ssrp self learning guide maths class 10 - in hindi

10 September 2015

lkekU; ifjp; vk S j mi;k sx d s rjhd s

xf.kr ds vn~Hkqr lalkj esa vkidk Lokxr gS!

xf.kr dh nqfu;k cgqr gh [kwclwjr gS& ;g fMtkbu ,oa iSVuZ ls Hkjh iM+h gSA xf.kr ,d izdkj dh

Hkk’kk gSA vad vkSj izrhd blds o.kZ ;k v{kj gSA

rqe rks tkurs gh gks& rqe bldh Hkk’kk dks le> ldrs gks vkSj bls fcuk fdlh dh lgk;rk ls

vuqHko dj ldrs gks vkSj rc rqe bldh lqUnjrk dh iz”kalk djksxsA ;gkW vkids fe= gS tks blesa

vkidh enn djsaxsa& jkspd iz”uksa dk gy] muesa tqM+ko] vo/kkj.kk cukuk] bR;kfnA

tc vki iz;kl djsxsa vkidks etk vk;sxk vkSj vki xyfr;k¡ djsaxsaA blls ?kcjk;s ughA D;k vki

tkurs gS fd ;s xyfr;k¡ vkidks csgrj lh[kus esa enn djsxhA

bl ;k=k esa vki dbZ vo/kkj.kkvksa dk irk yxkus tk jgs gSA ,d Mk;jh bl fdrkc ls tqM+h gq;h

gSA vki Mk;jh esa viuh] Hkkoukvksa] lansg ds loky bR;kfn fy[k ldrs gSaA vki bls vius nksLrksa

,oa f”k{kd ds lkFk lk>k dj ldrs gSA blls vkidks lh[kus es csgrj enn feysxhA

;g yfuZax xkbZM vkidks xf.kr dh cqfu;knh phtksa dks le>us esa vkidh enn djsxhA blls vki

vius vki xf.kr ¼;k dksbZ Hkh fo’k;½ lh[kus dk rjhdk le> tk;sxsaA

;gka nh xbZ lkexzh dks iwjk djus esa vkidks 45 ls 50 fnu yxsaxs ;fn vki gj fnu de ls 30&40

feuV dk;Z djsaxs] oSls vki tYnh Hkh iwjk dj ldrs gSaA vkSj ;fn blls T;knk le; Hkh yx jgk gS

rks dksbZ ckr ugha gS] ij bls le> ds lkFk gh iwjk djuk Bhd gksxkA

yfuZax xkbZM esa eksVs rkSj ij nks fgLLks gSa& Lkh[kus ds Lrjokj fcUnq vkSj muij vk/kkfjr Lrjokj

Lok/;k; vH;kl o vkdyu lq>koA

lh[ku s d s fcUn q

;gka N% Lrj ds yfuZax vkCtsfDVo fn, x, gSA gj ,d Lrj iwjk djus ds ckn vki ;gka fn, x,

y{;ksa ls feyku dj ldrs gSa fd ml Lrj esa fdruk lh[k ik, gSaA ;fn yxrk gS fd fdlh fcUnq

ds ckjs esa vkSj vH;kl djuk gS] rks LkkIrkfgd d{kk esa f”k{kd vkSj vU; lkfFk;ksa ls ckr djsaA

Lrj Lk h[ ku s d s fcUn q (yfu Z ax vkCt s fDVO k) 1 1- pkjksa cqfu;knh lafØ;kvksa dh xgjh le> vkSj muds varj lEc)rk dks tkuuk

2- pkjksa cqfu;knh lafØ;kvksa dks djus ds fy, fofHkUu rjhdksa dh le>

3- vius vki ls fu;ekas ,oa vo/kkj.kkvksa dks cukuk

4- la[;k iSVuZ dk vuqlj.k ,oa lkSan;Z dh ljkguk djuk

5- oMZ izkCye dks le>us vkSj mUgas gy djus ds fy, fofHkUu rjhdksa dk irk

yxkukA

SSRP LG Math-X 2014-15

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2" !

!

2 1- fHkUu] n”keyo ,oa izfr”kr ds vk/kkjHkwr fopkjksa dks izkIr djus esa

2- LFkkuh; eku ,oa iw.kkZdksa dh vo/kkj.kkvksa dks iqu% rktk cukus esa

3- lehdj.k ,oa lehdj.k ds larqyu ds vo/kkj.kk dks Li’V djuk

4- chtxf.kr ds vk/kkjHkwr vo/kkj.kkvksa dks le>us esa

5- xf.krh; izrhdksa ij lk/kkj.k izkstsDV djus esa

6- oxhZdj.k] iSVuZ irk djus esa vkSj vo/kkj.kkvksa dks fudkyus esa

3 1- fHkUu la[;kvksa ds chp lafØ;kvksa dh le>

2- fHkUu vkSj n”keyo la[;kvksa ds chp lafØ;kvksa esa vo/kkj.kkvksa dks dSls fl)

fd;k tkrk gS& blds ckjs esa le>A

3- fofHkUu vo/kkj.kkvksa ds chp ds vUrj&lEcU/k le>uk] tSls& xq.kk] foHkkT;rk

ds fu;e] y- l- ,oa e- l-A

4- iw.kkZadksa ds lkFk lafØ;k,a& lk/kkj.k igsfy;ka cukuk ,oa mudks gy djuk

5- f=Hkqt ds ckjs esa dqN vk/kkjHkwr fopkjksa dh le>A

4 1- Lka[;kvksa ds [ksy

2- Ikzfr”kr

3- okf.kT; xf.kr

4- Lkk/kkj.k C;kt

5 1- {ks=Qy] /kkfjrk vkSj vk;ru

2- vuqikr vkSj lekuqikrh Hkkx

3- lehdj.k

6 1- xzkQ

2- cgqinksa dk tksM vkSj ?kVkuk

3- cgqinksa dk xq.kk


!

3" !

!

Lok/;k; vH;kl vk S j xfrfof/ k;k a

gj ,d esa Lrj esa dh tkus okyh xfrfof/k;ksa ,ao vH;kl ds ckjs eas funsZ”k fn;s x, gSaA bls

tYnckth esa u djsa] cfYd ,d Lrj dk vH;kl iwjk djus esa 10 ls 12 fnu dk le; ysa] bl rjg

ls ;g vkB ls nl lIrkg dk dk;Z gks ldrk gSA

;gka ls vkids Lok/;k; vH;kl “kq# gksrs gSaA ;gka fn, x, vH;klksa dks ,d vyx uksVcqd esa djsaA

vkSj vius vuqHkoksa dks jkstuk viuh Mk;jh esa fy[krs jgsaA

;gka igys pkj Lrj ds fy, yfuZax xkbZM nh xbZ gSA bu ij dke djrs gq, vkxs ds lh[kus ds

y{;ksa ds ckjs esa T;knk Li’Vrk gksus ij f”k{kd ds lkFk feydj vkxs ds nks Lrj ¼Lrj 5 o 6½ dh

yfuaZax xkbZM fodflr gksxhA

Lok/;k; vH;kl vk S j xfrfof/ k;k a Lrj 1

1-1- la[;k ig syh % irk djks eSa dkSu g¡w\ v- eSa ,d rhu vadksa dh la[;k g¡wA

c - eSa ,d le la[;k g¡wA

l- rqe eq>s fcuk “ks’k ds 25 vkSj 50 ls foHkkftr dj ldrs gksA

n- eSa rhu vadh; lcls NksVh la[;k gWwA crkvks eS dkSu g¡w\

,slh gh dqN la[;k igsyh cukvksA D;k vkids fe= igsyh dk tokc ns ldrs gSa] muls iwNasA ,d

la[;k ds ckjs eas ,d ls vf/kd okD; fy[kus dh dksf”k”k djksA igsyh ds var esa dqN lqjkx nsuk er

HkwyukA

vc bl dke ds vk/kkj ij uhps fn, x, lokyksa dk mRrj fy[kks& 1- vkidks ;g djrs gq, dSlk yxk\

2- vkius vius fdu&fdu nksLrksa ds lkFk lk>k fd;kA mudks dSlk yxk \

3- D;k vius nksLr dh igsyh dk tokc ns ik, \ nksLrksa ds }kjk iwNh xbZ fdl igsyh dk

tokc nsus esa lcls T;knk eqf”dy gqbZ] D;ksa\

1 -2 - 'k Cnk s a dk oxh Zdj.k % D;k vki bckjrh ¼ftlesa o.kZu gks½ iz”uksa dks dfBu eglwl djrs

gks\ ;g xfrfof/k vkidh enn djsxhA bu “kCnksa dks i<+sa vkSj mUgsa uhps nh xbZ lkfj.kh esa

oxhZ—r djrs gqq, fy[ksa&

nsuk] ysuk] tkuk] vkuk] Hkkx] tksM+] “ks’k] T;knk] de] j[kuk] lewg] dqy] Hkjuk] mM+syuk] [kkyh

djuk] ckdh] Hkjk gqvk] cdk;k] vUrjA

tk sM + ? kVko x q. k k H k kx


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4" !

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D;k rqEgkjs fe= us Hkh mlh rjg oxhZ—r fd;k ftl rjg ls rqeus fd;k Fkk\ muds lkFk ppkZ

djksA



3- D;k vius nksLrksa us Hkh ;gh mRrj fy[kk \ irk djksA

1-3- tk sM + d s ckj s e s a % ;g vklku gS ;k eqf”dy\

300 $ 400] 450 $ 200] 325 $ 375] 260 $ 338

bu lokyksa ds ckjs esa D;k jk; gS\ D;k rqe fcuk isij vkSj isu dk mi;ksx fd;s tksM+ ldrs gks\

367 $ 438] 453 $ 249] 305 $ 157 $ 284

v- Åij iw.kZ djus ds ckn rqe bu lokyksa dks djus dk iz;kl djks&

26 $ 38] 36 $ 28] 24 $ 40 c- la[;kvksa ds cVokjs ls rqe vkjke ls budks tksM+ ldrs gks tSlk rqe pkgrs gksA ;gk¡ ,d

mnkgj.k gS& ;fn rqe 467 $ 378 dks tksM+uk pkgrs gks rks rqe la[;kvksa dks bl izdkj ck¡V

ldrs gks&

7867300400281735045087370460

+++

+++

+++

gk¡] bu rhuksa lokyksa dk ,d gh tokc gSA bu rhuksa esa ls dkSu lk ,d tksM+us esa vklku gS\ rhu

vadh; la[;kvksa blh izdkj fcuk dkih&isu ds tksM+us dk iz;kl djsaA



3- D;k rqe rhu vadh; la[;kvksa dks fcuk isij vkSj isu ds tksM+ ldrs gks\

4- D;k rqe dqN loky cuk ldrs gks ftudk ,d gh mRrj gks\


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5" !

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1-4- vf/ kdre vFk Z

ge 445 $ 440 ¾ 885 ls D;k le> ldrs gS\

⇒ 445 vkSj 440 feydj 885 cukrs gSaA

⇒ 440 vkSj 445 feydj Hkh 885 cukrs gSaA

⇒ ;fn ge 885 ls 445 dks fudkyrs gSa rks gesa 440 feysxkA

⇒ ;fn ge 440 dks 885 ls nwj ys tkrs gS rks gesa 445 feysxkA

mijksDr okD;ksa dks xf.krh; okD;ksa esa bl izdkj fy[krs gSa &

440445885445440885885445440885440445

=−

=−

=+

=+

;g mnkgj.k dsoy tksM ij vk/kkfjr gSA ,sls gh ?kVko] xq.kuQy vkSj Hkkx ds dqN mngkj.k cuk;s

,oa mues ,d ls vf/kd vFkZ irk yxkus dh dksf”k”k djsaA

vc bl dke ds vk/kkj ij uhps fn, x, lokyksa dk mRrj fy[kks&

1- vkidks ;g djrs gq, dSlk yxk\


3- tksM+ vkSj ?kVko dSls tqMs+ gq, gS\

4- xq.kk vkSj Hkkx dSls tqM+s gq, gS\ vius mRrj dks nksLr ds lkFk ppkZ djksA mnkgj.k ds lkFk

O;k[;k nksA

5- D;k vki xq.kk vkSj Hkkx es laca/k irk dj ldrs gks\

6- D;k vki vkSj cuk ldrs gS\

1 -5 - pk; oky s dk cgh[k krk

foØe dh gekjs Cykd vkWfQl ds ikl ,d pk; dh nqdku gSA Cykd ds deZpkjh ogk¡ ls pk; ihrs

gaS ysfdu iSlk os eghus ds vUr esa pqdkrs gaS] tc os osru ikrs gaSA blfy, foØe ,d cgh[kkrk

j[krk gSA

Okg la[;kvksa dks bl izdkj fy[krk gS& 15 $ 20 $ 35 $ 12 $ 74----mls 30 ls 40 rd la[;k

blh rjg fy[kuk gksrk gSa vkSj mls og mlh rsth ls tksM+ Hkh ysrk gSA

la[;kvksa dks {kSfrt esa fy[ksa vkSj tksM+us dk iz;kl djsaA ;g ,d vadh;] nks vadh;] rhu vkSj pkj

vadh; la[;kvksa ds lkFk djsaA


!

6" !

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3 -1 - ? kVkuk

;g xfrfof/k vkidks vklkuh ls ?kVkuk lh[kus esa enn djsxkA fuEufyf[kr lokyksa ds fy, mRrj

irk djsaA

........78156..........278356

=−

=−

.........10104........70164

=−

=−

..........16204

.........86274...........486674

=−

=−

=−

1- vkius blls D;k lh[kk\ bls fy[kus dk iz;kl djsa vkSj vius fe= ds lkFk ppkZ djsaA

;gk¡ ?kVkus dk ,d vkSj rjhdk gSA

...........278453 =−

igyh (cMh la[;k) l[;k dks rksM+sa (300+ 153)

278 dks 300 ls ?kVk;s (300-278=22)

vc 22 esa 153 dks tksM+s (153+22=175)

2- dqN iz”u fy[ksa vkSj nksukas rjhds ls mRrj izkIr djsaA dkSu lk rjhdk rqEgs T;knk ilan

vk;k\ ;k iz”u dh izd`fr ds vuqlkj lgh rjhdk pquuk T;knk lgh gS\

3- dqN iz”u fy[ksa vkSj fcuk isij vkSj isu ds mlds mRrj dks izkIr djus dk iz;kl djsaA

3 -2 - nk s l a fØ;k, a %&

efYydk ds ikl jk/kk dks nsus ds fy, 57 #i,] eksgu ds fy, 128 #i, ] m’kk ds fy, 375 #i,

vkSj vkfej ds fy, 250 #i;s gSA efYydk ds ikl dqy 1000 #i;s gSA bu lHkh dks #i;s nsus ds

ckn efYydk ds ikl dqy fdrus #i;s cpsxsa\ D;k rqe bls dsoy nks lafØ;k dk iz;ksx djrs gq,

gy dj ldrs gksA



3- D;k rqe bls nks lafØ;kvksa ds lkFk gy dj ik,\

4- D;k vki lokyksa dks i<+dj le> ikus esa l{ke gSa\

5- D;k vki ,sls loky dks vklku ikrs gS\


!

7" !

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4 -1 - [k qn d s fu;e cukuk

;g xfrfof/k vkidks dqN xf.krh; vo/kkj.kkvksa dks Lo;a irk yxkus esa enn djsxhA

1511513131551

=×

=×

=×

**;fn ge fdlh la[;k dk xq.kk 1 esa djrs gS rks ogh la[;k izkIr gksrh gSA** Åij fn, x, xq.kk ds

rF;ksa dks ns[k dj ge cgqr vPNh rjg ls ;g fu;e le> ldrs gaSA

fuEufyf[kr xq.kk ds rF;ksa dks ns[ksa vkSj fu;e cukus dk iz;Ru djsaA

**xq.kt ×!xq.kd ¾ ifj.kke** xq.kk dk ,d lkekU; fu;e gSA bl vk/kkj ij vki uhps fn, x,

xf.krh; dFkuksa ds fy, fu;e cukb,A bls vius uksVcqd esa fy[kuk er HkwyukA

1) 3003100 =× 120012100 =× 5005100 =×

2) 12342438

=×

=×

305660512

=×

=×

324864416

=×

=×

3) 30653056

=×

=×

8010880810

=×

=×

9015690615

=×

=×

4) )52413()2847()2446( =×==×+=×

)126718()70710()5678( =×==×+=×



3- D;k rqe fu;e cuk ik;s\

4- D;k rqe iz”u dks i<+dj le> ik;s\

5- D;k rqeus dksbZ u;k fu;e Lo;a ls cuk;k\

6- D;k rqeus vius fopkjksa dks fy[kk\


!

8" !

!

4 -2 - U kb Z x q. ku rk fydk

D;k rqe fdlh Hkh pkgs gq, la[;k dh xq.ku rkfydk cukuk pkgrs gks\ ;g xfrfof/k vkidh enn

djsxkA mnkgj.k ns[ksa&

162189126187541831081861441887218436182901851801810

=×

=×

=×

=×

=×

=×

=×

=×

=×

,d la[;k pqusA dksbZ nks vadh; la[;k vPNk jgsxkA xq.ku rkfydk cuk;saA isij] isu ;k dSydqysVj

dk iz;ksx u djsaA ;fn vkids ikl ?kM+h gks rks ns[ksa & ;g djus esa fdruk le; yxk\ vf/kd

la[;kvksa dk mi;ksx djds vH;kl djksA



2- D;k rqe u;k xq.ku rkfydk cuk ikus esa l{ke gks ldsa\

3- D;k rqeus le; dh x.kuk dh\

4- D;k rqeus dbZ la[;kvksa ds lkFk vH;kl fd;k] fdlds lkFk T;knk eqf”dy gqbZ\

4 -3 - ,d l s vu sd

xq.kuQy ds ,d rF; ds vk/kkj ij ge ,sls dbZ cuk ldrs gSaA ;gk¡ ,d mnkgj.k gS&

45153 =× bl rF; ls dj ge dbZ vkSj lokyksa dks Hkh gy dj ldrs gSaA tSls &

;fn rhl ckj iUnzg dks ge x.kuk djrs gSa rks 4501530 =×

2251515 =× (450 dk vk/kk 225)

;fn ge bu nksuksa rF;ksa dks tksM+s rks ge vPNh rjg ls 45 ckj 25 dh x.kuk dj ldrs gSaA

6751545 =×


!

9" !

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xq.kk ds rF;ksa ds lkFk xq.kuQy Hkjsa &

........1560........159

............1518

=×

=×

=×

,d xq.kk dk rF; dk pquko djsaA blds vk/kkj ij ,d ls vf/kd xq.kk ds rF;ksa dks cukus dk

iz;kl djksA le; dks fQDl djks rqe fdrus cuk ldrs gks\ vius fe= ds ckjs esa D;k [;ky gS\



2- D;k rqeus la[;kvksa dks nqxquk ;k vk/kk djrs le; vkjkenk;d eglwl fd;k\

3- ,d feuV esa rqe fdrus xq.kkRed rF;ksa dks cuk ik;s\

5 -1 - U k kprh l a[;k, a

D;k vkius eap ij urZfd;ksa ds lewg dks ukprs gq, ns[kk gS\ os iafDr;kW cuk ldrh gaSA dqN

urZfd;kW iafDr;ksa esa lkeus ls vkrh gSa vkSj dqN ihNs pyh tk;asxhA urZfd;ksa dh rjg ;gk¡ la[;k,¡

ukp dj jgh gSA fuEufyf[kr rF;ksa ds xq.kuQy dk irk yxk,W vkSj vadksa ds LFkku dks ns[ksaA

....................876923

....................676923....................1176923....................576923....................776923....................276923

=×

=×

=×

=×

=×

=×


2- D;k vki bl rjg ds fdlh xq.kuQy ls lacaf/kr jkspd iSVuZ dks tkurs gSa\

6 -1 - ,d l a fØ;k & nk s vFk Z

315÷ ,d loky gS] blls D;k irk pyrk gS\

⇒ 15 esa dqy fdrus 3 gSa\

⇒ ;fn ge 15 dks 3 cjkcj Hkkx esa ckWVs rks ,d Hkkx D;k gksxk\

⇒ D;k nksuksa ckrsa lgh gaS\

vc rqe 5315 =÷ “kkfCnd okD; ¼bckjrh loky½ esa dSls cnyksxs\


!

10" !

!


2- D;k vki xf.krh; okD;ksa dks “kkfCnd okD;ksa esa cny ikus esa l{ke gks lds\

6 -2 - fu;e cukuk

D;k rqEgsa ;kn gS fd dSls xq.kuQy ds dqN rF;ksa ij vk/kkfjr fu;e cukrs gS\ mlh izdkj ;gk¡ dqN

Hkkx ds rF;ksa ds dqN lsV gSaA dqN fu;e cukus dk iz;kl djksA **HkkT; » Hkktd ¾ HkkxQy**

vkidks fu;e cukus esa enn djsxkA

1) 12112 =÷ 18118 =÷ 33133 =÷

2) 5210 =÷ 10220 =÷ 20240 =÷

3) 20360 =÷ 10660 =÷ 51260 =÷

4) 51680 =÷ 5840 =÷ 5420 =÷

fHkUu ls Hkkx fdl izdkj lEcaf/kr gS\ Hkkx ds rF; ds pkSFks lsV dks ns[ksaA ;g le fHkUu ls fdl

izdkj lacaf/kr gS\


2- D;k rqe fu;e cukus esa l{ke gks lds\ gkW ;k uk

3- D;k rqe Kkr dj ldrs gks fd Hkkx dh fØ;k fdl izdkj xq.kk ls lacaf/kr gS\

4- D;k rqe la[;kvksa ds Hkkx ls lacaf/kr dqN vkSj fu;e dks irk dj ikus esa l{ke gks ldsa\

6 -3 - H k kx ;k x q. k k dh l q Unjrk\

379333376222373111

=÷

=÷

=÷

333937222637111337

=×

=×

=×

v- bl lwph dks c<+k;asA D;k ;g pkj vadh; la[;kvksa esa Hkh ykxw gS\

c - uhps nh xbZ la[;kvksa dk HkkxQy Kkr djksA blls D;k vk”p;Zpfdr gks jgs gks\ vius fe= ds

lkFk ppkZ djksA

..................18222222222...................9111111111

=÷

=÷


!

11" !

!


2- D;k rqeus fyLV dks c<+k;k\

3- D;k vkius vius mRrj dh tk¡p Hkktd vkSj HkkxQy dks xq.kk djds fd;k\

4- D;k rqeus bls vU; ds lkFk lk>k fd;k\

6 -4 - NqVh g qb Z l a[;k dk s irk djuk

^^HkkT; » Hkktd ¾ HkkxQy vkSj “ks’kQy** ds vk/kkj ij NwVh gqbZ la[;kvksa dk irk djksA

1- HkkT; gS 45 vkSj HkkxQy 9 gS] rks Hkktd gksxk --------------------------

2- Hkktd gS 8 vkSj HkkxQy gS 7 rks HkkT; gksx ---------------------------------

3- 47 dks 9 ls iwjk&iwjk foHkkftr ugha fd;k tk ldrk] “ks’kQy gksxk -------------------

vc vki **HkkT;] Hkktd] HkkxQy vkSj “ks’kQy** ds chp fdrus lEca/kksa dk irk yxk ldrs gaS\

vius fe= ds lkFk fudys fu’d’kksZa ds ckjs esa ppkZ djsaA


2- fdrus lEca/kksa dks vkius Kkr fd;k\

3- D;k rqeus vkSj mnkgj.kksa ij dke fd;k\


2-1 - ,d e s a nk s

1.1, 1.2, 1.3...

;|fi nks la[;k,¡ gaS] os vdsys xfrfof/k dh vksj ladsr djrh gSaA 1-1 eryc igys mnns”; esa igyh

xfrfof/kA 1-2 dk eryc gS igys mn~ns”; ds fy, nwljh xfrfof/kA blh izdkj fHkUuksa esa Hkh nks

la[;k,W gSA ysfdu os ,d la[;k dks iznf”kZr djrh gSaA ns[ksa ge fHkUu ls D;k le> ldrs gaSA

53

dk D;k eryc gksxk\

⇒ nksuksa 3 vkSj 5 ,d gh la[;k dk ladsr djrs gSaA

⇒ ,d la[;k ;k oLrq dks ik¡p cjkcj Hkkx esa ck¡Vk x;k gS vkSj muesa ls rhu Hkkx fy;k x;k

gSA


!

12" !

!

⇒ ;fn ge ik¡p cjkcj Hkkxksa esa ls rhu Hkkx ij fopkj djrs gaS rks tks cprk gS oks nks Hkkx gSA

bls ge ,sls mYys[k dj ldrs gSa 52

⇒ ;fn ge 52vkSj

53 dks tksM+rs gS rks ge iwjh oLrq ;k la[;k ik tk;saxsA

⇒ ;fn ge ik¡p Hkkx esa ls rhu Hkkxksa ij fopkj dj jgs gS rks bldk eku ,d ;kfu iw.kZ ls

de gaSA

⇒ fHkUu esa tks la[;k js[kk ls mij gksrh gS mls va”k dgrs gS vkSj tks la[;k js[kk ds uhps gksrh

gS mls gj dgk tkrk gSA

⇒ D;k fopkj gS ge fHkUu 74 ls D;k eryc fudky ldrs gS\ viuh uksV cqd esa fy[kksA vius

fopkj dks vius nksLrksa ds lkFk lk>k vkSj ppkZ djksA



2-2 - f H k Uuk s a dk le wg

31

74

311

55

743

89

36

34

75

1213

1211

710

1257

1224

714

1212

77

28

• fdrus fHkUuksa dk eku ,d ls vf/kd gS ¼iw.kZ½ • fdrus dk eku ,d ls de gS\ • fdrus fHkUUk ,d nwljs ds cjkcj gSa\

viuh uksVcqd esa rhuksa lokyksa ds vk/kkj ij rhu dkye cukvksa vkSj fn, x, fHkUuksa dks rhu lewg esa

ckWVksaA vius mRrj dks vius fe= ds mRrj ls tkWp djks ;fn vkidh #fp gks rks dqN vkSj fHkUuksa dks

fy[kksa vkSj mUkds lkFk ;gh dke nksLrksa ls feydj djus dk iz;kl djksA




!

13" !

!

2-3 - lgh dk s fVd djuk

uhps dqN xf.krh; dFku fn, tk jgs gSaA lgh ds lkeus ^lgh* vkSj xyr ds lkeus ^xyr* “kCn

fy[kks&

• fHkUu 52 esa ;fn gj leku jgs vkSj va”k dks Øe”k% 3] 4] 5 ----- c<+k;k tk; rks fHkUu dk

eku ?kVsxkA

• fHkUu 52

esa ;fn va”k leku jgs vkSj gj dks Øe”k% 6] 7] 8 ----- c<+k;k tk; rks fHkUu dk

eku leku jgsxkA

• ;fn va”k vkSj gj nksuks leku gks rks fHkUu dk eku ,d ds cjkcj gksxkA

• fdlh Hkh la[;k dks fHkUu ds :i esa iznf”kZr dj ldrs gaSA • la[;k 5 dks fHkUu ds :i esa fy[ksa rks mldk gj 1 gksxkA

• 55 ,d fefJr fHkUu gSA

• vius mRrj dh vius fe= ds mRrj ls tk¡p djksA

• 74 !dks

52 ds LFkku ij j[kdj mijksDr iz”uksa ds mRrj nsus ds iz;kl djksA



2 -4 - vk Sj H k h rjhd s

fHkUu esa ,d dk eryc gS **,d iw.kZ oLrq ;k la[;k** nks eryc ,d gh eku ds nks iw.kZ oLrq ;k

la[;kA fuEu fHkUuksa dks ns[kksA mlls rqe D;k fopkj fudky ldrs gks\

.....48,

36,

242

.....44,

33,

221

=

=!

1- ,slh rhu vkSj fHkUu fy[kks tks 3 ds cjkcj gks&

2- Åij dh vo/kkj.kk ds vk/kkj ij fuEu iz”ukas ds tokc nsus dk iz;kl djksA

a) 522− ! b)!

5214− !! c)!

523+ ! ! d)

542

534 +

!


!

14" !

!

bl dke ds vk/kkj ij uhps fn, x, lokyksa dk mRrj fy[kks&



2 -5 - Lkc fey s rk s ,d

31!dk D;k eryc gS\ gesa 15 dks rhu cjkcj Hkkx esa ckWVuk gS vkSj mlesa ls ,d Hkkx dks vyx

djuk gS] blfy, mRrj 5 gSA blh izdkj fuEufyf[kr lokyksa ds tokc Kkr djksA mRrj dk

fujh{k.k djksA fHkUuksa dks {kSfrt fy[ksA muds va”k vkSj gj dk fujh{k.k djsAA rqe vo/kkj.kk fudky

ldrs gksA

.............602010

.............60105

.............6063

.............6042

..............6021

=

=

=

=

=

of

of

of

of

of

gkW] ;fn ge leku la[;k ls va”k vkSj gj esa xq.kk djsa rks ge leku fHkUu ;k mlh fHkUu ds leku

eku dh nwljh fHkUu ik;ssaxsA

1- D;k va”k vkSj gj dks leku vad ls Hkkx nsus ij leku fHkUu izkIr gks ldrh gS\ 2- D;k ge leku fHkUu ikrs gS ;kfn ge leku la[;k dks fHkUu ds va”k vkSj gj ls tksMrs ;k

?kVkrs gaS\

3- D;k ge 53 ds leku 5 vkSj fHkUu fy[k ldrs gS\





!

15" !

!

2 -6 - dk Su Bhd \

Åij fn, x, fp= dks ysdj eqerkt vkSj r#.k us vius&vius rjhds ls fHkUu ds :i dks crk;kA

e qerkt% fHkUu tks nksuksa :iksa dks iznf”kZr djrh gS% 56 ;k

106

r:.k % ugh] rqe xyr gksA os nl Hkkxksaa esa foHkkftr gSaA mlesa ls 6 Hkkxksa dks jax fn;k x;k gS]

blfy, jaxs Hkkx ds fHkUu tks :i iznf”Zkr djrh gS og gS 106

vPNk vki gh crkb, &

1- dkSu lgh gS\ r:.k ;k eqerkt\ 2- vkids mRrj dk D;k dkj.k gS\ 3- D;k vkidk nksLr Hkh leku n`f’Vdks.k j[krk gS\

4- vki 311 dks fp= ls dSls iznf”kZr djsxsa\




2 -7 - tkn qb Z ox Z

QkmaMs”ku dSEi esa tknqbZ oxZ ds ckjs esa ckr gqbZ Fkh] D;k vkidks ;kn gS\ ;gkW oSLkk gh ,d nwljk

tknqbZ oxZ gSA ysfdu iz”u FkksM+k dfBu vkSj pqukSrhiw.kZ gSA D;k vius igys gh iz;kl esa bls lgh

iwjk dj ldrs gks\ fn, x, lokyksa ds vk/kkj ij lgh [kkus dks irk djks\

1 2 3

4 5 6

7 8 9

1- ;fn rqe bl la[;k esa 43 tksM+rs gks rks rqEgsa 4 feysxkA

2- rhu vk/ks fgLls fey dj bls cukrs gSSA


!

16" !

!

3- ;fn rqe 3 esa ls 41?kVkrs gks rks rqe bl la[;k dks ikvksxsA

4- bldk leku fHkUu 510 gSA

5- ;fn rqe bl fefJr fHkUu dks vleku fHkUu esa cnyrs gks rks rqEgsa 25izkIr gksxkA

6- 12 ds vk/ks dk vk/kk\

7- bl la[;k dk nqxquk 214 gS

8- )412()

216( ×+×

9- bl la[;k vkSj 2 dk varj 41 gSA

10- la[;kvksa dks {kSfrt [kMh+ vkSj frjNk tksM+ksaA D;k lHkh mRrj leku gS\ D;k rqe ,d u;k

tknqbZ oxZ cuk ldrs gks\ vius nksLrks ls ppkZ djksA ;fn vkidks enn dh vko”;drk gS

rks —i;k vius v/;kid ls iwNasA bl dke ds vk/kkj ij uhps fn, x, lokyksa dk mRrj fy[kks&



2 -8 - U k kprh l a[;k, a & fQj l s !

D;k vkidks ukprh la[;k,a ;kn gS& Lrj ,d okyh xfrfof/kA ;gka tokc fn, x, gSa] mudkss /;ku

ls ns[kks fQj uhps fn, x, iz”uksa dk mRrj nksA

153846 538461 384615 846153 461538 615384

1- lHkh Ng la[;kvksa dks ,d ckj esa fcuk xyrh ds i<us dh dksf”k”k djksA vc bls fcuk ns[ks

cksydj nksgjkvksA fdrus iz;klksa esa ;g dj ik,\ 2- fdl la[;k esa 1 dk LFkkuh; eku gtkj gS\ 3- vadksa ds LFkku dk eku dc c<+rk gS\ 4- vkSj ,sls gh 10 loky cukvksa ftlds mRrj bu la[;kvksa esa fNis gSa\


!

17" !

!

bl dke ds vk/kkj ij uhps fn, x, lokyksa dk mRrj fy[kks& 1- vkidks ;g djrs gq, dSlk yxk\


2 -9 - vk/ k kj g S 10

uhps fn, x, la[kkvksa ds iSVuZ dks /;ku ls nsf[k,&

1

10

100

1000

10000

100000

gka] tc la[;k esa dksbZ vad vius LFkku ls ck;ha vksj tkrk gS rks mldk eku c< tkrk gS &

• vius LFkku ls ,d vad ck;ha vksj tkus ij mldk eku fdrus xquk c<rk gS\

• nk;ha vksj ,d LFkku c<us ij bldk eku fdrus xquk de gksrk gS\

• la[;kvksa dk LFkkuh; eku 10 ls fdl izdkj lacaf/kr gS\



2 -10 - bdkb Z ] l a[;k vk S j v ad

la[;k 25067 esa fdruh bdkbZ;ka gS\

⇒ ;fn rqEgkjk mRrj 7 gS rks ;g xyr gSA lgh mRrj gS iPphl gtkj ljlB bdkbZ;kaA

⇒ ij rqEgkj tokc 7 Hkh fdlh loky dk mRRkj gS\ mRRkj 7 ikus ds fy, D;k gksxk\

⇒ ;fn rqe iwNrs gks** la[;k 25067 ds bdkbZ ds LFkku dk vad D;k gS\ rqe tokc ds :i esa

7 ikvksxsA

;gka dqN vkSj loky fn, x, gSa] mudks le>ks vkSj mRrj nksA

1- la[;k 25067 esas fdrus lSdM+s lfEefyr gSa\ 2- la[;k 25067 esas fdrus gtkj gSa\ 3- la[;k esa ngkbZ dh la[;k Kkr djusa dk lcls ljy rjhdk D;k gS\ 4- fdrus lSdMk feydj djksM+ cukrs gSa\ 5- ;fn fdlh la[;k essa 634 lSdM+k gS rks mlesa fdrus gtkj gksxsa\


!

18" !

!

6- dqN vkSj iz”u cukus dk iz;kl djks vkSj mldk mRrj vius fe=ksa ls iwNks



2 -11 - nk su k a s vk s j foLrkj

10000 1000

)1010000( ÷ 100

)101000( ÷ 10

)10100( ÷ 1

)1010( ÷ 101

)101( ÷ 1001

)1001( ÷

Åij nh xbZ lkfj.kh esa ,d iSVuZ gSA la[;k 1 okys dkye ds nksuksa vksj ,d [kkl Øe esa la[;kvksa

dk fOkLrkj gSA bl vk/kkj ij uhps fn, x, lokyksa dk mRrj nksA

1- rqe bls nksukas rjQ fdruk foLrkj dj ldrs gks\ 2- D;k ;g vkidks fiNys xfrfof/k esa iwNs x;s iz”uksa ds tokc ikus esa enn djrk gS\




2 -12 - Tkc LF k ku cnyrk g S !

ge tkurs gS fd la[;k esa vad dk LFkku cnyus ls bldk eku cny tkrk gSA ,d vad fofHkUu

inksa esa vyx&vyx vFkZ iznf”kZr djrk gSA 5 dh la[;k fn[kkbZ ns jgk gS mlds fofHkUu inksa dks

ns[kksA blds ckjs esa lkspks vkSj iz”u ds mRrj irk djus dk iz;kl djksA

5 51

65

65× 57× 52 ο5 -5 .5 +5

1- izR;sd la[;k esa ik¡p ds ckjs esa D;k irk pyrk gS\

2- muesa ls dkSu 105 ds cjkcj gS\

3- rqe 1005

dks n”keyo :i esa dSls fy[kksxs\

4- D;k rqe dksbZ vU; LFkku lksp ldrs gks ftlesa ik¡p dks LFkku fn;k tk lds\ bl dke ds vk/kkj ij uhps fn, x, lokyksa dk mRrj fy[kks&




!

19" !

!

2 -13 - fcUn q ;k v ad

ckrphr dks i<+ks vkSj le>ks vkSj uhps fn, x, lokyksa ds mRrj fy[kksA

fouk sn % ;fn rqe fdlh n”keyo la[;k dks 100 ls xq.kk djuk pkgrs gks rks fcUnq dks nks LFkku

nkfguh rjg ys tkvkasA 2.5743100432.57 =× lk su h % rqe xyr gksA ;g fcUnq ugha tks ?kwerk gSA fcUnq mlh txg jgrk gSA ;g rks la[;k dk

LFkkuh; eku gS tks cnyrk gSA ;fn rqe 100 ls xq.kk djrs gks rks ikp¡ ngkbZ feydj ik¡p gtkj gks

tkrs gSA lkr bdkbZ lkr lkS cu tkrk gSA

fouk sn % nwljs la[;kvksa esa D;k cnyko gksrk gS\

lk su h % eq>s ,d feuV lkspus nks\\

1- fdldk rØ lgh gS\ 2- D;k fcUnq cnyrk gS\ 3- lksuh us dSls O;k[;k fn;k fd la[;k dk eku cnyrk gS tks bdkbZ ;k nlosa LFkku ij gksrh

gSa\ 4- ;fn ge 57-432 dks 100 ls Hkkx nsa rks D;k mRrj izkIr gksxk\ 5- rqe bls dSls O;k[kk djksxs\




2 -14 - thjk s g S ghj k s ! bl la[;k 00500-01020 ds vk/kkj ij uhps fn, x, lokyksa dk mRrj D;k gksxk\

1- dkSu lk “kwU; la[;k dk eku c<+krk gS\ 2- vU; “kwU; D;ksa ugh\ 3- D;k rqe oks rhu LFkku crk ldrs gks tgka “kwU; ghjks gksrk gS\

2 -15 - i z fr”kr & 100 gj okyh f H k U U k l a[;k

^^Ikzfr”kr 100 gj ds lkFk ,d fHkUu gSA** D;k ;g dFku lgh gS\ bls dqN mnkgj.kksa ls irk djksA

rc fuEufyf[kr iz;kl djks

1- dqN izfr”kr la[;k,¡ yks vkSj mUgsa fHkUu esa cnyksA

2- blh rjg dqN fHkUu la[;kvksa dks izfr”kr eas cnyksA


!

20" !

!

Ikzfr”kr dks fHkUu esa cnyuk vkSj fHkUu dks izfr”kr eas cnyus ds fy, ge ,d vo/kkj.kk dks ykxw

djrs gS tks ge bl Lovf/kxe lkexzh esa fn;s xfrfof/k esa lh[k pqds gSaA



2 -16 - pkj vFk Z

uhps nh xbZ rkfydk esa [kkyh LFkku HkjksA vo/kkj.kkvksa ds chp lacU/k irk djks vkSj viuh dkih esa

fy[kksA

Fraction

21

41

...........

...........

411

..................

Percentage

%5010050

502501

==×

×

............................... ............................. 20% ......................... ...........................

Decimal

5.50.10050

==

................... .3 ..................... ................... ...................

Ratio 1: 2 ............. ............ ........... ......... .................




2 -17 - D;k l ar qyu cjkcj g S !


!

21" !

!

lkSjHk cktkj ls dqN lfCt;k¡ [kjhnus x;kA nqdkunkj us ,d fdyks I;kt rkSykA D;k nqdkunkj us

Hkkj dk lgh x.kuk fd;k\ D;k larqyu lgh gS\ lkSjHk dks “kd FkkA bls dSls tk¡ps\ mlus ,d

feuV lkspk vkSj ,d fopkj mlds eu esa vk;kA mlus nqdkunkj ls I;kt vkSj ck¡V ds rjktw dks

cnyus ds fy, dgkA

vc uhps fn, x, lokyksa dk mRrj nksA

1- lkSjHk us ,slk D;ksa dgk\

2- vki larqyu ds ckjs esa D;k dg ldrs gS\

3- rjktw dh lqbZ vkSj xf.kr dk ladsr cjkcj gSA D;k rqe lger gks\ dSls\

4- D;k rqEgsa fx¶V ckDl dk mnkgj.k ;kn gS ftlds ckjs esa dSEi esa ppkZ gqbZ Fkh\ og blls

dSls tqM+rk gS\

5- D;k rqe bl vo/kkj.kk dks ,d mnkgj.k ftlesa (3+ 2 = 5) la[;k dk iz;ksx gks O;k[;k

dj ldrs gks



2 -18 - eku yk s !

eku yks Ο=Δ rks uhps fn, x, dFkuksa esa ls dkSu lk dFku lgh gS\

a) )32(4 ++Ο=+Δ

b) 21

2×Ο=

Δ

c) 2%50 ×Ο=×Δ

d) )21

21(

4××Ο=

Δ

e) 15)10(5 −+Ο=−Δ

f) 105

21

+Ο=+Δ

g) Ο×=×Δ 55 h) Ο−=−Δ 55

1- mRrj ds ckjs esa nksLrksa ds lkFk ppkZ djksA

2- ;fn rqe viuk mRrj tk¡puk pkgrs gks rks rqe D;k djksxs\


!

22" !

!




2 -19 - chtxf.krh; dFku

Ckhtxf.kr ,oa xf.kr dh Hkk’kk ds ckjs esa ;kn djks ftlds ckjs esa dSEi esa ppkZ dh xbZ FkhA ;gk¡

dqN dFku gSaA bUgsa chtxf.krh; okD;ksa esa cnyus dk iz;kl djksA

• v** dks pkj ckj tksM+uk

• ^v* dks pkj ls xq.kk djks

• pkj ckj **v**

• ,d la[;k dk rhu xquk ,oa nwljs la[;k dk pkj xquk 24 gksrk gSA

• fdlh la[;k dk ,d frgkbZ 5 gS

• fdlh la[;k ds pkj xquk esa 5 tksM+us ij 30 feyrk gS

• la[;k dk vk/kk vkSj mldk ,d frgkbZ feydj 5 gksrk gSA

• ,sls ikap okD; vkSj cukdj mudks chtxfr.kh; okD; eas cnyksA




2 -20 - xf.kr dFkuk s a dh bckjr

fn, x, mnkgj.k dh rjg chtxf.krh; dFku dks bckjrh ¼”kkfCnd½ dFku esa cnyksA

aa =×1 : fdlh la[;k esa 1 ls xq.kk djus ij xq.kuQy og la[;k gh gksrh gSA

1) aa =÷1 2) aaaaa 4=+++ 3) acabcba +=+ )(

vc dqN lw= dks ;kn djks vkSj bldks “kkfCnd dFku esa cnyksA rc dFku dks vius nksLrksa ls

i<+okvksA fQj bu xf.krh; lw=ksa dks dkih eas fy[kksA




2 -21 - eku D;k \


!

23" !

!

uhps fn, x, chtxf.krh; dFkuksa dk Hkk’kkbZ dFkuksa esa cnydj fy[kksA fQj gj ,d ^a’!esa dk eku

Kkr djksA

1) 295 =+a

2) 152=

a

3) 362 =a

4) 752=+

aa

5) 433)5( =+×a nksLrksa ls ppkZ djks&

1- vki vius mRrj dh tk¡p dSls djsxsa\

2- 1 vkSj 0 dh le> dSls vkidks bu leL;kvksa dks lqy>kus es enn djrk gS\




2 -22 - eku dh vnyk&cnyh

;fn a = 4 vkSj b = 6 rks uhps fn, x, chtxf.krh; dFkuksa dk eku irk dhft,A vkidh enn ds

fy, ,d mnkgj.k fn;k x;k gSA

1064 =+=+ ba 1) ...........2 =+ ba

2) ........)21()

41( =×+× ba

3) .........=ab 4) ......2)43( =−+ ba

vc bUgsa Hkh iwjk djks&

1- dqN vU; chtxf.krh; dFku cukvks vkSj vius nksLrksa ls cnydj gy djksA

2- pjksa ds fy, vU; eku nksA

3- pj dk eku fHkUu la[;k ekudj Hkh ,sls dFku cukvks vkSj gy djksA




!

24" !

!

2 -23 - X k f. kr dh H k k k " k k (i z k st sDV dk; Z)

xf.kr la[;kvksa ,o izrhdks dh Hkk’kk gS A Hkk’kk esa tSls&TkSls vkidk “kCn HkaMkj c<+sxk vkidh

okrkZyki “kSyh lq/kjsxhA blh izdkj tSls&tSls ge xf.krh; ladsrksa ds izfr tkx:d gksrs gaS] ge

xf.kr dh vo/kkj.kkvksas dks Li’V ,oa vklkuh ls le> ldrs gSaA

uhps dqN dk;Z crk, x, gSa muds vuqlkj ;g izkstsDV iwjk gks ldrk gSA ;g dk;Z djus esa ,d

lIrkg yxsaxs] ;fn 30 ls 40 feuV jkst yxkvksA

1- xf.krh; ladsrksa dks ;kn djsa vkSj fy[ksa] tSls& etc...,, 〈÷× !

2- vius nksLrksa dks viuh fyLV fn[kkvks vkSj ns[kks rqe D;k Hkwy x, Fks\ !3- vc d{kk 7] 8 vkSj 9 dh ikB~;iqLrd ns[kks] D;k vkSj Hkh dqN ladsr gSa] mUgsa Hkh viuh

fyLV esa tksMksA !4- vc ;g Hkh fy[kks fd fdl ladsr dk mi;ksx fdlfy, gksrk gS vkSj mldk eryc D;k gS\!5- vc viuh QkbZuy fyLV dks Vhpj ls lk>k djksA !



2 -24 - rc D;k gk sr k\

cgqr xehZ gSA rkieku 43 fMxzh rd igq¡p x;kA vki dejs esa cSBs ilhuk cgk jgs gSaA vki bl

fLFkfr ds ckjs esa D;k lkspsaxs\ fuf”pr :i ls BaMh gokvksa ds ckjs esaA tc ge fdlh pht dks [kksrs

gSa rc gesa mlds ewY; dk irk pyrk gSA

blh rjg dh fLFkfr dh dYiuk djks vkSj vius fopkjksa ds ckjs esa vius fe= ds lkFk ppkZ djksA

1- ;fn gekjh la[;k iz.kkyh esa “kwU; ugha gksrk gesa D;k eqf”dy vkrh\ lkspks vkSj fyLV

cukvksA

2- ;fn udkjkRed la[;k,a u gksrha] rks ge xf.kr eas fdu fopkjksa dk O;Dr ugha dj ikrs\





!

25" !

!

2 -25 - ? k kV s e s a ? k kVk

D;k rqEgsa Lrj 1 dh xfrfof/k 1.2 ;kn gSA ml xfrfof/k esa geus dqN “kCnksa dks pkj vk/kkjHkwr

lafØ;kvksa ds vk/kkj ij lewg esa ck¡Vk FkkA vc dqN fLFkfr;ksa ij fopkj djksA D;k rqe bUgs

ldkjkRed ;k udkjkRed Js.kh esa ckWV ldrs gks\

1- iznhi ds ikl 10000 :i;s gSA ysfdu mlus bls vius fe= eksbu ls m/kkj fy;k

(+10000/ - 10000) 2- mlus NksVk lk O;kikj fd;k vkSj 2000 #i;s dek;sA (-2000/+2000) 3- mlus yksu ds fgLls ds Hkqxrku ds fy, 2000 :i;s iznhi dks fn;sA (-2000/+2000) 4- vxys o’kZ mlus vius O;kikj dks c<+kus dh lksphA mlus cSad ls 20000 dk yksu fy;kA 5- ml lky iznhi us 12000 dk ykHk dek;kA (-12000/+12000) 6- mlus cSad dks nl gtkj vkSj eksbu dks nks gtkj :i;s fn;sA

vc bu lokyksa ds mRrj crkvks& v- iznhi us fdruk #i;k m/kkj fy;k\

c- mlus fdruk ykHk dek;k\

l- orZeku esa mls yksu pqdkus ds fy, vkSj fdruk dekuk gksxk\


3-1 -1 - , slk D;k s gk sr k g S\

xf.kr dh d{kk esa lewg esa dk;Z gks jgk FkkA vafdr vkSj jsgkuk ,d lewg esa FksA mUgsa fHkUu la[;k esa

fHkUu dk xq.kk djuk FkkA

v a fdr% fHkUu ls fHkUu dk xq.kk djus ds fy, va”k vkSj gj nksuksa esa xq.kk djuk pkfg,A

j sg kuk % D;k ;g lgh gS\ rqe dSls tkurs gks\ D;k rqe bls fl) dj ldrs gks\

v a fdr% vPNk! rqeus <sjkas iz”u iwN fy;kA eq>s iz;kl djus nksA

vafdr us fy[kuk vkjEHk fd;kA

a

221

21

21

21

=+++

2214 =× (xq.kk] tksM dk gh laf{kIr :i gSA) !

2.............21

28

==× (4 ds LFkku ij mlds leku fHkUu la[;k j[kus ij½

j s s g kuk % eq>s [kkyh LFkku Hkjus nksA

mlus ,d {k.k lkspk vkSj viuh uksVcqd esa fy[kk 248

2218

21

28

==×

×=×

j sg kuk % D;k ;g lgh gS\


!

26" !

!

vafdr% fcydqy Bhd! D;k rqe xq.kd ds fy, mlds leku mlds leku dksbZ vkSj fHkUu j[kdj ;g

xq.k dj ldrh gksA jsgkuk] D;k rqe 21ds LFkku ij

63dk j[kd ;g gy dj ldrh gks\

jsgkuk% eq>s iz;kl djus nksA vkSj mlus gy djuk “kq: fd;kA

21224

6238

63

28

2..........63

28

==××

=×

==×

v a fdr% cgqr vPNk! Rkqe le> xbZA

j sg kuk % ysfdu vafdr] tc ge xq.kk djrs gS] lkekU;r% xq.kuQy] xq.kd vkSj xq.; ls cM+k gksxkA

D;k ;gka Hkh ,slk gh gS\ v a fdr% **izk;% gksrk gS ij gj le; ughaA bl xq.kuQy dks ns[kks] rqe Bhd ls le> tkvksxh eSa

D;k dguk pkgrk gw¡A

mlus jsgkuk dh uksV cqd esa xq.kuQy ds dqN rF; fy[ks&

1414

2214

4148241234

=×

=×

=×

=×

=×

jsgkuk gka] eSa le> xbZA ;fn xq.kd ,d ls de gS] rks xq.kuQy xq.; ls de gksxkA D;k eSa lgh gw¡\

vafdr% gka rqe Bhd dg jgh gksA

vc x q. k k dju s dh dk s f”k”k djk s& 1- ,d lefHkUu ,oa fo’ke fHkUu dk

2- ,d le fHkUu vkSj ,d fefJr fHkUu dk

3- nks vleku fHkUu

4- nks feJr fHkUu

vc bu lHkh esa xq.kd ds LFkku ij mlds leku dksbZ nwljh fHkUu la[;k j[kdj ;gh dk;Z djks]

vkSj vius mRrj dh tkap Hkh djksA

3 -1 -2 - D;k r qe fl) dj ldr s gk s !

;fn rqe fHkUu la[;k dks fHkUu ls Hkkx nsuk pkgrs gks rks bls nwljs fHkUu ds O;qRØe ¼myVs½ fHkUu ls

xq.kk djuk gksrk gSA rqe bl fu;e dks tkurs gksA ysfdu D;k bls dHkh VsLV djus dk iz;kl

fd;k\ rqe bls dSls fl) dj ldrs gks\ ;gkW ,d mnkgj.k gSA bls /;ku ls ns[kks vkSj le>ks!


!

27" !

!

⇒ igys bls ge xf.kr dh Hkk’kk esa j[krs gaSA

cd

ba

dc

ba

×=÷

⇒ vc vafdr ds crk, x, rjhds ls dks viukrs gSa &

⇒ ,d xf.krh; dFku dk p;u djks] tSls fd 6213 =÷ . ¼;gka rhu esa N% vk/ks fgLls gSa ;k N%

vk/ks fgLls feydj rhu cukrs gaS½

⇒ vc 3 ds LFkku ij mlds leku nwljh fHkUu la[;k j[kks 621

412

=÷

⇒ vc 21 ds O;qRØe ls xq.kk djks

424

12

412

=×

⇒ leku fHkUu ikus ds fy, va”k vkSj gj dks 4 ls Hkkx nks 644424=

÷÷

⇒ vc 21 ds LFkku ij blds leku nwljh dksbZ fHkUu j[kks 6...........

105

412

==÷

⇒ vc 21 ds O;qRØe fHkUu ls xq.kk djks

20120

510

412

=×

⇒ ;g gS rqEgkjs loky dk gy 616

202020120

==÷

÷

;gk¡ dqN fHkUu la[;k,a nh xbZ gSaA bu fHkUu la[;kvksa dks fdlh nwljh fHkUu la[;k ls Hkkx nksA Hkkx

ds fy, vyx&vyx rjg ds fHkUuksa dks pquksA mlds ckn ,d fHkUu pquks] bls leku fHkUu esa cnyks

vkSj fQj ls Hkkx djds ns[kksA vius mRrj dh tkap t:j djrs jgksA

416

75

1012

89

1512

653

vc fdlh fefJr fHkUu dks vleku fHkUu esa cnyks vkSj fQj Hkkx nksA vius mRrj dh tkWp Hkh

djksA

3 -1 -3 - nk s f H k Uuk s a d s chp dh fH k Uu

D;k rqe 31ls cM+h vkSj

21ls NksVh fHkUu fy[k ldrs gks\

^^leku fHkUu** vo/kkj.kk dh le> ls rqEgas blesa enn feysxhA ftlds ckjs esa rqe igys gh le>

pqds gksA D;k rqEgsa ;kn gS] dgka vkSj dc\


!

28" !

!

3 -2 -1 - n”keyo d s lkF k l a fØ;k, a ,d fHkUu la[;k dks izfr”kr] n”keyo vkSj vuqikr esa cnyk tk ldrk gSA ¼Lrj 2 esa ns[ksasa½ blfy, n”keyo la[;kvksa ds xq.kk ds fy, mUgsa fHkUu esa cnydj xq.kk fd;k tk ldrk gSA ns[ksa

mnkgj.k& 1.10.10010

102

1052.5. ===×=×

v- uhps fn, x, lokyksa dks ,sls gh xq.kk djds mRrj ns[kksA fQj mUgsa fcuk fHkUu esa cnys xq.kk

djus dk iz;kl djksA

002.005.02.005.002.05.02.05.2.05.

×

×

×

×

×

c- uhps nh xbZ n”keyo la[;kvksa ds Hkkx ds fy, leku rjhds dks viukvksA

03.015.03.15.315.3.5.135.15315

÷

÷

÷

÷

÷

=÷

igys ,d ;k nks n”keyo la[;kvksa dks fHkUu esa cnyks rc Hkkx nksA mRrj ikus ds ckn HkkxQy dh

tkap djus dk iz;kl djksA

l- D;k rqeus bu lokyksa ds fy, dqN j.kuhfr fudkyh\ mUgssa viuh dkih ij fy[kuk ,oa vius

fe= ds lkFk ppkZ djuk u HkwyukA

3 -2 -2 - n”keyo l[;kvk s a dk tk sM + ,o a ? kVkuk

n”keyo la[;kvksa dk tksM+&?kVkuk Hkh lkekU; la[;kvksa ds ftruk gh ljy gSA uhps fn, x, pj.kksa

dks viukrs gq, ,slk djks] rqe [kqn gh blds rjhds le> tkvksxs&

⇒ dqN n”keyo la[;kvksa dks fy[kks ...)007.0,3.3,45.0,02.1,5(.

⇒ muesa ls dksbZ nks la[;k,a pquks 02.1,5. ⇒ mUgsa vadksa ds LFkkuh; eku ds vuqlkj fy[kksA ;fn ,slk vklkuh ls ugha dj ik jgs gks rks

uhps fn, x, ckDl tSls ,d LFkkuh; eku ckDl cuk yksA vkSj mlesa lgh dkye esa


!

29" !

!

la[;kvkas dks fy[k yksA tSlk fd 0-5 vkSj 1-02 fy[kk x;k gSA vc rqe fdlh Hkh la[;k dk

LFkkuh; eku vklkuh ls tku ikvksxsA

⇒ rc la[;kvksa dks tksM+ks A dbZ ckj ,slk vH;kl gksus ds ckn rqe fcuk LFkkuh; ckDl ds Hkh

;g dk;Z dj ikvksxsA

v)- D;k ?kVkuk Hkh ,sls gh gks ldrk gS\ vkt+ek dj ns[kks] cl ,d dke djuk gksxk cMh la[;k

¼ftlesa ls ?kVk;k tkuk gS½ dks ckDl esa Åij fy[kuk gksxkA vkSj ?kVus okyh la[;k dks mlds uhps

okyh iafDr esaA

c)- ,d vk Sj vH;kl% dqN vkSj n”keyo la[;kvksa dks fy[k ysaA vc vuqeku djsa fd buesa ls

dkSu lh nks ;k rhu ,slh la[;k,a gksaxha ftudk tksM 50 vkSj 60 ds chp gks ldrk gSA vc tksM+dj

nsf[k,] vki mRrj ds fdrus djhc gSa\ ,slk dbZ ckj djds nsf[k, vki ik;saxs fd vkidk vuqeku

vlyh mRrj ds djhc gksrk tk jgk gSA

3 -3 -1 - x q. kt vk S j vioR; Z

vkidks igkM+k ;kn gS] uhps nsf[k, ;gka 2] 3] 4] vkSj ikap ds igkMs+ gh gSa !

2, 4, 6, 8, 10, 12, 14...

3, 6, 9, 12, 15, 18, 21... 4, 8, 121, 16, 20, 24, 28... 5, 10, 15, 20, 25, 30, 35...

gj ,d iafDr esa fy[kh la[;k,a ml iafDr dh igyh la[;k ds xq.kt gSaA ;s lHkh la[;k,a 2]3]4]5] dh

xq.kt gSaA budks D;k vkSj dqN Hkh dgrs gSa\ bUgsa vkSj Bhd rjhds ls dSls dg ldrs gSa\

uhps fn, x, xf.krh; dFkuksa dks ns[kksA le>us esa eqf”dy gks jgh gks rks vius vklikl fdlh ls

ppkZ djks] ;k fiNyh d{kk dh ikB~;iqLrdksa dks Hkh ns[k ldrs gksA ge bl ckjs esa vkSj fdrus vkSj

fdl izdkj ds dFku cuk ldrs gS\

⇒ ;fn abba =× rks “ab”, “a” vkSj “b” dk xq.kuQy gksxkA

⇒ “ab” dks “a” vkSj “b” ls fcuk “ks’kQy ds Hkkx ns ldrs gSA

bdkbZ 101

1001

5

1 0 2


!

30" !

!

⇒ “a” vkSj “b” dks “ab” dk vioR;Z dgk tkrk gSA

⇒ fdlh Hkh xq.kuQy esa nks ;k mlls vf/kd xq.kt ¼vioR;Z½ gksrs gSaA

⇒ fos”ks’k iz—fr dh la[;kvksa dk xq.kuQy Hkh fo”ks’k gksrk gSA ¼le la[;kvksa dk xq.kuQy \½

⇒ 2 ds lHkh xq.kt le gksaxsA

vc uhp s fn, x, i z”uk s a d s mRrj nk sA 1- D;k 4 ds lHkh vioR;Z le la[;k,a gS\

2- 1 dks dkeu QSDVj (lekioR;Z) D;kas dgrs gaS\

3- D;k 4 ds lHkh vioR;Z 2 ds Hkh vioR;Z gS\

4- D;k 10 ds lHkh vioR;Z 2 vkSj 5 ds Hkh vioR;Z gS\

5- 1] 2 ]3] 4] 5] vkSj 6 esa ls dkSu 12 dk viorZd ugh gSA

6- D;k 12] 12 dk vioR;Z ;k viorZad gS ;k nksuksa gS\

3 -3 -2 - H k kT; l a[;k ,o a vHk kT; x q. ku[k.M

tSlk fd rqe tkurs gks vHkkT; la[;kvksa ds nks xq.ku[k.M gksrs gaS% 1 vkSj Lo;a og la[;kA

mnkgj.k ds fy, 17 dks ge fcuk “ks’kQy ds dsoy 1 vkSj 17 ls foHkkftr dj ldrs gSA blhfy,

17 dks vHkkT; la[;k dgrs gSA

v- vc bu iz”uksa ds mRrj nsus dk iz;kl djksA

1- lcls NksVh vHkkT; la[;k dkSu gS\

2- D;k le la[;k,W vHkkT; la[;k gksrh gS\

3- 1 vkSj 100 ds chp esa fdruh vHkkT; la[;k,W gS\ 25 ;k 26\

4- D;k 1 vHkkT; la[;k gS\ vius mRrj dk dkj.k crkvksA

c- vc uhps fn, x, xf.krh; dFkuksa ij /;ku nks vkSj uhps fn, x, lokyksa ds mRrj nksA

7139132222296

33222725522100352260

32212

×=

×××××=

××××=

×××=

×××=

××=

1- vioR;kZs ds D;k fo”ks’k xq.k gksrs gSa\

2- bl izfØ;k dks D;k dgrs gSa\ D;k bldk dksbZ [kkl uke Hkh gS\

3- gesa ;g lh[kus dh D;ksa vko”;drk gS\


!

31" !

!

3 -3 -3 - ok fr Zd ;k bckjrh i z”u

uhps fn, x, bckjrh lokyksa dks i<ks] le>ks vkSj gy djksA bu okfrZd iz”uksa ds mRrj irk djksA

D;k rqEgs y/kqRke lekioR;Z dh vo/kkj.kk irk gS] rc rks ;s loky rqEgkjs fy, fcydqy gh vklku

gSa!

v- jatw ikWp fnu esa ,d ckj iqLrdky; tkrh gSA latw pkj fnuksa esa ,d ckj iqLrdky; tkrh

gSA vkt os iqLrdky; esa feysA os vxyh ckj dc feysaxs\

c- nknh ekW Vksdjh esa vke ys tk jgh FkhA ,d lkbfdy lokj us mUgsa VDdj ekj nhA Vksdjh

fxj iM+hA nksuks us vke dks bdV~Bk fd;kA vkneh us pqipki vkeksa dh fxurh dh vkSj

iwNk& rqEgkjh Vksdjh esa fdrus vke Fks\ nknh us dgk& eSa ugha tkurh ysfdu ;fn eS bls 6

ds lewg esa j[kwa rks blesa 5 cp tk;saxsA vkSj ;fn eS bls 5 ds lewg esa j[krh gwa rks pkj cp

tkrs gSaA ;fn bls eS pkj ds lewg esa j[krh gwW rks 3 cpsaxsA vkneh us lkspk vkSj cksyk&

geus lkjs vke bdV~Bs dj fy, gSA vc rks crkvks] Vksdjh esa fdrus vke Fks\ rqe rks crk

gh nksxs] rks tYnh crkvks& nknh dh Vksdjh eas fdrus vke Fks\

l- vkeksa dh D;k la[;k gksxh ;fn nknh ekW dgrh& ;fn eS bls 6 ds lewg esa j[kwW rks 4 cp

tkrs gaSA 5&5 ds lewg cukÅa rks 3 cp tkrs gS vkSj ;fn 4 dss lewg esa j[kwa rks 2 cprs gSa

vkSj 3 ds lewg esa j[kus ij 1 vke cp tkrk gSA

n- Vksdjh esa fdrus vke gksxsa ;fn nknh ekW dgrh& ;fn eSa lHkh vkeksa dks 6] 5] 4 ] 3 vkSj 2

ds lewg esa j[kwa rks 1 vke cprk gSA

3 -3 -4 - y?k qre lekior Zd (y-l-) vk S j egRre lekior Zd (e-l-)

60 vkSj 40 dk y- l- = 12032522 =××××


!

32" !

!

ftldk eryc gS fd 120 og NksVh ls NksVh la[;k gS tks 60 vkSj 40 ls iw.kZr;k foHkkftr gSA

60 vkSj 40 dk e- l = 20522 =×× ftldk eryc gS fd 20 og cM+h ls cM+h la[;k gS ftlls 60 vkSj 40 nksuks dks fcuk “ks’kQy ds

foHkkftr dj ldrs gksA

;gkW ,d vkSj rjhdk gS

rqEgkjs fy, dkSu lk rjhdk vklku gS\

v- ;gkW dqN la[;k,W nh x;h gSA muds y- l- vkSj e- l- nksuks rjhdksa ls Kkr djksA 48, 50, 75, 120, 96, 80, 150, 160...

c- nks ;k rhu la[;kvkas dks pquksA blds y- l- vkSj e- l- dks dk vuqeku djks] gy djks fQj vius

vuqeku ls feyku djksA

3 -3 -5 - D;k tku ik, \

v)- nks Øekxr la[;kvksa dks pquks] tSls 6] 7 ;k 20] 21- vc mudk y- l- vkSj e- l- Kkr djksA

⇒ rqeus blls D;k le>k\

c)- nks vHkkT; la[;k,a pquks] tSls 13] 17 ;k 19] 29 bR;kfnA vc mudk y- l- vkSj e- l- Kkr

djksA ⇒ rqeus blls D;k le>k\

l)- nks la[;k,¡ yksA mudk y- l- vkSj e- l- Kkr djksA vc mu nksuksa la[;kvksa dk xq.kuQy

Kkr djksA muds y- l- vkSj e- l- dk xq.kuQy Kkr djksA

⇒ nksuks xq.kuQy dks ns[kksA D;k ikrs gks\


!

33" !

!

l)- nks la[;kvksa dk e- l- gS 3

y- l- gS 120 ,d la[;k gS 24

nwljh la[;k D;k gksxh\

;fn rqe bl leL;k dks gy u dj ikvksa rks nks ;k rhu la[;kvksa dk tksM+k pquksA mudk y- l-

vkSj e- l- irk djksA vc y-l- vkSj e- l- dk xq.kuQy Kkr djksA la[;kvksa ds chp ds laca/kksa dks

Kkr djksA vc irk djks& D;k ;s lEca/k fdUgha Hkh nks ;k rhu la[;kvksa ds lsV ij ykxw gksrs gSaA ;k

lHkh l[;kvksa ds lkFk\ vc rks vkidks lqfuf”pr gks tkuk pkfg, fd vkidk gy lgh gSA

3 -3 -6 - foH k kT;rk d s fu;e

D;k vki 1001 ds xq.ku[k.M dks Kkr dj ldrs gSa\ ¼;k ,slh la[;k ftlls ge mls fcuk “ks’kQy

ds iwjk dk iwjk foHkkftr dj ik,aA½ ;g xfrfof/k ls vki la[;kvksa ds foHkkftr gksus ds fu;e dks tYnh gh le> tk;saxsA blls tYnh

ls irk djuk vklku gks tk,xk fd dksbZ la[;k iwjh iwjh foHkkT; gS ;k ugha\

10] 20] 30] 40] 50 --------;s 10 ds xq.kt gSA ;g iSVuZ Li’V gSA bdkbZ ds LFkku ij “kwU; la[;k gSA

blfy, ;fn fdlh la[;k ds bdkbZ ds LFkku ij “kwU; gksxk rks og la[;k 10 ls foHkkT; gks ldrh

gSA

vc fuEufyf[kr vioR;ksZa ij /;ku nks vkSj la[;kvksa dh foHkkT;rk ds fu;e cukvksA

5, 10, 15, 20, 25, 30...

2, 4, 6, 8, 10, 12, 14...

3, 6, 9, 12, 15, 18, 21...

9, 18, 27, 36, 45, 54...

rqe dqN vkSj fu;e Hkh cuk ldrs gksA ;gka ,d ladsr gS&

3 vkSj 9 ds vioR;ksaZ ds vadksa dks tksM+ksA ¼3 ds vioR;Z& 1$2] 1$5] 2$1 vkSj 9 ds& 1$8] 2+7½ D;k dksbZ iSVuZ feyrk gS\ vc fy[kksa ,slh la[;kvksa dh foHkkT;rk ds D;k fu;e gksaxs\

0] 121] 132] 143] 154----------- ds fu;e cukus ds fy, 11 ds vioR;ksZa ds Øfed vadksa dk tksM+ Kkr

djksA foHkkT;rk fu;eksa dks fy[kuk u HkwyukA

vc] fuEufyf[kr iz”uksa ds mRrj nks&

1- D;k 3 dk vioR;Z 1456 gS\ 2- bls 3 dk vioR;Z cukus ds fy, rqe blesa fdruk vkSj tksM+ksxs\ 3- bls 3 dk vioR;Z cukus ds fy, rqe blesa ls fdruk ?kVkvksxsa\


!

34" !

!

4- pkj ;k ik¡p rhu vadh; la[;kvksa dks fy[kksA bls 4 ls xq.kk djksA blds bdkbZ vkSj ngkbZ

ds LFkku ds vadksa dk fujh{k.k djksA rqeus D;k ik;k\ 3 -4 -1 - i w. k k Z ad k s a d s lkF k l a fØ;k, a

(v)- bu dFkuksa dks iwjk djks&

1- nks /kukRed la[;kvksa dk ;ksx-----------------------------------------------------------------------------------------------------------

2- nks xq.kkRed la[;kvksa dk ;ksx----------------------------------------------------------------------------------------------------------

3- ,d xq.kkRed vkSj ,d /kukRed la[;k dk ;ksx-------------------------------------------------------------------------

4- nks /kukRed la[;kvksa dk vUrj------------------------------------------------------------------------------------------------------------

(c)- Åij dh rjg xq.kk vkSj Hkkx dh fØ;kvksa ds fy, fu;e fy[kksA

(l)- fdlh ckr dks fl) djuk ,d vPNk xf.krh; dkS”ky gSA uhps nh xbZ xfrfof/k vkidks ;g

le>us esa enn djsxh& nks la[;kvksa dk xq.kuQy ,d /kukRed la[;k gksrh gSA

Ikgys dh rhu vo/kkj.kkvksa dks ;kn j[ksaA

aa ×=1 ;g 1 dh fo”ks’krk gS

acabcba +=+ )( forj.k dk fu;e

1, 2, 3, 4, 5.... = +1, +2, +3, +4, +5.... lHkh izk—frd la[;k,a ;k fxufr;ka /kukRed gksrs gSA

uhps lR;kiu fn;k x;k gS A izR;sd pj.k esa mi;ksx ykbZ x;h vo/kkj.kkvksa dks igpkuus dk iz;kl

djksA

1)11(0)1()1(0)1(.............

0)11(.............0)11()11(

)0(1)11(1011

0)1(1)1(1()0(1)11(1

0)11(

+=−×−

=−++

=−+

=+×−+

=+×−+−×−

−=+−−

=+−

=+×+−×

=+−

=+−

( )11()11( −×=+×− )


!

35" !

!

(n)- vc rqe fl) djus dh dksf”k”k djks 1)1()1( +=−÷−

3 -5 -1 - fdru s f=dk s . k

f=dks.k dh dqy la[;k Kkr djus dh dksf”k'k djksA vius mRrj dks nksLrksa ds lkFk lk>k djksA D;k

muds Hkh mRrj ogh gSa\

3 -5 -2 - cuk, a fp= ig syh

1- ,d f=Hkqt cuk;asA vc fdlh dks.k ls mlds lkeus okyh js[kk dks feykrs gq, ,d js[kk

[khasfp, ¼fp= 1½A vc f=Hkqtksa dh la[;k dh x.kuk djksA D;k ;g 3 gS\

2- vc nwljs fp= dks ns[kks blesa ,slh gh js[kk nks dks.kksa ls [khaph xbZ gS A f=Hkqtksa dh la[;k

dh x.kuk djksA D;k rqeus blesa Nqis gq, iSVuZ dks tkuk\

3- vc bUgha dks.kksa ls ,d&,d js[kk vkSj [khpksa ¼fp= 3½A irk djksa fd ;g iSVuZ lgh gS ;k

ugh\

(fp= 1) (fp= 2) (fp= 3)

4- fdlh f=Hkqt ds vk/kkj dks.k ls rhu js[kk, [khpksaA fp= dks vius fe= dks fn[kkvksa vkSj

dgks fd os dqy f=Hkqtksa dh la[;k irk djsaA


!

36" !

!

3 -5 -3 - f=H k qt d s ckj s e s a

;gk¡ f=Hkqt ls lEcaf/kr dqN dFku fn;s x;s gSaA muds lkeus lR; ;k vlR; fy[kksA fQj vius

fe=ksa ds lkFk mRrj dks lk>k djksA

1- f=Hkqt ds lHkh dks.k vf/kd dks.k gks ldrs gSaA

2- f=Hkqt ds rhu esa ls nks dks.k ledks.k gks ldrs gSaA

3- f=Hkqt dh yEckbZ vkSj mlds dks.kksa dh eki chp lEca/k gksrk gSA

4- rqe 3 lseh] 4 lseh vkSj 8 lseh Hkqtkvksa okys f=Hkqt cuk ldrs gksA

5- nks ledks.k f=Hkqt feydj ,d oxZ cukrs gSaA

6- ,sls gh dqN vkSj okD; cukvksA

.


!

37" !

!


4-1 -1 - la[;kvk s a e s a vk—fr;k a

1- tc ge fdlh oLrq ds ckjs esa lksprs gSa& dqN vk—fr;ka gekjs eu esa vkrh gSaA tSls tc ge

f[kM+dh ds ckjs esa lksprs gS rks vk;r dh vk—fr eu esa mHkjrh gSA blh rjg pwfM+;ksa ls

o`Rr] xsan ls xksyk bR;kfnA ysfdu tc ge fdlh la[;k ds ckjs esa lksprs gSa rks D;k dksbZ

vk—fr gekjs eu esa mHkjrh gS\ bu fp=ksa dks /;ku ls ns[ksa&

2- D;k ge 3] 6] 10--- dks f=Hkqt la[;k,aW dg ldrs gSa\ D;k vki vxyk f=Hkqt cuk ldrs gS\

D;k 22 ,d f=Hkqt laa[;k gS\

3- ;gka fNis ,d vkSj iSVuZ dk irk yxk ldrs gSa%

(1 + 2 = 3; 3 + 3 = 6; 6 + 4 = 10.....)

4- vc la[;kvksa ds vU; iSVuZ dks ns[ksaA D;k vki muds fy, mfpr vk—fr lksp ldrs gS\

4, 9, 16, 25...

5- ;fn 3] 6] 10------ dks f=Hkqt la[;k dgk tkrk gS rks 4] 9] 16------- dks ge D;k dgsxsa\

6- D;k vk;r ,d la[;k dks iznf”kZr dj ldrk gS\

7- D;k vki vk;r vkSj la[;kvksa ds chp dksbZ iSVuZ irk dj ldrs gS\

4 -1 -2 - xf.kr& Nk sV k s e s a cMh ckr

ge tkurs gS fd xf.kr vU; rjg dh **Hkk’kk** gSA ;g ges cM+s fopkjksa dks cgqr NksVs esa crkuk

fl[kkrk gSA **ikWp dk lkr ckj tksM+** dks xf.kr esa cgqr gh vklkuh ls vkSj NksVs :i eas fy[k

ldrs gSa% 7 ×!5

,d vkSj mnkgj.k ysaA **;fn 50 dks 5 leku fgLlksa esa ckWVrs gS rks ,d Hkkx 10 gksxk** dks xf.kr

dh Hkk’kk esa fy[k ldrs gSa%

105051

=×


!

38" !

!

mijksDr dh rjg ge “a x a x a x a x a x a x a x a?” dks dSls fy[k ldrs gaS\ vkSj bls ge

“a+a+a+a+a+a+a+a?” ls dSls vyx dj ldrs gaS\

bldk mRrj bl ckr ij fuHkZj gS fd 7 dks dgk¡ j[kk tkrk gSA rjhdk gS igys iz”u dks fy[kus

dk rjhdk gksxk “ 7a ” vk rks nwljs dk D;k eryc gksxk\

1- D;k vki fuEufyf[kr dk foLrkj dj ldrs gS\

...............................)5(..................................4....................................5....................................

4

5

5

=

=

=

=

bbbb

4 -1 -3 - chth; O; atdk s dk s tk sM +u k + vk S j ? kVkuk

;gk¡ dqN chtxf.krh; dFku gSaA mUgsa /;ku ls ns[ksaA uhps fn, pj.kokj funsZ”kksa ds vuqlkj vH;kl

djsa&

.........)()()()(

........)()()()(........23)()(.........)()()()(........23)()(

23

23

23

=+=×+××

=−=×−××

=−=+−++

=×=××××

=+=++++

aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

⇒ mRrj dk vuqeku yxkb;sA

⇒ pj “a” dk eku 2 j[ksaA

⇒ vius mRrj dk eku Kkr dhft,

⇒ vkids fdrus mRrj lgh gS\

⇒ D;k vki xyr mRrj okys lokyksa ds fy, nksckjk iz;kl djuk pkgrs gaS\

⇒ vki blls D;k lh[ks\

⇒ vius lkfFk;ksa ds lkFk vius lh[ks x, fu’d’kksZa ds ckjs esa ppkZ djsaA

gks ldrk gS vkids vfUre nks iz”uksa ds mRrj gksaA vxyh xfrfof/k esa blds ckjs esa vkSj csgrj

le>us esa enn feysxhA


!

39" !

!

4 -1 -4 - leku ,o a vleku in

abmnaa

bmnaba

32312

345

÷+×

−+

−

⇒ Åij ds dFku esa “ a5 ” , ,d in gSA

⇒ igys dFku esa nks in gSA nwljs esa rhu in rFkk rhljs esa pkj in gSA

vc fuEufyf[kr iz”uksa ds mRrj irk djus dk iz;kl djsa&

1 - 3 lkbfdy $ 2 “kVZ 2 - 6 cksry $ 4 feBkbZ 3 - 10 eksckby Qksu $ 5 eksckby Qksu

4 - ........34 =+ ba

5 - ......45 =+ aa

6 - .......52 33 =+ aa 7 - igys nks iz”u esa ge phtksa dks D;ksa ugh tksM+ ldrsa\ 8 - vfUre nks iz”uksa ds ckjs esa D;k fopkj gS\

9 - D;k 23,aa leku in gS ;k vleku in\

10 - ;g dSls irk djksxs\

D;k vki bl vo/kkj.kk ij vius fe= dh le> dks tkWpuk pkgrs gS\ mlds le{k rØ nsa fd 23,aa leku in gSa vkSj bl lanHkZ esa mlds rØ ij /;ku nsaA

4 -1 -5 - cgl

jksfgr us efyd ds lkFk ,d pky pyus dk fu.kZ; fy;kA ;|fi efyd vleatl esa Fkk fQj Hkh

og jkth gks x;kA nksuksa ds chp bl izdkj ckrphr gqbZ&

jk s fgr % eSa nks xf.kr ds dFku fy[kus tk jgk gw¡A D;k rqe crk ldrs gks fd muesa ls dkSsu lgh gS

vkSj dkSu xyr\

efyd% gk¡] crk ldrk gw¡A


!

40" !

!

jk s fgr% ;s nks dFku gSa 2

2

aaaaaa

=×

=+

efyd% igyk xyr gSA

jk s fgr% vPNk vc pj dk eku 2 j[kksa vkSj tkap djksA

E k fyd% vksg! nksuksa dk mRrj rks ,d gh gS vkSj lgh gSA

jk s fgr % rks D;k rqe lger gks fd igyk dFku lgh gS\

efyd% ;|fi mRrj lgh gS ysfdu igyk dFku xyr gS D;ksafd ----------

efyd us D;k dgk gksxk] \ blls rqEgkjs eu esa D;k ckrsa Li’V gqbZa] viuh uksVcqd esa fy[kksA

4 -1 -6 - x q. ku rjhd s

lkekU;r% xf.krK fdlh vutku dkjd dks iznf”kZr djus ds fy, “x” dk iz;ksx djrs gaSA ysfdu

;g ,d leL;k mRiUu djrk gSA

bls ns[kks] D;k ;g lUnsg ugha iSnk djrk\

xxxxx ××××

rc xf.krKksa us xq.kk dks iznf”kZr djus ds fy, ,d nwljk rjhdk lkspk tks nks pjksa ds chp fcUnq

j[kukA

“ cba .. ” dk eryc gS “ cba ×× ”

bl rjhds us Hkh Hkze iSnk fd;k [kkldj rc tc ge n”keyo dk iz;ksx djrs gSaA ge ¼5-4 xq.kk 3½ dks dSls fy[k ldrs gS\

“5.4.3?”

rc mUgksaus ,d vU; rjhds dk irk yxk;kA mUgksaus fu.kZ; fy;k& tc xq.kk ds izrhd ds LFkku ij

dks’Bd dk iz;ksx djsaA

5 (6) (4) = 5 x 6 x 4.

bl rjhds us Hkh leL;k [kM+h dj nhA tc Nk=ksa us bl rjg ds dFku 2 + (6+7) – 5 dks ns[kk]

mUgksaus FkksM+k eqf”dy eglwl fd;kA

vUr esa os ,d vU; jkspd rjhds ij vk;sA xq.kk ds fy, fdlh Hkh izrhd dk iz;ksx u djsaA

“abc” eryc “ cba ×× ”


!

41" !

!

blfy, xf.kr esa vktdy viuh bPNkuqlkj bu rhuksa rjhdksa dk iz;ksx djrs gSaA vc fuEufyf[kr

dFku dks foLrkj djus dk iz;kl djrs gaSA

a(b) = ............

a.b = .............

ab = .............

4.5 = ..............

vc bu lokykasa ij lkspks vkSj mRrj nks&

1- ;g djrs gq, vkius dSlk eglwl fd;k\

2- D;k vkius xq.kk djus ds fy, igys bu rhuksa fof/k;ksa dk iz;ksx fd;k Fkk\

3- D;k vki rhuksa rjhdks dk iz;ksx dj ldsA

4 -2 -1 - f H k Uu l s i z fr”kr

v- bl xfrfof/k dks djus ls igys iwoZ dh yfuZax xkbZM dk lanHkZ ysaA mu vo/kkj.kkvksa dks ;kn

djasA

1- ge fHkUu la[;kvksa ds ckjs esa dkSu&dkSu lh vo/kkj.kk,a le> ldrs gSa\

2- leku fHkUu dh vo/kkj.kk

3- fHkUu dks n”keyo] vuqikr esa cnyuk

4- izfr”kr dh ifjHkk’kk ¼,slk fHkUu ftldk gj 100 gks] izfr”kr dgykrk gSA½

c- ;g xfrfof/k vkidks fHkUu la[;kvksa dks izfr”kr esa cnyuk lh[kus esa enn djsxhA ;g

n”keyo fHkUu dks izfr”kr esa cnyus ds rjhdks dks irk djus esa Hkh enn djsxhA

ge tkurs gaS %2010020

= vkSj %3510035

= ...

bl vk/kkj ij ge 21dks izfr”kr esa dSls cnysxsa\

,d rjhdk gS **leku fHkUu** dh vo/kkj.kk dk iz;ksxA va”k vkSj gj esa 50 ls xq.kk djus ij gesa

10050

izkIr gksxkA rc ge vklkuh ls dg ldrs gS fd %5021=

ysfdu 31

dks izfr”kr esa dSls cnysaxs\


!

42" !

!

fuEufyf[kr xf.krh; lehdj.kksa dks /;ku ls ns[ksasA

%10010050

502501

21

==×

×= )

2100210050( =÷= %5050

2100

21001

===×

%2010020

205201

51

==×

×= )

5100510020( =÷= %2020

5100

51001

===×

%2510025

254251

41

==×

×= )

4100410025( =÷= %2525

4100

41001

===×

Åij fn;s x;s lehdj.kks ls ges ,d vo/kkj.kk dk irk pyrk gS& **fHkUu la[;kvksa dks izfr”kr esa

cnyus ds fy, muesa 100 ls xq.kk djuk gksxkA**

l- fuEu vc uhps fn, x, pj.kokj funsZ”kksZa dk ikyu djrs gq, iz”uksa ds mRrj nks&

1- izR;sd Js.kh dh nks fHkUu la[;k,a fy[kks& le fHkUu] fo’ke fHkUu vkSj feJ fHkUu

2- mUgsa izfr”kr esa cnyks

3- rc 5 n”keyo la[;k,a fy[kks

4- mUgsa izfr”kr esa cnyksA

5- mRrj dks ns[kksA D;k vo/kkj.kk cu jgh gS] viuh dkih esa fy[kksA

6- bl vo/kkj.kk ij vius fe= ds lkFk ppkZ djksA

4 -2 -2 - la[;k d s fuf”pr i z fr”kr

;g xfrfof/k vkidks la[;kvksa ds dqN fuf”pr izfr”kr ,oa mlds T;knk vFkZ ikus esa enn djsxkA

;kn jgs fd izfr”kr eryc ,d ,slh fHkUu la[;k gS ftldk gj 100 gSA blfy, fHkUu la[;kvksa ds

lHkh xq.k ,oa fØ;kfof/k;ka izfr”kr ij Hkh ykxw gksrh gSA

ge tkurs gS fd 60 dk vk/kk 30 gksrk gS vkSj vk/kk dk eryc gS 50 %.

blfy, 60 dk 50 % gksrk gS 30

xf.kr esa ge bls O;Dr djrs gSa 302160

1005060%5060 =×=×=×

154160

1002560%2560 =×=×=×

blfy, 60 dk 25 % gksxk 15

mijksDr ls ge D;k le> ikrs gSa!


!

43" !

!

60%10045%75

%75%25%100451560

%2541

6015

15%2560

=

=

=−

=−

==

=×

vc fuEufyf[kr iz”uksa dks gy djksA

................).........430%25.).........3

12.........60)2..........%1050)1

=×

=×

=×

=×

bu dFkukas ds ckjs esa vkSj irk djus dh dksf”k”k djks vkSj mRrj dks vius nksLrksa ls lk>k djksA

4 -2 -3 - i z fr”kr vk/ k k fjr l a fØ;k, a

;g xfrfof/k fdlh la[;k dk fuf”pr izfr”kr Kkr djus ds rjhds dks pquus esa vkidh enn djsxhA

fuEu dFkuksa dks ns[ksa vkSj fn, x, lokyksa dk mRrj nsa&

6%10606530

%10%5%50

=×

=÷

=÷

18%30601836

%303%10

=×

=×

=×

36%60606%106030%5060

%60%10%50

=×

=×

=×

=+

24630%40606%106030%5060

%40%10%50

=−=×

=×

=×

=−

1- 240 dk 55% Kkr djus dk lcls vklku rjhdk D;k gS\

2- 240 dk 80% rqe dSls Kkr djksxs\

3- fdrus rjhdksa ls rqe 240 dk 95% Kkr dj ldrs gks\

4- ,sls gh T;knk ls T;knk iz”u cukvks vkSj vius nksLrksa ds lkFk gy djrs gq, bu rjhdksa ij

ppkZ djksA

4 -3 -1 - ok f. kT; xf. kr

;g xfrfof/k vkids fopkj tks vkius ykHk vkSj gkfu ds ckjs esa lh[kk gS dks rktk djus esa vkidks

enn djsxhA vkSj lw=ksa ds vk/kkj ij vkidksa vius lw= cukus esa Hkh enn djsxhA

fuEu inksa dh ifjHkk’kk fy[kus djus dk iz;kl djks] tSls& Ø; ewY;] foØ; ewY;] ykHk] gkfu

bR;kfnA vius lkFkh ls bu ifjHkk’kkvksa dh tkWp djuk u HkwyukA

vius ifjHkk’kk ds vk/kkj ij vf/kdre iz”u cukus dk iz;kl djksA ;gkW dqN mnkgj.k gSa&


!

44" !

!

⇒ ;fn Ø; ewY;] foØ; ewY; ls vf/kd gS rks ge ik;saxs fd------

⇒ ;fn] foØ; ewY;] Ø; ewY; ls vf/kd gS rks ge ik;saxs------

⇒ ;fn gkfu dks fo+Ø; ewY; esa tksM+rs gSa rks ge ik;saxs-------

cuk; s x; s mRrj d s vk/ k kj ij l w= cuku s dk i z;kl djk sA

(CP – cost price; SP – selling price; P – profit; L – loss)

o foØ; ewY; & Ø; ewY; ¾ ykHk SP – CP = P

o Ø; ewY; & foØ; ewY; ¾ gkfu CP – SP = L

o Ø; ewY; $ykHk ¾ -------- ----------------------------

4 -3 -2 - dk Su lgh \

Qk:[k vkSj lq”khy ds chp ds okrkZyki dks i<+ksA cgl dks iwjk djus esa lq”khy dh enn djksA

Qk:[k % 100 izfr”kr eryc 100100

, D;k eS lgh gwa\

l q”k hy% gkW] rqe fcydqy lgh gksA

Qk:[k % ;fn fHkUu dk va”k vkSj gj leku gksrk gS rks eku 1 gksrk gSA

l q”k hy% gkW] blesa dksbZ lansg ughA

Qk:[k % blfy, lkS izfr”kr eryc gS 1- ;fn rqe xf.kr esa lkS izfr”kr vad ikrs gks rks bldk

eryc gS rqeus dsoy 1 vad izkIr fd;kA D;k rqe lger gks\

l q”k hy% D;k\ ugha] rqe xyr gks D;ksafd--------------------------------------------------------------------------------------------------------

;fn fdlh fu’d’kZ ij ugha igWqp ik jgs rks dksbZ ckr ughaA vxyh xfrfof/k vkidh enn djsxhA


!

45" !

!

4 -3 -3 - i z fr”kr e s a ykH k vk S j gk W fu

rkfydk ns[kksA fjDr LFkkuksa dh iwfrZ djus dk iz;kl djksA ;gkW dqN ladsr fn;s x;s gS

SPCP

SPPCPCP

=×

=+

=+

=×

%115%115%15100

%)15(%100%100

Cost price Profit Loss Selling pirce

400 15% .....................

550 10% .....................

.................... 15% 690

500 .................... 600

................... 15% 340

vius mRRkj dks nksLrksa ds lkFk lk>k djuk er HkwyukA

4 -5 -1 - Lk k / k kj. k C;kt

D;k rqeus **C;kt** “kCn lquk gS\ rqe blls D;k le>rs gks\ D;k rqEgkjs fe= Hkh ogh le>rs gSa]

tSlk rqe\ muds lkFk ppkZ djksA

**C;kt ,d fVdV ds leku gS tks ge ;k=k ds le; cl ;k Vsªu esa [kjhnrs gSA** bl dFku vkSj

C;kt ds ckjs esa rqEgkjs fopkjksa esa fdruh lekurk,W rqe <Ww< ldrs gks\

cl gekjh ugha gksrhA ge dqN le; bldk iz;ksx djrs gSA ge blds fy, dqN “kqYd vnk djrs gSA

blh izdkj ge fdlh ds iSls dk dqN le; ds fy, mi;ksx djrs gSa rks mlds fy, C;kt nsrs gSaA

;fn rqe yEch nwjh dh ;k=k djrs gSa rks fVdV esa vf/kd iSls yxsaxsA blh izdkj ;fn rqe fdlh ds

iSLkksa dk yEcs le; rd iz;ksx djrs gks rks blds fy, T;knk C;kt nsuk gksxkA

fdl Js.kh vkSj fdl izdkj ds cl esa ;k=k dj jgs gks] blds vuqlkj Hkh “kqYd cny tk;sxkA blh

izdkj C;kt esa Hkh O;fDr ;k cSad ftlus rqEgsa iSls fn;s gS muds vuqlkj cnyko gksxkA

blfy, C;kt dh x.kuk rhu phtksa ij fuHkZj djrk gS&

⇒ ftl iSls dk mi;ksx gksrk gS ¼ewy/ku½ ⇒ ftrus le; ds fy, iSls dk iz;ksx fd;k tkrk gS ¼fnu@eghuk@o’kZ dh la[;k½ ⇒ vkSj C;kt dh njA


!

46" !

!

C;kn dh nj izk;% izfr”kr esa O;Dr fd;k tkrk gSA ,d mnkgj.k ysrs gSaA C;kt dh nj 5% gSA blds rhu igyw gSa&

1- ge 100 #i;s dk mi;ksx dj jgs gS

2- ge bls ,d o’kZ ds fy, mi;ksx dj jgs gSA

3- tc ge 100 :i;s dks ,d o’kZ ckn okil djsxssa rks ge 5 #i;s vkSj nsus gksaxsA ¼;fn ge lkS

#i;s dks cSd esa tek djrs gS tks 5% dk C;kt nsrk gS rks ge ,d o’kZ ckn 5 #i;s vf/kd

ik;sxsa½

vc fuEufyf[kr i z”uk s a d s mRrj nk s&

1- **C;kt dh nj 8 izfr”kr gS** dFku ls D;k rkRi;Z gS\

2- ;fn ge 1000 #i, cSad esa tek djrs gSa tks 8 izfr”kr dk C;kt nsrk gS A rks ge ,d o’kZ

ds i”pkr~~ fdrus #i, ik;saxs\

3- ge 4000 #i, dk yksu cSad ls ysrs gSa] tks 9 izfr”kr dk C;kt ysrk gSA ,d o’kZ ds ckn

gesa fdruk C;kt okil djuk gksxk\

4- ge 2000 #i, dk yksu cSad ls ysrs gS] tks 9 izfr”kr dk C;kt ysrk gSA nks o’kZ ds ckn gesa

fdruk C;kt okil djuk gksxk\

5- ;fn ge 2000 #i;s dk yksu cSad ls 18 izfr”kr C;kt dh nj ls ysrs gSa rks] ,d o’kZ ds

ckn gesa fdruk C;kt okil djuk gksxk\

6- vki vfUre rhu iz”uksa ls D;k le>s\

7- ;g rhu la[;kvksa ds xq.kuQy ls dSls lacaf/kr gS\

8- D;k rqe C;kt dh x.kuk djus ds fy, lw= cuk ldrs gksa\

4 -5 -2 - vKkr dk irk yxkuk

ge tkurs gS fd C;kt dh x.kuk djus ds fy, ewy/ku] o’kZ vkSj C;kt dh nj dk xq.kk djuk gksrk

gSA ge bls ,sls j[k ldrs gSa&

“ PNRI = ”

;g xfrfof/k ges lw= esa NwVs gq, dks irk djus esa enn djsxhA

blds fy, D;k izfØ;k viuk,axs\ blds fy, ,d dFku ysrs gS

P = 1000 rupees; N = two years; R = 8%

ge tkurs gSa fd mRrj 160 #i;s gaSA

bl dFku dks mRrj lfgr ,sls fy[krs gSa% 821001000160 ××=


!

47" !

!

eku ysrs gSa fd le; ugha fn;k x;k gS rc lw= dks fuEu izdkj ls fy[krs gq, le; dk irk dj

ldrs gSa%

2212

80180

801160

80160810160

810160

=

=

×=

××=×

×=

××=

××=

NNN

N

NN

N

vc fuEufyf[kr dks gy djks&

82100

160

21001000160

××=

××=

P

R

D;k rqe vo/kkj.kk dks le> ik, ftlds dkj.k ge NwVs gq, ¼vKkr½ dk irk dj ik,A

,sls vkSj Hkh iz”u cukb;s] mRrj Kkr dhft, ,d lwpuk dks fNikb;s vkSj vius fe= ls mls irk

djus dks dgsaA

tc vki iz”u cuk jgsa gksa rks bls vkSj dfBu djus ds fy, fuEu rjhdksa dk iz;ksx djsaaA

v- o’kZ dh la[;k dks eghus esa cnfy,&

1 eghuk =121

o’kZ

2 eghuk =61

o’kZ

73 fnu =51

o’kZ

146 fnu = 52o’kZ bR;kfn

c- C;kt dh nj dks fefJr fHkUu esa cnydj loky cuk,a& 218 %


!

48" !

!

Lrj 5 vk S j Lrj 6

f”k{kd }kjk lq>k, x, funsZ”kksZa ds vuqlkj Lrj 5 vkSj 6 ds vH;kl djus gSaA

fiNys pkj Lrj ij dke djrs gq, rqeus vusd izdkj dh xf.krh; vo/kkj.kkvksa ds ckjs esa lkspk]

le>k gksxkA D;k vHkh Hkh ,slk yxkrk gS fd dqN phtksa ds ckjs esa vkSj tkuus&le>us dh t:jr

gSA

,sls Yk{;ksa] ftuds ckjs esa rqEgsa nksckjk vH;kl dh t:jr gskxh mUgsa ;gka fy[k yksA bl xkbZM dh

“kq#vkr esa Lrj 5 vkSj 6 ds fy, igys ls gh dqN y{; r; fd, x, gSaA bu nksuksa dks feykdj

Lrj 5 vkSj 6 ds fy, yfuZax xkbZM cusxhA

5 4- {ks=Qy] /kkfjrk vkSj vk;ru

5- vuqikr vkSj lekuqikrh Hkkx

6- lehdj.k

6 4- xzkQ

5- cgqinksa dk tksM+ vkSj ?kVkuk

6- cgqinksa dk xq.kk

vc vius f”k{kd ls feyksA mUgsa vius fy, Lrj 5 o 6 ds fy, fudkys x, lh[kus ds fcUnqvksa dks

fn[kkvksA os rqEgsa bu nksuksa gh Lrjksa ds fy, vH;kl r; djus esa enn nsaxsA

ssrp self learning guide maths class 10 - in hindi

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