ËZmVyZ `wŠbrS>À`m ^y{_VrMm emoY bmJbm.

`m ^mJmÀ`m eodQ>r nwÝhm EH$Xm _ybVËd, à_o` Am{U AZw_mZ `m _Yrb \$aH$m~Ôb COiUr H$ê$`m.

J{UVr {dYmZo Or ñd`§{gÕ AmhoV Am{U H$moUË`mhr nwamì`m{edm` J«mø _mZbr OmVmV Ë`mZm _ybVËdo åhUVmV.

J{UVr {dYmZo, Á`m§Mr gË`Vm {h AOyZ {gÜX ìhm`Mr Amho. Ë`m§Zm J¥{hVVËdo åhUVmV. Am{U Oo J{UVr {dYmZ VH©$ewÜX nÜXVrZo {gÜX H$aVm òVo Ë`mbm à_o` åhUVmV.

ñdmÜ`m` A 1.3 1. H$moUË`mhr VrZ H«$_dma g_g§»`m KoD$Z Ë`m§Mm JwUmH$ma H$am. CXmhaUmW© 2 × 4 × 6 = 48,

4 × 6 × 8 = 192 BË`mXr Aem JwUmH$mam~Ôb VrZ AZw_mZ V`ma H$am.

2. nwÝhm EH$Xm nmñH$bMm {ÌH$moU ¿`m.

n{hbr Amoi : 1 = 110

Xþgar Amoi : 11 = 111

{Vgar Amoi : 121 = 112

Mm¡Wr Amoi Am{U nmMdr Amoi `m~Ôb AZw_mZ V`ma H$am. AgoM AZw_mZ V`ma hmoD$ eH$Vo H$m nhm. Vw_Mo AZw_mZ 6 ì`m Amoirbmhr bmJy nS>Vo H$m ?

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3. nwÝhm EH$Xm (AmH¥$Vr A-1.2 ) {ÌH$moUr g§»`m nmhÿ. XmoZ g§b½Z {ÌH$moUr g§»`m§Mr ~oarO H$am CXm : T1 + T2 = 4, T2 + T3 = 9, T3+ T4 = 16 Va T4 + T5 Mr qH$_V {H$Vr? , Tn - 1 + Tn

~Ôb AZw_mZ V`ma H$am. 4. Imbrb Z_wZm nhm. 12 = 1 112 = 121 1112 = 12321 11112 = 1234321 111112 = 123454321 Imbrb àËòH$mMo AZw_mZ V`ma H$am.

111111

1111111

2

2

............

ho AZw_mZ gË` Amho H$m? ho Vnmgm.

5. `m nwñVH$mV dmnaboë`m nmM _ybVËdm§Mr `mXr H$am.

A 1.5 J{UVr {gÜXVm åhUOo H$m`?

AmVm AmnU {gÜXVo~ÔbÀ`m {d{dY ~mOy nmhÿ. àW_ AmnU nS>Vmim Am{U {gÜXVm `m _Yrb \$aH$ g_OmdyZ KoD$. `mnwdu Vwåhr J{UVmVrb H$mhr {gÜXVm {eH$bmV, _y»`Ëdo Vwåhmbm H$mhr {dYmZo nS>VmiÊ`mg gm§{JVbr OmVmV.

CXmhaUmW© : H$moUË`mhr XmoZ g_ g§»`m§Mm JwUmH$ma g_ AgVmo.

ho H$mhr g§»`m KoD$Z nS>VmiÊ`mg gm§{JVbo AgVo. åhUyZ Vwåhr H$moUË`mhr XmoZ g_ g§»`m ghOnUo {ZdSy> eH$Vm.

g_Om 24 Am{U 2006 `m§Mm JwUmH$ma Ho$ë`mg.

24 × 2006 = 48144 hr g_g§»`m {_iVo. AmVmn`ªV AemàH$maMr AZoH$ CXmhaUo Vwåhr Ho$br AgUmaM.

VgoM Vwåhmbm AZoH$ {ÌH$moU aMyZ Ë`m_Yrb Am§VaH$moZm§Mr ~oarO H$aÊ`mMm CnH«$_ dJm©_Üò H$am`bm gm§{JVbm hmoVm Agob. _moOÊ`mVrb H$mhr CUrdm dJiVm, {ÌH$moUmÀ`m {VÝhr Am§VaH$moZm§Mr ~oarO 180° òVo ho AmT>iyZ Ambo AgobM.

`m nÜXVr_Üò H$m` CUrd Amho? nS>VmiÊ`mÀ`m `m nÜXVr_Üò AZoH$ g_ñ`m AmhoV Am{U {Xbobo {dYmZ gË` Amho `mda {dídmg R>odÊ`mg, hoM Amnë`mbm _XV H$aVo nU AmnU Var gwÜXm ImÌrnyd©H$ gm§Jy eH$V Zmhr H$s Vo gd© KQ>Zm§gmR>r gË` Amho.

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156 J{UV

CXmhaUmW© : AZoH$ g_g§»`m§À`m OmoS>`m§Mm JwUmH$mamdê$Z AmnU ho AZw_mZ H$ê$ eH$Vmo XmoZ g_g§»`m§Mm JwUmH$ma hm g_g§»`mM AgVmo. VWmnr XmoZ g_ g§»`m§Mm JwUmH$ma g_ g§»`m AgVo `mMr ImÌr Zmhr. ^m¡{VH$[aË`m Vwåhr eŠ` Agboë`m gd© g_g§»`m§À`m OmoS>çm§Mm JwUmH$ma H$aVmZm Vnmgy eH$V Zmhr. Oa Vwåhr ì`§J{MÌm_Yrb _wbr gmaIo H$arV Agmb Va Vwåhr Vw_À`m Cd©[aV OrdZmV g_g§»`mÀ`m OmoS>`m§Mm JwU>mH$ma H$arV ahmb. `mà_mUoM Vwåhr AOyZ Ago {ÌH$moU aMbm Zgmb Á`m§À`m {VÝhr Am§VaH$moZm§Mr ~oarO 180° n`©V `oV Zmhr. eŠ` Agboë`m gd© {ÌH$moUm§À`m Am§VaH$moZm§Mr ~oarO H$ê$Z _moOy eH$V Zmhr.

`m {edm` gwadmVrbm nS>bmiUo ho AZoH$Xm MwH$sÀ`m _mJm©Zo hmoVo. CXmhaUmW©, nmñH$bÀ`m {ÌH$moUmdê$Z AZw_mZ H$aÊ`mMr BÀN>m Pmbr (ñdmÜ`m` A1.3 _Yrb à.2). Am{U _mJrb nS>VmiUr dê$Z 115 ⇒ 15101051 nU dmñVdmV 115 = 161051 Amho. åhUyZ Amåhmbm H$mhr KQ>Zm§À`m ~m~VrV nS>VmiUrda Adb§~yZ amhVm oV Zmhr. m{R>H$mUr AmUIr EH$ nÜXV Amho Vr åhUOo {dYmZmMr {gÜXVm XoUo.

VH©$ewÜX VËdm§Mm dmna H$ê$Z J{UVr {dYmZmMr gË`Vm {gÜX H$aÊ`mÀ`m nÜXVrbm "J{UVr {gÜXVm' Ago åhUVmV.

KQ>H$ A-1.2 _Yrb CXmhaU 2 gmoS>{dVmZm Vwåhr nmhÿ eH$Vm {H$ J{UVr {dYmZ AgË` Ago åhUVmV. ho EH$ {déÜX CXmhaU V`ma H$aÊ`mg Amdí`H$ AgVo.

åhUyZ Oar hOmamo KQ>Zm nS>VmiyZ nm{hbo Var J{UVr {dYmZmMr g~bVm {gÜX H$ê$ eH$V Zmhr. Voìhm EH$M {déÜX CXmhaU nwaogo AgVo Á`m_wio J{UVr MwH$sMo R>ê$ eH$Vo hm EH$M _wÔm AË`§V _hËdmMm Amho.

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J{UVr {dYmZ Amho ho XmI{dÊ`mgmR>r EH$M àË`wXmhaU CXmhaU emoYUo nyaogo Amho. åhUyZ (7 + 5) = 12 ho EH$ Ago {déÜX CXmhaU Amho Oo {gÜX H$aV Zmhr H$s XmoZ {df_ g§»`m§Mr ~oarO {df_ g§»`m AgVo.

AmVm AmnU {gÜXVo_Üò Agboë`m _yb^yV KQ>H$m§Mr `mXr H$ê$.

(i) à_o`mMr {gÜXVm XoÊ`mgmR>r AmnUmnmer EH$ ñWyb {dMma Agmdm H$s Ë`mMr H$er gwédmV H$amdr `m~Ôb WmoS>Š`mV _m{hVr Agmdr.

(ii) à_o`m_Üò àW_ {Xbobr _m{hVr (J¥hrV) ì`dñWrV g_OyZ KoD$Z dmnaVm `mdr.

CXmhaUmW© : à_o` A 1.2 Mr Om{Và{Vkm H$moUË`m{h XmoZ g_ g§»`m§Mm JwUmH$ma hm g_g§»`m AgVo åhUOoM Amåhmbm XmoZ g_ Z¡g{J©H$ g§»`m {Xboë`m AgVrb Va Amåhr Ë`m§À`m JwUY_m©Mm dmna Ho$bm nm{hOo.

Ad`d à_o` _Üò (àH$aU 2 _Üò) Vwåhmbm EH$ ~hÿnXr {Xbobr Amho Am{U gm§{JVbo Amho {H$ p(a) = 0 ho dmnê$Z Vwåhmbm (x – a) hm p (x) Mm Ad`d Amho ho XmIdm`Mo Amho.

Ë`mà_mUoM Ad`d à_o`mÀ`m ì`Ë`mg (converse) gwÜXm Oa (x – a) hm p (x) Mm Ad`d Agob Va darb J¥{hVH$ dmnê$Z Vwåhmbm p(a) = 0 ho {gÜX H$amd`mMo AgVo.

à_o`mMr {gÜXVm _m§S>VmZm H$mhr doiobm Vwåhmbm aMZm gwÜXm H$amdr bmJVo. CXmhaUmW© : {ÌH$moUmÀ`m {VÝhr Am§VaH$moZm§Mr ~oarO 180° Amho ho {gÜX H$aÊ`mgmR>r darb {eamoq~XÿVyZ nm`mbm g_m§Va aofm H$mTy>Z g_m§Va aofm§Mo JwUY_© dmnê$ eH$Vm.

(iii) {gÜXVm_Üò AmnU AZoH$ J{UVr {dYmZo H«$_mZo _m§Sy> eH$Vmo {gÜXVo _Yrb àËòH$ J{UVr {dYmZ ho {gÜXVo_ÜòM _mJo VH©$ewÜX nÜXVrZo V`ma Ho$boë`m {dYmZmdê$Z, qH$dm nyduÀ`m à_o`mÀ`m {gÜXVodê$Z qH$dm _ybVËd qH$dm J¥{hVH$mdê$Z _m§Sy> eH$Vm.

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158 J{UV

(iv) à_o`m_Üò Or JmoîQ> {gÜX H$amd`mMr Amho Vr {gÜX H$aÊ`mgmR>r. J{UVmVrb gË` {dYmZm§Mo {ZîH$f© ho VH©$ewÜX nÜXVrZo H«$_mZo _m§Sy>Z H$ê$ eH$Vmo.

darb gd© KQ>H$m§Mm Aä`mg H$aÊ`mgmR>r Amåhr à_o` A 1.1 Am{U Ë`mMr {gÜXVm `mMo n¥W…H$aU nmhÿ. hm à_o` Vwåhr `m nydu àH$aU 6 _Üò {eH$bm AmhmVM. à_o`mMr {gÜXVm _m§S>Ê`mgmR>r AmH¥$Vr H$mT>Uo \$ma _hËdmMo Amho. VWm{n {gÜXVo_Yrb àËòH$ {dYmZ ho VH©$ H$ê$ZM {gÜX H$amdo bmJVo. AZoH$Xm Amåhr Ago EoH$bo Amho {H$ {dÚmWu Aer {dYmZo H$aVmV H$s$ ho XmoZ H$moZ g_mZ AmhoV H$maU Vo AmH¥$VrV g_mZ {XgV AmhoV. qH$dm Vmo H$moZ 90° Mm Amho H$maU Ë`m XmoZ ~mOy EH$_oH$m§Zm b§~ AmhoV Aem {XgV AmhoV. AemàH$mao {XgÊ`mdê$Z \$gì`m JmoîQ>rnmgyZ gmdY ahm. (bjmV R>odm AmH¥${V A 1.3)

åhUyZ AmVm AmnU à_o` A 1.1 nmhÿ.

à_o` A1.1: {ÌH$moUmÀ`m {VÝhr Am§Va H$moZm§Mr ~oarO 180° AgVo.

{gÜXVm: ABC hm EH$ {ÌH$moU Amho (Am. A 1.4 nhm)

Amåhmbm ABC BCA CAB+ + = °180 Amho Ago {gÜX Ho$bo nmhrOo.

...... (1)

AmH¥$Vr A1.4

A _YyZ OmUmar BC bm g_m§Va AgUmar DE hr aofm H$mT>m. ...... (2)

DE aofm BC bm g_m§Va Amho Am{U AB {h N>o{XH$m Amho åhUyZ DAB Am{U ABC ho ì`wËH«$_ H$moZ AmhoV.

åhUyZ àH$aU 6 _Yrb à_o` 6.2 dê$Z Vo g_mZ AmhoV.

åhUOoM DAB ABC= ...... (3)

Ë`m à_mUoM CAE ACB= ...... (4)

`mdê$Z ABC BAC ACB DAB BAC CAE+ + = + + ...... (5)

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nU DAB BAC CAE+ + = °180 H$maU ho gai aofm V`ma H$aVmV ...... (6)

AmVm AmnU {gÜXVo_Yrb àËòH$ nmàrda MMm© H$ê$`m.

nmàr (1) : Amnbm à_o` hm {ÌH$moUmÀ`m JwUY_m© g§~§{YV Amho åhUyZ AmnU {ÌH$moUmnmgyZ gwédmV H$ê$`m.‹

nmàr (2) : hm nwT>rb JmoîQ>rVrb _m{hVr XoUmam EH$ _wÔm Amho qH$dm `mdê$Z à_o` {gÜX H$aÊ`mgmR>r H$go nwT>o Om`Mo `m ~Ôb kmZ XoVo. AZoH$Xm ^m¡{_VrH$ {gÜXVm XoVmZm aMZoMr JaO AgVo.

nmàr (3) Am{U (4) : g_m§Va aofm DE Am{U BC `m§À`m dñVwpñW{V dê$Z Amåhr hm {ZîH$f© H$mTy> eH$Vmo H$s

DAE ABC= Am{U CAE ACB= (aMZm) Am{U nwdu {gÜX Ho$bobm à_o` 6.2 Á`mMr Om{Và{Vkm Amho XmoZ g_m§Va aofm§Zm EH$ N>o{XH$m N>oXV Agob Va V`ma hmoUmao ì`wËH«$_ H$moZ g_mZ AgVmV.

nmàr 5 : `m {R>H$mUr Amåhr `yH${bS>Mo _ybVËd dmnê$ (àH$aU 5 nhm) Á`mMr ì`m»`m Amho g_mZ KQ>H$mV g_mZ KQ>H$ {_i{dë`mg CÎma g_mZ òVo. Oo {gÜX H$aVo ABC BAC ACB DAB BAC CAE+ + = + + = °180

åhUOoM {ÌH$moUmÀ`m {VÝhr Am§VaH$moZm§Mr ~oarO hr EH$ aofoda V`ma hmoUmè`m H$moZm§À`m ~oaOo BVH$s AgVo.

nmàr 6 : `m {R>H$mUr Amåhr g§b½Z H$moZm§Mr Omo‹S>r Mo _ybVËd àH$aU 6 _Yrb dmnad eH$Vmo. Á`mMr ì`m»`m Amho.

EH$m gai aofoda V`ma hmoUmÀ`m gd© H$moZm§Mr ~oarO 180° AgVo.

Omo {gÕ H$aVo ABC BAC ACB DAB BAC CAE+ + = + + = °180

nmàr 7 : òWo `wŠbrS>Mo _ybVËd dmnê$ eH$Vmo Á`mMr ì`m»`m Amho.

Oo KQ>H$ EH$mM KQ>H$mer g_mZ AgVmV Vo nañnam§er g_mZ AgVmV. `mdê$Z {ZîH$f© {_iVmo.

ABC BAC ACB DAB BAC CAE+ + = + + = °180

gyMZm bjmV R>odm nmàr 7 _Yrb VËd à_o`m_Üò dmnaVmo Vmo à_o` {gÜX Ho$bm Amho.

AmVm AmnU A 1.2 Am{U A 1.3 _Yrb ho XmoZ à_o` H$moUË`mhr n¥W…H$aUm {edm` {gÜX H$a eH$Vmo .

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160 J{UV

à_o` : A 1.2

H$moUË`mhr XmoZ g_g§»`m§Mm JwUmH$ma g_g§»`m§M AgVo.

{gÜXVm : g_Om x Am{U y `m XmoZ g_ g§»`m AmhoV. Amåhmbm ho {gÜX H$amd`mMo Amho {H$ x y hr g_g§»`m Amho.

x Am{U y m g_g§»`m AmhoV åhUyZ Ë`mZm 2 Zo mJ OmVmo. åhUyZ Ë`m§Zm x = 2m Am{U y = 2n ñdê$nmV _m§Sy> eH$Vmo. OoWo m d n `m H$moUË`mhr Z¡gJuH$ g§»`m AmhoV.

AmVm xy = 4mn

4mn bm 2 Zo ^mJ OmVmo åhUyZ x y bm 2 Zo ^mJ OmVmo

\x y hr g_g§»`m Amho.

à_o` : A 1.3 H$moUË`mhr VrZ H«$_dma Z¡g{J©H$ g_ g§»`mÀ`m JwUmH$mambmbm 16 Zo ^mJ OmVmo.

{gÜXVm: H$moUË`mhr VrZ H«$_dma g_ g§»`m 2n, 2n + 2 Am{U 2n + 4 `m ñdê$nmV Agy eH$VmV Á`m_Ü`o n hr H$moUVrhr Z¡g{J©H$ g§»`m Amho. Amåhmbm ho {gÕ H$amd`mMo Amho {H$ `m§Mm JwUmH$ma (2n) × (2n + 2) × (2n + 4) `m bm 16 Zo ^mJ OmVmo.

(2n) × (2n + 2) × (2n + 4) = 2n × 2 (n + 1) × 2 (n + 2) = 2 × 2 × 2n ( n + 1) (n + 2) 8n (n + 1) (n + 2)

AmVm Amnë`mH$S>o XmoZ KQ>Zm AmhoV. n {h g_ g§»`m Agob qH$dm {df_. AmVm AmnU àË`oH$ KQ>ZoMm Vnmg H$ê$.

g_Om n {h g_ g§»`m Amho :

åhUyZ Amåhr {bhÿ eH$Vmo n = 2m OoWo m hr EH$ Z¡g{J©H$ g§»`m Amho.

Am{U AmVm

2n × (2n + 2) (2n + 4) = 8(2m) (2m + 1) (2m + 2) = 16 m (2m +1) (2m + 2)

åhUyZ 2n (2n + 2) (2n + 4) bm 16 Zo ^mJ OmVmo.

nwT>o Oa n {h {df_ g§»`m Amho Va n + 1 {h {df_ g§»`m Amho.

åhUyZ Amåhr {bhÿ eH$Vmo n + 1 = 2r OoWo r {h H$moUVrhr Z¡gJuH$ g§»`m Amho.

\ åhUyZ 2n (2n + 2) (2n + 4) = 8n ( n + 1) (n + 2) = 8(2r –1) × 2r × (2r + 1) = 16r (2r – 1) (2r +1)

åhUyZ 2n (2n + 2) (2n + 4) bm 16 Zo ^mJ OmVmo.

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åhUyZ XmoÝhr KQ>Zm_Üò Amåhr {gÜX Ho$bo Amho {H$ H$moUË`mhr VrZ H«$_dma g_g§»`m§À`m JwUmH$mambm 16 Zo ^mJ OmVmo.

`mdê$Z Amåhr H$mhr {ZîH$f© ì`ŠV H$ê$ eH$Vmo {H$ Am¡nMm[aH$[aË`m {gÜXVm {bhUo Am{U J{UVmÀ`m nÜXVrZo {gÜXVm V`ma H$aUo `m_Yrb \$aH$ H$gm AgVmo. da Z_yX Ho$ë`m à_mUo àËòH$ {gÜXVobm EImXo H$maU AgVoM (H$mhr doiobm EH$mnojm OmñV). J{UVkmOdi Agbobo A§V…km©Z hoM EH$ H|$Ðq~Xÿ Amho Oo {dMma H$am`bm Am{U {ZîH$f© {ZpíMV H$am`bm _XV H$aVo. AZoH$Xm ào_o`mMr {gÜXVm {bhVmZm J{UVVkmOdi AZoH$ JmoîQ>r AgVmV nU Ë`m `mo½` H«$_mV ZgVmV. nU `mo½` {gÕVm {b{hV AgVmZm J{UVVk Amnë`m VH©$kmZmMm dmna H$ê$Z. ânya {dMma H$aV Am{U CXmhaUo KodyZ `mo½` H«$_mV _m§S>Ê`mMm à`moJ H$aVmo. Amnë`mH$S>o Agboë`m gO©Zerb H$boMm dmna H$ê$Z hmo gd© dmX {ddmXmË_H dmŠò EH$ÌrH$ê$Z {b{hë`mZ§VaM `mo½` {gÜXVm {_iVo.

Wmoa ^maVr` J{UVVk lr{Zdmg am_mZwOZ `m§Zr AZoH$ CÀM à{VÀ`m Šb¥ßË`m dmnê$Z AZoH$ J{UVr {dYmZo V`ma H$ê$Z R>odbr AmhoV Am{U Vr gË` AmhoV Agm Ë`mZr Xmdm Ho$bm Amho. Ë`mn¡H$s AZoH$ gË` AmhoV Ago {gÜX Pmbo AgyZ Vr à{gÜX à_o` åhUyZ Amåhr pñdH$mabo Amho. Var gwÜXm AmO hr g§nyU© OJâmVrb H$mhr J{UVk `m§Mo ho AZw_mZ gË` (qH$dm AgË`) R>a{dÊ`mgmR>r à`ËZ H$arV AmhoV.

ñdmÜ`m` – A 1.4

1. Imbrb {dYmZm§Zm AgË` {gÜX H$aÊ`mgmR>r {damoYr CXmhaUo (àË`wXmhaUo) Úm.

(i) Oa XmoZ {ÌH$moUmMo g§JV H$moZ g_mZ AgVrb Va Vo XmoZ {ÌH$moU EH$ê$n AgVmV.

(ii) Mmar ~mOy g_mZ AgUmè`m Mm¡H$moZmbm Mm¡ag åhUVmV.

(iii) Mmar H$moZ g_mZ AgUmè`m Mm¡H$moZmbm Mm¡ag åhUVmV.

(iv) Oa a Am{U b ho H$moUVohr XmoZ nyUmªH$ AgVrb Va 2 2+ = +a b a b

(v) n hr H$moUVrhr nyU© g§»`m Agob Va 2n2 + 11 hr _yi g§»`m Amho.

(vi) n2 – n + 41 hr _yi g§»`m Amho.

2 Vw_À`m AmdS>rMr H$moUVrhr EH$ {gÜXVm ¿`m Am{U darb mJm_Üò Ogo {Xbo Amho Vgo Ë`mMo àËòH$ nmàrbm {díbofU H$am. (J¥hrV åhUOo H$m`? à_o` åhUOo H$m`? H$m` {gÜX Ho$bo Amho, H$moUVo _ybVËd dmnabo BË`mXr)

lr{Zdmg am_mZwOZ(1887–1920)

AmH¥$Vr A1.5

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162 J{UV

3. H$moUË`mhr XmoZ {df_g§»`m§Mr ~oarO hr g_ AgVo ho {gÜX H$am.

4. H$moUË`mhr XmoZ {df_g§»`m§Mm JwUmH$ma hr {df_ g§»`mM AgVo ho {gÜX H$am.

5. H$moUË`mhr VrZ H«$_dma g_g§»`m§À`m ~oaOobm 6 Zo ^mJ OmVmo ho {gÜX H$am.

6. y = 2x `m g_rH$aUmÀ`m aofoda Ag§»` q~Xÿ AmhoV ho {gÜX H$am&

( Q>rn : H$moUË`mhr n nyUm©H$m§À`m ~m~VrV (n, 2n) hm EH$ q~Xÿ ¿`m)

7. Vwåhmbm Vw_À`m H$mhr {_ÌmZr EImXr g§»`m _ZmV Yam`bm gm§{JVbr AgUma Ë`m g§»òÀ`m ~m~VrV AZoH$ {H«$`m H$am`bm gm§{JVë`m AgUma Am{U _yig§»`m _m{hV ZgVmZmhr Vw_À`m _ZmVrb eodQ>Mr g§»`m gm§{JVbr AgUma Am{U Vo H$go eŠ` Amho `mMm Aä`mg H$am.

(i) EH$ g§»`m _ZmV Yam. {VMr XþßnQ> H$am. Ë`m_Üò ZD$ {_idm. nwÝhm Vw_À`m _ZmVrb g§»`m {_idm. Z§Va VrZZo ^mJm. nwÝhm Mma {_idm. Vw_À`m _ZmVrb g§»`m dOm H$am Vw_Mo CÎma òB©b gmV.

(ii) H$moUVrhr EH$ VrZ A§H$s g§»`m {bhm (CXmhaUmW© 425 VoM A§H$ nwÝhm EH$Xm {bhÿZ 6 A§H$s g§»`m V`ma H$am (425425). Vw_À`m ZdrZ g§»òbm 7, 11 Am{U 13 Zo ^mJ OmVmo.

A 1.6 gmam§e

`m AZw~§Y>_Üò Vwåhr Imbrb JmoîQ>tMo AÜ``Z Ho$bmV.

1. J{UVmV {dYmZ Oa Zoh_r gË` qH$dm AgË` Agob VaM Vo pñdH$maÊ`mOmoJo AgVo.

2. J{UVr {dYmZ AgË` Amho Ago XmI{dÊ`mgmR>r EH$ {déÜX CXmhaU nwaogo Amho.

3. _ybVËdo hr nwamì`m {edm` Iar _mZbobr {dYmZo AmhoV.

4. AZw_mZ ho {dYmZ Á`mda Amåhr J{UVr A§Vkm©ZmÀ`m AmYmamda {dídmg R>odyZ gË` _mZVmo na§Vy Vo Amåhr {gÜX H$amd`mMo AgVo.

5. Oo J{UVr {dYmZ VH©$ewÜX nÜXVrZo {gÜX H$aVm òVo Ë`mbm à_o` åhUVmV.

6. J{UVr {dYmZo {gÜX H$aÊ`mMo _w»` VH©$ewÜX gmYZ åhUOo AmZw_m{ZH$ `wpŠVdmX ({dMmagaUr) Amho.

7. {gÜXVm hr EH$m Z§Va EH$ òUmè`m J{UVr {dYmZm§À`m> H«$_mZo V`ma hmoVo. {gÜXVo_Yrb àËòH$ {dYmZ _mJrb _m{hVrÀ`m {dYmZmdê$Z VH©$ewÜXnUo _m§S>bobo AgVo. qH$dm nydu {gÜX Ho$boë`m à_o`mda qH$dm _ybVËdmda qH$dm J¥{hV à_o`mda AZw_m{ZV Ho$bobo AgVo.

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CÎma gyMr 163

CÎma gyMr

ñdmÜ`m` 1.1

1. hmo`, 0 0 001 2 3

= = = BË`mXr, àË`oH$ q bm XoIrb G U nyUmªH$ _mZbo OmD$ eH$Vo.

2. 3 Am{U 4 `m g§»`m§À`m Xaå`mZ AZoH$ n[a_o` g§»`m Agy eH$VmV Ë`m {_i{dÊ`mMr

EH$ nÜXV Amho. 2136 1

=+

Am{U 2846 1

=+

V|ìhm ghm g§»`m {_iVmV Ë`m åhUOo

22 23 24 25 26 27, , , , ,7 7 7 7 7, 7

3. 3 30 4 40,5 50 5 50

= = åhUyZ, nmM n[a_o` g§»`m 31 32 33 34 35, , , ,50 50 50 50 50

.

4. (i) gË` Amho. H$maU nyU© g§»`mÀ`m g§J«hmV gd© Z¡g{J©H$ g§»`m AgVmV.

(ii) AgË` Amho. CXm -2 hr EH$ nyU© g§»`m Zmhr.

(iii) AgË` Amho CXm. 12 hr n[a_o` g§»`m Agbr Var Vr nyUmªH$ g§»`m Zmhr.

ñdmÜ`m` 1.2

1. (i) gË` Amho, H$maU dmñVd g§»`mMm gQ> n[a_o` Am{U An[a_o` g§»`mZr ~Zbobm AgVmo.

(ii) AgË` Amho, H$maU H$moUVrhr G U g§»`m H$moUË`mhr Z¡g{J©H$ g§»`oMo dJ©_yi Agy eH$V Zmhr.

(iii) AgË` Amho, CXmhaUmW© 2 hr dmñVd g§»`m Agbr Var Vr An[a_o` Zmhr.

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164 J{UV

2. Zmhr, CXmhaUmW© 4 2= , EH$ n[a_o` g§»`m Amho.

3. Am. 1.8 _Ü`o XmIdboir {H«$`m AZoH$doim H$am. àW_ 4 {_idm Am{U Ë`mZ§Va 5 {_idm.

ñdmÜ`m` 1.3

1. (i) 0.36 gm§V

(ii) 0.09 AI§S> AmdVu

(iii) 4.125 gm§V

(iv) 0.230769 AI§S> AmdVu

(v) 0.18 AI§S> AmdVu

(vi) 0.8225 gm§V

2. 2 12 0.285714,7 7

= × = 3 13 0.4285717 7

= × =��

, 4 14 0.57142587 7

= × =��

,

5 15 0.7142857 7

= × =��

, 6 16 0.8571427 7

= × =

3. (i) [2 0.666......3

=x g_Om. åhUyZ 10 x = 6.666…… qH$dm 10 x = 6 + x qH$dm

x = 6 29 3

=

(ii) 4390 (ii)

1999

4. 1 g_Om. [ 0.9999......=x åhUyZ 10x + 9.9999........ qH$dm 10 x = 9 + x qH$dm 9 19

= =x

5. 0.0588235294117647

6. q À`m _wi Ad`dm_Ü`o Ho$di 2 Mm KmV qH$dm 5 Mm KmV qH$dm XmoÝhr KmV Agy eH$VmV.

7. 0.0100100010000 ........, 0.202002000200002.........., 0.003000300003........ .

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CÎma gyMr 165

8. 0.75075007500075000075......, 0.767076700767000767......, 0.80800800080008 .........

9. (i), Am{U (v) An[a_o` Amho, (ii), (iii) Am{U (iv) n[a_o` Amho.

ñdmÜ`m` 1.4

1. 2.665 H$arVm n[aÀN>oX 1.4 À`m AZwgma H¥$Vr H$am.

2. CXmhaU 11 À`m AZwgma H¥$Vr H$am.

ñdmÜ`m` 1.5

1. (i) An[a_o` g§»`m (ii) n[a_o` g§»`m

(iii) n[a_o` g§»`m (iv) An[a_o` g§»`m

(v) An[a_o` g§»`m

2. (i) 6 3 2 2 3 6+ + + (ii) 6

(iii) 7 2 10+ (iv) 3

3. `mMr H$moUVmhr {dg§JVr Zmhr. ho bjmV R>odm H$s, Ooìhm H$Yrhr EH$m nÅ>rZo qH$dm AÝ` H$moUË`mhr nÜXVrZo bm§~r _moOVm Voìhm Vwåhmbm Ho$di EH$ OdiMo n[a_o` à_mU àmßV hmoVo.åhUyZ c qH$dm d An[a_o` Amho ho Vwåhmbm H$iy eH$V Zmhr.

4. 1 AmH¥$Vr. 1.17 nhm

5. (i) 77

(ii) 7 6+ (iii) 5 23− (iv)

7 23+

ñdmÜ`m` 1.6

1. (i) 8 (ii) 2 (iii) 5

2. (i) 27 (ii) 4 (iii) 8

(iv) ( )1 1

3 13 31 125 5 55

− − − = =

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166 J{UV

3. (i) 13152 (ii) 21 (iii)

1411 (iv)

1256

ñdmÜ`m` 2.1

1. (i) AgË` : {dÚmWu ho Amnë`m S>moù`m§Zr nmhÿ eH$VmV.

(ii) AgË` : ho {dYmZ 2.1 `m J¥hrVH$mer {dg§JV Amho.

(iii) gË`. (_ybVËd 2)

(iv) gË`. Oa EH$m dVw©imÀ`m ì`mßV ^mJmg Xþgè`m ì`mßV ^mJmda R>odbo, Va Vo g_mZ

AgVrb åhUyZ `m§Mr H|$Ðo Am{U gr_m g§nm§Vr AgVmV åhUyZ Ë`m§À`m {ÌÁ`m g§nmVr

AgVmV.

(v) gË` : `wŠbrS>Mo n{hbo J¥{hV

3. {dÚmÏ`mªZm Á`m§Mr _m{hVr Agm`bm hdr Ago AZoH$ n[a^mfrH$ eãX AmhoV. H$maU `mV XmoZ

doJdoJù`m n[apñWVtMm Aä`mg Ho$bm OmVmo. åhUOoM (i) Oa A Am{U B Ago XmoZ q~Xÿ

{Xbobo AgVrb, Va Ë`m§À`m _Ü`o C q~Xÿ AgVmo, (ii) Oa A Am{U B {Xbo AgVrb Va

Vwåhr C q~Xÿ KoD$ eH$Vm Omo A Am{U B bm N>oXÿZ OmUmÀ`m aofoda pñWa ZgVmo. hr "J¥{hVHo$'

`wŠbrS>À`m J¥{hVH$mMo AZwgaU H$aV ZmhrV. Varhr Vo J¥{hVH$ 2.1 Mo AZwgaU H$aVmV.

4. AC = BC

åhUyZ AC + AC = BC + AC (g_mZ ~mOy {_i{dë`m)

åhUyZ 2 AC = AB [BC + AC ho AB Mo g§nmVr Amho]

åhUyZ AC = ½ AB

5. VmËnwaVo Ago g_Om H$s, C d D ho AB Mo XmoZ _Ü`q~Xÿ AmhoV. {OWo C d D doJdoJio q~Xÿ

ZmhrV ho XmI{dUma AmhmoV.

6. AC = BD ({Xbobo Amho) .............. (1)

AC = AB + BC (B q~Xÿ A d C À`m _Ü`o Amho) .............. (2)

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CÎma gyMr 167

BD = BC + CD (C q~Xÿ, B Am{U D À`m _Üò Amh .............. (3)

(1), (2) d (3) À`m qH$_Vr (1) _Üò R>odyZ ho XmI{dbm òVo.

AB + BC = BC + CD g_mZ ^mJ dOm H$ê$Z

7. OJmVrb H$moUË`mhr ^mJmVrb H$moUË`mhr dñVw g§X^m©V ho gË` Amho. Ë`m_wio Ë`mbm d¡pídH$

gË` g_Obo OmVo.

ñdmÜ`m` 2.2

1. {dÚmÏ`m©Zo _m§S>boÀ`m gwÌmÀ`m _mÝ`Vo~m~V dJm©V MMm© Ho$br nm{hOo.

2. gaimaofm l hr gaiaofm m Am{U n `m§Zm Aer N>oXVo H$s, N>oXrHo$À`m EH$m A§Jmg AgUmao

Am§VaH$moZ ho H$mQ>H$moZ AmhoV. nU `wŠbrS>À`m 5 ì`m J¥hrVH$mÀ`m g§H$ënZoZwgma `m aofm l

À`m ~mOwg N>oXV ZmhrV. AmVm AmnUmg _mhrV Amho. l aofmÀ`m Xþgè`m A§Jmg Agbobo

Am§Va H$moZ ho gwÜXm H$mQ>H$moZ AmhoV. aofm l À`m Xþgè`m ~mOwghr Ë`m aofm EH$_oH$m§g N>oXV

ZmhrV åhUyZ aofm m d n `m g_m§Va aofm AmhoV.

ñdmÜ`m` 3.1

1. 30°, 250° 2. 126°

4. EH$m q~Xÿdarb gd© H$moZm§Mr ~oarO = 360°

5. ∠ QOS = ∠ SOR + ∠ ROQ Am{U ∠ POS = ∠ POR – ∠ SOR.

6. 122°, 302°

ñdmÜ`m` 3.2

1. 130°, 130° 2. 126° 3. 126°, 36°, 54°

4. 60° 5. 50°, 77°

6. AmnmVr H$moZ = namdV©Z H$moZ. q~Xÿ B da BE ⊥ PQ H$mT>m. q~Xÿ C d CF ⊥ RS H$mT>m.

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168 J{UV

ñdmÜ`m` 3.3

1. 65° 2. 32°, 121° 3. 92°

4. 60° 5. 37°, 53°

6. ∆ PQR À`m gd©H$moZm§Mr ~oarO = ∆ QTR À`m H$moZm§Mr ~oarO Am{U PRS = QPR + PQR

ñdmÜ`m` 4.1

1. (i) Am{U (ii) EH$m MbnXrVrb ~hÿnXr Amho.

(v) VrZ MbnXrVrb EH$ ~hÿnXr Amho.

(iii), (iv) ~hþnXr Zmhr, H$maU `m ~hÿnXrMo KmV nyU© g§»`m ZmhrV.

2. (i) 1 (ii) −1 (iii) 2π

(iv) 0.

3. 3x35 – 4 ; 1002y (doJdoJù`m ghJwUH$mÀ`m ghmæ`mZo AmUIr H$m§hr ~hÿnXr Vwåhr {bhÿ

eH$Vm)

4. (i) 3 (ii) 2 (iii) 1 (iv) 0

5. (i) dJu` (ii) KZ~hÿnXr (iii) dJu` (iv) aofr` ~hÿnXr

(v) aofr` ~hþnXr (vi) dJu` (vii) KZ ~hÿnXr

ñdmÜ`m` 4.2

1. (i) 3 (ii) −6 (iii) −3

2. (i) 1, 1, 3 (ii) 2, 4, 4, (iii) 0, 1, 8 (iv) –1, 0, 3

3. (i) hmo` (ii) Zmhr (iii) hmo` (iv) hmo`

(v) hmo` (vi) hmo`

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CÎma gyMr 169

(vii) 13

− `m ~hþnXrMm EH$ ewÝ` Amho na§Vw 23 ~hþnXrMm ewÝ` Zmhr

(viii) Zmhr

4. (i) – 5 (ii) 5 (iii) 5

2−

(iv) 23

(v) 0 (vi) 0 (vii) −dc

ñdmÜ`m` 4.3

1. (i) 0 (ii) 278

(iii) 1 (iv) 3 23 3 1−π + π − π +

(v) 278

−

2. 5a

3. Zmhr, H$maU ~mH$s eyÝ` Zmhr.

ñdmÜ`m` 4.4

1. (x + 1) hm (i) Mm EH$ Ad`d Amho na§Vw (ii), (iii) Am{U (iv) `m ~hþnXrMm Ad`d Zmhr.

2. (i) hmo` (ii) Zmhr (iii) hmo`

3. (i) −2 (ii) ( )2 2− + (iii) 2 1− (iv) 32

4. (i) (3x – 1) (4x – 1) (ii) (x + 3) (2x + 1)

(iii) (2x + 3)(3x – 2) (iv) (x + 1) (3x – 4)

5. (i) (x – 2)(x – 1)(x + 1) (ii) (x + 1) (x + 1)(x – 5)

(iii) (x + 1) (x + 2)(x + 10) (iv) (y – 1)(y + 1)(2y + 1)

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170 J{UV

ñdmÜ`m` 4.5

1. (i) 2 14 40x x+ + (ii) 2 2 80x x− − (iii) 29 3 20x x− −

(iv) 4 94

y − (v) 29 4x−

2. (i) 11021 (ii) 9120 (iii) 9984.

3. (i) (3x + y)(3x + y) (ii) (2y – 1)(2y – 1)

(iii) .10 10y yx x + −

4. (i) 2 2 24 16 4 16 8+ + + + +x y z xy yz xz

(ii) 2 2 24 4 2 4x y z xy yz xz+ + − − +

(iii) 2 2 24 9 4 12 12 8x y z yz yz xz+ + − + −

(iv) 2 2 29 49 42 14 6+ + − + −a b c ab bc ac

(v) 2 2 24 25 9 20 30 12+ + − − +x y z xy yz xz

(vi) 2 2

116 4 4 2a b ab ab+ + − − +

5. (i) (2x + 3y – 4z)(2x + 3y – 4z)

(ii) ( )( )2 2 2 2 2 2x y z x y z− + + − + +

6. (i) 8x3 + 12x2 + 6x + 1

(ii) − + +( ) − + +( )2 2 2 2 2 2 2x y z x y z

(iii) 3 227 27 9 18 4 2

x x x+ + +

(iv) 2

3 3 28 4227 3

xyx y x y− − +

7. (i) 970299 (ii) 1061208 (iii) 994011992

8. (i) (2a + b)(2a + b)(2a + b) (ii) (2a – b) (2a – b)(2a – b)

(iii) (3 – 5a)(3 – 5a)(3 – 5a)

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CÎma gyMr 171

(iv) (4a – 3b)(4a – 3b)(4a – 3b) (v) 1 1 13 3 36 6 6

p p p − − −

10. (i) (3y + 52) ( )2 29 25 15y z yz+ −

(ii) ( )( )2 24 7 16 49 28m n m m mn− + +

11. (3x + y + z) ( )2 2 29 3 3x y z xy yz xz+ + − − − .

12. COì`m ~mOybm (RHS) gai ê$n ¿`m.

13. {ZË`g{_H$aU VIII _Üò x + y + z = 0 Kmbm.

14. (i) −1260.g_Om A = – 12, b = 7, c = 5 BWo a + b + c = 0 àíZ 13 _Üò {Xboë`m CÎmamMm dmna H$am.

15. (i) EH$ g§^mì` CÎma Amho : bm§~r = 5a – 3, é§Xr = 5a – 4 .

(ii) EH$ g§^mì` CÎma Amho : bm§~r = 7y−3,é§Xr = 5y + 4

16. (i) EH$ g§^mì` CÎma Amho : 3, x, Am{U x−4.

(ii) EH$ g§^mì` CÎma Amho : 4k, 3y + 5.Am{U y−1

ñdmÜ`m` 5.1

1. Vo g_mZ AmhoV. 6. BAC = DAE

ñdmÜ`m` 5.2

6. BCD = BCA + DCA = B + D 7. 45° àËòH$s

ñdmÜ`m` 5.3

3. (ii) (i) nmgyZ ABM = PQN

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172 J{UV

ñdmÜ`m` 5.4

4. BD OmoS>m d XmIdm B > D . AC OmoS>m d XmIdm A > C

5. Q + QPS > R + RPS , BË`mXr

ñdmÜ`m` 7.1

1. 36°, 60°, 108° Am{U 156°

6. (i) ∆ DAC Am{U ∆ BCA nmgyZ XmIdm DAC = BCA d ACD = CAB B.

(ii) à_o` 8.4 dmnê$Z XmIdm BAC = BCA

ñdmÜ`m` 7.2

2. PQRS hm g_m§Va^yO Mm¡H$moZ Amho ho XmIdm. VgoM PQ||AC Am{U PS||BD åhUyZ

P = 90°

5. AECF hm g_m§Va^yO Mm¡H$moZ Amho åhUyZ AF||CE B.

ñdmÜ`m`À A 1.1

1. (i) AgË`. dfm©_Ü`o 12 _{hZo AgVmV.

(ii) g§{X½Y : EH$m {Xboë`m dfu {Xdmir ewH«$dmar Agy eH$Vo Am{U Agy eH$V Q>r Zmhr.

(iii) g§{X½Y : 26°c dfm©V H$Yr H$Yr _mJmXrV Agy eH$Vo.

(iv) Zoh_rM gË`

(v) Zoh_rM AgË` : Hw$Ìo CSy> eH$V ZmhrV.

(vi) g§{X½Y EH$m {bn dfm©V \o$~«wdmar _{hÝ`mMo 29 {Xdg AgVmV.

2. (i) AgË` EH$ Mm¡agmÀ`m Am§VaH$moZm§Mr ~oarO 360° AgVo.

(ii) gË` (iii) gË` (iv) gË`

(v) AgË` CXm. 7 + 5 = 12 hr {df_g§»`m Zmhr.

3. (i) XmoZ nojm _moR>çm AgUmè`m gd© _yi g§»`m {df_ AgVmV.

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CÎma gyMr 173

(ii) Z¡g{J©H$ g§»`oMr XþßnQ> hr Zoh_rM g_ AgVo.

(iii) H$moUË`mhr x > 1 gmR>r, 3x + 1 > 4

(iv) H$moUË`mhr x≥0gmR>r x3≥0.

(v) EH$m g_^wO {ÌH$moUmV _Ü`Jm hr H$moZ Xþ^mOH$ AgVo.

ñdmÜ`m`ÀÀ A 1.2

1. (i) _mZdmbm nmR>rMm H$Um AgVmo..

(ii) Zmhr, {XZoe Amnbo Ho$g BVa H$moUË`mhr {Xder H$mny eH$bm AgVm.

(iii) JwbJ Mr Or^ bmb Amho.

(iv) JQ>mamMr g\$mB© VmËH$mb H$am`bm hdr Agm {ZîH$f© AmnU H$mT>Vmo.

(v) eo`Q>r AgUmao gd©M àmUr Hw$Ìo AgmdoV ho JaOoMo Zmhr. CXm. ~¡b, _mH$S>, _m§Oa `m àmÊ`m§Zmhr eonyQ> AgVo na§Vw Vo Hw$Ìo ZgVmV.

2. AmVm Vwåhr B Am{U 8 bm CbQ> H$ê$Z nhm. XþgarH$S>o ‘B’ Oa g_g§»`m Agob Va {Z`_^§J hmoVmo. Aem VèhoZo Oa XþgarH$S>o 8 da EH$ ñda Agob Va {Z`_^§J hmoVmo.

ñdmÜ`m`ÀÀÀ A 1.3

3. VrZ g§^mì` KQ>Zm KS>Ê`mMo `moJ Ago AmhoV.

(i) H$moUË`mhr VrZ H«$_dma g§»`m§Mm JwUmH$ma g_ AgVmo.

(ii) H$moUË`mhr VrZ H«$_dma g_ g§»`mÀ`m JwUmH$mamg 4 Zo ^mJ OmVmo.

(iii) H$moUË`mhr VrZ H«$_dma g_ g§»`m§À`m JwUmH$mamV 6 Zo ^mJ OmVmo.

2. Mm¡Wr Amoi : 1331 = 113 nmMdr Amoi : 14 641 = 114 Mm¡Ï`m d nmMì`m Amoir bmJy hmoVo Zmhr. H$maU 65 ≠ 15101051

3. T4 + T5 = 25 = 52, Tn−1 + Tn = n2

4. 1111112 = 12345654321

11111112 = 1234567654321.

5. {dÚmÏ`mªMo ñdV…Mo CÎma CXmhaUmgmR>r `wŠbrS>À`m J¥{hVH$ ¿`m.

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174 J{UV

ñdmÜ`m` A 1.4

1. (i) g_mZ H$moZ, na§Vw doJdoJù`m ~mOy AgUmao H$moUVohr XmoZ {ÌH$moU Agy eH$VmV.

(ii) g_^yO Mm¡H$moZmÀ`m ~mOy g_mZ AgVmV na§Vy Vmo Mm¡ag AgboM Ago Zmhr.

(iii) Am`VmMo gd© H$moZ g_mZ AgVmV na§Vy Vmo Mm¡ag AgboM Ago Zmhr.

(iv) a = 3 Am{U = 4 AgVmZm {dYmZ gË` Zmhr.

(v) n = 11 Agë`mg 2n2 + 11 = 253, hr _wi g§»`m Zmhr.

(vi) n = 41 Agë`mg n2 – n + 41 hr _yb g§»`m Zmhr.

2. {dÚmÏ`m©Mo ñdV…Mo CÎma

3. x d y `m XmoZ {df_ g§»`m Agë`mMo _mZbo. Va x = 2m + 1 {VWo m hr Z¡g{J©H$ g§»`m Amho. y = 2n + 1 VgoM n hr EH$ Z¡g{J©H$ g§»`m Amho. x + y = 2 ( m + n + 1) åhUyZ x + y bm 2 Zo ^mJ OmVmo d g_g§»`m Amho.

4. àíZ 3 nhm xy = (2m + 1) (2n +1 ) = 2 ( 2mn + m + n ) + 1 åhUyZ xy bm 2 Zo ^mJ OmUma Zmhr, åhUyZ Vr {df_ Amho.

5. g_Om 2n, 2n + 2 Am{U 2n + 4 `m VrZ H«$_mJV g_ g§»`m AmhoV. V|ìhm `m§Mr ~oarO 6 (n + 1) Amho Orbm 6 Zo ^mJ OmVmo.

7. (i) g_Om _yi g§»`m ‘n’ Amho. V|ìhm AmnU nyT>rb {H«$`hm H$aVmo.

2 2 9 2 9 3 9n n n n n n→ → + → + + = +

3 93 9 3

3nn n+→ + → = + →

3 4 7 7 7n n n→ + + = → + − =

(ii) bj Úm 7 11 13 1001× × = Amho VrZ A§H$s g§»`m g_Om a.b.c ¿`m. V|ìhm 1001 abc abcabc× = AgVo åhUyZ ghm A§H$s g§»`m abcabc bm 7, 11 Am{U 13 Zo

^mJ OmVmo.

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