03. root & square (math tutor by kabial noor) [www.itmona.com]

1. Math Tutor  1
Aa¨vq 03
eM©g~j I c~Y©eM© msL¨v
 03.01 eM© I eM©g~j Kx?
†Kvb msL¨v‡K GKB msL¨v Øviv ¸Y Ki‡j †h ¸Ydj cvIqv hvq, Zv‡K H msL¨vi eM© e‡j| †hgb- 4 †K 4 Øviv ¸Y
Ki‡j ¸Ydj 16 cvIqv hvq, GLv‡b 16 n‡”Q 4 Gi eM©| Ab¨w`‡K GLv‡b 4 n‡”Q 16 Gi eM©g~j| eM©g~jm~PK wPý
n‡”Q | MvwYwZKfv‡e, 4 Øviv 4 Gi eM©g~j‡K eySvq (A_©vr, 4 = 2)|
 03.02 c~Y©eM© msL¨v
†h mKj msL¨vi eM©g~j †Kvb c~Y©msL¨v ev fMœvsk n‡q _v‡K, Zv‡`i‡K c~Y©eM© msL¨v e‡j| †hgb-
c~Y© eM©msL¨v eM©g~j c~Y© eM©msL¨v eM©g~j
4 2 (2 GKwU c~Y© msL¨v)
4
9
2
3 (
2
3 GKwU fMœvsk)
9 3 (3 GKwU c~Y©msL¨v)
16
25
4
5 (
4
5 GKwU fMœvsk)
 ¸iæZ¡c~Y© K‡qKwU c~Y©eM© msL¨vi ZvwjKv gyL¯’ ivLybt
eM©g~j c~Y©eM© msL¨v eM©g~j c~Y©eM© msL¨v eM©g~j c~Y©eM© msL¨v
11 121 (11 × 11) 18 324 (18 × 18) 25 625 (25 × 25)
12 144 (12 × 12) 19 361 (19 × 19) 30 900 (30 × 30)
13 169 (13 × 13) 20 400 (20 × 20) 31 961 (31 × 31)
14 196 (14 × 14) 21 441 (21 × 21) 32 1024 (32 × 32)
15 225 (15 × 15) 22 484 (22 × 22) 41 1681 (41 × 41)
16 256 (16 × 16) 23 529 (23 × 23) 81 6561 (81 × 81)
17 289 (17 × 17) 24 576 (24 × 24) 144 20736 (144 × 144)
 g‡b ivLybt
(K) †Kvb c~Y©eM© msL¨vi GK¯’vbxq A¼ KL‡bv 2, 3, 7 I 8 n‡e bv| Dc‡iv³ Q‡K c~Y©eM© msL¨vi GKK ¯’v‡b Giv †bB|
cv‡ki wP‡Î jÿ¨ Kiæb, mvwii msL¨v Av‡Q 4 wU Ges cÖwZwU
mvwi‡Z gv‡e©‡ji msL¨vI Av‡Q 4 wU| Avcbv‡K hw` cÖkœ Kiv
nq, wPÎwU‡Z †gvU KqwU gv‡e©j Av‡Q? Avcwb `yB Dcv‡q †gvU
gv‡e©‡ji msL¨v †ei Ki‡Z cv‡ib| cª_gZ, GKwU GKwU K‡i
wn‡me K‡i †gvU gv‡e©‡ji msL¨v †ei Ki‡Z cv‡ib| wØZxqZ,
4 Gi mv‡_ 4 ¸Y K‡i †gvU gv‡e©‡ji msL¨v †ei Ki‡Z
cv‡ib| GLv‡b, 4 Gi mv‡_ 4 ¸Y Kiv‡K 4 Gi eM© e‡j|
e‡M©i msÁv Abymv‡i Avgiv Rvwb, †Kvb msL¨v‡K GKB msL¨v
Øviv ¸Y Ki‡j †h ¸Ydj cvIqv hvq, Zv‡K H msL¨vi eM©
e‡j| A_©vr, 4×4 = 16 GLv‡b, 16 n‡”Q 4 Gi eM© Ges
GKBfv‡e 4 n‡”Q 16 Gi eM©g~j, hv Avgiv Gfv‡eI
wjwL- 16 = 4|
1bs mvwi
2bs mvwi
3bs mvwi
4bs mvwi
cÖwZ mvwi‡Z gv‡e©j 4wU
†gvUmvwi4wU
me©‡gvU gv‡e©j = 4 × 4 =16wU
†ewmK, GgwmwKD I wjwLZ Av‡jvPbv 

2. 2  Math Tutor
(L) GKwU c~Y©eM© msL¨vi GKK ¯’vbxq A¼ 1, 4, 5, 6, 9 I 0 _v‡K| Dc‡iv³ QKwU †PK Kiæb, welqwU wK¬qvi n‡e|
wb‡Pi Q‡K welqwU Zz‡j aiv nj| Z‡e c~Y©eM© msL¨vi †k‡l k~b¨ Aek¨B †Rvo AvKv‡i _vK‡Z n‡e| †hgb- 100, 400|
c~Y©eM© msL¨vi GKK ¯’v‡b eM©g~j msL¨vi GKK ¯’v‡b c~Y©eM© msL¨vi GKK ¯’v‡b eM©g~j msL¨vi GKK ¯’v‡b
1 _vK‡j 1 A_ev 9 cv‡eb 6 _vK‡j 4 A_ev 6 cv‡eb
4 _vK‡j 2 A_ev 8 cv‡eb 9 _vK‡j 3 A_ev 7 cv‡eb
(M) c~Y©eM© msL¨v‡Z GK †Rvov k~b¨ (00) _vK‡j eM©g~‡j GKwU k~b¨ (0) nq| †hgb- 100 Gi eM©g~j 10|
(N) †Kvb c~Y©eM© msL¨vi eM©g~j cwRwUf (+) I †b‡MwUf (-) DfqB n‡q _v‡K| †hgb- 9 Gi eM©g~j = +3 A_ev -3 |
(+3) × (+3) = 9 Ges -3 × -3 = 9 | KviY, cøv‡m cøv‡m cøvm nq I gvBbv‡m gvBbv‡m cøvm nq| Zvn‡j x2
eM©g~j n‡e, x2 = ±x |
01. 215, 219, 325, 625 msL¨v¸wji g‡a¨ c~Y© eM©
†KvbwU? WvK wefv‡Mi †cv÷vj Acv‡iUi: 16
215 219
625 325 DËi: M
02. 169 msL¨vwUi eM©g~j KZ?- 34th
BCS; ¯^v¯’¨ I cwievi
Kj¨vY gš¿Yvj‡qi KwgDwbwU †nj_ †Kqvi †cÖvfvBWvi -18
11 13
15 17 DËi: L
03. 1024 Gi eM©g~j KZ? cÖv_wgK we`¨vjq mnKvix wkÿK : 19
22 52
42 32 DËi: N
04. 13169  n‡j ?121  35Zg wewmGm wjwLZ
(gb¯ÍvwË¡K)
14 16
11 12 DËi: M
05. 𝐱 𝟐 = ? cÖv_wgK we`¨vjq mnKvix wkÿK (4_© avc) 19
x -x ±x
x2
DËi: M
06. 289 Gi eM©g~j n‡jv- 10g †emiKvwi wkÿK wbeÜb I
cÖZ¨qb cixÿv 2014
g~j` Ag~j`
¯^vfvweK msL¨v c~Y© msL¨v DËi: K
289 Gi eM©g~j n‡”Q 17, hv GKwU ¯^vfvweK
msL¨v| †h‡Kvb c~Y©eM© msL¨vi eM©g~j memgq g~j`
msL¨v nq| Z‡e, †h‡Kvb †g․wjK msL¨vi eM©g~j
Ag~j` msL¨v nq|
07. wb‡Pi †Kvb msL¨v GKwU c~Y©eM© msL¨v n‡Z 1 Kg?
RbcÖkvmb gš¿Yvj‡qi cÖkvmwbK Kg©KZ©v : 2015
44941 98594
16899 75432 DËi: M
Ack‡b cÖ`Ë msL¨v¸‡jvi mv‡_ 1 †hvM Ki‡j †hwU
c~Y©eM© msL¨v n‡e †mwUB DËi| Pjyb ïiæ‡ZB 1 †hvM
K‡i †`Lv hvK- 44941 + 1 = 44942
98594 + 1 = 98595 16899 + 1 = 16900
Ges 75432 + 1 = 75433| Avgiv Rvwb,
†hme msL¨vi †k‡l 2, 3, 7, 8 _v‡K Zviv KL‡bv
c~Y©eM© msL¨v n‡Z cv‡i bv, ZvB Ackb I
ev`| 98595 = 313.9984 (c~Y©eM© msL¨v bq)
16900 = 130 (mwVK DËi) |
08. 2
(100) = ? MYc~Z© Awa. Dcmn. cÖ‡K․kjx (wmwfj): 17
10 100
1000 10000 DËi: L
2
(100) = 2
1
2
100)(

= 100
09. 11  Gi eM© KZ? RbcÖkvmb gš¿Yvj‡qi cÖkvmwbK
Kg©KZ©v: 15
2 4
3 2 1 DËi: L
22
1211 )()( 
41412 22
 )(
10. ?)77( 2
 †mvbvwj, RbZv I AMÖYx e¨vsK
Awdmvi- 2008
98 49
28 21 DËi: M
22
7277 )()( 
287472 22
 )(
11. ?)2525( 2
 Bangladesh Bank Assistant
mgvavb
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK
M
L
NK
N
M
LK
mgvavb
NM
LK
mgvavb
NM
LK
N
MLK
NM
LK
NM
LK
NM
LK
NM
LK

3. Math Tutor  3
Director-2010
50 20
100 125 DËi: M
100)10()55()2525( 222

12. 2
4)(3  wb‡Pi †KvbwUi mgvb? Lv`¨ Awa`߇ii
mnKvix Dc-Lv`¨ cwi`k©K : 09
25 14
49 †Kv‡bvwUB bq DËi: M
2
4)(3  = 72
= 49|
13. 2
(0.002) = KZ? cÖvK-cÖv_wgK mnKvix wkÿK (myigv): 13
0.004 0.0004
0.00004 0.000004 DËi: N
2
(0.002) = 0.002  0.002 = 0.000004
14. 2
(0.003) = KZ? cÖv_wgK we`¨vjq mnKvix wkÿK (cÙv):
12/ cÖv_wgK we`¨vjq mnKvix wkÿK (Ki‡Zvqv):10
0.000009 0.00009
0.0009 0.009 DËi: K
2
(0.003) = 0.003  0.003 = 0.000009
15. 2
(0.004) = KZ? cÖv_wgK we`¨vjq mnKvix wkÿK (cÙv):
12/ cÖv_wgK we`¨vjq cÖavb wkÿK (W¨v‡dvwWj): 12
0.016 0.000016
0.00016 0.0016 DËi: L
2
(0.004) = 0.004  0.004 = 0.000016
16. 2
(0.005) = KZ? cÖv_wgK we`¨vjq mn. wkÿK (†gNbv): 12
0.025 0.0025
0.000025 0.0000025 DËi: M
17. 999 msL¨vwUi eM© wbY©q Kiæb- cwi‡ek Awa`߇ii mn-
cwiPvjK (KvwiMi/cÖvkvmb) I wimvP© Awdmvi: 07/ cvm‡cvU© I
Bwg‡MÖkb Awa`߇ii mnKvix cwiPvjK: 07
999801 990001
998001 988001 DËi: M
9992
= (1000-1)2
[(a - b)2
m~Îvbymv‡i]
= 10002
- 2  1000 1 + 12
= 1000000 - 2000 + 1 = 998001
 03.03 †g․wjK Drcv`‡Ki mvnv‡h¨ eM©g~j wbY©q c×wZ
Drcv`‡Ki mvnv‡h¨ †Kvb c~Y©eM© msL¨vi eM©g~j wbY©‡qi avcmg~n Zz‡j aiv n‡jv-
(i) 4900 (ii) 64 (iii) 1089
 avc 01t cÖ_‡g cÖ`Ë msL¨vwU‡K †g․wjK Drcv`‡K we‡kølY Ki‡Z n‡e|
(i) 2 4900 (ii) 2 64 (iii) 3 1089
2 2450 2 32 3 363
5 1225 2 16 11 121
5 245 2 8 11
7 49 2 4
7 2
 avc 02t cÖwZ †Rvov Drcv`K‡K GKmv‡_ cvkvcvwk wjL‡Z n‡e|
4900 64 1089
= (22)(55)(77) = (22)(22)(22) = (33) (1111)
 avc 03t cÖwZ †Rvov Drcv`K †_‡K GKwU K‡i Drcv`K wb‡Z n‡e|
=(22)(55) (77) = (2×2)  (22)(22) = (33) (1111)
2  5  7 2  2  2 3  11
 avc 04t Gevi cÖwZ †Rvov †_‡K cÖvß GKwU K‡i Drcv`Kmg~n‡K ¸Y Ki‡Z n‡e Ges ¸Y K‡i cÖvß
¸YdjB n‡”Q eM©g~j|
= 257 = 222 = 311
= 70 = 8 = 33
ms‡ÿ‡c, 4900 = 70 (eM©g~j) ms‡ÿ‡c, 64 = 8 (eM©g~j) ms‡ÿ‡c, 1089 = 33 (eM©g~j)
mgvavb
NM
LK
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK
 hv g‡b ivL‡Z n‡e
1| †Kvb msL¨v c~Y©eM© n‡Z n‡j Aek¨B Zvi Drcv`Kmg~n †Rvovq †Rvovq _vK‡Z n‡e|
2| cÖwZ †Rvov Drcv`‡Ki Rb¨ eM©g~j n‡e GKwU Drcv`K|

4. 4  Math Tutor
 03.04 ¸Y/fvM K‡i †Rvo ‣Zwii gva¨‡g c~Y©eM© msL¨v •Zwii c×wZ
†Kvb msL¨vi Drcv`Kmg~n †Rvovq †Rvovq bv _vK‡j ev †Kvb Drcv`K wm‡½j _vK‡j 2 Dcv‡q c~Y©eM© msL¨v •Zwi
Kiv hvq| Pjyb wbqg `ywU †R‡b †bqv hvK| ïiæ‡Z GKwU msL¨v 24 †bqv hvK|
∴ 24 = 2223 [Drcv`‡K we‡kølY K‡i cÖvß Drcv`Kmg~n]
= (22) 2 (wm‡½j)3 (wm‡½j)
jÿ¨ Kiæb, 24 †_‡K cÖvß Drcv`Kmg~‡ni gv‡S †k‡li 2 I 3 wm‡½j, Giv †Rvo AvKv‡i †bB, Zvi gv‡b 24 c~Y©eM©
msL¨v bq| Avgiv PvB‡j 24 †K 2 Dcv‡q c~Y©eM© msL¨vq iƒcvšÍi Ki‡Z cvwi| Pjyb †R‡b †bqv hvK -
 Dcvq 01t Dc‡iv³ 24-Gi Drcv`Kmg~‡ni gv‡S GKwU 2 I GKwU 3 wm‡½j Av‡Q, Gevi 2 I 3 Øviv fvM
Ki‡j wm‡½j 2 I 3 ev` hv‡e Ges msL¨vwU c~Y©eM© msL¨vq iƒcvšÍi n‡e|
A_©vr,
32
3222


= 22 = 4 (4 n‡”Q c~Y©eM© msL¨v, †h‡nZz 4 Gi Drcv`K n‡”Q †Rvo msL¨K)
Gfv‡e GKwU msL¨vq hZwU wm‡½j Drcv`K _v‡K, wVK ZZwU Øviv fvM K‡i msL¨vwU‡K c~Y©eM© msL¨vq iƒcvšÍi
Ki‡Z nq| Dc‡iv³ D`vni‡Y wm‡½j Drcv`K wQj 2 I 3, ZvB Avgiv 2 × 3 (=6) Øviv fvM K‡i 24 †K
c~Y©eM© msL¨v 4-G iƒcvšÍi K‡iwQ|
 Dcvq 02t Dc‡iv³ 24 Gi Drcv`Kmg~‡ni gv‡S wm‡½j 2 I wm‡½j 3 -Gi mv‡_ GKwU K‡i 2 I 3 ¸Y Ki‡j
msL¨vwU c~Y©eM© msL¨vq iƒcvšÍi n‡e| A_©vr, 222233 = (22)(22)(33) =
144 (144 n‡”Q c~Y©eM© msL¨v, †h‡nZz 144 Gi Drcv`Kmg~n †Rvovq †Rvovq Av‡Q)
Gfv‡e GKwU msL¨vq hZwU wm‡½j Drcv`K _v‡K, wVK ZZwU wm‡½j Drcv`K Øviv ¸Y K‡i msL¨vwU‡K c~Y©eM©
msL¨vq iƒcvšÍi Ki‡Z nq| Dc‡iv³ D`vni‡Y wm‡½j Drcv`K wQj 2 I 3, ZvB Avgiv 2 × 3 (=6) Øviv ¸Y K‡i
24 †K c~Y©eM© msL¨v 144-G iƒcvšÍi K‡iwQ|
 Tips 01: Avcbv‡K hw` ejv nq- 24 msL¨vwU‡K KZ Øviv ¸Y Ki‡j msL¨vwU c~Y© eM©msL¨v n‡e? Avgiv cÖ_‡g
†`Le 24 Drcv`K KqwU wm‡½j Av‡Q, Zvici †mB KqwU wm‡½j msL¨v Øviv ¸Y Ki‡jB DËi cvIqv hv‡e| †hgb-
24-Gi wm‡½j Drcv`K Av‡Q 2 I 3 ZvB 23 ev 6 Øviv ¸Y Ki‡Z n‡e|
 Tips 02: Gevi Avcbv‡K hw` ejv nq- †Kvb ÿz`ªZg msL¨v Øviv 24 †K fvM Ki‡j c~Y©eM© cvIqv hv‡e? GeviI
Avgiv †`Le wm‡½j Drcv`K KqwU Av‡Q, †h KqwU _vK‡e †mB KqwU wm‡½j msL¨v Øviv fvM Kie| GeviI 24 Gi
†ÿ‡Î 23 ev 6 Øviv fvM Ki‡Z n‡e|
18. 2450 msL¨vwU‡K KZ Øviv ¸Y Ki‡j msL¨vwU c~Y©
eM©msL¨v n‡e? m~Îtwb¤œ gva¨wgKMwYZ,mßg†kÖwY-1997wkÿvel©
2450 Gi Drcv`K mg~n- 2(55)(77)|
GLv‡b 2 n‡”Q wm‡½j Drcv`K, ZvB 2 Øviv ¸Y
Ki‡jB msL¨vwU c~Y© eM©msL¨v n‡e|
19. Ggb jwNô c~Y©msL¨v wbY©q Ki, hv Øviv 450†K ¸Y
Ki‡j ¸Ydj GKwU c~Y© eM© msL¨v n‡e|m~Î: cvwUMwYZ-
hv`eP›`ª PµeZx©
mgvavb
GKwU welq jÿ¨ K‡i‡Qb?
Avgiv hLb fvM K‡i 24 †K c~Y©eM© msL¨vq iƒcvšÍi Kijvg, ZLb wm‡½j Drcv`K 2 I 3 (= 6) Øviv fvM
Kijvg| GKBfv‡e, hLb ¸Y K‡i 24 †K c~Y©eM© msL¨vq iƒcvšÍi Kijvg, ZLbI wm‡½j Drcv`K 2 I 3 (=6)
ØvivB ¸Y Kijvg! Zvi gv‡b welqwU wK `vov‡jv? n¨vu ejwQ- hLb †Kvb msL¨v‡K c~Y©eM© msL¨vq iƒcvšÍi Ki‡Z ejv
n‡e ZLb Dfq †ÿ‡ÎB wm‡½j Drcv`K hv _vK‡e Zv ØvivB ¸Y ev fvM Ki‡Z nq|

5. 6  Math Tutor
450 Gi Drcv`Kmg~n = 23355|
GLv‡b 2 n‡”Q wm‡½j Drcv`K, ZvB 2 Øviv ¸Y
Ki‡j ¸Ydj GKwU c~Y©eM© msL¨v n‡e| DËi: 2
20. †Kvb ÿz`ªZg msL¨v Øviv 4608 †K fvM Ki‡j c~Y©eM©
cvIqv hv‡e? wb¤œ gva¨wgKMwYZ,mßg†kÖwY-1997wkÿvel© (D`vniY)
4608Gi Drcv`Kmg~n- (22) (22) (2
2) (22) 2(33)| GLv‡b 2 n‡”Q
wm‡½j Drcv`K, ZvB 2 Øviv fvM Ki‡j msL¨vwU c~Y©
eM©msL¨v n‡e| DËi: 2
21. Ggb jwNó msL¨v wbY©q Ki, hØviv 968 †K fvM
Ki‡j fvMdj GKwU c~Y©eM© msL¨v n‡e| cvwUMwYZ-
hv`eP›`ª PµeZx©
968 Gi Drcv`Kmg~n = 2(22)(1111)
| GLv‡b 2 n‡”Q wm‡½j Drcv`K, ZvB 2 Øviv fvM
Ki‡j fvMdj GKwU c~Y©eM© msL¨v n‡e| DËi: 2
22. 2450 msL¨vwU‡K KZ Øviv ¸Y Ki‡j msL¨vwU c~Y©
eM©msL¨v n‡e? mvaviY exgv K‡c©v‡ikb, Rywbqi Awdmvi c`
cixÿv-16
2 5
7 11 DËit K
2450 = 25577, GLv‡b 2 wm‡½j,
ZvB 2 Øviv ¸Y Ki‡j msL¨vwU c~Y©eM© n‡e|
23. †Kvb& ÿz`ªZg msL¨v w`‡q 294 †K ¸Y Ki‡j Zv GKwU
c~Y©eM© n‡e? iƒcvjx e¨vsK wmwbqi Awdmvi-13
2 3
6 24 DËi: M
294 = 2377, GLv‡b 2 I 3 wm‡½j ZvB
23 = 6 Øviv ¸Y Ki‡j msL¨vwU c~Y©eM© n‡e|
24. †Kvb ÿz`ªZg msL¨v Øviv 1470 †K fvM Ki‡j fvMdj
GKwU c~Y©eM© msL¨v n‡e| cjøx Kg© mnvqK dvD‡Ûk‡bi
A¨vwmm‡U›U g¨v‡bRvi- 2014
5 6
15 30 DËi: N
1470 = 23577, GLv‡b 2, 3 I 5
wm‡½j, ZvB 235 = 30 Øviv fvM Ki‡j msL¨vwU
c~Y©eM© n‡e|
25.
5
(24) †K b~b¨Zg KZ Øviv ¸Y Ki‡j ¸Ydj GKwU
c~Y©eM© msL¨v n‡e? ¯^v¯’¨ cÖ‡K․kj Awa`߇ii mnKvix
cÖ‡K․kjx (wmwfj): 17
2 3
6 4 DËi: M
24 †K Drcv`‡K we‡kølY Ki‡j `vovq- 222
3| 24 †K c~Y©eM© Ki‡Z wm‡½j Drcv`K 2 I 3
†K †Rvovq iƒcvšÍi Ki‡Z n‡e| A_v©r, 23 = 6
Øviv ¸Y Ki‡Z n‡e| wKš‘ cÖ‡kœ ïay 24 †K c~Y©
eM©msL¨v Kivi K_v ejv nqwb, ejv n‡q‡Q
5
(24) †K
c~Y© eM©msL¨vq iƒcvšÍi Ki‡Z n‡e| g‡b ivLyb- †Kvb
msL¨v hw` c~Y© eM©msL¨v nq Zvn‡j H msL¨vi hZ
¸wYZK/¸Ydj •Zwi Kiv n‡e ev H msL¨vi hZ
cvIqviB _vKzK bv †Kb c~Y©eM© msL¨v n‡e| GRb¨
Avgiv ïay 24 †K 23 = 6 Øviv ¸Y K‡i c~Y©
eM©msL¨vq iƒcvšÍi K‡iwQ| †m‡ÿ‡Î c~Y©eM© msL¨vwU
n‡”Q 246 = 144| GKB fv‡e 5
(144) -I n‡”Q
c~Y©eM© msL¨v|
26. 72  75 
3
3 
3
4  8
2 †K b~b¨Zg KZ Øviv
¸Y Ki‡j ¸Ydj GKwU c~Y©eM© msL¨v n‡e? wewfbœ
gš¿Yvj‡qi Dc-mnKvix cÖ‡K․kjx (wmwfj): 17
4 5
2 3 DËi: M
72  75 
3
3 
3
4  8
2
72 Gi Drcv`Kmg~n- 22233 = 3
2 
2
3
75 Gi Drcv`Kmg~n- 355 = 3 
2
5 ;
3
4 Gi Drcv`Kmg~n- 222222 = 6
2 |
=
3
2 
2
3  3 
2
5 
3
3  6
2  8
2
=
863
2 

312
3 

2
5
=
17
2 
6
3 
2
5
GLv‡b 2 n‡”Q 17 wU hv †e‡Rvo msL¨K, ZvB 2 Øviv ¸Y
Ki‡Z n‡e|
27. †Kvb ÿz`ªZg eM©msL¨v‡K 10, 12, 15 Ges 18 Øviv
wbt‡k‡l fvM Kiv hvq? Indiabix.com
600 700
800 900 DËi: NNM
LK
mgvavb
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK
mgvavb
mgvavb
mgvavb

6. Math Tutor  7
(cÖ‡kœ Abymv‡i- Ggb GKwU ÿz`ªZg eM©msL¨v †ei
Ki‡Z n‡e, hv‡K 10, 12, 15 Ges 18 Øviv wbt‡k‡l fvM
Kiv hv‡e| GLv‡b cÖ‡kœ cÖ`Ë 10, 12, 15 Ges 18 Gi
j. mv. ¸ †h msL¨vwU †ei n‡e †mB msL¨vwU‡K 10, 12, 15
Ges 18 Øviv wbt‡k‡l fvM Kiv hv‡e| wKš‘ cÖvß msL¨vwU
eM©msL¨v n‡e wKbv †mwU wbf©i Ki‡e j.mv.¸i cÖvß
Drcv`K¸‡jv †Rvovq †Rvovq cvIqv hvq wKbv Zvi Dci|
Pjyb Av‡M j.mv.¸ K‡i †`Lv hvK|)
2 10, 12, 15, 18
3 5, 6, 15, 9
5 5, 2, 5, 3
1, 2, 1, 3
10, 12, 15 Ges 18 Gi j. mv. ¸
= 22335 = 180
j.mv.¸ †_‡K cÖvß 180 msL¨vwU 10, 12, 15 Ges 18
Øviv wbt‡k‡l wefvR¨ wKš‘ 180 eM©msL¨v bv| KviY,
j.mv.¸ †_‡K cÖvß Drcv`K¸‡jvi me KqwU †Rvovq
†Rvovq †bB, 5 Drcv`KwU wm‡½j Ae¯’vq Av‡Q|
wKš‘ Avgv‡`i cÖ‡kœ †P‡q‡Q cÖvß msL¨vwU eM©msL¨v n‡Z
n‡e| GRb¨ wm‡½j 5 Gi m‡½ 5 ¸Y Ki‡Z n‡e|
GLv‡b 180 (22335) †K eM©msL¨vq
iƒcvšÍi Ki‡Z n‡j 180 †K 5 ¸Y Ki‡Z n‡e|
∴ wb‡Y©q eM©msL¨vwU = 1805 = 900.
28. Ggb jwNó c~Y© eM©msL¨v wbY©q Ki, hvnv 10 Øviv, 16
Øviv Ges 24 Øviv wefvR¨| m~Î:cvwUMwYZ-hv`eP›`ª PµeZx©
10, 16, 24 Gi j.mv.¸ = (22)(22)5
3 = 240| GLv‡b 240 †K eM©msL¨vq iƒcvšÍi
Ki‡Z n‡j 240 †K 53 = 15 Øviv ¸Y Ki‡Z
n‡e| ∴ wb‡b©q eM©msL¨vwU = 240 15 = 3600|
29. GKwU ¯‹z‡j Qv·`i wWªj Kivevi mgq 8, 10 Ges 12
mvwi‡Z mvRv‡bv hvq| Avevi eM©vKv‡iI mvRv‡bv hvq|
H ¯‹z‡j Kgc‡ÿ KZRb QvÎ Av‡Q? 12Zg wewmGm;
wb¤œ gva¨wgK MwYZ-93 wkÿvel©, 7g †kÖwY, D`vniY 16
3600 2400
1200 3000 DËi: K
(cÖ‡kœ ¯‹y‡ji †gvU QvÎ msL¨v Rvb‡Z PvIqv n‡q‡Q,
hv‡`i‡K 8, 10 I 12 mvwi‡Z mvRv‡bv hv‡e| Avgiv Rvwb,
8, 10 I 12 †K j.mv.¸ Ki‡j †gvU QvÎ msL¨v cvIqv
hv‡e| Z‡e wØZxq k‡Z© ejv n‡q‡Q, Zv‡`i‡K eM©vKv‡iI
mvRv‡bv hv‡e| g‡b ivL‡Z n‡e 8, 10 I 12 Gi j.mv.¸
†_‡K cÖvß Drcv`K¸‡jv hw` †Rvovq †Rvovq _v‡K Zvn‡j
j.mv.¸wU‡K eM©vKv‡i mvRv‡bv hv‡e| wKš‘ hw` †Rvovq
†Rvovq bv _v‡K Zvn‡j Drcv`Kmg~n‡K †Rvo •Zwii
gva¨‡g j.mv.¸wU‡K eM© msL¨vq iƒcvšÍi Ki‡Z n‡e| Pjyb
Av‡M j.mv.¸ K‡i †`Lv hvK|)
2 8, 10, 12
2 4, 5, 6
2, 5, 3
∴ j.mv.¸ = 22253 = 120
j.mv.¸ = (22)253 =120 | GLv‡b
Drcv`Kmg~n †_‡K †`Lv hv‡”Q 2, 5 I 3 Drcv`Kmg~n
n‡”Q wm‡½j, ZvB 120 †K eM©vKv‡i mvRv‡bv hv‡e bv| wKš‘
eM©msL¨v nIqvi ev eM©vKv‡i mvRv‡bvi kZ© n‡”Q, Drcv`K-
mg~n‡K †Rvovq †Rvovq _vK‡Z n‡e, ZvB Drcv`Kmg~n‡K
†Rvovq †Rvovq mvRv‡Z n‡e| A_©vr, 120 †K 2 53
= 30 Øviv ¸Y Ki‡Z n‡e| e¨m, Gevi 12030 =
3600 †K 8, 10 I 12 mvwi‡ZI mvRv‡bv hv‡e Ges
eM©vKv‡iI mvRv‡bv hv‡e|
GLv‡b, 120 †K eM© msL¨vq iƒcvšÍi Ki‡Z 30
(253) Øviv ¸Y Ki‡Z n‡e| d‡j cÖvß
msL¨vwU‡K eM©vKv‡i mvRv‡bv hv‡e| myZivs, †gvU QvÎ
n‡e- 12030 = 3600|
 03.05 fv‡Mi mvnv‡h¨ eM©g~j wbY©q c×wZ
D`vni‡Yi mvnv‡h¨ eM©g~j wbY©‡qi c×wZ wb‡P †`Lv‡bv n‡jv-
mgvavb
NM
LK
mgvavb
mgvavb
eM©g~j wbY©‡q `vM Uvbvi wbqg
1| ALÐ As‡k GKK †_‡K µgvš^‡q evgw`‡K cÖwZ `yB A‡¼i Dci `vM w`‡Z n‡e| †hgb- | Z‡e hw` me©ev‡g †Rvo bv _v‡K
Zvn‡j GKwUi DciB `vM Uvb‡Z n‡e| †hgb- 92416|
2| `kwgK As‡k `kwgK we›`yi Wvbcv‡ki A¼ †_‡K ïiæ K‡i Wvbw`‡K µgvš^‡q †Rvovq †Rvovq `vM w`‡Z n‡e| †hgb- 26.5225
Giƒ‡c hw` †`Lv hvq me©‡k‡l gvÎ GKwU A¼ evwK Av‡Q, Z‡e Zvic‡i GKwU k~b¨ ewm‡q `yB A‡¼i Dci `vM w`‡Z n‡e| †hgb-
0. 251 GLv‡b 25 Gi Dci `vM †`qvi ci 1 evKx Av‡Q, ZvB Gi mv‡_ 0 ewm‡q 10-Gi Dci `vM w`‡Z n‡e| A_©vr, .2510 n‡e
1024

7. Math Tutor  7
D`vniYt 6561 Gi eM©g~j wbY©q Kiæb|
 avc-01t Wvbw`K †_‡K evg w`‡K cÖ‡Z¨K †Rvovi Dci
`vM Uvb‡Z n‡e Ges `vM Uvbvi ci
msL¨vi Wvbcv‡k GKwU Lvov `vM Uvb‡Z
n‡e|
 avc-02t msL¨vwUi G‡Kev‡i euv‡qi †Rvo ev GKK A¼
w`‡q MwVZ msL¨vwUi wVK Av‡Mi eM© msL¨vwU Gi wb‡P wj‡L
we‡qvM Kiv nq Ges Wvbcv‡ki Lvov `v‡Mi cv‡k
eM©msL¨vwUi eM©g~j †jLv nq| A_©vr, Lvov `v‡Mi Wvb cv‡k
Ggb GKwU eM©g~j wjL‡Z n‡e hvi
c~Y©eM© msL¨v cÖ_g †Rvo 65Gi mgvb ev
Zvi †P‡q †QvU n‡e| †hgb- Lvov `v‡M
Wvb cv‡k eM©g~j 8 wb‡j 8-Gi c~Y©eM©
msL¨v 64 n‡e, hv 65 Gi wb‡P wj‡L
we‡qvM Ki‡Z n‡e|
avc-03t cÖ_g †Rvo †_‡K cvIqv we‡qvMd‡ji Wv‡b
cieZx© A¼ †Rvo bvwg‡q wjL‡Z
n‡e| d‡j bZzb Av‡iKwU msL¨v •Zwi
n‡e hvi euv cv‡k GKwU Lvov `vM
w`‡q Wv‡bi (eM©g~j) msL¨vwU‡K wظY
K‡i GB Lvov `v‡Mi evu‡q wjL‡Z n‡e
†hgb- we‡qvMdj 1-Gi Wv‡b cieZx© A¼ †Rvo 61 bvwg‡q
†jLv n‡q‡Q Ges bZzb msL¨v 161 Gi evu‡q GKwU Lvov `vM
Uvbv n‡q‡Q| Zvici Wv‡bi 8 †K wظY 16 K‡i bZzb Lvov
`v‡Mi evu‡q †jLv n‡q‡Q|
 avc-04t wظY K‡i cÖvß 16 msL¨vwUi Wv‡b GKwU
Dchy³ A¼ ewm‡q Ggb GKwU msL¨v
•Zwi Ki‡Z n‡e, †h msL¨vwU‡K H
Dchy³ A¼wU Øviv ¸Y Ki‡j bZzb
K‡i cÖvß msL¨vwU n‡e c~‡e© cvIqv
161 Gi mgvb ev Gi †P‡q †QvU |
†hgb- 16-Gi Wv‡b 1 ewm‡q 161
cvIqv hvq, hvi m‡½ Avevi 1 ¸Y K‡i cvIqv hvq 161 †hwU
c~‡e© cvIqv 161 Gi mgvb| Gevi c~‡e©i 161 †_‡K bZzb
K‡i cÖvß 161 we‡qvM K‡i Ki‡Z n‡e|
 P~ovšÍ avct 16 Gi Wvbcv‡k
†h 1 A¼wU emv‡bv n‡q‡Q
(e„‡Ëi wfZi †`Lv‡bv n‡q‡Q)
†mwU Wvb cv‡ki Lvov `v‡Mi 8-
Gi cv‡k wjL‡Z n‡e| Gfv‡e
81 cvIqv | Gfv‡e cÖvß 81
n‡”Q 6561 Gi eM©g~j |
D`vniYt 31684 Gi eM©g~j wbY©q Kiæb|
 avc-01t Wvbw`K †_‡K evg w`‡K cÖ‡Z¨K †Rvovi Dci
`vM Uvbv nj, Z‡e me© ev‡g hw` †Rvo bv nq Zvn‡j GKwUi
DciB `vM Uvb‡Z n‡e| GLv‡b me© ev‡g
ïay 3 Gi Dci `vM Uvbv nj Ges `vM
Uvbvi ci 31684 msL¨vi Wvbcv‡k GKwU Lvov `vM Uvbv
nj|
 avc-02t msL¨vwUi G‡Kev‡i cÖ_g
GKK A¼ 3 msL¨vwUi wb‡P wVK Av‡Mi
eM© msL¨v 1 †K wj‡L we‡qvM Kiv nj
Ges Wvbcv‡ki Lvov `v‡Mi cv‡k
eM©msL¨vwUi eM©g~j 1 †jLv nj|
 avc-03t cÖ_g GKK A¼ †_‡K cvIqv we‡qvMdj 2 Gi
Wv‡b cieZx© A¼ †Rvo 16 bvwg‡q †jLvq
bZzb Av‡iKwU msL¨v 216 •Zwi nj, hvi
euv cv‡k GKwU Lvov `vM w`‡q Wv‡bi
msL¨v 1 †K wظY (1 ×2 =2) K‡i
GB Lvov `v‡Mi evu‡q †jLv nj|
 avc-04t wظY K‡i cÖvß 2 msL¨vwUi Wv‡b GKwU Dchy³
A¼ 7 ewm‡q 27 •Zwi Kiv nj Ges 27 †K 7 w`‡q ¸Y
K‡i 189 cvIqv †Mj| Zvici c~‡e©
cvIqv 216 †_‡K 189 we‡qvM
Kivq we‡qvMdj `vovj 27|
Gevi GB Dchy³ A¼ 7 †K Wv‡bi
Lvov `v‡Mi cv‡ki 1-Gi cv‡k †jLv
nj, d‡j msL¨vwU `vovj 17|
 avc-05t we‡qvMdj 27 Gi Wv‡b cieZx© A¼ †Rvo 84
bvwg‡q †jLv nj, d‡j bZzb Av‡iKwU
msL¨v 2784 •Zwi nj, hvi euv cv‡k
GKwU bZzb Lvov `vM Uvbv nj Ges
Wv‡bi Lvov `v‡Mi cv‡ki 17 †K
wظY K‡i GB bZzb Lvov `v‡Mi
cv‡k 34 †jLv nj|
P~ovšÍ avct wظY K‡i cÖvß 34 msL¨vwUi Wv‡b GKwU
Dchy³ A¼ 8 ewm‡q 348 •Zwi Kiv nj Ges 348 †K 8
w`‡q ¸Y K‡i 2784 cvIqv †Mj|
Zvici c~‡e© cvIqv 22784 †_‡K
2784 we‡qvM Kivq we‡qvMdj
`vovj 0| Gevi GB Dchy³ A¼
8 †K Wv‡bi Lvov `v‡Mi cv‡ki
17-Gi cv‡k †jLv nj, d‡j
msL¨vwU `vovj 178| GB 178 n‡”Q
31684 Gi eM©g~j|

8. 8  Math Tutor
D`vniYt fv‡Mi mvnv‡h¨ 92416 msL¨vwUi eM©g~j wbY©q
Kiæb|
 avc-01, 02 I 03 t GavcwU
c~‡e©i mgvavb `ywUi b¨vq n‡e|
cÖ‡qvR‡b Avevi †`‡L wb‡Z
cv‡ib|
 avc-4t wظY K‡i cÖvß 6
msL¨vwUi Wv‡b GKwU Dchy³ A¼
ewm‡q GKwU msL¨v ‣Zwi Ki‡Z n‡e
Ges D³ msL¨v‡K H Dchy³ A¼
w`‡q ¸Y Ki‡Z n‡e| Avgiv hw`
Dchy³ A¼wU 1 emvB, Zvn‡j
msL¨vwU n‡e 61 Ges 61 †K H 1 Øviv ¸Y Ki‡j ¸Ydj
n‡e 61, hv c~‡e© cvIqv 24 Gi †P‡q eo n‡q hvq, ZvB 1
emv‡bv hv‡e bv| hw` k~b¨ emvB Zvn‡j msLvwU n‡e 60 Ges
60 †K 0 Øviv ¸Y Ki‡j ¸Ydj n‡e 00, hv 24 Gi †P‡q
†QvU, ZvB Dchy³ A¼ 0 emv‡bv hv‡e| dvBbvwj 24 †_‡K
00 we‡qvM Ki‡j Gevi we‡qvMdj n‡e 24| Gevi GB
Dchy³ A¼ 0 †K Wv‡bi Lvov `v‡Mi cv‡ki 3-Gi cv‡k
wjL‡Z n‡e, d‡j msL¨vwU `vov‡e 30|
 k~‡b¨i avcwU Dc‡ii wP‡Îi gZ bv
K‡i Avcwb Gfv‡e fve‡Z cv‡ib-
Dci †_‡K †Rvo wb‡P †K‡U Avbvi
ciI fvM Kiv bv †M‡j Wvbcv‡ki
Lvov `v‡Mi cv‡ki msL¨vi mv‡_ I bZzb Lvov `v‡Mi evu‡qi
wظ‡Yi mv‡_ k~b¨ (0) emv‡Z nq| cv‡ki wP‡Î †`Lyb,
Wvbcv‡ki Lvov `v‡Mi cv‡ki 3-Gi mv‡_ 0 (AvÛvijvBb
Kiv) ewm‡q 30 Ges bZzb Lvov `v‡Mi evu‡qi wظY 6-Gi
mv‡_ 0 (AvÛvijvBb Kiv) ewm‡q 60 evbv‡bv n‡q‡Q|
 P~ovšÍ avct h_vixwZ 60 Gi Wv‡b 4 ewm‡q 604 cvIqv
hvq , hvi m‡½ 4 ¸Y K‡i cvIqv hvq 2416 Ges GB cÖvß
2416 msL¨vwU c~‡e© cÖvß 2416 Gi
mgvb, hv‡`i‡K we‡qvM Ki‡j
we‡qvMdj 0 nq| Gevi 60 Gi
Wv‡b emv‡bv 4 †K Wvbcv‡ki Lvov
`v‡Mi cv‡ki 30-Gi c‡i emv‡bv n‡q‡Q| Gevi cÖvß 304
msL¨vwUB n‡”Q 92416 Gi eM©g~j|
wb‡R PP©v Kiæb
(K) 2304 (L) 1444 (M) 55225
DËit (K) 48 (L) 38 (M) 235

9. Math Tutor  9
 03.06 Non Perfect Square Root Kivi c×wZ
†hme msL¨v c~Y©eM© msL¨v bq, †mme msL¨vi eM©g~j †ei Kiv wbqgvewj Av‡jvPbv Kiv n‡jv|
30. fv‡Mi mvnv‡h¨ 2 Gi eM©g~j wbY©q Kiæb|
 avc-01: Wvbcv‡ki Lvov `v‡Mi cv‡ki msL¨vq `kwgK ewm‡q wb‡Pi we‡qvMd‡ji mv‡_ k~b¨ emv‡bvi wbqgt-
2 | 1 
1
2 1 00
 avc-02: Lvov `v‡Mi Wvbcv‡ki `kwgKhy³ msL¨vi wظY I cÖwZ av‡c we‡qvMd‡j `ywU K‡i k~b¨ (00) †bqvi
wbqgt-
2 | 1  4
1
24 100
96
28 4 00
 avc-03: Ab¨vb¨ wbqg h_vixwZ AbymiY Ki‡Z n‡e|
2 | 1414 ...
1
24 100
96
281 400
281
2824 11900
11296
604 (Gfv‡e Pj‡Z _vK‡e...)
 PP©v-01 : (K) 3 (L) 5 (M) 6 (N) 7 (O) 8 (P) 10 -Gi eM©g~j wbY©q Kiæb|
(K) 3 1.73 (L) 5 2.23 (M) 6 2.44 (N) 7 2.64 (O) 8 2.82 (P) 10 3.16
1 4 4 4 4 9
27 200 42 100 44 200 46 300 48 400 61 100
189 84 176 276 384 61
343 1100 443 1600 484 2400 524 2400 562 1600 626 3900
1029 1329 1936 2096 1124 3756
71 271 464 304 476 144
(K) cÖ_g av‡c fvM w`‡q we‡qvM Kivi ci we‡qvMd‡ji cv‡k Dci †_‡K cieZx©
†Rvo †K‡U wb‡q Avm‡Z nq| wKš‘ †K‡U Avbvi gZ hw` †Kvb †Rvo bv _v‡K, †m‡ÿ‡Î
Wvbcv‡ki Lvov `v‡Mi cv‡ki msL¨vq `kwgK () ewm‡q wb‡Pi H we‡qvMd‡ji cv‡k
GK‡Rvov k~b¨ (00) emv‡Z nq| †hgb- Wvb cv‡ki Lvov `v‡Mi 1 Gici `kwgK (.)
ewm‡q wb‡Pi we‡qvMd‡ji cv‡k GK‡Rvov k~b¨ ewm‡q 100 Kiv n‡q‡Q|
(L) `kwgKhy³ fvMdj 1.4 Gi wظY †bqvi wbqg n‡”Q `kwgK ev` w`‡q 14-Gi
wظY †bqv | †hgb- cv‡ki mgvav‡b 1.4 Gi cwie‡Z© 14 Gi wظY 28 †bqv n‡q‡Q|
(M) Wvbcv‡ki Lvov `v‡Mi `kwg‡Ki ci GKevi e‡m †M‡j, wb‡Pi we‡qvMd‡j Dci
†_‡K †K‡U Avbvi my‡hvM bv _vK‡j cÖwZevi GK‡Rvov k~b¨ (00) emv‡Z nq| †hgb-
we‡qvMdj 4 Gici GK‡Rvov k~b¨ (00) emv‡bv n‡q‡Q| Gfv‡e Pj‡e..
(N) Gfv‡e hZÿY bv fvM‡kl wR‡iv (0) n‡”Q ZZÿY ch©šÍ `kwg‡Ki ci GK‡Rvov
k~b¨ we‡qvMd‡j hy³ Ki‡Z nq| Z‡e †g․wjK msL¨vi eM©g~j Amxg Ni ch©šÍ Pj‡Z
_vK‡j Avcbvi hZ Ni `iKvi ZZ Ni ch©šÍ mgvavb Ki‡jB Pj‡e|

10. 10  Math Tutor
 03.07 `kwgK fMœvs‡ki eM©g~j wbY©q Kivi c×wZ
c~Y©msL¨vi eM©g~j wbY©‡qi †h wbqg Avgiv AbymiY K‡iwQ, `kwgK fMœvs‡ki eM©g~j wbY©q Kivi †ÿ‡ÎI †mB GKB wbqg
AbymiY Kie| G‡ÿ‡Î AwZwi³ wbqgvewj Zz‡j aiv n‡jv-
(K) mvaviY wbq‡g eM©g~j wbY©‡qi cÖwµqvq ALÛ As‡ki KvR †kl K‡i `kwgK we›`yi c‡ii cÖ_g `yBwU A¼
bvgv‡bvi Av‡MB eM©g~‡j `kwgK we›`y w`‡Z nq|
(L) `kwgK we›`yi ci we‡qvMd‡j GK‡Rvov k~b¨ wb‡j eM©g~‡j `kwgK we›`yi ci GKwU k~b¨ w`‡Z nq|
31. 644.1444 Gi eM©g~j wbY©q Kiæb| wb¤œ gva¨wgK MwYZ,
7g †kÖwY, D`vniY 8
6 44 . 14 44 25.38
4
45 244
225
503 1914
1509
5068 40544
40544
0
32. 0.001936 Gi eM©g~j wbY©q Kiæb| [wb¤œ gva¨wgK MwYZ,
7g †kÖwY, D`vniY 9]
0.00 19 36 .044
16
84 3 36
3 36
0
33. 25.462 Gi eM©g~j `yB `kwgK ¯’vb ch©šÍ wbY©q Kiæb|
wb¤œ gva¨wgK MwYZ, 7g †kÖwY, D`vniY 10
( wZb `kwgK ¯’vb ch©šÍ eM©g~j wbY©q Ki‡Z n‡j msL¨vi
`kwgK we›`yi ci Kgc‡ÿ 6wU A¼ wb‡Z n‡e| `iKvi n‡j
Wvbw`‡Ki †kl A‡¼i ci cÖ‡qvRbg‡Zv k~b¨ emv‡Z nq|
G‡Z msL¨vi gv‡bi cwieZ©b nq bv|)
25.46 20 00 5.045
25
1004 46 20
40 16
10085 604 00
504 25
99 75
`yB `kwgK ¯’vb ch©šÍ DËi †jLvi wbqgt (K) eM©g~‡j
`kwg‡Ki Z…Zxq A¼wU 0, 1, 2, 3 ev 4 n‡j wØZxq A¼wU hv
wQj ZvB _vK‡e| (L) eM©g~‡j `kwg‡Ki Z…Zxq A¼wU 5, 6,
7, 8 ev 9 n‡j, wØZxq A‡¼i mv‡_ 1 †hvM Ki‡Z n‡e|
†hgb- cÖvß eM©g~j 5.045 Gi Z…Zxq A¼wU 5 nIqvq wØZxq
A¼wUi mv‡_ 1 †hvM Ki‡Z n‡e| myZivs mwVK DËi n‡e
5.05|
 PP©v-02 : (K) 0.1 (L) 0.01 (M) 0.001 (N) 0.0001 (O) 0.00001 I (P) 0.000001 -Gi eM©g~j
wbY©q Kiæb|
`kwgK w`‡q ïiæ †Kvb msL¨vi eM©g~j wbY©q Kivi Rb¨ fvMd‡ji ïiæ‡ZB `kwgK w`‡Z nq Ges evKx Ab¨vb¨ wbqg¸‡jv
h_vixwZ AbymiY Ki‡Z nq| (K), (M), (O) bs †e‡Rvo msL¨K nIqvq GKwU k~b¨ (0) w`‡q †Rvo Kiv n‡q‡Q| gRvi e¨vcvi
†`Lyb- (K), (M) I (O) Gi DËi c¨vUvb© GKB ai‡Yi Ges (L), (N) I (P) Gi DËi c¨vUvb©I wKš‘ GKB ai‡Yi|
 g‡b ivLybt- `kwgK msL¨vi eM©g~j wbY©q Kivi †ÿ‡Î Av‡M Wvbcv‡ki `v‡Mi cv‡k `kwgK w`‡q Zvici mgvavb Ki‡Z nq|
(K) .10 .316 (L) .01 .1 (M) .0010 .0316 (N) .0001 .01 (O) .000010 .00316 (P) .000001 .001
9 1 9 1 9 1
61 100 0 61 100 0 61 100 0
61 61 61
626 3900 626 3900 626 3900
3756 3756 3756
144 144 144
mgvavb
mgvavb
mgvavb
†Rvo †K‡U Avbvi ciI hw`
fvM bv hvq Zvn‡j cieZx© †Rvo
†K‡U Avb‡Z n‡e Ges Gi Av‡M
†K‡U Avbv †Rvovi Rb¨ eM©g~‡j
GKwU k~b¨ w`‡Z n‡e| †hgb-
46 †K‡U Avbvi ci fvM bv
hvIqvq eM©g~‡j 0 w`‡q cieZx©
†Rvo 20 †K‡U Avbv nj Ges
50 Gi wظY 100 †bqv nj|

11. Math Tutor  11
 PP©v-03: (K) 0.2 (L) 0.02 (M) 0.002 (N) 0.0002 (O) 0.00002 I (P) 0.000002 -Gi eM©g~j
wbY©q Kiæb|
(K) .20 .447 (L) .02 .141 (M) .0020 .0447 (N) .0002 .0141 (O) .000020 .00447 (P) .000002 .00141
16 1 16 1 16 1
84 400 24 100 84 400 24 100 84 400 24 100
336 96 336 96 336 96
887 6400 281 400 887 6400 281 400 887 6400 281 400
6209 281 6209 281 6209 281
191 119 191 119 191 119
 PP©v-04: (K) 0.9 (L) 0.09 (M) 0.009 (N) 0.0009 (O) 0.00009 I (P) 0.000009 -Gi eM©g~j
wbY©q Kiæb|
(K) .90 .948 (L) .09 .3 (M) .0090 .0948 (N) .0009 .03 (O) .000090 .00948 (P) .000009 .003
81 9 81 9 81 9
184 900 0 184 900 0 184 900 0
736 736 736
1888 16400 1888 16400 1888 16400
15104 15104 15104
1296 1296 1296
wb‡R wb‡R Kiæb
 (K) 0.3 (L) 0.03 (M) 0.003 (N) 0.0003 (O) 0.00003 (P) 0.000003 -Gi eM©g~j wbY©q Kiæb|
DËi: (K) .547 ... (L) .173 ... (M) .0547 ... (N) .0173 ... (O) .00547 ... (P) .00173 ...
 (K) 0.4 (L) 0.04 (M) 0.004 (N) 0.0004 (O) 0.00004 (P) 0.000004-Gi eM©g~j wbY©q Kiæb|
DËi: (K) .632 ... (L) .2 (M) .0632 ... (N) .02 (O) .00632 ... (P) .002
34. 2 Ag~j` msL¨vwUi Avmbœ gvb n‡e- Kvivv Awa`߇ii
Kviv ZË¡veavqK: 10
2.414 1.414
1.421 2.412 DËi: L
35. 0.00000625 = KZ? AvenvIqv Awa`߇ii mnKvix
AvenvIqvwe`: 00
0.0025 0.00025
0.000025 0.00625 DËi: K
.00000625 .0025
4
45 225
225
0
 kU©KvU t- .00000625 ai‡Yi msL¨v n‡j- cÖ_‡g
†`L‡eb k~b¨ eZxZ †h msL¨vwU Av‡Q †mwU c~Y©eM©
wKbv? †hgb- GLv‡b 625 Av‡Q †hwU c~Y©eM© Ges Gi
eM©g~j 25| Zvici wØZxqZ †`L‡eb, `kwg‡Ki ci
†gvU wWwRU¸‡jv †Rvo msL¨K wKbv? †hgb- GLv‡b
†gvU wWwRU Av‡Q 8wU| Giƒc‡ÿ‡Î .00000625-
Gi eM©g~j n‡e 8Gi A‡a©K 4 wWwR‡Ui| GB 4
wWwR‡Ui g‡a¨ c~Y©eM© msL¨vwUi eM©g~‡ji 2wU wWwRU
†bqvi ci evKx `ywU wWwRU 0 w`‡q c~iY Ki‡eb-
 = .0025 |
36. 0.09= KZ? mgvR‡mev Awa`߇ii cÖ‡ekb Awdmvi: 13
0.03 0.3
0.003 0.0003 DËi: L
k~Y¨mn `kwg‡Ki ci 2wU wWwRU Av‡Q (hv †Rvomgvavb
NM
LK
mgvavb
NM
LK
NM
LK
00 25

12. 12  Math Tutor
msL¨K) Ges 9 c~Y©eM© msL¨v ZvB `kwg‡Ki ci
eM©g~j 1 wWwR‡Ui n‡e, hv ïay Ackb †Z Av‡Q|
we¯ÍvwiZ mgvav‡bi Rb¨ PP©v-04 co–b|
37. 0.0009 = KZ? Lv`¨ Awa`߇ii Lv`¨ cwi`k©K: 00
0.03 0.3
0.003 0.0003 DËi: K
k~Y¨mn `kwg‡Ki ci 4wU wWwRU Av‡Q (hv †Rvo
msL¨K) Ges 9 c~Y©eM© msL¨v ZvB `kwg‡Ki ci
eM©g~j 2 wWwR‡Ui n‡e, hvi gv‡S 9 Gi eM©g~j (3)
†_‡K 1wU wWwRU Ges Avi 1wU wWwRU 0 emv‡Z n‡e|
GwU ïay Ackb †Z Av‡Q| we¯ÍvwiZ mgvav‡bi
Rb¨ PP©v-04 co–b|
38. 0.000009 = KZ? GbGmAvB Gi mn. cwiPvjK: 15
0.03 0.3
0.003 0.0003 DËi: M
k~Y¨mn `kwg‡Ki ci 6wU wWwRU Av‡Q (hv †Rvo
msL¨K) Ges 9 c~Y©eM© msL¨v ZvB `kwg‡Ki ci eM©g~j
3 wWwR‡Ui n‡e, hvi gv‡S 9 Gi eM©g~j (3) †_‡K 1wU
wWwRU Ges AviI 2wU wWwRU 0 emv‡Z n‡e| GwU ïay
Ackb †Z Av‡Q| we¯ÍvwiZ mgvav‡bi Rb¨ PP©v-04 co–b|
39. 0.1 Gi eM©g~j KZ? evwZjK…Z wewmGm/ cÖvK-cÖv_wgK
mnKvix wkÿK: 15
0.1 0.01
0.25 †Kv‡bvwUB bq DËi: N
40. 0.001 Gi eM©g~j KZ? cÖv_wgK we`¨vjq cÖavb wkÿK: 94
0.1 0.01
0.001 †Kv‡bvwUB bq DËi: N
41. 0.0001 Gi eM©g~j KZ? gva¨wgK I D”P wkÿv Awa`߇ii
wnmve mnKvix: 13/ gva¨wgK we`¨vjq mnKvix wkÿK: 00
0.1 0.01
0.001 1 DËi: L
42. 15.6025 = ? 36Zg wewmGm (wcÖwj.)
3.85 3.75
3.95 3.65 DËi: M
15.6025 3.95
9
69 660
621
785 3925
3925
0
43. 15.6323 = ? †ijc_ gš¿Yvj‡qi Dc-mn. cÖ‡K․kjx: 17
3.85 3.95
3.75 3.20 DËi: L
15.6325 3.95 (cÖvq)
9
69 663
621
785 4223
3925
298
44. 25.36 Gi eM©g~j KZ? ¯^v¯’¨ wkÿv I cwievi Kj¨vY
wefvM Awdm mnKvix Kvg Kw¤úDUvi : 19
5.036 5.03
5.6 3.5 DËi: L
25.36 5.03
25
1003 3600
3009
591
 wØZxq †Rvo 36 †K‡U wb‡P wb‡q Avmvi ci djvdj
5 Gi wظY 10 Gi ci wgwbgvg 1 emv‡jI 36 Gi
†P‡q eo n‡q hvq, ZvB 36 Gi mv‡_ GK‡Rvov ïb¨
ewm‡q wbqgvbyhvqx djvd‡jI GKUv k~b¨ (0) emv‡bv
n‡q‡Q| Gevi djvdj 50 Gi wظY 100 K‡i Ges
Zvici 3 ewm‡q mgvavb Kiv n‡q‡Q|
mgvavb
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK
NM
LK
NM
LK
NM
LK
M
mgvavb
NM
LK
K
mgvavb
NM
LK
L

13. Math Tutor  13
†R‡b wbb - 20
 eM© I eM©g~‡ji †h welq¸‡jv Rvbv Riæwi-
eM© I eM©g~j m¤úwK©Z ev¯Íe mgm¨vewj mgvavb Kivi Rb¨ †h welq¸‡jv Rvbv Riæwi
Zv wb‡P Zz‡j aiv n‡jv-
Zvn‡j †Zv eyS‡ZB cvi‡Qb, 4wU Mv‡Qi mvwi I cÖwZ Kjv‡g 4wU K‡i MvQ _vKvi A_© n‡”Q GwU GKwU ÔeM©Õ m¤úwK©Z
mgm¨v| A_©vr, GLv‡bi †gvU Mv‡Qi msL¨v n‡”Q 4 Gi eM© A_©vr 16 wU MvQ| (†hLv‡b, 4 × 4 = 16) A_ev aiæb
cÖkœwU Gfv‡e Kiv n‡jv- †Kv‡bv evMv‡b †gvU 16wU MvQ Av‡Q| evMv‡bi •`N¨© I cÖ‡¯’i Dfq w`‡Ki cÖ‡Z¨K mvwi‡Z
mgvb msL¨K MvQ _vK‡j cÖ‡Z¨K mvwi‡Z Mv‡Qi msL¨v wbY©q Kiæb| GLv‡b •`N¨© I cÖ‡¯’i Dfq w`‡K cÖ‡Z¨K mvwi‡Z
ÔmgvbÕ msL¨K MvQ _vKv gv‡b wKš‘ †mB eM©g~j‡KB eySv‡”Q! A_©vr, GLv‡bi cÖ‡Z¨K mvwii MvQ gv‡bB n‡”Q 16 Gi
eM©g~j 4|
 Avgiv wK wkLjvg? Avgiv hv hv wkLjvg Zv wb‡¤œ D‡jøL Kiv n‡jv- (GB Av‡jvPbvUzKz Aek¨B co–b)
(1) mvwi I Kjvg GKB msL¨v n‡j †KvbwKQzi †gvU n‡e mvwi A_ev Kjv‡gi eM©| †hgb-
Dc‡iv³ D`vni‡Y mvwi 4 I Kjvg 4, ZvB †gvU MvQ n‡”Q 4 Gi eM© 16 wU| GKBfv‡e,
•`N¨© I cÖ¯’ mgvb n‡j †gvU eM© GKK n‡e ‣`N©¨ ev cÖ‡¯’i eM©|
(2) †Kvb wKQzi †gvU †`qv _vK‡j Ges ÔcÖwZ mvwi‡Z KZ Rb QvÎ/‣mb¨ msL¨v Av‡Q?Õ PvIqv
n‡j DËi n‡e cÖ`Ë †gvU msL¨vwUi eM©g~j| A_v©r, ÔcÖwZ mvwi ev cÖwZ KjvgÕ ej‡Z
eM©g~j‡K eywS‡q _v‡K|
(3) ÔhZ Rb , ZZ UvKvÕ ej‡Z c~Y©eM© aviYv‡KB eywS‡q _v‡K| †hgb- GKwU K¬v‡m hZ QvÎ
Av‡Q, cÖ‡Z¨‡K ZZ UvKv K‡i cÖ`vb Ki‡j †gvU UvKv KZ n‡e? GLv‡b hZ QvÎ ej‡Z
hw` x nq, ZZ UvKv ej‡ZI x n‡e| Zvn‡j †gvU UvKv n‡e x Gi eM© x2
| wKš‘ Kxfv‡e?
aiv hvK- H K¬v‡m K I L 2 Rb QvÎ Av‡Q A_©vr QvÎ msL¨v 2 Rb| cÖkœvbymv‡i, QvÎ msL¨v hZ n‡e Zviv UvKvI
w`‡e ZZ K‡i| †h‡nZz QvÎ msL¨v †cjvg K I L 2 Rb, †m‡nZz K w`‡e 2 UvKv Ges L w`‡e 2 UvKv, Zvn‡j
†gvU UvKv n‡e 2 + 2 = 4 UvKv| MvwYwZK fvlvq, Avgiv †h msL¨v¸‡jv †hvM Kwi, †m msL¨v¸‡jv hw` evievi
GKB msL¨v nq Zvn‡j Gfv‡e 2 + 2 = 4 bv wj‡L ¸Y AvKv‡i wjL‡Z cvwi| †hgb- GLv‡b 2 Av‡Q 2 wU, Zvn‡j
2 Gi mv‡_ 2 ¸Y Ki‡j ¸Ydj n‡e 4| GRb¨B Avgiv hZ Rb, ZZ UvKv _vK‡j hZ = x Ges ZZ = x a‡i
A¼ Kwi| G‡ÿ‡Î †gvU UvKv nq x × x = x2
A_©vr x Gi eM©|
45. †Kv‡bv evMv‡b 8wU Mv‡Qi mvwi Av‡Q| cÖ‡Z¨K mvwi‡Z
8wU K‡i MvQ jvMv‡j KZwU Mv‡Qi cÖ‡qvRb n‡e? wb¤œ
gva¨wgK MwYZ, mßg †kÖwY-1997 wkÿvel©
(†h‡nZz Mv‡Qi mvwi 8 Ges cÖ‡Z¨K mvwi‡Z MvQ Av‡Q
8wU, ZvB †gvU Mv‡Qi msL¨v n‡e 8 Gi eM©!)
†gvU MvQ = mvwi 8wU × cÖwZ mvwi‡Z MvQ 8 wU
= 64wU
A_ev, HwKK w`‡qI mgvavb Ki‡Z cv‡ib|
1 mvwi‡Z Mv‡Qi msL¨v 8 wU
∴ 8 Ó Ó Ó 8 × 8 wU = 64 wU |
mgvavb
mvwi=4
Kjvg = 4
 Kjvg (column) I mvwi (Row)t Wvb w`K †_‡K evg w`‡K
cvkvcvwk Ae¯’vbiZ MvQ¸‡jv wg‡j GKwU mvwi Ges Dci †_‡K wb‡Pi
w`‡K MvQ¸‡jv wb‡q GKwU Kjvg •Zwi n‡q‡Q| wP‡Î Giƒc 4wU mvwi Ges
4wU Kjvg i‡q‡Q| GLv‡b, 4wU mvwi Ges 4wU Kjvg ¸Y Ki‡j †gvU
16wU MvQ cv‡eb|
 wPšÍb cªwµqvt
1| aiæb cÖkœ Kiv n‡jv- †Kv‡bv evMv‡b 4 wU Mv‡Qi mvwi Av‡Q| cÖ‡Z¨‡K
mvwi‡Z 4wU K‡i MvQ jvMv‡j KZwU Mv‡Qi cÖ‡qvRb n‡e?
Mv‡Qimvwi8wU
‣`N©¨
cÖ¯’

14. 14  Math Tutor
46. †Kv‡bv evMv‡b 1024wU bvwi‡Kj MvQ Av‡Q| •`N©¨ I
we¯Ív‡i Dfq w`‡Ki cÖ‡Z¨K mvwi‡Z mgvb msL¨K MvQ
_vK‡j cÖ‡Z¨K mvwi‡Z KZwU MvQ Av‡Q? wb¤œ gva¨wgK
MwYZ, mßg †kÖwY-1997 wkÿvel©
•`N¨© I we¯Ívi Dfq w`‡K Mv‡Qi msL¨v mgvb nIqvi
A_©B n‡”Q †gvU bvwi‡Kj Mv‡Qi eM©g~jB n‡”Q cÖ‡Z¨K
mvwii Mv‡Qi msL¨v|
1024 = 32|
myZivs, cÖ‡Z¨K mvwi‡Z 32wU Av‡Q|
47. †Kvb we`¨vj‡qi 2704 Rb wkÿv_x©‡K cÖvZ¨wnK
mgv‡ek Kivi Rb¨ eM©vKv‡i mvRv‡bv nj| cÖ‡Z¨K
mvwi‡Z wkÿv_x©i msL¨v wbY©q Kiæb? wb¤œ gva¨wgK MwYZ-
2013 wkÿvel©, 7g †kÖwY, Abykxjbx 1.2 Gi 16bs cÖkœ
(eM©vKv‡i mvRv‡bv nj gv‡b 2704 msL¨vwU c~Y©eM©
msL¨v| G‡ÿ‡Î cÖ‡Z¨K mvwii wkÿv_x©i msL¨v n‡e
2704 Gi eM©g~j)
2704 = 52|
myZivs, cÖ‡Z¨K mvwi‡Z 52 Rb K‡i wkÿv_©x Av‡Q|
48. †Kvb evMv‡b 1296wU AvgMvQ Av‡Q| evMv‡bi •`N©¨ I
cÖ‡¯’i Dfq w`‡Ki cÖ‡Z¨K mvwi‡Z mgvb msL¨K
AvgMvQ _vK‡j cÖ‡Z¨K mvwi‡Z Mv‡Qi msL¨v wbY©q
Kiæb| wb¤œ gva¨wgK MwYZ, mßg †kÖwY-13 wkÿvel©, Abykxjbx
1.2Gi 9 bs D`vniY
(•`N©¨ I cÖ‡¯’i Dfqw`‡K mgvb †h K_v mvwi I
Kjvg mgvb nIqv GKB K_v| GLv‡b cÖ‡Z¨K mvwi‡Z
AvgMv‡Qi msL¨v ej‡Z 1296 Gi eM©g~‡ji K_v ejv
n‡”Q|
1296 = 36|
myZivs, cÖ‡Z¨K mvwi‡Z AvgMv‡Qi msL¨v 36wU|
49. 20740 msL¨K •mb¨‡K eM©vKv‡i mvRv‡Z wM‡q 4 Rb
AwZwi³ nq| cÖwZ mvwi‡Z •mb¨ msL¨v- GKwU evwo
GKwU Lvgvi cÖK‡íi Dc‡Rjv mgš^qKvix: 17; GbGmAvB Gi
mnKvix cwiPvjK: 15
142 144
136 140 DËi: L
(20740 msL¨K •mb¨‡K eM©vKv‡i mvRv‡Z wM‡q †h‡nZz
4 Rb AwZwi³ nq, †m‡nZz 20740 †_‡K 4 ev` w`‡q
20736 †K c~Y©eM© msL¨v Ki‡Z n‡e| cÖ‡kœ cÖwZ mvwi‡Z
•mb¨ msL¨v PvIqv n‡q‡Q, ZvB 20736 Gi eM©g~jB n‡e
cÖwZ mvwii ‣mb¨ msL¨v)
eM©vKvi msL¨v = 20740 - 4 = 20736
2 07 36 144
1
24 1 07
96
284 11 36
11 36
0
myZivs, cÖwZ mvwi‡Z •mb¨ msL¨v 144 Rb|
50. †Kv‡bv evMv‡b 1800 wU PvivMvQ eM©vKv‡i jvMv‡Z
wM‡q 36wU Pviv †ewk n‡jv| eM©vKv‡i mvRv‡bvi c‡i
cÖwZwU mvwi‡Z Pvivi msL¨v KZ? wb¤œ gva¨wgK MwYZ-2013
wkÿvel©, 7g †kÖwY, Abykxjbx 1.2 Gi 18 bs cÖkœ, 12Zg
we‡RGm (mnKvix RR) cÖv_wgK cixÿv 2018
eM©vKvi msL¨v = 1800 - 36 = 1764
17 64 42
16
82 1 64
1 64
0
myZivs, cÖwZ mvwi‡Z Pvivi msL¨v = 42 wU|
51. GK †mbvcwZ 63009 Rb •mb¨ wb‡q eM©vKv‡i
mvRv‡Z wM‡q †`L‡jb 8 Rb •mb¨ DØ„Ë i‡q‡Q|
cÖ‡Z¨K mvwi‡Z KZ Rb •mb¨ wQj? wb¤œ gva¨wgK MwYZ,
mßg †kÖwY-1997 wkÿvel©, cÖkœgvjv 1.4 Gi 15 bs D`vniY
eM©vKvi msL¨v = 63009 - 8 = 63001
myZivs, 63001 = 251 (DËi)
†R‡b wbb- 21
 ÔhZ --- ZZÕ _vK‡j, †gvU n‡e = x × x = x2
(52 I 53 bs A¼ †`Lyb)
 ÔhZ --- ZZ 10 UvKv / hZ --- ZZ 10 cqmvÕ _vK‡j, †gvU n‡e = x × x × 10 = 10x2
( 54 †_‡K 62 bs
ch©šÍ A¼ †`Lyb)
 ÔhZ --- ZZ UvKvi †P‡q AviI 10 UvKv †ewk / hZ --- ZZ cqmvi †P‡q AviI 10 cqmv †ewkÕ _vK‡j,
†gvU n‡e = x (x +10) = x2
+ 10x | (63 I 64 bs A¼ †`Lyb)
mgvavb
mgvavb
mgvavb
NM
LK
mgvavb
mgvavb
mgvavb

15. Math Tutor  15
 g‡b ivLyb:
(K) hLb †Kvb mgm¨v mgvav‡b ARvbv ivwk a‡i wb‡eb, ZLb H ARvbv ivwki gvb †ei Kivi Rb¨ Aek¨B
ÔcÖkœg‡ZÕ ev ÔkZ©g‡ZÕ Zzjbv Ki‡Z n‡e|
(L) a‡i †bqv ivwki wfwˇZ cÖvß †gvU Pvu`vi mgvb n‡e cÖ‡kœ cÖ`Ë †gvU Puv`v, hv memgq Avgiv ÔkZ©g‡ZÕ ev ÔcÖkœg‡ZÕ
AvKv‡i †jL‡Z nq|
(M) cÖkœg‡Z/kZ©g‡Z GK‡Ki mgZv Avbv Avek¨K t- ÔcÖkœg‡ZÕ mgvb wP‡ýi Dfq cv‡k GKB ai‡Yi GKK
e¨envi Ki‡Z nq| †Kvb A‡¼i kZ©g‡Zi Dfqc‡ÿ GKB ai‡Yi GKK bv _vK‡j GK‡Ki mgZv K‡i wb‡Z n‡e|
(56, 57, 58, 63 I 64 bs A¼ †`Lyb)
52. GKwU †kÖwY‡Z hZRb evjK wQj cÖ‡Z¨‡K ZZ UvKv
K‡i Puv`v w`‡j 100 UvKv nq| evj‡Ki msL¨v KZ?
wbe©vPb Kwgkb mwPevj‡q mnKvix cwiPvjK: 95
10 100
25 35 DËi: K
(hZ -- ZZ AvBwWqvwU GKwU c~Y©e‡M©i AvBwWqv, ZvB
hZ = x Ges ZZ = x n‡j, †gvU Pvu`v n‡e x2
)
g‡bKwi, †kÖwYi evjK msL¨v = x Rb
Ges cÖ‡Z¨K evj‡Ki Puv`v = x UvKv |
 †gvU Puv`v = x × x = x2
cÖkœg‡Z, x2
= 100
ev, x2
= 102
∴ x = 10
myZivs, †kÖwYi evj‡Ki msL¨v = 10 Rb|
 kU©KvUt hZ Rb evjK n‡”Q 100 Gi eM©g~j, ZvB
mivmwi 100-Gi eM©g~j 10-B n‡”Q evj‡Ki msL¨v|
53. GKwU †kÖYx‡Z hZRb QvÎ Av‡Q cÖ‡Z¨‡K ZZ UvKv
K‡i cÖ`vb Ki‡j †gvU 6561 UvKv nq| QvÎ msL¨v
KZ? Kg©ms¯’vb e¨vsK A¨vwmm‡›UU Awdmvi t 01/ wbev©Pb
Kwgkb mwPevj‡qi mnKvix cwiPvjK t 95; WvK,
†Uwj‡hvMv‡hvMIZ_¨cÖhyw³gš¿Yvj‡qiAwdmmnKvix Kvg Kw¤úDUvi
gy`ªvÿwiK2018(wjwLZ)
92 75
91 81 DËi: N
g‡bKwi, †kÖwYi evjK msL¨v = x Rb
Ges cÖ‡Z¨K Qv‡Îi UvKv = x UvKv |
 †gvU UvKv = x × x = x2
cÖkœg‡Z, x2
= 6561
ev, x2
= 812
 x = 81
myZivs, †kÖwY‡Z QvÎ msL¨v = 81 Rb|
 kU©KvUt 6561 = 81|
54. †Kv‡bv ¯’v‡b hZ †jvK wQj cÖ‡Z¨‡K ZZ cuvP UvKv
K‡i Puv`v †`qvq †gvU 4500 UvKv Av`vq n‡jv|
GLv‡b †jvKmsL¨v KZ? wewfbœ gš¿Yvj‡qi mnKvix
†gBb‡Ub¨vÝ BwÄwbqvi: 17/ cwi‡ek Awa`߇ii mnKvix cwiPvjK
(KvwiMwi): 11
750 900
800 †Kv‡bvwUB bq DËi: N
g‡bKwi, †jvK msL¨v = x Rb
Ges cÖ‡Z¨K †jv‡Ki Pvu`v = x 5 UvKv |
 †gvU Pvu`v nj = x x 5 = 5x2
cÖkœg‡Z, 5x2
= 4500
ev, x2
=
5
4500
ev, x2
= 900
∴ x = 900 = 30
myZivs, †jvK msL¨v = 30 Rb|
 kU©KvUt Gai‡Yi mgm¨v mgvavb Gfv‡e- ZZ Gi
mgvavb
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK
evgc‡ÿ Wvbc‡ÿ evgc‡ÿ Wvbc‡ÿ
UvKv n‡j UvKv n‡e wgUvi n‡j wgUvi n‡e
cqmv n‡j cqmv n‡e N›Uv n‡j N›Uv n‡e
wK‡jvwgUvi n‡j wK‡jvwgUvi n‡e †m‡KÛ n‡j †m‡KÛ n‡e

16. 16  Math Tutor
mv‡_ †h msL¨v _vK‡e †mwU w`‡q †gvU msL¨v‡K fvM
Kiæb Ges cÖvß msL¨vwUi eM©g~j wbY©q Kiæb|
 cÖ_‡g fvM Kiæb:
5
4500
= 900 Ges Zvici
eM©g~j wbY©q Kiæb: 900 = 30 Rb|
55. hZ `vZv cÖ‡Z¨‡K ZZ 10 cqmv †`qv‡Z 250 cqmv
n‡jv| `vZvi msL¨v KZ? evsjv‡`k wUGÛwU †ev‡W©i
mnKvix cwiPvjK : 95
5 10
20 25 DËi: K
 kU©KvUt  250  10 = 25
 `vZvi msL¨v = 25 = 5|
56. †Kvb †kÖwY‡Z hZRb wkÿv_©x cÖ‡Z¨‡K ZZ `k cqmv
K‡i Puv`v †`qvq beŸB UvKv msMÖn nj| H †kÖwY‡Z
wkÿv_©xi msL¨v- mgvR‡mev Awa. mgvRKj¨vY msMVb: 05
90 Rb 60 Rb
30 Rb 15 Rb DËi: M
g‡bKwi, †kÖwY‡Z wkÿv_x©i msL¨v = x Rb
Ges cÖ‡Z¨K wkÿv_x©i Puv`v = x  10 cqmv |
 †gvU Pvu`v nj = x x 10 = 10x2
cqmv
cÖkœg‡Z, 10x2
= 90 100 (100¸YK‡icqmvKivnj)
ev, x2
=
10
10090
ev, x2
= 900
∴ x = 900 = 30
myZivs, wkÿv_x©i msL¨v = 30 Rb|
 kU©KvUt  (90100)  10 = 900
 wkÿv_x©i msL¨v = 900 = 30|
57. †Kvb ¯’v‡b hZ †jvK Av‡Q ZZ cuvP cqmv Rgv Kivq
†gvU 31.25 UvKv Rgv nj| H ¯’v‡b KZ †jvK wQj?
`yb©xwZ `gb ey¨‡iv wbe©vPbx cixÿv: 84
25 55
125 †Kv‡bvwUB bq DËi: K
g‡bKwi, †jvK msL¨v = x Rb Ges cÖ‡Z¨K †jvK
Rgv Kij = x  5 cqmv K‡i|
 †gvU Rgv nj = x x 5 = 5x2
cqmv|
cÖkœg‡Z, 5x2
= 31.25  100
ev, 5x2
= 3125 (31.25  100 = 3125 cqmv)
ev, x2
=
5
3125
ev, x2
= 625
∴ x = 625 = 25 |
myZivs, wb‡Y©q †jvK msL¨v = 25 Rb|
 kU©KvUt  3125 ÷ 5 = 625
 †jvK msL¨v = 625 = 25
58. †Kv‡bv †kÖYx‡Z hZRb QvÎ wQj Zv‡`i cÖ‡Z¨‡K ZZ
cuvP cqmv K‡i Puv`v †`Iqvq †gvU 125 UvKv nj| H
†kÖwY‡Z †gvU KZRb QvÎ wQj? wb¤œ gva¨wgK MwYZ, mßg
†kÖwY-1997 wkÿvel©, cÖkœgvjv 1.4 Gi 7 bs cÖkœ; 24Zg wewmGm
wjwLZ
wjwLZ Dc‡ii wbq‡g mgvavb Kiæb|
 kU©KvUt (125 UvKv 100 = 12500 cqmv)
 12500  5 = 2500
 †gvU QvÎ = 2500 = 50 (DËi)
59. GKwU mgevq mwgwZi hZRb m`m¨ wQj cÖ‡Z¨‡K ZZ
20 UvKv K‡i Puv`v †`Iqvq †gvU 20480 UvKv n‡jv|
H mwgwZi m`m¨msL¨v wbY©q Ki| wb¤œ gva¨wgK MwYZ,
mßg †kÖwY-2013 wkÿvel©, Abykxjbx 1.2 Gi 20 bs cÖkœ
 kU©KvUt  20480  20 = 1024
 mwgwZi m`m¨v msL¨v = 1024 = 32 (DËi)
60. GKwU QvÎvev‡m hZRb QvÎ _v‡K, Zv‡`i cÖ‡Z¨‡Ki
gvwmK LiP Zv‡`i †gvU msL¨vi `k¸Y| H QvÎvev‡mi
mKj Qv‡Îi †gvU gvwmK LiP 6250 UvKv n‡j H
QvÎvev‡m KZRb QvÎ _v‡K? evsjv‡`k K…wl Dbœqb
K‡c©v‡ik‡bi mnvKvix cÖkvmwbK Kg©KZ©v: 17; wb¤œ gva¨wgK MwYZ,
mßg †kÖwY-1997 wkÿvel©, cÖkœgvjv 1.4 Gi 11 bs cÖkœ
15 25
35 45 DËi: L
g‡b Kwi, †gvU QvÎ msL¨v = x Ges cÖ‡Z¨‡Ki
gvwmK LiP = x Gi 10 ¸Y = 10x |
†gvU gvwmK LiP = x  10x = 10x2
|
cÖkœg‡Z, 10x2
= 6250
ev, x2
=
10
6250
mgvavb
NM
LK
mgvavb
mgvavb
NM
LK
mgvavb
NM
LK
NM
LK

17. Math Tutor  17
ev, x2
= 625
∴ x = 625 = 25
myZivs, †gvU QvÎ msL¨v = 25 Rb|
 kU©KvUt  6250 ÷ 10 = 625
 QvÎ msL¨v = 625 = 25
61. †Kv‡bv †evwW©s-G hZ †jvK _v‡Kb, cÖ‡Z¨‡K †evW©v‡ii
†jvK msL¨vi 4 ¸Y UvKv Pvu`v †`Iqvq †gvU 4900
UvKv Pvu`v D‡V| KZRb †jvK _v‡Kb? wb¤œ gva¨wgK MwYZ,
mßg †kÖwY-1997 wkÿvel©, cÖkœgvjv 1.4 Gi 10 bs cÖkœ
g‡b Kwi, †evwW©s-Gi †jvK msL¨v = x Ges
cÖ‡Z¨‡Ki Puv`v = x Gi 4 ¸Y = 4x|
†gvU Puv`v = x4x = 4x2
|
cÖkœg‡Z, 4x2
= 4900
ev, x2
=
4
4900
ev, x2
= 1225
∴ x = 1225 = 35
myZivs, †jvK msL¨v = 35 Rb|
 kU©KvUt  4900 ÷ 4 = 1225
 †jvK msL¨v = 1225 = 35
62. GKwU avb‡ÿ‡Zi avb KvU‡Z kÖwgK †bIqv n‡jv|
cÖ‡Z¨K kÖwg‡Ki •`wbK gRywi Zv‡`i msL¨vi 10 ¸Y|
•`wbK †gvU gRywi 6250 UvKv n‡j kÖwg‡Ki msL¨v
†ei Kiæb| wb¤œ gva¨wgK MwYZ, mßg †kÖwY-2013 wkÿvel©,
Abykxjbx 1.2 Gi 20 bs cÖkœ
 6250  10 = 625
 kÖwgK msL¨v = 625 = 25 (DËi)
63. GKwU †kÖwY‡Z hZRb QvÎ-QvÎx c‡o, cÖ‡Z¨‡K ZZ
cqmvi †P‡qI AviI 20 cqmv †ewk K‡i Puv`v †`qvq
†gvU 48 UvKv DVj| H †kÖwY‡Z QvÎ-QvÎxi msL¨v
KZ? ciivóª gš¿Yvj‡qi e¨w³MZ Kg©KZv© : 06
50 Rb 55 Rb
60 Rb 70 Rb DËi: M
g‡bKwi, QvÎ-QvÎxi msL¨v = x Rb Ges cÖ‡Z¨‡Ki
Pvu`v = x + AviI 20 cqmv = x + 20 cqmv|
 †gvU Puv`v = x (x+20) cqmv|
cÖkœg‡Z, x (x+20) = 48 × 100
ev, x2
+ 20x = 4800
ev, x2
+ 20x - 4800 = 0
ev, x2
+ 80x - 60x - 4800 = 0
ev, x (x + 80) - 60 ( x + 80) = 0
ev, (x + 80) (x - 60) = 0
∴ x = - 80 ev x = 60, wKš‘ x= -80 MÖnY‡hvM¨ bq|
myZivs, QvÎ-QvÎxi msL¨v = 60|
 kU©KvUt †gvU †jvK/QvÎ-QvÎx/kÖwgK msL¨v × cÖ‡Z¨‡Ki
ZZ Puv`v/gRywi/LiP = †gvU Pvu`v/gRywi/LiP
50 × (50 + 20) = 3500 
55 × (55 + 20) = 4125 
60 × (60 + 20) = 4800 
70 × (70 + 20) = 6300 
64. GKwU †kÖwY‡Z hZRb QvÎ-QvÎx Av‡Q cÖ‡Z¨‡K ZZ
cqmvi †P‡q AviI 25 cqmv †ewk K‡i Puv`v †`qvq
†gvU 75 UvKv DVj| H †kÖwYi QvÎ-QvÎxi msL¨v KZ?
34Zg wewmGm
70 85
75 100 DËi: M
g‡b Kwi, QvÎ-QvÎxi msL¨v = x Rb Ges
cÖ‡Z¨‡Ki Pvu`v = x + AviI 25 cqmv|
 †gvU Puv`v = x(x+25) cqmv|
cÖkœg‡Z, x(x+25) = 75 × 100
ev, x2
+ 25x – 7500 = 0
ev, x2
+ 100x – 75x – 7500 = 0
ev, x(x + 100) – 75(x + 100) = 0
ev, (x - 75) (x + 100) = 0
nq, x - 75 = 0 A_ev, x + 100 = 0
∴ x = 75 ∴ x = - 100 [hv MÖnY‡hvM¨ bq]
myZivs, QvÎ-QvÎxi msL¨v = 75 Rb|
 kU©KvUt Ackb cÖ‡kœi kZ© wm× K‡i|
75 × (75 + 25) = 7500
 02.08 †hvM ev we‡qv‡Mi gva¨‡g c~Y©eM© msL¨v •Zwii wbqgt
(K) we‡qv‡Mi gva¨‡g c~Y©eM© msL¨v •Zwit
†R‡b wbb-22
M
M
mgvavb
NM
LK
N
M
L
K
mgvavb
NM
LK
mgvavb
mgvavb

18. 18  Math Tutor
 hw` cÖ‡kœ we‡qvM K‡i †Kvb msL¨v‡K c~Y©eM© Kivi K_v e‡j, Zvn‡j fvM‡klB DËi n‡e |
65. 18 †_‡K †Kvb ÿz`ªZg msL¨v we‡qvM Ki‡j we‡qvMdj
GKwU c~Y©eM© msL¨v n‡e?
†hvM ev we‡qv‡Mi gva¨‡g c~Y©eM© wbY©q Kivi Rb¨
fvM cÖwµqv AbymiY Kiv nq|
18 4
16
2
 18 msL¨vwU c~Y©eM© bq| msL¨vwU †_‡K 2 we‡qvM
Ki‡j c~Y©eM© n‡e|
AZGe, 18 †_‡K 2 ÿz`ªZg msL¨vwU we‡qvM Ki‡j
we‡qvMdjwU GKwU c~Y©eM© msL¨v n‡e|
66. 49289 †_‡K †Kvb ÿz`ªZg msL¨v we‡qvM Ki‡j
we‡qvMdj c~Y©eM© msL¨v n‡e? wb¤œ gva¨wgK MwYZ -97
wkÿvel©, Abykxjbx 1.1 Gi 6 bs cÖkœ
4 92 89 222
4
42 92
84
442 889
884
5
 49289 msL¨vwU c~Y©eM© b‡n| msL¨vwU †_‡K 5
we‡qvM Ki‡j c~Y©eM© n‡e|
AZGe, 49289 †_‡K 5 ÿz`ªZg msL¨vwU we‡qvM
Ki‡j we‡qvMdjwU GKwU c~Y©eM© msL¨v n‡e|
67. 8655 †_‡K †Kvb ÿz`ªZg msL¨v we‡qvM Ki‡j we‡qvMdj
GKwU c~Y©eM© msL¨v n‡e? R¡vjvwb I LwbR m¤ú` gš¿Yvj‡qi
wbivcËv Kg©KZv© 2019
99 6
16 55 DËi: L
86 55 93
81
183 5 55
5 49
6
myZivs, 8655 †_‡K 6 we‡qvM Ki‡j msL¨vwU c~Y©eM© nq|
68. 9220 Rb •mb¨ †_‡K Kgc‡ÿ KZRb •mb¨ mwi‡q ivL‡j
•mb¨`j‡K eM©vKv‡i mvRv‡bv hvq? wb¤œ gva¨wgK MwYZ -97
wkÿvel©, Abykxjbx 1.1 Gi 6 bs cÖkœ
(Ô9220 Rb •mb¨ †_‡K Kgc‡ÿ KZRb •mb¨ mwi‡q
ivL‡j •mb¨`j‡K eM©vKv‡i mvRv‡bv hv‡eÕ Ges Ô9220
†_‡K †Kvb ÿz`ªZg msL¨v we‡qvM Ki‡j we‡qvMdj c~Y©eM©
msL¨v n‡eÕ evK¨ `ywU Øviv g~jZ GKB AvBwWqv‡K
eySv‡”Q, ZvB Dc‡ii wbq‡gB mgvavb Ki‡Z n‡e|)
92 20 96
81
186 11 20
11 16
4
myZivs, 9220 Rb¨ •mb¨‡K eM©vKv‡i mvRv‡bv hv‡e bv|
GB ‣mb¨ msL¨v †_‡K 4 Rb •mb¨ mwi‡q ivL‡j •mb¨-
`j‡K eM©vKv‡i mvRv‡bv hv‡e|
∴ 9220 Rb •mb¨ †_‡K Kgc‡ÿ 4 Rb •mb¨ mwi‡q
ivL‡j •mb¨`j‡K eM©vKv‡i mvRv‡bv hv‡e|
69. 47080 Rb •mb¨ †_‡K Kgc‡ÿ KZ Rb •mb¨ mwi‡q
wb‡j •mb¨ `j‡K eM©vKv‡i mvRv‡bv hv‡e? moK I Rb -
c_ Awa`߇ii Dc-mnKvwi cÖ‡K․kjx: 10/ cÖv_wgK we`¨vjq
mnKvix wkÿK (PÆMÖvg wefvM): 02
124 224
424 504 DËi: M
4 70 80 216
4
41 70
41
426 2980
2556
424
myZivs, 47080 Rb¨ •mb¨‡K eM©vKv‡i mvRv‡bv hv‡e
bv| GB ‣mb¨msL¨v †_‡K 424 Rb •mb¨ mwi‡q ivL‡j
•mb¨`j‡K eM©vKv‡i mvRv‡bv hv‡e|
∴ 47080 Rb •mb¨ †_‡K Kgc‡ÿ 424 Rb •mb¨
mwi‡q ivL‡j •mb¨`j‡K eM©vKv‡i mvRv‡bv hv‡e|
70. 4639 †_‡K †Kvb ÿz`ªZg msL¨v we‡qvM Ki‡j we‡qvMdj
GKwU c~Y©eM© msL¨v n‡e? wb¤œ gva¨wgK MwYZ-2013
wkÿvel©, 7g†kÖwY, Abykxjbx 1.1 Gi 5 bs cÖkœ DËit 15
71. 8655 †_‡K †Kvb ÿz`ªZg msL¨v we‡qvM Ki‡j we‡qvMdj
GKwU c~Y©eM© msL¨v n‡e? wb¤œ gva¨wgK MwYZ-2013 wkÿvel©,
7g †kÖwY,Abykxjbx 1.1 Gi 4 bs &D`vniY DËit 6
72. 7428 †_‡K †Kvb ÿz`ªZg msL¨v we‡qvM Ki‡j
we‡qvMdj GKwU c~Y© eM©msL¨v n‡e? wb¤œ gva¨wgK MwYZ -
9 7 wkÿvel©, Abykxjbx 1.1 Gi 6 bs D`vniY DËit 32
mgvavb
NM
LK
mgvavb
mgvavb
NM
LK
mgvavb
mgvavb

19. Math Tutor  19
(L) †hv‡Mi gva¨‡g c~Y©eM© msL¨v •Zwit
†R‡b wbb-23
 †Kvb msL¨vi (18) Av‡M I c‡ii eM©g~j wbY©q †KŠkj:
4 × 4 = 16 ----[18]--- 5 × 5 = 25
jÿ¨ Kiæb, 18 Gi c~‡e©i c~Y©eM© 16, hvi eM©g~j 4 Ges 18 Gi c‡ii c~Y©eM© 25, hvi eM©g~j 5| A_©vr 18 Gi
Av‡Mi I c‡ii c~Y©e‡M©i eM©g~j `ywU n‡”Q ci¯úi µwgK msL¨v| c~‡e©i eM©g~j 4 †c‡j Gi mv‡_ 1 †hvM K‡i c‡ii
c~Y©eM© msL¨vi eM©g~j 5 cvIqv hvq Ges GB 5 †K eM© Ki‡j c‡ii c~Y©eM© 25 cvIqv hvq|
 KZ †hvM K‡i c~Y©eM© †ei Ki‡Z n‡e? ejv _vK‡j Avgiv cÖ_‡g cÖvß eM©g~j-Gi mv‡_ 1 †hvM K‡i †h †hvMdj cve
†mwUi eM© †ei Kie, Zvici D³ eM©msL¨v †_‡K cÖ‡kœ cÖ`Ë msL¨vwU we‡qvM Ki‡ev| e¨m, DËi P‡j Avm‡e| †hgb-
73. 18 Gi mv‡_ †Kvb ÿz`ªZg msL¨v †hvM Ki‡j †hvMdj GKwU c~Y©eM© msL¨v n‡e?
18 4  cÖvß eM©g~j 4 Gi mv‡_ 1 †hvM Ki‡Z n‡e 4 + 1 = 5 Ges 5 †K eM© Ki‡Z n‡e-
16 52
= 25| Gevi 25 †_‡K cÖ‡kœ cÖ`Ë msL¨v 18 we‡qvM Ki‡Z n‡e- 25 - 18 = 7|
2  18 Gi mv‡_ 7 †hvM Ki‡j †hvMdj GKwU c~Y©eM© msL¨v n‡e|
74. 651201 Gi mv‡_ †Kvb ÿz`ªZg msL¨vwU Gi mv‡_
†hvM Ki‡j †hvMdj GKwU c~Y© eM©msL¨v nq| c~Y©
eM©msL¨vwUi eM©g~j KZ? wb¤œ gva¨wgK MwYZ -97 wkÿvel©,
cÖkœgvjv 1.1 Gi 7 bs D`vniY
65 12 01 806
64
1606 1 12 01
96 36
15 65
651201 msL¨vwU c~Y©eM© bq| Gi mv‡_ GKwU ÿz`ªZg
msL¨v †hvM Ki‡j †hvMdj GKwU c~Y©eM© msL¨v n‡e| H
c~Y©eM© msL¨vwUi eM©g~j n‡e, 806+1 = 807 Ges c~Y©eM©
msL¨vwU n‡e 807 Gi eM© A_©vr, 8072
= 651249 |
 651201 Gi mv‡_ †hvM Ki‡Z n‡e
651249 - 651201 = 48| (DËi)
75. 6558 Gi mv‡_ †Kvb ÿz`ªZg msL¨v †hvM Ki‡j †hvMdj
GKwU c~Y© eM©msL¨v n‡e? wb¤œ gva¨wgK MwYZ -97 wkÿvel©,
cÖkœgvjv 1.1 Gi 7 bs cÖkœ , cÖwZiÿv gš¿Yvj‡qi GWwgwb‡÷ªkb
I cv‡m©vbvj Awdmvi : 06
65 58 80
64
16 1 58
6558 msL¨vwU c~Y©eM© bq| Gi mv‡_ GKwU ÿz`ªZg msL¨v
†hvM Ki‡j †hvMdj GKwU c~Y©eM© msL¨v n‡e| H
c~Y©eM© msL¨vwUi eM©g~j n‡e, 80 + 1 = 81 Ges c~Y©eM©
msL¨vwU n‡e 81 Gi eM© A_©vr 812
= 6561|
AZGe, 6558Gi mv‡_ †hvM Ki‡Z
n‡e 6561 - 6558 = 3|
 02.09 ARvbv msL¨v wbY©q
76. GKwU msL¨vi e‡M©i mv‡_ 4 †hvM Ki‡j †hvMdj 40
nq| msL¨vwU KZ? MYc~Z© Awa`߇ii wnmve mnKvix-06
4 5
6 8 DËi: M
g‡bKwi, msL¨vwU = x
cÖkœg‡Z, x2
+ 4 = 40 ev, x2
= 40 - 4
ev, x2
= 36 ∴ x = 36 = 6
 kU©KvU t (Option Test) 42
+ 4 = 20
(mwVK bq) 52
+ 4 = 29 (mwVK bq) 62
+ 4 = 40 (mwVK) 82
+ 4 = 68 (mwVK bq)
77. †Kvb abvZ¥K msL¨vi wظ‡Yi e‡M©i mv‡_ 15 †hvM
Ki‡j †hvMdj 415 n‡e? GbweAvi-12/ 13Zg †emiKvwi
cÖfvlK wbeÜb I cÖZ¨qb cixÿv (K‡jR/mgch©vq)-16
11 10
9 12 DËi: L
g‡bKwi, msL¨vwU = x
cÖkœg‡Z, (2x)2
+ 15 = 415 ev, 4x2
= 415 - 15
ev, 4x2
= 400 ev, x2
=100 ev, x = 100 = 10
 kU©KvU t Ackb cÖ‡kœi kZ©‡K wm× K‡i-
(102)2
+ 15 = 400 + 15 = 415 |
L
mgvavb
NM
LK
N
ML
K
mgvavb
NM
LK
mgvavb
mgvavb
mgvavb