Mathematics26_126162401920091224

cv_wgK MwYZ 1

RvZxq wkvg I cvVcyZK evW KZK 2006 wkveQi _K

cg kYxi cvVcyZKic wbavwiZ

cv_wgK MwYZ cg kYx

iPbv W. gywbeyi ingvb Payix Av.g.g. Avnmvb Djvn

nvwg`v evby eMg gvt iwdKzj Bmjvg

mv`bv Av.d.g. Lv`v`v` Lvb

RvZxq wkvg I cvVcy K evW, XvKv

cv_wgK MwYZ 2

RvZxq wkvg I cvVcy ZK evW 69-70, gwZwSj evwYwRK GjvKv, XvKv-1000

KZK cKvwkZ|

[ cKvkK KZK me ^Z msiwZ ]

cixvg ~jK msiY

c_g gy` Y : Avei, 2004 c ~bgy` Y : m^i, 2009

cPQ` I wPvb gvt Ave y`j gvgb wgb

KwcDUvi KcvR jRvivb wjt

wWRvBb RvZxq wkvg I cvVcy ZK evW

meRbxb cv_wgK wkv Kgm ~wPi AvIZvq miKvi KZK webvg~j weZiYi Rb

gy` Y :

cv_wgK MwYZ 3

f ~wgKv cv_wgK wkvi YMZ gvb Dbqbi j cv_wgK wkv Dbqb cK (2q chvq) Ges RvZxq wkvg I cvVcyZK evW, XvKv GKwU weZZ KgcwiKbv MnY Ki| G cwiKbv wkvg, cvVcyZK I wkv mvgMxi g~jvqb I cwigvRb Kiv nq Ges wewfb welqi bZzb cvVcyZK I Abvb cVb-cvVb mvgMx cYqb Kiv nq| cv_wgK ii wkvg cg kYxi MwYZ welqi Rb wPwZ kYxwfwK ARbvchvMx hvMZvi wfwZ RvZxq wkvg I cvVcyZK evW, XvKv KZK wbevwPZ Ges MYcRvZx evsjv`k miKvii cv_wgK I MYwkv gYvjq KZK Abygvw`Z jLKMY cg kYxi Rb bZzb MwYZ cvVcy ZK iPbv Kib| RvZxq Kgkvjvi gvag kYx wkK, wkK cwkK, welq wekl I wkvg weklMYi mnvqZvq eBwUi hwK g~jvqb mb Kiv nq| MvwYwZK avb aviYv ek RwUj| ZvB MvwYwZK aviYv cZi Kivi Rb welqeZy Dcvcb cqvRbxq evLv I Qwe eenvi Kiv nqQ| wkLb wkv_xì AvMn mw I cvV MnY mnR Kivi Rb Abykxjbi eev ivLv nqQ| ZvQvov cwZwU Aavqi mgmvjv mnR _K KwVb ixwZ AbymiY Ki mvRvbv nqQ Ges Gjv mgvavb wkv_xiv hvZ D`vMx nZ cvi mw`KI wekl `w Ìqv nqQ| DciZy DcvwcZ welqeZyi hme Z_ wkv_xì Rvbvi Rb Acwinvh Zv j Kwi I ei ga mwbek Kiv nqQ| Avkv Kiv hvq, GZ wkv_xì MvwYwZK aviYvi myz weKvk NUe Ges MwYZi cwZ AnZzK fxwZ ~ì ne| G cvVcy ZK evsjv GKvWgx Ges GbwmwUwe-Gi evbvb ixwZ AbymiY Kiv nqQ| Dfq cwZvbi evbvb ixwZ GKB| Ze hyvi mijxKiY mwKZ GbwmwUwei evbvb ixwZB cvVcyZK AbymZ nqQ| wkvg Dbqb GKwU avivevwnK cwqv| ZvB, cvVeBqi Dbqbi Rb fwelZ G cwqv AevnZ _vKe| mswk eweMi mhZ cqvm I mZKZv mI cyZKwUZ wKQy ywU wePzwZ _K hZ cvi| cieZx msiY cvVcyZKwU ywUgy Kivi Pv AevnZ _vKe| G eBwU iPbv, mcv`bv I g ~jvqbmn cKvkbvi wewfb chvq huviv mnvqZv KiQb, Zuvì mevBK RvbvB AvwiK KZZv I abev`| hvì Rb eBwU cYxZ nj, Zviv DcKZ nj Avgvì mKj cqvm mdj ne ej Avwg gb Kwi|

cdmi gv: gvZdv KvgvjDwb Pqvigvb RvZxq wkvg I cvVcyZK evW XvKv|

MwYZ 1

m~wPc Aavq welqe cv

Aavq-1 hvM I weqvM 1 9 Aavq-2 Y I fvM 10 19 Aavq-3 cv_wgK Pvi wbqg mwKZ mgmvewj 20 23 Aavq-4 HwKK wbqg 24 31 Aavq-5 mij 32 35 Aavq-6 Mo 36 40 Aavq-7 M.mv.. I j. mv. . 41 50 Aavq-8 MvwYwZK cZxK I evK 51 57 Aavq-9 mvaviY fMvsk 58 100 Aavq-10 `kwgK fMvsk 101 115 Aavq-11 kZKiv 116 127 Aavq-12 cwigvc 128 148 Aavq-13 mgq 149 164 Aavq-14 Rgv LiP I Kvk ggv 165 171 Aavq-15 Dcv web KiY 172 175 Aavq-16 jLwP 176 180 Aavq-17 RvwgwZ 181 192 Aavq-18 KvjKzjUi eenvi 193 197 Aavq-19 RbmsLv 198 203 Digvjv 204 215

MwYZ 1

c_g Aavq

hvM I weqvM D`vniY 1| hvM Ki : 23041, 10532, 31210, 22000, 2104 I 102|

mgvavb : 2 3 0 4 1 A m k ` G 1 0 5 3 2 2 3 0 4 1 3 1 2 1 0 1 0 5 3 2 2 2 0 0 0 3 1 2 1 0 2 1 0 4 2 2 0 0 0 1 0 2 2 1 0 4 8 8 9 8 9 1 0 2 8 8 9 8 9 Di : 88989|

D`vniY 2| hvM Ki : 312024, 103001, 150310, 201200, 131141 I 102203|

mgvavb : 3 1 2 0 2 4 j A m k ` G 1 0 3 0 0 1 3 1 2 0 2 4 1 5 0 3 1 0 1 0 3 0 0 1 2 0 1 2 0 0 1 5 0 3 1 0 1 3 1 1 4 1 2 0 1 2 0 0 1 0 2 2 0 3 1 3 1 1 4 1 9 9 9 8 7 9 1 0 2 2 0 3 9 9 9 8 7 9 Di : 999879|

msLvjvK GKKi wbP GKK, `kKi wbP `kK, kZKi wbP kZK Gfve ci ci mvwRq jLv nqQ|

GKKi Ni _K hvM Kiv iy nqQ| Kvb A nvZ bv _vKvq cwZwU Nii hvMdj wbP wbP jLv nqQ|

MwYZ 2

D`vniY 3| hvM Ki : 572104, 650489, 948531, 886350, 95448 I

749065|

mgvavb : 5 7 2 1 0 4 6 5 0 4 8 9 9 4 8 5 3 1 8 8 6 3 5 0 9 5 4 4 8 7 4 9 0 6 5 3 9 0 1 9 8 7 3901987|

wb j A m k ` G

3 45 6 9 8 7

37 5 4 8 9 4

12 0 8 6 5 9

2 1 4 5 3 4 0

2 0 8 3 5 4 6

4 9 1 0 8 5

Di : 3 9 0 1 9 8 7

msLvjvK QKi ga mvwRq jLv nqQ| GKKi Nii Ajvi hvMdj 27| 27 nPQ 2 `kK 7 GKK, ZvB 27 Gi

7 iLvi wbP GKKi Ni jLv nqQ Ges 27 Gi 2, `kKi Ni Ici Mvji ga jLv nqQ|

`kKi Nii Ajv hvM Ki GKBfve nvZi msLv kZKi Ni Ici Mvji ga jLv nqQ|

kZK, mnm, ARyZ I ji Nii Ajvi hvM GKBfve Kiv nqQ| ji Nii Ajv hvM Ki nvZi msLv wbhyZi Ni Mvji ga jLv

nqQ|

wbhyZi Ni Avi Kvbv A bv _vKvq Mvji wfZii AwU hvMdj wbhyZi Ni jLv nqQ|

D`vniY 4| cvkvcvwk wjL hvM Ki : (K) 31420, 50261, 47523, 9431, 23084 I 62100|

(L) 425731, 653802, 749243, 86010, 72301 I 930211|

MwYZ 3

mgvavb : (K) 31420 + 50261 + 47523 + 9431 + 23084 + 62100

= 223819

Di : 223819|

(L) 425731 + 653802 + 749243 + 86010 +72301 + 930211 = 2917298 Di : 2917298|

msLvjvK hvM wP (+) w`q cvkvcvwk jLv nqQ| msLvjv hvM Ki wbPi jvBb mgvb wP (=) w`q hvMdj jLv nqQ| cwZwU msLvi GKK, `kK, kZK, ...........Ajv chvqg hvM Kivi

AvM Ajvi Ici `vM ( ) `Iqv nqQ| hvM Kivi mgq Kvbv Nii nvZi msLv cieZxZ evgcvki Ai mv_

hvM Kiv nqQ|

gZe : Kvbv Nii Ajv hvM Kivi mgq mB Ajvi Ici `vM ( ) w`q bIqv myweavRbK|

hvM Ki (GKwU Ki `Lvbv nj) : (K) 6 3 2 0 5 4 (L) 4 6 8 5 3 (M) 5 3 2 8 6 1 (N) 7 3 2 8 4 6 2 8 4 7 1 8 7 2 0 4 1 7 0 4 1 0 5 8 5 3 9 2 4 0 5 6 5 3 5 3 9 0 5 2 9 5 0 3 1 9 0 8 7 0 5 5 2 0 8 9 0 2 0 7 1 4 3 2 0 5 2 4 6 2 4 1 5 7 7 4 1 5 0 6 1 6 3 0 0 8 1 3 0 1 6 5 7 0 2 3 4 8 1 3 2 3 2 8 1 4 6 7 9 0 6 3 3 3 9 8 0 5 3 LvwjNi ciY Ki : (K) 645317 + 8297 + 53086 + 264309 + 708321 = (L) 526394 + 230087 + 630801 + 319500 + 460009 + 809625 =

MwYZ 4

D`vniY 5| wgRvb mvne wZbwU Rwg _K h_vg 25087 UvKv, 16920 UvKv I 30725 UvKvi avb Ges Ab GKwU Rwg _K 9872 UvKvi Qvjv weq Kib| wgRvb mvne gvU KZ UvKvi dmj weq Kib ? mgvavb : wZbwU Rwgi avb weq Kib : 2 5 0 8 7 UvKv 1 6 9 2 0 UvKv 3 0 7 2 5 UvKv Qvjv weq Kib : 9 8 7 2 UvKv gvU 8 2 6 0 4 UvKv (hvM Ki) Di : wgRvb mvne gvU 82604 UvKvi dmj weq Kib| D`vniY 6| 839642 _K 713241 weqvM Ki| mgvavb : 8 3 9 6 4 2 j A m k ` G 7 1 3 2 4 1 8 3 9 6 4 2 weqvRb 1 2 6 4 0 1 7 1 3 2 4 1 weqvR 1 2 6 4 0 1 weqvMdj Di : 126401|

D`vniY 7| 758346 _K 94273 weqvM Ki|

mgvavb : 7 5 8 3 4 6 9 4 2 7 3 6 6 4 0 7 3 664073|

j A m k ` G 7

+1 5+10

9 84

3 2+1

4+10 7

6 3

Di : 6 6 4 0 7 3

D`vniY 8| 960752 _K 784906 weqvM Ki|

mgvavb : 9 6 0 7 5 2 7 8 4 9 0 6 1 7 5 8 4 6175846|

j A m k ` G 9

7+1 6+10

8+1 0+10

4+1 7+10

9 50+1

2+10 6

Di : 1 7 5 8 4 6

D`vniY 9| cvkvcvwk wjL weqvM Ki : (K) 650263 _K 95742 (L) 836041 _K 579406

MwYZ 5

mgvavb : (K) 650263 95742 = 554521 GLvb weqvRb 650263, weqvR 95742 Ges weqvMdj 554521|

Di : 554521|

(L) 836041 579406 = 256635 GLvb weqvRb 836041, weqvR 579406 Ges weqvMdj 256635 | Di : 256635|

weqvRb I weqvRi gvS weqvM wP () Ges weqvMdji AvM mgvb wP (=) Ìqv nqQ|

weqvM Kivi myweav_ cZKwU msLvi GKK, `kK, kZK BZvwì Ici `vM ( ) Ìqv AvQ|

weqvM Ki (GKwU Ki `Lvbv nj) :

(K) 8 5 3 2 0 4 (L) 6 5 0 4 8 7 (M) 7 2 8 5 1 3 (N) 8 6 0 5 2 1 4 7 8 5 2 6 7 3 2 9 5 5 4 3 2 7 9 3 9 4 0 7 6 3 7 4 6 7 8 LvwjNi ciY Ki (GKwU Ki `Lvbv nj) : (K) 628713 372946 = 255767 (L) 462051 94608 = (M) 750432 364853 = (N) 804706 525376 = D`vniY 10| gxbvì wjPz MvQ _K 12804 wU wjPz cvov nq| gxbvi evev H wjPz _K 7925 wU wjPz wew Kib| Avi KZwU wjPz iBj? mgvavb : wjPz cvov nq : 1 2 8 0 4 wU wew Kiv nq : () 7 9 2 5 wU iBj : 4 8 7 9 wU Di : 4879 wU wjPz Aewk iBj|

MwYZ 6

D`vniY 11| wZbwU msLvi hvMdj 845076| Gi ga `yBwU msLv 321674 I 286539| ZZxq msLvwU KZ? mgvavb : y`BwU msLv 3 2 1 6 7 4 2 8 6 5 3 9 y`BwU msLvi hvMdj 6 0 8 2 1 3 wZbwU msLvi hvMdj 8 4 5 0 7 6 Ges y`BwU msLvi hvMdj 6 0 8 2 1 3 Aci msLvwU 2 3 6 8 6 3 (weqvM Ki) Di : ZZxq msLvwU 236863| D`vniY 12| Avyj jwZdi 621345 UvKv wQj| wZwb H UvKv _K eo gqK 85924 UvKv, QvU gqK 84790 UvKv I QjK eo gq Acv 9830 UvKv ewk w`jb| evwK UvKv xK w`jb| Zuvi x KZ UvKv cveb? mgvavb : eo gq cvq 8 5 9 2 4 UvKv QvU gq cvq 8 4 7 9 0 UvKv y`BRb cvq 1 7 0 7 1 4 UvKv (hvM Ki) Qj cvq eo gqi _K 9830 UvKv ewk| 8 5 9 2 4 UvKv (+) 9 8 3 0 UvKv

Qj cvq 9 5 7 5 4 UvKv

wZbRbi GK nq 1 7 0 7 1 4 UvKv (+) 9 5 7 5 4 UvKv 2 6 6 4 6 8 UvKv gvU UvKv wQj 6 2 1 3 4 5 UvKv wZbRbi GK 2 6 6 4 6 8 UvKv evwK _vK 3 5 4 8 7 7 UvKv (weqvM Ki) Di : Avyj jwZdi x cveb 354877 UvKv|

MwYZ 7

Abykxjbx- 1 1| hvM Ki : (K) 8 3 5 7 2 (L) 2 3 0 1 4 0 (M) 5 3 6 7 0 9 4 0 2 1 3 3 1 2 1 0 3 4 7 0 9 8 5 2 1 0 0 4 1 2 4 3 1 5 3 2 5 0 7 1 6 5 3 2 2 0 0 2 1 0 6 0 8 2 1 3 8 4 5 0 1 1 2 3 0 1 2 8 9 0 5 0 (N) 1 3 2 0 2 3 (O) 6 4 8 7 5 3 (P) 8 7 2 0 9 4 2 1 0 2 4 0 2 6 9 1 0 7 4 3 5 8 0 9 3 0 1 1 0 2 8 5 3 2 6 4 5 6 9 2 7 5 1 2 3 0 1 2 5 9 8 6 0 3 3 2 5 6 8 0 1 2 0 3 0 0 8 7 5 6 2 6 5 0 9 4 2 1 0 2 3 1 2 6 9 8 7 5 7 9 8 0 5 6

2| LvwjNi ciY Ki : (K) 43570 + 26095 + 40591 + 9207 + 8754 = (L) 560829 + 427108 + 82075 + 6429 + 325790 + 53008 = (M) 325708 + 246805 + 508362 + 730096 + 92346 + 64008 = (N) 410527 + 356807 + 512934 + 239006 + 614500 + 800357 =

3| weqvM Ki : (K) 865497 _K 53205 (L) 798546 _K 462136 (M) 875320 _K 96837 (N) 962053 _K 479568 (O) 927405 _K 658379

MwYZ 8

4| LvwjNi ciY Ki :

(K) 763569 51407 = (L) 972403 48705 = (M) 860523 285306 = (N) 732645 428759 = 5| LvwjNi ciY Ki :

(K) 5 3 6 1 4 (L) 8 2 7 5 3

2 7 9 8 5 6 9 8 6 7 6 9 5 4 6 5 4 7 9 8 7 6 8 9 3 7 5 6 9

2 2 8 2 6 4 2 9 7 1 5 0

(M) 8 7 5 0 6 (N) 9 6 4 2 5

9 6 7 8 5 7 4 6 8 7 7 8 2 4 3 9 4 5 7 6| ivRy`i Avg evMvb GKwU MvQ _K 240851wU, Ab GKwU MvQ _K

92105wU Ges Aci GKwU MvQ _K 55008 wU Avg cvov nq| H evMvb _K gvU KZwU Avg cvov nj ?

7| weqvRb 985214 Ges weqvR 97465 nj, weqvMdj KZ ? 8| Kvb msLv _K 68975 weqvM Kij weqvMdj 794768 nq| msLvwU KZ ? 9| Qq Ai z `Zg msLv _K cuvP Ai eng msLv weqvM Ki| 10| Pvi Ai eng msLvi mv_ KZ hvM Kij hvMdj 9 j ne ? 11| 6, 8, 9, 5, 0, 4 Ajvi cZKwU GKevi eenvi Ki MwVZ Qq Ai

eng I z `Zg msLvi weqvMdj KZ ? 12| ivRyi `v`v 25830 UvKvi avb, 30645 UvKvi Mg I 9786 UvKvi Qvjv weq

Kijb Ges 45927 UvKvi cvU wKbjb| Zuvi wbKU KZ UvKv iBj ?

MwYZ 10

wZxq Aavq

Y I fvM

D`vniY 1| Y Ki : 5 3 8 1 2 5 mgvavb : 5 3 8 Y 1 2 5 YK 2 6 9 0 5 3 8 5 1 0 7 6 0 5 3 8 2 `k 5 3 8 0 0 5 3 8 1 kZ 6 7 2 5 0 Ydj Di : 67250| D`vniY 2| 2465 K 234 viv Y Ki|

mgvavb : 2 4 6 5 Y 2 3 4 YK 9 8 6 0 2 4 6 5 4 7 3 9 5 0 2 4 6 5 3 `k 4 9 3 0 0 0 2 4 6 5 2 kZ 5 7 6 8 1 0 Ydj Di : 576810| D`vniY 3| 3271 K 420 viv Y Ki| mgvavb : 3 2 7 1 Y 4 2 0 YK 6 5 4 2 0 3 2 7 1 2 `k 1 3 0 8 4 0 0 3 2 7 1 4 kZ 1 3 7 3 8 2 0 Ydj

Di : 1373820|

YKi GKKi Ni kb (0) _vKvq GKKi Y `Lvbv nqwb| GRb c_g avc GKKi Ni kb ewmq `kKi Y GK Ni evg _K jLv nqQ Ges wZxq avc GKK I `kKi Ni kb ewmq kZKi Y y`B Ni evg _K jLv nqQ|

MwYZ 11

D`vniY 4| 6582 K 308 viv Y Ki|

mgvavb : 6 5 8 2 3 0 8 5 2 6 5 6 6 5 8 2 8 1 9 7 4 6 0 0 6 5 8 2 3 kZ 2 0 2 7 2 5 6 Di : 2027256| YKi `kKi Ni kb (0) _vKvq `kKi Y `Lvbv nqwb| GRb wZxq avc GKK I `kKi Ni kb ewmq kZKi Y y`B Ni evg _K jLv nqQ|

D`vniY 5| 7596 K 600 viv Y Ki|

mgvavb : 7 5 9 6 6 0 0 4 5 5 7 6 0 0 7 5 9 6 6 kZ Di : 4557600|

YKi GKK I `kKi Ni kb (0) _vKvq GKK I `kKi Y `Lvbv nqwb| ZvB GKK I `kKi Ni kb ewmq kZKi Y y`B Ni evg _K jLv nqQ|

D`vniY 6| Y Ki : 7 6 9 D`vniY 7| Y Ki : 4 5 8 6 2 4 8 4 7 9 mgvavb : 7 6 9 mgvavb : 4 5 8 6 2 4 8 4 7 9 6 1 5 2 4 1 2 7 4 3 0 7 6 0 3 2 1 0 2 0 1 5 3 8 0 0 1 8 3 4 4 0 0 1 9 0 7 1 2 2 1 9 6 6 9 4

Di : 190712| Di : 2196694|

ge : wkv_xiv cPwjZ wbqg Y A Kie| Zv`i wbKU Kvb evLv PvIqv ne bv|

MwYZ 12

D`vniY 8| Y Ki : 3 7 9 2 D`vniY 9| Y Ki : 9 2 0 7 8 5 0 6 4 5 mgvavb : 3 7 9 2 mgvavb : 9 2 0 7 8 5 0 6 4 5 1 8 9 6 0 0 4 6 0 3 5 3 0 3 3 6 0 0 3 6 8 2 8 0 3 2 2 3 2 0 0 5 5 2 4 2 0 0 5 9 3 8 5 1 5

Di : 3223200| Di : 5938515|

D`vniY 10| Y Ki : 4 9 0 8 D`vniY 11| Y Ki : 8 0 7 5 4 0 0 7 0 0 mgvavb : 4 9 0 8 mgvavb : 8 0 7 5 4 0 0 7 0 0 1 9 6 3 2 0 0 5 6 5 2 5 0 0 Di : 1963200| Di : 5652500|

D`vniY 12| mnR cwZZ Y Ki : (K) 9999 425 (L) 8254 999 mgvavb : (K) 9999 425 (L) 8254 999 = (100001) 425 = 8254 (10001) = (10000 425) (1425) = (82541000) (82541) = 4250000 425 = 8254000 8254 = 4249575 = 8245746 Di : 4249575| Di : 8245746 |

MwYZ 13

D`vniY 13| GK ewi `wbK Avq 216 UvKv| GK eQi Zuvi Avq KZ?

[1 eQi = 365 w`b]

mgvavb : 1 eQi = 365 w`b 1 w`b Avq Kib 216 UvKv 365 w`b Avq Kib (216 365) UvKv|

GLvb, 3 6 5 2 1 6 2 1 9 0 3 6 5 0 7 3 0 0 0 7 8 8 4 0 1 eQi Avq Kib 78840 UvKv| Di : 78840 UvKv| ge : Y Kivi myweav_ QvU msLvwUK YK aiv nqQ|

D`vniY 14| GKwU evMvb 2605wU mycvwi MvQ AvQ| Gic 316wU evMvb KZwU mycvwi MvQ AvQ ?

mgvavb : 1wU evMvb mycvwi MvQ AvQ 2605wU| 316wU " " " (2605 316) wU

GLvb, 2 6 0 5 3 1 6 1 5 6 3 0 2 6 0 5 0 7 8 1 5 0 0 8 2 3 1 8 0

Di : 823180 wU mycvwi MvQ AvQ|

MwYZ 16

D`vniY 5| 98027 K 132 viv fvM Ki| D`vniY 6| 49735 K 245 viv fvM Ki| mgvavb : 132) 9 8 0 2 7 (742 245) 4 9 7 3 5 (203

9 2 4 4 9 0 5 6 2 7 3 5 5 2 8 7 3 5 3 4 7 0 2 6 4 Di : fvMdj 203| 8 3 Di : fvMdj 742, fvMkl 83|

10 A_ev 100 w`q fvM

D`vniY 7| 24657 K 10 w`q fvM Ki|

mgvavb : 10) 2 4 6 5 7 (2465 2 0 4 6 4 0 6 5 6 0 5 7 5 0 7 Di : fvMdj 2465, fvMkl 7|

fvRK 1 Gi Wvb GKwU kb (0) AvQ| fvRi Wvbw`K _K GKNi ci Kgv emvj cvB 2465, 7| GLvb Kgvi evgw`Ki msLvwU fvMdj Ges Wvbw`Ki msLvwU fvMkl| KvRB, 10 w`q h Kvb msLv fvM Kivi mgq Gfve Kgv ewmq fvMdj I fvMkl wbYq Kiv mnR nq|

D`vniY 6 G fvMi wZxq avc 73, fvRK 245 _K QvU| ZvB fvMdj `kKi Ni kb (0) emvbv nqQ| Gici fvRi 5, 73 Gi cvk wjL 735 nqQ Ges 735, 245 viv wbtkl wefvR nqQ|

MwYZ 18

D`vniY 12| fvR 37024, fvMdj 89 Ges fvMkl 0 nj fvRK KZ? mgvavb : 89) 3 7 0 2 4 (416 3 5 6 1 4 2 8 9 5 3 4 5 3 4 0 Di : fvRK 416| wbtkl wefvRi

fvR fvRK = fvMdj fvR fvMdj = fvRK fvRK fvMdj = fvR

wbtkl wefvR bv nj fvR = fvRK fvMdj + fvMkl fvRK = (fvR fvMkl) fvMdj fvMdj = (fvR fvMkl) fvRK

D`vniY 13| Kvb fvM A fvR 28087, fvRK 264 Ges fvMkl 103 nj fvMdj KZ?

mgvavb : Avgiv Rvwb, fvMdj = (fvR fvMkl) fvRK GLvb, fvR fvMkl = 28087 103 = 27984 fvMdj = 27984 264 = 106 Di : fvMdj 106| D`vniY 14| 325 KwR Pvji `vg 5200 UvKv nj cwZ KwR Pvji `vg KZ? mgvavb : 325) 5 2 0 0 (16 3 2 5 1 9 5 0 1 9 5 0 0 cwZ KwR Pvji `vg 16 UvKv| Di : 16 UvKv|

MwYZ 20

ZZxq Aavq

cv_wgK Pvi wbqg mwKZ mgmvewj

D`vniY 1| 8 WRb LvZvi `vg 2400 UvKv| 1wU LvZvi `vg KZ?

mgvavb : 1 WRb = 12wU

8 WRb = (8 12 ) wU = 96 wU GLvb, 8 WRb A_vr 96wU LvZvi `vg 2400 UvKv

1wU LvZvi `vg (2400 96) UvKv GLb, 96) 2 4 0 0 (25 1 9 2 4 8 0 4 8 0 0 1wU LvZvi `vg 25 UvKv Di : 25 UvKv

D`vniY 2| wcZv I Kbvi eqmi mgw 80 eQi| wcZvi eqm Kbvi eqmi 4 Y nj, Kvi eqm KZ?

mgvavb : Kbvi eqm = Kbvi eqmi 1 Y wcZvi eqm = Kbvi eqmi 4 Y wcZv I Kbvi eqmi mgw = Kbvi eqmi 5 Y Kbvi eqmi 5 Y = 80 eQi

Kbvi eqm = 80 eQi 5 = 16 eQi| wcZvi eqm = 16 eQi 4 = 64 eQi| Di : wcZvi eqm 64 eQi Ges Kbvi eqm 16 eQi|

MwYZ 21

D`vniY 3| Kvb fvM A fvRK 78, fvMdj 25 Ges fvMkl 63, fvR KZ?

mgvavb : Avgiv Rvwb, fvR = fvRK fvMdj + fvMkl = 78 25 + 63 = 1950 + 63 = 2013 Di : fvR 2013|

D`vniY 4| 6wU Pqvi I 4wU Uweji gj GK 5570 UvKv| GKwU Pqvii gj 675 UvKv nj, GKwU Uweji g j KZ?

mgvavb : 1wU Pqvii gj 675 UvKv 6wU Pqvii gyj (675 6)UvKv = 4050 UvKv 6wU Pqvi I 4wU Uweji gj GK = 5570 UvKv Q 6wU Pqvii gj = 4050 UvKv 4wU Uweji g j = 1520 UvKv (weqvM Ki) 1wU Uweji g j = (1520 4) UvKv = 380 UvKv Di : 380 UvKv|

D`vniY 5| wgbv I wibvi GK 7532 UvKv AvQ| wgbvi wbKU wibv Acv 560 UvKv ewk _vKj wgbv I wibvi UvKvi cwigvY KZ?

mgvavb : gvU UvKv _K wgbvi ewk UvKv ev` w`j y`BRbi UvKvi cwigvY mgvb ne|

7532 UvKv- 560 UvKv = 6972 UvKv. wibvi AvQ (6972 2) UvKv ev, 3486 UvKv myZivs, wgbvi AvQ (3486 + 560 ) UvKv ev, 4046 UvKv Di : wgbvi 4046 UvKv Ges wibvi 3486 UvKv|

MwYZ 22

D`vniY 6| AvjZvd mvnei gvwmK eZb 9870 UvKv| wZwb cwZ gvm evwo fvov 3800 UvKv I msmvi LiP eve` 5650 UvKv eq Kib| evwK UvKv evsK Rgv ivLb| AvjZvd mvnei eQi KZ UvKv Rgv nq?

mgvavb : evwo fvov I msmvi LiP eve` eq nq (3800 + 5650) UvKv ev, 9450 UvKv cwZ gvm evsK Rgv ivLb (9870 9450) UvKv = 420 UvKv 1 eQi Rgv nq (420 12) UvKv = 5040 UvKv Di : 5040 UvKv|

Abykxjbx- 3

1| 9 WRb cwji `vg 1620 UvKv| 1wU cwji `vg KZ?

2| 38 nvwj Avgi `vg 1216 UvKv| 1wU Avgi `vg KZ?

3| 12 WRb LvZvi `vg 2304 UvKv| 8wU LvZvi `vg KZ?

4| wcZv I cyi eqmi mgw 96 eQi| wcZvi eqm cyi eqmi 3 Y nj, Kvi eqm KZ?

5| GKwU SzwoZ 168 wU Avg AvQ| Gic 15wU SzwoZ hZjv Avg AvQ Zv _K gxbvK 780 wU Ges ivwYK 750 wU Avg `Iqv nj| evwK Avg ivRyK `Iqv nj, ivRy KZwU Avg cj?

6| Kvb fvM A fvR 8903, fvRK 87 Ges fvMkl 29 nj fvMdj KZ?

7| Kvb fvM A fvRK fvMkli 12 Y Ges fvR 9896| fvMkl 8 nj fvMdj KZ?

8| y`BwU msLvi Ydj 6272, GKwU msLvi 4 Y 256 nj Aci msLvwU KZ?

9| 2wU Mi I 3wU Lvwmi gj GK 9924 UvKv| 1wU Lvwmi gj 1280 UvKv nj 1wU Mi i g j KZ?

MwYZ 24

PZz_ Aavq HwKK wbqg

D`vniY 1| 1 wU LvZvi `vg 7 UvKv nj 5 wU LvZvi `vg KZ ?

mgvavb : 1wU LvZvi `vg 7 UvKv 5 (7 5) UvKv = 35 UvKv Di : 35 UvKv|

j Kwi : 1 wU LvZvi `vg 7 UvKvK 5 w`q Y Ki 5 wU LvZvi `vg cvIqv Mj| 1 wU wRwbmi `vgK wbw` msLv w`q Y Kij, H wbw` msLK wRwbmi `vg cvIqv hvq|

mwVK Di () wUK wP `vI : (1) 1 wU cbwmji `vg 3 UvKv nj 6 wU cbwmji `vg KZ? K. 2 UvKv L. 9 UvKv M. 18 UvKv N. 12 UvKv (2) 1 wU LvZvi `vg 5 UvKv nj 10 wU LvZvi `vg KZ? K. 15 UvKv L. 50 UvKv M. 2 UvKv N. 20 UvKv

D`vniY 2| 3 wU LvZvi `vg 21 UvKv nj, 1 wU LvZvi `vg KZ ? mgvavb : 3 wU LvZvi `vg 21 UvKv

1 321 UvKv

= 7 UvKv Di : 7 UvKv|

j Kwi : 3 wU LvZvi `vg 21 UvKvK 3 w`q fvM Ki 1wU LvZvi `vg cvIqv Mj|

GK RvZxq KqKwU wRwbmi `vg, IRb, `N `Iqv _vKj ZvK wRwbmi msLv w`q fvM Kij, 1wU wRwbmi `vg, IRb, `N cvIqv hvq|

MwYZ 25

mwVK Di () (wUK) `vI| (1) 6 wU cwji `vg 12 UvKv nj 1 wUi `vg KZ? K. 72 UvKv L. 3 UvKv M. 2 UvKv N. 24 UvKv (2) 5 wU Avgi `vg 35 UvKv nj 1 wUi `vg KZ? K. 7 UvKv L. 175 UvKv M. 15 UvKv N. 3 UvKv (3) 3 KwR Pvji `vg 60 UvKv nj 1 KwRi `vg KZ? K. 15 UvKv L. 20 UvKv M. 30 UvKv N. 12 UvKv

D`vniY 3| 10 wU cbwmji `vg 30 UvKv nj 8 wU cbwmji `vg KZ ? mgvavb : 10 wU cbwmji `vg 30 UvKv 1 (30 10) UvKv = 3 UvKv 8 (3 8) UvKv = 24 UvKv Di : 24 UvKv| j Kwi : (1) 10 wU cbwmji `vgK 10 w`q fvM Ki c_g 1wU cbwmji `vg wbYq Kiv

nqQ| ci 1 wU cbwmji `vgK 8 w`q Y Ki 8 wU cbwmji `vg wbYq Kiv nqQ|

(2) GLvb cbwmji `vg wbYq KiZ ne ZvB A mvRvbvi mgq `vgK mekl ivLv nqQ|

Gfve c_g GKwUi `vg ei Ki mgmvi mgvavb Kiv nq ej GB cwZK HwKK wbqg ejv nq| HwKK wbqg h ivwkwU ei KiZ ne m ivwkwU AekB A mvRvbvi mgq c_g jvBbi kli w`K (Wvbw`K) ivLZ ne|

D`vniY 4| 9wU eBqi `vg 108 UvKv nj 12wU eBqi `vg KZ ? mgvavb : 9wU eBqi `vg 108 UvKv 1 (108 9) UvKv = 12 UvKv 12 (12 12) UvKv = 144 UvKv Di : 144 UvKv|

MwYZ 26

D`vniY 5| GKk wjPzi `vg 80 UvKv nj 25 wU wjPzi `vg KZ ?

mgvavb : 100 wU wjPzi `vg 80 UvKv ev 8000 cqmv 1 (8000 100) cqmv = 80 cqmv 25 (8025) cqmv = 2000 cqmv = 20 UvKv

Di : 20 UvKv|

D`vniY 6| cwZwU mgvb IRbi 1 WRb Avcji IRb 72 Mvg nj Gic 9wU Avcji IRb KZ ?

mgvavb : 12wU Avcji IRb 72 Mvg

1 (7212) Mvg = 6 Mvg 9 (69) Mvg = 54 Mvg Di : 54 Mvg|

D`vniY 7| GKRb kwgK mvn 875 UvKv Avq Kib| 12 w`b wZwb KZ UvKv Avq Kieb ?

mgvavb : 7 w`b Avq Kib 875 UvKv

1 (8757) UvKv = 125 UvKv

12 (125 12) UvKv = 1500 UvKv Di : 1500 UvKv|

MwYZ 27

D`vniY 8| GKwU Qvvevm 40 Rb Qvi 20 w`bi Lv` AvQ| hw` Qvvevm AviI 10 Rb bZzb Qv hvM `q, Ze H Lv` Zv`i KZ w`b Pje ?

mgvavb : 10 Rb bZzb Qv Avmvq gvU Qv msLv = (40+10) = 50|

40 Rb Qvi Pj 20 w`b

1 (2040) w`b = 800 w`b

50 (80050) w`b = 16 w`b

Di : 16 w`b|

j Kwi : GKB cwigvY Lv` LvIqvi Rb Qvi msLv Kgj w`bi msLv evo, ZvB wZxq avc Y Kiv nqQ| Avevi Qvi msLv evoj w`bi msLv Kg ZvB ZZxq avc fvM Kiv nqQ|

D`vniY 9| 6 Rb jvK 21 w`b Kvb Rwgi dmj KvUZ cvi| 18 Rb jvK KZ w`b H Rwgi dmj KvUZ cvie ?

mgvavb : 6 Rb jvK dmj KvUZ cvi 21 w`b

1 (21 6) w`b = 126 w`b

18 (126 18) w`b = 7 w`b

Di : 7 w`b|

MwYZ 28

D`vniY 10| GKwU cyKzi Lbb KiZ 200 Rb jvKi 15 w`b jvM| cyKziwU 10 w`b Lbb KiZ PvBj AwZwi KZRb wbqvM Kiv cqvRb ?

mgvavb : 15 w`b cyKziwU Lbb Ki 200 Rb jvK

1 (200 15) Rb jvK = 3000 Rb jvK

10 (3000 10) Rb jvK = 300 Rb jvK|

AwZwi jvK cqvRb (300 200) Rb = 100 Rb

Di : AwZwi 100 Rb jvK wbqvM KiZ ne|

Abykxjbx- 4

1| mwVK DiwUK wUK () wP `vI : (1) 1wU LvZvi `vg 8 UvKv nj 4wU LvZvi `vg wbYq KiZ ne; 8 Avi 4 hvM

Ki / Y Ki/ fvM Ki/ weqvM Ki|

(2) 1 Rvov KeyZii `vg 100 UvKv nj 3 Rvov KeyZii `vg wbYq KiZ ne 100 Avi 3 Y Ki /fvM Ki/ hvM Ki/ weqvM Ki|

(3) 8wU cbwmji `vg 24 UvKv nj 1wU cbwmji `vg wbYq KiZ ne 24K 8 w`q Y Ki / hvM Ki/ weqvM Ki/ fvM Ki|

(4) 6wU LvZvi `vg 24 UvKv nj 3wU LvZvi `vg KZ ne Zv wbYq KiZ nj 6wU LvZvi `vg _K c_g 2wU /5wU /1wU / 3wU LvZvi `vg ei KiZ ne|

(5) 6wU LvZvi `vg 24 UvKv nj 1wU LvZvi `vg Kg ne / mgvb ne / ewk ne / wY ne|

MwYZ 29

(6) 3wU SzwoZ 48 wU Avg ai| 9wU SzwoZ KZjv Avg aie Zv wbYq KiZ nj c_g 3wU SzwoZ aiv msLv _K 2wU/1wU/3wU/9wU SzwoZ aiv Avgi msLv ei Ki wbZ ne|

(7) 3wU SzwoZ aiv 48 wU Avg _K 1wU SzwoZ aiv Avgi msLv wbYq KiZ ne 48 K 3 w`q fvM Ki / hvM Ki/ Y Ki/ weqvM Ki|

2| kbvb ciY Ki : (1) 1wU wRwbmi `vgK wbw` msLv w`q Kij H wbw ` msLK

wRwbmi `vg cvIqv hvq|

(2) 1 WRb Kjvi `vg 30 UvKv nj 3 WRb Kjvi `vg UvKv| (3) 10wU wWgi `vg 40 UvKv nj 1wU wWgi `vg UvKv| (4) 6wU cbwmji `vg 24 UvKv nj 1wU cbwmji `vg UvKv| (5) 1wU cbwmji `vg 4 UvKv nj 3wU cbwmji `vg UvKv| (6) 3wU SzwoZ ai 48 wU Avg| 1wU SzwoZ ai wU Avg| (7) 1wU SzwoZ ai 8wU Avg| 9wU SzwoZ ai wU Avg| (8) Szwoi msLv KgvZ Avgi msLv hve| (9) Szwoi msLv evovZ Avgi msLv hve| 3| mwVK Di () wUK wP `vI (1) 5wU cbwmji `vg 20 UvKv nj 1wU cbwmji `vg (K) 15 UvKv (L) 4 UvKv (M) 25 UvKv (N) 12 UvKv (2) 1wU LvZvi `vg 8 UvKv nj 8wUi `vg (K) 1 UvKv (L) 16 UvKv (M) 64 UvKv (N) 20 UvKv (3) 8wU cbwmji `vg 24 UvKv nj 4wU cbwmji `vg (K) 12 UvKv (L) 32 UvKv (M) 16 UvKv (N) 28 UvKv (4) 1 WRb wWgi `vg 48 UvKv nj 6wUi `vg (K) 24 UvKv (L) 42 UvKv (M) 72 UvKv (N) 64 UvKv

MwYZ 30

4| 15 KwR PvDji `vg 300 UvKv nj, 8 KwR PvDji `vg KZ ?

5| 12 wgUvi Kvcoi `vg 120 UvKv nj, 15 wgUvi Kvcoi `vg KZ ?

6| 4 wU LvZvi `vg 28 UvKv nj, 11 wU LvZvi `vg KZ ?

7| GK WRb Kjvi `vg 60 UvKv nj, 22 wU Kjvi `vg KZ ?

8| GKk wjPzi `vg 90 UvKv nj, 20 wU wjPzi `vg KZ ?

9| 24 KwR Wvji `vg 1200 UvKv nj, 13 KwR Wvji `vg KZ ?

10| 5 KwR Lvwmi Mvi `vg 600 UvKv nj, 8 KwR Lvwmi Mvi `vg KZ ?

11| 6 wU Pqvii `vg 1200 UvKv nj, 9wU Pqvii `vg KZ ?

12| 20 Rb jvK GKwU KvR 15 w`b KiZ cvi| 15 Rb jvK H KvR KZ w`b KiZ cvie ?

13| 16 Rb jvK GKwU KvR 5 w`b KiZ cvi| H KvR 20 Rb jvK KZ w`b KiZ cvie ?

14| 8 Rb jvK 21 w`b Kvb Rwgi dmj KvUZ cvi| 14 Rb KZ w`b H Rwgi dmj KvUZ cvie ?

15| 200 Rb jvK h Lv` 20 w`b LZ cvi, KZRb jvK m Lv` 40 w`b LZ cvie ?

16| GKRb kwgK mvn 490 UvKv Avq Kib| 1050 UvKv Avq KiZ Zuvi KZ w`b jvMe ?

17| 45 Rb jvKi h PvDj 20 w`b Pj KZRb jvKi H PvDj 24 w`b Pje ?

18| GKwU Qvvevm 16 Rb Qvi 25 w`bi Lv` AvQ| KqKRb bZzb Qv Avmvq 20 w`b Lv` kl nq Mj| bZzb Qvi msLv KZ ?

19| Kvb wkwei 1200 mbi 28 w`bi Lv` AvQ| H wkwei nZ 400 mb Pj Mj evwK mbi H Lv` KZ w`b Pje ?

MwYZ 32

cg Aavq

mij

D`vniY 1| mij Ki : 48 65 + 378 + 39 116 7 147 + 248 mgvavb : 48 65 + 378 + 39 116 7 147 + 248 = (48 + 378 + 39 + 248) (65 + 116 + 7 + 147) = 713 335 = 378 Di : 378|

j Kwi : (K) hvMevaK msLvjvK GK hvM Kiv nqQ ; 48 + 378 + 39 + 248 = 713 (L) weqvMevaK msLvjvK GK hvM Kiv nqQ ; 65 + 116 + 7 + 147 = 335 (M) c_g hvMdj _K wZxq hvMdj weqvM Kiv nqQ ; 713 335 = 378 (N) GKB wPhy msLvK GKwZ Kivi Rb ( ) ebx eenvi Kiv nqQ| D`vniY 2| mij Ki : 12 + 29 64 8 3 52 13 8 30 6 mgvavb : 12 + 29 64 8 3 52 13 8 30 6 = 12 + 29 8 3 4 8 5 = 12 + 232 3 4 40 = (12 + 696) (4 + 40) = 708 44 = 664 Di : 664|

j Kwi : AwUZ hvM, weqvM, Y I fvMi KvR iqQ| c_g fvMi I ci Yi Ges mekl hvM I weqvMi KvR Kiv nqQ|

MwYZ 33

D`vniY 3| mij Ki : 5 24 8 3 + 5 17 + 18 3 mgvavb : 5 24 8 3 + 5 17 + 18 3 = 5 3 3 + 5 17 + 6 = 5 9 + 5 17 + 6 = 45 + 5 17 + 6 = (45 +5 + 6) 17 = 56 17 = 39 Di : 39| D`vniY 4| mij Ki : 25 5 [5 {24 (18 15)}] mgvavb : 25 5 [5 {24 (18 15)}] = 25 5 [5 {24 3}] = 25 5 [5 8] = 25 5 40 = 5 40 = 200 Di : 200|

j Kwi : AwUZ c_g ebx ( ), wZxq ebx { }, ZZxq ebx [ ], fvM, Y I

weqvMi KvR iqQ| c_g c_g ebx, ci wZxq ebx Ges kl ZZxq ebxi KvR Kiv nqQ| D`vniY 5| mij Ki : 38 [13 (73 46 2 3) {45 3 (7 + 21 7)}] mgvavb : 38 [13 (73 46 2 3) {45 3 (7 + 21 7)}] = 38 [13 (73 23 3) {45 3 (7 + 3)}] = 38 [13 (73 69) {45 3 10}] = 38 [13 4 {45 30}] = 38 [13 4 15] = 38 [52 15] = 38 37 = 1 Di : 1|

MwYZ 34

D`vniY 6| mij Ki : 78 [56 + {165 (48 6 9) 2}] mgvavb : 78 [56 + {165 (48 6 9) 2}] = 78 [56 + {165 (8 9) 2}] = 78 [56 + {165 72 2}] = 78 [56 + {165 144}] = 78 [56 + 21] = 78 77 = 1 Di : 1|

D`vniY 7| mij Ki : 48 [4 + 28 {4 + 12 (7 3)}] mgvavb : 48 [4 + 28 {4 + 12 (7 3)}] = 48 [4 + 28 {4 + 12 4}] = 48 [4 + 28 {4 + 3}] = 48 [4 + 28 7] = 48 [4 + 4] = 48 8 = 6 Di : 6|

Abykxjbx-5

mij Ki : 1| 80 + 47 43 + 24 49 2| 688 589 + 321 301 + 81 3| 56 7 9 + 121 11 4| 2 3 18 6 45 9 5| 16 4 + 168 8 65 + 48 6| 13 4 + 132 12 7 52 13 12 3 5 7| 49 7 64 8 + 81 9 42 6 8| 12 14 + 85 5 41 4 6 3

MwYZ 35

9| 32 2 4 + 12 3 6 5 7 + 15 8 3 10| 12 6 6 2 + 15 5 11| 16 4 + 15 3 2 12| 9 37 6 42 + 4 82 3 32 + 8 2 3 13| (18 9) 5 + (20 + 3 15) 14| (36 3) {4 (5+4 8 + 1)} 15| 98 [25 + 3 (11 + 3) {36 3 (12 7)}] 16| 405 {146 + (62 4 93 + 15 + 3)} 17| 7 {502 (25 5 5 23)} + 251 18| 36 6 {7 (15 9 11 + 7)} 19| 65 [25 + 2 (13 + 8) {25 2 (13 8)}] 20| 81 [6 + 39 {4 + 90 (33 23)}] 21| 270 5 [4 + 81 {93 (60 5) (35 28)}] 22| 107 + 91 13 {77 (85 5 3 + 21)} 105 23| 82 + [3 {(99 11) (18 6)}] 140 24| 49 [5 + 24 {5 + 70 (33 23)}] 25| 82 [13 (73 46 2 3) {45 3 (7 + 21 7)}] 26| [105 + 91 13 {77 (85 5 3 + 21)}] 106 27| 130 5 [4 + 81 {93 (60 5) (35 28)}] 28| [36 {2 + 28 (4 + 12 4)}] 6

MwYZ 36

l Aavq

Mo

Icii QweZ c_g SzwoZ 11wU, wZxq SzwoZ 6wU, ZZxq SzwoZ 13wU Avg `Lv hvQ| eo GKwU SzwoZ H wZbwU Szwoi Avg GK ivLv nj; GZ eo SzwoZ gvU (11+6+13) wU = 30wU Avg nj| hw` wZbwU Szwoi cZKwUZ 10wU Ki Avg _vKZ, ZvnjI Szwo wZbwUi Avgi gvU msLv 10+10+10 = 30 nZ| G Aevq Avgiv ewj, Szwo wZbwUZ Mo 10wU Avg wQj| GLvb 10 = 30 3 = Szwo wZbwUZ Avgi gvU msLv Szwoi msLv| mvaviYfve,

GKB RvZxq GKvwaK ivwki Mo = ivwkjvi hvMdj ivwkjvi msLv Icii D`vniY wZbwU SzwoZ gvU Avg wQj 30 wU Ges SzwojvZ Mo wQj 10wU Avg| GLvb 103 = 30| mvaviYfve,

Mo ivwkjvi msLv = ivwkjvi hvMdj

11 6 13

30

10 10 10

1g Szwo 2q Szwo 3q Szwo

MwYZ 37

Mo wbYq D`vniY 1| wgbv, wgVy, iyby, iwb I bxjvi eqm h_vg 10, 11, 12, 13 I 14 eQi|

Zvì Mo eqm KZ ?

mgvavb : GLvb ivwkjvi hvMdj = (10+11+12+13+14) eQi = 60 eQi ivwkjvi msLv = 5 myZivs eqmi Mo = 60 eQi 5 = (60 5) eQi = 12 eQi| Di : wbYq Mo eqm 12 eQi|

D`vniY 2| GKRb wKU Ljvqvoi QqwU IqvbW gvPi wmwiR msMnxZ ivb nQ 23, 12, 37, 3, 14, 13| wZwb H wmwiR Mo KZ ivb KiQb?

mgvavb : gvU ivb = 23+12+37+3+14+13 = 102 gvPi msLv = 6 myZivs ivbi Mo = 102 6 = 17 Di : wZwb H wmwiR Mo 17 ivb KiQb|

D`vniY 3| evsjv`k Kvb vbi 2004 mvji evi gvm ewcvZi cwigvY wbgic:

gvm ewcvZi cwigvYRvbyqvwi 23 wg.wgdeyqvwi 06 wg.wg

gvP 18 wg.wgGwcj 57 wg.wg

g 31 wg.wgRyb 108 wg.wg

RyjvB 296 wg.wgAvM 271 wg.wgmî 104 wg.wgAvei 33 wg.wgbfi^ 21 wg.wgwWmî 04 wg.wg

H vb 2004 mvj gvwmK Mo ewcvZi cwigvY KZ ?

MwYZ 38

mgvavb : ewcvZi gvU cwigvY = (23+06+18+57+31+108+296+271+104+33+21+04) wg.wg

= 972 wg.wg gvmi msLv = 12 myZivs ewcvZi gvwmK Mo = (972 wg.wg) 12 = (972 12) wg.wg. = 81 wg.wg. Di : H vb 2004 mvj gvwmK Mo ewcvZi cwigvY 81 wg.wg.|

D`vniY 4| GKwU h_ Kvievi njb 15,000 UvKv, Kvgvj 20,500 UvKv, Wvwjg 18,000 UvKv, wZZvm 13,500 UvKv Ges wkwki 17,000 UvKv wewbqvM Kij| Kvievi Zviv Mo KZ UvKv wewbqvM Kij ?

mgvavb : 15,000 20,500 18,000 13,500 17,000 hvMdj : 84,000 myZivs gvU wewbqvM = 84,000 UvKv| wewbqvMKvixi msLv = 5 myZivs Mo wewbqvM = 84,000 UvKv 5 = (84,000 5) UvKv = 16,800 UvKv| Di : 16,800 UvKv|

D`vniY 5| cuvPwU gvPi GKwU U wmwiRi wKU Ljvq mdiKvix `ji QqRb evUmgvbi ivbi Mo 76, PviRb evjvii ivbi Mo 21| H wmwiR H Ljvqvoiv Mo KZ ivb KiwQjb ?

evUmgvb`i cuvPwU Ljvq msMnxZ gvU ivb = 6765 = 7630 = 2280 evjvi`i cuvPwU gvP msMnxZ gvU ivb = 4215 = 2120 = 420 megvU ivb = 2280+420 = 2700

MwYZ 39

Mo wbYq weeP ivwki msLv = Ljvqvoi msLv gvPi msLv = = 10 5 = 50

myZivs wbYq Mo = 2700 50 = 54 Di : H wmwiR mdiKvix `ji Ljvqvoiv cuvPwU Ljvq Mo 54 ivb KiwQjb|

D`vniY 6| ZiwU msLvi hvMdj 1924| G`i mvZwU msLvi Mo 172| Ab QqwU msLvi Mo KZ? meMyjv msLvi Mo KZ ?

mgvavb : 7wU msLvi hvMdj = 1727 = 1204 myZivs Ab 6wU msLvi hvMdj = 19241204 = 720 AZGe 6wU msLvi Mo = 720 6 = 120 mejv, A_vr H 13wU msLvi Mo = 1924 13 = 148 Di : Ab QqwU msLvi Mo 120| mejv msLvi Mo 148|

D`vniY 7| ebfvRb hvevi Dk GKwU KviLvbvi 60 Rb KgPvixi cZK fvovi UvKv mgvbfve enb Kivi kZ 2400 UvKvq GKwU evm fvov Kiv nj| wKQy KgPvix ebfvRb bv hvIqvq gv_v wcQy fvov 8 UvKv ewk coj| KZRb KgPvix ebfvRb wMqwQjb ?

mgvavb : 60 Rb KgPvixi mevB ebfvRb Mj gv_v wcQy fvov coZ 2400 UvKv 60 = 40 UvKv|

wKQy KgPvix bv hvIqvq gv_v wcQy fvov coj 40 UvKv + 8 UvKv = 48 UvKv| gv_v wcQy fvov ebfvRb AskMnYKvixi msLv = 2400 UvKv| myZivs ebfvRb AskMnYKvix KgPvixi msLv = 2400 48 = 50 48 ) 2400 ( 50 2400 0 Di : 50 Rb KgPvix ebfvRb wMqwQjb|

MwYZ 40

Abykxjbx- 6

1| Mo wbYq Ki : (K) 23, 37, 47, 61 (L) 22, 46, 60, 72 (M) 364 UvKv, 541 UvKv, 775 UvKv| (N) 47 m.wg., 49 m.wg., 54 m.wg., 52 m.wg., 53 m.wg.| (O) 42 KwR, 32 KwR, 37 KwR, 29 KwR, 41 KwR, 35 KwR| (P) 90 wjUvi, 92 wjUvi, 89 wjUvi, 86 wjUvi, 93 wjUvi, 85 wjUvi, 95 wjUvi| 2| wjwj wewfb `vgi AvUwU wdZv wKbj| wdZvjvi Mo g ~j 5 UvKv 75 cqmv

nj, wjwj KZ UvKvi wdZv wKbwQj ? 3| gwj evwlK cixvq evsjvq 68, MwYZ 96, BsiwRZ 81, cwiek cwiwPwZ-

wevb 77 Ges cwiek cwiwPwZ-mgvR 73 bî cqQ| m cwZ welq Mo KZ bî cqQ ?

4| Kwj, Wwj, cwj, gwj I wjwji DPZv h_vg 123 m.wg., 131 m.wg., 135 m.wg., 126 m.wg. Ges 130 m.wg.| Zvì Mo DPZv KZ ?

5| Acy I `xcyi Mo eqm 22 eQi, `xcy I wUcyi Mo eqm 23 eQi| Acyi eqm 21 eQi; `xcyi I wUcyi eqm KZ ?

6| mvZwU msLvi hvMdj 401| c_g wZbwU msLvi Mo 56 Ges kli wZbwU msLvi Mo 58| PZz_ msLvwU KZ ?

7| GMviwU msLvi ga c_g QqwU msLvi Mo 87 Ges kli cuvPwU msLvi Mo 131| mejv msLvi Mo KZ ?

8| GKwU BDwbqbi cuvPwU Mvgi RbmsLv h_vg 1327, 1872, 2187, 2516 Ges 2943| Mvgjvi Mo RbmsLv KZ ?

9| evwlK cixvq `k Rb Qv-Qvxi MwYZi bî wbgic : 76, 61, 87, 56, 42, 64, 73, 68, 50, 73| Gì Mo bî KZ ?

10| wZb mZvb I Zvì wcZvi Mo eqm 17 eQi| H wZb mZvb I Zvì gvZvi Mo eqm 13 eQi| gvi eqm 22 eQi nj wcZvi eqm KZ ?

11| Avlvp gvmi c_g `k w`b XvKvq `wbK Mo 23 wg.wg. wZxq `k w`b `wbK Mo 27 wg.wg., ZZxq `k w`b `wbK Mo 29 wg.wg. Ges gvmi kl w`b 16 wg.wg. ewcvZ nj| H gvm XvKvq `wbK Mo ewcvZi cwigvY KZ ?

MwYZ 41

mg Aavq

M.mv.. I j.mv..

Mwi mvaviY YbxqK (M.mv..)

18, 24, 30 msLv wZbwUi YbxqKjv weePbv Kwi :

18 Gi mKj YbxqK : 1 , 2 , 3 , 6 , 9, 18

24 Gi mKj YbxqK : 1 , 2 , 3 , 4, 6 , 8, 12, 24

30 Gi mKj YbxqK : 1 , 2 , 3 , 5, 6 , 10, 15, 30

AZGe, msLv wZbwUi mvaviY YbxqKjv nQ 1, 2, 3, 6| Gì ga 6 mePq eo| myZivs 18, 24, 30 Gi Mwi mvaviY YbxqK ev M.mv.. nQ 6| msLv wZbwUi mvaviY gwjK YbxqK nQ 2 I 3, hvì Ydj nQ M.mv..i mgvb|

GKvwaK msLvi mvaviY YbxqKjvi ga mePq eowU Zvì Mwi mvaviY YbxqK (M.mv..)

GKvwaK msLvi M.mv.. = Gì mvaviY gwjK YbxqKjvi Ydj, msLvjvi Kvb mvaviY gwjK YbxqK bv _vKj Zvì M.mv.. 1

YbxqKi Aci bvg Drcv`K|

gwjK Drcv`Ki mvnvh M.mv.. wbYq

D`vniY 1| 40, 60 I 75 Gi M.mv.. wbYq Ki|

mgvavb : 40 = 220 = 2210 = 2225 60 = 320 = 3225 75 = 325 = 355

`Lv hvQ h, 40, 60, 75, GB wZbwU msLvi GKgv gwjK Drcv`K (ev YbxqK) 5|

myZivs Gì M.mv.. 5|

Di : 5|

MwYZ 42

D`vniY 2| 48, 72 I 168 Gi M.mv.. wbYq Ki|

mgvavb : 48 = 68 = 23222 = 22223 72 = 89 = 22233 168 = 821 = 22237 `Lv hvQ h, 48, 72, 168 msLv wZbwUi mvaviY gwjK YbxqK (ev

Drcv`K) nQ 2, 2, 2, 3| myZivs; msLv wZbwUi M.mv. = 2223 = 24 Di : 24|

D`vniY 3| 24, 30 I 77 Gi M.mv. wbYq Ki|

mgvavb : 24 = 38 = 3222 = 2223 30 = 56 = 523 = 235 77 = 711 `Lv hvQ h, 24, 30, 77 msLv wZbwUi Kvb mvaviY gwjK Drcv`K

bvB| wKy 1 hKvb msLviB YbxqK| myZivs c` msLvjvi GKgv mvaviY YbxqK 1, myZivs G`i M.mv.. 1|

Di : 1|

GKvwaK msLvi Kvb mvaviY gwjK YbxqK (ev Drcv`b) bv _vKj Zv`i M.mv. 1

fvM-cwqvi gvag M. mv. . wbYq

18 I 84 Gi M. mv. . wbYqi c_g avc Avgiv eo msLvwUK QvU msLv w`q fvM Kwi; GLvb 84 K 18 w`q fvM Ki fvMkl cvB 12| wZxq avc QvU msLvwUK cv fvMkl 12 viv fvM Kwi; GLb fvMkl nj 6| ZZxq avc c eeZx avci fvM kl 12 K H avci fvMkl 6 viv fvM Kwi| 12 = 62 ej fvMkl kb nj Ges

MwYZ 43

fvM cwqvi mgvw NUj| mekl fvRKB c` msLv y`BwUi M.mv..| wbP GB

cwqvwU cPwjZ AvwK `Lvbv nj| 18) 84 (4 72 12) 18 (1 12 6) 12 (2 12 0 evLv : me`v fvR = fvMdj fvRK + fvMkl GB D`vniYi wZbwU avc nQ : (K) 84 = 418 +12 (L) 18 = 112 + 6 (M) 12 = 26 (M) _K `wL h 6, 12 Gi GKwU YbxqK| myZivs (L) _K `wL h 6, 18 Gi GKwU YbxqK| Gici (K) _K `wL h 6, 84 Gi GKwU YbxqK| AZGe 6, 18 I 84 Gi mvaviY YxqK|

wecixZg, (K) _K evSv hvq h 84 I 18 Gi cZK mvaviY YbxqK 12 GiI YbxqK| myZivs (L) `L evSv hvq h mwU 6 GiI YbxqK| A_vr 18 I 84 Gi h Kvb mvaviY YbxqK 6-Gi YbxqK| AZGe 6 nQ 18 I 84 Gi Mwi mvaviY YbxqK ev M. mv. .| fvM cwqvi gvag M. mv. . wbYqi cwqv BDwKWxq cwqv bvg cwiwPZ| D`vniY 4| 171 I 561 Gi M. mv. . wbYq Ki| mgvavb :

171) 561 (3 513 48) 171 (3 144 27) 48 (1 27 21 ) 27 (1 21 6 ) 21 (3 18 3 ) 6 (2 6 0

wbYq M. mv. . = mekl fvRK = 3Di : 3|

MwYZ 44

BDwKWxq cwqvq M. mv. . wbYq wewfb avc cv fvMdjjvi Kvb fwgKv bvB; mekl fvMklB nQ wbYxZe M. mv. .| BDwKWxq cwqvi GKwU cqvM Kej y`BwU msLvi M. mv. . wbYq Kiv hvq|

D`vniY 5| 143, 231 I 385 Gi M. mv. . wbYq Ki|

mgvavb : c_g 231 I 385 Gi M. mv. . wbYq Kwi| 231 ) 385 (1 231 154 ) 231 (1 154 77 ) 154 (2 154 0 myZivs 231 I 385 Gi M. mv. . 77| GLb 77 I 143 Gi M. mv. . wbYq Kwi| 77 ) 143 (1 77 66 ) 77 (1 66 11 ) 66 (6 66 0 Di : 11| evLv : 143, 231 I 385 msLv wZbwUi h Kvb mvaviY YbxqK 77 I 358 GiI YbxqK| Avevi, 77 I 385 Gi h Kvb YbxqK 143, 231 I 385 GiI YbxqK| AZGe, 143, 231 I 385 Gi M. mv. . = 77 I 358 Gi M. mv. .| ge| D`vniY 5| AviI y`Bfve mgvavb Kiv hZ : c_g 143 I 231 Gi M. mv. . wbYq Ki; ev c_g 143 I 358 Gi M. mv. . wbYq Ki| Avgv`i mgvavb c_g eo msLv y`BwUi M.mv.. ei Kivq mvweK weePbvq KvR mnR nqQ|

MwYZ 45

D`vniY 6| 221, 304 I 418 Gi M. mv. . wbYq Ki| mgvavb : c_g 304 I 418 Gi M. mv. . wbYq Kwi| 304 ) 418 (1 304 114) 304 (2 228 76) 114 (1 76 38) 76 (2 76 myZivs 304 I 418 Gi M. mv. . 38| 0 GLb 38 I 231 Gi M. mv. . wbYq Kwi| 38 ) 231 (6 228 3 ) 38 (12 3 8 6 2) 3 (1 2 1) 2 (2 2 0 myZivs 38 I 231 Gi M. mv. . 1| AZGe c` msLv wZbwUi M. mv. . 1| Di : 1| jwN mvaviY wYZK 18, 24 I 27 Gi wYZKjvi KqKwU jLv : 18 Gi wYZK : 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, 198, 216, 234, BZvw`| 24 Gi wYZK : 24, 48, 72, 96, 120, 144, 168, 192, 216, 240, BZvw`| 27 Gi wYZK : 27, 54, 81, 108, 135, 162, 189, 216, 243, BZvw`| wYZKi mvwi wZbwUi cZKwUZ 216 `LZ cvwQ| myZivs 216 msLvwU 18, 24 I 30 Gi mvaviY wYZK| wZbwU mvwii KvbwUZB 216 Gi Pq QvU Ggb Kvb msLv bvB, hv wZbwU mvwiZB AvQ| myZivs 216 msLvwU 18, 24 I 27 Gi z `Zg ev jwN mvaviY wYZK (j.mv.)|

MwYZ 46

gwjK Drcv`Ki mvnvh j.mv.. wbYq Ki|

D`vniY 7| 18, 24 I 30 Gi j.mv.. wbYq Ki| 18 = 2 3 3 24 = 2 2 2 3 30 = 2 3 5

wZbwU msLvi gwjK Drcv`Kjvi ga 2 AvQ mevwaK wZb evi (24 Gi Drcv`K wnme) Ges 3 AvQ 2 evi (18 Gi Drcv`K wnme)| GQvov 5 AvQ GKevi (30 Gi Drcv`K wnme)| myZivs 18, 24, 30 Gi j.mv..

= 2 2 2 3 3 5 = 4 9 10 = 360

Di : 360|

msw cwZZ j.mv.. wbYq GB cwZZ c` msLvjvi mejvi mKj gwjK Drcv`K GKKwU Ki ei Kiv nq| GBme gwjK Drcv`Kjvi YdjB c` msLvjvi j.mv..|

D`vniY 8| 18, 24 I 40 Gi j.mv.. wbYq Ki|

mgvavb : 2 18, 24, 40

2 9, 12, 202 9, 6, 10

3 9, 3, 5 3, 1, 5

myZivs wbYq j.mv.. = 222335 = 360 (Di)| evLv : gwjK msLv 2 w`q 24 fvM Kiv MQ wZbevi; kl fvMdj nqQ 3| hv gwjK msLv| myZivs 24 = 2223| 2 w`q 18 K fvM Kiv MQ GKevi, fvMdj 9 K gwjK msLv 3 w`q y`Bevi; kl fvMdj nqQ 1| myZivs 18 = 233|

MwYZ 47

2 w`q 40 K fvM Kiv MQ wZbevi; kl fvMdj nqQ 5, hv gwjK msLv| myZivs 40 = 2225| AZGe, 18, 24 I 40 Gi j.mv.. = 222335 = 360| j.mv.. wbYqi msw cwZ gwjK Drcv`Ki mvnvh j.mv.. wbYqiB GKxfZ ic| msw cwZZ cZK avc msLvjvi ga AZ: y`BwU msLvi Drcv`K, Ggb GKwU gwjK msLv w`q fvM Kiv nq| Kvb msLv H avc bqv gwjK Drcv`K viv fvM bv Mj mwU AcwiewZZfve cii avc bvgvZ ne| mekl avci msLvjvi Kvb mvaviY gwjK Drcv`K _vKe bv|

D`vniY 9| 16, 24, 30, 42, 45 Gi j.mv.. wbYq Ki|

mgvavb t 2 16, 24, 30, 42, 452 8, 12, 15, 21, 45

2 4, 6, 15, 21, 453 2, 3, 15, 21, 45

5 2, 1, 5, 7, 15 2, 1, 1, 7, 3

myZivs wbYq j.mv.. = 22235273 = 5040 Di : 5040| M.mv.. I j.mv.My. msv mnR mgmv D`vniY 10| Kvb& eng msLv viv 138, 215, 457 K fvM w`q h_vg 3, 5, 7

Aewk _vK ? mgvavb : 138 3 = 135, 215 5 = 210, 457 7 = 450| myZivs ck u`vovQ, Kvb& eng msLv viv 135, 210 I 450 wbtkl wefvR| GB msLvwU nQ 135, 210 I 450 Gi M.mv.|

210) 450 (2 420 30) 210 (7 210

210 I 450 Gi M.mv.. 30| 0 myZivs wbYq eng msLv 15| Avevi, 30 I 135 Gi M.mv.. 15| Di : 15|

30) 135(4 120 15) 30 (2 30 0

MwYZ 48

D`vniY 11| PviwU Nv c_g GK eR cwZ 6, 9, 12 I 15 mK AZi evRZ jvMj| b~ bZg KZY ci Nvjv Avevi GK evRe ?

mgvavb : wbYq bbZg mgq nQ 6, 9, 12 I 15 Gi j.mv.. hZ ZZ mK| 3 6, 9, 12, 15

2 2, 3, 4, 5 1, 3, 2, 5

myZivs msLv PviwUi j.mv. = 32325 = 1810 = 180| AZGe, Nv PviwU b bZg 180 mK = 3 wgwbU ci Avevi GK evRe| Di : 3 wgwbU| D`vniY 12| Ggb GKwU z `Zg msLv wbYq Ki hvK 6, 10, 15 I 21 viv fvM

w`j cwZ 4 Aewk _vK| mgvavb : wbYq msLvwU nQ 6, 10, 15 I 21 Gi j.mv.. _K 4 ewk|

3 6, 10, 15, 212 2, 10, 5, 7

5 1, 5, 5, 7 1, 1, 1, 7

myZivs msLvjvi j.mv..= 3257 = 2110 = 210| AZGe, wbYq z `Zg msLv = 210+4 = 214| Di : 214| D`vniY 13| Kvb& z `Zg msLvK 4, 6, 10, 16 viv fvM Kij h_vg 2, 4, 8,

14 Aewk _vK|

mgvavb : 42 = 2, 64 = 2, 108 = 2, 1614 = 2 A_vr cwZ, fvRK fvMkl = 2; myZivs wbYq msLvwU 4, 6, 10 I 16 Gi j.mv.. Acv 2 Kg|

2 4, 6, 10, 162 2, 3, 5, 8

1, 3, 5, 4myZivs msLvjvi j.mv.. = 22354 = 1220 = 240| AZGe, wbYq z `Zg msLv = 2402 = 238| Di : 238|

MwYZ 49

D`vniY 14| GKwU AvqZvKvi Nii `N 7.20 wgUvi Ges c 4.40 wgUvi| mevwaK Kvb& mvBRi eMvKvi gvej cv_i viv Nii gS euvavbv hve, hvZ Kvb cv_i fvv bv co ?

mgvavb : Nii `N = 7.20 wg. = 72 W.wg., c = 4.40 wg. = 44 W.wg. cv_i eMvKvi nIqvq Ges Kvb cv_i bv fvvi kZ _vKvq, Gi evi `Ni cwigvc evwUi `N i Ges ci cwigvci mvaviY YbxqK nZ ne| AZGe, 72 Ges 44 M.mv.. hZ, mevwaK mvBRi eMvKvi cv_ii evi `N ZZ WwmwgUvi|

72 = 89 = 22233 44 = 411 = 2211 myZivs 72 I 44 Gi M.mv.. = 22 = 4 4 W.wg. = 40 m.wg.

Di : mevwaK mvBRi eMvKvi cv_ii evi `N 40 m.wg.|

Abykxjbx-7

1| gwjK Drcv`Ki mvnvh M.mv.. wbYq Ki : (K) 144, 192 (L) 160, 275 (M) 112, 343 (N) 24, 60, 72 (O) 45, 75, 120 (P) 22, 77, 110 (Q) 48, 72, 160,240 (R) 35, 56, 84, 119 (S) 30, 50, 90, 140 (T) 36, 48, 84, 156

2| BDwKWxq cwqvq M.mv.. wbYq Ki : (K) 90, 325 (L) 76, 361 (M) 115, 810 (N) 154, 484 (O) 336, 730 (P) 1769, 2378 (Q) 77, 231, 352 (R) 117, 183, 303 (S) 56, 147, 357 (T) 75, 225, 390

3| gwjK Drcv`Ki mvnvh j.mv.. wbYq Ki : (K) 51, 85 (L) 76, 95 (M) 42, 112 (N) 32, 48, 80 (O) 35, 55, 75 (P) 28, 42, 70

MwYZ 50

4| msw cwZZ j.mv.. wbYq Ki : (K) 24, 36, 40 (L) 15, 33, 45 (M) 25, 45, 85 (N) 21, 35, 49, 56 (O) 24, 36, 48, 72 (P) 18, 27, 45, 54 (Q) 26, 44, 77, 99 (R) 12, 18, 24, 30, 42 (S) 15, 25, 35, 45, 55 5| Kvb& eng msLv viv 76, 114 Ges 228 K fvM w`j Kvb fvMkl _vKe

bv? 6| 60 wU Avg I 150 wU wjPz mevwaK KZRb evjK-evwjKvi ga wbtkl fvM Ki

`Iqv hve? cZK KqwU Avg I KqwU wjPz cve ? 7| y`BwU Wvgi aviY gZv h_vg 228 wjUvi I 348 wjUvi| mevwaK KZ aviY

gZvi Kjm cYmsLK evi cvwb w`q Wvg y`BwU fiv hve? Kvb& Wvg KZ Kjm cvwb ai ?

8| Kvb& eng msLv viv 137, 212 I 452 K fvM w`j cwZ 2 Aewk _vK?

9| Kvb& eng msLv viv 129, 236 I 364 K fvM w`j h_vg 3, 5 I 7 Aewk _vK?

10| GKwU AvqZvKvi njNii `N 1225 wgUvi Ges c 770 wgUvi| mevwaK Kvb& mvBRi eMvKvi Uvwj viv NiwUi gS euvavbv hve, hvZ Kvb Uvwj fvv bv co ?

11| Kvb& z `Zg msLv 18, 24, 30, 36 viv wbtkl wefvR ? 12| Kvb& z `Zg msLvK 16, 24, 32, 40 viv fvM Kij cwZ 6 Aewk

_vKe ? 13| Kvb& z `Zg msLvK 12, 18, 30 viv fvM Kij h_vg 6, 12, 24 Aewk

_vKe ? 14| PviwU Nv GK eR ci 5, 7, 12 Ges 15 mK A i evRZ jvMj|

b bZg KZY ci Nvjv cybivq GK evRe ? 15| KZKjv PvivMvQ cwZ mvwiZ 3, 5, 6, 8, 10 I 15 wU Ki jvMvj cwZevi

y`BwU Pviv evwK _vK| PvivMvQi b bZg msLv KZ ? 16| Kvb& z `Zg msLvi m 7 hvM Kij hvMdj 15, 18, 20, 24 I 32 viv

wbtkl wefvR ne ?

MwYZ 51

Ag Aavq

MvwYwZK cZxK I evK MwYZ wewfb cKvii cZxK eenvi Kiv nq| hgb, msLv cZxK

0 1 2 3 4 5 6 7 8 9

kb GK y`B wZb Pvi cuvP Qq mvZ AvU bq

GB `kwU cZxKi mvnvh mKj msLv jLv hvq| msLv jLvq eeZ msLv cZxKK A ejv nq| D`vniY^ic 786 GKwU wZb Ai msLv, hvi GKKi A 6, `kKi A 8 I kZKi A 7| jYxq : 1, 2, 3, 4, 5, 6, 7, 8, 9 Ajvi cZK GK Ai msLv|

cwqv cZxK +

hvM wP weqvM wP Y wP fvM wP

mK cZxK = > < =/ >/

MwYZ 52

mgvavb : (K) ARvbv msLvwUi Rb cZxK wbj mgmvwU u`vovq : + 19 = 72 nj, = KZ ?

weqvM hvMi wecixZ cwqv weavq msLvwU ne 72 _K 19 Gi weqvMdj| = 72 19 = 53 Di : 53|

(L) ARvbv msLvwUi Rb Ai cZxK K wbj mgmvwU u`vovq : K 7 = 15 nj, K = KZ ? Y, fvMi wecixZ cwqv weavq K msLvwU ne 15 I 7 Gi Ydj| K = 15 7 = 105 Di : 105| msLvivwk KwZcq msLvK cwqv wP Ges cqvRb ebx viv hy Kij GKwU

msLvivwk Zwi nq| hgb,

(36 4) 5 7 wekl cZxK ev Ai cZxK m^wjZ MvwYwZK evKK Lvjv evK ejv nq|

D`vniY 2| Ai cZxK eenvi Ki wbPi evK nZ MvwYwZK Lvjv evK MVb Ki| 33wU wjPy wbq KqKRb evjK-evwjKvi cZKK QqwU wjPz `qvi ci wZbwU wjPz Aewk _vKj| mgvavb : GLvb evjK-evwjKvi msLv ARvbv| gb Kwi evjK-evwjKvi msLv K| cZK cqQ 6wU wjPz; myZivs K Rb cqQ 6 K wU wjPz| ck Abyhvqx 6 K + 3 = 33 D 3wU wjPz ev` w`j _vK 33 3 = 30 wU wjPz| myZivs 6 K = 30 30 nQ K Gi 6 Y; myZivs K nQ 30 Gi 6 fvMi 1 fvM| A_vr K = 30 6 = 5 Di : wbYq Lvjv evK : 6 K + 3 = 33 K Gi gvb (evjK-evwjKvi msLv) = 5|

MwYZ 53

D`vniY 3| >/ (eo bq) Gi mgv_K cZxK (QvU A_ev mgvb) Ges Ai cZxK eenvi Ki wbPi evK nZ MvwYwZK Lvjv evK MVb Ki| AZtci Ai cZxKi Kvb& Kvb& gvbi Rb evKwU mZ, wbYq Ki|

cuvPRb wki ga 21wU jR mgvb msLvq fvM Ki `qv nj|

mgvavb : gb Kwi cZK wkK K msLK jR `qv nj; GLvb K 1, 2, 3,... nZ cvi| myZivs cuvPRb wk cj 5 K wU jR| ck Abymvi GB msLvwU 21 Gi Pq QvU ev 21 Gi mgvb| A_vr, 5 K 21|

GLb 5 1 = 5, 5 2 = 10, 5 3 = 15 Ges 5 4 = 20, GB wYZKMyjv cZK 21 Gi Pq QvU| wK 5 5 = 25, 5 6 = 30,...; AZGe 5 Gi Abvb wYZK 21 Gi Pq eo| myZivs Lvjv evKwU K-Gi aygv 1, 2, 3, 4 GB PviwU msLvi Rb mZ|

Di : wbYq Lvjv evK 5 K 21| GwU mZ hw` Ges Kej hw` K = 1, 2, 3 ev 4 nq| D`vniY 4| ebx cZxK eenvi Ki wb `wkZ msLvivwk MVb Ki Ges ivwkwU

wbYq Ki : (K) 72 K 8 w`q fvM Ki fvMdjK 3 w`q Y Kiv nj| (L) 72 K 8 Ges 3 Gi Ydj w`q fvM Kiv nj|

mgvavb : (K) Afx msLvivwk (72 8) 3| gvb wbYq : 72 8 = 9 Ges 9 3 = 27 (Di) (L) Afx msLvivwk 72 (8 3)| gvb wbYq : 8 3 = 24 Ges 72 24 = 3|

MvwYwZK evK, Lvjv evK MvwYwZK evK nQ msLv, cZxK, ivwk ev MvwYwZK aviYv m^wjZ Ggb GKwU Dw, hv mZ bv wg_v wbtmskq ejv hvq| hgb, (72 8) 3 = 72 (8 3) D`vniY 4 nZ Avgiv `wL, GB MvwYwZK evKwU wg_v| cZK wfzR wZbwU ev I wZbwU KvY _vK, GwU mZ MvwYwZK evKi D`vniY|

MwYZ 54

mywbw ` Ki mZ bv wg_v ejv hvq, MwYZ welqK Ggb Kvb DwK MvwYwZK Dw ejv nq| hgb,

Dw mZ/wg_v

87 GKwU gwjK msLv wg_v

wPi wfyRwU mgKvYx wg_v

81 GKwU eMmsLv mZ|

Ai cZxK, ev ARvbv msLv ev ivwk wb `k Ki Ggb cZxK m^wjZ MvwYwZK evKK Lvjv evK ej| hgb, 72 (8 3) = (K 8) 3 D`vniY 5| Ai cZxKi Kvb& gvbi Rb Lvjv evK (K 8) 3 = 72 (8 3) mZ MvwYwZK evK ne? 72 (8 3) = 72 24 = 3| myZivs (K 8) 3 = 3 nZ ne 1 3 = 3 weavq K 8 = 1 nZ ne| 8 8 = 1| myZivs K = 8 Di : 8 avivevwnK hvM

D`vniY 6| 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = KZ ?

mgvavb : wbYq hvMdj = K awi| GKw`K K = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 Abw`K K = 15 + 13 + 11 + 9 + 7 + 5 + 3 + 1 Wvbci y`BwU ivwkK c`vbywgKfve hvM Ki cvB K + K = 16 + 16 + 16 + 16 + 16 + 16 +16 + 16 ev, 2 K = 16 8 = 128 myZivs K = 128 2 = 64 Di : 64 |

MwYZ 55

evLv : GB D`vniYi ivwkwU GKwU aviv| gj avivwUi c`jvK wecixZg mvRvbv GKB avivi c`wji m c`vbywgKfve hvM Kivq cwZ evi GKB msLv 16 cvIqv MQ; m avivq cì msLv = g j avivq cì msLv = 8| AZGe, 2 K = 16 8| j Kwi h c` avivi cZK c` ceeZx c` _K 2 ewk; A_vr cZK c` I Zvi ceeZx cì Ai GKwU wbw ` msLv| Gic avivK mgvi aviv ejv nq| h Kvb mgvi avivi c`jvi mgw c` wkZ cvq mnRB ei Kiv hvq|

D`vniY 7| 1 + 3 + 9 + 27 + 81 = KZ?

mgvavb : wbYq hvMdj = K awi| dj K = 1 + 3 + 9 + 27 + 81 myZivs, 3 K = 3 + 9 + 27 + 81 + 243 Avevi 1 K = 1 + 3 + 9 + 27 + 81 weqvM Ki cvB, 3 K 1 K = 1 + 243 [ mgvb c`jv eRb Ki ] ev 2 K = 242 myZivs, K = 242 2 = 121 Di : 121|

evLv : GB D`vniYi cZK c` ceeZx cì 3 Y| GRb 3 K avivwU MVb Ki G _K K = 1 K avivwU weqvM `qv nqQ, hvi dj 3 K avivi kl c` _K K avivi c_g c`wU weqvM KiZ nqQ| Kbbv aviv `yBwUi Ab mKj c` wb `wkZ g mvRvbv|

GB aibi avivK Yvi aviv ejv nq| h Kvb Yvi avivi c`jvi mgw c` wkZ cvq Kiv hvq|

MwYZ 56

Abykxjbx- 8

1| cZxKi mvnvh cKvk Ki :

(K) wZb j wk|

(L) cuvP KvwU evqvb j mi nvRvi mvZk mvZ|

(M) wZbk ZZvwj kK mvZ w`q fvMdj _K Ebwk weqvM| (N) Pwjki AvUYK GKvb I Qvweki weqvMdj w`q fvM| (O) AvU nvRvi cuvPk mvZvwk _K mvZ nvRvi bqk evnvi weqvM Ki

weqvMdjK GKk mvZvk I weivwk hvMdj w`q Y|

2| cwqv cZxK I mK cZxK eenvi Ki cKvk Ki : (K) AvUk AvVvi _K wZbk mvZPwjk I Pvik ZBki hvMdji

weqvMdj mgvb AvUk AvVvi _K ci ci wZbk mvZPwjk I Pvik ZBki weqvMdji mgvb|

(L) GKk mvZvkK ZZvwjk I lvji hvMdj w`q Y Ki cv msLv GKk mvZvk I ZZvwj ki Ydji m GKk mvZvk I lvji Ydji hvMdji mgvb|

(M) cPviK cuvP w`q fvM Ki fvMdjK wZb w`q Y w`q cv msLv, cPviK cuvP I wZbi Ydj w`q fvM Ki cv msLvi mgvb bq|

3| wbPi MvwYwZK evK (Dw)jvi mZZv hvPvB Ki Ges Dwi kl mZ/wg_v jL :

(K) 347 (189 65) = 347 189 + 65 (L) (285 5) 3 = 285 (5 3) (M) (134 + 78 + 49) 7 = 134 7 + 78 7 + 49 7 4| Ai cZxKi Kvb& Kvb& gvbi Rb wbPi Lvjv evKjv mZ ne?

(K) (K + 15) 3 = 90 (L) 7 K < 50 (M) (K 6) + 5 = 12

MwYZ 57

5| ARvbv msLvwUi RvqMvq Ai cZxK ewmq Lvjv evK MVb Ki Ges ARvbv msLvwU wbYq Ki|

(K) gUzi KqKwU gvej wQj| gUz Zv _K PviwU gvej wcUzK w`j| GZ gUzi AvUwU gvej _vKj|

(L) wcqvi wKQy UvKv AvQ| Zvi gv ZvK AviI cvk UvKv w`jb| GZ Zvi h UvKv nj, AvMi UvKvi wYi Pq Zv `k UvKv ewk|

(M) ZzlviK Zvi evev evRvi Kivi Rb wKQy UvKv w`jb| Zvi gv ZvK AviI cvk UvKv w`jb| Zzlvi evRvi _K GKk lvU UvKvi gvQ, cPvi UvKvi gyiMx Ges AvUwk UvKvi ZwiZiKvwi Avbj| evRvi kl Zzlvii KvQ mvZvk UvKv _vKj|

6| avivevwnK hvMdj wbYq Ki :

(K) 1 + 4 + 7 + 10 + 13 + 16 + 19

(L) 2 + 6 + 10 + 14 + 18 + 22 + 26 + 30

(M) 3 + 8 + 13 + 18 + 23 + 28

(N) 1 + 2 + 4 + 8 + 16 + 32 + 64

(O) 2 + 6 + 18 + 54 + 162

MwYZ 58

beg Aavq

mvaviY fMvsk

mgni wewk fMvsk

53 GKwU fMvsk hvi je 3 Ges ni 5|


y`BwU fMvskiB ni 5| Giv mgni wewk fMvsk|

53 I 6

1 fMvsk y`BwUi ni wfb wfb| G`i mgni wewk fMvsk cKvk

Kiv hvq|

53 = 6 5

6 3 = 30

18

61 = 5 6

5 1 = 30

5

fMvsk `yBwU 30 ni wewk fMvsk cwiYZ nj|

D`vniY 1| 43 , 8

7 I 1211 K 48 ni wewk fMvsk cKvk Ki|

mgvavb : 48 4 = 12 43 = 124

12 3 = 48

36

48 8 = 6 87 = 68

6 7 = 48

42

48 12 = 4 1211 = 412

4 11 = 48

44 |

Di : 43 = 48

36 , 87 = 48

42 , 1211 = 48

44 |

GLvb 48 c` fMvskjvi nii GKwU mvaviY wYZK|

MwYZ 59

mgje wewk fMvsk



y`BwU fMvskiB je 6| Giv mgje wewk fMvsk|

75 I 8

3 fMvsk y`BwUi je wfb wfb| G`i mgje wewk fMvsk cKvk

Kiv hvq|

75 = 3 7

3 5 = 21

15

83 = 5 8

5 3 = 40

15

fMvsk y`BwU 15 je A_vr mgje wewk fMvsk cwiYZ nj|

D`vniY 2| 92 , 10

3 , 174 K 12 je wewk fMvsk cKvk Ki|

mgvavb : 12 2 = 6 92 = 69

62 = 54

12

12 3 = 4 103 = 410

43 = 40

12

12 4 = 3 174 = 317

34 = 51

12

Di : 92 = 54

12 , 103 = 40

12 , 174 = 51

12

GLvb 12 c` fMvskjvi jei GKwU mvaviY wYZK|

MwYZ 60

fMvski jwN ic

`Lv hvQ, 86 = 4

3

A_vr 86 I 4

3 mgZzj fMvsk|

86 Gi je I nii mvaviY Drcv`Kjv eRb Ki cvB

86 = 2 2 2 1

3 2 1/

/ = 22 3 = 4

3

86 Gi jwN ic 4

3

fMvski je I nii mvaviY Drcv`Kjv eRb Ki fMvskwUi jwN ic cvIqv Mj|

D`vniY 3| 7248 K jwN AvKvi cKvk Ki|

mgvavb : 7248 = 332 2 2

322 2 2////////

= 32

Di : 32 |

86

43

MwYZ 61

D`vniY 4| 32 , 6

5 , 94 I 12

7 K jwN mgni wewk fMvsk cKvk Ki|

mgvavb : GLvb fMvskjvi ni 3, 6, 9 I 12 |

GLb, 2 3, 6, 9, 123 3, 3, 9, 6

1, 1, 3, 2

3, 6, 9 I 12 Gi j.mv.. = 2 3 3 2 = 36 36 3 = 12 3

2 = 123122

= 36

24

36 6 = 6 65 = 66

65 = 36

30

36 9 = 4 94 = 49

44 = 36

16

36 12 = 3 127 = 312

37 = 36

21

Di : 32 = 36

24 , 65 = 36

30 , 94 = 36

16 , 127 = 36

21 |

GKvwaK fMvski jwN mgni Zv`i nii mvaviY wYZK| ZvB c_g nijvi j.mv.. ei Kiv nqQ| AZtci GB j.mv.. K mvaviY ni ai fMvskjvK mgni wewk fMvsk cwiYZ Kiv nqQ|

MwYZ 62

fMvski Zzjbv

61

63

61 Ges 6

3 DfqiB ni 6; wKZy je 3 > je 1|

ni GKB nj h fMvski je eo mB fMvskwU eo|

D`vniY 5| 245 I 24

11 fMvskqi KvbwU eo, cZxK eenvi Ki `LvI|

mgvavb : GLvb fMvsk y`BwUi ni GKB| G`i je 5 I 11|

hnZz 11 > 5 myZivs 2411 > 24

5

D`vniY 6| 127 I 4

3 Gi ga KvbwU eo cZxK eenvi Ki `LvI|

mgvavb : GLvb fMvsk `yBwUi ni 12 I 4| GLb 2 12, 4

2 6, 23, 1

12 I 4 Gi j.mv.. = 2 2 3 = 12 12 12 = 1 12

7 = 112 17 = 12

7

12 4 = 3 43 = 34

33 = 12

9

hnZz 9>7 myZivs 129 > 12

7 A_vr 43 >12

7

Di : 43 > 12

7 |

fMvsk `yBwUK mgni wewk fMvsk icvi Ki G`i jejvi Zzjbv Kiv nqQ| hwUi je eo mBwUB eo fMvsk|

63 > 6

1

MwYZ 63

D`vniY 7| 97 , 12

5 , 1811 fMvskjv gvbi Eag Abymvi mvRvI|

mgvavb : fMvskjvi ni 9, 12 I 18| GLb, 2 9, 12, 18

3 9, 6, 93 3, 2, 3

1, 2, 1 9, 12 I 18 Gi j.mv.. = 2 3 3 2 = 36 36 9 = 4 9

7 = 4947

= 36

28

36 12 = 3 125 = 312

35 = 36

15

36 18 = 2 1811 = 218

211 = 36

22

hnZz 15

MwYZ 64

42 14 = 3 149 = 314

39 = 42

27

42 21 = 2 215 = 221

25 = 42

10

hnZz 27>18>10 myZivs 4227 > 42

18 > 4210

ev, 149 > 7

3 > 215

myZivs gvbi Aatg Abymvi mvwRq cvB 149 , 7

3 , 215 |

Di : 149 , 7

3 , 215 |

Qwe j Kwi : 43 > 16

3 |

43 I 16

3 Gi je GKB A_vr je 3 |

43 Gi ni 4, 16

3 Gi ni 16 Ges 4 29

4 |

Di : 134 > 29

4 |

43

163

MwYZ 65

D`vniY 10| 98 , 33

8 , 158 , 19

8 fMvskjv gvbi Eag Abymvi mvRvI|

mgvavb : fMvskjvi je 8 Ges ni 9, 33, 15 I 19|

hnZz 9

MwYZ 66

5| wbPi fMvskjv gvbi Aatg Abymvi mvRvI :

K) 7549

2521

1513

54 ,,, L) 78

612615

3925

139 ,,,

M) 172

9314

412

237

101 ,,,, N) 14

33920

2312

112

94 ,,,,

6| Acy GKwU evKvi GuK Gi 51 Ask bxj, 4

1 Ask njy` Ges 2011 Ask

jvj is Kij| wUZ Kvb is ewk RvqMv Ryo AvQ? 7| ejvj mvne wKQy UvKv wbq mKvj evmv _K ei njb| wZwb Zvi UvKvi

83 Ask w`q `ycyi Lvevi Ljb, 24

1 Ask wfzKK `vb Kijb Ges 31 Ask

wivfvov w`jb| Avi wKQy UvKv iq Mj| wZwb Kvb KvR mePq ewk UvKv LiP Kijb?

8| Rwgi Avjx mvne GK L Rwgi 41 Ask UgUv, 12

5 Ask Avjy Ges 31

Ask gjvi Pvl Kijb| wZwb Kvb me&wR mePq Kg RwgZ Pvl Kijb? 9| 1bs, 2bs I 3bs GB wZbwU nuvmi Lvgvii mv`v nuvmi msLv h_vg gvU

nuvmi msLvi 161 , 32

3 I 716 Ask| Kvb Lvgvi mv`v nuvmi msLv mePq

Kg ?

10| GKwU Mvgi DcvRbg jvKi gvU msLvi 53 Ask Kwl KvR Kib,

101 Ask PvKzix Kib, 50

3 Ask Abvb ckvq wbqvwRZ I 256 Ask eKvi

AvQb| H Mvg Kvb ckvi jvK mePq ewk?

MwYZ 68

983 = 9 9

2

wgk I AcKZ fMvski cvicwiK icvZi

1 mg Z 3 ZZxqvsk

1 mg Z 3 ZZxqvsk

31 1 ZZxqvsk

gvU 2 mg Z 31 gvU 7 ZZxqvsk

ev 2 31 ev 3

7

`Lv hvQ, 2 31 = 3

7

j Kwi : 37 3) 7 ( 2 2 mg Z

6 1 1 ZZxqvsk AcKZ fMvsk I icvwiZ wgk fMvski m hy cKZ fMvski ni GKB| AcKZ fMvski jeK Gi ni w`q fvM Ki h fvMdj cvIqv hvq Zv icvwiZ wgk fMvski cY Ask| Avi fvM kl nQ icvwiZ wgk fMvski m hy cKZ fMvski je|

D`vniY 1| wbPi AcKZ fMvskjv wgk fMvsk icv i Ki :

K) 983 L) 15

79 M) 1991 N) 27

92 O) 3297

mgvavb : K) 9 ) 8 3 ( 9 8 1 2 L) 15 ) 7 9 ( 5 7 5 4

1579 = 5 15

4

37 = 2 3

1

MwYZ 69

1991 = 4 19

15

2792 = 3 27

11

3297 = 3 32

1

M) 19 ) 9 1 ( 4 7 6 1 5

N) 27 ) 9 2 ( 3 8 1 1 1

O) 32 ) 9 7 ( 3 9 6 1

7 = 17 , 4 = 1

4 , 11 = 111 BZvw`|

h Kvb cYmsLvK 1 ni wewk fMvski AvKvi jLv hvq|

j Kwi :

1 23

212 1

21

221

21

2121

21 1

1 2112

1 =+=+=+=+=+=

2 511

5152

51

552

51

5152

51 1

25125

1 =+=+=+=+=+=

7 431

4347

43

447

43

4147

43 1

7 43 74

3 =+=+=+=+=+=

msc, 1 ;; 511

5152

512 22

1 2 1 21 3 =+==+= 7 4

3 = 4 3 47 + = 4

31

wgk fMvsk = ni jeniAsk cY +

= AcKZ fMvsk

D`vniY 2| wbPi wgk fMvskjv AcKZ fMvsk cKvk Ki :

K) 614 L) 11

78 M) 1345 N) 10

36 O) 17115

MwYZ 70

mgvavb : K) 625

616 4

614 =+=

L) 1195

117118

1178 =+=

M) 1369

134135

1345 =+=

N) 1063

103106

1036 =+=

O) 1796

1711175

17115 =+=

Di : K) 625 L) 11

95 M) 1369 N) 10

63 O) 1796

Abykxjbx 9 (L)

1| wbPi fMvskjv _K cKZ, AcKZ I wgk fMvsk c_K Ki :

257971

1123841

42760

321723

131514

427 ,,,,,,,, |

2| wbPi AcKZ fMvskjv wgk fMvsk icvi Ki :

K) 529 L) 9

85 M) 1178 N) 37

89 O) 1523

P) 5173 Q) 59

147 R) 25103 S) 43

521 T) 2031416

3| wbPi wgk fMvskjv AcKZ fMvsk icvi Ki :

K) 1633 L) 44

52 M) 4871 N) 16

310

O) 7388 P) 11

7112 Q) 231127 R) 33

2998

S) 491100 T) 19

18103

MwYZ 72

fMvski hvM I weqvM

fMvski hvM

D`vniY 1| =+++ 7523

7513

754

751 KZ?

mgvavb : 7523

7513

754

751 +++

= 752313 4 1 +++

= 7541

Di : 7541

mgni wewk KZjv fMvski hvMdji ni fMvskjvi mvaviY ni| GB hvMdji je fMvskMyjvi jei hvMdj|

D`vniY 2| 2511

157

52 ++ = KZ ?

mgvavb : fMvskjvi ni 5, 15, 25| GLb, 5 5, 15, 25

1, 3, 5 5, 15 I 25 Gi j.mv.. = 5 3 5 = 75 75 5 = 15 75

3015 5152

52 =

= 75 15 = 5 75

355 1557

157 =

= 75 25 = 3 75

333 25311

2511 =

= 75

337535

7530

2511

157

52 ++=++

= 75333530 ++

= 7598 = 1 75

23 Di : 1 7523

hvM Kivi ce fMvskjvK mgni wewk fMvsk cwiYZ KiZ nq|

MwYZ 73

D`vniY 3| 910

913

911 ++ = KZ?

mgvavb : 910

913

911 ++

= 9101311 ++

= 934

= 3 97

Di : 3 97

hvMdj AcKZ fMvsk nj mvaviYZ GK wgk fMvsk wnmve cKvk Kiv nq|

D`vniY 4| 115211

241113 ++ = KZ ?

mgvavb : 115211

241113 ++

= 1127

1146

1134 ++

= 11274634 ++

= 11107

= 9 118

Di : 9 118

wgk fMvskjv AcKZ fMvsk icvi Ki hvM Kiv nqQ|

MwYZ 74

evLv : GLvb ni 4, 3 I 9| GLb, 3 4, 3, 9

4, 1, 3 4, 3 I 9 Gi j.mv.. = 3 4 3 = 36 36 4 = 9 36 3 = 12 36 9 = 4

evLv : GLvb ni 4, 2 I 8| GLb, 2 4, 2, 8

2 2, 1, 4 1, 1, 2

4, 2 I 8 Gi j.mv.. = 2 2 2 = 8 8 4 = 2 8 2 = 4 8 8 = 1

D`vniY 5| 9513

23415 ++

= 9513

23415 ++

= 914

311

421 ++

= 3641412119 21 ++

= 3656132 189 ++

= 36377

= 10 3617

Di : 10 3617 |

D`vniY 6| 8322

15412 ++ = KZ?

mgvavb : 8322

15412 ++

= 819

211

49 ++

= 81194 112 9 ++

= 819 44 18 ++

= 881

= 10 81

Di : 10 81 |

MwYZ 75

D`vniY 7| weRq evey evRvi wMq Zvi UvKvi 21 Ask Pvj, 10

3 Ask ZwiZiKvix

Ges 201 Ask w`q jeY wKbjb| wZwb UvKvi KZ Ask LiP Kijb?

mgvavb : 201

103

21 ++

= 201610 ++

= 2017

weRq evey UvKvi 2017 Ask LiP Kijb|

Di : 2017 Ask|

fMvski weqvM

D`vniY 8| 2811

2819 = KZ?

mgvavb : 2811

2819

= 2811 19

= 728

28/

= 72

Di : 72

j Kwi : mgni wewk fMvsk y`BwUi weqvMdji ni G`i mvaviY ni| Avi

weqvMdji je G`i jei weqvMdj| fMvski hvMdj ev weqvMdj me mgq jwN AvKvi cKvk KiZ nq|

MwYZ 76

D`vniY 9| 79

34 = KZ?

mgvavb : 79

34

= 7494

3949

= 2836

2736

= 2827

36

= 1

36

Di : 1

36|

D`vniY 10| 617

237 = KZ?

mgvavb : 617

237 =

61237

387 = 5

37

Di : 537|

D`vniY 11| 418 2

35 = KZ?

mgvavb : 418 2

35

= 338

138

= 335 13 8

40

= 165104

40

= 6140 = 1

2140

Di : 12140

evLv : fMvsk y`BwU mgni wewk bq| Gì mgni wewk Kiv nqQ| Gì ni 9 I 4 Ges nii A_vr 9 I 4 Gi j. mv. . 36 36 9 = 4 36 4 = 9 evLv : GLvb fMvsk y`BwUi ni 8 I 5 Ges Gì j. mv. = 40 40 8 = 5 40 5 = 8

MwYZ 77

D`vniY 9| kixd wgqv Zvi Rwgi 41 Ask cvU I 3

1 Ask avb Pvl Kijb|

Aewk RwgZ Mg Pvl Kijb| wZwb gvU Rwgi KZ Ask Mg Pvl Kijb?

mgvavb : 1243

31

41 +=+

= 127

kixd wgqv Zvi Rwgi 127 Ask cvU I avb Pvl Kijb|

Aewk iBj (1 Rwg 1 Rwgi 127 Ask) A_vr gvU Rwgi (1 12

7 ) Ask

= 12712

Ask

= 125 Ask

wZwb gvU Rwgi 125 Ask Mg Pvl Kijb|

Di : 125 Ask|

Abykxjbx 9 (M) 1| hvMdj wbYq Ki :

K) 2512

257

251 ++ L) 18

11185

181 ++

M) 41

31

21 ++ N) 10

743

51 ++

O) 149

74

41 ++ P) 9

23919

910 ++

MwYZ 78

Q) 2281

2245

2223 ++ R) 8

568328

14 ++ S) 3

21613

165312

710877 ++++

T) 310125

44324

11 ++++ U) 1 + 36

1318512

17923 +++

2| weqvMdj wbYq Ki :

K) 176

179 L) 45

144519 M) 32

1543 N) 40

112013

O) 4219

149 P) 6

13641 Q) 10

7910

103 R) 125412

75 S) 9

74916 T) 5

131094 U) 5

22435 V) 1 15

7

W) 6 1392

3| mij Ki :

K) 127

65

21 + L) 2

15214

36 + M) 4

1181

515 N) 24

1116144

37 O) 15

4210745

38 + 4| wgbv GKwU eB cov iy Ki c_g w`b eBwUi 5

2 Ask Ges ci w`b 158 Ask

coj| m y`B w`b eBwUi KZ Ask coj ?

5| GKwU euvki 51 Ask Kvj, 6

1 Ask jvj Ges 103 Ask meyR is Kiv AvQ|

euvkwUi gvU KZ Ask is Kiv AvQ?

6| 240wU Avgi GKwU Szwo wKb `Lv Mj gvU Avgi 161 Ask cP MQ|

SzwoZ KqwU fvj Avg AvQ?

MwYZ 79

7| GKRb KlK Zvi GKwU Rwgi 83 Ask evwo, 5

2 Ask evMvb Ges evwK Ask

cyKzi Lbb Kijb| cyKzi KiZ H Rwgi KZ Ask wZwb eenvi Kijb?

8| GKRb gvwj Zvi evMvbi 52 Ask Mvjvc, 15

1 Ask iRbxMv Ges 253

Ask Rev dzji Pvl Kijb| wZwb evMvbi gvU KZ Ask dzji Pvl Kijb?

9| Rwjj wgqvi wbKU 200 UvKv wQj| wZwb Zvi UvKvi 52 Ask KvgvjK Ges 4

1

Ask gyKzjK w`jb| Zvi wbKU Avi KZ UvKv iBj?

fMvski Y fMvskK cY msLv w`q Y

98

98

92

92

92

92 =+++

92 4 = 9

8 = 9

2 4

j Kwi : =+++ 92

92

92

92 9

2222 +++

= 942

= 98

fMvsk cY msLv = fMvski je cY msLvfMvski ni

MwYZ 80

D`vniY 1| 87 5 = KZ?

mgvavb : 87 5 = 8

5 7 = 8

35

= 4 83

Di : 4 83 |

D`vniY 2| 613 7 = KZ?

mgvavb : 613 7 = 6

7 13 = 6

91 = 15 61

Di : 15 61 |

D`vniY 3| 6 101 8 = KZ?

mgvavb : 6 101 8 = 10

61 8 = 10

8 61

=

510

244488

= 5244

= 48 54

Di : 48 54 |

MwYZ 81

fMvskK fMvsk w`q Y

* eMwUi dj = 1 eM wgUvi * eMwU 15wU QvU QvU AvqZ wef

nqQ|

cwZwU AvqZ&i dj = 151

eM wgUvi

* Mvp Aski dj 5

43

2 eM wgUvi|

* Mvp Ask 8wU QvU AvqZ iqQ|

* Mvp Aski dj = 15

1 8 eMwgUvi

= 15

8 eMwgUvi

54

32

= 158

j Kwi : 54 3

2 = 158 A_vr 5

4 32 = 35

24

GKwU fMvskK Ab GKwU fMvsk w`q Y KiZ nj fMvsk y`BwUi niK Y Ki Ydj ni wnme Ges fMvsk y`BwUi jeK Y Ki Ydj je wnme eenvi KijB fMvsk y`BwUi Ydj cvIqv hve|

D`vniY 4| 65 8

3 = KZ?

mgvavb : 65 8

3 = 826

135// = 16

5

Di : 165 |

D`vniY 5| 711 9

4 = KZ?

mgvavb : 711 9

4 = 97411

= 6344

Di : 6344 |

c

=3

2wgUvi

1 wg Uv i

1 wgUvi

`N = 5

4 wgUvi

MwYZ 82

D`vniY 6| 1013 17

25 = KZ?

mgvavb : 1013 17

25 = 17210

52513

= 3465 = 1 34

31

Di : 1 3431 |

D`vniY 7| 4 121 7

3 = KZ?

mgvavb : 4 121 7

3

= 1712 4

3497

// (AcKZ fMvsk icvi Ki)

= 47

= 1 43

Di : 1 43 |

D`vniY 8| 17228

36 = KZ?

mgvavb : 17228

36 = 1

17

936

28

351/ = 2

27 = 13 21

Di : 13 21 |

MwYZ 83

Gi Gi A_

j Kwi : * wU gvU 15wU mgvb QvU QvU wef nqQ|

* is Kiv Ask = wUi 52 Ask

* Mvp is Kiv Ask = is Kiv Aski 32

= wUi 52 Aski 3

2

wK y Mvp is Kiv Ask = wUi 154 Ask

52 Aski 3

2 = 154

msc 52 Gi 3

2 = 154

wKy 154 = 35

22

52 Gi

32 = 35

22

D`vniY 9| cg kYxi MwYZ eBqi 98 Ask cvwUMwYZ| cvwUMwYZi 16

5 Ask

fMvsk| MwYZ eB Gi KZ Ask fMvsk?

mgvavb : 165 Gi 16

5 =

21695

18/

= 2951

= 185

MwYZ eB Gi 185 Ask fMvsk |

Di : 185 Ask|

MwYZ 84

D`vniY 10| GK wgUvi Kvcoi 43 Ask jvj| KvcowUi jvj Aski 2

1 Ask w`q

GKwU igvj Zwi Kiv nj| igvj Zwi KiZ GK wgUvi Kvcoi KZ Ask eeZ nqQ?

mgvavb : igvj Zwi KiZ GK wgUvi Kvcoi ( 43 Gi 2

1 ) Ask eeZ nqQ|

GLb 43 Gi 2

1 = 2413

= 83

igvj Zwi KiZ GK wgUvi Kvcoi 83 Ask eeZ nqQ|

Di : 83 Ask |

wecixZ fMvsk

cZK fMvskiB je Ges ni AvQ| Kvb fMvski jeK ni Ges niK je Kij mB fMvski wecixZ fMvsk cvIqv hvq|

43 GKwU fMvsk hvi ni 4 Ges je 3| 3 Ges 4 Gi vb e`j Ki cvIqv hvq

34 |

43 Gi wecixZ fMvsk 3

4

A_ev, 34 Gi wecixZ fMvsk 4

3

j Kwi : 43

34 = 1

Kvb fMvsk H fMvski wecixZ fMvsk = 1

MwYZ 85

D`vniY 11| 95 , 6

1 , 1011 , 7, 5

9 fMvskjvi wecixZ fMvsk jL|

mgvavb : 95 Gi wecixZ fMvsk 5

9

61

16 ev 6

1011

1110

7 71

59

95

Di : 59 , 6, 11

10 , 71 , 9

5 |

fMvski fvM

wUK 20 wU mgvb QvU QvU fvM Kiv nqQ| 20wU QvU i 12 wU is Kiv nqQ|

A_vr is Kiv Ask = wUi

520

312 ev, 5

3 Ask

Mvp is Kiv Ask wUi 20 fvMi 3 Ask|

A_vr, Mvp is Kiv Ask = wUi 203 Ask

Avevi, Mvp is Kiv Ask is Kiv Aski 4 fvMi 1 Ask| A_vr, Mvp is Kiv Ask

= is Kiv Ask 4 = wUi 5

3 Ask 4 = wUi ( 5

3 4) Ask

MwYZ 86

Avevi, Mvp is Kiv Ask

= is Kiv Aski 41

Ask

= wUi 53

Aski 41

Ask

= wUi ( 53

41 ) Ask

53 4 = 5

3 4

1

`Lv hvQ, 53 K 4 w`q fvM Kiv Avi 4

1 w`q Y Kiv GKB|

ZvB 53 K 4 w`q fvM KiZ 4

1 w`q Y Kiv nqQ|

Kvb fMvskK Aci GKwU fMvsk w`q fvM KiZ nj c_g fMvskK wZxq fMvski wecixZ fMvsk w`q Y KiZ nq|

D`vniY 12| 253

4 = KZ?

mgvavb : 253 4 = 25

3 14

= 253

( 14 Gi wecixZ fMvsk)

= 253 4

1

= 253

41

= 1003

Di : 1003

|

MwYZ 87

D`vniY 13| 78 9 = KZ?

mgvavb : 78 9 = 7

8 19

= 78 ( 1

9 Gi wecixZ fMvsk)

= 78 9

1

= 9718

= 638

Di : 638

|

D`vniY 14| 3 53 15 = KZ?

mgvavb : 3 53 1

15

= 518 ( 1

15 Gi wecixZ fMvsk)

= 5

618

5151

= 256

Di : 256 |

D`vniY 15| 13 2 41 = KZ?

mgvavb : 13 2 41 = 1

13 49

= 113 9

4

= 952

= 5 97

Di : 5 97 |

MwYZ 88

D`vniY 16|

1147 = KZ?

mgvavb :

1147 = 7 11

4

= 411

17

= 4

77

= 1941

Di : 1941

|

D`vniY 17|

2071813

= KZ?

mgvavb :

2071813

= 1813

207

= 91813

7

1020

= 63130

= 2634

Di : 2634

|

MwYZ 89

D`vniY 18|

53

6

1511

= KZ?

mgvavb :

53

6

1511

=

5331511

= 533

1511 =

333

15

315

111 /

= 33

1 = 9

1

Di : 91

|

D`vniY 19| evbi eqm fvBqi eqmi 43

Ask| fvBqi eqm 16 eQi nj

evbi eqm KZ?

mgvavb : fvBqi eqm 16 eQi|

evbi eqm = fvBqi eqmi 43

Ask

= 143416 eQi

= 12 eQi

Di : 12 eQi|

MwYZ 90

Abykxjbx 9 (N) 1| Ydj wbYq Ki :

K) 1285 L) 6

137 M)

83

65

N) 1712

2411 O)

149

516 P)

223

1655

158

Q) 51

4143

3 R) 351

3143

8 S) 231

71813

7

2| fvMdj wbYq Ki :

K) 643 L)

1315

25 M) 1 7201 N) 5

43

3

O) 31

2 12 P) 1039

2513 Q)

3546

7523 R)

71

153

5

S) 143

373

11 3| gvb wbYq Ki :

K) 34

Gi 1511

L) 125

Gi 89

M) 53

Gi 3225

N)

2310

O) 344

17

P)

41

5

21

3

4| 21

1 WRb wWgi `vg 21 UvKv| 21

5 WRb wWgi `vg KZ?

5| GKwU Mvwo Nvq 45 wKjvwgUvi hZ cvi| 31

4 Nvq MvwowU KZ

wKjvwgUvi hve?

6| GKwU mwi 158

Aski gj 48,000 UvKv| H mwi 31

Aski gj

KZ?

MwYZ 91

7| GKwU wcjvii 81

Ask gvwUi wbP Ges evwK Ask gvwUi Dci AvQ| gvwUi

wbP 1 wgUvi wcjvi _vKj gvwUi Dci KZ wgUvi AvQ?

8| GKwU LuywUi 61

Ask Kv`vq, 21

Ask cvwbZ Ges evwK Ask cvwbi Dci

AvQ| cvwbi Dcii Aski `N 2 wgUvi nj cvwbZ hZUzKz LuywU AvQ Zvi `N KZ?

9| GKwU msLvK 21

5 w`q fvM Kij fvMdj 21

20 nq| msLvwU KZ?

10| GKwU wdZvi 32

Ask jvj Ges evwK Ask meyR| meyR Aski `N 3 wgUvi

nj wdZvwUi `N KZ?

11| y`BwU msLvi Ydj 5461

| GKwU msLv 641

nj, AciwU KZ?

fMvski mij

D`vniY 1| mij Ki: 53

152

271

132

4 + mgvavb :

53

152

271

132

4 +

= 58

512

178

3

214 +

= 58

512

316 +

= 15

243680 +

= 15

36104 =

1568

Di : 158

4 |

* c_g avc wgk fMvskK AcKZ fMvsk icvi Kiv nqQ Ges Y Kiv nqQ

* wZxq avc hvM I weqvM GKB m Kiv nqQ|

MwYZ 92

D`vniY 2| mij Ki : 53

941

1143

72

1 Gi 32

21

321

81

3 +

mgvavb : 53

941

1143

72

1 Gi 32

21

321

81

3 +

= 15

648

45

143

79

/ Gi 32

27

21

18

525 +/

= 32

27

21

3045

143

79 +

= 32

27

21

6301

451

3

214

1793 +////

= 32

47

241

6 +

= 24

16 42 1 144 +

= 24

17 186

= 24169

= 7241

Di : 7241

|

* c_g Gi Gi KvR Kiv nqQ| Gici fvMi KvR Ges Yi KvR Kiv

nqQ| * mekl hvM I weqvM GK Kiv nqQ|

MwYZ 93

D`vniY 3| 43

43

1 Gi 41

221

385

31

+

mgvavb : 43

43

1 Gi 41

221

385

31

+

= 4

1347 / Gi

49

27

85

131

+/

= 49

27

85

41

47 +

= 49

27

5

28

141

47 +//

= 49

27

52

47 +

= 20

45 70 8 35

+

= 20

8 7045 35

+

= 20

78 80

=1020

12 /

= 101

Di : 101

|

MwYZ 94

D`vniY 4| 31

143

354

4 Gi 32

141

31

21

1 )(

mgvavb : 31

143

354

4 Gi 32

141

31

21

1 )(

= 34

415

524 Gi

35

123 4

23

)(

= 13

24

415

524 Gi

35

121

12

13

= 35

121

2 4

15524

= 53

121

21

4

315

15242

= 209

Di : 209

|

* c_g ( ) ebxi KvR Kiv nqQ| Zvici Gi Gi KvR Kiv nqQ|

* AZtci c_g fvMi KvR I ci Yi KvR Kiv nqQ|

* kl avc cuvPwU ivwki ga Y wP _vKvq Yi KvR GK m Kiv nqQ|

ge : cvkvcvwk Y I fvMi KvR _vKj c_g fvMi KvR I ci Yi KvR KiZ nq|

MwYZ 95

D`vniY 5| 21

341

5 Gi 2)139

222913

+ (

mgvavb : 21

341

5 Gi 2)139

222913

+ (

= 27

421 Gi 2)

1335

22913

+ (

= 27

421 Gi )

21

1335

22913

+ (

= 27

421 Gi )

2635

22913

+ (

= 27

421 Gi )

26 3552

2913

+ (

= 27

421 Gi

2687

2913

= 5891

421

2687

= 13915829

24213

2687

=2687

2687

=26

87 87

= 260

= 0 Di : 0|

MwYZ 96

D`vniY 6| 31

1

81

8

92

7

75

6

87

5 + Gi

32

2 (61

2 Gi 81

263 + )

mgvavb : 31

1

81

8

92

7

75

6

87

5 + Gi

32

2 (61

2 Gi 81

263 + )

= 34

865965

747847

+ Gi 38

(26

13 1 / Gi 226

3 1/ )81+

= 34

865

965

747

847

+ Gi )(81

41

38

+ =

34

865

965

747

847

+ Gi )(8

1238

+

= 34

865

965

747

847

+ Gi 83

38

=

83

932

865

965

747

847

+

= 18

3

39324

1658

9651

1477

8471

+

=34

98

87

+ =72

966463

+

=72

96127

=

7231

Di : 7231

|

MwYZ 97

Abykxjbx 9 (O)

1| 241

61

121

41

31

++

2| 2716

235

53

43

21

+

3| 6325

71

251

471

83

+ 4| 75

53

73

6 Gi 21

218

5| 87 Gi

43

54

Gi 95

21

109

6| 21

243

3 Gi 1103

54

191

32

1 +

7| 81

121 5 Gi 2

31

53331

1 +

8| 31

2 Gi 121

2 32

32

441 6 +

9| 31

721

2 41 11

21

13 + Gi 83

449

10| 91

161

31

52

22513

+ )(

11| (21

2 21 3

21

121

2 21 3 () Gi )

21 1

12| 20 Gi 151

{( 154

153

32

31

4 + }) 13|

83

1165

42320

1 {[ Gi )}](143

75

21

5 +

14| 31

2 21 5

32

343

131

7 + ({[ Gi )}]43

21 1 +

MwYZ 98

15| [10 {501

132

2091

11101

3 (+ Gi )}]21

Documents

Mathematics26_126162401920091224