Mathematics17_126162399620091224

cÖv_wgK MwYZ 1

RvZxq wk¶vµg I cvV¨cy¯ZK †evW© KZ…©K 2005 wk¶veQi †_‡K

PZy_© †kªYxi cvV¨cy¯ZKiƒ‡c wbav©wiZ

cªv_wgK MwYZ PZz_© †kªYx

iPbv W. gywbeyi ingvb †PŠayix

Av.g.g. Avnmvb Dj−vn

nvwg`v evby †eMg

Avw¤qv LvZzb

m¤úv`bv Av.d.g. †Lv`v`v` Lvb

RvZxq wk¶vµg I cvV¨cy —K †evW©, XvKv

2 cÖv_wgK MwYZ

RvZxq wk¶vµg I cvV¨cy ZK †evW© 69-70, gwZwSj evwYwR¨K GjvKv, XvKv-1000

KZ©„K cÖKvwkZ|

[ cÖKvkK KZ„©©K me© ^Z¡ msiw¶Z ]

cix¶vg ~jK ms¯‹iY

cÖ_g gy ªY : A‡±vei, 2004

c ~bg©y ªY : A‡±vei, 2008

c ~bg©y ªY : †m‡Þ¤^i, 2009

cÖPQ` I wPÎv¼b †gvt Ave yj †gv‡gb wgëb

Kw¤cDUvi K‡¤cvR †jRvi ¯‹¨vb wjt

wWRvBb RvZxq wk¶vµg I cvV¨cy ZK †evW©

me©Rbxb cÖv_wgK wk¶v Kg©m ~wPi AvIZvq miKvi KZ…©K webvg~‡j¨ weZi‡Yi Rb¨

gy ª‡Y :

cÖv_wgK MwYZ 3

f~wgKv cÖv_wgK wk¶vi ¸YMZ gvb Dbœq‡bi j‡¶¨ cÖv_wgK wk¶v Dbœqb cÖKí (2q ch©vq) Ges RvZxq wk¶vµg I cvV¨cy¯ZK †evW©, XvKv GKwU we¯Z…Z Kg©cwiKíbv MÖnY K‡i| G cwiKíbv Abyhvqx cÖv_wgK ¯Z‡ii wk¶vµg, cvV¨cy¯ZK I wk¶v mvgMÖxi g ~j¨vqb I cwigvR©b Kiv nq Ges G Kvh©µ‡gi wfwË‡Z cÖv_wgK ¯Z‡i cvV¨ cÖ‡Z¨KwU wel‡qi cvV¨cy¯—K Ab¨vb¨ cVb-cvVb mvgMÖx cÖYqb Kiv nq| MwYZ wel‡qi Rb¨ wPwýZ †kªYxwfwËK AR©‡bvc‡hvMx †hvM¨Zvi wfwË‡Z RvZxq wk¶vµg I cvV¨cy¯ZK †evW©, XvKv KZ©©„K wbe©vwPZ Ges MYcÖRvZš¿x evsjv‡`k miKv‡ii cÖv_wgK I MYwk¶v gš¿Yvjq KZ©„K Aby‡gvw`Z †jLKMY PZy_© †kªYxi Rb¨ bZzb MwYZ cvV¨cy ZK iPbv K‡ib| Kg©kvjvi gva¨‡g †kªYx wk¶K, wk¶K cÖwk¶K, welq we‡klÁ I wk¶vµg we‡kl‡Ái mnvqZvq eBwUi †hŠw³K g~j¨vqb m¤cbœ Kiv nq| AZtci cÖ‡dkbvj KwgwU Gbwmwmwm Ges cÖv_wgK I MYwk¶v gšÎYvjq cy¯ZKwUi P~ovš— Aby‡gv`b †`q| MvwYwZK a¨vb aviYv †ek RwUj| ZvB MvwYwZK aviYv ¯cóZi Kivi Rb¨ welqe¯Zy Dc¯’vc‡b cÖ‡qvRbxq e¨vL¨v I Qwe eënvi Kiv n‡q‡Q| wkL‡b wk¶v_x©‡ì AvMÖn m„wó I cvV MÖnY mnR Kivi Rb¨ Abykxj‡bi eë¯’v ivLv n‡q‡Q| ZvQvov cÖwZwU Aa¨v‡qi mgm¨v¸‡jv ÔmnR †_‡K KwVbÕ ixwZ AbymiY K‡i mvRv‡bv n‡q‡Q Ges G¸‡jv mgvav‡b wk¶v_x©iv hv‡Z D‡`¨vMx n‡Z cv‡i †mw`‡KI we‡kl `„wó †`qv n‡q‡Q| DcišZ Dc¯’vwcZ welqe¯Zyi †hme Z_¨ wk¶v_x©‡ì Rvbvi Rb¨ Acwinvh© Zv Ôj¶ KwiÕ I e‡·i g‡a¨ mwbœ‡ek Kiv n‡q‡Q| Avkv Kiv hvq, G‡Z wk¶v_x©‡ì MvwYwZK aviYvi myôz weKvk NU‡e Ges MwY‡Zi cÖwZ A‡nZzK fxwZ ~i n‡e| Avgiv Rvwb, wk¶vµg Dbœqb GKwU avivevwnK cÖwµqv Ges Gi wfwË‡Z cvV¨ iwPZ nq| Kv‡RB cvV¨cy¯ZK Av‡iv Dbœq‡bi Rb¨ †h‡Kv‡bv MVbg~jK I hyw³m½Z civgk© ¸i“‡Z¡i mv‡_ we‡ewPZ n‡e| 2021 mv‡j ¯vaxbZvi myeY© Rqš—x‡Z cÖZ¨vwkZ mg„× evsjv‡`k Movi wbiš—i cÖ‡Póvi Ask wn‡m‡e wk¶v_©x‡ì weÁvbgb¯‹ K‡i †Zvjvi j‡¶¨ eZ©gvb ms¯‹i‡Y wKQz cwigvR©b Kiv n‡q‡Q| mswk−ó e¨w³e‡M©i mhZœ cÖqvm I mZK©Zv m‡Ë¡I cy¯ZKwU‡Z wKQy Î“wU wePz¨wZ †_‡K †h‡Z cv‡i| cieZ©x ms¯‹i‡Y cvV¨cy¯ZKwU Î“wUgy³ Kivi †Póv Ae¨vnZ _vK‡e| G eBwU iPbv, m¤cv`bv I g ~j¨vqbmn cÖKvkbvi wewfbœ ch©v‡q huviv mnvqZv K‡i‡Qb, Zuv‡ì mevB‡K RvbvB Avš—wiK K…ZÁZv I abëv`| hv‡ì Rb¨ eBwU cÖYxZ nj, Zviv DcK…Z n‡j Avgv‡ì mKj cÖqvm mdj n‡e e‡j Avwg g‡b Kwi|

cÖ‡dmi †gv: †gv¯Zdv KvgvjDwÏb †Pqvig¨vb RvZxq wk¶vµg I cvV¨cy¯ZK †evW© XvKv|

cÖv_wgK MwYZ 1

m~wPcÎ Aa¨vq welqe¯‘ c„ôv

cÖ_g msL¨v 1 − 22

wØZxq †hvM I we‡qvM 23 − 31

Z…Zxq ¸Y 32 − 37

PZz_© fvM 38 − 43

cÂg mnR mgm¨v 44 − 47

lô mijxKiY 48 − 51

mßg ¸YYxqK I ¸YxZK 52 − 66

Aóg MvwYwZK cÖZxK 67 − 71

beg mvaviY fMœvsk 72 − 90

`kg `kwgK fMœvsk 91 − 108

GKv`k cwigvc 109 − 121

Øv`k mgq 122 − 128

Î‡qv`k DcvË msMÖn I web¨¯ZKiY 129 − 132

PZz ©k †jLwPÎ 133 − 138

cÂ`k R¨vwgwZ 139 − 152

DËigvjv 153−160

cÖv_wgK MwYZ 1

cÖ_g Aa¨vq

msL¨v msL¨v co I †jL (A‡¼ I K_vq) (GKwU K‡i †`Lv‡bv nj)

msL¨v (A¼)

nvRvi kZK `kK GKK

msL¨v (K_vq) AhyZ mnmª

5021 5 0 2 1 cuvP nvRvi GKzk

990

4672

7903

6550

3989

8095

8704

9008

9999

10000

2 cÖv_wgK MwYZ

`k, kZ I nvRv‡ii `j ˆZwi Ki

cÖv_wgK MwYZ 3

MYbv Ki I co

1 nvRvi 5 kZ 3 `k 7 1537

7 nvRvi 4 kZ 8 `k 9 7489

10 nvRvi

10000

40 nvRvi 6 kZ 1 `k 5

40615

nvRvi `k nvRvi kZ `k

nvRvi kZ kZ kZ kZ kZ kZ

`k `k `k

nvRvi nvRvi nvRvi nvRvi nvRvi nvRvi nvRvi

kZ kZ kZ kZ

`k `k `k `k `k `k `k `k

`k nvRvi

nvRvi nvRvi nvRvi nvRvi nvRvi

nvRvi nvRvi nvRvi nvRvi nvRvi

`k nvRvi `k nvRvi `k nvRvi `k nvRvi

kZ kZ kZ kZ kZ kZ `k

4 cÖv_wgK MwYZ

MYbv Ki I co

57 nvRvi 3 kZ 2

57302

73 nvRvi 2 kZ 4 `k

73240

100 nvRvi

1 j¶

100000

`k nvRvi `k nvRvi `k nvRvi `k nvRvi `k nvRvi

nvRvi nvRvi nvRvi nvRvi nvRvi nvRvi nvRvi

kZ kZ kZ

kZ


nvRvi nvRvi nvRvi`k nvRvi `k nvRvi

kZ kZ kZ kZ kZ



cÖv_wgK MwYZ 5

co

msL v (A‡¼)

†KvwU j¶ nvRvi

kZK `kK GKK msL v (K_vq) wbhyZ j¶ AhyZ mnmª

10001 1 0 0 0 1 `k nvRvi GK10009 1 0 0 0 9 `k nvRvi bq50716 5 0 7 1 6 cÂvk nvRvi

mvZk †lvj99999 9 9 9 9 9 wbivbeŸB

nvRvi bqkwbivbeŸB

100000 1 0 0 0 0 0 GK j¶ 780612 7 8 0 6 1 2 mvZ j¶

Avwk nvRvi Qqk

evi 4509325 4 5 0 9 3 2 5 cuqZvwj−k

j¶ bq nvRvi

wZbk cuwPk 9999999 9 9 9 9 9 9 9 wbivbeŸB

j¶ wbivbeŸB

nvRvi bqkwbivbeŸB

10000000 1 0 0 0 0 0 0 0 GK †KvwU j¶ Kwi : (1) 1 AhyZ = 10 nvRvi| (2) 1 wbhyZ = 10 j¶| (3) covi mgq 1 AhyZ Gi cwie‡Z© 10 nvRvi ejv nq| (4) covi mgq 1 wbhyZ Gi cwie‡Z© 10 j¶ ejv nq|

6 cÖv_wgK MwYZ

co, A‡¼ I K_vq †jL (GKwU K‡i †`Lv‡bv nj)

msL v (A‡¼) †KvwU wbhyZ j¶ AhyZ mnmª kZK `kK GKK msL v (K_vq)

907865 9 0 7 8 6 5 bq j¶ mvZ nvRvi AvUk cuqlwÆ

12934 78050 420199 2877602 8909009 10532040

co I A‡¼ †jL (GKwU K‡i †`Lv‡bv nj)

msLv (K_vq) †KvwU wbhyZ j¶ AhyZ mnmª kZK `kK GKK msLv

(A‡¼) cuvP j¶ beŸB nvRvi mvZk Pvi

5 9 0 7 0 4 590704

cuwPk nvRvi bqk AvUlwÆ

Qq j¶ cuPvËi nvRvi `yBk wÎk

†PŠÏ j¶ Pvi nvRvi AvUPwj −k

wQqvËi j¶ `k nvRvi PzivbeŸB

wbivbeŸB j¶ bq nvRvi beŸB

GK †KvwU wÎk j¶ cÂvk nvRvi mvZ

cÖv_wgK MwYZ 7

ÔKgvÕ eënvi K‡i msL¨v co

msL v (A‡¼)

Kgv eënvi K‡i

msL v (K_vq) †KvwU j¶ nvRvi

†Kv wb j A m k ` G

45798 4 5, 7 9 8 cuqZvwj−k nvRvi mvZk

AvUvbeŸB

97060 9 7, 0 6 0 mvZvbeŸB nvRvi lvU

642109 6, 4 2, 1 0 9 Qq j¶ weqvwj−k nvRvi

GKk bq

8908005 8 9 0 8, 0 0 5 EbbeŸB j¶ AvU

nvRvi cuvP

16543217 1, 6 5, 4 3, 2 1 7 GK †KvwU cuqlwÆ j¶

†ZZvwj−k nvRvi yBk m‡Zi

j¶ Kwi :

• Wvb w`K †_‡K wZbwU A‡¼i ci cÖ_g Kgv †Ìqv n‡q‡Q nvRvi †evSv‡bvi

Rb¨|

• cÖ_g Kgvi ev‡g yBwU A¼ †i‡L wØZxq Kgv †Ìqv n‡q‡Q j¶ †evSv‡bvi

Rb¨|

• wØZxq Kgvi ev‡g yBwU A¼ †i‡L Z…Zxq Kgv †Ìqv n‡q‡Q †KvwU †evSv‡bvi

Rb¨|

• 10 kZ‡K 1 nvRvi, 100 nvRv‡i 1 j¶, 100 j‡¶ 1 †KvwU|

8 cÖv_wgK MwYZ

ÔKgvÕ eënvi K‡i msL¨v †jLv

msL¨v (K_vq) Kgv eënvi K‡i msL¨v (A‡¼)

†KvwU j¶ nvRvi k ` G

0, 00, 00, 0 0 0

cÂvbœ nvRvi bqk mvZvËi 55, 9 7 7 55977

wbivbeŸB nvRvi AvUk cuqlwÆ 99, 8 6 5 99865

AvU j¶ AvUwÎk nvRvi Qqk PweŸk 8, 38, 6 2 4 838624

cuPvwk j¶ cuvP nvRvi cÂvbœ 85, 05, 0 5 5 8505055

yB †KvwU bq j¶ evi nvRvi mvZk bq 2, 09, 12, 7 0 9 20912709

j¶ Kwi :

• nvRvi wjL‡Z GKwU Kgv, j¶ wjL‡Z yBwU Kgv I †KvwU wjL‡Z wZbwU Kgv eënvi

Kiv n‡q‡Q|

• Wvb w`K †_‡K cÖ_‡g GKK, `kK I kZ‡Ki A¼ emv‡bv n‡q‡Q|

• Wvb w`K †_‡K kZ‡Ki A¼ emv‡bvi ci cÖ_g Kgv w`‡q nvRv‡ii A¼ emv‡bv n‡q‡Q|

• Wvb w`K †_‡K nvRv‡ii A¼ emv‡bvi ci wØZxq Kgv w`‡q j‡¶i A¼ emv‡bv n‡q‡Q|

• Wvb w`K †_‡K j‡¶i A¼ emv‡bvi ci Z…Zxq Kgv w`‡q †KvwUi A¼ emv‡bv n‡q‡Q|

• cÖ‡qvRb Abyhvqx 0 emv‡bv n‡q‡Q|

cÖv_wgK MwYZ 9

ÔKgvÕ eënvi K‡i msL¨v co I K_vq †jL (GKwU K‡i †`Lv‡bv nj) msL¨v (A‡¼) Kgv eënvi K‡i msL¨v (K_vq) 8205690 82,05,690 weivwk j¶ cuvP nvRvi Qqk beŸB 68704 942059 4935200 34406070

ÔKgvÕ eënvi K‡i msL¨v co I A‡¼ †jL (GKwU K‡i †`Lv‡bv nj)

msL¨v (A‡¼) Kgv eënvi K‡i msL¨v (K_vq) mvZvbœ j¶ cuqZvwj−k nvRvi AvUk †lvj

57,45,816 5745816

beŸB nvRvi bqk beŸB

Qq j¶ cuPvËi nvRvi Pvik wQqvwk

AvUvwk j¶ wZb nvRvi bqk yB

cuvP †KvwU GKvbeŸB j¶ Pvi nvRvi cuvPk Pwj−k

10 cÖv_wgK MwYZ

¯’vbxq gvb Qwe †`‡L msL¨v †jL ¯’vbxq gvb †jL (GKwU K‡i †`Lv‡bv nj) (GKwU K‡i †`Lv‡bv nj)

7 GKK = 7 6 `kK = 60 5 kZK = 500 8 mnmª = 8000

8 5 6 7

7 9 0 2

4 3 9 9

9 0 4 8

8 8 8 8

mnmª kZK `kK GKK 7385

mnmª kZK `kK GKK

mnmª kZK `kK GKK

mnmª kZK `kK GKK

mnmª kZK `kK GKK

cÖv_wgK MwYZ 11

¯’vbxq gvb

wbhyZ j¶ AhyZ mnmª kZK `kK GKK




†KvwU wbhyZ j¶ AhyZ mnmª kZK `kK GKK

4 2 7 1 5 3

6 5 0 7 0 3 1

8 0 4 6 2 0 0

6 2 3 8 5 1 4

1 0 0 0 0 0 0

3 GKK = 35 `kK = 501 kZK = 1007 mnmª = 70002 AhyZ = 200004 j¶ = 400000

1 GKK = 13 `kK = 300 kZK = 07 mnmª = 70000 AhyZ = 05 j¶ = 5000006 wbhyZ = 6000000



0 GKK = 00 `kK = 00 kZK = 00 mnmª = 00 AhyZ = 00 j¶ = 00 wbhyZ = 01 †KvwU = 10000000

12 cÖv_wgK MwYZ

¯’vbxq gvb Qwe †`‡L msL¨v †jL ¯’vbxq gvb †jL (GKwU K‡i †`Lv‡bv nj) (GKwU K‡i †`Lv‡bv nj)

wbhyZ j¶ AhyZ mnmª kZK kK

5046013

7 3 0 2 8 5 0

2 4 3 9 6 0 1

5 8 0 6 7 0 2

8 0 0 4 6 5 0

9 9 0 9 9 9 5

cÖv_wgK MwYZ 13

¯’vbxq gvb

†KvwU wbhyZ j¶ AhyZ mnmª kZK `kK GKK msL¨v (K_vq)

1 GK

1 0 `k

1 0 0 GKkZ

1 0 0 0 GK nvRvi

1 0 0 0 0 `k nvRvi

1 0 0 0 0 0 GK j¶

1 0 0 0 0 0 0 `k j¶

1 0 0 0 0 0 0 0 GK †KvwU

¯’vbxq gvb msL¨vi Wvb w`K †_‡K cov nq :

10 GK‡K 1 `kK

10 `k‡K 1 kZK

10 kZ‡K 1 nvRvi ev 1 mnmª

10 nvRv‡i 1 AhyZ

10 Ahy‡Z ev 100 nvRv‡i 1 j¶

10 j‡¶ 1 wbhyZ

10 wbhy‡Z ev 100 j‡¶ 1 †KvwU

14 cÖv_wgK MwYZ

¯’vbxq gvb wbY©q Ki Ges ¯’vbxq gvb¸‡jv †hvM Ki ( yBwU K‡i †`Lv‡bv nj)

7 Gi ¯’vbxq gvb 76 Gi ¯’vbxq gvb 603 Gi ¯’vbxq gvb 3005 Gi ¯’vbxq gvb 50002 Gi ¯’vbxq gvb 200009 Gi ¯’vbxq gvb 9000008 Gi ¯’vbxq gvb 8000000 8925367

9 Gi ¯’vbxq gvb 90008 Gi ¯’vbxq gvb 8000000 Gi ¯’vbxq gvb 01 Gi ¯’vbxq gvb 100006 Gi ¯’vbxq gvb 60000004 Gi ¯’vbxq gvb 42 Gi ¯’vbxq gvb 20 6819024

8 Gi ¯’vbxq gvb 9 Gi ¯’vbxq gvb 5 Gi ¯’vbxq gvb 0 Gi ¯’vbxq gvb 2 Gi ¯’vbxq gvb 6 Gi ¯’vbxq gvb 1 Gi ¯’vbxq gvb






8925367 6819024

7586923 1620598

3805792 9863054

85369746532871

cÖv_wgK MwYZ 15

msL¨vi Zzjbv eo †QvU wbY©q I cÖZxK e¨envi K‡i †jL

7865, 8743 7865 = 7 nvRvi 8 kZ 6 `k 5 8743 = 8 nvRvi 7 kZ 4 `k 3

∴7865 †QvU, 8743 eo ev, 7865 < 8743 ; < †QvU wPý covi wbqg : 7865, 8743 †_‡K †QvU| Avevi, Gfv‡eI †jLv hvq 8743 > 7865 ; > eo wPý covi wbqg : 8743, 7865 †_‡K eo|

Pvi A‡¼i `yBwU msL¨vi g‡a¨ †h msL¨vwU‡Z nvRv‡ii A¼ eo, †mB msL¨vwU eo|

9754, 9745 9754 = 9 nvRvi 7 kZ 5 `k 4 9745 = 9 nvRvi 7 kZ 4 `k 5

∴ 9754 eo, 9745 †QvU ev, 9754 > 9745

Pvi A‡¼i `yBwU msL¨vi g‡a¨ nvRv‡ii A¼ GKB n‡j, kZK, `kK I GK‡Ki A¼ †`‡L eo †QvU wbY©q Ki‡Z nq|

46059, 47182 46059 = 4 AhyZ 6 mnmª 0 kZ 5 `k 947182 = 4 AhyZ 7 mnmª 1 kZ 8 `k 2

∴ 46059 †QvU, 47182 eo ev, 46059 < 47182

cuvP A‡¼i `yBwU msL¨vi g‡a¨ Ahy‡Zi A¼ GKB n‡j, mnmª, kZK, `kK I GK‡Ki A¼ †`‡L †QvU eo wbY©q Ki‡Z nq|

100000, 99999 100000 = 1 j¶ 99999 = 9 AhyZ 9 mnmª 9 kZ 9 `k 9

∴ 100000 eo, 99999 †QvU ev, 100000 > 99999

yBwU msL¨vi g‡a¨ GKwU Qq A‡¼i I GKwU cuvP A‡¼i n‡j, Qq A‡¼i msL¨vwU me mgqB eo|

16 cÖv_wgK MwYZ

msL¨vi Zzjbv eo †QvU wbY©q Ki I cÖZxK e¨envi K‡i †jL ( yBwU K‡i †`Lv‡bv nj)

8539, 8607

8539 †QvU, 8607 eo

8539 < 8607

6789, 7654

7654 eo, 6789 †QvU

7654 > 6789

4385, 4702

⎯⎯ †QvU, ⎯⎯ eo

7825, 7852

⎯⎯ eo, ⎯⎯ †QvU

9628, 9569

⎯⎯ eo, ⎯⎯ †QvU

35706, 43082

⎯⎯ †QvU, ⎯⎯ eo

56789, 56817

⎯⎯ eo, ⎯⎯ †QvU

85398, 85389

⎯⎯ †QvU, ⎯⎯ eo

78029, 79103

⎯⎯ eo, ⎯⎯ †QvU

99999, 88888

⎯⎯ eo, ⎯⎯ †QvU

cÖv_wgK MwYZ 17

msL¨v Abyµ‡g mvRv‡bv eo †_‡K †QvU I †QvU †_‡K eo µ‡g mvRv‡bv 4309, 8142, 7214, 8124, 6037, 7218 GLv‡b,4309 = 4 nvRvi 3 kZ 0 `k 9 8142 = 8 nvRvi 1 kZ 4 `k 2 7214 = 7 nvRvi 2 kZ 1 `k 4 8124 = 8 nvRvi 1 kZ 2 `k 4 6037 = 6 nvRvi 0 kZ 3 `k 7 7218 = 7 nvRvi 2 kZ 1 `k 8 nvRvi, kZK, `kK I GK‡Ki A¼¸‡jv †`‡L msL¨v¸‡jv‡K eo †_‡K †QvU µ‡g mvwR‡q cvB, 8142, 8124, 7218, 7214, 6037, 4309 ev, 8142 > 8124 > 7218 > 7214 > 6037 > 4309| Avevi, †QvU †_‡K eo µ‡g mvwR‡q cvB, 4309, 6037, 7214, 7218, 8124, 8142 ev, 4309 < 6037 < 7214 < 7218 < 8124 < 8142|65904, 891254, 98536, 890397, 7706253, 6814300, 98563 GLv‡b, 65904 = 65 nvRvi 9 kZ 0 k 4 891254 = 8 j¶ 91 nvRvi 2 kZ 5 k 4 98536 = 98 nvRvi 5 kZ 3 k 6 890397 = 8 j¶ 90 nvRvi 3 kZ 9 k 7 7706253 = 77 j¶ 6 nvRvi 2 kZ 5 k 3 6814300 = 68 j¶ 14 nvRvi 3 kZ 98563 = 98 nvRvi 5 kZ 6 k 3 j¶, nvRvi, kZK, kK I GK‡Ki A¼ ‡jv † ‡L msL v ‡jv‡K eo †_‡K †QvU µ‡g mvwR‡q cvB, 7706253, 6814300, 891254, 890397, 98563, 98536, 65904 ev, 7706253 > 6814300 > 891254 > 890397 > 98563 > 98536 > 65904 Avevi, †QvU †_‡K eo µ‡g mvwR‡q cvB, 65904, 98536, 98563, 890397, 891254, 6814300, 7706253 ev, 65904 < 98536 < 98563 < 890397 < 891254 < 6814300 < 7706253|

18 cÖv_wgK MwYZ

msL¨v Abyµ‡g mvRv‡bv eo †_‡K †QvU I †QvU †_‡K eo µ‡g mvRvI (GKwU K‡i †`Lv‡bv nj)

msL¨v eo †_‡K †QvU †QvU †_‡K eo 65032, 8973, 26940, 53278, 80149, 84256, 9856

84256, 80149, 65032,53278, 26940, 9856, 8973| 84256 >80149>65032>53278>26940>9856> 8973|

8973, 9856, 26940, 53278, 65032, 80149, 84256| 8973 < 9856 < 26940 < 53278 < 65032 < 80149 < 84256

8265, 9702, 8814, 7523, 9537, 7532

65068, 72498, 66253, 72085, 54987, 39875, 55210

543210, 35721, 68057, 54321, 759002, 752083

889531, 902671, 888530, 725094, 875320, 923510, 732189

992570, 8548071, 8877510, 999999, 8888888, 632571, 8509724

5746800, 3587260, 889006, 4370582, 5526897, 5742810, 64085913

cÖv_wgK MwYZ 19

e„nËg I ¶z ªZg msL¨v MVb GKB A¼ gvÎ GKevi e¨envi K‡i e„nËg I ¶z`ªZg msL¨v MVb cÖ Ë A¼ : 5, 2, 7, 9 9>7>5>2 2<5<7<9

∴e„nËg msL¨v : 9752 ¶z`ªZg msL¨v : 2579

cÖ Ë A¼ : 6, 0, 8, 5, 3 8>6>5>3>0 0<3<5<6<8

∴ e„nËg msL¨v : 86530 ¶z`ªZg msL¨v : 30568

cÖ Ë A¼ : 4, 5, 0, 8, 7, 2 8>7>5>4>2>0 0<2<4<5<7<8

∴ e„nËg msL¨v : 875420 ¶z`ªZg msL¨v : 204578

cÖ Ë A¼ : 3, 6, 7, 4, 5, 8 8>7>6>5>4>3 3<4<5<6<7<8

∴ e„nËg msL¨v : 876543 ¶z ªZg msL¨v : 345678

GK ev GKvwaK A‡¼i e„nËg I ¶z ªZg msL¨v cÖ Ë A¼ e„nËg msL¨v ¶z`ªZg msL¨v

GK A¼ wewkó msL¨vi yB A¼ wewkó msL¨vi

wZb A¼ wewkó msL¨vi Pvi A¼ wewkó msL¨vi cuvP A¼ wewkó msL¨vi Qq A¼ wewkó msL¨vi

9 99 999 9999 99999 999999

1 10 100 1000 10000 100000

j¶ Kwi : • cÖ Ë A¼¸‡jv‡K cÖZx‡Ki mvnv‡h¨ eo †_‡K †QvU Ges †QvU †_‡K eo µ‡g

wj‡L e„nËg I ¶z ªZg msL¨v MVb Kiv n‡q‡Q| • cÖ Ë A‡¼ Ô0Õ _vK‡j me‡P‡q †QvU A¼wUi Wv‡b Ô0Õ ewm‡q †QvU †_‡K

eo A¼ wj‡L ¶z ªZg msL¨v MVb Kiv n‡q‡Q| • GK ev GKvwaK A‡¼i e„nËg msL¨v MV‡bi Rb¨ wbw ©ó msL¨K Ô9Õ †jLv

n‡q‡Q| • Ô1Õ Gi Wv‡b cÖ‡qvRbxq msL¨K Ô0Õ w`‡q wbw ©ó A‡¼i ¶z`ªZg msL¨v MVb

Kiv n‡q‡Q|

20 cÖv_wgK MwYZ

†ivgvb msL¨v cÖZxK

cv‡ki QwewU GKwU †Ìqvj Nwoi|

Nwoi Wvqv‡ji Dci mgq wb‡ ©kK

msL¨v¸‡jv †ivgvb msL¨v cÖZx‡K †jLv

Av‡Q| mvaviYZ †kªYxK‡¶i ìRvi

Dc‡i †ivgvb msL¨v cÖZxK eënvi Kiv

n‡q _v‡K|

j¶ Kwi : • 1 Gi †ivgvb msL¨v cÖZxK I ; 5 Gi †ivgvb msL¨v cÖZxK V ; 10 Gi

†ivgvb msL¨v cÖZxK X • I, V I X GB wZbwU cÖZxK eënvi K‡i 12 ch©š� msL¨v †ivgvb c×wZ‡Z

†jLv n‡q‡Q| • I, V I X cÖZx‡Ki Wvbcv‡k I emv‡j GK †ewk nq| †hgb, II = 1 + 1 = 2; III = 2 + 1 = 3 ; VI = 5 + 1 = 6 ; XI =

10 + 1 = 11 • V ev X cÖZx‡Ki evgcv‡k I emv‡j GK Kg nq| †hgb, IV = 5 - 1 = 4 ; IX = 10 - 1 = 9| †ivgvb msL¨v cÖZxK †`‡L msL¨v †jL : II = ⎯, IV = ⎯, X = ⎯ VII = ⎯. XII = ⎯. msL¨v¸‡jvi †ivgvb msL¨v cÖZxK †jL : 3 = ⎯, 5 = ⎯, 8 = ⎯ 11 = ⎯, 12 = ⎯|

msL¨v msL¨v msL¨v (†ivgvb (A‡¼) (K_vq) c×wZ‡Z) 1 GK I 2 yB II 3 wZb III 4 Pvi IV 5 cuvP V 6 Qq VI

msL¨v msL¨v msL¨v (†ivgvb (A‡¼) (K_vq) c×wZ‡Z) 7 mvZ VII 8 AvU VIII 9 bq IX 10 `k X 11 GMvi XI 12 evi XII

cÖv_wgK MwYZ 21

Abykxjbx- 1 1| ÔKgvÕ e¨envi K‡i wb‡Pi msL¨v¸‡jv co I K_vq †jL : 25734, 47099, 880539, 5704602, 8055505, 8888888, 10000000| 2| wb‡Pi msL¨v¸‡jv A‡¼ †jL : cuqZvwj−k nvRvi mvZk evi; AvUlwÆ nvRvi bqk wZb; Qq j¶ cÂvk nvRvi

wZbk bq; mvZvk j¶ Pvi nvRvi EblvU; AvUvwk j¶ mvZk Avwk; wbivbeŸB j¶ bq nvRvi bqk beŸB; GK †KvwU|

3| k �b¨¯’vb c�iY Ki : 5908726 GB msL¨vwU‡Z

(K) 9 Gi ¯’vbxq gvb ⎯⎯

(L) 8 Gi ¯’vbxq gvb ⎯⎯

(M) 7 Gi ¯’vbxq gvb ⎯⎯

(N) 6 Gi ¯’vbxq gvb ⎯⎯

(O) 5 Gi ¯’vbxq gvb ⎯⎯

(P) 0 Gi ¯’vbxq gvb ⎯⎯

(Q) 2 Gi ¯’vbxq gvb ⎯⎯ 4| 555555 msL¨vwU‡Z cÖ‡Z¨KwU Ô5Õ Gi ¯’vbxq gvb †jL| 5| wb‡Pi msL¨vi †Rvov¸‡jv †_‡K cÖZx‡Ki mvnv‡h¨ eo †QvU wbY©q Ki : 8326, 8362 ; 7506, 8021 ; 6798, 6789 ; 85320, 76203 ; 58249, 58492| 6| wb‡Pi msL¨v¸‡jv‡K eo †_‡K †QvU I †QvU †_‡K eo µ‡g mvwR‡q †jL : (K) 4321, 3270, 4902, 3357, 4287, 5027| (L) 8506, 5786, 6210, 5768, 6218, 8560| (M) 72570, 72607, 83259, 60892, 60829, 83592| (N) 90062, 83001, 90257, 78253, 78235, 83251, 79805|

22 cÖv_wgK MwYZ

7| GKwU A¼ gvÎ GKevi e¨envi K‡i wb‡Pi A¼¸‡jv w`‡q e„nËg I ¶z`ªZg msL¨v MVb Ki : (K) 3, 9, 8 (L) 9, 0, 7 (M) 4, 3, 8, 5 (N) 6, 0, 3, 8 8| cuvP A‡¼i e„nËg I ¶z ªZg msL¨v †jL| 9| 5, 7, 3, 2, 8 A¼¸‡jv Øviv MwVZ e„nËg msL¨vwU‡Z 8, 7 I 3 Gi ¯’vbxq

gvb †jL| 10| wZb A‡¼i I cuvP A‡¼i e„nËg msL¨vi c‡ii msL¨vwU †jL| 11| Pvi A‡¼i I Qq A‡¼i ¶z ªZg msL¨vi Av‡Mi msL¨vwU †jL| 12| wb‡Pi msL¨v¸‡jv †ivgvb c×wZ‡Z †jL : 2, 5, 8, 9, 11, 12 13| wb‡Pi †ivgvb msL¨v cÖZxK¸‡jv‡K msL¨vq cÖKvk Ki : IV, VII, IX, X, VI, XII 14| `vM †U‡b msL¨vi mv‡_ †ivgvb msL¨v cÖZx‡Ki wgj Ki :

6 XI 9 V 4 VIII 11 IX 8 VI 10 IV 5 VII 7 X

cÖv_wgK MwYZ 23

wØZxq Aa¨vq

†hvM I we‡qvM D`vniY 1| †hvM Ki : 2403, 3214, 1052, 2310|

mgvavb : 2 4 0 3 2 nvRvi 4 kZ 0 `k 3 3 2 1 4 3 nvRvi 2 kZ 1 `k 4 1 0 5 2 1 nvRvi 0 kZ 5 `k 2 + 2 3 1 0 2 nvRvi 3 kZ 1 `k 0 8 9 7 9 8 nvRvi 9 kZ 7 `k 9 DËi : 8979| = 8979 D`vniY 2| †hvM Ki : 3257, 4062, 2103, 5130, 1436| mgvavb : 3 2 5 7 3 nvRvi 2 kZ 5 `k 7 4 0 6 2 4 nvRvi 0 kZ 6 `k 2 2 1 0 3 2 nvRvi 1 kZ 0 `k 3 5 1 3 0 5 nvRvi 1 kZ 3 `k 0 + 1 4 3 6 1 nvRvi 4 kZ 3 `k 6 1 5 9 8 8 15 nvRvi 8 kZ 17 k 18

= 15 nvRvi 8 kZ 18 k 8

DËi : 15988| = 15 nvRvi 9 kZ 8 `k 8

= 15988

j¶ Kwi : • msL¨v¸‡jv‡K GK‡Ki wb‡P GKK, `k‡Ki wb‡P `kK, kZ‡Ki wb‡P kZK

Gfv‡e mvwR‡q †jLv n‡q‡Q|

• GK‡Ki Ni †_‡K †hv‡Mi KvR ïiy Kiv n‡q‡Q|

• †Kvb N‡ii A‡¼i †hvMdj yB A‡¼i n‡j, ïay Wvb cv‡ki A¼wU †mB N‡i emv‡bv n‡q‡Q| evgcv‡ki A¼wU evg w ‡Ki N‡ii A¼¸‡jvi mv‡_ †hvM Kiv n‡q‡Q|

24 cÖv_wgK MwYZ

D`vniY 3| †hvM Ki : 25007, 40018, 14023, 3625, 16241 mgvavb : 2 5 0 0 7 4 0 0 1 8 1 4 0 2 3 3 6 2 5 + 1 6 2 4 1 9 8 9 1 4 DËi t 98914| j¶ Kwi : • GK‡Ki N‡ii A¼¸‡jvi †hvMdj 24| 24 Gi 4 †iLvi wb‡P GK‡Ki N‡i

e‡m‡Q Ges 2 `k `k‡Ki N‡i Dc‡i †jLv n‡q‡Q| • `k‡Ki N‡ii A¼¸‡jv †hvM K‡i GKBfv‡e nv‡Zi msL¨v kZ‡Ki N‡i Dc‡i

†jLv n‡q‡Q| • Abyiƒcfv‡e, kZK, mnmª I Ahy‡Zi N‡ii A¼¸‡jvi †hvM GKBfv‡e Kiv

n‡q‡Q| D`vniY 4| cvkvcvwk wj‡L †hvM Ki : 2304, 1620, 40521, 12032, 21350| mgvavb : 2304 + 1620 + 40521 + 12032 + 21350 = 77827 DËi t 77827| j¶ Kwi : • msL¨v¸‡jv‡K †hvM wPý (+) w`‡q cvkvcvwk †jLv n‡q‡Q| • msL¨v¸‡jvi cv‡k mgvb wPý (=) w`‡q †hvMdj †jLv n‡q‡Q| • cÖwZwU msL¨vi GKK, `kK, kZK, nvRvi, AhyZ A¼¸‡jv ch©vqµ‡g †hvM

Kivi Av‡M A¼¸‡jvi Dci `vM (/) †`Iqv n‡q‡Q| • †hvM Kivi mgq †Kvb N‡ii nv‡Zi msL¨v cieZ©x‡Z evgcv‡ki A‡¼i mv‡_

†hvM Kiv n‡q‡Q| gš�e : wk¶v_©xiv cÖPwjZ wbq‡g †hvM A¼ Ki‡e| Zv‡`i wbKU †Kvb e vL v PvIqv n‡ebv|

A m k ` G

+1

2

4

1

1

5

0

4

3

6

+1

0

0

0

6

2

+2

0

1

2

2

4

7

8

3

5

1

9 8 9 1 4

cÖv_wgK MwYZ 25

†hvM Ki (GKwU K‡i †`Lv‡bv nj) : 3 2 0 7 9 2 5 7 2 4 2 1 4 0 2 4 6 0 7 1 8 2 0 4 1 0 3 9 1 5 3 7 9 1 3 8 5 2 5 6 2 5 3 8 0 4 1 6 8 0 2 1 5 0 3 4 2 0 5 3 1 1 3 5 6 2 0 5 3 1 3 0 7 9 1 1 4 3 6 2 4 9 2 1 1 6 2 4 8 9 0 8 0 1 LvwjNi c �iY Ki :

2160 + 1532 + 1425 + 2054 =

35046 + 1725 + 2908 + 26871 + 10425 =

15270 + 26085 + 14326 + 13572 + 30628 =

D`vniY 5| GKwU we`¨vj‡q 3528 Rb wk¶v_©x wQj| eQ‡ii cÖ_‡g H we`¨vj‡q AviI 1675 Rb wk¶v_©x fwZ© nj| GLb H we`¨vj‡q †gvU KZRb wk¶v_©x nj?

mgvavb : wk¶v_©x wQj 3528 Rb fwZ© nj 1675 Rb

∴†gvU wk¶v_©x 5203 Rb DËi : 5203 Rb|

D`vniY 6| GKwU MvQ †_‡K 2590 wU wjPz cvov n‡q‡Q| Ab¨ GKwU MvQ †_‡K

3255wU Ges Avi GKwU MvQ †_‡K 2834wU wjPz cvov n‡q‡Q| wZbwU MvQ †_‡K †gvU KZwU wjPz cvov n‡q‡Q?

mgvavb : wZbwU MvQ †_‡K wjPz cvov nq 2590 wU

3255 wU 2834 wU wZbwU MvQ †_‡K †gvU wjPz cvov nq 8679 wU

∴ †gvU wjPzi msL¨v 8679

DËi : 8679 wU|

26 cÖv_wgK MwYZ

D`vniY 7| 8574 †_‡K 6142 we‡qvM Ki| mgvavb : 8 5 7 4 8 nvRvi 5 kZ 7 `k 4 we‡qvRb

− 6 1 4 2 − 6 nvRvi 1 kZ 4 `k 2 we‡qvR¨ 2 4 3 2 2 nvRvi 4 kZ 3 `k 2 we‡qvMdj = 2432 DËi : 2432| D`vniY 8| 94632 †_‡K 7583 we‡qvM Ki| mgvavb : 9 4 6 3 2 −7 5 8 3 8 7 0 4 9 GLv‡b, 94632 we‡qvRb, 7583 we‡qvR¨ Ges 87049 we‡qvMdj| DËi : 87049| D`vniY 9| 63205 †_‡K 20847 we‡qvM Ki| mgvavb : 6 3 2 0 5 - 2 0 8 4 7 4 2 3 5 8 DËi : 42358| D`vniY 10| 52314 †_‡K 16729 cvkvcvwk we‡qvM Ki| mgvavb : 52314 - 16729 = 35585 GLv‡b, 52314 we‡qvRb, 16729 we‡qvR¨ Ges 35585 we‡qvMdj| DËi : 35585| we‡qvRb - we‡qvR¨ = we‡qvMdj we‡qvRb = we‡qvMdj + we‡qvR¨

we‡qvRb - we‡qvMdj = we‡qvR¨ j¶ Kwi : • we‡qvRb I we‡qv‡R¨i gv‡S we‡qvM (−) wPý Ges †k‡l mgvb (=) wPý † Iqv n‡q‡Q| • cÖ‡Z¨KwU msL vi GKK, kK, kZK BZ vw wP‡b we‡qvM Kivi myweav‡_© msL vi Dc‡i ch©vqµ‡g (/) `vM †`Iqv n‡q‡Q|

AhyZ mnmª kZK `kK GKK 9

−+1

4+10 7

6 5+1

3+10 8+1

2+10 3

8 7 0 4 9

AhyZ mnmª kZK `kK GKK 6

−2

3 0+1

2+10 8+1

0+10 4+1

5+10 7

4 2 3 5 8

cÖv_wgK MwYZ 27

we‡qvM Ki (GKwU K‡i †`Lv‡bv nj) : 9 6 0 2 5 6 2 5 4 8 4 3 0 6 2 6 5 3 1 − 3 8 2 7 6 − 3 0 7 9 − 5 7 2 8 3 −8 6 0 7 5 7 7 4 9 LvwjNi c �iY Ki (GKwU K‡i †`Lv‡bv nj) : 4635 − 1586 = 4503 − 3821 = 92507 − 73058 = 86013 − 58320 = D`vniY 11| we‡qvRb 8506 Ges we‡qvR¨ 3789 n‡j, we‡qvMdj KZ? mgvavb : we‡qvRb 8 5 0 6 we‡qvR¨ 3 7 8 9 we‡qvMdj 4 7 1 7 DËi : 4717| D`vniY 12| `yBwU msL¨vi †hvMdj 7209| GKwU msL¨v 915 n‡j, Aci

msL¨vwU KZ? mgvavb : yBwU msL¨vi †hvMdj 7 2 0 9 GKwU msL¨v − 9 1 5 ∴ Aci msL¨vwU 6 2 9 4 DËi : 6294| D`vniY 13| wZbwU msL¨vi †hvMdj 57304| Zv‡`i g‡a¨ yBwU msL¨v 13528 I 4952| Z…Zxq msL¨vwU KZ? mgvavb : yBwU msL¨v 1 3 5 2 8 + 4 9 5 2 1 8 4 8 0 wZbwU msL¨vi †hvMdj 5 7 3 0 4 yBwU msL¨vi †hvMdj − 1 8 4 8 0 ∴ Z…Zxq msL¨vwU 3 8 8 2 4 DËi : 38824|

3049

28 cÖv_wgK MwYZ

D`vniY 14| gwR‡`i wbKU 40895 UvKv Av‡Q| Kvwb‡Ri wbKU gwR` A‡c¶v 5389 UvKv Kg Av‡Q| dv‡Zgvi wbKU KvwbR A‡c¶v 987 UvKv †ewk Av‡Q| dv‡Zgvi KZ UvKv Av‡Q? mgvavb : gwR‡`i Av‡Q 40895 UvKv

− 5389 UvKv

∴ Kvwb‡Ri Av‡Q 35506 UvKv Avevi, Kvwb‡Ri Av‡Q 35506 UvKv + 987 UvKv

∴ dv‡Zgvi Av‡Q 36493 UvKv DËi : dv‡Zgvi 36493 UvKv Av‡Q| D`vniY 15| wcZv I cy‡Îi eZ©gvb eq‡mi mgwó 125 eQi| 10 eQi Av‡M cy‡Îi eqm 36 eQi wQj| 8 eQi c‡i wcZvi eqm KZ n‡e? mgvavb : 10 eQi Av‡M cy‡Îi eqm wQj 36 eQi + 10 eQi cy‡Îi eZ©gvb eqm 46 eQi wcZv I cy‡Îi eZ©gvb eq‡mi mgwó 125 eQi

cy‡Îi eZ©gvb eqm −46 eQi

∴ wcZvi eZ©gvb eqm 79 eQi wcZvi eZ©gvb eqm 79 eQi + 8 eQi

∴ 8 eQi c‡i wcZvi eqm 87 eQi DËi : 87 eQi|

cÖv_wgK MwYZ 29

Abykxjbx - 2

1| †hvM Ki : (K) 6 4 0 2

2 5 7 1 3 0 8 9

7 9 6

(L) 4 5 8 62 1 9 05 0 3 41 7 0 8

(M) 1 5 2 67 0 8 52 6 0 04 9 5 43 1 9 7

(N) 4 0 5 95 3 2 18 6 4 72 9 0 53 5 7 4

(O) 2 3 5 7 0 4 6 0 8 9

5 9 0 7 2 6 5 3 3 7 9 2

(P) 3 0 7 9 12 8 0 5 61 3 8 0 72 6 0 8 5

(Q) 1 2 6 5 31 7 2 0 62 4 0 8 93 2 5 7 01 0 2 3 4

(R) 2 5 0 4 61 7 2 8 01 3 7 0 52 0 4 5 12 1 6 0 3

2| LvwjNi c�iY Ki : (K) 2357 + 2506 + 723 + 406 = (L) 3240 + 1528 + 4039 + 1247 = (M) 2470 + 1305 + 2167 + 3280 + 1075 = (N) 13264 + 20315 + 13204 + 15023 + 398 = (O) 12304 + 15360 + 20471 + 19052 + 31052 = 3| we‡qvM Ki :

(K) 8 5 0 4 7 2 6

(L) 7 5 2 42 8 3 0

(M) 9 2 5 0 38 6 0 7

(N) 6 1 3 8 29 3 0 7

(O) 9 1 6 0 5 3 27 3 8

(P) 6 0 3 2 71 4 5 6 9

(Q) 5 1 6 0 81 2 5 3 2

(R) 7 2 5 3 02 2 0 7 5

4| LvwjNi c �iY Ki : (K) 6325 Ñ 896 = (L) 7104 Ñ 1368 = (M) 7052 Ñ 1695 = (N) 85213 Ñ 7359 = (O) 96021 Ñ 57234 = (P) 63205 Ñ 24307 =

30 cÖv_wgK MwYZ

5| we‡qvM Ki I cv‡ki LvwjNi c�iY Ki : (K) 9 4 0 3 GLv‡b, we‡qvRb 4 3 8 7 we‡qvR¨ we‡qvMdj (L) 7 3 2 5 1 GLv‡b, we‡qvR¨ 5 3 7 4 we‡qvMdj we‡qvRb (M) 6214-2578 = GLv‡b, we‡qvMdj we‡qvRb we‡qvR¨ (N) 5302-1805 = GLv‡b, we‡qvR¨ we‡qvRb we‡qvMdj 6| GKwU MÖv‡g 2310 Rb †jvK evm K‡i| Ab¨ GKwU MÖv‡g 4096 Rb Ges

Avi GKwU MÖv‡g 985 Rb †jvK evm K‡i| wZbwU MÖv‡g †gvU KZRb †jvK evm K‡i?

7| `yBwU msL¨vi †hvMdj 83502| GKwU msL¨v 2408 n‡j, Aci msL¨vwU KZ?

8| GKwU we‡qvM A‡¼i we‡qvMdj 5240 Ges we‡qvR¨ 759| we‡qvRb KZ? 9| Qq A‡¼i ¶z`ªZg msL¨v †_‡K cuvP A‡¼i e„nËg msL¨v we‡qvM Ki| 10| yBwU msL¨vi we‡qvMdj 4215| we‡qvRb 8350 n‡j, we‡qvR¨ KZ? 11| GKwU MvQ †_‡K 3825 wU Avg cvov n‡q‡Q| Aci GKwU MvQ †_‡K 876 wU

Avg Kg cvov n‡q‡Q| yBwU MvQ †_‡K †gvU KZwU Avg cvov n‡q‡Q? 12| GKwU we`¨vj‡q 4876 Rb wk¶v_©x wQj| eQ‡ii cÖ_‡g 512 Rb wk¶v_©x

we`¨vjq †_‡K P‡j †Mj Ges 1954 Rb wk¶v_©x bZyb fwZ© nj| H we`¨vj‡q wk¶v_©x msL¨v KZ nj?

32 cÖv_wgK MwYZ

Z…Zxq Aa¨vq

¸Y D`vniY 1| ¸Y Ki : mgvavb :

DËi : 10353| e¨vL¨v : 3 5 7 2 9 7 9 5 `k 2 `k 3 kZ GLb, 3 5 7 × 9 7 × 9 = 63 = 63 5 `k × 9 = 45 `k = 450 3 kZ × 9 = 27 kZ = 2700 3213 Ges 3 5 7 × 2 `k 7 × 2 `k = 14 `k = 140 5 `k × 2 `k = 50 × 2 `k = 100 `k = 1000 3 kZ × 2 `k = 300 × 2 `k = 600 `k= 6000 7140 AZGe, 357 × 9 = 3213 357 × 2 `k = 7140 357 × 29 = 10353

357×29

357×29

32137140

10353

357×29

32137140

10353

357 × 9 357 × 2 `k

35 × 9 = 315

¸Y¨ ¸YK ¸Ydj

cÖv_wgK MwYZ 33

D`vniY 2| ¸Y Ki : 532 × 37 mgvavb : 532 532 × 37 × 37 3724 3724 532 × 7 15960 15960 532 × 3 `k 19684 19684 DËi : 19684| D`vniY 3| ¸Y Ki : 513 × 127 mgvavb : 5 1 3 5 1 3 × 1 2 7 × 1 2 7 3 5 9 1 3 5 9 1 513 × 7 1 0 2 6 0 1 0 2 6 0 513 × 2 `k 5 1 3 0 0 5 1 3 0 0 513 × 1 kZ 6 5 1 5 1 6 5 1 5 1 DËi : 65151| D`vniY 4| ¸Y Ki : 409 × 235 mgvavb : 4 0 9 4 0 9 × 2 3 5 × 2 3 5 2 0 4 5 2 0 4 5 409 × 5 1 2 2 7 0 1 2 2 7 0 409 × 3 `k 8 1 8 0 0 8 1 8 0 0 409 × 2 kZ 9 6 1 1 5 9 6 1 1 5 DËi : 9 6 1 1 5|

34 cÖv_wgK MwYZ

D`vniY 5| 236 †K 150 Øviv ¸Y Ki|

mgvavb : 2 3 6

× 1 5 0

1 1 8 0 0 2 3 6 × 5 `k

2 3 6 0 0 2 3 6 × 1 kZ 3 5 4 0 0

e¨vL¨v : 236 × 5 `k = 1180 `k = 1 1 8 0 0

236 × 1 kZ = 236 kZ = 2 3 6 0 0

∴ 236 × 150 = 3 5 4 0 0 DËi : 35400|

j¶ Kwi : ¸Y‡Ki GK‡Ki N‡i k�b¨ (0) _vKvq GK‡Ki ¸Y †`Lv‡bv nqwb| ZvB cÖ_g av‡c GK‡Ki N‡i k�b¨ ewm‡q `k‡Ki ¸Y GK Ni evg †_‡K †jLv n‡q‡Q Ges wØZxq av‡c GKK I `k‡Ki N‡i k�b¨ ewm‡q kZ‡Ki ¸Y yB Ni evg †_‡K †jLv n‡q‡Q|

D`vniY 6| 235 †K 102 Øviv ¸Y Ki| mgvavb : 2 3 5

× 1 0 2

4 7 0 235 × 2

2 3 5 0 0 235 × 1 kZ 2 3 9 7 0

DËi : 23970|

j¶ Kwi : ¸Y‡Ki `k‡Ki N‡i k�b¨ (0) _vKvq `k‡Ki ¸Y †`Lv‡bv nqwb| ZvB wØZxq av‡c GKK I `k‡Ki N‡i k�b¨ ewm‡q kZ‡Ki ¸Y yB Ni evg †_‡K †jLv n‡q‡Q|

gš�e : wk¶v_©xiv cÖPwjZ wbq‡g ¸Y A¼ Ki‡e| Zv‡`i wbKU †Kvb e vL v PvIqv n‡e bv|

cÖv_wgK MwYZ 35

D`vniY 7| ¸Y Ki : 1 4 5 2 D`vniY 8| ¸Y Ki : 3 5 2 6

× 3 2 × 2 3 mgvavb : 1 4 5 2 mgvavb : 3 5 2 6

× 3 2 × 2 3 2 9 0 4 1 0 5 7 8 4 3 5 6 0 7 0 5 2 0 4 6 4 6 4 8 1 0 9 8

DËi : 46464| DËi : 81098| D`vniY 9| ¸Y Ki : 3 84 D`vniY 10| ¸Y Ki : 1000

× 100 × 100 mgvavb : 384 mgvavb : 1000

× 100 × 100 38400 100000 DËi : 38400| DËi : 100000| D`vniY 11| mnR c×wZ‡Z ¸Y Ki :

(K) 245 × 99 (L) 52 × 98

mgvavb : (K) 245 × 99 (L) 51 × 98

= 245 × (100 −1) = 51 × (100 − 2)

= (245 × 100) − (245 × 1) = (51 × 100) − (51 × 2)

= 24500 − 245 = 5100 − 102

= 24255 = 4998

DËi : 24255 | DËi : 4998|

36 cÖv_wgK MwYZ

¸‡Yi wewbgq wewa D`vniY 12| ¸Y Ki : (K) 7 3 (L) 2 5 6

× 2 5 6 × 7 3 mgvavb : (K) 7 3 (L) 2 5 6

× 2 5 6 × 7 3 4 3 8 7 6 8 3 6 5 0 1 7 9 2 0 1 4 6 0 0 1 8 6 8 8 8 6 8 8 DËi : 18688| DËi : 18688| j¶ Kwi : ¸Y¨ I ¸Y‡Ki ¯’vb wewbgq K‡i ¸Y Ki‡j ¸Ydj GKB _v‡K| D`vniY 13| kvwK‡ji Kv‡Q 324 UvKv Av‡Q| iwe‡bi Kv‡Q kvwK‡ji UvKvi 27 ¸Y Av‡Q| iwe‡bi KZ UvKv Av‡Q ? mgvavb : kvwK‡ji 324 UvKv Av‡Q| iwe‡bi Av‡Q (324 × 27) UvKv| GLb, 3 2 4 × 2 7 2 2 6 8 6 4 8 0 8 7 4 8 DËi : iwe‡bi 8748 UvKv Av‡Q| D`vniY 14| 1 †KwR Av‡gi `vg 65 UvKv| Giƒc 30 †KwR Av‡gi `vg KZ? mgvavb : 1 †KwR Av‡gi `vg 65 UvKv ∴ 30 †KwR Av‡gi `vg (65 × 30) UvKv GLb, 65 × 30 1950 DËi : 1950 UvKv|

38 cÖv_wgK MwYZ

PZz_© Aa¨vq

fvM D`vniY 1| 964 †K 4 Øviv fvM Ki| mgvavb : 4 ) 9 6 4 ( 241 8 16 16 4 4 0 DËi : fvMdj 241| e¨vL¨v : 964 = 9 kZ 6 `k 4 4 ) 9 kZ 6 `k 4 ( 2 kZ 4 `k 1

8 kZ 1 6 `k [KviY, 1 kZ 6 `k = 16 `k] 1 6 `k 4 4 0 j¶ Kwi : fvM A¼wU‡Z fvM‡kl k�b (0), A_©vr 964 msL vwU 4 Øviv wbt‡k‡l wefvR¨| D`vniY 2| 932 †K 6 Øviv fvM Ki| mgvavb : 6 ) 932 ( 155 e¨vL¨v : 932 = 9 kZ 3 `k 2 6 6 ) 9 kZ 3 `k 2 (1 kZ 5 `k 5 33 6 kZ 30 33 `k [KviY, 3 kZ 3 `k = 33 `k] 32 30 `k 30 32 [KviY, 3 `k 2 = 32] 2 30 DËi : fvMdj 155, fvM‡kl 2| 2 j¶ Kwi : A¼wU‡Z fvM‡kl 2| myZivs fvR¨, fvRK Øviv wbt‡k‡l wefvR¨ bq|

cÖv_wgK MwYZ 39

D`vniY 3| 2842 †K 14 Øviv fvM Ki| mgvavb : 14 ) 2842 ( 203 e¨vL¨v : 2842 = 2 nvRvi 8 kZ 4 `k 2 28 = 28 kZ 4 `k 2 42 14) 28 kZ 4 `k 2 ( 2 kZ 0 `k 3 42 28 kZ 0 4 `k 0 `k 42 4 0 DËi : fvMdj 203| D`vniY 4| 6257 †K 17 Øviv fvM Ki| mgvavb : 17 ) 6257 ( 368 51 115 102 137 136 1 DËi : fvMdj 368, fvM‡kl 1| D`vniY 5| 2409 †K 48 Øviv fvM Ki| mgvavb : 48 ) 2409 ( 50 240 9 0 9 DËi : fvMdj 50, fvM‡kl 9| j¶ Kwi : fv‡Mi wØZxq av‡c fv‡R¨i 9 bvgv‡bv n‡q‡Q| GLb 9, fvRK 48 †_‡K †QvU| ZvB fvMd‡j GK‡Ki N‡i GKwU k�b¨ (0) emv‡bv n‡q‡Q| fv‡R¨ Avi †Kvb A¼ bv _vKvq fvM Kiv †kl n‡q‡Q Ges fvM‡kl 9 i‡q‡Q| ∴ 2409, 48 Øviv wbt‡k‡l wefvR¨ bq|

40 cÖv_wgK MwYZ

¸Y I fv‡Mi m¤cK© j¶ Kwi : ¸Y¨ × ¸YK = ¸Ydj

wbt‡k‡l wefv‡R¨i †¶‡Î ;

fvR¨ ÷ fvRK = fvMdj myZivs ejv hvq, wbt‡k‡l fvM n‡”Q ¸‡Yi wecixZ cÖwµqv| Avevi j¶ Kwi : fvRK fvR¨ fvMdj

GLv‡b, 67 = 9 ×7+4 9 ) 67 ( 7 63 4 fvM‡kl D`vniY 6| †Kvb fvM A‡¼ fvRK 45, fvMdj 23 I fvM‡kl 29| fvR¨ KZ?

mgvavb : Avgiv Rvwb, fvR¨ = fvRK × fvMdj + fvM‡kl

GLv‡b, fvRK × fvMdj = 45 × 23 = 1035

∴ fvR¨ = 1035 + 29 = 1064

DËi : 1064|

¸Y¨ × ¸YK = ¸Ydj ¸Ydj ÷ ¸Y¨ = ¸YK ¸Ydj ÷ ¸YK = ¸Y¨

fvR¨ ÷ fvRK = fvMdj fvRK × fvMdj = fvR¨ fvR¨ ÷ fvMdj = fvRK

fvR¨ = fvRK ×fvMdj +fvM‡kl fvRK = (fvR¨−fvM‡kl) ÷ fvMdj

9 × 7 = 63

63 ÷ 9 = 7

63 ÷ 9 = 7 63 ÷ 7 =9

9 × 7 = 63 63 ÷ 7 =9

cÖv_wgK MwYZ 41

D`vniY 7| †Kvb fvM A‡¼ fvR¨ 1335, fvMdj 22 Ges fvM‡kl 15 n‡j, fvRK KZ? mgvavb : Avgiv Rvwb, fvRK = (fvR¨ - fvM‡kl) ÷ fvMdj ∴ fvRK = (1335 − 15) ÷ 22 = 1320 ÷ 22 = 60 DËi : 60| D`vniY 8| 25 †KwR Pv‡ji `vg 475 UvKv| 1 †KwR Pv‡ji `vg KZ? mgvavb : 25 ) 475 ( 19 25 225 225 0 DËi : 19 UvKv| D`vniY 9| 1 †KwR AvUvi `vg 17 UvKv| 408 UvKvq KZ †KwR AvUv cvIqv hv‡e? mgvavb : 17 ) 408 ( 24 34 68 68 0 DËi : 24 †KwR| D`vniY 10| Pvi A‡¼i ¶z ªZg msL¨v‡K 64 w`‡q fvM Ki| mgvavb : Pvi A‡¼i ¶z`ªZg msL¨v = 1000 64 ) 1000 ( 15 64 360 320 40 DËi : fvMdj 15, fvM‡kl 40|

42 cÖv_wgK MwYZ

D`vniY 11| cuvP A‡¼i e„nËg msL¨v‡K 64 w`‡q fvM Ki| mgvavb : cuvP A‡¼i e„nËg msL¨v = 99999 64 ) 99999 ( 1562 64 359 320 399 384 159 128 31 DËi : fvMdj 1562, fvM‡kl 31| 10 A_ev 100 w`‡q fvM D`vniY 12| 624 †K 10 w`‡q fvM Ki| mgvavb : 10 ) 624 ( 62 60 24 20 4 DËi : fvMdj 62, fvM‡kl 4| j¶ Kwi : fvR‡Ki †k‡l GKwU k�b¨ (0) Av‡Q, fv‡R¨i Wvbw`K †_‡K GK Ni ci Kgv emv‡j cvB 62, 4| Kgvi evg w`‡Ki msL¨vwUB fvMdj Ges Wvb w`‡Ki msL¨vwU fvM‡kl| 10 w`‡q fvM Kivi mgq Gfv‡e fvMdj I fvM‡kl wbY©q Kiv mnR| D`vniY 13| 4257 †K 100 w`‡q fvM Ki| mgvavb : fvRK 100 Gi Wv‡b yBwU k�b¨ Av‡Q| myZivs fv‡R¨i Wvbw`K †_‡K yB Ni c‡i Kgv ewm‡q cvIqv hvq 42,57

DËi : fvMdj 42, fvM‡kl 57|

cÖv_wgK MwYZ 43

D`vniY 14| 4700 †K 100 w`‡q fvM Ki| mgvavb : GLv‡b fvRK 100| fv‡R¨i Wvbw`K †_‡K yB A¼ ci Kgv ewm‡q cvIqv hvq 47,00| DËi : fvMdj 47, fvM‡kl 0| j¶ Kwi : 10 ev 100 w`‡q mn‡R fvM Ki‡Z n‡j fvR‡K hZ¸‡jv k �b¨ Av‡Q fv‡R¨i Wvbw`K †_‡K ZZ¸‡jv A‡¼i ci Kgv emv‡j Kgvi evgw`‡Ki msL¨vwU fvMdj Ges Wvbw`‡Ki msL¨vwUB n‡e fvM‡kl|

Abykxjbx- 4 1| fvMdj I fvM‡kl †ei Ki : (K) 525 ÷ 5 (L) 462 ÷ 6 (M) 637 ÷ 7 (N) 835 ÷ 9 (O) 4248 ÷ 4 (P) 5676 ÷ 6 (Q) 2436 ÷ 7 (R) 7530 ÷ 8 (S) 7665 ÷ 15 (T) 6336 ÷ 24 (U) 5488 ÷ 37 (V) 8001 ÷ 65 2| mnR c×wZ‡Z fvMdj I fvM‡kl †ei Ki : (K) 650 ÷ 10 (L) 740 ÷ 10 (M) 815 ÷ 10 (N) 5462÷ 100 (O) 6300÷100 (P) 7035 ÷ 100 3| (K) fvR¨ 3640, fvRK 70, fvMdj KZ? (L) fvR¨ 3414, fvRK 43, fvM‡kl 17, fvMdj KZ? (M) fvR¨ 7363, fvMdj 49, fvM‡kl 13, fvRK KZ? (N) fvR¨ 3579, fvRK 47, fvM‡kl KZ? (O) fvRK 72, fvMdj 123, fvM‡kl 6, fvR¨ KZ? 4| 950 UvKv w`‡q 25 WRb ej‡cb †Kbv hvq| 1 WRb ej‡c‡bi `vg KZ? 5| 85 †KwR Pv‡ji `vg 1445 UvKv| 1 †KwR Pv‡ji `vg KZ? 6| 1716 wgUvi j¤v wdZv‡K mgvb 78 UzKiv Kiv nj| cÖwZ UzKivi ˆ`N© KZ? 7| Pvi A‡¼i ¶z`ªZg msL¨v‡K 16 w`‡q fvM Ki| 8| Pvi A‡¼i e„nËg msL¨v‡K 78 w`‡q fvM Ki| 9| 2, 3, 5, 7 A¼ KqwU GKevi K‡i wb‡q MwVZ e„nËg msL¨vwU‡K yB A‡¼i

e„nËg msL¨v w`‡q fvM Ki| 10| 1, 4, 0, 8 A¼ KqwU GKevi K‡i wb‡q MwVZ ¶z`ªZg msL¨vwU‡K yB A‡¼i

e„nËg msL¨v w`‡q fvM Ki| 11| 7642 UvKv 52 R‡bi g‡a¨ mgvb fv‡M fvM K‡i w`‡j cÖ‡Z¨‡K KZ UvKv

K‡i cv‡e? KZ UvKv evwK _vK‡e? 12| 9702 UvKv 63 R‡bi g‡a¨ mgvb fv‡M fvM K‡i w`‡Z n‡e| cÖ‡Z¨‡K KZ

UvKv K‡i cv‡e?

44 cÖv_wgK MwYZ

cÂg Aa¨vq

mnR mgm¨v (†hvM, we‡qvM, ¸Y I fvM msµvš�)

D`vniY 1| 5wU †cbwm‡ji `vg 60 UvKv| Giƒc 9wU †cbwm‡ji `vg KZ ?

mgvavb : 5wU †cbwm‡ji `vg 60 UvKv

∴ 1wU †cbwm‡ji `vg (60÷5) UvKv

= 12 UvKv

∴ 9wU †cbwm‡ji `vg (12 × 9) UvKv

= 108 UvKv

DËi : 108 UvKv

D`vniY 2| GKwU Kjv evMv‡b 9wU mvwi‡Z 216wU KjvMvQ jvMv‡bv n‡q‡Q|

Giƒc 15 mvwi‡Z KZ¸‡jv KjvMvQ jvMv‡bv hv‡e ?

mgvavb : 9 mvwi‡Z KjvMv‡Qi msL¨v 216

∴ 1 mvwi‡Z KjvMv‡Qi msL¨v (216 ÷ 9)

= 24

∴ 15 mvwi‡Z KjvMv‡Qi msL¨v (24 × 15)

= 360

DËi : 360wU

D`vniY 3| GKwU Sywo‡Z 165wU Avg Av‡Q| Giƒc 12wU Sywo‡Z hZ¸‡jv Avg

Av‡Q Zv †_‡K ¯cb‡K 280wU Ges iweb‡K 210wU Avg †Ìqv nj| evwK Avg Ab¨ 5

R‡bi g‡a¨ mgvb fv‡M fvM K‡i †Ìqv nj| Zv‡ì cÖ‡Z¨‡K KqwU K‡i Avg †cj?

mgvavb : GKwU Szwo‡Z Avg Av‡Q 165wU

∴ 12 Szwo‡Z Avg Av‡Q (165 × 12)wU

= 1980wU

cÖv_wgK MwYZ 45

¯cb I iweb‡K †`Iqv nj (280 + 210)wU

= 490wU|

Aewkó Avg Av‡Q (1980 - 490) wU

= 1490wU

5 Rb cvq 1490 wU Avg

∴ 1 Rb cvq (1490 ÷ 5) wU = 298 wU Avg

DËi : 298 wU|

D`vniY 4| iƒcv I gwbi GK‡Î 875 UvKv Av‡Q| iƒcvi UvKv A‡c¶v gwbi 125

UvKv †ewk Av‡Q| Zv‡`i Kvi KZ UvKv Av‡Q ?

mgvavb : gwbi 125 UvKv †ewk Av‡Q| †gvU UvKv †_‡K 125 UvKv ev` w`‡j

yBR‡bi UvKvi cwigvY mgvb nq|

(875-125) UvKv = 750 UvKv|

∴ iƒcvi UvKvi cwigvY = (750 ÷ 2) = 375

∴ gwbi UvKvi cwigvY (375 + 125) = 500

DËi : iƒcv 375 UvKv, gwb 500 UvKv|

D`vniY 5| gvZv I Kb¨vi eq‡mi mgwó 55 eQi| gvZvi eqm Kb¨vi

eq‡mi 4 ¸Y| Kvi eqm KZ ?

mgvavb : Kb¨vi eqm = Kb¨vi eq‡mi 1 ¸Y

gvZvi eqm = Kb¨vi eq‡mi 4 ¸Y

∴ gvZv I Kb¨vi eq‡mi mgwó = Kb¨vi eq‡mi 5 ¸Y

∴ Kb¨vi eqm (55 ÷ 5) eQi

= 11 eQi

∴ gvZvi eqm = (11 × 4) eQi

= 44 ermi

DËi : gvZvi eqm 44 eQi, Kb¨vi eqm 11 eQi|

46 cÖv_wgK MwYZ

Abykxjbx- 5 1| GKwU Szwo‡Z 75 wU Kgjv Av‡Q| Giƒc 8 wU Szwo‡Z KqwU Kgjv Av‡Q ? 2| 169wU bvi‡Kj 14 Rb‡K mgvb fv‡e fvM K‡i †`Iqv nj| cÖ‡Z¨‡K KqwU

K‡i bvi‡Kj †cj ? KqwU bvi‡Kj Aewkó iBj? 3| 8 Rb K…lK avb‡¶‡Z KvR K‡i 1 w`‡b 560 UvKv gRyix cvq| 1 Rb

K…l‡Ki 1 w`‡bi gRyix KZ ? 4| 1 Rb kªwg‡Ki 1 w`‡bi gRyix 154 UvKv n‡j, 28 Rb kªwg‡Ki 1 w`‡bi

†gvU gRyix KZ ? 5| wgbvi eZ©gvb eqm 25 eQi| wibv Zvi †_‡K 6 eQ‡ii †QvU| wibvi eqm

†_‡K wgVzi eqm 11 eQi †ewk| wgbv, wibv I wgVzi eq‡mi mgwó KZ? 6| Kvjv evRvi †_‡K 145 UvKvi gvQI 75 UvKvi KvuPv mewR wKb‡jv| †m

†`vKvb`vi‡K 500 UvKvi GKwU †bvU w`j| †`vKvb`vi Zv‡K KZ UvKv †diZ †`‡e?

7| †mv‡nj evRvi †_‡K 18 UvKv `‡i 15 †KwR Pvj, 150 UvKv w`‡q GKwU Bwjk gvQ Ges 12 UvKv `‡i 3 †KwR Avjy wKbj| †m †gvU KZ UvKvi evRvi Kij?

8| GK e¨w³i gvwmK †eZb 7500 UvKv| wZwb cÖwZgv‡m 7275 UvKv LiP K‡ib| 10 gvm ci KZ UvKv Rgv n‡e?

9| 10 e¨vM evjyi IRb 8 e¨vM wm‡g‡›Ui IR‡bi mgvb| hw` GK e¨vM wm‡g‡›Ui IRb 50 †KwR nq, Zvn‡j GK e¨vM evjyi IRb KZ ?

10| †Kvb cÖv_wgK we`¨vj‡q QvÎQvÎxi msL¨v 575| eQ‡ii ïiy‡Z 216 Rb fwZ© nj Ges 35 Rb we`¨vjq †Q‡o P‡j †Mj| eZ©gv‡b H we`¨vj‡qi QvÎQvÎxi msL¨v KZ?

11| yBwU msL¨vi ¸Ydj 912| GKwU msL¨vi 5 ¸Y 95 n‡j, Aci msL¨vwU KZ?

12| yBwU msL¨vi fvMdj 25| GKwU msL¨vi 4 ¸Y 96 n‡j, Aci msL¨vwU KZ?

cÖv_wgK MwYZ 47

13| wcZv I cy‡Îi eq‡mi mgwó 68 eQi| wcZvi eqm cy‡Îi eq‡mi 3 ¸Y|

eZ©gv‡b Kvi eqm KZ? 5 eQi ci Zv‡ì cÖ‡Z¨‡Ki eqm KZ n‡e?

14| wPby A‡c¶v wgbyi 75 UvKv †ewk Ges ivey A‡c¶v 35 UvKv Kg Av‡Q|

wgbyi 215 UvKv _vK‡j, wPby I iveyi cÖ‡Z¨‡Ki KZ UvKv Av‡Q? wZbR‡bi

†gvU KZ UvKv Av‡Q?

15| fvRK fvM‡k‡li 4 ¸Y Ges fvMdj fvR‡Ki 6 ¸Y| fvM‡kl 3 n‡j,

fvR¨ KZ?

16| 4wU gyiwM I 3wU nvu‡mi `vg GK‡Î 639 UvKv| GKwU nuv‡mi `vg 85

UvKv| GKwU gyiwMi `vg KZ?

17| KZK¸‡jv †cqviv 45 Rb †Q‡j‡g‡qi g‡a¨ mgvb fv‡M fvM K‡i †Ìqv

nj| cÖ‡Z¨‡K 14wU K‡i †cqviv †cj Ges 25wU †cqviv Aewkó iBj|

Zvn‡j †gvU KZ¸‡jv †cqviv wQj?

18| 310wU wjPz 50 Rb †Q‡j‡g‡qi g‡a¨ fvM K‡i †Ìqv nj| cÖ‡Z¨K †Q‡j

5wU Ges cÖ‡Z¨K †g‡q cÖ‡Z¨K †Q‡ji †P‡q 2wU †ewk †cj| †g‡q‡ì I

†Q‡j‡ì msL¨v KZ?

19| Ave`yi iwng mv‡ne 25 †KwR Pvj, 126 UvKvi Wvj, 215 UvKvi †Zj Ges

105 UvKvi Ab¨vb¨ wRwbm wKb‡jb| cÖwZ †KwR Pv‡ji `vg 18 UvKv| wZwb

†`vKvb`vi‡K yBwU 500 UvKvi †bvU w`‡jb| †`vKvb`vi Zuv‡K KZ UvKv

†diZ †`‡e?

20| gwb I wjwci GK‡Î 936 UvKv Av‡Q| gwbi wjwc A‡c¶v 208 UvKv Kg

Av‡Q| cÖ‡Z¨‡Ki KZ UvKv Av‡Q?

21| †Kvb& msL¨vi 14 ¸‡Yi mv‡_ 75 †hvM Ki‡j †hvMdj 859 n‡e?

22| †Kvb mwgwZ‡Z m`m¨ msL¨v 105 Rb| 15 Rb bZzb m`m¨ †hvM w`j|

eb‡fvR‡bi Rb¨ cÖ‡Z¨K m`m¨ 75 UvKv K‡i Puv`v †`q| †gvU KZ UvKv

Puv`v cvIqv †Mj?

48 cÖv_wgK MwYZ

lô Aa¨vq

mijxKiY

D`vniY 1| mij Ki : 154 − 64 + 27 + 59 − 76 − 84

mgvavb : 154 − 64 + 27 + 59 − 76 − 84

= 90 + 27 + 59 − 76 − 84

= 117 + 59 − 76 − 84

= 176 − 76 − 84

= 100 − 84 = 16 DËi : 16| j¶ Kwi : (K) ivwkwU‡Z ïay †hvM I we‡qv‡Mi KvR i‡q‡Q| (L) evg w`K †_‡K av‡c av‡c †hvM I we‡qvM K‡i mij Kiv n‡q‡Q|

154 − 64 = 90 ; 90 + 27 = 117 ; 117 + 59 = 176;

176 − 76 = 100 ; 100 − 84 = 16

weKí mgvavb : 154 − 64 + 27 + 59 − 76 − 84

= (154 + 27 + 59) − (64 + 76 + 84)

= 240 − 224 = 16 j¶ Kwi : (K) †hvM‡evaK msL¨v¸‡jv‡K GK‡Î †hvM Kiv n‡q‡Q : 154 + 27 + 59 = 240 (L) we‡qvM‡evaK msL¨v¸‡jv‡K GK‡Î †hvM Kiv n‡q‡Q ; 64 + 76 + 84 = 224 (M) cÖ_g †hvMdj †_‡K wØZxq †hvMdj we‡qvM Kiv n‡q‡Q; 240 − 224 = 16 (N) GKB wPýhy³ msL¨v‡K GKwÎZ Kivi Rb¨ ( ) eÜbx e¨envi

Kiv n‡q‡Q| weKí c×wZ‡Z mgvavb Kiv myweavRbK|

cÖv_wgK MwYZ 49

D`vniY 2| mij Ki : 225 − 45 − 60 + 35 + 68 − 123

mgvavb : 225 − 45 − 60 + 35 + 68 − 123

= (225 + 35 + 68) − (45 + 60 + 123)

= 328 − 228

= 100

DËi : 100|

D`vniY 3| mij Ki : 27 − 28 + 110 − 120 − 54 + 23 + 105

mgvavb : 27 − 28 + 110 − 120 − 54 + 23 + 105

= (27 + 110 + 23 + 105) − (28 + 120 + 54)

= 265 − 202

= 63

DËi : 63|

D`vniY 4| mij Ki : 2 × 3 × 4 × 5 × 6 × 7

mgvavb : 2 × 3 × 4 × 5 × 6 × 7

= 6 × 4 × 5 × 6 × 7

= 24 × 5 × 6 × 7

= 120 × 6 × 7

= 720 × 7

= 5040

DËi : 5040|

j¶ Kwi : (K) A¼wU‡Z ïay ¸‡Yi KvR i‡q‡Q|

(L) evgw`K †_‡K cici ¸Y K‡i mij Kiv n‡q‡Q|

50 cÖv_wgK MwYZ

D`vniY 5| mij Ki : 45 × 4 − 28 × 3 − 15 × 5 + 32 × 2

mgvavb : 45 × 4 − 28 × 3 − 15 × 5 + 32 × 2

= 180 − 84 − 75 + 64

= (180 + 64) − (84 + 75)

= 244 − 159 = 85 DËi : 85| j¶ Kwi : (K) A¼wU‡Z †hvM, we‡qvM I ¸‡Yi KvR i‡q‡Q| (L) mgvavb Kivi mgq cÖ_‡g ¸‡Yi KvR Kiv n‡q‡Q| c‡i †hvM‡evaK

msL¨v¸‡jv‡K GK‡Î †hvM Kiv n‡q‡Q Ges we‡qvM‡evaK msL¨v¸‡jv‡K GK‡Î †hvM Kiv n‡q‡Q|

(M) cÖ_g †hvMdj †_‡K wØZxq †hvMdj we‡qvM Kiv n‡q‡Q| mij A‡¼ †hvM, we‡qvM I ¸‡Yi KvR _vK‡j, ¸‡Yi KvR Av‡M Ki‡Z n‡e|

D`vniY 6| mij Ki : 22 × 22 − 3 × 25 × 2 + 4 × 16 − 160 − 5 × 19

mgvavb : 22 × 22 − 3 × 25 × 2 + 4 × 16 − 160 − 5 × 19

= 484 − 75 × 2 + 64 − 160 − 95

= 484 − 150 + 64 − 160 − 95

= (484 + 64) − (150 + 160 + 95)

= 548 − 405

= 143

DËi : 143|

cÖv_wgK MwYZ 51

Abykxjbx- 6

mij Ki :

1| 75 + 36 − 32 − 39 + 25

2| 772 − 528 + 188 − 222 − 85

3| 85 − 215 + 529 + 50 − 249

4| 455 + 25 − 36 − 157 + 105 − 111 − 82

5| 502 + 112 − 325 − 40 − 101 + 52 − 98

6| 38 − 55 + 278 + 29 − 115 − 8 − 147

7| 2 × 3 × 6 × 8 × 9

8| 4 × 8 × 3 × 5 × 7

9| 6 × 2 × 8 × 7 × 9

10| 5 × 12 − 6 × 9 + 18 × 11 − 37 × 2 − 40 × 3

11| 4 × 9 × 21 + 7 − 41 × 3 − 32 × 12 − 16 × 16

12| 3 × 5 × 7 − 43 × 4 + 6 × 11 × 2 − 13 × 5 + 3 × 2 + 4

13| 7 × 9 × 8 − 3 × 6 × 9 + 42 − 25 × 8 + 23 × 4 − 5 × 5 × 11 + 3

14| 103 × 10 + 7 − 25 × 6 × 5 − 5 × 5 × 9 + 3 × 2 − 13 × 5

15| 25 − 4 × 2 + 11 × 5 − 6 × 7 × 3 + 2 × 4 × 8

52 cÖv_wgK MwYZ

mßg Aa¨vq

¸YbxqK I ¸wYZK ¸YbxqK

1 Rb wk¶v_©xi Rb¨ 12 wU wjPz|

cÖ‡Z¨‡K cvq 12 wU K‡i wjPz;

12 × 1 = 12|


cÖ‡Z¨‡K cvq 6 wU K‡i wjPz ;

6 × 2 = 12|



4 × 3 = 12



3 × 4 = 12|

cÖv_wgK MwYZ 53

6 Rb wk¶v_©xi Rb¨ 12 wU wjPz| cÖ‡Z¨‡K cvq 2 wU K‡i wjPz ;

2 × 6 = 12|

12 Rb wk¶v_©xi Rb¨ 12 wU wjPz| cÖ‡Z¨‡K cvq 1 wU K‡i wjPz ;

1 × 12 = 12|

j¶ Kwi : • 1, 2, 3, 4, 6 ev 12 Rb wk¶v_©xi g‡a¨ 12 wU wjPz cÖ‡Z¨K‡K mgvb msL¨K

fv‡M fvM Kiv hvq|

• 12 msL¨vwU 1, 2, 3, 4, 6 I 12 Øviv wefvR¨|

• 1, 2, 3, 4, 6, 12 Gi cÖwZwU 12 Gi ¸YbxqK ev Drcv`K|

• 5 Rb wk¶v_©xi g‡a¨ 12wU wjPz cÖ‡Z¨K‡K mgvb fv‡M fvM K‡i †`Iqv hvq bv|

• 5 msL¨vwU 12 Gi ¸YbxqK ev Drcv`K bq|

†Kvb msL¨v †h †h msL¨v Øviv wefvR¨, †m¸‡jvB H msL¨vi ¸YbxqK ev

Drcv`K| ¸YbxqK‡K Drcv`KI ejv n‡q _v‡K|

54 cÖv_wgK MwYZ

4 msL¨vwU 1, 2 I 4 Øviv wefvR¨|

∴ Giv cÖ‡Z¨‡K 4 Gi ¸YbxqK| 4 msL¨vwU 3 Øviv wefvR¨ bq| ZvB 3, 4 Gi ¸YbxqK ev Drcv`K bq|

∴ 4 Gi mKj ¸YbxqK ev Drcv`K : 1, 2, 4|8 msL¨vwU 1, 2, 4 I 8 Øviv wefvR¨|

∴ Giv cÖ‡Z¨‡K 8 Gi ¸YbxqK ev Drcv`K| 8 msL¨vwU 3, 5, 6 ev 7 Øviv wefvR¨ bq| ZvB, G msL¨v¸‡jvi †KvbwU 8 Gi ¸YbxqK ev Drcv`K bq|

∴ 8 Gi mKj ¸YbxqK ev Drcv K : 1, 2, 4, 8| 12 msL¨vwU 1, 2, 3, 4, 6 I 12 Øviv wefvR¨|

∴ Giv cÖ‡Z¨‡K 12 Gi ¸YbxqK ev Drcv`K| 12 msL¨vwU 5, 7, 8, 9, 10 ev 11 Øviv wefvR¨ bq| ZvB G msL¨v¸‡jvi †KvbwU 12 Gi ¸YbxqK ev Drcv`K bq|

∴ 12 Gi mKj ¸YbxqK ev Drcv`K : 1, 2, 3, 4, 6, 12| 14 msL¨vwU 1, 2, 7 I 14 Øviv wefvR¨; wKš�y 14 Gi †P‡q †QvU Ab¨ †Kvb msL¨v Øviv wefvR¨ bq|

∴ 14 Gi mKj YbxqK ev Drcv K :1, 2, 7, 14| 17 msL¨vwU 1 I 17 Øviv wefvR¨; wKš�y 17 Gi †P‡q †QvU Ab¨ †Kvb msL¨v ØvivB 17 wefvR¨ bq|

∴ 17 Gi ¸YbxqK ev Drcv`K : 1 I 17|

j¶ Kwi : • cÖ‡Z¨K msL¨v wb‡RB Zvi GKwU ¸YbxqK| • 1 †h‡Kvb msL¨viB GKwU ¸YbxqK|

4 = 1 × 4 = 2 × 2

8 = 1 × 8 = 2 × 4

12 = 1 × 12 = 2 × 6

= 3 × 4

14 = 1 × 14 = 2 × 7

17 = 1 × 17

cÖv_wgK MwYZ 55

• 1 Gi GKgvÎ ¸YbxqK 1 wb‡RB|

• †Kvb msL¨vi †h‡Kvb ¸YbxqK ev Drcv`K msL¨vwUi mgvb ev Zvi †P‡q †QvU n‡e| • 1 Gi †P‡q eo †Kvb msL¨vi msL¨vwU ev‡` Ab¨ †h‡Kvb ¸YbxqK msL¨vwUi

A‡a©‡Ki mgvb ev Zvi †P‡q †QvU n‡e| • 2, 3, 5, 7, 11, 13, 17 msL¨v¸‡jvi cÖ‡Z¨‡Ki gvÎ yBwU ¸YbxqK i‡q‡Q,

Zvnj 1 Ges msL¨vwU wb‡R| • GKwU msL¨vi mKj ¸YbxqK ev Drcv`K wbY©q Ki‡Z n‡j, msL¨vwU‡K hZ

cÖKv‡i m¤¢e yBwU msL¨vi ¸Ydj iƒ‡c wjL‡Z n‡e| • †h‡Kvb msL¨vi ¶z ªZg ¸YbxqK 1 Ges e„nËg ¸YbxqK msL¨vwU wb‡RB| ¸YbxqK wbY©q D`vniY 1| 32 Gi mKj ¸YbxqK wbY©q Ki| mgvavb : 32 = 1 × 32 = 2 × 16 = 4 × 8 DËi : 32 Gi mKj ¸YbxqK : 1, 2, 4, 8, 16, 32| D`vniY 2| 49 Gi mKj ¸YbxqK wbY©q Ki| mgvavb : 49 = 1 × 49 = 7 × 7 DËi : 49 Gi mKj ¸YbxqK : 1, 7, 49| D`vniY 3| 60 Gi mKj ¸YbxqK wbY©q Ki| mgvavb : 60 = 1 × 60 = 2 × 30 = 3 × 20 = 4 × 15 = 5 × 12 = 6 × 10 DËi : 60 Gi mKj ¸YbxqK : 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60|

56 cÖv_wgK MwYZ

D`vniY 4| 175 Gi mKj ¸YbxqK wbY©q Ki| mgvavb : 175 = 1 × 175 = 5 × 35 = 7 × 25 DËi : 175 Gi mKj ¸YbxqK : 1, 5, 7, 25, 35, 175| D`vniY 5| 200 Gi mKj ¸YbxqK wbY©q Ki| mgvavb : 200 = 1 × 200 = 2 × 100 = 4 × 50 = 5 × 40 = 8 × 25 = 10 × 20 DËi : 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 200| †gŠwjK msL¨v, K…wÎg msL¨v 1 Gi †P‡q eo †h me msL¨vi 1 I msL¨vwU Qvov Ab¨ †Kvb ¸YbxqK †bB, †m¸‡jv †gŠwjK msL¨v| 1 Gi †P‡q eo †h me msL¨vi 1 I msL¨vwU Qvov Aš�Z GKwU ¸YbxqK Av‡Q, †m¸‡jv K…wÎg msL¨v|

msL¨v 1 I msL¨vwU Qvov ¸YbxqK Av‡Q wKbv †kªYx wefvM 2 3 4 5 6 7 8 9 10 11

bvB bvB Av‡Q, †hgb 2 bvB Av‡Q, †hgb 2 bvB Av‡Q, †hgb 2 Av‡Q, †hgb 3 Av‡Q, †hgb 2 bvB

†gŠwjK msL¨v †gŠwjK msL¨v K…wÎg msL¨v †gŠwjK msL¨v K…wÎg msL¨v †gŠwjK msL¨v K…wÎg msL¨v K…wÎg msL¨v K…wÎg msL¨v †gŠwjK msL¨v

cÖv_wgK MwYZ 57

msL¨v 1 I msL¨vwU Qvov ¸YbxqK Av‡Q wKbv †kªYx wefvM 12 13 14 15 16 17 18 19 20

Av‡Q, †hgb 2 bvB Av‡Q, †hgb 2 Av‡Q, †hgb 3 Av‡Q, †hgb 2 bvB Av‡Q, †hgb 2 bvB Av‡Q, †hgb 2

K…wÎg msL¨v †gŠwjK msL¨v K…wÎg msL¨v K…wÎg msL¨v K…wÎg msL¨v †gŠwjK msL¨v K…wÎg msL¨v †gŠwjK msL¨v K…wÎg msL¨v

j¶ Kwi : • Ab �aŸ© 20 Gi †gŠwjK msL¨v¸‡jv ; 2, 3, 5, 7, 11, 13, 17, 19|

• 2 ¶z ªZg †gŠwjK msL¨v|

• 2 ev‡` Ab¨vb¨ †gŠwjK msL¨v we‡Rvo msL¨v| wKš�y mKj we‡Rvo msL¨v †gŠwjK bq|

• K…wÎg msL¨vi ¸Ybxq‡Ki msL¨v Kgc‡¶ wZb|

• 1 †gŠwjK msL¨v bq, K…wÎg msL¨vI bq|

wefvR¨Zv cix¶v (K) 2 Øviv wefvR¨ 2 × 0 = 0, 2 × 1 = 2, 2 × 2 = 4, 2 × 3 = 6, 2 × 4 = 8,

2 × 5 = 10, 2 × 6 = 12, 2 × 7 = 14, 2 × 8 = 16, 2 × 9 = 18| AZGe, †h‡Kvb msL¨v‡K 2 Øviv ¸Y Ki‡j ¸Yd‡ji GKK ¯’vbxq A¼wU n‡e 0, 2, 4, 6 ev 8| myZivs †Kvb msL¨vi GKK ¯’vbxq A¼ 0, 2, 4, 6 ev 8 n‡j, msL vwU 2 Øviv wefvR¨ n‡e| Giƒc msL¨v‡K Avgiv †Rvo msL¨v e‡j Rvwb| (L) 3 Øviv wefvR¨ j¶ Kwi, 10 = 3 × 3 + 1

100 = 3 × 33 + 1

1000 = 3 × 333 + 1|

58 cÖv_wgK MwYZ

myZivs 261 = 200 + 60 + 1

= 2 × 100 + 6 × 10 + 1

= (3 × 66 + 2) + (3 × 18 + 6) + 1

= (3 × 66 + 3 × 18) + (2 + 6 + 1) Wvbc‡¶i eÜbxfy³ cÖ_g ivwk 3 Øviv wefvR¨ (fvMdj : 66 + 18)| myZivs 261 msL¨vwU 3 Øviv wefvR¨ n‡Z n‡j Wvbcv‡k eÜbxfy³ Aci ivwk 2+6+1, hv wKbv 261 msL¨vwUi A¼¸‡jvi mgwó, ZvI 3 Øviv wefvR¨ n‡Z n‡e| G‡¶‡Î 2+6+1= 9, 3 Øviv wefvR¨| myZivs 261 msL¨vwU 3 Øviv wefvR¨|

†Kvb msL¨vi A¼¸‡jvi mgwó 3 Øviv wefvR¨ n‡j, msL¨vwU 3 Øviv wefvR¨ n‡e|

418 msL¨vwU 3 Øviv AwefvR¨, †Kbbv 4 + 1 + 8 = 13, 3 Øviv AwefvR¨| 672 msL¨vwU 3 Øviv wefvR¨, †Kbbv 6 + 7 + 2 = 15, 3 Øviv wefvR¨| (M) 5 Øviv wefvR¨

0 × 5 = 0, 1 × 5 = 5, 2 × 5 = 10, 3 × 5 = 15, 4 × 5 = 20,

5 × 5 = 25, 6 × 5 = 30, 7 × 5 = 35, 8 × 5 = 40, 9 × 5 = 45|

†h‡Kvb msL¨v‡K 5 w`‡q ¸Y Ki‡j ¸Yd‡ji GKK ¯’vbxq A¼wU n‡e 0 ev 5| myZivs †Kvb msL¨vi GKK ¯’vbxq A¼ 0 ev 5 n‡j, msL¨vwU 5 Øviv wefvR¨ n‡e|

D`vniY 6| 87 msL¨vwU †gŠwjK bv K…wÎg?

mgvavb : 8 + 7 = 15, 3 Øviv wefvR¨|

∴ 87 K…wÎg msL¨v| DËi : 87 msL¨vwU K…wÎg| D`vniY 7| 745 msL¨vwU †gŠwjK bv K…wÎg?

mgvavb : msL¨vwUi GKK ¯’vbxq A¼ 5|

∴ msL¨vwU 5 Øviv wefvR¨| DËi : 745 msL¨vwU K…wÎg|

cÖv_wgK MwYZ 59

D`vniY 8| 43 msL¨vwU †gŠwjK bv K…wÎg? mgvavb : 43 we‡Rvo msL¨v weavq 2 Øviv wefvR¨ bq| 4 + 3 = 7, 3 Øviv AwefvR¨| myZivs 43 msL¨vwU 3 Øviv wefvR¨ bq| 43 Gi GKK ¯’vbxq A¼ 0 ev 5 bq| myZivs 43 msL¨vwU 5 Øviv wefvR¨ bq|

5 Gi c‡ii †gŠwjK msL¨v 7| wKš�y 7 × 7 = 49, 43 Gi †P‡q eo| myZivs 7 ev Gi †P‡q eo †Kvb †gŠwjK msL¨v w`‡q 43 Gi wefvR¨Zv cix¶v Kivi cÖ‡qvRb †bB|

∴ 43 †gŠwjK msL¨v| DËi : 43 msL¨vwU †gŠwjK| D`vniY 9| 89 msL¨vwU †gŠwjK bv K…wÎg? mgvavb : 89 we‡Rvo msL¨v weavq 2 Øviv wefvR¨ bq| 8 + 9 = 17, 3 Øviv AwefvR¨| myZivs 89 msL¨vwU 3 Øviv wefvR¨ bq| 89 Gi GKK ¯’vbxq A¼ 0 ev 5 bq| myZivs 89 msL¨vwU 5 Øviv AwefvR¨| mivmwi fvM K‡i †`wL †h,

89 msL¨vwU 7 Øviv wefvR¨ bq|

7 Gi c‡ii †gŠwjK msL¨v 11|

wKš�y 11 × 11 = 121, 89 Gi †P‡q eo| myZivs 7 Gi †P‡q eo †Kvb †gŠwjK msL¨v w`‡q 89 Gi wefvR¨Zv cix¶v Kivi cÖ‡qvRb †bB|

∴ 89 †gŠwjK msL¨v| DËi : 89 msL¨vwU †gŠwjK| j¶ Kwi : • †Kvb msL¨v †gŠwjK ev K…wÎg wKbv wbY©‡qi Rb¨ Ggb mKj †gŠwjK msL¨v w`‡q

msL¨vwUi wefvR¨Zv hvPvB KivB h‡_ó, hv‡`i eM© msL¨vwUi †P‡q eo bq|

• 20 Gi Ab �aŸ© †gŠwjK msL¨v¸‡jv Av‡MB kbv³ Kiv n‡q‡Q| Ab �aŸ© 100 Ges 20 Gi †P‡q eo †gŠwjK msL¨v¸‡jv n‡”Q :

23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97|

7) 89 (12 7 19 14 5

60 cÖv_wgK MwYZ

†gŠwjK Drcv`K 12 Gi Drcv`K 1, 2, 3, 4, 6 I 12|

G‡`i g‡a¨ 2 I 3 †gŠwjK Drcv`K| 12 msL¨vwU †h‡Kvb fv‡e `yBwU Drcv`‡Ki

¸Ydjiƒ‡c cÖKvk K‡i cÖwZev‡iB 12 †K 2 I 3, GB `yBwU †gŠwjK Drcv`‡Ki

gva¨‡g cÖKvk Kiv nq| †hgb ;

12 = 2 × 6 = 2 × 2 × 3

12 = 3 × 4 = 3 × 2 × 2 = 2 × 2 × 3

2 × 2 × 3 †K 12 Gi †gŠwjK Drcv`‡K we‡k −lY ejv nq|

36 Gi †gŠwjK Drcv`‡K we‡k−lY Kgc‡¶ Pvi fv‡e Kiv hvq :

36 = 2 × 18 = 2 × 2 × 9 = 2 × 2 × 3 × 3

36 = 3 × 12 = 3 × 2 × 6 = 3 × 2 × 2 × 3 = 2 × 2 × 3 × 3

36 = 4 × 9 = 2 × 2 × 3 × 3

36 = 6 × 6 = 2 × 3 × 2 × 3 = 2 × 2 × 3 × 3

†hfv‡eB Gi ïiy †nvK bv †Kb, 36 Gi †gŠwjK Drcv`‡K we‡k −lY wn‡m‡e GKwUB

DËi cvIqv hv‡e, Zv n‡”Q 2 × 2 × 3 × 3|

12 = 2 × 6 12 = 3 × 4

2 × 3 2 × 2

36 = 2 × 18 36 = 3 × 12 36 = 4 × 9 36 = 6 × 6

2 × 9 2 × 6 2 × 2 3 × 3 2 × 3 2 × 3

3 × 3 2 × 3

cÖv_wgK MwYZ 61

†gŠwjK Drcv`‡K we‡k−lY Ki (GKwU K‡i †`Lv‡bv nj) : msL¨v we‡k−lY †gŠwjK Drcv`‡K we‡k−lY 175 5 × 35

5 × 7

5 × 5 × 7

120 135 273 343 472

mvaviY ¸YbxqK, Mwiô mvaviY ¸YbxqK 12 Gi mKj ¸YbxqK : 1, 2, 3, 4, 6, 12| 18 Gi mKj ¸YbxqK : 1, 2, 3, 6, 9, 18|

∴ 1, 2, 3, 6 GB msL¨v¸‡jv 12 Ges 18 Df‡qiB ¸YbxqK; Giv msL¨v yBwUi mvaviY ¸YbxqK| GB mvaviY ¸YbxqK¸‡jvi g‡a¨ me‡P‡q eo nj 6; 6 n‡”Q 12 I 18 Gi Mwiô mvaviY ¸YbxqK ev, M.mv.¸.|

yBwU msL¨vi mvaviY ¸YbxqK¸‡jvi g‡a¨ me‡P‡q eowU Mwiô mvaviY ¸YbxqK (ms‡¶‡c M.mv.¸.)|

12 Gi ¸YbxqK 18 Gi ¸YbxqK

12 36

9 18

4 12

62 cÖv_wgK MwYZ

D`vniY 10| 24 I 36 Gi M.mv.¸. wbY©q Ki| mgvavb : 24 Gi mKj ¸YbxqK : 1, 2, 3, 4, 6, 8, 12, 24| 36 Gi mKj ¸YbxqK : 1, 2, 3, 4, 6, 9, 12, 18, 36| ∴ 24 I 36 Gi mvaviY ¸YbxqK 1, 2, 3, 4, 6, 12| ∴ 24 I 36 Gi Mwiô mvaviY ¸YbxqK = 12| D`vniY 11| 15 I 22 Gi M.mv.¸. wbY©q Ki| mgvavb : 15 Gi mKj ¸YbxqK 1, 3, 5, 15| 22 Gi mKj ¸YbxqK 1, 2, 11, 22| ∴ 15 I 22 Gi GKgvÎ mvaviY ¸YbxqK 1|

∴ 15 I 22 Gi Mwiô mvaviY ¸YbxqK = 1|

yBwU msL¨vi M.mv.¸. †¶Î we‡k‡l 1 n‡Z cv‡i|

D`vniY 12| 16 I 64 M.mv.¸. wbY©q Ki| mgvavb : mvaviYZ GLv‡b eo msL¨vwU †QvU msL¨vwU Øviv wefvR¨| myZivs †QvU msL¨vwUB wb‡Y©q M.mv.¸.| DËi : 16 I 64 Gi M.mv.¸. = 16| yBwU msL¨vi M.mv.¸. †¶Î we‡k‡l †QvU msL¨vwUI n‡Z cv‡i|

†gŠwjK Drcv`‡K we‡k−l‡Yi gva¨‡g M.mv.¸. wbY©q :

D`vniY 13| 225 I 455 Gi M.mv.¸. wbY©q Ki|

mgvavb : 225 = 5 × 45

= 5 × 5 × 9

= 5 × 5 × 3 × 3

= 3 × 3 × 5 × 5

455 = 5 × 91

= 5 × 7 × 13 225 I 455 Gi †gŠwjK ¸YbxqK¸‡jvi g‡a¨ (GKgvÎ) mvaviY ¸YbxqK 5|

∴ wb‡Y©q M.mv.¸. = 5

5) 225 (45 20 25 25 0

5) 455(91 45 5 5 0

16) 64 (4 64 0

cÖv_wgK MwYZ 63

D`vniY 14| 12, 30 I 36 Gi M.mv.¸. wbY©q Ki| mgvavb : 36, 12 Øviv wefvR¨ weavq 12 I 30 Gi M.mv.¸.B wb‡Y©q M.mv.¸.|

12 = 2 × 6 = 2 × 2 × 3

30 = 2 × 15 = 2 × 3 × 5

12 I 30 Gi †gŠwjK we‡k−lY yBwUi mvaviY Ask 2 × 3|

∴ wb‡Y©q M.mv.¸. = 2 × 3 = 6| D`vniY 15| 45, 60 I 75 Gi M.mv.¸. wbY©q Ki|

mgvavb : 45 = 3 × 15 = 3 × 3 × 5

60 = 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5

75 = 3 × 25 = 3 × 5 × 5

∴ 45, 60 I 75 Gi †gŠwjK Drcv`‡K we‡k−lY wZbwUi mvaviY Ask 3 × 5|

∴ wb‡Y©q M.mv.¸. = 3 × 5 = 15|

¸wYZK GKwU msL¨v †Kvb msL¨v Øviv wefvR¨ n‡j, cÖ_g msL¨vwU‡K Ab¨ msL¨vwUi ¸wYZK ejv nq| †Kbbv Ab¨ msL¨vwU‡K fvMdj Øviv ¸Y Ki‡j cÖ_g msL¨vwU cvIqv hvq| †hgb;

21 = 3 × 7 weavq 21 msL¨vwU 3 Gi MywYZK, Avevi 7 GiI ¸wYZK| AZGe, †h‡Kvb msL¨v Zvi cÖwZwU ¸Ybxq‡Ki ¸wYZK| GKwU msL¨v‡K †h‡Kvb msL¨v Øviv ¸Y K‡i Zvi GKwU ¸wYZK cvIqv hvq| †hgb;

7 × 1 = 7, 7 × 2 = 14, 7 × 3 = 21, 7 × 4 = 28, 7 × 5 = 35 BZ¨vw`| cÖ‡Z¨K msL¨v 7 Gi ¸wYZK| j¶ Kwi • GKwU msL¨v †Kvb msL¨v Øviv wefvR¨ n‡j, cÖ_g msL¨vwU fvRK I fvMdj

cÖ‡Z¨‡Ki ¸wYZK| • cÖ‡Z¨K msL¨v 1 Ges Zvi wb‡RiI ¸wYZK| • cÖ‡Z¨K msL¨vi AmsL¨ ¸wYZK i‡q‡Q| • cÖ‡Z¨K msL¨vi ¶z ªZg ¸wYZK msL¨vwU wb‡RB|

64 cÖv_wgK MwYZ

D`vniY 16| 8 Gi `kwU ¸wYZK wbY©q Ki|

mgvavb : 8 × 1 = 8 8 × 2 = 16

8 × 3 = 24 8 × 4 = 32

8 × 5 = 40 8 × 6 = 48

8 × 7 = 56 8 × 8 = 64

8 × 9 = 72 8 × 10 = 80

∴ 8 Gi (cÖ_g) `kwU ¸wYZK : 8, 16, 24, 32, 40, 48, 56, 64, 72, 80|

mvaviY ¸wYZK, jwNô mvaviY ¸wYZK 6 Gi ¸wYZK : 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, BZ¨vw`| 8 Gi ¸wYZK : 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, BZ¨vw`| j¶ Kwi : 24, 48 BZ¨vw` 6 I 8 Df‡qi ¸wYZK| myZivs 24 I 48 cÖ‡Z¨‡K 6 I 8 Gi mvaviY ¸wYZK| GB mvaviY ¸wYZK¸‡jvi g‡a¨ 24 ¶z ªZg ev jwNô|

∴ 6 I 8 Gi jwNô mvaviY ¸wYZK = 24|

jwNô mvaviY ¸wYZK‡K ms‡¶‡c j.mv.¸. ejv nq|

D`vniY 17| 10 I 30 Gi j.mv.¸. wbY©q Ki|

mgvavb : 30, 10 Gi ¸wYZK, †Kbbv 30 = 3 × 10 Avevi, 30 Gi ¶z`ªZg ¸wYZK 30 wb‡RB|

∴ 10 I 30 Gi j.mv.¸. = 30|

yBwU msL¨vi j.mv.¸. †¶Î we‡k‡l eo msL¨vwUI n‡Z cv‡i|

†gŠwjK Drcv`‡K we‡k−l‡Yi gva¨‡g j.mv.¸. wbY©q : D`vniY 18| 18 I 24 Gi j.mv.¸. wbY©q Ki|

mgvavb : 18 = 2 × 9 = 2 × 3 × 3

24 = 2 × 12 = 2 × 2 × 6 = 2 × 2 × 2 × 3 18 I 24 Gi †gŠwjK we‡k−lY wgwj‡q †gŠwjK Drcv`K 2 Av‡Q me©vwaK wZb evi †gŠwjK Drcv`K 3 Av‡Q me©vwaK yB evi|

∴ wb‡Y©q j.mv.¸. = 2 × 2 × 2 × 3 × 3 = 8 × 9 = 72

cÖv_wgK MwYZ 65

D`vniY 19| 35 I 50 Gi j.mv.¸. wbY©q Ki|

mgvavb : 35 = 5 × 7

50 = 2 × 25 = 2 × 5 × 5 35 I 50 Gi †gŠwjK Drcv`‡K we‡k −lY wgwj‡q †gŠwjK Drcv`K 2 Av‡Q me©vwaK GKevi, †gŠwjK Drcv`K 5 Av‡Q me©vwaK yBevi Ges †gŠwjK Drcv`K 7 Av‡Q me©vwaK GKevi|

∴ wb‡Y©q j.mv.¸. = 2 × 5 × 5 × 7 = 10 × 35 = 350| D`vniY 20| 12, 18, 24 Gi j.mv.¸. wbY©q Ki|

mgvavb : 24 = 2 × 12 msL¨vwU 12 Gi ¸wYZK| myZivs 18 I 24 Gi j.mv.¸.B wb‡Y©q j.mv.¸.| 18 I 24 Gi j.mv.¸. = 72 (D`vniY 18)

∴ wb‡Y©q j.mv.¸. = 72| D`vniY 21| 16, 24 I 36 Gi j.mv.¸. wbY©q Ki|

mgvavb : 16 = 2 × 8 = 2 × 2 × 4 = 2 × 2 × 2 × 2

24 = 2 × 12 = 2 × 2 × 6 = 2 × 2 × 2 × 3

36 = 2 × 18 = 2 × 2 × 9 = 2 × 2 × 3 × 3 16, 24 I 36 Gi †gŠwjK Drcv`‡K we‡k −lY wgwj‡q †gŠwjK Drcv`K 2 Av‡Q me©vwaK Pvievi, †gŠwjK Drcv`K 3 Av‡Q me©vwaK yBevi|

∴ wb‡Y©q j.mv.¸. = 2 × 2 × 2 × 2 × 3 × 3 = 16 × 9 = 144|

66 cÖv_wgK MwYZ

Abykxjbx- 7

1| wb‡Pi msL¨v¸‡jvi mKj ¸YbxqK wbY©q Ki : (K) 16 (L) 30 (M) 42 (N) 45 (O) 78 (P) 105 (Q) 144 (R) 189 (S) 216 (T) 324 2| wb‡Pi msL¨v¸‡jvi †Kvb&wU †gŠwjK, †Kvb&wU K…wÎg, kbv³ K‡i †jL : (K) 28 (L) 31 (M) 47 (N) 87 (O) 91 3| wb‡Pi msL¨v¸‡jv‡K †gŠwjK Drcv`‡K we‡k−lY Ki : (K) 60 (L) 72 (M) 93 (N) 112 (O) 135 (P) 222 (Q) 255 (R) 320 (S) 484 (T) 512 4| wb‡Pi msL¨v¸‡jvi †Kvb&wU 2 Øviv, †Kvb&wU 3 Øviv, †Kvb&wU 5 Øviv wefvR¨ †jL : (K) 69 (L) 80 (M) 95 (N) 165 (O) 255 (P) 360 (Q) 435 (R) 555 (S) 717 (T) 777 5| wb‡Pi msL¨v¸‡jvi cÖwZwUi cuvPwU K‡i ¸wYZK †jL : (K) 6 (L) 9 (M) 12 (N) 16 (O) 21 (P) 28 (Q) 35 (R) 42 (S) 44 (T) 50 6| cÖwZwU msL¨vhyM‡ji mKj mvaviY ¸YbxqK wbY©q Ki : (K) 18, 42 (L) 30, 75 (M) 36, 45 (N) 24, 64 (O) 48, 56 (P) 60, 100 7| M. mv. ¸. wbY©q Ki : (K) 16,24 (L) 28,42 (M) 35,49 (N) 45,60 (O) 12, 18, 30 (P) 16, 24, 40 (Q) 51, 63, 99 (R) 48, 60,72 8| j.mv.¸. wbY©q Ki : (K) 12,16 (L) 18,24 (M) 20,50 (N) 15,35 (O) 36, 45, (P) 42, 56 (Q) 46, 69 (R) 38, 57 (S) 12, 18, 24 (T) 20, 25, 30 (U) 32, 48, 72

cÖv_wgK MwYZ 67

Aóg Aa¨vq

MvwYwZK cÖZxK msL¨v cÖZxK : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 GB `kwU cÖZx‡Ki mvnv‡h¨ mKj msL¨v †jLv hvq e‡j G‡ì msL¨v cÖZxK ejv nq| msL¨v †jLvq eëüZ msL¨v cÖZxK‡K A¼ ejv nq|

cÖwµqv cÖZxK : + (†hvM wPý), − (we‡qvM wPý), × (¸Y wPý), ÷ (fvM wPý) g�j PviwU MvwYwZK cÖwµqv wb‡ ©k K‡i| m¤cK© cÖZxK : = (mgvb wPý), > (e„nËi wPý ev eo wPý), < (¶z ªZi wPý ev †QvU wPý) yBwU msL¨vi cvi¯cwiK m¤cK© wb‡ ©k K‡i|

†hgb : 7 × 12 = 84, 27 + 11 > 38-7

36 ÷ 9 < 3 × 2 wecixZ m¤cK© cÖZxK

≠ (mgvb bq) 16 ≠ 27 > (eo bq) 23 > 32 < (†QvU bq) 32 < 23

†hgb : 6 × 15 ≠ 16 × 7

35 − 17 > 15+12 40 + 5 < 60-35

72 ÷ 8 ≠ 3 × 2 MvwYwZK evK¨ (Dw³) Dc‡ii `kwU D`vni‡Yi cÖ‡Z¨KwU GKwU MvwYwZK evK¨ ev Dw³| MvwYwZK Dw³ mZÏ n‡Z cv‡i, wg_¨vI n‡Z cv‡i| Dc‡ii cÖ‡Z¨KwU Dw³ mZ¨| wg_¨v Dw³i D`vniY :

10 ÷ 2 = 6

2 × 13 > 25 + 11

27 − 12 < 4 × 3

68 cÖv_wgK MwYZ

†Lvjv evK¨ + 5 = 12 GwUI GK ai‡bi MvwYwZK evK¨| GLv‡b GKwU Awbw ©ó msL¨v †evSv‡Z cÖZxK eënvi Kiv n‡q‡Q| NiwU‡Z GKwU mywbw ©ó msL¨v bv emv‡bv ch©š� evK¨wU mZ¨ bv wg_¨v, wKQyB ejv hv‡e bv| Giƒc MvwYwZK evK¨‡K †Lvjv evK¨ ejv nq| †Lvjv ev‡K¨ cÖZx‡Ki RvqMvq GKwU (mywbw ©ó) msL¨v emv‡j GKwU MvwYwZK Dw³ cvIqv hvq| †mwU mZÏ n‡Z cv‡i, wg_¨vI n‡Z cv‡i| †hgb : 7 + 5 = 12 mZ¨ Dw³ 6 + 5 = 12 wg_¨v Dw³ 9 + 5 = 12 wg_¨v Dw³

ev¯�weK, + 5 = 12, †Lvjv evK¨wU‡Z cÖZx‡Ki RvqMvq ïaygvÎ 7 emv‡jB mZ¨ Dw³ cvIqv hv‡e| msL¨v ivwk KwZcq msL¨v‡K cÖwµqv cÖZxK Øviv mshy³ Ki‡j GKwU msL¨vivwk ˆZwi nq| †hgb : 26 ÷ 2 + 3 × 7 − 15 eÜbx wPý msL¨v ivwk‡Z †Kvb& cÖwµqv Av‡M m¤cbœ Ki‡Z n‡e, Zv eyS‡Z eÜbx wPý ( ) eënvi Kiv nq| †hgb : 37 − (13 + 8) GB ivwkwU‡Z cÖ_‡g †hvM cÖwµqv m¤cbœ Ki‡Z n‡e Ges Gi ci 37 †_‡K H †hvMdj we‡qvM Ki‡Z n‡e| D`vniY 1| †`LvI †h, (37-13) − 8 = 37 − (13 + 8) mgvavb : evgc¶ = (37 − 13) − 8 = 24 − 8 = 16 Wvbc¶ = 37 − (13 + 8) = 37 − 21 = 16 ∴evgc¶ = Wvbc¶| D`vniY 2| msL¨v cÖZxK I wecixZ m¤cK© cÖZxK eënvi K‡i cÖKvk Ki : (K) mvZk EbbeŸB, bqk bq †_‡K eo bq| (L) Pvi nvRvi Pvi Ges 40004 mgvb bq| mgvavb : (K) mvZk EbbeŸB = 789

bqk bq = 909 DËi : 789 > 909 (L) Pvi nvRvi Pvi = 4004 DËi : 4004 ≠ 40004|

cÖv_wgK MwYZ 69

D`vniY 3| N‡i m¤cK© wPý ewm‡q mZ¨ Dw³ MVb Ki|

(K) 137 − 69 47 + 14 (L) 16 × 12 24 × 8 (M) (72 ÷ 9) × 4 (72 × 4) ÷ 9 (N) (225 − 169) − 33 347 − (211 + 56) (O) (43 + 26) × 7 43 × 7 + 26 × 7 mgvavb : (K) 137 − 69 = 68 57 + 14 = 61 ∴ 68 > 61 DËi : 137 − 69 > 47 + 14

mgvavb : (L) 16 × 12 = 192 Avevi, 24 × 8 = 192 ∴ 16 × 12 = 24 × 8 DËi : 16 × 24 = = 24 × 8

mgvavb : (M) (72 ÷ 9) × 4 = 8 × 4 = 32 Avevi, (72 × 4) ÷ 9 = 288 ÷ 9 = 32 ∴ (72 ÷ 9) × 4 = (72 × 4) ÷ 9 DËi : (72 ÷ 9) × 4 = = (72 × 4) ÷ 9

mgvavb : (N) (225 − 169) − 33 = 56 − 33 = 23 Avevi, 347 − (211 + 56) = 347 − 267 = 80 GLb, 23 < 80 ∴ (225 − 169) − 33 < 347 − (211 + 56) DËi : (225 − 169) − 33 < 347 − (211 + 56)

mgvavb : (O) (43 + 26) × 7 = 69 × 7 = 483 Avevi, 43 × 7 + 26 × 7 = 301 + 182 = 483 ∴ (43 + 26) × 7 = 43 × 7 + 26 × 7 DËi : (43 + 26) × 7 = = 43 × 7 + 26 × 7

70 cÖv_wgK MwYZ

D`vniY 4| cÖZx‡Ki mvnv‡h¨ cÖkœwU cÖKvk Ki Ges ARvbv msL¨vwU wbY©q Ki| (K) †Kvb msL¨v †_‡K 17 we‡qvM Ki‡j we‡qvMdj 12 nq? (L) †Kvb& msL¨v‡K 9 Øviv ¸Y Ki‡j ¸Ydj 108 nq? mgvavb : (K) ARvbv msL¨vwUi Rb¨ cÖZxK wb‡j cÖkœwU uvovq : − 17 = 12 n‡j, = KZ? we‡qvM †hv‡Mi wecixZ cÖwµqv weavq msL vwU n‡e 12 I 17 Gi †hvMdj| ∴ = 12 + 17 = 29 DËi : 29| (L) ARvbv msL¨vwUi Rb¨ cÖZxK wb‡j cÖkœwU uvovq : × 9 = 108 n‡j, = KZ ? fvM ¸‡Yi wecixZ cÖwµqv weavq msL¨vwU n‡e 108 †K 9 Øviv fv‡Mi fvMdj| ∴ = 108 ÷ 9 = 12 DËi : 12|

Abykxjbx- 8 1| cÖZx‡Ki mvnv‡h¨ cÖKvk Ki : (K) GKk mvZvk †hvM wZbk wZb| (L) Pvik cuqlwÆ we‡qvM yBk EbwÎk| (M) AvUvwk ¸Y mvZ| (N) yBk EbbeŸB fvM m‡Zi| (O) evnvËi I AvUwÎ‡ki we‡qvMdj‡K GMvi Øviv ¸Y| 2| cÖZx‡Ki mvnv‡h¨ cÖKvk Ki : (K) wZbk mvZvbœ †_‡K yBk evi Gi we‡qvMdj cÂvk †_‡K eo| (L) wZbk lvU I AvVvi Gi fvMdj GKzk Gi mgvb bq| (M) GKzk I †lvj Gi †hvMd‡ji Qq ¸Y, GKzk Gi Qq¸Y I †lvj Gi Qq

¸‡Yi †hvMd‡ji mgvb| (N) †ZBk Gi †ZBk ¸Y cuvPk Gi †P‡q eo| (O) GKk Pzqvwj −k‡K evi w`‡q fvM K‡i fvMdj‡K †lvj w`‡q ¸Y Ki‡j

¸Ydj nq GKk weivbeŸB|

cÖv_wgK MwYZ 71

3| wb‡Pi Dw³¸‡jvi †KvbwU mZ¨, †KvbwU wg_¨v, hvPvB Ki : (K) 76 − 47 − 14 = 195 ÷ 13 (L) 537 + 482 = 689 + 320 (M) 59 + 47 > 63 + 39 (N) (380 ÷ 19) × 17 =/ (391 ÷ 17) × 19 (O) (187 + 121) + 97 < 187 + (121 + 97) 4| N‡i =, >, < Gi †h‡Kvb GKwU m¤cK© cÖZxK emvI †hb cÖvß Dw³wU mZ nq : (K) 49 + 28 103 − 37 (L) 23 × 1 1 × 24 (M) (189 − 27) − 69 189 − (27 + 69) (N) (13 × 11) × 7 13 × (11 × 7) (O) (343 ÷ 49) × 6 343 ÷ (7 × 7) 5| N‡i Giƒc cÖwµqv wPý emvI, †hb cÖvß Dw³ mZ¨ nq : (K) 71 35 = 9 4 (L) 36 24 = 18 48 (M) 7 5 = 105 3 (N) 87 38 = 7 7 (O) 324 18 = 19 18 6| wb‡Pi cÖ‡Z¨KwU †Lvjv ev‡K¨ Ggb msL¨v emvI †hb cÖvß Dw³wU mZ¨ nq| (K) − 7 = 32 − 17 (L) 6 × = 27 × 2 (M) ÷ 11 = 6 + 7 (N) 72 ÷ = 24 ÷ 3 (O) − 17 − 11 = 39 7| cÖZx‡Ki mvnv‡h¨ cÖ‡Z¨KwU cÖkœ cÖKvk Ki Ges ARvbv msL¨vwU wbY©q Ki : (K) †Kvb& msL¨vi m‡½ 23 †hvM Ki‡j †hvMdj 191 nq? (L) †Kvb& msL¨v †_‡K 39 we‡qvM Ki‡j we‡qvMdj 86 nq? (M) †Kvb& msL¨v‡K 13 Øviv ¸Y Ki‡j ¸Ydj 169 nq? (N) †Kvb& msL¨v‡K 19 Øviv fvM Ki‡j fvMdj 15 nq? (O) †Kvb& msL vi m‡½ 8 †hvM Ki‡j †hvMdj 17 I 9 Gi we‡qvMd‡ji mgvb nq?

72 cÖv_wgK MwYZ

beg Aa¨vq

mvaviY fMœvsk

j¶ Kwi : je 23 GKwU `vM †U‡b `v‡Mi wb‡P hZ fvM Kiv n‡q‡Q Ges `v‡Mi Dc‡i hZ fvM Mvp

ni Kiv n‡q‡Q Zv †jLv n‡q‡Q| wb‡Pi msL¨vwU ni Ges Dc‡ii msL¨vwU je| 23,

58,

34,

712 fMœvsk¸‡jv‡Z je ni A‡c¶v †QvU|

yB Z…Zxqvskev wZb fv‡Mi `yB

cuvP Aóvgvskev, AvU fv‡Mi cuvP

wZb PZz_©vskev, Pvi fv‡Mi wZb

mvZ Øv`kvskev, evi fv‡Mi mvZ

23

58

34

712

cÖv_wgK MwYZ 73

j¶ Kwi : 22,

33,

44,

55 cÖ‡Z¨‡K 1 Gi mgvb|

GLv‡b, je = ni|

12

24

36

48

510

612

12

24

36

48

510

612

mgZzj fMœvsk

74 cÖv_wgK MwYZ

12 =

1×22×2

= 24;

12 =

1×32×3

= 36

12 =

1×42×4

= 48;

12 =

1×52×5

= 510

12 =

1×62×6

= 612

j¶ Kwi : 12 =

24 =

36 =

48 =

510 =

612

∴ 12 =

24 =

36 =

48 =

510 =

512 mgZzj fMœvsk|

35 =

3×25×2

= 610;

35 =

3×35×3

= 915

35 =

3×45×4

= 1220;

35 =

3×55×5

= 1525

∴ 35,

610,

915,

1220,

1525 mgZzj fMœvsk|

GKwU fMœvs‡ki ni I je‡K GKB msL¨v Øviv ¸Y Ki‡j fMœvskwUi GKwU mgZzj fMœvsk cvIqv hvq|

Avevi j¶ Kwi :

915 =

9÷315÷3

= 35

1220 =

12÷420÷4

= 35

1525 =

15÷525÷5

= 35

GKwU fMœvs‡ki ni I je‡K GKB msL¨v Øviv fvM Ki‡j fMœvskwUi mgZzj fMœvsk cvIqv hvq|

cÖv_wgK MwYZ 75

fMœvs‡ki jwNô AvKvi 2472 =

24÷272÷2

= 1236

GLv‡b fMœvskwUi ni I je‡K Zv‡`i mvaviY ¸YbxqK 2 Øviv fvM Kiv n‡q‡Q| GB fvM Kiv Avi ni I j‡ei mvaviY ¸YbxqK 2 eR©b Kiv GKB K_v|

Avevi, 1236 =

12÷236÷2

= 618

618 =

6÷218÷2

= 39

Ges 39 =

3÷39÷3

= 13

j¶ Kwi : 1236,

618,

39,

13 Gi †¶‡Î fMœvs‡ki ni g�j fMœvs‡ki n‡ii †QvU|

∴ 2472 =

1 × 2 × 2 × 2 × 31× 2 ×2 × 2 × 3 × 3

= 13 GwU g �j fMœvs‡ki jwNô AvKvi|

j¶ Kwi : ni I j‡ei Mwiô mvaviY ¸YbxqK Øviv Dfq‡K fvM Ki‡j fMœvskwUi

jwNô AvKvi cvIqv hvq : 2472 =

24÷2472÷24

= 13

D`vniY 1| 2634 †K jwNô AvKv‡i cÖKvk Ki|

mgvavb : 2634 =

1 × 2 × 131× 2 ×17

= 1317

13 I 17 Gi 1 Qvov †Kvb mvaviY ¸YbxqK bvB|

∴ 2634 Gi jwNô AvKvi

1317

j¶ Kwi : †Kvb fMœvs‡ki jwNô AvKv‡i ni I j‡ei M.mv.¸. 1|

76 cÖv_wgK MwYZ

D`vniY 2| jwNô AvKv‡i cÖKvk Ki

(K) 5481 (L)

4896

(K) mgvavb : 5481 =

1 × 2 × 3 × 3 × 31 × 3 × 3 × 3 × 3

= 23

DËi : 23|

(L) mgvavb : 4896 =

48÷496÷4

= 1224 Avevi,

1224 =

12÷1224÷12

= 12

∴ 4896 Gi jwNô AvKvi

12

DËi : 12|

j¶ Kwi : fMœvskwUi je I ni‡K µ‡g µ‡g G‡`i mvaviY ¸YbxqK Øviv fvM K‡i jwNô AvKvi cvIqv †M‡Q|

mgni wewkó fMœvsk

932 I

1532 fMœvsk yBwUi ni GKB| G¸‡jv mgni wewkó fMœvsk|

78 I

35 mgni wewkó fMœvsk bq| G‡`i mgni wewkó fMœvs‡k cÖKvk Kiv hvq|

78 =

7×58×5

= 3540

35 =

3×85×8

= 2440

Avevi 78 =

7×108×10

= 7080

35 =

3×165×16

= 4880

Dfq‡¶‡Î ni 40 Kiv n‡q‡Q|

Dfq‡¶‡Î ni 80 Kiv n‡q‡Q|

cÖv_wgK MwYZ 77

fMœvsk¸‡jvi n‡ii †h‡Kvb mvaviY ¸wYZK‡K ni K‡i Zv‡`i mgni Kiv hvq| Z‡e mvaviYZ nimg�‡ni j.mv.¸. †K ni K‡i jwNô mgni wewkó fMœvsk ˆZwi Kiv nq|

D`vniY 3| 56 I

34 †K 48 mgni wewkó fMœvs‡k cÖKvk Ki|

mgvavb : 48 ÷ 6 = 8 ∴ 56 =

5×86×8

= 4048

48 ÷ 4 = 12 ∴ 34 =

3×124×12

= 3648

DËi : 4048 I

3648

D`vniY 4| 512 I

78 †K jwNô mgni wewkó fMœvs‡k cÖKvk Ki|

mgvavb : fMœvsk yBwUi ni 12 I 8; G‡`i j.mv.¸. 24|

24 ÷ 12 = 2 ∴ 512 =

5×212×2

= 1024

24 ÷ 8 = 3 ∴ 78 =

7×38×3

= 2124

DËi : 1024 I

2124

j¶ Kwi : cÖ‡Z¨K fMœvs‡ki ni Øviv ni¸‡jvi j.mv.¸. †K fvM K‡i †mB fvMdj Øviv je I ni‡K ¸Y K‡i mgni wewkó fMœvsk cvIqv hvq|

jwNô mgni K‡i LvwjNi c�iY Ki (GKwU K‡i †`Lv‡bv nj) :

(K) 35 =

2135 (L)

56 =

27 =

1035

49 =

(M) 78 = (N)

34 =

56 =

56 =

78 cÖv_wgK MwYZ

(O) 23 = (P)

47 =

58 =

58 =

316=

514=

D`vniY 5| Lvwj N‡i mwVK msL¨vwU emvI :

(K) 56 =

30 (L) 611 =

24

mgvavb : (K) GLv‡b, 30 ÷ 6 = 5 (L) GLv‡b, 24 ÷ 6 = 4

∴ 56 =

5×56×5

= 2530 ∴

611 =

6×411×4

= 2444

DËi : 56 =

25

30 DËi : 611 =

24

44

fMœvs‡ki Zzjbv

58

48

58 eo

48 †_‡K; A_©vr

58 >

48

Avevi, 48 †QvU

58 †_‡K; A_©vr

48 <

58

cÖv_wgK MwYZ 79

29

59

29 †QvU

59 †_‡K; A_©vr,

29 <

59

Avevi, 59 eo

29 †_‡K; A_©vr,

59 >

29

j¶ Kwi : 59 Ges

29 Gi ni GKB| wKš‘ je 5 > je 2 ∴

59 >

29

611 = 11 fv‡Mi 6 fvM 6 < 7 ∴

611 <

711

711 = 11 fv‡Mi 7 fvM

1017 = 17 fv‡Mi 10 fvM 10 > 6 ∴

1017 >

617

617 = 17 fv‡Mi 6 fvM

ni GKB n‡j †h fMœvs‡ki je eo †mB fMœvskwU eo

D`vniY 6| 710 I

310 fMœvsk yBwUi †KvbwU eo? MvwYwZK cÖZx‡Ki mvnv‡h¨ †`LvI|

mgvavb : 710 = 10 fv‡Mi 7 fvM

310 = 10 fv‡Mi 3 fvM

7 > 3 ∴ 710 >

310

DËi : 710 >

310

80 cÖv_wgK MwYZ

D`vniY 7| 516 I

916 fMœvsk yBwUi †KvbwU †QvU?

mgvavb : `yBwU fMœvs‡ki ni 16; G‡`i je 5 I 9|

5 > 9 ∴ 516 <

916

DËi : 516 †QvU|

311

35

j¶ Kwi : 35 I

311 Gi je GKB|

11 > 5∴ < 311 <

35

38

316

8 < 16 ∴ 38 >

316

je GKB n‡j, †h fMœvs‡ki ni †QvU †mB fMœvskwU eo|

D`vniY 8| 718 I

715 Gi g‡a¨ †Kvb&wU eo, MvwYwZK cÖZx‡Ki mvnv‡h¨ †`LvI|

mgvavb : fMœvsk yBwUi je GKB; ni 18 I 15

18 > 15 ∴ 715 >

718

DËi : 715 >

718

cÖv_wgK MwYZ 81

D`vniY 9| 315 I

45 Gi g‡a¨ †KvbwU eo, cÖZxK eënvi K‡i †`LvI|

mgvavb : fMœvsk yBwUi ni 15 I 5| G‡ì j.mv.¸. 15|

15 ÷ 15 = 1 ∴ 315 =

3×115×1

= 315

15 ÷ 5 = 3 ∴ 45 =

4×35×3

= 1215

12 > 3 ∴ 1215 >

315

A_©vr , 45 >

315

DËi : 45 >

315

j¶ Kwi : fMœvsk yBwU‡K mgni wewkó fMœvs†k iƒcvš�i Kiv n‡q‡Q| c‡i G‡ì je Zzjbv K‡i eo fMœvskwU †ei Kiv n‡q‡Q|

D`vniY 10| 78,

1316,

1124 fMœvsk¸‡jv‡K gv‡bi DaŸ©µg Abymv‡i mvRvI|

mgvavb : fMœvsk¸‡jvi ni 8, 16, 24| G‡ì j.mv.¸. = 48

48 ÷ 8 = 6 ∴ 78 =

7×68×6

= 4248

48 ÷ 16 = 3 ∴ 1316 =

13×316×3

= 3948

48 ÷ 24 = 2 ∴ 1124 =

11×224×2

= 2248

22 < 39 < 42 ∴ 2248 <

3948 <

4248

ev, 1124 <

1316 <

78

DËi : DaŸ©µg Abymv‡i mvwR‡q cvB, 1124 ,

1316 ,

78 |

82 cÖv_wgK MwYZ

fMœvs‡ki †hvM

(K) 37

17

47 Qwe‡Z j¶ Kwi :

37 +

17 =

47 =

3+17

59

29

19

(L)

89 Qwe‡Z j¶ Kwi :

59 +

29 +

19 =

89 =

5+2+19

j¶ Kwi : mgni wewkó KZK¸‡jv fMœvs‡ki †hvMd‡ji ni fMœvsk¸‡jvi mvaviY ni Ges GB †hvMd‡ji je fMœvsk¸‡jvi j‡ei †hvMdj|

D`vniY 11| †hvM Ki : 313 +

713 +

213

mgvavb : 313 +

713 +

213 =

3+7+213 =

1213

DËi : 1213|

D`vniY 12| †hvM Ki : 211 +

522

mgvavb : 211 =

2×211×2

= 422 ∴

211 +

522 =

422 +

522 =

4+522 =

922

DËi : 922|

j¶ Kwi : fMœvsk yBwU mgni wewkó bq| G‡`i mgni wewkó fMœvs‡k iƒcvš�i K‡i †hvM Kiv n‡q‡Q|

cÖv_wgK MwYZ 83

D`vniY 13| †hvM Ki : 18 +

16 +

512

mgvavb : fMœvsk¸‡jvi ni 8, 6, 12| G‡`i j.mv.¸. 24|

18 =

1×38×3

= 324 (KviY, 24 ÷ 8 = 3)

16 =

1×46×4

= 424 (KviY, 24 ÷ 6 = 4)

512 =

5×212×2

= 1024 (KviY, 24 ÷ 12 = 2)

∴ 18 +

16 +

512 =

324 +

424 +

1024 =

3+4+1024 =

1724

DËi : 1724|

D`vniY 14| †hvM Ki : 25 +

310 +

115

mgvavb : fMœvsk¸‡jvi ni 5, 10, 15 Ges G‡`i j.mv.¸. 30|

∴ 25 =

2×65×6

= 1230 (KviY, 30÷5 = 6)

310 =

3×310×3

= 930 (KviY, 30÷10 = 3)

115 =

1×215×2

= 230 (KviY, 30÷15 = 2)

∴ 25 +

310 +

115 =

1230 +

930 +

230 =

12×9+230 =

2330

DËi : 2330|

84 cÖv_wgK MwYZ

fMœvs‡ki we‡qvM

Qwe‡Z j¶ Kwi : 67 −

47 =

27 =

6−47

†`Lv hv‡”Q, fMœvsk `yBwUi we‡qvMdj ni G‡`i mvaviY ni Ges we‡qvMd‡ji je G‡`i j‡ei we‡qvMdj

Qwe‡Z j¶ Kwi : 1113 −

313 =

813 =

11−313

D`vniY 15| we‡qvM Ki : 79 −

59

mgvavb : 79 −

59 =

7−59 =

29

DËi : 29|

D`vniY 16| 725 −

325 = KZ ?

mgvavb : 725 −

325 =

7−325 =

425

DËi : 425|

1113

67

4 7

27

8 13

313

cÖv_wgK MwYZ 85

D`vniY 17| 825 †_‡K

15 we‡qvM Ki|

mgvavb : 825 −

15 ; fMœvsk yBwUi ni 25 I 5 Ges G‡ì j.mv.¸. 25

∴ 825 =

8×125×1

= 825 [KviY, 25 ÷ 25 = 1]

15 =

1×55×5

= 525 [KviY, 25 ÷ 5 = 5]

∴ 825 −

15 =

825 −

525 =

8−525 =

325

DËi : 325|

D`vniY 18| 512 −

14 = KZ?

mgvavb : GLv‡b fMœvsk yBwUi ni 12 I 4 Ges G‡ì j.mv.¸. 12

∴ 512 =

5×112×1

= 512 [KviY, 12 ÷ 12 = 1]

14 =

1×34×3

= 312 [KviY, 12 ÷ 4 = 3]

∴ 512 −

14 =

512 −

312 =

5−312 =

212 =

16

DËi : 16 |

j¶ Kwi : fMœvsk yBwU mgni wewkó bq| G‡ì mgni K‡i we‡qvM Kiv n‡q‡Q| c‡i we‡qvMdj‡K jwNô AvKv‡i cÖKvk Kiv n‡q‡Q|

D`vniY 19| 1 − 34 = KZ ?

mgvavb : 1 − 34 =

44 −

34 [1 I

34 †K mgni wewkó fMœvs‡k iƒcvš�i K‡i]

= 4−34 =

14

DËi : 14|

1 6

86 cÖv_wgK MwYZ

Lvwj N‡i mwVK msL¨v emvI (GKwU K‡i †`Lv‡bv nj)

(K) 25 +

15 =

35 (L)

16 +

23 = (M)

1724 +

112 =

(N) 1728 −

314 = (O)

1522 −

311 = (P)

715 −

25 =

(Q) 65+

14 = (R)

34−

310 =

fMœvs‡ki mij

D`vniY 20| mij Ki : 19 +

13 −

29

mgvavb : fMœvsk¸‡jvi ni 9, 3, 9 Ges G‡ì j.mv.¸. 9

19 =

1×19×1

= 19 [KviY, 9 ÷ 9 = 1]

13 =

1×33×3

= 39 [KviY, 9 ÷ 3 = 3]

29 =

2×19×1

= 29

∴ 19 +

13 −

29 =

19 +

39 −

29 =

1+3−29 =

4−29 =

29

DËi : 29|

j¶ Kwi : fMœvsk¸‡jv mgni wewkó bq| G‡ì mgni wewkó fMœvs‡k iƒcvš�i K‡i mij Kiv n‡q‡Q|

D`vniY 21| mij Ki : 914 +

37 −

528

mgvavb: fMœvsk¸‡jvi ni 14, 7, 28| G‡ì j.mv.¸. 28

∴ 914 +

37 −

528 =

1828 +

1228 −

528 =

18+12−528 =

30−528 =

2528

DËi : 2528 |

cÖv_wgK MwYZ 87

Abykxjbx- 9 (K) 1| wb‡Pi LvwjNi c�iY Ki : (K) 7 je I 10 ni n‡j fMœvskwU = (L) 5 je I 25 ni n‡j fMœvskwU = (M) 9 je I 15 ni n‡j fMœvskwU = (N) je 24 I ni 72 n‡j fMœvskwU = 2| wb‡Pi cÖwZwU fMœvs‡ki mgZzj fMœvsk wbY©q Ki (wZbwU K‡i) :

(K) 56 , (L)

47 , (M)

38

3| jwNô AvKv‡i cÖKvk K‡i Lvwj N‡i msL¨v emvI :

(K) 2540 = (L)

5472 = (M)

1545 =

(N) 3485 = (O)

3876 = (P)

2478 =

jwNô mgni iƒ‡c cÖKvk Ki (cÖkœ 4-9) :

4| 710 I

115 5|

1324 I

56 6|

56 ,

67 I

78

7| 115 ,

445 I

730 8|

57 ,

314 I

821 9|

38 ,

12 ,

34 I

916

N‡i mwVK msL¨v emvI (cÖkœ 10−15) :

10| 45 =

20 11| 78 =

49

12|

321 =

42

13| 815 =

24

14|

81 = 89 15|

49

=

712

wb‡Pi fMœvsk¸‡jvi g‡a¨ †Kvb&wU eo ev †QvU cÖZx‡Ki mvnv‡h¨ cÖKvk Ki :

16| (K) 35,

34 (L)

914,

710 (M)

1730 ,

715 (N)

311 ,

733

88 cÖv_wgK MwYZ

†hvM Ki : (cÖkœ 17 − 21)

17| 59 +

16 18|

215 +

475 19|

12 +

16 +

312

20| 928 +

314 +

13 21|

312 +

29 +

16

we‡qvM Ki : (cÖkœ 22 − 26)

22| 312 −

29 23|

13 −

211 24|

14 −

317 25|

713 −

926 26| 1 −

45

mij Ki : (cÖkœ 27 − 34)

27| 16 +

13 +

14 28|

16 +

34 −

18 29|

35 −

12 +

23 30|

514 +

12 −

27

31| 25 +

425 +

310 32|

215 +

13 +

415 33|

29 +

56 −

13 34|

58 −

56 +

512

fMœvs‡ki mgm¨v D`vniY 1| Avw`‡ei Kv‡Q KZK¸‡jv wjPz wQj| †m Zvi †evb iywg‡K †gvU wjPzi

Ask 38 w`j| Zvi Kv‡Q †gvU wjPzi KZ Ask iBj ?

mgvavb : Zvi †evb iywg‡K w`j †gvU wjPzi 38 Ask

∴ evwK iBj †gvU wjPzi (1 − 38 ) Ask

= ( 88 −

38 ) Ask

= 8−38 Ask =

58 Ask

DËi : 58 Ask|

cÖv_wgK MwYZ 89

D`vniY 2| GKwU evMv‡bi 37 As‡k AvgMvQ,

27 As‡k KuvVvjMvQ Av‡Q| evwK

As‡k RvgMvQ Av‡Q| evMv‡bi KZ As‡k RvgMvQ Av‡Q?

mgvavb : 37 +

27

= 3+27 =

57

∴ AvgMvQ I KuvVvjMvQ Av‡Q evMv‡bi 57 As‡k|

evwK iBj evMv‡bi (1 − 57 ) Ask|

= ( 77 −

57 ) Ask =

7−57 Ask =

27 Ask

DËi : evMv‡bi 27 As‡k RvgMvQ Av‡Q|

D`vniY 2| GK e¨w³ Zvi m¤cwËi 18 Ask ¯ �x‡K,

12 Ask cyÎ‡K Ges

14 Ask

Kb¨v‡K `vb Ki‡jb| Zuvi Avi KZUzKz m¤cwË evwK iBj ?

mgvavb : �x‡K w`‡jb 18 Ask, cyÎ‡K w`‡jb

12 Ask, Kb¨v‡K w`‡jb

14 Ask

†gvU `vb Ki‡jb (18 +

12 +

14 ) Ask ∴ =

1+4+28 Ask

78 =

Ask evwK iBj = (1 − 78 ) Ask = (

88 −

78 ) Ask =

8−78 Ask =

18 Ask

DËi : Zvi m¤cwZi 18 Ask evwK iBj |

j¶ Kwi : H e¨w³i m¤cwË 1 aiv n‡q‡Q|

A_©vr , 1 = 88 (8 fv‡Mi 8 Ask A_©vr m¤c�Y© Ask)

Abykxjbx- 9 (L)

1| GKwU m¤cwËi 12 As‡ki gvwjK †Rev,

13 As‡ki gvwjK iyYv, evwK As‡ki

gvwjK Rwj| Rwj H m¤cwËi KZ As‡ki gvwjK?

2| GKwU euv‡ki 25 Ask Kv`vq,

15 Ask cvwb‡Z Ges evwK Ask cvwbi Dc‡i

Av‡Q| cvwbi Dc‡i KZ Ask Av‡Q ?

90 cÖv_wgK MwYZ

3| GKRb K…lK Zuvi †gvU Rwgi 113 As‡k cvU,

12 As‡k avb Ges

326 As‡k

Mg Pvl K‡i‡Qb| wZwb Zuvi †gvU Rwgi KZ As‡k Pvl K‡i‡Qb ?

4| GK e¨w³ Zuvi †gvU UvKvi 35 Ask ¯Îx‡K w`‡jb,

415 Ask `vb Ki‡jb| Zuvi

wbKU †gvU UvKvi KZ Ask Aewkó iBj?

5| iweb Zvi QywUi w`b¸‡jvi 716 Ask evwo‡Z,

316 Ask gvgvi evwo‡Z KvUv‡jv,

evwK mgq †m ¯‹vDU K¨v‡¤c wQj| †m Zvi QywUi KZ Ask ¯‹vDU K¨v‡¤c wQj?

6| AvdRvj mv‡n‡ei Kv‡Q wKQy UvKv wQj| wZwb H UvKvi 13 Ask �x‡K w`‡jb,

115 Ask `vb Ki‡jb| evwK UvKv Zvi †Q‡j‡g‡q‡K w`‡jb| wZwb Zuvi UvKvi

KZ Ask †Q‡j‡g‡q‡K w`‡jb ?

7| wkcjy Zvi c‡KUgvwbi 23 Ask w`‡q M‡íi eB,

16 Ask w`‡q GKwU AvBmwµg

wKbj| wkcjy c‡KUgvwbi KZ Ask LiP Kij ?

8| iwdK mv‡ne Zuvi evMv‡bi 24 As‡k †m¸bMvQ,

14 As‡k kvjMvQ Ges evwK As‡k

Rviyj MvQ jvwM‡q‡Qb| H evMv‡bi KZ As‡k Rviyj MvQ jvMv‡bv n‡q‡Q?

9| GKwU euv‡ki 15 Ask Kv`vq,

12 Ask cvwb‡Z Ges evwK Ask cvwbi Dc‡i

Av‡Q| euvkwUi KZ Ask cvwbi Dci Av‡Q ? 10| MwY wgqv Zuvi cyKz‡i gvQ Pvl K‡i wKQy UvKv jvf Ki‡jb| wZwb GB UvKvi

37 Ask FY cwi‡kva Ki‡jb

314 Ask w`‡q gv‡Qi Lvevi wKb‡jb| Zuvi wbKU

jv‡fi KZ Ask Aewkó iBj ? 11| GKwU evwjKv we`¨vj‡qi QvÎx‡`i g‡a¨ hZRb Gm,Gm,wm cix¶vq Ask MÖnY

Kij Zv‡`i g‡a¨ 1

24 Ask A+ (G c−vm), 712 Ask A (G) Ges

1748 Ask A−

(G gvBbvm) †c‡q cvk Kij| Aewkó cix¶v_©x †dj Kij| †gvU cix¶v_©xi KZ Ask †dj Kij?

cÖv_wgK MwYZ 91

`kg Aa¨vq

`kwgK fMœvsk wb‡Pi Mvp Ni¸‡jv j¶ Kwi :

GK `kgvsk = 110 = .1 (`kwgK GK)

`yB `kgvsk = 210 = .2 (`kwgK yB)

wZb `kgvsk = 310 = .3 (`kwgK wZb)

Pvi `kgvsk = 410 = .4 (`kwgK Pvi)

cvuP `kgvsk = 510 = .5 (`kwgK cvuP)

Qq `kgvsk = 610 = .6 (`kwgK Qq)

mvZ `kgvsk = 710 = .7 (`kwgK mvZ)

AvU `kgvsk = 810 = .8 (`kwgK AvU)

bq `kgvsk = 910 = .9 (`kwgK bq)

j¶ Kwi : G‡`i cÖ‡Z¨‡K GKwU fMœvsk hvi ni 10| wb‡Pi Mvp Ni¸‡jv j¶ Kwi :

GK kZvsk = 1

100 cvuP kZvsk = 5

100

92 cÖv_wgK MwYZ

GK kZvsk = 1

100 = .01 (`kwgK k~b¨ GK)

cvuP kZvsk = 5

100 = .05 (`kwgK k~b¨ cuvP)

Qq kZvsk = 6

100 = .06 (`kwgK k~b¨ Qq)

---------------------------

bq kZvsk = 9

100 = .09 (`kwgK k~b¨ bq)

`k kZvsk = 10100 = .10 (`kwgK GK k~b¨)

GMvi kZvsk = 11

100 = .11 (`kwgK GK GK)

†lvj kZvsk = 16100 = .16 (`kwgK GK Qq)

wek kZvsk = 20100 = .20 (`kwgK yB k ~b¨)

-----------------------------

lvU kZvsk = 60100 = .60 (`kwgK Qq k~b¨)

wbivbeŸB kZvsk = 99100 = .99 (`kwgK bq bq)

j¶ Kwi : .01 = 1

100, .03 = 3

100, .05 = 5

100, .16 = 16100,

.19 = 19100, .99 =

99100,

G‡`i cÖ‡Z¨‡K GK GKwU fMœvsk hvi ni 100|

cÖv_wgK MwYZ 93

D`vniY 1| LvwjNi c~iY Ki ( yBwU K‡i †`Lv‡bv nj) :

K) 710 = .7 L)

18100 = .18

910 =

25100 =

510 =

40100 =

310 =

75100 =

110 =

2100 =

12 Ni Mvp is Ki :

†h fMœvs‡ki n‡i 10 ev Gi NvZ †hgb 100, 1000 BZ¨vw` msL¨v _v‡K, Zv‡`i ms‡¶‡c `kwgK we› y (.) e¨envi K‡i cÖKvk Kiv nq| G¸‡jvB `kwgK fMœvsk|

wgkª `kwgK fMœvsk : 2.3 msL¨vwU‡Z 2 c~Y© msL¨v Ges .3 nj `kwgK fMœvsk| GwU GKwU wgkª `kwgK fMœvsk| j¶ Kwi : wgkª `kwgK fMœvs‡k GKwU c~Y© msL¨v I GKwU `kwgK fMœvsk _v‡K|

94 cÖv_wgK MwYZ

`kwgK fMœvsk cV‡bi wbqg `kwgK fMœvsk cvV Ki‡Z ejv n‡j cÖ_‡g Ò`kwgKÓ K_vUv e‡j, c‡i we› yi Wv‡bi A¼¸‡jv cvV Ki‡Z n‡e| fMœvskwU wgkª n‡j A_©vr c �Y© msL¨v Ges `kwgK Ask _vK‡j, c�Y© msL¨vsi Ask mvaviY msL¨vi b¨vq cvV Ki‡Z n‡e|

`kwgK msL¨v (A‡¼) `kwgK msL¨v (K_vq) .01 `kwgK k~b¨ GK .10 `kwgK GK k ~b¨ .99 `kwgK bq bq .99.09 wbivbeŸB `kwgK k ~b¨ bq 80.08 Avwk `kwgK k~b¨ AvU

j¶ Kwi : .10 †K `kwgK `k cov hv‡e bv| .99 †K `kwgK wbivbeŸB cov hv‡e bv|

A‡¼ K_vq

8.46 AvU `kwgK Pvi Qq 12.429 205.001 555.505

K_vq †jLv `kwgK msL¨v A‡¼ †jL :

`kwgK msL¨v (K_vq) `kwgK msL¨v (A‡¼) c‡bi `kwgK yB mvZ 15.27 mvZ `kwgK k �b¨ wZb 7.03 evnvËi `kwgK bq AvU 72.98 wZbk `kwgK k �b¨ wZb 300.03 cuvP UvKv cuvP cqmv 5.05 UvKv

cÖv_wgK MwYZ 95

K_vq †jLv `kwgK msL¨v A‡¼ †jL (cÖ_gwU K‡i †`Lv‡bv nj) :

K_vq A‡¼ `k `kwgK k~b¨ wZb 10.03 c‡bi `kwgK wZb `yB GK

GKk wZb `kwgK k~b¨ GK wZb

GKk yB `kwgK bq bq

`k UvKv cuvP cqmv

¯’vbxq gvb : msL¨v †jLvi Rb¨ Avgiv 0, 1, 2, 3, 4, 5, 6, 7, 8, 9- GB `kwU cÖZxK e¨envi Kwi| ZvB Avgv‡`i msL¨v †kªYxi wfwË nj `k| GB wfwËi MYbv ïiy nq GKK ¯_vb †_‡K| wb‡Pi wPÎwU j¶ Kwi : GKK ¯_vb †_‡K evgw`‡Ki ¯_vb¸‡jvi gvb 10 ¸Y wnmv‡e †e‡o †M‡Q; Aci w`‡K GKK ¯_vb †_‡K Wvbw`‡Ki ¯_vb¸‡jvi gvb `k fv‡Mi GK Ask wnmv‡e K‡g †M‡Q| evg cv‡ki ¯_vb¸‡jv‡K `kK, kZK, mnmª BZ¨vw` ejv nq Ges Wvb cv‡ki ¯_vb¸‡jv‡K `kgvsk, kZvsk, mnmªvsk BZ¨vw` ejv nq| GKK ¯_v‡bi Wv‡b GKwU we›`y (.) ¯_vcb K‡i c�Y© msL¨vsk I `kwgKvsk c„_Kfv‡e †Pbv hvq| GB we› ywU nj `kwgK we› y| `kwgKvs‡ki `kgvsk, kZvsk, mnmªvsk BZ¨vw` ¯_vb¸‡jvi gvb h_vµ‡g .1, .01, .001 BZ¨vw` wj‡L cÖKvk Kiv nq| ¯’vbxq gvb : 36.94

mnmª kZK `kK GKK `kgvsk kZvsk mnmªvsk 1000 100 10 1 1

10 ev .1 1

100 ev .011

1000ev .001

3 6 9 4

96 cÖv_wgK MwYZ

GLv‡b, K_vq A‡¼ Gi ¯_vbxq gvb 3 `kK ev, 30 Gi ¯_vbxq gvb 6 GKK ev, 6 Gi ¯_vbxq gvb 9 `kgvsk ev, .9 Gi ¯_vbxq gvb 4 kZvsk ev, .04

3 6 .9 4

¯_vbxq gvb †jL (cÖ_gwU K‡i †`Lv‡bv nj) : 18.05 msL¨vq 1 Gi ¯’vbxq gvb 1 `kK ev, 10 8 ÕÕ ÕÕ 8 GKK ev, 8 0 ÕÕ ÕÕ 0 `kgvsk ev, .0 5 ÕÕ ÕÕ 5 kZvsk ev, .05 (K) 28.64, (L) 42.50, (M) 64.64, (N) 8.04, (O) 0.03 mvaviY fMœvsk‡K `kwgK fMœvs‡k iƒcvš—i

12 =

12 × 1 =

12 ×

55 =

510 = .5

15 =

15 × 1 =

15 ×

22 =

210 = .2

14 =

14 × 1 =

14 ×

2525 =

25100 = .25

Avevi, 15 =

15 × 1 =

15 ×

2020 =

20100 = .20

950 =

950 × 1 =

950 ×

22 =

18100 = .18

j¶ Kwi : ni‡K 10 ev 100 Kivi Rb¨ ni I je‡K GKB msL¨v w`‡q ¸Y Kiv n‡q‡Q| 15 = .2,

15 = .20, myZivs .20 = .2

A_©vr, `kwgK msL¨vi †k‡l k~b¨ bv wjL‡jI P‡j|

cÖv_wgK MwYZ 97

mvaviY fMœvsk‡K `kwgK fMœvs‡k i ~cvš—i Ki (cÖ_gwU K‡i †`Lv‡bv nj) : 320 =

320 × 1 =

320 ×

55 =

15100 = .15

720 =

910 =

3750 =

340 =

`kwgK fMœvsk‡K mvaviY fMœvs‡k i ~cvš—i

D`vniY 2 : (K) .06 = 6

100 = 350 (L) .5 =

510 =

12

(M) .45 = 45100 =

920 (N) .76 =

76100 =

3850 =

1925

j¶ Kwi : GLv‡b `kwgK we›`yi c‡ii msL¨v‡K je a‡i wb‡P n‡ii RvqMvq cÖ_‡g 1 †jLv n‡q‡Q| c‡i 1 Gi Wv‡b `kwgK we› yi ci hZ¸‡jv A¼ Av‡Q ZZ¸‡jv k�b¨ †jLv n‡q‡Q|

`kwgK fMœvsk‡K mvaviY fMœvs‡k iƒcvš�i Ki (cÖ_gwU K‡i †`Lv‡bv nj) :

.26 = 26100 =

1350

.4 = .25 = .64 = .04 = .48 = .75 =

1

2

3

50

9

20

38

50

13

50

19

25

98 cÖv_wgK MwYZ

`kwgK fMœvs‡ki Zzjbv

210 = .2

.2 .3 .110 = .1 =

310 = .3

wPÎwU‡Z j¶ Kwi : 210 >

110 ev, .2 > .1,

310 >

210 ev, .3 > .2

Abyiƒcfv‡e, .4 > .3 .5 > .4 BZ¨vw`

wb‡Pi wPÎ yBwU j¶ Kwi,

24100 ev .24

38100 ev .38

GLv‡b, .38 > .24

.1

cÖv_wgK MwYZ 99

D`vniY 3| .53 I .56 Gi g‡a¨ †QvU eo wbY©q Ki|

mgvavb : .53 = 53100 Ges .56 =

56100

53100 <

56100 myZivs .53 < .56

j¶ Kwi : †h `kwgK fMœvs‡ki `kgvsk eo, Zv eo msL¨v| Avevi `kgvsk

GKB n‡j hvi kZvsk eo, Zv eo|

Zzjbv Ki (cÖ_gwU K‡i †`Lv‡bv nj) :

.34 I .43 GLv‡b, .43 > .34 KviY, `kgvsk 4 > `kgvsk 3

.50 I .05 GLv‡b, ... ... ... KviY, ... ... ... ...

.35 I .53 GLv‡b, ... ... ... KviY, ... ... ... ...

.39 I .09 GLv‡b, ... ... .. KviY, ... ... ... ...

.57 I .59 GLv‡b, ... ... ... KviY, ... ... ... ...

.65 I .68 GLv‡b, ... ... ... KviY, ... ... ... ...

Abykxjbx- 10 (K)

1| K_vq †jL :

(K) 5.73 (L) 8.09 (M) 8.58 (N) 25.25 (O) 9.9 (P) 9.01

2| A‡¼ †jL :

(K) mvZ `kwgK k�b¨ bq (L) AvU `kwgK k�b¨ GK

(M) cuvP `kwgK cuvP cuvP (N) bq `kwgK bq bq|

3| wb‡Pi msL¨v¸‡jvi cÖ‡Z¨KwU A‡¼i ¯_vbxq gvb †jL :

(K) 4.34 (L) 15.05 (M) 18.26 (N) 17.52

(O) 13.03 (P) 20.02 (Q) 8.89

100 cÖv_wgK MwYZ

4| wb‡Pi mvaviY fMœvsk¸‡jv‡K `kwgK fMœvs‡k cÖKvk Ki :

(K) 910 (L)

110 (M)

810 (N)

710 (O)

320 (P)

720

(Q) 920 (R)

425 (S)

925 (T)

1350 (U)

2150

5| wb‡Pi `kwgK fMœvsk¸‡jv‡K mvaviY fMœvs‡k iƒcvš�i Ki :

(K) .3 (L) .7 (M) .9 (N) .13 (O) .27 (P) .39 (Q) .07

(R) .99 (S) .89 (T) .98 (U) .01

6| wb‡Pi `kwgK fMœvsk¸‡jv‡K mvaviY fMœvs‡ki jwNô AvKv‡i cÖKvk Ki :

(K) .4 (L) .04 (M) .5 (N) .20 (O) .25 (P) .40

(Q) .50 (R) .60 (S) .75 (T) .80

7| wb‡Pi fMœvsk hyM‡ji g‡a¨ >,< wPý w`‡q Zv‡`i gv‡bi Zzjbv Ki :

(K) .38, .46 (L) .58, .72 (M) .09, .90 (N) .30, .03

(O) .86, .87 (P) .64, .60 (Q) .5, .25 (R) .37, .32|

cÖv_wgK MwYZ 101

`kwgK fMœvs‡ki †hvM I we‡qvM D`vniY 4| †hvM Ki : 72.36 I 34.21 mgvavb : 72.36 e¨vL¨v : 34.21 106.57 DËi : 106.57| j¶ Kwi : kZvsk †hvM K‡i kZvs‡ki wb‡P Ges `kgvsk †hvM K‡i `kgvs‡ki wb‡P emvb n‡q‡Q| GB aviv c�Y© As‡ki Rb¨I mgfv‡e cÖ‡qvM Kiv n‡q‡Q| D`vniY 5| †hvM Ki : 4.9 I 3.55 mgvavb : 4.90 e¨vL¨v : 3.55 8.45 DËi : 8.45| j¶ Kwi : 9 `kgvsk I 5 `kgvsk GK‡Î 14 `kgvsk n‡q‡Q| `kgvs‡ki N‡i 4 `kgvsk †jLv n‡q‡Q Ges 1 GKK GK‡Ki N‡ii 4 Gi mv‡_ †hvM Kiv n‡q‡Q| †hvM Ki (cÖ_gwU K‡i †`Lv‡bv nj) : 5.8 .04 1.4 .53 .21 .53 .03 2.30 6.01 .1.40 .98 3.03 3.07 .18 4.83

kZK `kK GKK `kgvsk kZvsk73

24

3 2

6 1

1 0 6 5 7

kZK `kK GKK `kgvsk kZvsk4+13

9 5

0 5

8 4 5

102 cÖv_wgK MwYZ

D`vniY 6| we‡qvM Ki : 7.3 †_‡K 5.7| mgvavb : 7.3 Ñ 5.7 e¨vL¨v : 7 GKK 3 `kgvsk = 6 GKK 13 `kgvsk 1.6 5 GKK 7 `kgvsk = 5 GKK 7 `kgvsk 1 GKK 6 `kgvsk Ab¨fv‡e, DËi : 1.6| j¶ Kwi : 3 `kgvsk †_‡K 7 `kgvsk we‡qvM Kiv hvq bv| ZvB 3 `kgvs‡ki

mv‡_ 10 `kgvsk †hvM K‡i 13 `kgvsk Kiv n‡q‡Q| †h `kgvsk †ewk aiv n‡q‡Q

1 GKK wnmv‡e we‡qvR¨ 5 Gi mv‡_ †hvM K‡i Dfq msL¨v‡K mgvb ivLv n‡q‡Q|

ZvB 7 †_‡K 6 we‡qvM Kiv n‡q‡Q|

D`vniY 7| we‡qvM Ki : 3.41 †_‡K 2.54|

mgvavb : 3.41

2.54

0.87

DËi : 0.87

e¨vL¨v : 3 GKK 4 `kgvsk 1 kZvsk

2 GKK 5 `kgvsk 4 kZvsk 3 GKK 3 `kgvsk 11 kZvsk

2 GKK 5 `kgvsk 4 kZvsk




kZK `kK GKK `kgvsk kZvsk75+1

3+10 7

1 6

cÖv_wgK MwYZ 103

Ab¨fv‡e,

we‡qvM Ki (cÖ_gwU K‡i †`Lv‡bv nj) :

8.4 9.43 12.75 0.62 5.05 7.42 6.9 5.18 8.97 .49 0.56 1.39 1.5

Abykxjbx-10 (L) 1| †hvM Ki : (K) 0.5 (L) 0.7 (M) 2.36 (N) 3.64 0.8 3.05 4.63 5.87 (O) 5.02 (P) 4.76 (Q) 4.03 (R) 9.09 6.39 2.77 6.38 6.66 2| we‡qvM Ki : (K) 5.4 (L) 7.44 (M) 5.58 (N) 8.64 2.9 3.19 .07 2.46 (O) 12.1 (P) 12.1 (Q) 5.01 2.46 .01 3.15

kZK `kK GKK `kgvsk kZvsk 3

2+1 4+10

5+1 1+10

4 .0 8 7

104 cÖv_wgK MwYZ

`kwgK fMœvs‡ki ¸Y D`vniY 8| ¸Y Ki : 3.2 × 4 mgvavb : 3.2

× 4 12.8 DËi : 12.8| e¨vL¨v :

j¶ Kwi : cÖ_‡g 2 `kgvsk‡K 4 w`‡q ¸Y Kivq ¸Ydj 8 `kgvsk n‡q‡Q| 3 GKK †K 4 w`‡q ¸Y K‡i 12 GKK cvIqv †Mj| 12 GKK nj 1 `kK I 2 GKK| D`vniY 9| ¸Y Ki : 12.25 × 16 mgvavb : 12.25 × 16 73.50 122.50 196.00 DËi : 196.00| ¸Y Ki : (cÖ_gwU K‡i ‡`Iqv nj)

3.42 .4 12.12 0.03 4.04×4 ×8 ×6 ×12 ×15

13.68 D`vniY 10| ¸Y Ki : 5 × 8.3 mgvavb : 5 × 8.3 1.5 40.0 41.5 DËi : 41.5|

cybivq, 8.3×5 8.3 ×5 41.5 DËi : 41.5|

e¨vL¨v : cÖ_‡g ¸Y¨‡K ¸Y‡Ki GK‡Ki N‡ii A¼ 6 w`‡q ¸Y Kiv n‡q‡Q| c‡i ¸Y‡Ki `k‡Ki N‡ii A¼ 1 w`‡q ¸Y¨‡K ¸Y K‡i GK Ni ev‡g mwi‡q †jLv n‡q‡Q|

†hvM Kwi3.23.23.23.2

12.8`kK GKK `kgvsk kZvsk 3 2

× 4 2 8

cÖv_wgK MwYZ 105

j¶ Kwi : 5 × 8.3 Ges 8.3 × 5 GKB| myZivs 5 †K 8.3 w`‡q ¸Y Kivi cwie‡Z© 8.3 †K 5 Øviv ¸Y KivB myweavRbK| ¸Y Ki : (cÖ_gwU K‡i †`Iqv nj) (K) 6 × 7.6 = 7.6 × 6, (L) 12 × 2.3, (M) 8 × 2.05, (N) 15 × 4.2 ∴ 7.6

× 6 45.6

wP‡Î j¶ Ki : .1 × .1 = .1 Gi GK `kgvsk = 1 kZvsk = .01

Gfv‡e .3 × .5 = 15 kZvsk = .15 .6 × .7 = 42 kZvsk = .42

.2 × .1 = .2 Gi GK `kgvsk = 2 kZvsk

= .02

.3 × .2 = .6 Gi GK `kgvsk = 6 kZvsk = .06

.01

.1 .2

.02

.06.3

106 cÖv_wgK MwYZ

D`vniY 11| ¸Y Ki : .5 × .9 mgvavb : .5 e¨vL¨v : .5

× .9 × .9

.45 A_©vr .5 × .9 = 45 kZvsk = .45

j¶ Kwi : mvaviY ¸‡Yi b¨vq ¸Y Kiv n‡q‡Q| Gici ¸Y¨ I ¸Y‡Ki `kwgK

we› yi Wvbcv‡ki A‡¼i msL¨v †hvM K‡i ¸Yd‡ji Wvbw`K †_‡K †mB msL¨K

A‡¼i ci `kwgK we› y ewm‡q ¸Ydj cvIqv †M‡Q|

.9 .7 .8 .9 .7

× .4 × .5 × .8 × .5 × .8

D`vniY 12| Rvwn` evRv‡i wM‡q 62.25 UvKvi Pvj Ges 25.40 UvKvi ZiKvwi

wKb‡jb| wZwb †gvU KZ UvKv LiP Ki‡jb ?

mgvavb : Pvj 62.25 UvKv

ZiKvwi 25.40 UvKv

†gvU LiP 87.65 UvKv

DËi : 87.65 UvKv|

D`vniY 13| Abyi wbKU 72.75 UvKv I Aw`wZi wbKU 48.50 UvKv Av‡Q;

Abyi wbKU Aw`wZi †P‡q KZ UvKv †ewk Av‡Q ?

mgvavb : Abyi Av‡Q 72.75 UvKv

Aw`wZi Av‡Q 48.50 UvKv

Abyi †ewk Av‡Q 24.25 UvKv

DËi : 24.25 UvKv|

cÖv_wgK MwYZ 107

D`vniY 14| 1 wgUvi 39.37 BwÂi mgvb n‡j, 8 wgUv‡i KZ BwÂ? mgvavb : 39.37

× 8 314.96 DËi : 314.96 BwÂ| D`vniY 15| †mZv I mywg cÖ‡Z¨‡K GK Kzwo Av‡gi .5 Ask K‡i wbj| cÖ‡Z¨‡K KqwU K‡i Avg †cj ? mgvavb : GK Kzwo = 20

cÖ‡Z¨‡K †cj 20 × .5 20

× .5 10.0 = 10 DËi : 10wU| D`vniY 16| eKzj 100 UvKv wb‡q evRv‡i †M‡jb| wZwb 35.60 UvKvi gvQ

Ges 16.25 UvKvi ZiKvwi wKb‡jb| Zuvi wbKU Avi KZ UvKv iBj ?

mgvavb : gvQ 35.60 UvKv 100.00 UvKv

ZiKvwi 16.25 UvKv − 51.85 UvKv

†gvU LiP 51.85 UvKv 48.15 UvKv DËi : 48.15 UvKv| D`vniY 17| GK wjUvi `y‡ai `vg 21.75 UvKv| mvgmy 2 wjUvi `ya wK‡b yaIqvjv‡K 50 UvKv w`j| mvgmy KZ UvKv †diZ cv‡e ?

mgvavb : 2 wjUvi `y‡ai `vg (21.75 × 2) UvKv 21.75 UvKv 50.00 UvKv

× 2 − 43.50 UvKv

43.50 6.50 UvKv

DËi : 6.50 UvKv|

108 cÖv_wgK MwYZ

Abykxjbx- 10 (M)

1| ¸Ydj wbY©q Ki : (K) 0.7 × 8 (L) 10.25 × 4 (M) 0.04 × 12 (N) 1.03 × 9

(O) 2.75 × 5 (P) 5 × 7.4 (Q) 6 × 8.6 (R) 7 × 7.7

(S) 15 × 5.3 (T) 0.6 × 0.9 (U) 0.03 × 0.5 (V) 0.05 × 0.04 2| Rvgv‡ji wbKU 36.25 UvKv Ges Kvgv‡ji wbKU 52.40 UvKv Av‡Q| yB

R‡bi GK‡Î KZ UvKv Av‡Q? 3| MxZv evRv‡i wM‡q 28.70 UvKvi gvQ Ges 13.25 UvKvi ZiKvwi wKb‡jb|

wZwb evRv‡i KZ UvKv LiP Ki‡jb ? 4| mv‡n` 75.00 UvKv wb‡q evRv‡i †Mj| †m evRv‡i wM‡q 30.25 UvKvi gvQ

Ges 14.75 UvKvi ZiKvwi wKbj| Zvi Kv‡Q Avi KZ UvKv iBj ?

5| GKwU kn‡ii †g gv‡m Mo DòZv 33.6° †mjwmqvm Ges wW‡m¤i gv‡m Mo

DòZv 24.8° †mjwmqvm| wW‡m¤i gv‡mi Mo DòZv †g gv‡mi Mo DòZv †_‡K KZ Kg wQj ?

6| iywe AveŸvi KvQ †_‡K 100 UvKv Ges Av¤§vi KvQ †_‡K 50 UvKv †cj| †m 45.75 UvKv w`‡q GKwU cvRvgv Ges 80.25 UvKv w`‡q GKwU Rvgv wKbj| Zvi wbKU Avi KZ iBj ?

7| GK Kzwo wjPzi .7 Ask ixbv Ges .3 Ask iy‡kv wbj| †K KqUv K‡i wjPz †cj ?

8| GK Kzwo wW‡gi `vg 80 UvKv| gwj .8 Kzwo Ges iywg .5 Kzwo wWg wKbj| †K KqUv K‡i wWg wKbj Ges KZ UvKv `vg w`j ?

9| 1 wgUvi 39.37 BwÂi mgvb n‡j, 15 wgUv‡i KZ BwÂ ? 10| 1 Lvbv LvZvi `vg 3.45 UvKv n‡j, 24 Lvbv LvZvi `vg KZ ? 11| GKwU †ijMvwo NÈvq 36.5 wK‡jvwgUvi hvq| 3.6 NÈvq KZ wK‡jvwgUvi

hv‡e ?

cÖv_wgK MwYZ 109

GKv`k Aa¨vq

cwigvc ˆ`N©¨ cwigvc

10 wg. wg. = 1 †m. wg. ˆ`N© cwigv‡ci †gwUªK GKKvewj

10 wgwjwgUvi (wg.wg.) = 1 †mw›UwgUvi (†m.wg.) 10 †mw›UwgUvi (†m.wg.) = 1 †WwmwgUvi (†Wwmwg.) 10 †WwmwgUvi (†Wwmwg.) = 1 wgUvi (wg.) 10 wgUvi (wg.) = 1 †WKvwgUvi (†WKvwg.) 10 †WKvwgUvi (†WKvwg.) = 1 †n‡±vwgUvi (†n.wg.) 10 †n‡±vwgUvi (†n.wg.) = 1 wK‡jvwgUvi (wK.wg.)

†gwUªK GK‡Ki cvi¯cwiK m¤cK©

1000 wgwjwgUvi = 1 wgUvi 100 †mw›UwgUvi = 1 wgUvi 1000 wgUvi = 1 wK‡jvwgUvi

Lvwj Ni c �iY Ki ( yBwU K‡i †`Lv‡bv nj) : 19 †m.wg. = wg.wg. 7 wg. 9 †m.wg. = †m.wg. 52 †m.wg. = wg.wg. 5 wg. 88 †m.wg. = †m.wg. 27 †m.wg. 7 wg.wg. = wg.wg. 3 wK.wg. 822 wg. = wg. 69 †m.wg. 4 wg.wg. = wg.wg. 8 wK.wg. 450 wg. = wg.

ˆ`N© cwigv‡ci GKK : wgUvi 1 wgUvi = 100 †mw›UwgUvi

GKwU wgUvi †¯‹‡ji Qwe

190 709

110 cÖv_wgK MwYZ

D`vniY 1| 15 wK‡jvwgUvi‡K wgUv‡i cÖKvk Ki| mgvavb : 1 wK‡jvwgUvi = 1000 wgUvi|

∴ 15 wK‡jvwgUvi = 15 × 1000 wgUvi = 15000 wgUvi DËi : 15000 wgUvi| D`vniY 2| 9 wK‡jvwgUvi 225 wgUvi‡K wgUv‡i cÖKvk Ki| mgvavb : 1 wK.wg. = 1000 wg.

∴ 9 wK.wg. 225 wg.

= 9 × 1000 wg. + 225 wg. = 9000 wg. + 225 wg. = 9225 wg.

DËi : 9225 wgUvi| D`vniY 3| 6 wK‡jvwgUvi 375 wgUvi 50 †mw›UwgUvi‡K †mw›UwgUv‡i cÖKvk Ki|

mgvavb : 6 wK.wg. 375 wg. 50 †m.wg.

= 6 × 1000 wg. + 375 wg. + 50 †m.wg. ( 1000 wg. = 1 wK.wg.) = 6000 wg. + 375 wg. + 50 †m.wg. = 6375 wg. + 50 †m.wg. = 6375 × 100 †m.wg. + 50 †m.wg. ( 100 †m. wg. = 1 wg.) = 637500 †m.wg. + 50 †m.wg. = 637550 †m.wg.

DËi : 637550 †mw›UwgUvi| D`vniY 4| 3 wK‡jvwgUvi 215 wgUvi 90 †mw›UwgUvi 6 wgwjwgUvi‡K wgwjwgUv‡i cÖKvk Ki| mgvavb : 3 wK.wg. 215 wg. 90 †m.wg. 6 wg.wg.

= 3 × 1000 wg. + 215 wg. + 90 †m.wg. + 6 wg.wg. (∴1000 wg. = 1 wK. wg.)

= 3000 wg. + 215 wg. + 90 †m.wg. + 6 wg.wg. = 3215 wg. + 90 †m.wg. + 6 wg.wg.

cÖv_wgK MwYZ 111

= 3215 × 100 †m.wg. + 90 †m.wg. + 6 wg.wg. ( 100 †m. wg. = 1 wg.) = 321500 †m.wg. + 90 †m.wg. + 6 wg.wg.

= 321590 †m.wg. + 6 wg.wg.

= 321590 × 10 wg.wg. + 6 wg. wg.

= 3215900 wg.wg. + 6 wg.wg. ( 10 wg.wg. = 1 †m. wg.)

= 3215906 wg.wg.

DËi : 3215906 wgwjwgUvi|

D`vniY 5| 980 wgUvi 56 †mw›UwgUvi 5 wgwjwgUvi‡K wgwjwgUv‡i cÖKvk Ki|

mgvavb : 980 wg. 56 †m.wg. 5 wg.wg.

= 980 × 100 †m.wg. + 56 †m.wg. + 5 wg.wg. ( 100 †m.wg.=1 wg.)

= 98000 †m.wg. + 56 †m.wg. + 5 wg.wg.

= 98056 †m.wg. + 5 wg.wg.

= 98056 × 10 wg.wg. + 5 wg.wg. ( 10 wg.wg. = 1 †m.wg.)

= 980560 wg.wg. + 5 wg.wg.

= 980565 wg.wg.

DËi : 980565 wgwjwgUvi|

ˆ`N©¨ cwigv‡ci e„wUk GKKvewj †gwUªK I e„wUk cwigv‡ci m¤cK©

12 BwÂ = 1 dzU 3 dzU = 1 MR 1760 MR = 1 gvBj

1 wgUvi = 39.37 BwÂ (cÖvq) 1 BwÂ = 2.54 †m.wg. (cÖvq) 1 gvBj = 1.61 wK.wg. (cÖvq) 1 wK.wg. = 0.62 gvBj (cÖvq)

112 cÖv_wgK MwYZ

IRb cwigvc

wewfbœ evULvivi Qwe IRb cwigv‡ci †gwUªK GKKvewj

10 wgwjMÖvg (wg.MÖv.) = 1 †mw›UMÖvg (†m.MÖv.) 10 †mw›UMÖvg (†m.MÖv.) = 1 †WwmMÖvg (†WwmMÖv.) 10 †WwmMÖvg (†WwmMÖv.) = 1 MÖvg (MÖv.) 10 MÖvg (MÖv.) = 1 †WKvMÖvg (†WKvMÖv.) 10 †WKvMÖvg (†W.MÖv.) = 1 †n‡±vMÖvg (†n. MÖv.) 10 †n‡±vMÖvg (†n. MÖv.) = 1 wK‡jvMÖvg (wK.MÖvg ev †KwR)

ˆ`N©¨ cwigv‡ci †gwUªK GKKvewj‡Z ÔwgUviÕ Gi cwie‡Z© ÔMÖvgÕ emv‡j IRb cwigv‡ci †gwUªK GKKvewj cvIqv hvq|

†gwUªK GK‡Ki cvi¯cwiK m¤cK© 1000 wgwjMÖvg = 1 MÖvg 100 †mw›UMÖvg = 1 MÖvg 1000 MÖvg = 1 wK‡jvMÖvg

Lvwj Ni c �iY Ki (GKwU K‡i †`Lv‡bv nj) : 21 †m.MÖv. = wg.MÖv. 15 MÖvg = †m.MÖv. 49 †m.MÖv. = wg.MÖv. 25 †KwR = MÖvg 62 †m.MÖv. 5 wg.MÖv. = wg.MÖv. 3 wK.MÖv. 70 MÖv. = MÖvg 79 †m.MÖv. 9 wg.MÖv. = wg.MÖv. 4 †KwR 725 MÖvg = MÖvg

IRb cwigv‡ci GKK : MÖvg 1000 MÖvg = 1 wK‡jvMÖvg ev 1

1 †KwR 500 MÖvg 200 MÖvg 100 MÖvg 50 MÖvg

cÖv_wgK MwYZ 113

D`vniY 6| 27 †mw›UMÖvg 9 wgwjMÖvg‡K wgwjMÖv‡g cÖKvk Ki| mgvavb : 27 †mw›UMÖvg 9 wgwjMÖvg

= 27 × 10 wgwjMÖvg + 9 wgwjMÖvg ( 1 †mw›UMÖvg = 10 wgwjMÖvg) = 270 wgwjMÖvg + 9 wgwjMÖvg = 279 wgwjMÖvg DËi : 279 wgwjMÖvg| D`vniY 7| 325 MÖvg 4 †mw›UMÖvg 5 wgwjMÖvg‡K wgwjMÖv‡g cÖKvk Ki|

mgvavb : 325 MÖvg 4 †mw›UMÖvg 5 wgwjMÖvg

= 325 × 10 †WwmMÖvg + 4 †mw›UMÖvg + 5 wgwjMÖvg ( 10 †WwmMÖvg = 1 MÖvg) = 3250 †WwmMÖvg + 4 †mw›UMÖvg + 5 wgwjMÖvg

= 3250 × 10 †mw›UMÖvg + 4 †mw›UMÖvg + 5 wgwjMÖvg ( 10 †m. MÖv. = 1 †WwmMÖvg) = 32500 †mw›UMÖvg + 4 †mw›UMÖvg + 5 wgwjMÖvg = 32504 †mw›UMÖvg + 5 wgwjMÖvg

= 32504 × 10 wgwjMÖvg + 5 wgwjMÖvg ( 10 wg. MÖvg = 1 †mw›UMÖvg) = 325040 wgwjMÖvg + 5 wgwjMÖvg = 325045 wgwjMÖvg DËi : 325045 wgwjMÖvg| D`vniY 8| 3 †KwR 578 MÖvg 9 †WwmMÖvg †K †mw›UMÖv‡g cÖKvk Ki|

mgvavb : 3 †KwR 578 MÖvg 9 †WwmMÖvg

= 3 × 1000 MÖvg + 578 MÖvg + 9 †WwmMÖvg ( 1000 MÖvg = 1 †KwR) = 3000 MÖvg + 578 MÖvg + 9 †WwmMÖvg = 3578 MÖvg + 9 †WwmMÖvg

= 3578 × 10 †WwmMÖvg + 9 †WwmMÖvg ( 10 †WwmMÖvg = 1 MÖvg)

114 cÖv_wgK MwYZ

= 35780 †WwmMÖvg + 9 †WwmMÖvg = 35789 †WwmMÖvg

= 35789 × 10 †mw›UMÖvg ( 10 †m. MÖvg = 1 †WwmMÖvg) = 357890 †mw›UMÖvg DËi : 357890 †mw›UMÖvg|

D`vniY 9| 13 †KwR 825 MÖvg‡K MÖv‡g cÖKvk Ki| mgvavb : 13 †KwR 825 MÖvg

= 13 × 1000 MÖvg + 825 MÖvg ( 1000 MÖvg = 1 †KwR) = 13000 MÖvg + 825 MÖvg = 13825 MÖvg

DËi : 13825 MÖvg|

IRb cwigv‡ci †`kxq GKKvewj †gwUªK I †`kxq cwigv‡ci m¤cK© 5 †Zvjv = 1 QUvK 16 QUvK = 1 †mi 40 †mi = 1 gY

1 †Zvjv = 11.63 MÖvg (cÖvq) 1 gY = 37.32 †KwR (cÖvq) 1 †KwR = 1 †mi 1 QUvK 1 †Zvjv (cÖvq)

Zij c`v‡_©i AvqZb cwigvc

wjUvi gvcwb

Zij c v‡_©i AvqZb cwigv‡ci GKK : wjUvi 1000 wgwjwjUvi = 1wjUvi

cÖv_wgK MwYZ 115

Zij c`v‡_©i AvqZb cwigv‡ci †gwUªK GKKvewj

10 wgwjwjUvi (wg.wj.) = 1 †mw›UwjUvi(†m.wj.) 10 †mw›UwjUvi (†m.wj.) = 1 †WwmwjUvi (†Wwm. wj.) 10 †WwmwjUvi (†Wwmwj. = 1 wjUvi (wj.) 10 wjUvi (wj.) = 1 †WKvwjUvi (†WKvwj.) 10 †WKvwjUvi (†W.wj.) = 1 †n‡±vwjUvi 10 †n‡±vwjUvi (†n.wj.) = 1 wK‡jvwjUvi (wK.wj.)

ˆ`N© cwigv‡ci †gwUªK GKKvewj‡Z ÔwgUviÕ Gi cwie‡Z© ÔwjUviÕ emv‡j Zij c`v‡_©i AvqZb cwigv‡ci †gwUªK GKKvewj cvIqv hv‡e| D`vniY 10| 7 wjUvi‡K wgwjwjUv‡i cÖKvk Ki| mgvavb : 7 wjUvi = 7 × 1000 wgwjwjUvi ( 1000 wgwjwjUvi = 1 wjUvi) = 7000 wgwjwjUvi| DËi : 7000 wgwjwjUvi| D`vniY 11| 25 wjUvi 350 wgwjwjUvi‡K wgwjwjUv‡i cÖKvk Ki| mgvavb : 25 wjUvi 350 wgwjwjUvi = 25 × 1000 wgwjwjUvi + 350 wgwjwjUvi ( 1000 wgwjwjUvi = 1 wjUvi) = 25000 wgwjwjUvi + 350 wgwjwjUvi = 25350 wgwjwjUvi DËi : 25350 wgwjwjUvi| D`vniY 12| 75 wjUvi 9 †mw›UwjUvi‡K wgwjwjUv‡i cÖKvk Ki| mgvavb : 75 wjUvi 9 †mw›UwjUvi = 75 × 10 †WwmwjUvi + 9 †mw›UwjUvi = 750 †WwmwjUvi + 9 †mw›UwjUvi = 750 × 10 †mw›UwjUvi + 9 †mw›UwjUvi = 7500 †mw›UwjUvi + 9 †mw›UwjUvi = 7509 †mw›UwjUvi = 7509 × 10 wgwjwjUvi = 75090 wgwjwjUvi DËi : 75090 wgwjwjUvi|

116 cÖv_wgK MwYZ

g‡b ivLvi myweav‡_© ˆ`N©¨, IRb I Zij c`v_© cwigv‡ci †gwUªK GKKmg �n wb‡Pi Q‡K †`Lv‡bv nj :

Abykxjbx- 11 (K)

1| wgwjwgUv‡i cÖKvk Ki :

(K) 75 †mw›UwgUvi 7 wgwjwgUvi (L) 62 †mw›UwgUvi 3 wgwjwgUvi

(M) 94 †mw›UwgUvi (N) 16 wgUvi 88 †mw›UwgUvi

(O) 11 wgUvi (P) 13 wgUvi 6 wgwjwgUvi|

2| †mw›UwgUv‡i cÖKvk Ki :

(K) 250 wgUvi 75 †mw›UwgUvi (L) 345 wgUvi

(M) 95 wgUvi 36 †mw›UwgUvi (N) 2 wK‡jvwgUvi 12 †mw›UwgUvi

(O) 8 wK‡jvwgUvi 125 wgUvi 60 †mw›UwgUvi (P) 15 wK‡jvwgUvi 320 wgUvi 90 †mw›UwgUvi

3| wgUv‡i cÖKvk Ki :

(K) 70 wK‡jvwgUvi (L) 16 wK‡jvwgUvi 425 wgUvi

(M) 9 wK‡jvwgUvi 275 wgUvi (N) 98 wK‡jvwgUvi

(O) 21 wK‡jvwgUvi 5 wgUvi (P) 39 wK‡jvwgUvi 155 wgUvi|

cÖv_wgK MwYZ 117

4| wgwjMÖv‡g cÖKvk Ki :

(K) 45 MÖvg (L) 18 MÖvg 9 †mw›UMÖvg

(M) 32 MÖvg 21 †mw›UMÖvg 3 wgwjMÖvg (N) 65 MÖvg 8 wgwjMÖvg

(O) 2 wK‡jvMÖvg 25 MÖvg (P) 5 wK‡jvMÖvg 30 MÖvg 12 †mw›UwgUvi|

5| †mw›UMÖv‡g cÖKvk Ki :

(K) 236 MÖvg 25 †mw›UMÖvg (L) 52 MÖvg

(M) 122 MÖvg 19 †mw›UMÖvg (N) 72 MÖvg 8 †WwmMÖvg|

6| †WwmMÖv‡g cÖKvk Ki :

(K) 19 MÖvg 5 †WwmMÖvg (L) 2 †KwR 10 MÖvg 3 †WwmMÖvg

(M) 3 ‡KwR (N) 5 †KwR 16 MÖvg |

7| MÖv‡g cÖKvk Ki :

(K) 25 †KwR (L) 12 †KwR 160 MÖvg

(M) 8 †KwR 990 MÖvg (N) 35 †KwR 248 MÖvg|

8| wgwjwjUv‡i cÖKvk Ki :

(K) 6 †WwmwjUvi 4 †mw›UwjUvi (L) 9 †WwmwjUvi 5 †mw›UwjUvi

(M) 68 wjUvi (N) 19 wjUvi 250 wgwjwjUvi|

(O) 82 wjUvi (P) 27 wjUvi 895 wgwjwjUvi|

cÖv_wgK MwYZ 119

AvqZ‡¶Î I eM©‡¶‡Îi †¶Îdj :

Qwe‡Z K L M N GKwU AvqZ‡¶Î| Gi ˆ`N© 4 †m.wg. Ges cÖ ’ 2 †m.wg.| ˆ`N© I cÖ _ eivei 1 †m.wg. cici †iLv †U‡b AvqZ †¶ÎwU‡K wef³ Kiv n‡q‡Q| G‡Z • AvqZ‡¶ÎwU †gvU 2 × 4 = 8wU fv‡M wef³ n‡q‡Q| • cÖwZwU †QvU †¶‡Îi ˆ`N© 1 †m.wg. Ges cÖ _ 1 †m.wg. • cªwZwU †QvU †¶Î GKwU eM©‡¶Î hvi †¶Îdj 1 eM© †m.wg. myZivs AvqZ‡¶ÎwUi †¶Îdj = 8 wU †QvU eM©‡¶‡Îi †¶Îdj = (8 × 1) eM© †m.wg. = (4 × 2 × 1) eM© †m.wg. = (4 × 2) eM© †m.wg. = 8 eM© †m.wg. j¶ Kwi : ˆ`‡N© i cwigvc‡K cÖ‡¯_i cwigvc w`‡q ¸Y K‡i AvqZ‡¶‡Îi †¶Îd‡ji cwigvc cvIqv †M‡Q|

†¶Îd‡ji cwigvc = ˆ`‡N© i cwigvc × cÖ‡¯_i cwigvc

ms‡¶‡c, †¶Îdj = ˆ`N© × cÖ _ ªóe¨ : †¶Îdj wbY©‡qi mgq hw` ˆ`N© I cÖ _ wfbœ GK‡K _v‡K Z‡e ˆ`N© I

cÖ‡¯_i cwigvc‡K GKB GK‡K cwiYZ K‡i wb‡Z nq Ges †¶Îdj eM© GK‡K cÖKvk Ki‡Z nq| D vniY 1| GKwU AvqZvKvi †¶‡Îi `N© 12 †m.wg. I cÖ _ 8 †m.wg.| Gi †¶Îdj KZ? mgvavb : GLv‡b, AvqZvKvi †¶‡Îi ˆ`N© 12 †m.wg. Ges cÖ _ 8 †m.wg. Rvbv Av‡Q, †¶Îdj = ˆ`N©¨ × cÖ _ ∴ AvqZvKvi †¶‡Îi †¶Îdj = (12 × 8) eM© †m.wg

= 96 eM© †m.wg. DËi : 96 eM© †m.wg.|

LK 4 †mwg

1 †mwg

1 †mwg

2 †mwg

1 †mwg 1 †mwg 1 †mwg 1 †mwg MN

120 cÖv_wgK MwYZ

D`vniY 2| GKwU N‡ii ˆ`N© 19 wgUvi Ges cÖ _ 9 wgUvi| NiwUi †g‡Si †¶Îdj KZ? mgvavb : GLv‡b, NiwUi ˆ`N© 19 wgUvi Ges cÖ¯_ 9 wgUvi Rvbv Av‡Q, †¶Îdj = ˆ`N©¨ × cÖ _ ∴ NiwUi †g‡Si †¶Îdj = (19 × 9) eM©wgUvi = 171 eM©wgUvi| DËi : 171 eM©wgUvi| D`vniY 3| GKwU AvqZvKvi evMv‡bi ˆ`N© 45 MR Ges cÖ _ 25 MR| Gi †¶Îdj KZ ? mgvavb : GLv‡b, evMv‡bi ˆ`N© 45 MR Ges cÖ _ 25 MR Rvbv Av‡Q, †¶Îdj = ˆ`N© × cÖ _ ∴ evMv‡bi †¶Îdj = (45 × 25) eM©MR = 1125 eM©MR DËi : 1125 eM©MR| D`vniY 4| GKwU AvqZvKvi Rwgi †¶Îdj 5760 eM©wgUvi| Gi ˆ`N© 96 wgUvi n‡j cÖ¯_ KZ? mgvavb : GLv‡b, †¶Îdj = 5760 eM©wgUvi Ges ˆ`N© = 96 wgUvi Rvbv Av‡Q, †¶Îdj = ˆ`N©¨ × cÖ _ ∴ cÖ¯_ = †¶Îdj ˆ`N©¨ ∴ RwgwUi cÖ _ = (5760 96) wgUvi = 60 wgUvi DËi : 60 wgUvi| D`vniY 6| GKwU eM©vKvi †¶‡Îi ˆ`N© 25 wgUvi| Gi †¶Îdj KZ? mgvavb : Rvbv Av‡Q, eM©‡¶‡Îi ˆ`N© I cÖ _ mgvb| GLv‡b, eM©vKvi †¶‡Îi ˆ`N© = 25 wgUvi ∴ eM©vKvi †¶‡Îi cÖ _ = 25 wgUvi ∴ eM©vKvi †¶‡Îi †¶Îdj = (25 × 25) eM©wgUvi = 625 eM©wgUvi DËi : 625 eM©wgUvi|

cÖv_wgK MwYZ 121

Abykxjbx- 11 (L) 1| AvqZvKvi †¶Î¸‡jvi †¶Îdj wbY©q Ki :

(K) ˆ`N© 30 †m.wg. I cÖ _ 22 †m.wg. (L) ˆ`N© 56 wgUvi I cÖ¯_ 48 wgUvi

(M) ˆ`N© 2 wgUvi I cÖ _ 55 †m.wg. (N) ˆ`N© 4 wgUvi I cÖ¯_ 90 †m.wg.

(O) ˆ`N© 38 MR I cÖ _ 27 MR (P) ˆ`N© 62 dzU I cÖ _ 43 dzU|

2| eM©‡¶‡Îi evûi ˆ`N© †`Iqv Av‡Q| †¶Îdj wbY©q Ki|

(K) 72 †m.wg. (L) 29 wgUvi (M) 3 MR

3| GKwU AvqZvKvi Rwgi ˆ`N© 75 wgUvi| Gi †¶Îdj 1725 eM©wgUvi n‡j,

cÖ _ KZ ?

4| GKwU AvqZvKvi gv‡Vi †¶Îdj 1806 eM©wgUvi| Gi ˆ`N© 86 wgUvi n‡j,

cÖ ’ KZ ?

5| GKwU AvqZvKvi †¶‡Îi †¶Îdj 1092 eM©MR| Gi cÖ _ 13 MR n‡j,

ˆ`N© KZ ?

122 cÖv_wgK MwYZ

Øv`k Aa¨vq

mgq Rvbyqvwi 2005

kwb iwe †mvg g½j eya e„n¯cwZ ïµ1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

2005 mv‡ji Rvbyqvwi gv‡mi 1 ZvwiL kwbevi| Rvbyqvwi gv‡mi 31 ZvwiL †mvgevi| Rvbyqvwi gv‡mi iweev‡ii ZvwiL¸‡jv n‡”Q 2, 9, 16, 23 I 30| Rvbyqvwi gv‡m eyaevi Av‡Q 4wU| Rvbyqvwi gv‡mi †kl ïµevi 28 ZvwiL| Rvbyqvwi gv‡mi 3, 10, 17, 24 I 31 ZvwiL †mvgevi| Rvbyqvwi gv‡mi ïµev‡ii ZvwiL¸‡jv 7 Øviv wefvR¨ Ges †m¸‡jv n‡”Q 7, 14, 21, 28| LvwjNi c �iY Ki (GKwU K‡i †`Lv‡bv nj) : 2005 mv‡ji Rvbyqvwi gv‡mi 2 ZvwiL evi|

2005 mv‡ji Rvbyqvwi gv‡mi 18 ZvwiL evi|

2005 mv‡ji Rvbyqvwi gv‡mi Z…Zxq †mvgevi ZvwiL|

2005 mv‡ji Rvbyqvwi gv‡m cÖ_g eyaevi ZvwiL|

2005 mv‡ji Rvbyqvwi gv‡m kwbevi Av‡Q wU|

2005 mv‡ji Rvbyqvwi gv‡mi 6, 13, 20, 27 ZvwiL¸‡jv n‡”Q evi|

iwe

cÖv_wgK MwYZ 123

fv ª 1411 kwb iwe †mvg g½j eya e„n¯cwZ ïµ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

1411 mv‡ji fv ª gv‡mi 30 ZvwiL g½jevi|

1411 mv‡ji fv ª gv‡m iweevi Av‡Q 4wU|

1411 mv‡ji fv ª gv‡mi †kl e„n¯cwZevi 25 ZvwiL|

1411 mv‡ji fv ª gv‡mi PZz_© eyaevi 24 ZvwiL|

1411 mv‡ji fv ª gv‡mi 6, 13, 20, 27 ZvwiL¸‡jv n‡”Q kwbevi|

1411 mv‡ji fv ª gv‡mi 2 ZvwiL g½jevi|

1411 mv‡ji fv ª gv‡mi 5, 12, 19, 26 ZvwiL ïµevi|

LvwjNi c �iY Ki (GKwU K‡i †`Lv‡bv nj) :

1411 mv‡ji fv ª gv‡mi cÂg 29 ZvwiL|

1411 mv‡ji fv ª gv‡mi 14 ZvwiL evi|

1411 mv‡ji fv ª gv‡mi †kl ïµevi ZvwiL|

1411 mv‡ji fv ª gv‡mi 7, 14, 21, 28 ZvwiL¸‡jv Øviv wefvR¨|

1411 mv‡ji fv ª gv‡mi wØZxq kwbevi ZvwiL|

1411 mv‡ji fv ª gv‡m g½jevi Av‡Q wU|

1411 mv‡ji fv ª gv‡mi cÖ_g e„n®cwZevi ZvwiL|

†mvgevi

124 cÖv_wgK MwYZ

Bs‡iwR evi gv‡mi bvg I w`‡bi msL¨v gvm w`‡bi msL¨v gvm w`‡bi msL¨v

Rvbyqvwi 31 RyjvB 31 †deªyqvwi 28 AvM÷ 31 gvP© 31 †m‡Þ¤^i 30 GwcÖj 30 A‡±vei 31 †g 31 b‡f¤i 30 Ryb 30 wW‡m¤i 31

2004 mv‡ji †deªyqvwi gvm wQj 29 w`‡b| 2004 msL¨vwU 4 Øviv wefvR¨| GwU GKwU wjc Bqvi| wjc Bqvi n‡j †deªyqvwi gvm 29 w`‡b nq| 2005 msL¨vwU 4 Øviv wefvR¨ bq| ZvB 2005 mvj wjc Bqvi bq| wjc Bqvi bv n‡j †deªyqvwi gvm 28 w`‡b nq| ZvB 2005 mv‡ji †deªyqvwi gvm 28 w`‡b| evsjv evi gv‡mi bvg I w`‡bi msL¨v

gvm w`‡bi msL¨v gvm w`‡bi msL¨v ˆekvL 31 KvwZ©K 30 ˆR¨ô 31 AMÖnvqY 30 Avlvp 31 †cŠl 30 kªveY 31 gvN 30 fv ª 31 dvêyb 30

Avwk¦b 30 ˆPÎ 30

mgq cwigv‡ci GKKvewj :

60 †m‡KÛ = 1 wgwbU 60 wgwbU = 1 NÈv 24 NÈv = 1 w`b 7 w`b = 1 mßvn

30 w`b = 1 gvm 12 gvm = 1 eQi 12 eQi = 1 hyM 100 eQi = 1 kZvãx

365 w`b = 1 eQi

cÖv_wgK MwYZ 125

D`vniY 1| 6 mßvn 5 w`b‡K NÈvq cÖKvk Ki|

mgvavb : 6 mßvn 5 w`b

= 6 × 7 w`b + 5 w`b ( 7 w`b = 1 mßvn)

= 42 w`b + 5 w`b

= 47 w`b

= 47 × 24 NÈv ( 24 NÈv = 1 w`b)

= 1128 NÈv

DËi : 1128 NÈv|

D`vniY 2| 17 mßvn 3 w`b‡K NÈvq cÖKvk Ki|

mgvavb : 17 mßvn 3 w`b

= 17 × 7 w`b + 3 w`b ( 7 w`b = 1 mßvn)

= 119 w`b + 3 w`b

= 122 w`b

= 122 × 24 NÈv ( 24 NÈv = 1 w`b)

= 2928 NÈv

DËi : 2928 NÈv |

D`vniY 3| 12 w`b 22 NÈv‡K wgwb‡U cÖKvk Ki|

mgvavb : 12 w`b 22 NÈv

= 12 × 24 NÈv + 22 NÈv ( 24 NÈv = 1 w`b)

= 288 NÈv + 22 NÈv

= 310 NÈv

= 310 × 60 wgwbU ( 60 wgwbU = 1 NÈv)

= 18600 wgwbU

DËi : 18600 wgwbU|

126 cÖv_wgK MwYZ

D`vniY 4| 4 w`b 23 NÈv‡K wgwb‡U cÖKvk Ki| mgvavb : 4 w`b 23 NÈv = 4 × 24 NÈv + 23 NÈv (∴24 NÈv = 1 w`b) = 96 NÈv + 23 NÈv = 119 NÈv = 119 × 60 wgwbU (∴ 60 wgwbU = 1 NÈv) = 7140 wgwbU DËi : 7140 wgwbU| D vniY 5| 19 w b 15 NÈv 50 wgwbU Gi mv‡_ 12 w b 17 NÈv 35 wgwbU †hvM Ki| mgvavb : w`b NÈv wgwbU 19 15 50 12 17 35 32 9 25 DËi : 32 w`b 9 NÈv 25 wgwbU| j¶ Kwi : wgwb‡Ui N‡i 50+35 = 85| 60 wgwbU = 1 NÈv; myZivs 85 wgwbU = 1 NÈv 25 wgwbU| Kv‡RB wgwb‡Ui N‡i 25 e‡m‡Q| 1 NÈv NÈvi N‡i †hvM n‡q †mLv‡b 15+17+1 = 33 n‡q‡Q| 24 NÈv = 1 w`b| myZivs 33 NÈv = 1 w`b 9 NÈv| ZvB NÈvi N‡i 9 emv‡bv n‡q‡Q| w`‡bi N‡i 1 †hvM n‡q 19 + 12 + 1 = 32 w`b n‡q‡Q| D`vniY 6| †hvM Ki : 28 mßvn 5 w`b 16 NÈv 48 wgwbU Ges 16 mßvn 6 w`b 12 NÈv 25 wgwbU| mgvavb : mßvn w`b NÈv wgwbU 28 5 16 48 16 6 12 25 45 5 5 13 DËi : 45 mßvn 5 w`b 5 NÈv 13 wgwbU| D vniY 7| 32 w b 9 NÈv 47 wgwbU †_‡K 15 w b 22 NÈv 43 wgwbU we‡qvM Ki| mgvavb : w`b NÈv wgwbU w`b NÈv wgwbU 32 9 47 31 33 47 15 22 43 15 22 43 16 11 4 DËi : 16 w`b 11 NÈv 4 wgwbU|

cÖv_wgK MwYZ 127

j¶ Kwi : NÈvi N‡i 9 †_‡K 22 we‡qvM Kiv hvq bv| GRb¨ 32 w`b †_‡K 1 w`b ev 24 NÈv wb‡q 24 + 9 = 33 NÈv Kiv n‡q‡Q| d‡j NÈvi N‡i 33 NÈv n‡q‡Q Ges w`‡bi N‡i 31 w`b n‡q‡Q| D`vniY 8| 45 mßvn 4 w`b 10 NÈv 28 wgwbU †_‡K 29 mßvn 5 w`b 6 NÈv 53 wgwbU we‡qvM Ki| mgvavb : mßvn w`b NÈv wgwbU mßvn w`b NÈv wgwbU 45 4 10 28 44 11 9 88 29 5 6 53 29 5 6 53 15 6 3 35

DËi : 15 mßvn 6 w`b 3 NÈv 35 wgwbU| weKí mgvavb : mßvn w`b NÈv wgwbU 45 4+7 10 28+60 29+1 5 6+1 53 15 6 3 35 e¨vL¨v : 28 wgwbU †_‡K 53 wgwbU we‡qvM Kiv hvq bv| ZvB 28 wgwb‡Ui mv‡_ 1 NÈv ev 60 wgwbU †hvM Kiv n‡q‡Q| †h‡nZz 1 NÈv †ewk aiv n‡q‡Q| †m‡nZz we‡qv‡Rï 6 NÈvi mv‡_ 1 NÈv †hvM Kiv n‡q‡Q| Abyiƒcfv‡e, 4 w`b †_‡K 5 w`b we‡qvM Kiv hvq bv, ZvB 4 w`‡bi mv‡_ 1 mßvn ev 7 w`b †hvM Kiv n‡q‡Q Ges we‡qv‡Rï 29 mßv‡ni mv‡_ 1 mßvn †hvM K‡i we‡qv‡Mi KvR m¤cbœ Kiv n‡q‡Q| D`vniY 9| Kbv 3 w`b 16 NÈv 30 wgwbU Ges g �mv 4 w`b 21 NÈv 40 wgwbU †Uª‡b ågY Kij| Zv‡ì yB R‡bi åg‡Y †gvU KZ mgq †j‡MwQj ? mgvavb : w`b NÈv wgwbU 3 16 30 4 21 40 8 14 10 Zv‡ì åg‡Y †gvU mgq †j‡MwQj 8 w`b 14 NÈv 10 wgwbU| DËi : 8 w`b 14 NÈv 10 wgwbU|

cÖv_wgK MwYZ 129

Î‡qv`k Aa¨vq

DcvË msMÖn I web¨¯�KiY eKzj Zvi wbR¯ cwi‡ek †_‡K wbæwjwLZ Z_¨¸‡jv msMÖn K‡i‡Q|

Ni †jvKmsL¨v MvQcvjv M„ncvwjZ cï

wbR evwo 3 20 1221 16

cv‡ki cÖ_g evwo 5 35 1420 30

cv‡ki wØZxq evwo 4 30 1340 25

cv‡ki Z…Zxq evwo 4 32 1182 30

†ngv‡qZcyi miKvwi cÖv_wgK we`¨vj‡qi wb‡P ewY©Z Z_¨¸‡jv eKzj msMÖn K‡i‡Q;

cÖ_g †kªYx wØZxq †kªYx Z…Zxq †kªYx PZz_© †kªYx cÂg †kªYx

(K) wk¶v_©x 100 80 60 50 40

(L) †eÂ 25 †Rvov 20 †Rvov 15 †Rvov 13 †Rvov 10 †Rvov

(M) Rvbvjv 5wU 4wU 4wU 4wU 3wU

U¨vwj :

D`vniY 1| wb‡P 50 Rb wk¶v_©xi MwY‡Z cÖvß b¤i †`Iqv Av‡Q| cÖvß

b¤iavix wk¶v_©xi msL¨v U¨vwj wP‡ýi mvnv‡h¨ cÖKvk Ki|

b¤i wk¶v_©x msL¨v 52 10 57 8 60 11 62 12 65 9

130 cÖv_wgK MwYZ

mgvavb : b¤i wk¶v_©x msL¨v wb‡Y©q U¨vwj wPý 52 10 57 8 60 11 62 12 65 9

j¶ Kwi : • ⏐ wPýwU U¨vwj| • 52 b¤i †c‡q‡Q 10 Rb wk¶v_©x| 1 Rb wk¶v_©xi Rb¨ GKwU

U¨vwj ⏐ wPý †Ìqv n‡q‡Q| Gfv‡e PviwU U¨vwj wPý †Ìqvi ci cÂg U¨vwj wPýwU Avjv`vfv‡e bv w`‡q Av‡Mi PviwU wPý Ry‡o AvovAvwo fv‡e †Ìqv n‡q‡Q| Gici GKUy duvK w`‡q Av‡iv cuvPwU wPý GKBfv‡e †Ìqv n‡q‡Q|

D`vniY 2| 20 Rb wk¶v_©xi evwoi euvk Sv‡o wb‡æv³ msL¨K euvk Av‡Q (euv‡ki msL¨v wb‡P †Ìqv nj)| DcvËmg�‡ni mviYx ˆZwi Ki| 702, 712, 727, 717, 737, 707, 744, 734, 726, 730, 739, 728, 714, 709, 719, 747, 729, 704, 750, 715|

mgvavb : euv‡ki msL¨vi †kªYx wefvM U¨vwj wk¶v_©x msL¨v

701 -705 ⏐⏐ 2 706 -710 ⏐⏐ 2 711 -715 ⏐⏐⏐ 3 716 -720 ⏐⏐ 2 721 -725 0 726 -730 5 731 -735 ⏐ 1 736 -740 ⏐⏐ 2 741 -745 ⏐ 1 746 -750 ⏐⏐ 2

cÖv_wgK MwYZ 131

j¶ Kwi : • euv‡ki msL¨v¸‡jv‡K 5 Gi eëwa‡Z †kªYx wefvM Kiv n‡q‡Q|

• GKB eëwa‡Z KqRb wk¶v_©xi euvk Av‡Q Zv U¨vwj wPý w`‡q cÖKvk Kiv n‡q‡Q|

• U¨vwj wPý Abyhvqx wk¶v_©x msL¨v †jLv n‡q‡Q|

Abykxjbx- 13 1| †Zvgv‡ì wbR cwiev‡ii RbmsL¨v, M„ncvwjZ cïi msL¨v, M„ncvwjZ cvwLi

msL¨v, MvQcvjv (Avg, Rvg, KuvVvj, bvwi‡Kj, mycvwi, Zvj I †LRyi MvQ) Gi Z_¨ (msL¨v) mieivn Ki|

2| wb‡Pi Dcv‡Ëi mviYx ˆZwi Ki : 20 Rb wk¶v_©xi evwl©K cix¶vq MwYZ wel‡q cÖvß b¤i (†ivj b¤i Abyhvqx)

63 62 41 52 45 44 55 32 58 38 65 69 29 48 57 68 49 60 40 47

3| wb‡P 50 Rb wk¶v_©xi evsjv wel‡q cÖvß b¤i †Ìqv Av‡Q| U¨vwj wPý w`‡q cÖKvk Ki :

b¤i wk¶v_©xi msL¨v50 9 57 7 62 14 65 12 70 8

132 cÖv_wgK MwYZ

4| wb‡P cuvPwU †kªYxi †eÂ (wmU I nvB)- Gi msL¨v †`Iqv nj| †e‡Âi msL¨v U¨vwj wPý w`‡q cÖKvk Ki|

†kªYx †e‡Âi msL¨vcÖ_g †kªYx 25 †Rvov wØZxq †kªYx 20 †Rvov Z…Zxq †kªYx 15 †Rvov PZz_© †kªYx 13 †Rvov cÂg †kªYx 10 †Rvov

5| wb‡Pi DcvË¸‡jvi msL¨v U¨vwj wPý w`‡q cÖKvk Ki| 20 Rb wk¶v_©xi lvb¥vwmK cix¶vq MwYZ wel‡q cÖvß b¤i : 65 72 52 67 79 52 35 62 75 70 42 52 68 75 45 69 48 74 50 70

cÖv_wgK MwYZ 133

PZz`©k Aa¨vq

†jL wPÎ cjvk miKvwi cÖv_wgK we`¨vj‡qi wewfbœ †kªYxi QvÎx msL¨v wb‡P †`Iqv nj|

†kªYx QvÎx msL¨vcÖ_g †kªYx 100 wØZxq †kªYx 90 Z…Zxq †kªYx 70 PZz_© †kªYx 60 cÂg †kªYx 50

Q‡Ki Z_¨‡K wP‡Îi mvnv‡h¨ cÖKvk Kiv hvq|

(cÖwZ N‡ii gvb 10) wPÎwU cjvk miKvwi cÖv_wgK we`¨vj‡qi wewfbœ †kªYxi QvÎx msL¨vi ¯�¤¢¢‡jL|

100

90

80

70

60

50

40

30

20

10

M

K L1g †kªYx 2q †kªYx 3q †kªYx 4_© †kªYx 5g †kªYx

134 cÖv_wgK MwYZ

we`¨vjqwUi wewfbœ †kªYxi QvÎxi DcvË¸‡jv wbæiƒc wP‡Î cÖKvk Kiv hvq : cÖ_g †kªYx wØZxq †kªYx

Z…Zxq †kªYx PZz_© †kªYx cÂg †kªYx

(GKwU cÖZxK `k Rb aviY K‡i)

cÖv_wgK MwYZ 135

j¶ Kwi : • K L I K M yBwU j¤vjw¤ †iLv AuvKv n‡q‡Q|

• K L †iLvi Dci 1 †m.wg. Aš�i `vM †K‡U Ni AvuKv n‡q‡Q|

• K L †iLvi 1 Ni eëav‡b GK Ni‡K 1wU †kªYx aiv n‡q‡Q|

• K M †iLvi cÖwZ Ni‡K 10 Rb QvÎx a‡i `v‡Mi cv‡k 10, 20, 30, 40, 50, .................., 100 ch©š� †jLv n‡q‡Q|

• †kªYxi Ic‡i QvÎxi msL¨v Abymv‡i ¯�¤¢ AvuKv n‡q‡Q|

• ¯�¤¢‡jLwU my›ì, ¯có I AvKl©Yxq Kivi Rb¨ ¯�¤¢¸‡jv fivU Kiv n‡q‡Q|

• ¯�¤¢ †jLwU AvuKvi Rb¨ wb‡Pi wnmve Kiv n‡q‡Q| †kªYx QvÎxi msL¨v cÖwZ N‡ii gvb Ni msL¨v 1g 100 10 10 2q 90 10 9 3q 70 10 7 4_© 60 10 6 5g 50 10 5 D`vniY 1| †Kvb GKwU _vbvi wewfbœ mv‡ji RbmsL¨v (nvRv‡i) wb‡P †Ìqv nj| DcvË¸‡jv AbymiY K‡i GKwU ¯�¤¢ †jL AvuK Ges DcvË¸‡jv wP‡Î cÖKvk Ki : mvj RbmsL¨v 1941 50 nvRvi 1951 60 nvRvi 1961 70 nvRvi 1974 80 nvRvi 1984 90 nvRvi mgvavb : cÖ_‡g KL I KM yBwU j¤vjw¤ †iLv AvuwK| K L †iLvi Dci 1 †m.wg. ` �‡i ` �‡i `vM KvwU| GKNi eëav‡b 1941, 1951, 1961, 1974, 1984 mvj awi|

136 cÖv_wgK MwYZ

K M †iLvi cÖwZ Ni‡K 10 nvRvi a‡i `v‡Mi cv‡k 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 †jwL| Zvici ¯�¤¢‡jL AuvKvi Av‡M ¯�‡¤¢i ˆ`N©¨ wnmve K‡i wbB|

mvj 1941 1951 1961 1974 1984

RbmsL¨v50 nvRvi 60 nvRvi 70 nvRvi 80 nvRvi 90 nvRvi

cÖwZ N‡ii gvb10 nvRvi 10 nvRvi 10 nvRvi 10 nvRvi 10 nvRvi

Ni msL¨v 5 6 7 8 9

mv‡ji Ici RbmsL¨v Abymv‡i Ni ¸‡Y ¯�¤¢ AvuwK| †jLwPÎwUi ¯�¤¢¢ ‡jv Kv‡jv Kvwj‡Z fivU Kwi| GwUB RbmsL¨vi wb‡Y©q ¯�¤¢¢ †jL|

109

8

7

6

5

4

3

2

1

K 1941 1951 1961 1974 1984 L

cÖv_wgK MwYZ 137

Abykxjbx- 14 1| j²xcyi miKvwi cÖv_wgK we`¨vj‡qi wewfbœ †kªYxi wk¶v_©xi msL¨v wb‡Pi Q‡K

†Ìqv nj| Q‡Ki Z_¨ AbymiY K‡i GKwU ¯�¤¢ †jL AvuK Ges wP‡Î cÖKvk Ki| †kªYx wk¶v_©xi msL¨v

cÖ_g †kªYx wØZxq †kªYx Z…Zxq †kªYx PZz_© †kªYx cÂg †kªYx

120 Rb 100 Rb 90 Rb 80 Rb 70 Rb

2| †Zvgv‡ì †kªYx‡Z MZ mßv‡n (kwbevi eÜ wQj) Abycw¯_Z wk¶v_©xi msL¨v wb‡P †`Lv‡bv nj| DcvË¸‡jv AbymiY K‡i GKwU ¯�¤¢‡jL AvuK|

evi Abycw¯_Z wk¶v_©xi msL¨v iweevi †mvgevi g½jevi eyaevi

e„n¯cwZevi

3 2 4 5 7

3| evsjv‡`‡ki QqwU wefv‡Mi MÖv‡gi msL¨v (AvbygvwbK) †Ìqv nj| msL¨v¸‡jv‡K ¯�¤¢‡jL G‡uK cÖKvk Ki|

wefv‡Mi bvg MÖv‡gi msL¨v ivRkvnx Lyjbv

ewikvj XvKv

PÆMÖvg wm‡jU

18 nvRvi 8 nvRvi 5 nvRvi 19 nvRvi 15 nvRvi 4 nvRvi

138 cÖv_wgK MwYZ

4| evwl©K cix¶vq 5 Rb QvÎQvÎx MwY‡Z wbæiƒc b¤i †c‡q‡Q| ¯�¤¢‡jL Gu‡K

b¤i¸‡jv cÖKvk Ki|

QvÎQvÎxi bvg b¤i

kwdK

Aiƒc

Avdmvbv

wgwb

Dlv

100

80

70

80

60

5| evsjv‡`‡ki K‡qKwU kn‡ii ZvcgvÎv wbæiƒc cvIqv †Mj| ZvcgvÎv¸‡jv

†`wL‡q GKwU ¯�¤¢‡jL AvuK|

kn‡ii bvg ZvcgvÎv (†mjwmqvm)

XvKv

Lyjbv

ivRkvnx

PÆMÖvg

iscyi

31°

36°

38°

24°

20°

6| GKwU cvVvMv‡ii wewfbœ cÖKvi eB‡qi msL¨v wbæiƒc| eB¸‡jvi ¯�¤¢‡jL AvuK

Ges wP‡Î cÖKvk Ki|

eB‡qi cÖKvi msL¨v

mvwnZ¨

Rxebx

ag©xq

Mí

150

160

130

140

cÖv_wgK MwYZ 139

cÂ`k Aa¨vq

R¨vwgwZ

we›`y, †iLv I Zj cv‡ki wP‡Î KvM‡Ri Dci †cbwm‡ji miy AMÖfvM w`‡q GKUy Pvc †`Iqv n‡q‡Q| KvM‡R GKwU †QvU †duvUv ˆZwi n‡q‡Q| GB †duvUvwU GKwU we› yi cÖwZiƒc ev g‡Wj| mvaviYZ †duvUv w`‡qB we› y †evSvb nq| hw`I ev¯�weK c‡¶ we› y AvuKv hvq bv| KviY, we› yi †Kvb gvÎv †bB A_©vr Gi ˆ`N© ev cÖ _ ev †ea †bB| Gi †Kej wbw ©ó Ae¯_vb Av‡Q|

†iLv, †iLvsk, iwk¥

KvM‡Ri Dci K I L Øviv wb‡ ©wkZ yBwU we› y we‡ePbv Kwi| we› y yBwUi Dci GKwU mij iyjvi †i‡L miy †cbwmj w ‡q iyjvi eivei K †_‡K L ch©š� `vM Uvwb| K L GKwU mij‡iLvi As‡ki cÖwZiƒc A_©vr K L GKwU †iLvsk| †iLvskwU‡K Dfq w ‡K GKB eivei hZ �i Lywk evov‡jB GKwU mij‡iLvi cÖwZiƒc cvIqv hv‡e| GK UzKiv KvM‡Ri Dci Avgiv mij‡iLv AvuK‡Z cvwi bv| KviY †Kvb †iLvs‡ki wbw ©ó cÖvš�we› y I ˆ`N© Av‡Q, wKš�y mij‡iLvi †Kvb wbw ©ó cÖvš�we› y ev ‰`N© †bB| K L mij‡iLvi cÖwZiƒc| mij‡iLvi †Kvb cÖ _ †bB|

wP‡Î K †_‡K L Gi w`‡K †iLvwUi mxgvnxb Ask GKwU iwk¥| G‡K K L iwk¥ ejv nq|

K L

L K

K L

cÖv_wgK MwYZ 141

KZK¸‡jv Bs‡iwR A¶i †iLv w`‡q †jLv nq :

j¶ Kwi : • cÖ‡Z¨K †iLv, †iLvsk I iwk¥ Gi AmsL¨ we› y Av‡Q|

• cÖ‡Z¨K †iLv, †iLvsk I iwk¥i yBwU we› yi g‡a¨ AmsL¨ we› y Av‡Q| Zj : Avgv‡`i Pvicv‡ki wewfbœ e¯�yi DcwifvM †_‡K Avgiv Z‡ji aviYv cvB| †hgb, †Uwe‡ji DcwifvM, GKwU KvM‡Ri c„ô‡`k, BU| †Uwe‡ji DcwifvM ev KvMRwU‡K ev B‡Ui DcwifvM mKj w`‡K mxgvnxbfv‡e evov‡j GKwU R¨vwgwZK mgZ‡ji cÖwZiƒc cvIqv hvq| GB mgZ‡ji †h‡Kvb `yBwU we›`y w`‡q mij‡iLv we‡ePbv Ki‡j †iLvwU H Z‡jB _v‡K| GKwU BU j¶ Ki‡j †`Lv hvq †h, Gi †gvU QqwU Zj Av‡Q|

GKwU dzUe‡ji DcwifvMI GKwU Zj| GB Zj mgZj bq| GwU eµZj|

• cÖ‡Z¨K Z‡j AmsL¨ we› y Av‡Q|

• cÖ‡Z¨K Z‡j AmsL¨ †iLv AvuKv hvq|

BU

142 cÖv_wgK MwYZ

`vM †U‡b bv‡gi mv‡_ wP‡Îi wgj Ki (GKwU K‡i †`Lv‡bv nj) :

1bs 2bs 3bs 4bs Dc‡ii wP‡Î e· PviwU j¶ Kwi| cÖwZ e‡· yBwU iwk¥i cÖvš� we› y K I L Av‡Q| 1 bs Ges 2 bs e‡·i iwk¥i cÖvš� we› y Avjv`v Avjv`v| wKš�y 3 bs Ges 4 bs e‡·i iwk¥i cÖvš� we›`y K I L wgwjZ| iwk¥ yBwUi wgwjZ we› y‡Z †KvY Drcbœ n‡q‡Q|

we› y mij‡iLv ‡iLvsk iwk¥ eµ‡iLv eµZj mgZj

K L

K L

K L

K

L L K

K L

K L

144 cÖv_wgK MwYZ

wP‡Î m¤¢e †KvY¸‡jvi bvg †jL (cÖ_gwU K‡i †`Lv‡bv nj) :

†KvY cwigvc

K L iwk¥ I KM iwk¥ yBwU wecixZ iwk¥| GB iwk¥ `yBwU w`‡q ˆZwi ∠ MKL

GKwU mij †KvY| mij‡Kv‡Yi cwigvc 180° aiv nq| mij‡Kv‡Yi 180 fv‡Mi

1 fv‡Mi cwigvc 1 wWwMª| GK wWwMª‡K 1° †jLv nq| †KvY cwigv‡ci GKK wWwMÖ|

M

K

c

e

g∠ cKe, ∠ eKg, ∠ cKg,

K

K

ZL

`b

LZ M

bjN

1

2

3

K L

cÖv_wgK MwYZ 145

†KvY cwigv‡ci Rb¨ Puv`v e¨envi Kiv nq|

Dc‡ii wPÎwU j¶ Kwi| GwU GKwU Puv`v| NLM Puv`vi e¨vm Ges L-Gi †K›`ª we› y| Puv`vi Dcwifv‡Mi †iLvq yB iK‡gi msL¨v wj‡L `vM Uvbv n‡q‡Q| G¸‡jv †Kv‡Yi cwigvc wb‡ ©k K‡i| wb‡Pi msL¨v¸‡jv Wvbw`K †_‡K evgw`‡K Ges Dc‡ii msL¨v¸‡jv evgw`K †_‡K Wvbw`‡K| Puv`vi mvnv‡h¨ †Kv‡Yi cwigvc wbY©q :

Puv`vi mvnv‡h¨ ∠ LKM Gi cwigvc wb‡P †`Lv‡bv nj| Puv`vi †K› ª we› y‡K †Kv‡Yi kxl© we› y K Gi Dci emvB| Zvici Puv`vi e¨vm‡K

†Kv‡Yi KM evû eivei ivwL| ZLb †Kv‡Yi Aci evû KL Puv`vi Dci wjwLZ 60° `v‡Mi Dci coj| myZivs ∠ LKM Gi cwigvc 60°|

cÖv_wgK MwYZ 147

j¤ ` ~iZ¡ : K L GKwU mij‡iLv| c GKwU we›`y| Puv`vi e¨vm‡K K L †iLv eivei †i‡L Puv`vwU‡K Gw`K Iw`K mivB| Puv`vi 90° wb‡ ©kK iwk¥ c we› y‡Z Avmj| ZLb Puv`vi †K› ªwe› ywUi Ae¯_vb g nj| Zvn‡j g c iwk¥ K L †iLvi Dci j¤| c g †iLvs‡ki ˆ`N ©B c †_‡K K L †iLvi j¤ �iZ¡| mgvš�ivj †iLv : KL I MN `yBwU mij‡iLv| KL †iLvi †h‡Kvb we›`y c †_‡K MN †iLvi j¤ ` �iZ¡ cg| KL †iLvi Ab¨ GKwU we› y j †_‡K MN †iLvi j¤ ` �iZ¡ je| GB j¤ �iZ¡ yBwU mgvb| KL I MN †iLv yBwU mgvš�ivj|

`yBwU mij‡iLvi GKwUi †h‡Kvb ywU we› y †_‡K AciwUi j¤ ` �iZ¡ ci¯ci mgvb n‡j Giv mgvš�ivj| yBwU mgvš�ivj mij‡iLv KLbI †Q` K‡ibv| j¶ Kwi : †ij jvBb yBwU mgvš�ivj mij‡iLvi cÖwZiƒc|

150 cÖv_wgK MwYZ

PZyfy©R

PZzfz©R : PviwU †iLvsk Øviv Ave× wPÎ PZzfz©R|

KLMN GKwU PZzfz©R| KL, LM, MN, NK †iLvsk PviwU PZzfz©RwUi evû| KM I LN †iLvsk `yBwU Gi KY©|

KLMN PZzfz©RwUi KL Gi wecixZ evû NM mgvš�ivj Ges KN Gi wecixZ evû LM mgvš�ivj| GB PZzfz©RwU GKwU mvgvš�wiK| GLv‡b KL evû = NM evû Ges KN evû = LM evû|

mvaviYfv‡e ejv hvq, †h PZzfz©‡Ri wecixZ evû¸‡jv mgvb I mgvš�ivj ZvB mvgvš�wiK|

KLMN PZzfz©RwUi KL evû = NM evû| Gi cÖ‡Z¨KwU †KvY mg‡KvY| GwU GKwU AvqZ| GLv‡b KY© KM = KY© LN| KL I NM mgvš�ivj Ges KN I LM mgvš�ivj| myZivs KL MN AvqZwU GKwU mvgvš�wiK|

mvaviYfv‡e ejv hvq, †h PZzfz©‡Ri wecixZ evû¸‡jv mgvb I mgvšZivj Ges †KvY¸‡jv mg‡KvY ZvB AvqZ|

K L M N PZzfz©RwUi K L evû = L M evû = M N evû = N K evû| GwU GKwU i¤m| GLv‡b K L I N M mgvš�ivj Ges N K I M L mgvš�ivj|

mvaviYfv‡e ejv hvq, †h PZzfz©‡Ri cÖ‡Z¨KwU evû mgvb ZvB i¤m|

KL

MN

PZzfy©R

mvgšÍwiK

AvqZ LK

MN

K L

MN

eM©

cÖv_wgK MwYZ 151

K L M N PZzfy©RwUi K L evû = L M evû = M N evû = N K evû Ges cÖ‡Z¨KwU †KvY mg‡KvY| GwU GKwU eM©| GLv‡b K L I N M mgvš�ivj Ges N K I M L mgvš�ivj|

mvaviYfv‡e ejv hvq, †h PZyf©y‡Ri cÖ‡Z¨KwU evû mgvb Ges †KvY¸‡jv mg‡KvY ZvB eM©| `vM †U‡b bv‡gi mv‡_ wP‡Îi wgj Ki : (cÖ_gwU K‡i †`Lv‡bv nj) AvqZ i¤m eM© mvgvš�wiK PZzfz©R

e„Ë : L M N GKwU Ave× eµ‡iLv| Gi wfZ‡i K GKwU we› y| K †_‡K †iLvwUi mKj we› yi �iZ¡ mgvb| L M N eµ‡iLvwU GKwU e„Ë| K GB e„‡Ëi †K›`| †K› ª †_‡K e„‡Ëi †h‡Kvb we› yi �iZ¡ e„ËwUi e¨vmva©| GLv‡b K L e¨vmva©|

K

L

N

M

K L

MN

i¤^m

cÖv_wgK MwYZ 153

DËigvjv Abykxjbx-1

1| cuwPk nvRvi mvZk †PŠwÎk| mvZPwj−k nvRvi wbivbeŸB| AvU j¶ Avwk nvRvi cuvPk EbPwj−k| mvZvbœ j¶ Pvi nvRvi Qqk yB| Avwk j¶ cÂvbœ nvRvi cuvPk cuvP| AvUvwk j¶ AvUvwk nvRvi AvUk AvUvwk| GK †KvwU|

2| 45712; 68903; 650309; 270459; 8800780; 9909990; 10000000| 3| (K) 900000 (L) 8000 (M) 700 (N) 6 (O) 5000000 (P) 0 (Q) 20| 4| 1g 5 Gi ¯_vbxq gvb 5 2q 5 '' '' '' 50 3q 5 '' '' '' 500 4_© 5 '' '' '' 5000 5g 5 '' '' '' 50000 6ô 5 '' '' '' 500000 5| 8326 < 8362; 7506 < 8021; 6798 > 6789; 85320 > 76203;

58249 < 58492| 6| (K) 5027, 4902, 4321, 4287, 3357, 3270 3270, 3357, 4287, 4321, 4902, 5027| (L) 8560, 8506, 6218, 6210, 5786, 5768 5768, 5786, 6210, 6218, 8506, 8560 (M) 83592, 83259, 72607, 72570, 60892, 60829| 60829, 60892, 72570, 72607, 83259, 83592| (N) 90257, 90062, 83251, 83001, 79805, 78253, 78235| 78235, 78253, 79805, 83001, 83251, 90062, 90257| 7| (K) e„nËg msL¨v : 983 (L) e„nËg msL¨v : 970 ¶z`ªZg msL¨v : 389 ¶z`ªZg msL¨v : 709 (M) e„nËg msL¨v : 8543 (N) e„nËg msL¨v : 8630 ¶z`ªZg msL¨v : 3458 ¶z`ªZg msL¨v : 3068| 8| e„nËg msL¨v : 99999 ¶z`ªZg msL¨v : 10000

154 cÖv_wgK MwYZ

9| e„nËg msL¨v : 87532 8 Gi ¯_bxq gvb : 80000 7 Gi ¯_vbxh gvb : 7000 3 Gi ¯_bxh gvb : 30 10| 1000 I 100000| 11| 999 I 99999| 12| II, V, VIII, IX, XI, XII 13| 4, 7, 9, 10, 6, 12| 14| wb‡R Ki|

Abykxjbx- 2

1| (K) 12858 (L) 13518 (M) 19362 (N) 24506 (O) 82011 (P) 98739 (Q) 96752 (R) 98085| 2| (K) 5992 (L) 10054 (M) 10297 (N) 62204 (O) 98239 3| (K) 7778 (L) 4694 (M) 83896 (N) 52075 (O) 58867 (P) 45758 (Q) 39076 (R) 50455 4| (K) 5429 (L) 5736 (M) 5357 (N) 77854 (O) 38787 (P) 38898 5| (K) 5016, we‡qvRb 9403, we‡qvR¨ 4387, we‡qvMdj 5016| (L) 67877, we‡qvR¨ 5374, we‡qvMdj 67877, we‡qvRb 73251 (M) 3636, we‡qvMdj 3636, we‡qvRb 6214, we‡qvR¨ 2578 (N) 3497, we‡qvR¨ 1805, we‡qvRb 5302, we‡qvMdj 3497| 6| 7391 Rb| 7| 81094, 8| 4481, 9| 1 10| 4135 11| 6774wU, 12| 6318 13| 21591 14| 7978 15| e„nËg msL¨v : 97530 16| 99001 17| 78 eQi| ¶z ªZg msL¨v : 30579 18| 41400 †ewk| 19| 2302 UvKv| 20| 37429 UvKv|

Abykxjbx-3 1| 26979 2| 18144 3| 29232 4| 30114 5| 34726 6| 99042 7| 26220 8| 23793 9| 63625 10| 64320 11| 45500 12| 97935 13| 29007 14| 90288 15| 77670 16| 24750 17| 98901 18| 98010 19| 20592 wU 20| 83300 †KwR 21| 9430 UvKv 22| 8125 UvKv 23| 13875 UvKv 24| 17500 UvKv|

cÖv_wgK MwYZ 155

Abykxjbx-4 1| (K) fvMdj 105 (L) fvMdj 77 (M) fvMdj 91 (N) fvMdj 92, fvM‡kl 7 (O) fvMdj 1062 (P) fvMdj 946 (Q) fvMdj 348 (R) fvMdj 941, fvM‡kl 2 (S) fvMdj 511 (T) fvMdj 264 (U) fvMdj 148, fvM‡kl 12 (V) fvMdj 123, fvM‡kl 6| 2| (K) fvMdj 65 (L) fvMdj 74 (M) fvMdj 81, fvM‡kl 5 (N) fvMdj 54, fvM‡kl 62 (O) fvMdj 63 (P) fvMdj 70, fvM‡kl 35| 3| (K) 52 (L) 79 (M) 150 (N) 7 (O) 8862| 4| 38 UvKv 5| 17 UvKv 6| 22 wgUvi 7| fvMdj 62, fvM‡kl 8 8| fvMdj 128, fvM‡kl 15, 9| fvMdj 76, fvM‡kl 8 10| fvMdj 24, fvM‡kl 92| 11| cÖ‡Z¨‡K 146 UvKv cv‡e, 50 UvKv evwK _vK‡e|

12| cÖ‡Z¨‡K 154 UvKv cv‡e|

Abykxjbx-5 1| 600wU 2| cÖ‡Z¨‡K †cj 12wU, Aewkó iBj 1wU| 3| 70 UvKv 4| 4312 UvKv 5| 74 eQi 6| 280 UvKv, 7| 456 UvKv 8| 2250 UvKv 9| 40 †KwR 10| 756 Rb 11| 48 12| 600 13| cy‡Îi eqm 17 eQi, wcZvi eqm 51 eQi 5 eQi ci cy‡Îi eqm n‡e 22

eQi Ges wcZvi eqm n‡e 56 eQi| 14| wPbyi 140 UvKv, iveyi 250 UvKv| wZbR‡bi †gvU 605 UvKv| 15| 867 16| 96 UvKv 17| 655 wU 18| †g‡q 30 Rb, †Q‡j 20 Rb 19| 104 UvKv 20| gwbi 364 UvKv, wjwci 572 UvKv| 21| 56 22| 9000 UvKv|

Abykxjbx-6 1| 65 2| 125 3| 200 4| 199 5| 102 6| 20 7| 2592 8| 3360 9| 6048 10| 10 11| 0 12| 10 13| 4 14| 3 15| 10

Abykxjbx-7 1| (K) 1, 2, 4, 8, 16 (L) 1, 2, 3, 5, 6, 10, 15, 30 (M) 1, 2, 3, 6, 7, 14, 21, 42 (N) 1, 3, 5, 9, 15, 45 (O) 1, 2, 3, 6, 13, 26, 39, 78 (P) 1, 3, 5, 7, 15, 21, 35, 105

156 cÖv_wgK MwYZ

(Q) 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144 (R) 1, 3, 7, 9, 21, 27, 63, 189 (S) 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216 (T) 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, 162, 324 2| (K) K…wÎg (L) †gŠwjK (M) †gŠwjK (N) K…wÎg (O) K…wÎg| 3| (K) 2×2×3×5 (L) 2×2×2×3×3 (M) 3×31 (N) 2×2×2×2×7 (O) 3×3×3×5 (P) 2×3×37 (Q) 3×5×17 (R) 2×2×2×2×2×2×5 (S) 2×2×11×11 (T) 2×2×2×2×2×2×2×2×2 4| (K) 3 Øviv wefvR¨ (L) 2, 5 Øviv wefvR¨| (M) 5 Øviv wefvR¨ (N) 3, 5 Øviv wefvR¨ (O) 3, 5 Øviv wefvR¨ (P) 2, 3, 5 Øviv wefvR¨ (Q) 3, 5 Øviv wefvR¨ (R) 3, 5 Øviv wefvR¨| (S) 3 Øviv wefvR¨ (T) 3 Øviv wefvR¨| 5| (K) 6, 12, 18, 24, 30 (L) 9, 18, 27, 36, 45 (M) 12, 24, 36, 48, 60 (N) 16, 32, 48, 64, 80 (O) 21, 42, 63, 84, 105 (P) 28, 56, 84, 112, 140 (Q) 35, 70, 105, 140, 175 (R) 42, 84, 126, 168, 210 (S) 44, 88, 132, 176, 220 (T) 50, 100, 150, 200, 250 6| (K) 1, 3, 6 (L) 1, 3, 5, 15 (M) 1, 3, 9 (N) 1, 2, 4, 8 (O) 1, 2, 4, 8 (P) 1, 2, 4, 5, 10 7| (K) 8 (L) 14 (M) 7 (N) 15 (O) 6 (P) 8 (Q) 3 (R) 12 8| (K) 48 (L) 72 (M) 100 (N) 105 (O) 180 (P) 168 (Q) 138 (R) 114 (S) 72 (T) 300 (U) 288

Abykxjbx-8 1| (K) 127 + 303 (L) 465 − 229 (M) 88 × 7 (N) 289 ÷ 17 (O) (72 − 38) × 11 2| (K) 357 − 212 > 50 (L) 360 ÷ 18 =/ 21

cÖv_wgK MwYZ 157

(M) (21 + 16) × 6 = 21 × 6 + 16 × 6 (N) 23 × 23 > 500 (O) (144 ÷ 12) × 16 = 192 3| (K) mZ¨ (L) wg_¨v (M) wg_¨v (N) mZ¨ (O) wg_¨v 4| (K) > (L) < (M) = (N) = (O) > 5| (K) − , × (L) ×, × (M) ×, ÷ (N) −, ÷ (O) +, × 6| (K) 22 (L) 9 (M) 143 (N) 9 (O) 67 7| (K) 168 (L) 125 (M) 13 (N) 285 (O) 0

Abykxjbx- 9 (K)

1| (K) 710 (L)

525 (M)

915 (N)

2472

2| (K) 1012 ,

1518 ,

2024 (L)

814 ,

1221 ,

1628 (M)

716 ,

924 ,

1232

3| (K) 85 (L)

34 (M)

13 (N)

25 (O)

12 (P)

413

4| 2130,

230 5|

1324 ,

2024 6|

140168 ,

140168 ,

147168

7| 690,

890 ,

2190 8|

3042 ,

942 ,

1642 9|

616 ,

816 ,

1216 ,

916

10| 16 11| 56 12| 6 13| 45 14| 72 15| 84

16| (K) 35 <

34 (L)

914 <

710 (M)

1730 >

715 (N)

311 >

733

17| 1318 18|

1475 19|

1112 20|

7384 21|

2336

22| 1

36 23| 533 24|

568 25|

526 26| 2

27| 34 28|

1924 29|

2330 30|

47 31|

4350

32| 1115 33|

1318 34|

524

158 cÖv_wgK MwYZ

Abykxjbx- 9 (L)

1| 16 Ask 2|

25 As‡k 3|

913 As‡k 4|

215 Ask 5|

38 Ask

6| 35 Ask 7|

16 Ask 8|

14 As‡k 9|

310 Ask

10|415 Ask 11|

148 Ask|

Abykxjbx- 10 (K) 1| (K) cuvP `kwgK mvZ wZb (L) AvU `kwgK k�b¨ bq (M) AvU `kwgK cuvP AvU (N) cuwPk `kwgK yB cuvP (O) bq `kwgK bq (P) bq `kwgK k �b¨ GK 2| (K) 7.09 (L) 8.01 (M) 5.55 (N) 9.99 3| (K) 4 GKK 3 `kgvsk 4 kZvsk (L) 1 `kK 5 GKK 0 `kgvsk 5 kZvsk (M) 1 `kK 8 GKK 2 `kgvsk 6 kZvsk (N) 1 `kK 7 GKK 5 `kgvsk 2 kZvsk (O) 1 `kK 3 GKK 0 `kgvsk 3 kZvsk (P) 2 `kK 0 GKK 0 `kgvsk 2 kZvsk (Q) 8 GKK 8 `kgvsk 9 kZvsk 4| (K) 0.9 (L) 0.1 (M) 0.8 (N) 0.7 (O) 0.15 (P) 0.35 (Q) 0.45 (R) 0.16 (S) 0.36 (T) 0.26 (U) 0.58

5| (K) 310 (L)

710 (M)

910 (N)

13100 (O)

27100 (P)

39100

(Q) 7

100 (R) 99100 (S)

89100 (T)

98100 (U)

1100

6| (K) 25 (L)

125 (M)

12 (N)

15 (O)

14

(P) 25 (Q)

12 (R)

35 (S)

34 (T)

45

7| (K) .38 < .46 (L) .58 < .72 (M) .09 < 90 (N) .3 > .03 (O) .86 < .87 (P) .64 > .60 (Q) .5 > .25 (R) .37 > .32

cÖv_wgK MwYZ 159

Abykxjbx- 10 (L)

1| (K) 1.3 (L) 3.75 (M) 6.99 (N) 9.51 (O) 11.41 (P) 7.53 (Q) 10.41 (R) 15.75 2| (K) 2.5 (L) 4.25 (M) 5.51 (N) 6.18 (O) 9.64 (P) 12.09 (Q) 1.86

Abykxjbx- 10 (M)

1| (K) 5.6 (L) 41 (M) 0.48 (N) 9.27 (O) 13.75 (P) 37 (Q) 51.6 (R) 53.9 (S) 79.5 (T) 0.54 (U) 0.015 (V) 0.0020 2| 88.65 UvKv 3| 41.95 UvKv 4| 30.00 UvKv 5| 8.80 †mjwmqvm 6| 24 UvKv 7| ivbv 14wU, iy‡kv 6wU 8| gwj 16wU wWg wKbj Ges 64 UvKv

`vg w`j; iywg 10wU wWg wKbj Ges 40 UvKv w`j| 9| 590.55 BwÂ 10| 82.80 UvKv 11| 131.40 wK‡jvwgUvi|

Abykxjbx- 11 (K)

1| (K) 757 wgwjwgUvi (L) 623 wgwjwgUvi (M) 940 wgwjwgUvi (N) 16880 wgwjwjUvi (O) 11000 wgwjwjUvi (P) 13006 wgwjwjUvi| 2| (K) 25075 †mw›UwgUvi (L) 34500 †mw›UwgUvi (M) 9536 †mw›UwgUvi (N) 200012 †mw›UwgUvi (O) 812560 †m: wgUvi (P) 1532090 †m: wg: 3| (K) 70000 wgUvi (L) 16425 wgUvi (M) 9275 wgUvi (N) 98000 wgUvi (O) 21005 wgUvi (P) 39155 wgUvi 4| (K) 45000 wgwjMÖvg (L) 18090 wgwjMÖvg (M) 32213 wgwjMÖvg (N) 65008 wgwjMÖvg (O) 2025000 wgwjMÖvg (P) 5030120 wgwjMÖvg 5| (K) 23625 †mw›UMÖvg (L) 5200 †mw›UMÖvg (M) 12219 †mw›UMÖvg (N) 7280 †mw›UMÖvg 6| (K) 195 †WwmMÖvg (L) 20103 †WwmMÖvg (M) 30000 †WwmMÖvg (N) 50160 †WwmMÖvg

Mathematics17_126162399620091224

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