MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkœ I DËi : eûwbe©vPwb 1

gƒj ˆUÕ¡ ˆccvim-‰i AwZwiÚ Ask

wbe©vPwb cix¶vi cÖkc
œ ò I DËi-2015: eûwbe©vPwb
mKj ˆevGWÆi kxlÆÕ©vbxq ài‚Z½cƒYÆ Õ•zGji 2015 mvGji wbeÆvPwb cixÞvi cÉkc
² ò I Dîi
205. dqRyi ingvb AvBwWqvj BwbwÕ¡wUDU, XvKv welq ˆKvW : 1 0 9
mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
..
14. a  3 a +1 = 0 nGj
1
a + 3 ‰i gvb KZ? 30. {x ô : 9 <x<10} ‰i…c ˆmUGK Kx eGj?
1. 0.45 ‰i mvgvbÅ f™²vsk wbGPi ˆKvbwU?
2 3

4 9 5 9
a K wbGœQ` ˆmU L Amxg ˆmU
K 1 L0 M3 N4
K L M N M duvKv ˆmU N mmxgGmU
9 20 11 11 15. ‰KwU AvqZGÞGòi `yBwU mw®²wnZ evüi Š`NÆÅ 2
2. A = {a, b, c} ‰i cÉK‡Z DcGmU KqwU?
I 1 nGj ‰i KGYÆi Š`NÆÅ KZ nGe? A
M
B
K 3 L6 M7 N8 K 5 L3
3. a, b, c evÕ¦e msLÅv ‰es a<b nGj-
M 5 N3 2 O
i. ac < bc; hLb c > 0 16. mƒßGKvGYi cƒiK ˆKvY ˆKvbwU?
ii. ac > bc; hLb c  0
K mijGKvY L Õ©ƒjGKvY C D
iii. ab > bc; hLb c  0 M mgGKvY N mƒßGKvY N
wbGPi ˆKvbwU mwVK? 17. mgGKvYx wòfzGRi mƒßGKvY«¼Gqi ˆhvMdj KZ? DcGii wPòwU jÞ Ki ‰es 31 - 33 bs cÉGk²i Dîi
K i I ii L i I iii K 45 L 81 M 90 N 180 `vI|
M ii I iii N i, ii I iii 18. a + b = 1 ; ab = 4 nGj (ab)2 ‰i gvb KZ? O ˆK±`Ê wewkÓ¡ ABCD e†Gî AD I BC eÅvm,
4. mylg lofzGRi KqwU cÉwZmvgÅ ˆiLv iGqGQ? K 15 L 7 MNAB, AB = 8cm ‰es ON = 3cm
K 3 L4 M5 N6 M 9 N 12 31. AM evüi Š`NÆÅ KZ?
5. (x) = x2  3x + 2 nGj x ‰i ˆKvb gvGbi RbÅ 19. A = 30 nGj tanA. sin 2A ‰i gvb KZ? K 4 L5
(x) = 0 nGe? K
1
L
1
M1 N3
M 6 N8
K 0 L2 2 3 32. e†îwUi eÅvmvaÆ KZ ˆm.wg.?
M (1,2) N (1,2) 20. wcZv I cyGòi eZÆgvb eqm h^vKÌGg 40 eQi ‰es 10 K 3 L4 M5 N6
wbGPi Z^Å ˆ^GK 6, 7, I 8 bs cÉGk²i Dîi `vI: eQi| 5 eQi ci ZvG`i eqGmi AbycvZ KZ nGe? 33. e†îwUi ˆÞòdj KZ?
`yB Aâ wewkÓ¡ ‰KwU msLÅvi `kK Õ©vbxq K 4:1 L 3:1 M 1:4 N 1:3 K 75.4 L 78.54
Aâ ‰KK Õ©vbxq AGâi w«¼àY| 2 + 5 + 8 + 11 ....... avivwU jÞÅ Ki ‰es 21, 22 I M 83.44 N 85.48
6. ‰KK Õ©vbxq Aâ y nGj msLÅvwU KZ? 23 bs cÉGk²i Dîi `vI| 34. A
K 2y L 3y M 21y N 12y 21. avivwUi mvaviY A¯¦i KZ?
7. A⫼q Õ©vb wewbgq KiGj msLÅvwU KZ nGe? K 3 L3 M5 N7 LuywU
K 12y L 4y 22. avivwUi `kg c` KZ? 60
M 3y N 21y K 29 L 31 B C
10
8. y = 2 nGj, gƒj msLÅvi mvG^ Õ©vb wewbgqK‡Z M 35 N 37 DcGii wPGò LyuwUi Š`NÆÅ KZ wgUvi?
msLÅvi cv^ÆKÅ KZ nGe? 23. avivwUi 1g AvUwU cG`i mgwÓ¡ KZ? 10
K 20 L 18 M 34 N 36 K 200 L 124 K 5 L
3
9. P I Q GK±`Ê wewkÓ¡ `yBwU e†Gîi eÅvmvaÆ«¼q M 100 N 92
M 5 3 N 10 3
h^vKÌGg r1 I r2 nGj- 24. ‰KwU mgw«¼evü wòfzGRi f„wg x ‰es mgvb
35. jvf-ÞwZi ˆÞGò
i. e†î«¼q ewntÕ©fvGe Õ·kÆ KiGj PQ = r1 + r2 nGe evü«¼Gqi Š`NÆÅ y nGj- wòfzRwUi ˆÞòdj KZ?
4 4 i. KÌqgƒjÅ  weKÌqgƒjÅ = ÞwZ
ii. e†î«¼q A¯¦tÕ©fvGe Õ·kÆ KiGj PQ = r1 ~ r2 nGe K x2 4y2 L 4y2 x2
x x ii. weKÌqgƒjÅ KÌqgƒjÅ = jvf
iii. e†î«¼q ciÕ·iGK ˆQ` KiGj x x iii. jvf ev ÞwZ KÌqgƒGjÅi Dci wbfÆj KGi
r1 + r2 < PQ < r1  r2 nGe M 4y2 x2 N x24y2
4 4 wbGPi ˆKvbwU mwVK?
wbGPi ˆKvbwU mwVK? 25. log10x = 3 nGj, x ‰i gvb KZ? K i I ii L i I iii
K i, ii I iii L ii I iii K 30 L 10
M ii I iii N i, ii I iii
M i I iii N i I ii M x3 N 103
26. mƒPGKi ˆÞGò- 36. a21 + 2b  b2 ‰i Drcv`K KZ?
10. ‰KwU UÇvwcwRqvGgi Aã KqwU? K (a+ b + 1) (a b +1) L (a+ b + 1) (a+b 1)
i. (ab)m =am.bm
K AvUwU L `kwU ii. a0 = 1, (a0) M (a+ b + 1) (a + b+ 2) N (a+ b1) (ab + 1)
M PviwU N bqwU am 37. (3,1) ‰es (3, 1) we±`y `yBwUi AeÕ©vb QK
11. KtL = 3:4; L: M = 5 : 4 nGj K: L : M = KZ? iii. n = am + an
a KvMGRi ˆKvb PZzfÆvGM?
K 15 : 20 : 16 L 16 : 15 : 20 wbGPi ˆKvbwU mwVK? K 2q I 3q L 2q I 4^Æ
M 16 : 20 : 15 N 20 : 16 : 15 K i I ii L ii I iii M 1g I 4^Æ N 3q I 2q
12. 2wU KÌwgK msLÅvi eGMÆi A¯¦i 23 nGj msLÅv«¼q KZ? M i I iii N i, ii I iii 38. kƒbÅ gvòvi mîv ejv nq ˆKvbwUGK?
K 10,11 L 12,13
M 9,10 N 11,12
27. 8, 9, 7, 15, 10, 15, 11, 8, 10, 9, 8 DcvîàGjvi K ˆiLv L Zj
13. i. 3, 9, 4 ‰i PZz^Æ mgvbycvZx = 12 cÉPziK KZ? M we±`y N ˆiLvsk
ii. `yBwU msLÅvi AbycvZ 3:5 ‰es ZvG`i mgwÓ¡
K 8 L9 39. gƒj` msLÅv ˆKvbwU?
M 10 N 15
24 nGj àYdj 135 K 16 L 13
28. ‰KwU evüi Š`NÆÅ ˆ`Iqv ^vKGj ˆKvbwU Aâb
iii. a : b = c : d = e : f nGj ‰G`i mvaviY M 15 N 14
Kiv mÁ¿e?
40. ‰KwU eMÆGÞGòi ˆÞòdj 169 eMÆ ˆm. wg. nGj-
AbycvZ a + b  e K mvgv¯¦wiK L UÇvwcwRqvg
b+d a ‰i cwimxgv KZ ˆm.wg.?
M AvqZGÞò N eMÆGÞò K 13 L 26
wbGPi ˆKvbwU mwVK? 29. 1 + 2 + 3 + 4 + .......... + 100 = KZ? M 52 N 65
K i, ii I iii L ii I iii K 4750 L 4950
M i I iii N i I ii M 5050 N 5150

1 M 2 M 3 K 4 N 5 N 6 M 7 K 8 L 9 N 10 M 11 K 12 N 13 N 14 L 15 M 16 N 17 M 18 K 19 K 20 L
Dîi

21 L 22 K 23 M 24 M 25 N 26 K 27 K 28 N 29 M 30 M 31 K 32 M 33 L 34 N 35 N 36 N 37 L 38 M 39 K 40 M
2 MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkœcò I DËi : eûwbe©vPwb

206. avbgw´£ Mft eGqR nvB Õ•zj, XvKv welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1. m + n = 8 ‰es mn = 15 nGj, (m  n) ‰i gvb
2
wPGò 2AB = BC nGj, wbGPi ˆKvbwU mwVK?
KZ? i. BAC = 60 K i I ii L i I iii
K 2 L 4 ii. BAC = ACB = 45 M ii I iii N i, ii I iii
M 34 N 94 iii. ACB = 30
wbGPi ˆKvbwU mwVK? 28. wbGPi ˆKvb mƒòwU mwVK?
2. a3  3a2b + 2b3 ‰i Drcv`K  K tan2  = 1  sec2 
i. a  b K i I ii L i I iii L cosec2   tan2  = 1
ii. a + 2b M ii I iii N i, ii I iii M sin2   cos2  = 1
iii. a2 + ab + b2 16. ˆKvGbv e†Gîi AwaPvGc A¯¦wbÆwnZ ˆKvYwU? 1 1
wbGPi ˆKvbwU mwVK? N + =1
K mƒßGKvY L mgGKvY cosec2  sec2 
K i L i I iii M Õ©ƒjGKvY N cÉe†«¬ ˆKvY wbGPi wPGòi AvGjvGK 29 I 30 bs cÉGk²i Dîi `vI :
M ii I iii N i, ii I iii 17. e†Gîi ‰KwU we±`yGZ KqwU Õ·kÆK AuvKv mÁ¿e? P
3. p2  1 = 5p nGj
1
p3  3 = KZ? K 1wU L 2wU
p 5
M 3wU N AmsLÅ
K 0 L 2 5 18. RÅvwgwZK Dccv`Å cÉgvGY mvaviYZ KqwU avc 3
M 3 5 N 8 5 ^vGK? Q  R
4. 0.45 ‰i mvgvbÅ f™²vsk wbGPi ˆKvbwU? K 4 L 3
4 9 M 2 N 1 29. cos = KZ?
K L
9 20 19. ‰KwU eGMÆi mGeÆvœP KZwU cÉwZmvgÅ ˆiLv AvGQ? 3 3
5 9 K 2 L 3 K L
M N 5 4
11 11 M 4 N 5 4 5
5. p, q, r evÕ¦e msLÅv ‰es p < q nGj, 20. A M
5
N
4
i. pr < qr hLb r > 0 tan2  + 1
ii. pr > qr hLb r < 0
30. 2
sin   1
= KZ?
iii. pr > qr hLb r  0 D E
K
35
L  2.44
wbGPi ˆKvbwU mwVK? 8
M 1 N 1.56
K i I ii L i I iii B C
15
M ii I iii N i, ii I iii wPGò BC || DE ‰es AB = 8cm, BC = 6cm nGj 31. tan = nGj ‰i
18
AwZfzR KZ?
1 1 i. DE = 3 cm K 8 L 15
6. 1+ ....... avivwUi AÓ¡g c` KZ? ii. AD = 4 cm
2 2 M 17 N 23.43
K  16 L 8 iii. ABC I ADE m`†k 32. wbGPi ˆKvbwU mwVK?
M 8 N 32 wbGPi ˆKvbwU mwVK? sin 1
K i I ii L ii I iii K cot = L sin =
7. wbGPi ˆKvbwU gƒj` msLÅv? cos sec
6 M i I iii N i, ii I iii 1 1
K L M tan = N cot =
11
3 21. KqwU Õ¼Z¯¨ Dcvî ^vKGj ‰KwU PZzfzÆR AuvKv cot cosec
8 27 mÁ¿e? 1
33. cos = 2 nGj, tan = KZ?
M N K 4 L 5
7 48 K 2 L
M 6 N 7 3
8. ˆKvb kGZÆ a 0
=1
wbGPi wPò Abyhvqx 22 I 23 bs cÉGk²i Dîi `vI: 3 1
K a=0 L a0 M N
A 2 2
M a>1 N a<1 5
9. a  10 n nGjv ˆKvGbv msLÅvGK ŠeævwbK AvKvGi 2 ‰KK 34. cosec + cot =
6
nGj, cosec  cot = KZ?
cÉKvwkZ i…c, ˆhLvGb  K
1
L
5
K 1 < a < 10 L 1 < a  10 D E 6 6
M 1  a  10 N 1  a < 10 6
M 1 N
1 5
10. x = 5 + 2 6 nGj x3 
x3
‰i gvb wbGPi ˆKvbwU? B C 35. 15, 17, 24, 21, 16, 17, 23, 18, 20, 22
K 2 2 L 18 3 AD = BD, AE = CE, CE = 2.5 ‰KK DcvîmgƒGni gaÅK ˆKvbwU?
M 372 2 N 396 6 22. BC = KZ ‰KK? K 17 L 17.5
K 3 L 4 M 18.5 N 19
11. mKj gƒj` msLÅv I Agƒj` msLÅvGK Kx eGj? 36. AvqZGjL ˆ^GK Kx Aâb Kiv hvq?
M 5 N 6
K Õ¼vfvweK msLÅv L ˆgŒwjK msLÅv
23. DE = KZ ‰KK? K MY msLÅv eüfzR L AwRf ˆiLv
M cƒYÆ msLÅv N evÕ¦e msLÅv K 3 L 2.5 M cvBwPò N `´£wPò
12. hw` A = {a, b, c, d} nq, ZGe A ‰i cÉK‡Z M 2 N 1.5 37. 70-80 ‰KwU ˆkÉwY ‰i cÉK‡Z wbg³mxgv KZ?
DcGmU KqwU? 24. mƒßGKvGYi cƒiK ˆKvY ˆKvbwU? K 70.5 L 80.5
K 12 L 16 K mijGKvY L Õ©ƒjGKvY M 69.5 N 79.5
M 15 N 17 38. 5 ˆ^GK 100 chƯ¦ «¼viv wefvRÅ msLÅvàGjvi
M mgGKvY N mħGKvY 5
13. 2x  3 + 5 = 2 ‰i mgvavb wbGPi ˆKvbwU? 25. mvgv¯¦wiGKi ˆÞòdj mgvb ˆKvbwU? Mo KZ?
K 6 L 4 K 32.5 L 42.5
K KYÆ«¼Gqi àYdj L Š`NÆÅ  cÉÕ©
M 2 N  M 52.5 N 62.5
1
14. a + b = 7 ‰es a  b = 3 nGj, ab = KZ? M f„wg  DœPZv N  f„wg  DœPZv 39. AwRf ˆiLv wKi…c?
2
K 10 L 5 26. ˆKvb ‰KwU wòfzRGK mgw«¼LwíZ KGi? K Da»ÆMvgx L wbg³Mvgx
M 3 N 1
K f„wg M mij‹iwLK N e†îvKvi
15. 40. DcvîmgƒGni cÉwZwbwaZ½ KGi ˆKvbwU?
L gaÅgv
A
M DœPZv K ˆkÉwY L MYmsLÅv
N kxlÆ ˆKvGYi mgw«¼L´£K M ˆKw±`Êq gvb N cwimi
27. wbGPi Z^ÅàGjv jÞ Ki :
i. sin2  = 1 + cos2
ii. sec2  = 1 + tan2
B C iii. cosec2  = 1 + cot2
1 L 2 K 3 N 4 L 5 K 6 L 7 N 8 L 9 N 10 N 11 N 12 M 13 N 14 N 15 L 16 K 17 K 18 K 19 M 20 N
Dîi

21 L 22 K 23 N 24 N 25 M 26 L 27 M 28 N 29 M 30 L 31 N 32 M 33 L 34 N 35 N 36 K 37 M 38 M 39 K 40 M
MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkœcò I DËi : eûwbe©vPwb 3

207. we ‰ ‰d kvnxb KGjR, XvKv welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1.
..
0.33 ‰i mvgvbÅ f™²vsk ˆKvbwU? 14. sin + cos = 1 nGj, sin.cos ‰i gvb KZ? wbGPi ˆKvbwU mwVK?
33 33 30 1 K 1 L
1
M 0 N 1 K i I ii L i I iii
K
100
L
90
M
99
N
3
2 M ii I iii N i, ii I iii
2. f(x) = x3 + kx2  4x  8 nGj k ‰i ˆKvb gvGbi
15. f(x) abvñK gvòvi eüc`x nGj ‰es f(x) ˆK    
(2x + 3) «¼viv fvM KiGj fvMGkl KZ nGe? 28. cos sin tan cot ‰i gvb KZ?
RbÅ f(2) = 0 nGe? 4 4
1
4
1
4
3 2

3.
K 2 L 1
x2 + 1 = 2x nGj
M 2 N 3 K f ()2
Lf () 3
K
2
L
2
M 1 N 2
3 2 29. mgevü wòfzGRi evüi Š`NÆÅ 8 ˆm.wg. nGj ‰i
1
i. x + = 2
x
1 1
ii. x + = x2 + 2
x x
M f ( ) 2
Nf  ( ) 3 DœPZv KZ ˆm.wg.?
1 1 16. `yB AâwewkÓ¡ ˆKvb msLÅvi AsK«¼Gqi A¯¦i 4| K 2 3 L4 3
iii. x2 + 2 = x3  3 ‰KK Õ©vbxq AsK, `kK Õ©vbxq AsK AGcÞv
x x M 16 3 N 32 3
wbGPi ˆKvbwU mwVK? 4 ˆewk| msLÅvwU KZ? 30. 10 cm
K i I ii L i I iii K 73 L 51 M 37 N 95 A D
M ii I iii N i, ii I iii 17. A = {4, 5, 6, 7} ˆmUwUGK MVb c«¬wZGZ wjLGj
4. x + y = 6, x  y = 4 nGj, x2 + y2 ‰i gvb wbGPi ˆKvbwU nGe?
K A = {x  ô : 4 < x < 8} M
ˆKvbwU?
K 10 L 20 M 26 N 52 L A = {x  ô : 4 < x  8} B C
1 M A = {x  ô : 4  x < 8} ABCD ‰KwU eMÆ ‰es BMC ‰KwU Puv`v nGj
5. n  ô nGj, ˆKvb kGZÆ a = n nGe?
−n
a N A = {x  ô : 3 < x  8} eGMÆi ˆÞòdj I Puv`vi ˆÞòdGji AbycvZ
K a=0 L a1 18. ‰KwU NbGKi mÁ·ƒYÆ c†GÓ¤i ˆÞòdj 96 eMÆ ˆKvbwU?
M a 0 N a0 wgUvi| ‰i c†Ó¤ZGji KGYÆi Š`NÆÅ KZ wgUvi? K 1: L 2: M 4: N 8:
6. A K 4.578 L 5.656 31. 10x
M 6.928 N 64
120 19. 4x  3 + 6 = 5 mgxKiYwUi mgvavb ˆmU wbGPi A
ˆKvbwU?
B D 4x 4x
C K  L 1 M {1} N {}
.. ..
ABC mƒßGKvYx wòfzGR 20. 3.89 + 2.178 = ? B C
i. AB + AC > BC .. . . DcGii wPGò x ‰i gvb KZ?
ii. AB – AC < BC K 5.267 L 6.068
K 30 L 45 M 90 N 180
iii. A + B < 60 .. ..
wbGPi ˆKvbwU mwVK? M 6.068 N 6.07 7 32. ‰KwU mvgv¯¦wiGKi `yBwU mw®²wnZ evüi Š`NÆÅ
K i I ii L i I iii 21. tanA =
5
nGj cosecA = KZ? h^vKÌGg 10 cm I 20 cm nGj ‰i AaÆcwimxgvi
M ii I iii N i, ii I iii
12 gvb KZ?
5 12 13 13 K 10 cm L 20 cm M 30 cm N 60 cm
wbGPi wPGòi AvGjvGK 7 I 8 bs cÉGk²i Dîi `vI : K L M N
13 13 12 5
33. D C
A 22. A

30 O
C B
O O A B
7. AOB ‰i mÁ·ƒiK ˆKvGYi gvb KZ? ABCD iÁ¼mwUi cwimxgv KZ?
K 30 L 45 M 60 N 150 B C K 12 cm L 16 cm
8. wPGò wbG`ÆwkZ cÉe†«¬ ˆKvY ‰es AOB ‰i M 20 cm N 28 cm
mÁ·ƒiK ˆKvGYi mgwÓ¡ KZ? ABC mgevü wòfzR nGj BOC ‰i gvb KZ? wbGPi ZG^Åi AvGjvGK (34 I 35) bs cÉGk²i Dîi `vI:
K 180 L 210 M 330 N 480 K 30 L 60 M 90 N 120 A = {x  ô : x  6 ‰es ˆgŒwjK msLÅv}
9. A = {1, 2}, B = {3, 4} nGj R = A  B A®¼qwUi 23. 4a2 + 25b2 ‰i mvG^ KZ ˆhvM KiGj ˆhvMdj B = {x  ô : 2 < x < 7 ‰es ˆRvo msLÅv}
ˆWvGgb KZ? cƒYÆeMÆ nGe? 34. A  B = ˆKvbwU?
K A L B M AB N AB K 10ab L 20ab M 100ab N 200ab K  L {4, 6}
10. ‰KwU AvqZGÞGòi Š`NÆÅ 5 àY ‰es cÉÕ© 4 àY 24. 2y = 32 nGj, y ‰i gvb ˆKvbwU? M {2, 3, 5} N {2, 3, 4, 5, 6}
nGj ˆÞòdj KZ àY nGe? 2 5 35. A  B ‰i cÉK‡Z DcGmGUi msLÅv KZ?
K 8 L 10 M 18 N 20 K L 2 M N 5
5 2 K 10 L 25 M 31 N 32
11. A = {a, b}, B = {a, b} nGj A  B KZ? D C 36. wbGPi ˆKvbwU gƒj` msLÅv?
K {a, b} L 0 M {} N  3 5 9 8
12. K t L = 3 : 4, L t M = 4 : 5, M t N = 5 : 6 O K
3
L
3
M
16
N
7
8cm

nGj K t N = KZ? ˆkÉwY 41-50 31-40 51-60 61-70

K 1 : 10 L 1:2 M 6:3 N 2:1
MYmsLÅv 6 4 8 12
13. 10 ˆm.wg. A P B DcGii mviwY AejÁ¼Gb (37-40) bs cÉGk²i
wP‰úi Av‰jv‰K (25-26) bs c¥¤‰kßi Dîi `vI :
6 ˆm.wg. D͹i `vI : 37. cÉPziK ˆkÉwY ˆKvbwU?
K 31-40 L 41-50
ABCD ‰KwU eMÆ|
M 51-60 N 61-70
DcGii AvqZwUGK e†nîg evüi Pviw`GK ˆNvivGj 25. e†îwUi eÅvm KZ ˆm.wg.? 38. DcvGîi gaÅK ˆkÉwY ˆKvbwU?
ˆh NbeÕ§ ŠZwi nq Zvi K 4 2 L 4 3 M 8 2 N 8 3 K 31-40 L 41-50
i. f„wgi eÅvmvaÆ 6 ˆm.wg. 26. AOD ‰i ˆÞòdj KZ eMÆ ˆm.wg.? M 51-60 N 61-70
ii. DœPZv 16 ˆm.wg.
K 8 L 16 M 32 N 64 39. gaÅK ˆkÉwYi gaÅgvb KZ?
iii. AvqZb 1130.976 Nb ˆm.wg.
27. ‰KwU NbGKi avi 2 ‰KK nGj K 35.5 L 45.5 M 55.5 N 65.5
i. ‰i KGYÆi Š`NÆÅ 2 ‰KK 40. cÉ`î DcvGîi cÉPziK ˆkÉwYi wbÁ²mxgv KZ?
wbGPi ˆKvbwU mwVK? K 31 L 41 M 51 N 61
K i I ii L i I iii ii. AvqZb 2 2 Nb ‰KK
iii. mgMÉZGji ˆÞòdj 12 eMÆ ‰KK
M ii I iii N i, ii I iii

1 N 2 M 3 K 4 M 5 M 6 K 7 N 8 N 9 K 10 N 11 N 12 L 13 L 14 M 15 M 16 M 17 M 18 L 19 K 20 N
Dîi

21 N 22 N 23 L 24 N 25 M 26 L 27 M 28 L 29 L 30 N 31 K 32 M 33 M 34 K 35 M 36 M 37 N 38 M 39 M 40 N

pp 04 \\prepress47\E\Data 2015-2016\SSC Testpaprts Link 2015\mth 26.10.15\mthtqo.doc 4th proof

4 MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkœcò I DËi : eûwbe©vPwb

208. ˆm´Ÿ ˆMÉMix nvBÕ•zj, XvKv welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1. 2
f(x) = x  2xa + (a + b) (a  b) nGj, f(x) = 0 16. log12 2 3  log12 = KZ? 27. ‰KwU eMÆGÞGòi KqwU cÉwZmvgÅ ˆiLv AvGQ?
nGe hw` 2 K 1 wU L 2 wU
K 0 L1
i. x = a + b nq M 3 wU N 4 wU
1 3
ii. x = a  b nq M
2
N
2 28. NƒYÆb cÉwZmgZv wbYÆGqi RbÅ KqwU welq jÞÅ
iii. x = (a + b)2 nq 17. a  Ñ ‰es b  Ñ I ab = 0 nGj, ivLGZ nq?
wbGPi ˆKvbwU mwVK? i. a = 0 nGe K 1 wU L 2 wU
K i I ii L i I iii ii. b = 0 nGe M 3 wU N 4 wU
M ii I iii N i, ii I iii iii. a = 0 ‰es b = 0 nGe 29. ABC ‰ AB = 10 ˆm.wg., BC = 6 ˆm.wg. ‰es
2. A = {5, 6, 7} ‰i cÉK‡Z DcGmGUi msLÅv KZ? wbGPi ˆKvbwU mwVK? ABC = 30 nGj, wòfzRwUi ˆÞòdj KZ eMÆ
K 3 L6 M7 N8K i I ii L i I iii ˆm.wg.?
3. 5x  1 + 3 = 5 ‰i mwVK mgvavb ˆmU ˆKvbwU? M ii I iii N i, ii I iii K 240 L 120
K {1} L {1} M {2} N { } M 60 N 15
18. 2x2  4ax = 0 mgxKiGYi mgvavb ˆmU ˆKvbwU?
4. 5 + a + b + 625 àGYvîi avivfzÚ nGj a + b = K {0} L {2a}
30. iÁ¼Gmi ˆÞòdj 120 eMÆ ˆm.wg. ‰es ‰i ‰KwU
KZ? M {0, 2a} N {0,  2a} KYÆ 10 ˆm.wg. nGj, Aci KGYÆi Š`NÆÅ KZ ˆm.wg.
K 30 L 50 M 150 N 155 19.  ˆKvGYi cƒiK ˆKvY ‰es wecÉZxc ˆKvY ciÕ·i nGe?
5. 4.9 = KZ? mgvb nGj, tan = KZ? K 24 L 18 M 12 N 6
K
1
L
49 1 31. ‰KwU NbGKi mÁ·ƒY Æ c† GÓ¤i ˆÞòdj 96 eMÆ
5 90 K 0 L wgUvi nGj, evüi Š`NÆÅ KZ wgUvi?
49 3
M N5 K 2 L4 M6 N8
99 M 1 N 3
D
6. 729 ‰i jM 4 nGj, wfwî KZ? 20. ABC ‰ A = 50 ‰es B I C ‰i C
K 6 3 L6 M3 3 N3 mgw«¼LíK«¼q O we±`yGZ wgwjZ nq| BOC =
7. PZzfzÆR AâGY KqwU AbbÅ DcvGîi cÉGqvRb KZ?
nq? K 65 L 75
B
K 3wU L 4wU M 115 N 125 A E
M 5wU N AmsLÅ 21. ˆKvb kGZÆ , a1 x + b1 y = c 1 ‰es a2 x + b2 y = c 2 wPGò ABCD UÇvwcwRqvGg AB = 6 ˆm.wg.,
8. Zj KZ gvwòK? mgxKiY ˆRvGUi AmsLÅ mgvavb ^vKGe? DC = 4 ˆm.wg., AB || DC ‰es DE = 4 ˆm.wg.|
K 0 L1 M2 N3 K 
a1 b1
L = =
a1 b1 c1
DcGii ZG^Åi AvGjvGK 32-34 bs cÉGk²i Dîi `vI:
a2 b2 a2 b2 c2
A a1 b1 c1 a1 b1 c1 32. UÇvwcwRqvgwUi ˆÞòdj KZ eMÆ ˆm.wg.
M =  N   K 40 L 20
a2 b2 c2 a2 b2 c2
1 2 3 M 14 N 10
3 22. , , , ..... AbyKÌGgi mvaviY c` wbGPi ˆKvbwU?
2 3 4 33. AD ‰es BC evüi gaÅwe±`y h^vKÌGg E ‰es F
C  B K
n1
,nô L
n+1
,nô
nGj, EF = KZ ˆm.wg.?
n n1 K 20 ˆm. wg. L 12 ˆm. wg.
3
DcGii wPGòi AvGjvGK 9-11 bs cÉGk²i Dîi `vI : M
n1
,nô N
n
,nô M 10 ˆm. wg. N 5 ˆm. wg.
n+1 n+1
9. ABC ‰ ACB =  = KZ? 34. BD ‰es AC KGYÆi gaÅwe±`y h^vKÌGg P ‰es Q
23. cÉ^g 20wU Õ¼vfvweK msLÅvi ˆhvMdj KZ?
K 30 L 45 M 60 N 66 K 105 L 200
nGj, PQ = KZ ˆm.wg.?
10. sin + cos = KZ? K 1 L5 M 10 N 12
M 210 N 420
1 3 A
35. x4  x2 + 1 = 0 nGj
K 2 3 L 1
2 i. x2 + 3 = 1
x
3+1
M N 2 1 2
11.
2
AB : BC : AC = KZ? 100
ii. ( )
4
x+
x
=3
iii. x = (x + 1) (x  1)
K 1:2:3 L 3:2:1 50
B C wbGPi ˆKvbwU mwVK?
M 3:1: 2 N 3:1:2 D
K i I ii L i I iii
12. A = {x : x abvñK cƒYÆmsLÅv ‰es x2 < 5} nGj A ABC ‰ AD, BAC ‰i mgw«¼LíK| AB = 6
M ii I iii N i, ii I iii
‰i ZvwjKv c«¬wZ wbGPi ˆKvbwU? ˆm.wg., AC = 5 ˆm.wg., ABC = 50, ADB = 100
K {2, 1, 0, 1, 2} L {2, 1} DcGii ZG^Åi AvGjvGK 24-26bs cÉGk²i Dîi `vI : 36. 27.37 msLÅvwUi jGMi cƒYÆK KZ?
M {0, 1, 2} N {1, 2} K 1 L0
24. ACB = KZ? M 1 N2
13. y = 2x dvskGbi ˆjLwPò K 50 L 60
i. gƒjwe±`y w`Gq hvGe M 70 N 80
37. ax2 + bx + c = 0 mgxKiYwU w«¼NvZ mgxKiY bv
ii. (1, 2) we±`yMvgx 25. BD t DC = KZ?
nGj, a ‰i gvb KZ nGe?
K 2 L 1 M0 N1
iii. x AÞGK (3, 0) we±`yGZ
ˆQ` KiGe| K 5t6 t2 L1
38. e†Gîi DcPvGci Dci `ívqgvb ˆKvY wK ˆKvY?
wbGPi ˆKvbwU mwVK? M 2t1 t5 N6
K mħGKvY L mgGKvY
K i I ii L i I iii 26. CAD ‰i mÁ·ƒiK ˆKvGYi ‰K Z‡Zxqvsk mgvb M Õ©ƒjGKvY N mijGKvY
M ii I iii N i, ii I iii i. ABC ‰i mgvb
39. ˆKw±`Êq cÉeYZvi cwigvc KqwU?
14. a  b = 2 ‰es ab = 24 nGj (a + b) = KZ? ii. ADB ‰i AGaÆK
K 1 wU L 2 wU
K 10 L 10 iii. ABD ‰es ADB ‰i mgwÓ¡i mgvb
M 3 wU N 4 wU
M  10 N 5 2 wbGPi ˆKvbwU mwVK? 40. 14, 8, 11, 6, 9 DcvGîi cwimi KZ?
15. 2 2 ‰i 2 wfwîK jM KZ? K i I ii L i I iii K 7 L8 M9 N 10
2 2 3 M ii I iii N i, ii I iii
K 2 2 L M N
2 3 2
1 K 2 M 3 L 4 M 5 N 6 M 7 M 8 M 9 M 10 M 11 N 12 N 13 K 14 M 15 N 16 N 17 K 18 M 19 M 20 M
Dîi

21 L 22 N 23 M 24 M 25 N 26 K 27 N 28 N 29 N 30 K 31 L 32 L 33 N 34 K 35 N 36 M 37 M 38 K 39 M 40 M
MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkc
œ ò I DËi : eûwbe©vPwb 5

209. ˆZRMuvI miKvwi weævb KGjR, XvKv welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1. 0.0025 ‰i eMÆgƒj ˆKvb aiGbi msLÅv? 27. 4
A + sin A = 1 nGj
2

K mmxg `kwgK L Amxg `kwgK 16. ( )(3

5 5 ) ‰ 5 ‰i kwÚ KZ?
sin
i. sin2 A = cosA
M Õ¼vfvweK N Agƒj` ii. tanA = cosecA
3 5
K 5 L 5 M N5 iii. tanA cosecA = 1
.. 6
2. 0.81 ‰i mvG^ KZ àY KiGj àYdj 3 nGe? wbGPi ˆKvbwU mwVK?
17. jGMi gvb KZ?
. . K i I ii L i I iii
K 3.6 L 3.6 M 3.7 N 3.7
K
5
L
3
5 M 5 N5 M ii I iii N i, ii I iii
3. ˆKvGbv ˆmGUi kwÚ ˆmGUi Dcv`vb msLÅv 32 6
28. cot = 1 nGj cos = ?
nGj, H ˆmGUi Dcv`vb msLÅv KZ? 18. 2x  3 =  3 mgxKiGYi 1 1
K 2 L3 M5 N 32 K 0 L1 M N
i. mgvavb ˆmU = {6} 2 2
4. A = {x  N : x, 42 ‰i àYbxqK} nGj, ii. ˆKvb mgvavb ˆbB 2x + 3y = 18 nGj x : y = KZ?
i. A = {1, 2, 3, 6, 7, 14, 21, 42} 29.
iii. mgvavb ˆmU =  3x + 5y 29
ii. B = {x  N : x, 21 ‰i àYbxqK}, A ‰i K 4:5 L 3:4 M 2:3 N 4:3
wbGPi ˆKvbwU mwVK?
‰KwU cÉK‡Z DcGmU K i I ii L i I iii 30. 2x  2y = 8 ‰i Ici wbfÆikxj mgxKiY wbGPi
iii. A I B ˆmU `yBwU Ab¯¦ ˆmU M ii I iii N i, ii I iii ˆKvbwU?
wbGPi ˆKvbwU mwVK? 19. ‰KwU wòfzGRi ˆKvYàGjvi AbycvZ 1 : 2 : 3 nGj K 2x  3y = 5 L 5x + 3y = 3
K i I ii L i I iii M x+y=4 N 3x  3y = 12
wòfzRwU ˆKvb aiGbi?
M ii I iii N i, ii I iii K mgGKvYx L mħGKvYx
31. ‰KwU cÉK‡Z f™²vsk x nGj
y
5. (x + y, 0) = (1, x  y) nGj (x, y) = KZ? M Õ©ƒjGKvYx N mgevü i. y  x nGe
K (0, 0) L (1, 1)
wbGPi ZG^Åi wfwîGZ (20-22) bs cÉGk²i Dîi `vI: ii. x  y nGe
1 1
M (2, 2) N
( )

2 2 A M B
x
iii. + 1 ‰KwU AcÉK‡Z f™²vsk
y
wbGPi ZG^Åi AvGjvGK (6-8) bs cÉGk²i Dîi wbGPi ˆKvbwU mwVK?
`vI: O K i I ii L i I iii
f(x) = ax2 + b2x.
M ii I iii N i, ii I iii
6. f(2) = KZ?
K 4a + 2b2 L 2x2 + 4x
C N D 32. 3 + 10 + 17 + 24 +  avivi AÓ¡g c` KZ nGe?
M 6ab2 N 4ab2 K 35 L 42 M 52 N 62
O ˆK±`ÊwewkÓ¡ e†Gî AD ‰es BC `yBwU eÅvm| 33.
7. a ‰i ˆKvb gvGbi RbÅ f(1) = 0 nGe?  12 + 4  4 +  avivwUi ˆKvb c`  4 ?
b2 MN  AB| 3 27
2 2
K b L b M b N K 4 L5 M6 N7
a AB = 8 ˆm.wg. ‰es ON = 3 ˆm.wg.|
20. AM = KZ ˆm.wg.?
wbGPi ZG^Åi AvGjvGK (34-36) bs cÉGk²i Dîi `vI:
8. a = b nGj f(2) = KZ?
2

K 6b2 L 4b2 M 4a N 5a K 4 L5 M6 N8 ‰KwU avivi cÉ^g c` a, mvaviY AbycvZ r, PZz^Æ c` 2.

21. e†îwUi eÅvmvaÆ KZ ˆm.wg.? 34. wbGPi ˆKvbwU mwVK mgxKiY?
9. x  2 6  5 = 0 nGj x + 1 ‰i gvb ˆKvbwU?
2
a(rn1  1)
x K 3 L4 M5 N6 K ar4 = 2 L =2
K 10 L2 3 M2 2 N1 22. e†Gîi ˆÞòdj KZ ˆm.wg.? r1
10. a + b = 2, a2  ab + b2 = 8 nGj a3 + b3 = ? a(1  rn)
K 75.4 L 78.54 M ar3 =  2 N =2
1 M 83.44 N 85.48 1r
K 4 L2 2 M2 N 35. beg c` 8 2 nGj r = KZ?
4 23. B
11. 35  2x  x2 ‰i ‰KwU Drcv`K wbGPi ˆKvbwU? K  2 L 2 M2 N 2
K 5+x L x7 36. cÉ^g c`wUi gvb KZ?
M 5x N x2 + 7 x 2x 1 1
A C K  L 2 M N 2
12. 10% nvi gybvdvq 200 UvKvi 3 eQGii PKÌe†w«¬ O 2 2
gybvdv 66 UvKv nGj 37. ‰KwU eMÆGÞGòi cwimxgv wg. nGj ‰i 32 2
i. mij gybvdv ˆÞòdj KZ eMÆwgUvi nGe?
ii. PKÌe†w«¬ gybvdvq me†w«¬ gƒjab 266.2 UvKv O ˆK±`ÊwewkÓ¡ e†Gî ACB ‰i mÁ·ƒiK ˆKvY K 128 L 164
iii. PKÌe†w«¬ gybvdv I mij gybvdvi cv^ÆKÅ 6.2 UvKv KZ wWwMÉ? M 180 N 185
wbGPi ˆKvbwU mwVK? K 120 L 130 M 140 N 150 38. ‰KwU e†Gîi NƒYÆb cÉwZmgZvi gvòv KZ?
24. O ˆK±`ÊwewkÓ¡ e†Gî `yBwU Õ·kÆK PQ I PR nGj K 1 L2 M3 N Amxg
K i I ii L i I iii
M ii I iii N i, ii I iii PQR Kx aiGbi wòfz R ? 39. KÌ g GhvwRZ MYmsLÅv mviwY
4 K mgGKvYx L mgevü i. eÅZxZ Mo wbYÆq Kiv mÁ¿e bq
13. 6 ivwkwUi wfwî wbGPi ˆKvbwU?
() ii. gaÅgv wbYÆGqi RbÅ AvekÅK
9 M mgw«¼evü N welgevü
K
9
L
3
M
2
N4 wbGPi ZG^Åi AvGjvGK 25 I 26 bs cÉGk²i Dîi `vI: iii. AwRf ˆiLv AâGbi RbÅ AvekÅK
6 2 3 wbGPi ˆKvbwU mwVK?
14. 400 ‰i jM 4 nGj wfwî KZ? A K i I ii L ii I iii
K 2 5 L5 2 M 5 5 N 25  13 M i I iii N i, ii I iii
wbGPi ZG^Åi wfwîGZ (15-17) bs cÉGk²i Dîi 40. MYmsLÅv eüfzR AâGbi RbÅ AvqZGjGLi
`vI: B C AvqZmgƒGni f„wgi mgv¯¦ivj wecixZ evüi ˆKvb
12
we±`ymgƒn wbaÆviY Kiv nq?
log5 ( )
3
5 ( 5) 25. sin = KZ?
K cÉv¯¦we±`y«¼q
15. ‰LvGb jGMi wfwî KZ? K
4
L
12
M
13
N
13
13 13 12 5 L gaÅwe±`y
K 5 L 5
3
26. cot2 = KZ? M KYÆ«¼Gqi ˆQ`we±`y
M 5 N
5 144 25 25 144 N f„wgi DciÕ© ˆhGKvb we±`y
K L M N
6 169 169 144 25
1 K 2 L 3 M 4 K 5 N 6 K 7 M 8 K 9 L 10 K 11 M 12 N 13 M 14 K 15 K 16 M 17 K 18 M 19 K 20 K
Dîi

21 M 22 L 23 K 24 M 25 L 26 M 27 K 28 N 29 L 30 N 31 L 32 M 33 L 34 M 35 K 36 M 37 K 38 N 39 L 40 L
6 MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkœcò I DËi : eûwbe©vPwb

210. XvKv KGjwRGqU Õ•zj, XvKv welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1. n  ô ‰i RbÅ ˆKvbwU weGRvo msLÅv? wbGPi ˆKvbwU mwVK? ‰GÞGò AC ‰i gvb KZ?
K n+2 L n+1 K i I ii L ii I iii K 20 wgUvi L 20 3 wgUvi
M 2n + 1 N 2n
M i I iii N i, ii I iii 20
2. GKvGbv f™²vsGki je I niGK Õ¼vfvweK msLÅvq wbGPi ZG^Åi AvGjvGK 16 I 17 bs cÉGk²i Dîi `vI: M wgUvi N 40 wgUvi
3
cÉKvk KiGZ cviGj, ‰ f™²vskwU nGe wbGPi
`yB Aâ wewkÓ¡ ‰KwU msLÅvi `kK Õ©vbxq 27. `yBwU msLÅvi AbycvZ 3 : 4 ‰es M. mv. à 5 nGj,
ˆKvbwU?
Aâ ‰KKÕ©vbxq AGâi wZbàY| msLÅv `yBwU KZ?
K gƒj` msLÅv L Agƒj` msLÅv
16. ‰KK Õ©vbxq Aâ x nGj msLÅvwU KZ? K 15, 20 L 30, 40
M Õ¼vfvweK msLÅv N `kwgK msLÅv M 45, 60 N 60, 80
K 31x L 21x M 3x N 2x
3. x, y, z evÕ¦emsLÅv ‰es x < y nGj 17. x = 2 nGj, gƒj msLÅvi mvG^ Õ©vb wewbgqK‡Z 28. x + y = 4, x  y = 2 nGj, (x, y) ‰i gvb wbGPi
i. xz > yz hLb z > 0 msLÅvi cv^ÆKÅ KZ? ˆKvbwU?
ii. xz > yz hLb z < 0 K 26 L 16 M 36 N 46 K (2, 4) L (4, 2) M (3, 1) N (1, 3)
iii. x(y + z) = xy + xz 18. mijGiLv ‰KwU ˆmU nGj Zvi Dcv`vb ˆKvbwU? 29. ‰KwU mgv¯¦i avivi cÉ^g c` 5 ‰es mvaviY
wbGPi ˆKvbwU mwVK? K we±`y L ˆiLvsk A¯¦i 3 nGj DÚ avivi x Zg c` KZ?
K i I ii L i I iii M Zj N iwkÄ K 2x  2 L 3x  3
M ii I iii N i, ii I iii 19. A M 3x + 2 N 3x + 5
4. hw` A = {a, b, c} I B = {d, e, f} nq, P(A  B) wbGPi Z^Å ˆ^GK 30 I 31 bs cÉGk²i Dîi `vI:
‰i m`mÅ msLÅv KZ? log2 + log4 + log8 + ........
K 9 L8 D E 30. avivwUi mvaviY A¯¦i ˆKvbwU?
M 7 N6 K 2 L4 M log2 N 2log2
B C
5. A = {2, 3, 4}, B = {1, 2, 9}, C = {2, 4, 9} nGj, 31. avivwUi mµ¦g c` KZ?
ABC ‰ DE || BC ‰es ADE = 60 nGj
(A  B)  C ˆmU ˆKvbwU? K log32 L log64
K {2} L {2, 4} B = KZ? M log128 N log256
K 30 L 120
M {2, 3, 4, 9} N {1, 2, 3, 4, 9} 32. `yB wòfzGRi gGaÅ ˆKvb Dcv`vbàGjv mgvb nIqv
wbGPi Z^ÅwUi AvGjvGK 6 I 7 bs cÉGk²i Dîi `vI: M 60 N 50
2 2 20. wbGPi ˆKvb PZzfzÆGRi KYÆ«¼q ciÕ·iGK mGî½I wòfzR«¼q meÆmg nGe bv?
A = {x : x  5x + 6 = 0}, B = {y : y  7y + 12 = 0}
K wZb ˆKvY L `yB evü A¯¦fzÆÚ ˆKvY
6. A ‰i ˆmU ˆKvbwU? mgGKvGY mgw«¼LwíZ KGi?
K {1, 3} L {2, 4} K AvqZGÞò L mvgv¯¦wiK M wZb evü N `yB ˆKvY ‰K evü
M {2, 3} N {3, 4} M iÁ¼m N UÇvwcwRqvg 33. GKvGbv ‰KwU mgevü wòfzGRi ˆÞòdj mgevü
7. B ‰i ˆmU ˆKvbwU? 21. wòfzGRi cwimxgvi mgvb nGj wòfzRwUi ‰K
D
K {1, 2} L {2, 4} C evüi Š`NÆÅ KZ?
M {2, 3} N {3, 4}
K 2 3 L3 3 M4 3 N5 3
1
8. x = 2  3 nGj ‰i gvb KZ nGe?
x 60 34. ‰KwU PvKvi eÅvm 4.2 wgUvi nGj, ‰i cwiwa
A
K 1 L2 3
B KZ?
M 2 3+2 N 2+ 3 wPòwU nGZ ABC ‰i gvb KZ? K 13.19 wgUvi (cÉvq) L 13.18 wgUvi (cÉvq)
9. (a  b)2 = 3 nGj (a  b)3 = KZ? K 140 L 150 M 130 N 120
M 13.16 wgUvi (cÉvq) N 13.07 wgUvi (cÉvq)
K 3 L6 M3 3 N9
22. `yBwU e†î ciÕ·iGK mGeÆvœP KqwU we±`yGZ ˆQ` 35. ‰KwU eMÆGÞGòi KGYÆi Š`NÆÅ 4 wgUvi nGj, ‰i
10. I = Pnr ‰i ˆÞGò
KiGZ cvGi? evüi Š`NÆÅ KZ wgUvi?
K 1wU L 2wU M 3wU N 4wU
i. I = mij gybvdv K
1
L
1
M
1
N2 2
23. 2
ii. P = gƒjab A 2 2 2
iii. c = ‰KK mgq 36. wbGPi Z^ÅàGjv jÞ Ki:
wbGPi ˆKvbwU mwVK? 1 i. e†Gîi eÅvm < cwiwa
K i I ii L ii I iii ii. eMÆGÞGòi KYÆ > eMÆGÞGòi evü
M i I iii N i, ii I iii B C iii. ˆKvYGKi ˆnjvGbv D®²wZ > DœPZv

3 1
3 wbGPi ˆKvbwU mwVK?
11. ‰i gvGbi ˆÞGò wbGPi ˆKvbwU mwVK? tan ‰i gvb KZ? K i I ii L i I iii
64
1 1 1 3 K
1
L
1
M
3
N 3 M ii I iii N i, ii I iii
K L M N 2 2
8 9 4 64 3 37. GKvbwU ˆKw±`Êq cÉeYZvi cwigvc bq?
24. sin + cos = 1 nGj  = KZ? K Mo L gaÅK
12. log5{( ) } 3
5 ( 5) ‰i gvb KZ? K 30 L 45 M 60 N 90 M cÉPziK N MYmsLÅv
K
7
L
5
M
6
N
6 25. A = 30 nGj 38. wbGPi ˆKvbwU AwewœQ®² PjK?
6 6 5 7 i. sin2A = cos2A ii. sinA = cos2A
1 K Zvcgvòv L eqm
13. x + = 2 nGj, x ‰i gvb KZ? iii. tanA =
1
x
3 M IRb N meàGjv
1
K 1 L1 M
2
N2 wbGPi ˆKvbwU mwVK? 39. 2, 10, 8 ˆKvGbv mviwYi MYmsLÅv nGj ‰G`i
14. ‰KwU Õ¼vfvweK msLÅvi eGMÆi mvG^ msLÅvwU ˆhvM K i I ii L ii I iii KÌGgvGhvwRZ MYmsLÅv KZ?
M i I iii N i, ii I iii K 10 L 18 M 20 N 22
KiGj wbGPi ˆKvbwU nGe?
K x2 + x L x2 + 2x 26. 40. mviwYfzÚKiGYi ˆÞGò cÉ^Gg ˆKvbwU wbYÆq
M x2 + 1 N x2 + 2 A KiGZ nq?
15. wbGPi Z^ÅàGjv jÞÅ Ki: K ˆkÉwY msLÅv L ˆkÉwY eÅeavb
i. x + 3 = 10 ‰KwU mgxKiY 1
M cwimi N MYmsLÅv
ii. x + 3 = 10 mgxKiYwUi mgvavb ˆmU {7}
60
iii. x2 + x + 12 ‰KwU mgxKiY C B
20 wgUvi

1 M 2 K 3 M 4 K 5 N 6 M 7 N 8 N 9 M 10 K 11 M 12 L 13 L 14 K 15 K 16 K 17 M 18 K 19 M 20 M
Dîi

21 N 22 L 23 K 24 N 25 L 26 L 27 K 28 M 29 M 30 M 31 M 32 K 33 M 34 K 35 N 36 N 37 N 38 N 39 M 40 M
MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkc
œ ò I DËi : eûwbe©vPwb 7

211. ebvbx we`Åv wbGKZb, XvKv welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1. sin =
1
nGj, cot ‰i gvb KZ? 16. 4x + 9 ‰i mvG^ KZ ˆhvM KiGj ˆhvMdj cƒYÆeMÆ
4

2 nGe? wPGò AB = 10 cm
K 2 L 3 M
3
N
1 K 12x2 L 8x2 M 6x2 N 4x2 ABCD PZzfzÆR ˆÞòwUi ˆÞòdj KZ eMÆ ˆm.wg.?
2 3 17. ‰KwU eBGqi gƒjÅ 320 UvKv| ‰B gƒjÅ cÉK‡Z K 30 L 40 M 50 N 60
2. x x
+ 3 =  3 nGj, x ‰i gvb KZ?
gƒGjÅi 80% | eBwUi cÉK‡Z gƒjÅ KZ? 30. wPGò ABC ‰i
6 3 K 400 UvKv L 360 UvKv i. cwimxgv 36 ˆm.wg.
A 15 ˆm.wg.
K 120 L 60 M 36 N 6
M 350 UvKv N 340 UvKv ii. ˆÞòdj 54 eMÆ ˆm.wg.
1
3. , 1, 3 AbyKÌgwUi mvaviY AbycvZ KZ? 18. f(x) = (x  2)2 nGj, f(a  2) ‰i gvb KZ? iii. BAC > ABC B 
C
3
1
K x2 L a2 wbGPi ˆKvbwU mwVK? 12 ˆm.wg.
K i I ii L i I iii
2 2
K  3 L 1 M N 3 M (a  2) N (a  4)
3
19. a 
1 1
= 0 nGj, a3 + 3 ‰i gvb KZ? M ii I iii N i, ii I iii
4. 0.00435 msLÅvwUi jGMi cƒYÆK KZ? a
  
a 31. wbGPi ˆKvb evüàGjv «¼viv mgGKvYx wòfzR AuvKv
K 0 L 1 M2 N 3 K 0 L2 M 5 N4 hvq?
5. ‰KwU NbGKi mÁ·ƒYÆ c†GÓ¤i ˆÞòdj 96 eMÆwgUvi A
20. A = 25x  1, B = 5x + 1 nGj, B = KZ? K 1 ˆm.wg., 2 ˆm.wg. I 3 ˆm.wg.
nGj, ‰i avi KZ? K 5 x1
L5 x+1 x
M 5  1N 5  1 2x L 2 ˆm.wg., 3 ˆm.wg. I 4 ˆm.wg.
K 4 wg. L 6 wg. M 3 ˆm.wg., 4 ˆm.wg. I 5 ˆm.wg.
M 6.928 wg. N 8 wg.
21. 2x  5 + 3 = 4 ‰i mwVK mgvavb ˆmU ˆKvbwU?
K {3} L {3} M {3} N  N 4 ˆm.wg., 5 ˆm.wg. I 6 ˆm.wg.
6. U = {1, 2, 3, 4, 5}, A = {2, 4, 5}, B = {1, 3, 5}
22. ax2 + bx + c = 0 [ˆhLvGb a, b, c aË‚eK] mgxKiYwU 32. ‰KwU mgevü wòfzGRi evüi Š`NÆÅ 25 ˆm.wg.|
nGj w«¼NvZ nIqvi kZÆ wbGPi ˆKvbwU? cÉwZwU evüi Š`NÆÅ 4% KgvGj, cwimxgvi Kxi…c
i. A ˆmGU ˆgŒwjK msLÅv 2wU K a=0 L b = 0, a = 0 cwieZÆb nGe?
ii. P(A) ‰i Dcv`vb msLÅv 4 M c = 0, a = 0 N a0 K AcwiewZÆZ ^vKGe L 3 ˆm.wg. nËvm cvGe
iii. A  B = A 23. E A F M 3 ˆm.wg. e†w«¬ cvGe N 24 ˆm.wg. e†w«¬ cvGe
wbGPi ˆKvbwU mwVK? D
wbGPi ZG^Åi AvGjvGK 33 I 34bs cÉGk²i Dîi `vI:
K i I ii L i I iii AvqZvKvi ‰KwU NGii ˆgGSi Š`NÆÅ, cÉÕ© AGcÞv 4
M ii I iii N i, ii I iii wgUvi ˆewk ‰es ˆgGSi cwimxgv 88 wgUvi|
B
7. a + b = 3 ‰es a  b = 2 nGj C
33. NiwUi ˆgGSi Š`NÆÅ KZ wgUvi?
1 wPGò BE || CF, BC || AD ‰es AB || CD nGj K 24 L 20 M 16 N 12
i. ab = ii. a2 + b2 = 5
4 i. ABC = ABD
ii. ABC = BCD
34. NiwUi ˆgGSi ˆÞòdj KZ eMÆwgUvi?
iii. a2  b2 = 6 K 192 L 320 M 480 N 488
iii. BCD = ACF
wbGPi ˆKvbwU mwVK? wbGPi ˆKvbwU mwVK? 35.
K i I ii L i I iii x 0 2 4
K i I ii L i I iii
M ii I iii N i, ii I iii y 4 0 4
M ii I iii N i, ii I iii wbGPi ˆKvbwUi RbÅ cvGki QKwU mwVK?

8. 0.81 ‰i mvgvbÅ f™²vsk wbGPi ˆKvbwU? 24. D C
K y=x4 L y=8x
4 9 5 9 M y = 4  2x N y = 2x  4
K L M N 5 ˆm.wg.
9 20 11 11 36. a1x + b1y = c1, a2x + b2x = c2 mgxKiY ˆRvU
wbGPi ZG^Åi AvGjvGK 9 I 10 bs cÉGk²i Dîi `vI :
mvgémÅ nGe hw`
ˆkÉwY 6-10 11-15 16-20 21-25 A B
wPGò ABCD eGMÆi KGYÆi Š`NÆÅ KZ ˆm.wg.? a1 b1
NUb msLÅv 4 10 15 20 i. 
a2 b2
K 4 2 L4 3 M5 2 N5 3
9. Z‡Zxq ˆkÉwYi cÉK‡Z DœP mxgv KZ? wbGPi ZG^Åi wfwîGZ 25 I 26 bs cÉGk²i Dîi `vI :
a1 b1 c1
ii. = 
K 20.5 L 20 M 19.5 N 18 a2 b2 c2
10. DcvGîi cÉPziK wPGò AB = BC = CA = 4 A a1 b1 c1
K 20 L 21 M 22 N 23 ˆm.wg.| D, E, F h^vKÌGg AB, iii. = =
a2 b2 c2
D F
wbGPi ZG^Åi AvGjvGK 11 I 12 bs cÉGk²i Dîi `vI : BC I AC evüi gaÅwe±`y| wbGPi ˆKvbwU mwVK?
wPGò AB = 5 ˆm.wg., AO = 3 ˆm.wg. K i I ii L i I iii
B C
A E M ii I iii N i, ii I iii
D 25.  ˆÞò DEF ‰i ˆÞòdj KZ eMÆ ˆm.wg.? 37. ABCD PZzfzÆGRi ˆKvYàGjvi AbycvZ 3 : 4 : 5 :
90 K 3 3 L2 3 M 3 N
3 6 nGj, e†nîg I Þz`ËZg ˆKvGYi A¯¦i KZ?
2 K 60 L 40 M 30 N 10
O 26. ABC-‰i cwie†Gîi eÅmvaÆ KZ ˆm.wg.? 38. n msLÅK evüwewkÓ¡ mylg eüfzGRi ˆÞòdj
B C
11. ABCD PZzfzÆGRi cwimxgv KZ ˆm.wg.? K
4
L
2
M
1
N
3 wbYÆGqi mƒò wbGPi ˆKvbwU?
3 3 3 4
na2 na2
12.
K 20 L 16 M 12 N 8
ABCD PZzfzÆR ˆÞGòi ˆÞòdj KZ eMÆ ˆm.wg.? 27. sec + tan =
1
2
nGj, sec  tan ‰i gvb KZ?
K
4 ( )
cot
180
n
L( )
4
tan
180
n
K 48 L 25 M 24 N 20 a2 a2
13. A ˆmU ˆ^GK B ˆmGU R ‰KwU A®¼q nGj wbGPi K
1
9
L
1
4
M
1
2
N2 M
4 ( )
cot
180
n ( )
N tan
4
180
n
wbGPi ZG^Åi AvGjvGK 39 I 40 bs cÉGk²i Dîi `vI :
ˆKvbwU mwVK? 28. A we±`yGZ B we±`yi AebwZ ˆKvY KZ?
K R A B L R A B A
D A
M A B R N A BR
14. S = {(3, 8}, (2, 3), (1, 0), (0, 1), (1, 0), (2, 3)} 30
nGj, S ‰i
i. ˆWvGgb {3, 2, 1, 0, 1} B C
D
ii. ˆié = {8, 3, 0, 1} B C wPGò, BC = 10 wgUvi, AD = 4 wgUvi|
iii. ‰KB Dcv`vGbi ‰KvwaK cÉwZœQwe ˆbB K 30 L 45 M 60 N 90 39. ABC ‰i DœPZv I f„wgi AbycvZ KZ?
wbGPi ˆKvbwU mwVK? 29. 1 4 2 5
D 6cm C K L M N
K i I ii L i I iii 2 5 5 4
M ii I iii N i, ii I iii 40. ABD : ADC = KZ?
15. x3  3xy2 + 2y3 ‰i ‰KwU Drcv`K wbGPi ˆKvbwU? 5cm K 3:7 L 7:3 M 2:5 N 5:2
K x+y L xy M x  2yN x  3y A
F
B
E
1 L 2 M 3 K 4 N 5 K 6 N 7 L 8 N 9 K 10 M 11 K 12 M 13 L 14 M 15 L 16 K 17 K 18 N 19 L 20 M
Dîi

21 K 22 N 23 K 24 M 25 M 26 K 27 N 28 M 29 L 30 K 31 M 32 L 33 K 34 M 35 N 36 L 37 K 38 K 39 M 40 K
8 MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkœcò I DËi : eûwbe©vPwb

212. wewmAvBwm KGjR, XvKv welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1. f(x) = x + 5x  3 nGj, f(1) ‰i
5
gvb KZ? wPGò ABCD mvgv¯¦wiK nGj,
K 9 L 7 M3 N7 i. BE = 8 3 wg. wPGò ABC ‰i ˆÞòdj KZ?
2. mƒPGKi ˆÞGò ii. ABCD ‰i ˆÞòdj 800 eMÆwgUvi K 12 3 L 6 3 M 24 3 N 19
i. (ab) m = am. bm ii. a0 = 1 (a  0) iii.  ˆÞò ABE = 32 3 eMÆ wg. 1
am m n 28. a = 3 + 2 nGj, a + a ‰i gvb KZ?
iii. n = a + a
a
wbGPi ˆKvbwU mwVK?
K i I ii L i I iii K 3 3 L 6 M2 3 N2 2
wbGPi ˆKvbwU mwVK? 29. e†Gîi
K i I ii L ii I iii M ii I iii N i, ii I iii
i. eÅvmB e†nîg RÅv
M i I iii N i, ii I iii 15. hw` DcvGîi msLÅv n nq ‰es n weGRvo msLÅv
ii. mKj RÅv ˆK±`Ê ˆ^GK mg`ƒieZÆx
3. nq, ZGe gaÅK nGe ˆKvb cG`i gvb?
D A n n+1 iii. ˆK±`Ê ˆ^GK mg`ƒieZÆx mKj RÅv ciÕ·i mgvb
60 K Zg L Zg wbGPi ˆKvbwU mwVK?
2 2
n n+2 K i I ii L i I iii
B
90
C
M
2
+ 1 Zg ( )
N
2
Zg M i, ii I iii N ii I iii
16. 7x + y = 5 mgxKiGY x = 1 emvGj mgxKiYGK 30. (3, 1) ‰es (3, 1) we±`y `yBwUi AeÕ©vb QK
A we±`yGZ B we±`yi AebwZ ˆKvGYi cwigvY KZ? KvMGRi ˆKvb PZzfÆvGM?
wm«¬Kvix we±`ywU ˆKvb PZzfÆvGM coGe?
K 90 L 60 M 45 N 30 K 2q I 3q L 2q I 4^Æ
K 1g L 2q M 3q N 4^Æ
4. log10 x = 3 nGj, x ‰i gvb KZ? . M 1g I 4^Æ N 1g I 2q
K 30 L 10 Mx 3
N 10 3
17. 0.84 ‰i mvgvbÅ f™²vsGk cÉKvk ˆKvbwU? 31. 1 + 2 + 3 + 4 + .... + 100 = KZ?
K 4750 L 4950
5. cosec + cot = 56 nGj cosec  cot = KZ? K
84
99
L
84
90
M
38
48
N
84
100 M 5050 N 5150
K
1
L
6
M1 N
5 18. A 32. 3x  5y = 7, 6x  10y = 15 mgxKiY ˆRvGUi
6 5 6 mgvavb?
6. 4x + 6y = 10, 8x + 12y = 20 mgxKiY ˆRvGUi K AmsLÅ L AbbÅ
cÉK‡wZ Kx? B C M `yBwU N mgvavb ˆbB
K mãwZcƒYÆ L wbfÆikxj 33. ‰KwU eGMÆi mGeÆvœP KZwU cÉwZmvgÅ ˆiLv AvGQ?
wPGò C = 2A nGj A ‰i gvb KZ?
M AmãwZcƒYÆ N AwbfÆikxj K 4 L5 M2 N3
K 10 L 30 M 45 N 60
a2  3a + 1 = 0 ‰LvGb a> 1
19. 34. wbGPi ˆKvbwU wewœQ®² PjK?
DcGii ZG^Åi AvGjvGK 7 I 8 bs cÉGk²i Dîi `vI: A
D K eqm L DœPZv
1 M IRb N RbmsLÅv
7. a + a2 ‰i gvb KZ?
2 50
35. ‰KwU wòfzGRi wZbevüi Š`NÆÅ h^vKÌGg 5 ˆm.wg.,
K 6 L7 M9 N 10 B 6 ˆm.wg., 7 ˆm.wg. ‰i ˆÞòdj KZ?
C
8. a2  a12 ‰i gvb wbGPi ˆKvbwU? K 15.7 eMÆ ˆm.wg. L 15 eMÆ ˆm.wg.
wPGò BCD ‰i gvb KZ? M 14.7 eMÆ ˆg.wg. N 13.7 eMÆ ˆm.wg.
K 45 L 40 M 3 5 N 3 5 K 50 L 130 36. O
A M 40 N 120
10
20. 24 ˆKwR wgkÉGbi ˆgŒjmgƒGni AbycvZ 12 :
2 1
:
3 4 A
6
B
O
AbycvGZi mij AbycvZ wbGPi ˆKvbwU? D
B C K 1:2:1 L 2:3:4
M 6:8:3 N 3:6:8 wPGò AB ‰i Š`NÆÅ KZ?
9. ABC ‰KwU mgevü wòfzR BOC = KZ? K 8 L 12
K 90 L 100 M 120 N 130 21. 0.00123 msLÅvwUi ŠeævwbK AvKvi ˆKvbwU? M 16 N 20
10. x, y, z evÕ¦e msLÅv ‰es x < y nGj, K 123  103 L 12.3  10 4
37. AwRf ˆiLvi ˆÞGò wbGPi ˆKvbwU mwVK?
M 1.23  103 N 0.123  103
i. xz < yz (hLb z  0) K Da»ÆMvgx L wbÁ²Mvgx
ii. xz  yz (hLb z < 0)
22. NbGKi avGii Š`NÆÅ 3 wgUvi nGj, cÉwZ ZGji M mgv¯¦ivj N DjÁ¼
iii. x(y + z) = xy + xz ˆÞòdj KZ? 38. D A
wbGPi ˆKvbwU mwVK? K 18 eMÆ wg. L 3 eMÆ wg.
K i I ii L i I iii M 3 eMÆ wg. N 3 3 eMÆ wg. 1

M ii I iii N i, ii I iii 23. 5 sin A = 3 nGj tan A ‰i gvb KZ? C


B
wbGPi ZG^Åi AvGjvGK 11 I 12bs cÉGk²i Dîi `vI: K
4
L
5
M
3
N
4 3
A
5 4 4 3 wPGò BAC ‰i  ˆKvY ‰i gvb KZ?
13
24. K 30 L 45

x M 60 N 90
B C
70
wbGPi Z^Å AbymvGi 39 I 40bs cÉGk²i Dîi `vI:
12 70
11. sin = KZ? D

K
4
L
12
M
13
N
5 wPGò x ‰i gvb KZ? B 8
13 13 12 13 K 50 L 70 M 120 N 110 x x
12. cot2 = KZ? 25. ‰KwU NƒYÆb cÉwZmgeÕ§i NƒYÆb ˆKvY 40 cƒYÆ NƒYÆb 6
K
144
L
25
M
25
N
144 mÁ·ƒYÆ KiGZ KqwU Õ©vGb eÕ§wUi AvK‡wZGZ O
E
169 100 144 25
cwieZÆb NUGe bv?
13. hw` x, y, z KÌwgK mgvbycvwZK nq, ZGe ˆKvbwU K 9 L 11 M4 N7
A C
mwVK?2 26. 1 ˆ^GK 22 chƯ¦ 3 «¼viv wefvRÅ msLÅvàGjvi ‰LvGb AB || DC
K x = yz L y2 = xz 39. wbGPi ˆKvbwU mwVK?
M z2 = xy N x2 = y2z gaÅK KZ?
K 9 L 12 M 15 N 18 OA OB OD OC
14. K = L =
A 100 wg. 27. A
OD OC OB OA
D AB OD OC AB
M = N =
16 wg. 4 CD OA OB CD
30 60 40. CE : OE = KZ?
B C K 4:1 L x:x
B 3 3 M 3:4 N 8:6
E C

1 M 2 K 3 N 4 N 5 L 6 L 7 L 8 M 9 M 10 N 11 L 12 M 13 L 14 K 15 L 16 N 17 M 18 L 19 L 20 M
Dîi

21 M 22 M 23 M 24 N 25 K 26 L 27 N 28 M 29 L 30 L 31 M 32 N 33 K 34 N 35 M 36 M 37 K 38 K 39 K 40 M
MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkc
œ ò I DËi : eûwbe©vPwb 9

213. AvigvwbGUvjv miKvwi DœP we`Åvjq, XvKv welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
10 5 14. GKvb `yBwU mgGKvYx wòfzGRi ˆKvY? 26. a : b = 4 : 7 ‰es b : c = 5 : 6 nGj
1.  = KZ?
18 12 K 45, 50 L 35, 45 a : b : c = KZ?
3 M 40, 50 N 43, 57 K 4:7:6 L 20 : 35 : 24
K L 3 15. C M 20 : 35 : 42 N 24 : 35 : 30
5
2 3 27. 5x + 2y = 0 ‰es 5x  2y = 0 mgxKiY ˆRvGUi
M N RbÅ ˆKvbwU mZÅ?
3 2
. O K mgvavb AbbÅ L mgvavb ˆbB
2. 5.6 ‰i mvgvbÅ f™²vsk ˆKvbwU? A B M mgvavb AmsLÅ N mgvavb 2wU
13 15
K
3
L
7 28. ˆKvb BsGiwR eYÆwUi cÉwZmvgÅ ˆiLv bvB?
19 17 O ˆK±`ÊwewkÓ¡ e†Gî ACB = 45 nGj OAB = KZ? K ‰ L we
M N M wm N wU
5 3 K 45 L 50
3. wbGPi Z^ÅàGjv jÞ Ki: M 60 N 90 29. ‰KwU mgw«¼evü wòfzGRi mgvb mgvb evüi Š`NÆÅ
i. ˆh me Õ¼vfvweK msLÅv 10 ‰i eo ZvG`i ˆmU 16. 90 ˆKvGYi cƒiK ˆKvY KZ? 9 cm ‰es f„wgi Š`NÆÅ 8 cm nGj, ˆÞòdj KZ?
A = {x  ô : x > 10} K 90 L 60 K 30.25 e. ˆm. wg. L 32.25 e. ˆm. wg.
ii. ˆh me Õ¼vfvweK msLÅv 16 ‰i àYbxqK ZvG`i M 45 N 0 M 34.25 e. ˆm. wg. N 36.25 e. ˆm. wg.
x 17. ‰KwU wòfzGRi KqwU gaÅgv AvuKv hvGe? 30. Zvcgvòv, eqm, DœPZv, IRb BZÅvw`i RbÅ
ˆmU B = { xô:
16
cƒYÆmsLÅv
K 5wU } L 4wU eÅen†Z PjK ˆKvbwU?
iii. ˆh me Õ¼vfvweK msLÅv 7 ‰i àwYZK ZvG`i M 3wU N AmsLÅ K AwewœQ®² PjK L wewœQ®² PjK
x 18. wbGP wòfzGRi evüi Š`NÆÅ ˆ`Iqv AvGQ| wòfzRwU M Õ¼vaxb PjK N Aaxb PjK
{
ˆmU C = x  ô : 7 cƒYÆmsLÅv }
AvuKv mÁ¿e bq? 31. 5, 3, 0, 7, 8, 7, 10, 2, 6 ‰i Mo KZ?
wbGPi ˆKvbwU mwVK? K 1.88 L 1.87
K 3, 5, 7 ˆm. wg. L 2, 3, 5 ˆm. wg.
K i I ii L ii I iii M 1.78 N 1.68
M 4, 5, 6 ˆm. wg. N 5, 6, 8 ˆm. wg.
M i I iii N i, ii I iii 32. 5 + 11 + 17 + 23 + --------- + 59 = KZ?
19. ˆKvGbv e†Gîi `yBwU RÅv ciÕ·iGK mgw«¼LwíZ K 320 L 220
4. (x + y, 0) = (1, x  y) nGj, (x, y) ‰i gvb
KiGj ZvG`i ˆQ`we±`y ˆKv^vq nGe? M 420 N 520
ˆKvbwU?
K e†îwUi ˆKG±`Ê 33.
1 1
1 + + + -------- avivwUi 5g c` KZ?
K (1, 1) L (1, 1) 3 9
1 1 1 L e†îwUi ˆKG±`Êi evwnGi
M 1 ( )
2
N 
2 2 ( ) M ˆh ˆKvGbv RÅvGqi Dci K
1
27
L
1
54
5. (x) = x2  5x + 6 nGj, x ‰i ˆKvb gvGbi RbÅ N eÅvGmi Dci 1 1
M N
(x) = 0 nGe? 20. 81 243
K 2, 3 L 2, 3 O 34. 13 + 23 + 33 + ---------- + 63 = KZ?
M 2, 3 N 5, 6 K 225 L 441
M 768 N 729
6. x y=2 ‰es xy = 63 nGj, x2 + y2 ‰i gvb A B
M 35.
KZ?
K 160 L 140 O
M 130 N 120 DcGii wPò nGZ cvB
1 3 1 i. M, AB ‰i gaÅwe±`y A B
7. p + = nGj p2 + 2 = KZ? D
p 2 p ii. OMB = OMA = ‰K mgGKvY wPGò AB = 8 ˆm. wg., OD = 3 ˆm. wg. nGj
3 17
K
2
L
4 iii. OB = 4 ˆm.wg. nGj OA = 4 ˆm. wg. OB = KZ?
1 wbGPi ˆKvbwU mwVK? K 7 ˆm. wg. L 6 ˆm. wg.
M N 5
4 K i I ii L i I iii M 5 ˆm. wg. N 4 ˆm. wg.
8. ax + b ivwkwU (x) ‰i Drcv`K nGj ˆKvbwU M ii I iii N i, ii I iii
mwVK? 3 36. a = 3  1, b = 3 + 1 nGj ab = KZ?
K (a) = 0 L (b) = 0
21. tan =  ‰es cos abvñK nGj sec ‰i gvb
4
3+1 31
K 0 L1
b ˆKvbwU?
M  

 ( )
=0 N ( ) a
=0
K 
5
L
4
M
5
N
4 37.
M 2 3 N2 2
x + y = 4, xy = 0 nGj x3 + y3 = KZ?
9. ‰KwU KjGgi gƒjÅ 10 UvKv| 30% jvGf 4 5 4 3
K 141 L 101
 
KjgwUi weKÌqgƒjÅ KZ? 22. sec2  tan2
3 3
‰i gvb ˆKvbwU? M 81 N 64
K 13 UvKv L 14 UvKv 38. x + y = 2 ‰es x2 + y2 = 4 nGj xy = KZ?
K 2 3 L3 K 7 L4
M 15 UvKv N 16 UvKv
M 3 3 N1 M 3 N0
10. 32x = 27 nGj x ‰i gvb KZ? 39. ‰KwU Mvwoi KÌqgƒjÅ x UvKv| MvwowU KZ gƒGjÅ
1 3
23.  ˆKvGYi cƒiK ˆKvY KZ?
K
3
L
2
M3 N2 K 90 L 90 +  wewKÌ KiGj y% jvf nGe?
M 90   N 70 +  xy y
11.
4
81 ‰i gvb KZ? 24. sin 45 ‰i gvb wbGPi ˆKvbwUi mgvb?
K
100
L x x+ ( 100 )
K 3 L 1 K cos45 L tan45 y y
M 3 N AwbGYÆq M cot45 N sec45 (
M x 1+
x+4
)
100
N
100
12. log2 5 400 = x nGj x ‰i gvb KZ? 25. ‰KwU euvGki ˆMvov ˆ^GK 60 wgUvi `ƒGi f„wgi 40. 1
K 2 L3 M4 N6 ‰KwU we±`yGZ euvkwUi D®²wZ ˆKvY 45 | euvGki
()9
= 1 nGj x = KZ?
K 3 L 4
13.
x+2
= 3 ‰i mgvavb ˆmU wbGPi ˆKvbwU? DœPZv KZ? 1
x M  N4
K 60 wg. L 60 3 wg. 3
K {0} L {1}
M {1} N {2} M 30 3 wg. N 20 3 wg.

1 M 2 N 3 M 4 N 5 K 6 M 7 M 8 N 9 K 10 L 11 N 12 M 13 M 14 M 15 K 16 N 17 M 18 L 19 K 20 N
Dîi

21 M 22 N 23 M 24 K 25 K 26 M 27 K 28 M 29 L 30 K 31 M 32 K 33 M 34 L 35 M 36 L 37 N 38 N 39 M 40 L
10 MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkœcò I DËi : eûwbe©vPwb

214. evsjvG`k wkÞK mwgwZ (AaÅÞ gynvÁÃ` Kvgi‚ävgvb), XvKv welq ˆKvW : 1 0 9
mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1. sin 60º + cos 60º = KZ?
2 2
16. 81 ˆK x n
AvKvGi cÉKvk KiGj cvIqv hvGe@ 31. e†Gîi ˆK±`ÊMvgx RÅvGK Kx eGj?
1 K 43 L 34 K eÅvm L eÅvmvaÆ
K L 1
2 M 25 N 52 M cwie†î N Pvc
M
4+ 3
N
3
(4 + 3)
17. a4 + a2b2 + b4 = 3 ‰es a2  ab + b2 = 3 nGj 32. e†Gîi eÅvmvaÆ r nGj@
4 4 a2 + ab + b2 = KZ? i. e†Gîi cwiwa = 4r
2. tan A = 3 nGj cot2A = KZ? K 0 L 1
ii. e†Gîi ˆÞòdj = r2
1 3 M 2 N 9
K L 
3 2 18. 25 x + 1 = 625 nGj x ‰i gvb KZ? iii. e†î Kjvi ˆÞòdj = 360  r2
1 K 1 L 2
M 3 N
3
wbGPi ˆKvbwU mwVK?
M 3 N 4
K i I ii L i I iii
2cos (A + B) = 1 = 2sin ( A  B) nGj 3 I 4 bs 19. 2x  3 + 5 = 2 mgxKiYwUi mgvavb ˆmU M ii I iii N i, ii I iii
cÉGk²i Dîi `vIt ˆKvbwU?
3. A ‰i gvb KZ? K {  3} L {3}
33. cot (90º  ) = 43 nGj cos = KZ?
K 45º L 60º M {6} N {} 3 3
M 90º N 0º K L
20. log3 log2 log 81 = KZ? 5 4
4. sec2B  tan2B = KZ? 3 4 5
K 1 L 4 M N
K 15º L 30º 3 3
M 1 N 0.5 M 81 N log 81 34. 3 + 6 + 9+........avivi KZZg c` 30?
5. sin + cos = 2 nGj tan  = ? 21. x2  5x + 6 = 0 mgxKiGYi gƒj«¼q ˆKvbwU? K 10 L 12
K 0 L 2 K (3,  2) L (3, 2) M 14 N 16
3 M (1, 6) N (3, 2) 35.  mƒßGKvY nGj@
M N 1 22. 5x  3y = 7 ‰es 10x  6y = 14 `ywU mgxKiY@ i. sin + cos > 1
2
mgxKiY ˆRvU mãwZcƒYÆ ii. sin2 + cos2 = 1
6. AvqZ ˆjGLi DœPZv ˆKvbwU? i.
iii. sec2  tan2 = 1
K MYmsLÅv L KÌgGhvwRZ MYmsLÅv ciÕ·i wbfÆikxj
ii.
wbGPi ˆKvbwU mwVK?
M gaÅgvb N GkÉwY eÅeavb iii. AmsLÅ mgvavb AvGQ
K i I ii L i I iii
7. 2x + 2x2 ‰i Drcv`K ˆKvbwU? wbGPi ˆKvbwU mwVK? M ii I iii N i, ii I iii
K 2 L 2x K i I ii L i I iii 36. mKj exRMvwYwZK mƒòB Kx nGe?
M 2x N 2x M ii I iii N i, ii I iii K mgxKiY L ˆf`
8. ˆKvbwU wewœQ®² PjK? 23. ‰KwU mgv¯¦i avivi n Zg c` 5n
+ 3 ‰i M AGf` N PjK
K Zvcgvòv L eqm mvaviY A¯¦i KZ? 37.
M DœPZv N RbmsLÅv K 2 L 5
A
wbGPi ZvwjKvi AvGjvGK 9 I 10 bs cÉGk²i Dîi `vI: M 6 N 8
cÉvµ¦ bÁ¼i 1 -10 11-20 21-30 31-40 41-50 wbGPi Z^Å ˆ^GK 24 I 25 bs cÉGk²i Dîi `vI:
MYmsLÅv 7 10 16 18 9 gƒjab 600 UvKv ‰es mgq 4 eQi gybvdvi nvi 4% 4cm 5cm
9. gaÅK ˆkÉwY ˆKvbwU? 24. mij gybvdvi ˆÞGò me†w«¬ gƒj KZ nGe?
K 11 - 20 L 21 - 30 K 650 UvKv L 578 UvKv
M 31 - 40 N 41 - 50 B C
M 686 UvKv N 696 UvKv
10. cÉPziK ˆKvb ˆkÉwYGZ AvGQ? 25. PKÌe†w«¬ gƒj KZ? wPGòi AvGjvGK sin A + cos C ‰i gvb KZ?
K 2q L 3q
K 701.9 UvKv L 710 UvKv 6 7
M 4^Æ N 5g K
5
L
5
M 709. 1 UvKv N 778.9 UvKv
11. A = {1, 2, 3} nGj A ˆmGUi cÉK‡Z DcGmU KqwU? 8 29
K 8 L 7 26. NbGKi avGii Š`NÆ Å 3 wgUvi| cÉ w Z ZGji M
5
N
15
M 9 N 10 ˆÞòdj KZ? 38.
12. 3 I 4 ‰i gvGS cYƒ msLÅv KqwU? K 3 e. wgUvi L 3 e. wgUvi A
K 1 L 2 M 3 3 e. wgUvi N 18 e. wgUvi
M 3 N AmsLÅ 27. ‰KwU eMÆGÞGòi KGYÆi Š`NÆÅ I cwimxgvi
13. kƒbÅmn mKj abvñK msLÅvi bvg Kx? AbycvZ KZ? 2x
x
K gƒj` msLÅv L cƒYÆmsLÅv B D
K 2 2 :1 L 1: 2 C
M AFYvñK msLÅv N Õ¼vfvweK msLÅv
M 1: 2 2 N 1:4 ABC ‰i gvb KZ?
wbGPi ZG^Åi AvGjvGK 14 I 15 bs cÉGk²i Dîi `vIt K 50º L 60º
1
28. AaÆ e † G îi KqwU cÉ w ZmvgÅ ˆiLv iGqGQ?
2x2  5x = M 120º N 30º
3 K 1wU L 2wU
39. e†Gî A¯¦wjÆwLZ ABCD PZzfzÆGRi A = 60º ‰es
1 M 3wU N 4wU
14. 2x  3x = KZ? wecixZ C = KZ wWMÉx?
29. ‰KwU iÁ¼Gmi KYÆ `yBwUi Š`NÆÅ 12cm I 16cm |
1 1 K 60º L 90º
K
5
L
3 iÁ¼Gmi cwimxgv KZ? M 120º N 180º
M 3 N 5 K 20 cm L 28 cm 40. mgGKvYx wòfzGRi mƒßGKvY«¼q ciÕ·i@
M 40 cm N 56 cm
1
15. 8x  27x3 = KZ?
3 K mgvb L cƒiK
30. ÷ay cwimxgv Rvbv ^vKGjB ˆKvbwU AuvKv mÁ¿e?
K 115 L 125 M mÁ·ƒiK N wecÉZxc
K AvqZ L eMÆ
M 135 N 158
M mvg¯¦wiK N iÁ¼m

1 L 2 N 3 K 4 M 5 N 6 K 7 L 8 N 9 L 10 M 11 L 12 L 13 M 14 N 15 M 16 L 17 L 18 K 19 N 20 K
Dîi

21 N 22 N 23 L 24 N 25 K 26 L 27 M 28 K 29 M 30 L 31 K 32 M 33 K 34 K 35 N 36 M 37 K 38 L 39 M 40 L
MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkc
œ ò I DËi : eûwbe©vPwb 11

215. wLjMuvI Mft nvB Õ•zj ‰´£ KGjR, XvKv welq ˆKvW : 1 0 9
mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1. ˆKvbwU Agƒj` msLÅv? wbGPi ˆKvbwU mwVK? 28. GKvGbv eGMÆi evüi Š`NÆÅ 10% e†w«¬ ˆcGj ‰i
3 5 K i I ii L i I iii ˆÞòdj kZKiv KZ e†w«¬ cvq?
K 8 L
2 M ii I iii N i, ii I iii K 10% L 11%
M 4 N 25 wbGPi wPò ˆ^GK 18 I 19bs cÉGk²i Dîi `vI: M 20% N 21%
2. ˆKvbwU mnGgŒwjK? A 29. `yBwU msLÅvi AbycvZ 3 : 4 ‰es ZvG`i
M. mv. à
K 5, 10 L 12, 15 z
5 nGj, msLÅv `yBwUi j. mv. à ˆKvbwU?
x
M 6, 21 N 4, 9 K 60 L 20
3. GKvGbv ˆmGUi m`mÅ msLÅv n nGj, Zvi cÉK‡Z B
45
y
M 15 N5
y C D
DcGmGUi msLÅv KZwU nGe? 18. BAC ‰i gvb ˆKvbwU? 30. ‰KwU AvqZGÞGòi Š`NÆÅ I cÉÕ© cÉGZÅKwU w«¼àY
K n2 L 2n K 30 L 45 nGj ‰i ˆÞòdj KZ àY nGe?
M 2n  1 N 2n  1 M 60 N 90 K 2 àY L 3 àY
4. (x) = x2  5x + 6 nGj, x ‰i ˆKvb gvGbi RbÅ 19. z = KZ? M 4 àY N 8 àY
(x) = 0 nGe? K x2 + y2 L y2  x2 31. x + y = 4, x  y = 2 nGj, (x, y) ˆKvbwU?
K 2, 3 L 2, 3 M 2
x + 4y 2
N x2  4y2 K (1, 3) L (3, 1)
M 6, 1 N 6, 1 M (2, 4) N (4, 2)
20. PZzfzÆR AvuKvi RbÅ `yBwU evü I KqwU ˆKvGYi
5.
1
nGj, x + x ‰i gvb ˆKvbwU? 32. 1 + 3 + 5 + ------ avivwUi nZg c` ˆKvbwU?
x= 2+1 cÉGqvRb?
K n L 2n  1
K 1 L2 K 1wU L 2wU M 2n + 1 N 2n  1
M 2 2 N 2 M 3wU N 4wU 33. 5 + a + 45 + ------- àGYvîi avivfzÚ nGj a = KZ?
6. a2  1 + 2b  b2 ‰i Drcv`GK weGkÏwlZ i…c ˆKvbwU? 21. e†Gîi ewntÕ© ˆKvb we±`y ˆ^GK H e†Gî meÆvwaK K 15 L 40
K (a  b  1) (a + b  1) KZwU Õ·kÆK AvuKv hvq? M 45 N 30
L (a  b + 1) (a + b  1) K 1wU L 2wU 34. a  a + a  a + --------- avivwUi
M (a + b + 1) (a  b  1)
M 3wU N 4wU i. mvaviY AbycvZ 1
N (a  b + 1) (a  b  1)
22. e† G îi ˆKvGbv Pvc «¼ viv Drc®² ˆK±`Ê Õ© ˆKvY ii. (2n + 1) cG`i mgwÓ¡ a
7. x  1x = 2 nGj, x3  x13 ‰i gvb ˆKvbwU? e†îÕ© ˆKvGYi  iii. (2n + 1) Zg c` a
K 2 L6 M8 N 14 K mgvb L w«¼àY wbGPi ˆKvbwU mwVK?
M mgvbycvwZK N eÅÕ¦vbycvwZK K i I ii L i I iii
8. 5 ab5 ‰i gvb ˆKvbwU?
23. M ii I iii N i, ii I iii
1 1
K a5b L ab M ab 5 N a5b
P
35. ABC I DEF m`†kGKvYx nGj
9. m + n = 2 nGj, (1)n  (1)m  (1)2 ‰i gvb O
B
i. A = D, B = E, C = F
A AB AC BC
ˆKvbwU? ii. =
DE DF EF
=
K 2 L1 M 1 N 2 Q
iii. ABC  DEF
10. log22 2 ‰i gvb ˆKvbwU? wPGò wbGPi ˆKvbwU mwVK?
K
3
L
2
M2 N 2 i. A, O, B ‰KB mijGiLvq AewÕ©Z K i I ii L i I iii
2 3 ii. AOP = BOP
11. 1234 msLÅvwUGZ cƒYÆK ˆKvbwU? M ii I iii N i, ii I iii
iii. PQ, AB ‰i mgw«¼LíK
K 1 L2 36. ABC ‰ AB = AC = BC = 4 ‰KK ‰es
wbGPi ˆKvbwU mwVK?
M 3 N4 ADBC nGj, ABC ‰i ˆÞòdj ˆKvbwU?
K i L ii
12. 5x  6 + 5 = 2 ‰i mgvavb ˆmU ˆKvbwU? I ii N i, ii I iii
3
M i K
4
eMÆ ‰KK L 3 eMÆ ‰KK
K {6} L {5}
M {3} N {}
wbGPi wPò Abyhvqx 24 I 25 bs cÉGk²i Dîi `vI: M 2 3 eMÆ ‰KK N 4 3 eMÆ ‰KK
D
13. `yBwU KÌwgK msLÅvi eGMÆi A¯¦i 199 nGj, msLÅv 37. ˆKvGbv e†Gîi eÅvmvaÆ 4 ‰KK nGj Dnvi cwiwai
`yBwU ˆKvbwU? C Š`NÆÅ ˆKvbwU?
K 101, 102 L 100, 101 A K 2 ‰KK L 4 ‰KK
M 99, 100 N 98, 99
M 8 ‰KK N 16 ‰KK
14. ‰KwU AcÉK‡Z f™²vsGki je I nGii mgwÓ¡ 11
B 38. ‰KwU NYGKi Š`NÆÅ 5 ˆm.wg. nGj, ‰i
‰es A¯¦idj 3 nGj f™²vskwU ˆKvbwU?
6 7 8 4
BAD = 95 mgMÉZGji ˆÞòdj ˆKvbwU?
K L M N 24. ABC + ADC = KZ wWMÉx? K 10 eMÆ ˆm.wg. L 100 eMÆ ˆm.wg.
5 4 3 7
K 90 L 105
15. ˆKvGbv mgGKvYx wòfzGRi mƒßGKvY«¼Gqi cv^ÆKÅ M 125 eMÆ ˆm.wg. N 150 eMÆ ˆm.wg.
M 180 N 265
6 nGj, Þz`ËZg ˆKvGYi gvb ˆKvbwU? wbGPi ZG^Åi AvGjvGK 39 I 40 bs cÉGk²i Dîi
25. BCD ‰i cwigvY ˆKvbwU?
K 38 L 41 K 85 L 90
`vI:
M 42 N 49 M 95 N 105 Rvbyqvix gvGmi 1g 10 w`Gbi Zvcgvòv
16. mgGKvYx wòfzGRi AwZfzR msj™² ˆKvY `yBwUi 26. sec = x2 + 1 nGj, tan ‰i gvb ˆKvbwU? 14, 14, 13, 12, 10, 9, 9, 8, 7, 13
cÉGZÅGK 2 2 39. cÉ^g 10 w`Gbi Mo Zvcgvòv ˆKvbwU?
K 1x L x 1
K mƒßGKvY L Õ©ƒjGKvY K 14 L 10.9
1
M Nx M 9.1 N 13
M mgGKvY N mijGKvY x
1 1 40. Zvcgvòvi gaÅK ˆKvbwU?
17. ABC-‰ AB = AC nGj 27. tan = nGj, cot ‰i gvb ˆKvbwU? K 9 L 9.5
i. ABC = ACB 3
1 M 10 N 11
ii. ACB = BAC K L 3
iii. A = 180  2B 3
M 1 N0

1 L 2 N 3 N 4 K 5 M 6 L 7 N 8 K 9 L 10 K 11 M 12 N 13 M 14 L 15 M 16 K 17 L 18 L 19 M 20 M
Dîi

21 L 22 L 23 M 24 M 25 K 26 N 27 K 28 N 29 K 30 M 31 L 32 L 33 K 34 N 35 K 36 N 37 M 38 N 39 L 40 N
12 MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkœcò I DËi : eûwbe©vPwb

216. evsjvevRvi miKvwi evwjKv DœP we`Åvjq, XvKv welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
4 16. 1wU ˆKvY I 1wU evüi mvG^ KqwU ˆKvY ˆ`Iqv wbGPi ˆKvbwU mwVK?
1. ( a) ‰i gvb ˆKvbwU? ^vKGj wòfzR AuvKv hvq? K i I ii L ii I iii
1
K 1 L2 M3 N4 M i I iii N i, ii I iii
K ( a) 4
L a4
17. wbGPi Z^ÅàGjv jÞ Ki: 28. ‰KwU evGÝi Š`NÆÅ, cÉÕ© I DœPZv h^vKÌGg 5, 4,
1
4
M ( a) 4 N i. ‰KB ˆKvGYi wòGKvYwgwZK AbycvZmgƒGni I 3 nGj ‰i meGPGq eo `yBwU c†GÓ¤i ˆÞòdGji
a cÉGZÅKwU aË‚eK mgwÓ¡ KZ eMÆ ˆm. wg.?
2. AvGjvi ˆeM cÉwZ ˆmGKG´£ 300000000 wg. nGj ii. cot sec21 =1 K 30 L 40 M 45 N 48
‰i ŠeævwbK i…Gc cÉKvk wbGPi ˆKvbwU? iii. cƒiK ˆKvGYi sinC = ˆKvGYi cotangent 29. L = 41, f1 = 5, f2 = 20, h = 10 nGj cÉPziK KZ?
K 3 108 L 3 107 wbGPi ˆKvbwU mwVK? K 40 L 43 M 44 N 46
M 3 106 N 3 105 K i I ii L i I iii
30. 1wU iÁ¼Gmi KYÆ«¼q 12 Gm. wg. I 13 ˆm. wg.
3. AaÆe†î AGcÞv eo PvcGK Kx eGj? M ii I iii N i, ii I iii
‰es DœPZv 6 ˆm. wg. nGj f„wg KZ ˆm.wg?
K 6 L 12 M 13 N 18
K DcPvc L AwaPvc 18. iÁ¼Gmi cwimxgv ‰i cÉwZwU evüi KZ àY?
31. 1wU wiKkvi mvgGbi PvKvi eÅvmvaÆ 30 Gm. wg.
M AaÆPvc N e†î K 2 L3 M4 N5
19. nGj ‰K AveZÆGb mvBGKj/(wiKkvwU) KZ
4. A
ˆm.wg. `ƒiZ½ AwZKÌg KiGe?
O K 20 L 30 M 40 N 60
A B
D E 32. 1wU mge†îf„wgK wmwj´£vGii eÅvmvaÆ 6 ˆm.wg. I
D
DœPZv 3 ˆm. wg. nGj eKÌc†GÓ¤i ˆÞòdj KZ eMÆ
B C ˆm.wg.?
wPGò BOD =? F
K 112 L 113 M 114 N 115
K 30 L 45 M 60 N 90
wPGò ‰es D, E I F ZvG`i
AB = BC = CA
gaÅwe±`y nGj, Afů¦ i Õ© wòfz R àGjv Kx aiGbi 33. mswÞµ¦ c«¬wZGZ Mo wbYÆGqi ˆÞGò-
5. `yBwU e†Gîi ˆK±`Ê«¼q Õ·kÆGKi wecixZ cvGk¼Æ fixi
nGe? i. mƒòwU nGjv: x = a + h
^vKGj ZvGK Kx eGj? n
K mgw«¼evü L mgevü
K mij mvaviY Õ·kÆK L wZhÆK Õ·kÆK ii. cÉGZÅK ˆkÉwYi gaÅgvb ˆ^GK AvbygvwbK Mo
M welgevü N mgGKvYx
M A¯¦Õ·kÆK N ewn:Õ·kÆK weGqvM KGi ZvGK eÅvwµ¦ «¼viv fvM KGi avc
20. 13+ 23 + 33 + ........... n3 = KZ?
6. wbGPi ˆKvbwU gƒj` msLÅv? n(n +1) n(n+ 1) (2n + 1)
wePzÅwZ wbYÆq Kiv nq
K
2
L
6 iii. wePzÅwZi Mo wbYÆq KGi ‰i mvG^ AvbygvwbK
K 196 L 27 M 13 N 5
n(n+1) 2 Mo ˆhvM KGi Mo cvIqv hvq|
7. evÕ¦e msLÅvGK mvaviYZ KqwUGZ fvM Kiv hvq?
K 2 L3 M4 N5
M
2 { N n3} wbGPi ˆKvbwU mwVK?
21. sin2 cosec ‰i gvb KZ? K i I ii L i I iii
8. y2 = 3y mgxKiGYi mgvavb ˆmU ˆKvbwU?
K cosec L cos M sin2 N sin M ii I iii N i, ii I iii
K {0,3} L {0,2} M {0,1} N {0, 3} 22. i. e†Gîi cÉGZÅK ˆQ`GKi ˆQ`we±`y«¼Gqi A¯¦eZxÆ 34. ‰KwU NbGKi avi 2 Gm. wg. nGj mKj avGii
x3
9. x  3 =
x
mgxKiGYi mgvavb ˆmU wbGPi mKj we±`y e†Gîi Afů¦Gi ^vGK Š`GNÆÅi mgwÓ¡ KZ ˆm. wg.?
ii. cÉGZÅKwU ˆQ`K e†îGK `yBwU we±`yGZ ˆQ` KGi K 12 L 18 M 24 N 30
ˆKvbwU bq? wbGPi wPòvbyhvqx 35 I 36 bÁ¼i cÉGk²i Dîi `vI:
iii. e†Gîi eÅvmvaÆ e†Gîi 1wU Õ·kÆK
K {1,3} L {3,1} A
M {1,3} N {1,3}
wbGPi ˆKvbwU mwVK?
K i I ii L i I iii C
10. wbGPi ˆKvbwU wewœQ®² PjGKi D`vniY:
M ii I iii N i, ii I iii
K cixÞvi bÁ¼i L Zvcgvòv
23. wòfzGRi wZbwU ˆKvGYi AbycvZ 3: 4 : 5 nGj,
M eqm N IRb
Þz`ËZg ˆKvYwUi e†îxqgvb KZ?
11. wbGPi Z^ÅàGjv jÞ Ki: c c c c
60
B
K L M N D
i. AbycvZ ‰KwU f™²vsk 180 8 4 2 8 wg.
ii. 16 : 9 ‰i w«¼fvwRZ AbycvZ 256 : 81 24. ˆmÝUÅv´Ÿ bvgK hG¯¨i mvnvGhÅ Kx cwigvc Kiv nq? wPGò, AC I CD ‰KB wRwbGmi wf®² AeÕ©vb
iii. 5 : 7 ‰i eÅÕ¦ AbycvZ = 7 : 5 K jÁ¼ L DœPZv M `ƒiZ½ N ˆKvY 35. BCD ‰i cwigvY KZ?
wbGPi ˆKvbwU mwVK? 25. C
K 30 L 45
M 60 N 90
K i I ii L i I iii O 36. i. AB ‰i Š`NÆÅ 24 3 wgUvi
A B
M ii I iii N i, ii I iii ii. CD ‰i Š`NÆÅ 16 wgUvi
12. AvqZGjL eÅZxZ MYmsLÅv eüfzR AâGbi RbÅ iii. BC ‰i Š`NÆÅ 8 3 wgUvi
D
wbGPi ˆKvbwU wbYÆq KiGZ nq? wbGPi ˆKvbwU mwVK?
wPGò, AOD ˆK Kx ˆKvY ejv nq?
K ˆkÉwY wbÁ²mxgv L ˆkÉwY DœPmxgv K i I ii L i I iii
K mgGKvY L D®²wZ ˆKvY
M KÌgGhvwRZ MYmsLÅv N ˆkÉwYi gaÅwe±`y M ii I iii N i, ii I iii
M AebwZ ˆKvY N mƒßGKvY
13. KÌgGhvwRZ MYmsLÅv wbGekGbi cÉ^g ˆkÉwY (31-35) 26. mgGiL bq ‰gb KqwU we±`y w`Gq 1wU gvò e†î 37. y3 =1 nGj y2 + y + 1 = KZ?
‰es MYmsLÅv 4 nGj KÌgGhvwRZ MYmsLÅv KZ? K 0 L1 M2 N3
AuvKv hvq? 38. 5570 msLÅvwUi jGMi cƒYÆK wbGPi ˆKvbwU?
K 1 L2 M3 N4 K 1 L2 M3 N4
K 2 L3 M4 N5
14. evGÝi c†Ó¡Zj Kx aiGbi? 27. wbGPi Z^ÅàGjv jÞ Ki: 39. ˆKvbwU gƒj` msLÅv?
K eKÌZj L mgZj
3
M AuvKvevKuv N fvGRi gZ K 5 L 4 M 3 N 5
15. (x, y) = (3, 2) ˆKvb mgxKiGYi ˆRvUwUGK wm«¬ KGi? 6 3

K 3x + 4y = 14, 4x 3y = 2
40. ()
5
= wbGPi ˆKvbwU?
L 3x  2y = 5, 2x + y = 8 i. wPòwUi ˆiLv cÉwZmgZvi gvb-2 63 6 5 63 63
K L M N
5 3 53
M 3x  2y = 2, 5x 3y = 5 ii. ˆiLv cÉwZmgZv I cÉwZdjb cÉwZmgZv ‰KB 53
x y y iii. E eYÆwUi ˆiLv cÉwZmgZv we`Åvgvb
N + = 3, x + = 3
2 3 6

1 L 2 K 3 L 4 N 5 L 6 K 7 K 8 N 9 N 10 K 11 L 12 N 13 N 14 L 15 L 16 K 17 K 18 M 19 L 20 M
Dîi

21 N 22 K 23 M 24 N 25 N 26 M 27 L 28 L 29 L 30 M 31 N 32 L 33 M 34 K 35 K 36 M 37 K 38 L 39 L 40 M
MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkc
œ ò I DËi : eûwbe©vPwb 13

217. mvfvi KÅv´ŸbGg´Ÿ cvewjK Õ•zj I KGjR, XvKv welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1. a, b, c I d àGYvîi avivi KÌwgK c` nGj wbGPi 14. ‰KwU mylg cçf„GRi NƒYÆb ˆKvGYi cwigvY KZ? 26. ‰KwU eGMÆi evüi Š`NÆÅ 3 àY e†w«¬ Kiv nGj,
ˆKvbwU mwVK? K 60 L 62 ˆÞòdj KZ àY nGe?
c+d b+c M 90 N 72 K 4 àY L 8 àY
K b= L a=
2 2 15. ‰KwU mylg AÓ¡fzGRi NƒYÆb cÉwZmgZvi gvòv M 6 àY N 9 àY
M d=
c+a
N c = bd KZ? ‰KwU iÁ¼Gmi evüi Š`NÆÅ 5 ˆm.wg. ‰es ‰KwU KGYÆi
2 K 3 L4 Š`NÆÅ 8 ˆm.wg.
wbGPi avivwUi wfwîGZ 2 I 3 bÁ¼i cÉGk²i Dîi `vI : M 8 N6 DcGii ZG^Åi AvGjvGK (27-28) bs cÉGk²i Dîi
log 3 + log 9 + log 27 + ...................
2. avivwUi mvaviY A¯¦i ˆKvbwU? 16. BsGiwR eYÆ Z ‰i ˆÞGò `vI:
K 3 L2 i. ÷ay ŠiwLK cÉwZmgZv AvGQ 27. iÁ¼mwUi Aci KGYÆi Š`NÆÅ KZ?
M log3 N 2log3 ii. cÉwZmvgÅ ˆiLvi msLÅv 0 K 4 ˆm.wg. L 6 ˆm.wg.
3. avivwUi 5g c` KZ? iii. NƒYÆb cÉwZmgZvi gvòv 2 M 6 2 ˆm.wg. N 4 2 ˆm.wg.
K log81 L log243 DcGii ZG^Åi AvGjvGK wbGPi ˆKvbwU mwVK? 28. iÁ¼mwUi ˆÞòdj KZ eMÆ ˆm.wg.?
M log128 N log254 K i I ii L i I iii K 20 L 48
4. Õ¼vfvweK msLÅvi ˆÞGò M 40 N 24
M ii I iii N i, ii I iii
n(n + 1) D
29. ˆh PjGKi gvb ˆh ˆKvGbv evÕ¦e msLÅv nGZ cvGi
i. 1 + 2 + 3 + 4 + ....... + n =
2 ZvGK Kx eGj?
ii. 1 + 3 + 5 + ..... + (2n  1) = n2 K AwewœQ®² PjK L wewœQ®² PjK
9 †m.wg.
iii. 1 + 2 + 4 + ....... n msLÅK c` chƯ¦ = 2n  1 A B M AwebÅÕ¦ PjK N webÅÕ¦ PjK
DcGii ZG^Åi AvGjvGK wbGPi ˆKvbwU mwVK? 8cm 30. wbGPi ˆKvbwU ˆKw±`Êq cÉeYZvi cwigvc?
K i I ii L i I iii 10 cm
K ˆkÉwYi gaÅwe±`y L GkÉwYeÅvwµ¦
M ii I iii N i, ii I iii C M MYmsLÅv N gaÅK
 
5. 0.5 + 0.6 = KZ? 31. 4x + 2x + 1 ˆ^GK KZ weGqvM KiGj weGqvMdj
  DcGii ZG^Åi AvGjvGK (17-18)bs cÉGk²i Dîi `vI: cƒYÆeMÆ nGe?
K 0.11 L 1.1 17. wPGò Mvp AsGki ˆÞòdj KZ? K 2x L 2x
  K 36m2 L 18m2 M 4x N 4x
M 1.2 N 1.11
M 9m2 N 4.5m2
6. wbGPi ˆKvbwU Agƒj` msLÅv? 32. x = 8 nGj, x = KZ?
2 3

3 18. mgMÉ AsGki ˆÞòdj KZ?

K 5 L K 2 L 2
5 K 137.10m2 L 80.55m2
M 52.27m2 N 38.14m2 M ± 2 N 2
M
27
N 50 19. wbGPi ˆKvb kGZÆ 33.
ax + by + c = 0 I px + qy + r x2  y + 2y  1 ‰i Drcv`GK

weGkÏwlZ i…c
3
= 0 mgxKiY ˆRvUwU mãwZcƒYÆ I ciÕ·i
wbGPi ˆKvbwU?
1 1 K (x + y  1)(x  y + 1)
7. x
x
= 0 nGj, x3 + 2 = KZ?
x wbfÆikxj nGe? L (x + y + 1)(x  y  1)
K 0 L1 a b a b c M (x  y + 1)(x  y + 1)
K  L = =
M 2 N3 p q p q r N (x + y + 1)(x  y + 1)
8. 3 ˆm.wg. eÅvmvaÆ wewkÓ¡ ˆMvjGKi c†GÓ¤i ˆÞòdj a b c
M = 
a b
N = 34. 5.2n  4.2n  2 = KZ?
KZ eMÆ ˆm.wg.? p q r p q
K 1 L 3
K 9 L 36 ‰KwU ˆbŒKv `uvo ˆeGq ˆmÉvGZi AbyK„Gj N´Ÿvq 15 M 2n N 2n + 2
M 12 N 6 wK.wg. hvq ‰es ˆmÉvGZi cÉwZK„Gj hvq N´Ÿvq 5 wK.wg.| 35. log 0.00045 ‰i cƒYÆK KZ?
‰KwU wmwj´£vGii f„wgi ‰es DœPZv 6 DcGii ZG^Åi AvGjvGK (20 I 21) bs cÉGk²i Dîi
eÅvmvaÆ 3 ˆm.wg. K 2 L 3
ˆm.wg.| `vI: M 4 N 4
DcGii ZG^Åi AvGjvGK (9-10) bs cÉGk²i Dîi `vI: 20. ˆbŒKvi ˆeM KZ? 36. 2 x+3
= 64 nGj, x = KZ?
K 3 L 4
9. wmwj´£vGii AvqZb KZ Nb ˆm.wg.? K 10 wK.wg. L 20 wK.wg.
M 5 N 6
K 18 L 36 M 15 wK.wg. N 5 wK.wg. 37. we±`yi gvòv KqwU?
M 54 N 27 21. ˆmÉvGZi ˆeM KZ? K 1 L2
10. wmwj´£vGii eKÌc†GÓ¤i ˆÞòdj KZ eMÆ ˆm.wg.? K 10 wK.wg. L 5 wK.wg. M 3 N0
K 36 L 18 38. 120 ˆKvGYi mÁ·ƒiK ˆKvY KZ wWwMÉ?
M 2 wK.wg. N 7.5 wK.wg.
M 54 N 27 K 30 L 60
11. a  b = 5 ‰es ab = 4 nGj, a + b ‰i gvb 22. K : L = 2 : 3, L : M = 3 : 4 ‰es M : N = 4 : 6 M 120 N 150
KZ? nGj, K : N = KZ? 39. sin = 1 nGj, cos = KZ?
K 1:2 L 2:1
K ±6 L 5 K 1 L0
M 1:3 N 3:1 1 1
M ±3 N 2
ABC ‰i B = 2x, C = x, A = y
‰es A = M
2
N
12. ˆK x  1 «¼viv fvM KiGj fvMGkl
4x2  7x + 4 B + C.
2
wbGPi ˆKvbwU? 40.
DcGii ZG^Åi AvGjvGK (23-24)bs cÉGk²i Dîi `vI: A
K 0 L1
M 2 N4
23. x = KZ?
K 30 L 60
13. O ˆK±`Ê wewkÓ¡ e†Gîi eÅvmvaÆ r ‰es Pvc S M 90 N 45
ˆKG±`Ê  ˆKvY Drc®² KiGj 24. y = KZ?
r
i. S= K 90 L 60 
180 M 45 N 30
ii. e†îwUi ˆÞòdj = 4r 2
a c
B C
 25. = nGj wbGPi ˆKvbwU ‰Kv¯¦iKiY agÆ? sin = x nGj, sin(90  ) = KZ?
iii. e†îvsGki ˆÞòdj =  r2 b d
1 1
360 a d b c K L
K = L =
DcGii ZG^Åi AvGjvGK wbGPi ˆKvbwU mwVK? b c a d x 1  x2
K i I ii L i I iii b d
M =
a b
N = 1  x2
a c c a M 1  x2 N
M ii I iii N i, ii I iii x

1 N 2 M 3 L 4 N 5 M 6 N 7 M 8 L 9 M 10 K 11 M 12 L 13 L 14 N 15 M 16 M 17 N 18 N 19 L 20 K
Dîi

21 L 22 M 23 K 24 K 25 N 26 N 27 L 28 N 29 K 30 N 31 K 32 M 33 K 34 N 35 N 36 K 37 N 38 L 39 L 40 M
14 MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkœcò I DËi : eûwbe©vPwb

218. ivbx wejvmgwY miKvwi evjK DœP we`Åvjq, MvRxcyi welq ˆKvW : 1 0 9
mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1.

0.2 7 ‰i gƒj`xq f™²vsk ˆKvbwU? 13. ˆKw±`Êq cÉeYZvi cwigvc KqwU? 27. mvaviYZ n ‰i wfwî wbGPi ˆKvbwU?
3 3 K1 L2 Ke L 10
K L M3 N4 M1 N0
10 11
5 14. wbGÁ²i ˆKvbwU AwewœQ®² ˆkÉYx eÅvwµ¦? 28. ˆKvb e†Gîi AwaPvGc A¯¦wjÆwLZ ˆKvY 
M
18
N ˆKvbwUB bq
K05 L 04 K mgGKvY L Õ©ƒjGKvY
2. wbGPi ˆKvbwU Agƒj` msLÅv cÉKvk Ki? 5  10 59
M cƒiK ˆKvY N mƒßGKvY
10  15 10  14
K 5.23 L
6
M meKqwU N ˆKvbwUB bq 29. mKj Õ¼vfvweK msLÅvi ˆmU ‰KwU-
3
1 K Amxg ˆmU L mmxg ˆmU
M
27
N
5 15. logx = log y
2
nGj, logx = KZ?
2
M cƒiK ˆmU N msGhvM ˆmU
48 20 Ky L logy 5
3. `yBwU KÌwgK ˆRvo msLÅvi àYdj- 30. sec(90 ) = 3 nGj, sin ‰i gvb KZ?
M log y Nx
i. weGRvo msLÅv 16. `yBwU msLÅvi AbycvZ 2 : 3 I M.mv.à 4 3 5
K L
ii. 4 «¼viv wbtGkGl wefvRÅ 5 4
nGj, e†nîg msLÅvwU KZ? 4 5
iii. 8 «¼viv wbtGkGl wefvRÅ K 12 L 10 M
3
N
3
wbGPi ˆKvbwU mwVK? M6 N4 31. ˆKvGbv DcvGîi cwimi 17 ‰es ˆkÉYx
K i I ii L i I iii 17. e†îÕ© mvgv¯¦wiK ‰KwU wK? eÅeavb 3 nGj ˆkÉYx msLÅv KqwU nGe?
M ii I iii N ˆKvbwUB bq K eMÆGÞò L iÁ¼m K5 L6
4. B = {a, b, c, d, e} ˆmGUi DcGmU msLÅv M AvqZGÞò N UÇvwcwRqvg M7 N8
KZ? 18. e†îÕ© PZzfzÆGRi wecixZ ˆKvY«¼Gqi mgwÓ¡ 32. kx3  6x2  4x + 3 ‰i ‰KwU Drcv`K 2x
K5 L 25 KZ? + 1 nGj, k ‰i gvb KZ?
M 32 N 64 K 360 L 180 K 28 L 14
5. 22c+1 = 128 nGj, c ‰i gvb ˆKvbwU? M 270 N 120 M 14 N ˆKvGbvwUB bq
K5 L4 19. ‰KwU wòfzGRi gaÅgvòq cici mgvb nGj, 33. nGj cos9B = sinB ‰es B < 10 nGj cot5B
M3 N2 wòfzRwU wK? = KZ?
6. 2x  3 + 5 = 2 mgxKiGYi mgvavb ˆmU K mgevü L mgw«¼evü K0 L 0.5
ˆKvbwU? M welgevü N meKqwU M1 N 3
K {0} L {6} 20. e†Gîi ˆKvb wbw`ÆÓ¡ we±`yGZ KqwU Õ·kÆK 34. 0.00054 ‰i mvaviY jGMi cƒYÆK KZ?
M {4} N2 AuvKv hvq? K 4 L 3
7. ˆmU cÉKvGki c«¬wZ KqwU? K1 L2 M 2 N 3
K1 L2 35. e†Gîi e†nîg RÅvGK wK eGj?
M3 N4
M3 N4
21. ‰KwU mgevü wòfzGRi cÉwZwU evüi Š`NÆÅ 8 K eÅvmvaÆ L cwiwa
8. NbeÕ§i gvòv KqwU? M eÅvm N ˆKvGbvwUB bq
wgUvi nGj, ‰i ˆÞòdj KZ eMÆwgUvi?
K1 L2
M3 N4 K8 3 L4 3 36. sin59, sec31 = KZ?
K 1 L0
9. 3x  4y = 0 ‰es 2x  3y = 1 mgxKiY M 16 3 N
3
M 0.5 N1
4
ˆRvGUi mgvavb ˆKvbwU? 37. ÷ay cwimxgv RvbGjB wbGPi ˆKvbwU AuvKv
22. AvqZGjL ‰i DœPZv ˆKvbwU?
K (1, 2) L (2, 1)
K MYmsLÅv L KÌgGhvwRZ mÁ¿e?
M (3, 4) N (4, 3)
MYmsLÅv K AvqZ L eMÆ
10. ABCD ‰KwU eMÆ hvi-
M ˆkÉYx gaÅgvb N ˆkÉYx eÅeavb M mvgv¯¦wiK N iÁ¼m
i. cÉGZÅK ˆKvY mgGKvY 2
ii. 4wU cÉwZmvgÅ ˆiLv AvGQ
23. n weGRvo nGj, gaÅK wbYÆGqi mƒò ˆKvbwU? 38. cosec = nGj  = ‰i gvb KZ?
n1 n+2 3
iii. ˆÞòdj (evü) eMÆ ‰KK K
2
Zg c` L
2
Zg c` K 30 L 45
wbGPi ˆKvbwU mwVK? n M 60 N 90
M Zg c` N ˆKvbwUB bq
K i I ii L i I iii 2 39.  = 0 nGj, sin + cos = KZ?
M ii I iii N i, ii I iii 24. wbGPi ˆKvbwU AwewœQ®² PjK? K
7
L
5
K Zvcgvòv L eqm 5 7
11. 1 + 2 + 3 + ...................... + 10 = ?
7
K 49 L 50 M IRb N meàGjv M1 N
25
M 55 N 110 25. loga 1 = 0 ‰i kZÆ ˆKvbwU? 40. ˆKvb e†Gîi AwaPvGc A¯¦wjÆwLZ ˆKvY-
12. mgGKvYx wòfzGRi mgGKvY msj™² evü«¼q 4 K a0, a>1 L a>0, a=1
K Õ©ƒjGKvY L mgGKvY
ˆm.wg. I 6 ˆm.wg. nGj ˆÞòdj KZ eMÆ M a>0, a1 N a=0, a>1
M mħGKvY N mijGKvY
ˆm.wg.? 26. 0.0245 ‰i ŠeævwbK i…c ˆKvbwU?
K 10 L 12 K 2.45  10 1 L 2.45  102
M 24 N 48 M 2.45103 N 2.4510 4

1 M 2 L 3 M 4 M 5 M 6 K 7 L 8 M 9 N 10 N 11 M 12 L 13 M 14 K 15 L 16 K 17 M 18 L 19 K 20 K
Dîi

21 M 22 K 23 N 24 N 25 M 26 L 27 K 28 N 29 K 30 K 31 L 32 N 33 M 34 K 35 M 36 N 37 L 38 M 39 M 40 M
MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkc
œ ò I DËi : eûwbe©vPwb 15

219. RqG`ecyi miKvwi evwjKv DœP we`Åvjq, MvRxcyi welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1. ˆh ˆKvb mvweÆK ˆmU ( ‰i RbÅ- 14. x + y = 4, xy = 4 ‰es x > 0, y > 0 nGj, y ‰i 29. ˆKvb e†Gîi eÅvm 14 ˆm.wg. e†îwUi A¯¦wjÆwLZ
i. A \ A =  gvb KZ? e†nîg eMÆGÞGòi ˆÞòdj KZ eMÆ ˆm.wg.?
ii. A \ (A \ A) = A K 102 L 100
K 4 L3 M2 N1
iii. A \ (A \ A) =  M 98 N 96
wbGPi ˆKvbwU mwVK? 15. ‰KwU mgw«¼evü mgGKvYx wòfzGRi cwimxgv
wgUvi| ‰i AwZfz G Ri Š`NÆ Å KZ? 30. wbGPi Z^ÅàGjv jÞÅ Ki:
K i I ii L i I iii (6 + 6 2 )
i. 2x  y = 0 I x  2y = 0 mgxKiGYi ciÕ·i
M ii I iii N i, ii I iii K 6 L1 M0 N 5
16. avivwUi mvaviY A¯¦ i- wbfÆikxj
2. ‰KwU msLÅvGK a  10 AvKvGi ˆjLvi RbÅ kZÆ
n log2 + log4 + log8 + ....
ii. x  2y + 3 = 0 mgxKiGYi ˆjLwU (3, 0) we±`yMvgx
K 2 L4
ˆKvbwU?
M log 2 N 2log 2 iii. 3x + 4y = 1 mgxKiGYi ˆjL ‰KwU mijGiLv
K 1  a  10 L 1 < a  10
M 1  a < 10 N 1 < a < 10 17. wZbwU ˆiLvsGki Š`NÆÅ ˆm.wg. ‰ ˆ`Iqv AvGQ| wbGPi ˆKvbwU mwVK?
3. y = 2x dvskGbi- ˆKvb ˆÞGò wòfzR Aâb mÁ¿e bq? K i I ii L i I iii
i. ˆjLwPGòi DcGi ‰KwU we±`y (1, 2) K 1, 2, 3 L 3, 4, 5 M ii I iii N i, ii I iii
M 5, 6, 7 N 7, 8, 9 31. e†îÕ© mvgv¯¦wiK ‰KwU−
ii. ˆjLwPòwU gƒj we±`yMvgx mijGiLv
18. wPGò− K AvqZ L iÁ¼m
iii. GjLwPòwU y-AGÞi mgv¯¦ivj
C D M eMÆ N UÇvwcwRqvg
wbGPi ˆKvbwU mwVK?
K i I ii L i I iii 32.  13 x  y = 0
M ii I iii N i, ii I iii x
x  3y = 0 mgxKiY ˆRvUwU
4. evÕ¦e msLÅvi ˆÞGò-
i. mãwZcƒYÆ
i. weGRvo msLÅvi eMÆ ‰KwU weGRvo msLÅv B A
ii. AwbfÆikxj
ii. `ywU ˆRvo msLÅvi àYdj 4 ‰i àwYZK
iii. ˆKvb mgvavb ˆbB
iii. cƒYÆeMÆ bq ‰gb msLÅvi eMÆgj
ƒ gƒj` msLÅv i. ABD = 90
wbGPi ˆKvbwU mwVK? ii. ABD = 90 − X wbGPi ˆKvbwU mwVK?
K i I ii L i I iii
iii. ABC  ABD = X K i I ii L i I iii
wbGPi ˆKvbwU mwVK? M ii I iii N i, ii I iii
M ii I iii N i, ii I iii
K i I ii L ii I iii 33. ‰KwU `Ë e Å 20% jvGf weKÌq Kiv nj, weKÌqgƒjÅ
3
5. (p + q)2 = 27 ‰es pq = 0 nGj- M i I iii N i, ii I iii ‰es KÌqgƒGjÅi AbycvZ ˆKvbwU?
i. p + q = 3 19. cÉe†«¬ ˆKvY  cwigvY ˆKvbwU? K 5:4 L 6:5
ii. p3q + pq3 = 0 K 0 <  < 90 L 90 <  < 180 M 5:6 N 4:5
iii. p2 + q2 = 3 M 180 <  < 360 N  > 360 34. 225 = 2z nGj, z ‰i gvb KZ ˆm.wg.?
wbGPi ˆKvbwU mwVK? 20. 63, 64, 69, 65, 61 DcvîàGjvi gaÅK KZ? 32
K L 6:4
K i L i I ii K 63 L 63.5 M 65 N 64
3
M i I iii N i, ii I iii M 7.5 N 7.25
21. cos ‰i mGeÆvœP gvb KZ?
6. 10% ÞwZGZ 60 UvKvi ‰KwU LvZvi weKÌq gƒjÅ 35. tan 1  sin2 = KZ?
K 1 L0 M1 N2
KZ UvKv? K sec L cot
22. cçf„GRi kxlÆ we±`yàGjvi ˆKvGYi mgwÓ¡- M sin N tan
K 54 L 50 M 44 N 40
K Pvi mgGKvY L cuvP mgGKvY .
7. a + b =  c nGj, a2 + 2ab + b2 ˆK c ‰i gvaÅGg 36. 0.51 ‰i mvgvbÅ f™²vsk ˆKvbwU?
M Qq mgGKvY N AvU mgGKvY
cÉKvk KiGj wbGPi ˆKvbwU cvIqv hvGe? 23 51
23. 2x  3 + 4 = 3 mgxKiYwUi mgvavb ˆKvbwU? K L
K c2 L c2 M ac N bc 45 100
8. mgGKŒwYK kxlÆ ˆ^GK AwZfzGRi gaÅwe±`yi `ƒiZ½ K  L {3}
M
46
N
23
KZ? M {3} N {6} 90 99
24. i. ˆKvY AâGYi ˆÞGò f„wg > jÁ¼ nGe
30 37. mgGKvYx wòfzGRi ‰KwU ˆKvY 40
nGj Aci
ˆKvY AâGbi ˆÞGò f„wg = jÁ¼ nGe
ii. 45 mƒßGKvGYi gvb KZ wWMÉx?
6cm K 30 L 45
iii. 60 ˆKvY AâGbi ˆÞGò f„wg < jÁ¼ nGe
M 50 N 60
wbGPi ˆKvbwU mwVK? 38. KgcGÞ KqwU evüi Š`NÆÅ Rvbv ^vKGj weGkl
K i I ii L i I iii ˆÞGò PZzfzÆR AuvKv mÁ¿e?
8cm M ii I iii N i, ii I iii K 1 L2
K 5 cm L 6 cm
M 8 cm N 10 cm
25. wbGPi ˆKvb eYÆàGjvi NƒYÆb cÉwZmgZvi gvòv M 3 N4
mgvb? 39. ‰KwU eMÆGÞGòi KGYÆi Š`NÆÅ ‰es cwimxgvi
1 1
9. a + = 5 nGj a  ‰i gvb wbGPi ˆKvbwU?
a a K Z.P.O L A.T.T AbycvZ KZ?
K 51 L 41 M 31 N 21 M Z.H.S N B.H.M K 1:2 2 L 1: 2
10. ˆnbwi weËMm KZ mvGj 10 wfwîK jM AvweÓ•vi 26. mƒGhÆi D®²wZ ˆKvY 60 nGj wgbvGii Qvqvi Š`NÆÅ M 1:2 N 1:4
KGib? 240 wgUvi nq| wgbviwUi DœPZv KZ? 40. wbGPi Z^ÅàGjv jÞÅ Ki:
K 425 wg. L 415.69 wg. n h
K 1614 L 1616 M 1618 N 1624
11. mgGKvYx wòfzGRi mƒßGKvY«¼Gqi cv^ÆKÅ 6 nGj, M 325.10 wg. N 315.69 wg.
i. cÉPziK = L +
2
 Fc 
Jm ( )
J1
Þz`ËZg ˆKvGYi gvb KZ? 27. f„ZjÕ© we±`y ˆ^GK wgbviwUi kxlÆ we±`yi `ƒiZ½ ii. gaÅK = L +
J1 + J2
h
K 38 L 41 M 42 N 49 KZ? k
12. log 4096 = KZ? K 100 L 400
1
iii. MvwYwZK Mo = +  Jixi
2 2 n i= 1
K 4 L6 M7 N8 M 125 N 500
13. x2  (a + b)x + ab = 0 mgxKiGYi mgvavb ˆmU 28. PQRS mvgv¯¦wiGKi P I
Q ‰i wbGPi ˆKvbwU mwVK?
wbGPi ˆKvbwU? mgw«¼L´£K«¼q KZ ˆKvGY ciÕ·iGK ˆQ` KGi? K i L ii
K {a, b} L {a, b} K 60 L 90 M iii N i I ii
M {a, b} N {a, b} M 100 N 120

1 K 2 M 3 K 4 K 5 N 6 K 7 L 8 K 9 N 10 N 11 M 12 N 13 K 14 M 15 K 16 M 17 K 18 L 19 M 20 N
Dîi

21 M 22 M 23 K 24 N 25 L 26 L 27 N 28 L 29 M 30 M 31 K 32 K 33 L 34 M 35 M 36 K 37 M 38 K 39 K 40 N
16 MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkœcò I DËi : eûwbe©vPwb

220. NvUvBj KÅv´ŸbGg´Ÿ cvewjK Õ•zj I KGjR, UvãvBj welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1. ˆKvY cwigvc Kiv nq ˆKvb hG¯¨i mvnvGhÅ? 21. wbGPi Z^Åàwj jÞÅ Ki : iii. wòfzGRi ˆh ˆKvb ‰KwU evüi gaÅwe±`y w`Gq
K IGWvwgUvi L ˆcwiGÕ•vc 1
i. x = 5 + 2 nGj, = 5 + 2 Aci mgv¯¦ivj mijGiLv Z‡Zxq evüi
M ˆmÝGU´Ÿ N ˆUwjGÕ•vc x
ii. x = b  c, y = c  a, z = a  b nGj, x + y + z
gaÅwe±`yMvgx
2. wbGPi ˆKvbwU «¼viv cƒYÆ msLÅvi ˆmU cÉKvk Kiv wbGPi ˆKvbwU mwVK?
=a+b+c
nq? iii. x2 + 8x  20 = (x + 4)2  62 K i, ii I iii L ii I iii
K R LZ MQ NN wbGPi ˆKvbwU mwVK? M i I iii N iii
3. ‰KwU evüi Š`NÆÅ ˆ^GK ˆKvbwU AvuKv hvGe? K i I ii L ii I iii
K mvgv¯¦wiK L wòfzR 30. ˆKvGbv e†Gîi
M i, ii I iii N iii
M eMÆ N iÁ¼m i. ˆKvGbv we±`yGZ `yBwU gvò Õ·kÆK AuvKv hvq
22. i. r eÅvm wewkÓ¡ e†Gîi cwiwa = r
4. ˆKvbwU AwewœQ®² PjK? ii. Õ·kÆK we±`yGZ Õ·kÆGKi Dci AwâZ jÁ¼
ii. a evü wewkÓ¡ eGMÆi KGYÆi Š`NÆÅ = 3a
K ˆkÉwYGZ Qvò msLÅv ˆK±`ÊMvgx
iii. `yBwU eMÆGÞGòi evüi Š`NÆÅ h^vKÌGg a I b
L ˆKvb cwievGii wk÷ msLÅv iii. ˆK±`Ê ˆ^GK ‰i ˆKvGbv Õ·kÆGKi Dci
nGj ZvG`i ˆÞòdGji AbycvZ a2 : b2
M evRvGi PvGji gƒjÅ wbGPi ˆKvbwU mwVK? AwâZ jÁ¼ Õ·kÆwe±`y w`Gq hvq
N ˆKvb cyKzGi gvGQi msLÅv K i L ii I iii wbGPi ˆKvbwU mwVK?
5. ˆKvbwU RÅvwgwZK ˆKvY bq? M i, ii I iii N i I iii K i, ii I iii L ii I iii
K 1 mgGKvY L 0 23. i. a, b, c KÌwgK mgvbycvZx nGj, ZvG`i AbycvZ M i I iii N i I ii
M 360 N 450 a:b=b:c 31. P I Q ˆK±`ÊwewkÓ¡ `yBwU e†Gîi eÅvmvaÆ«¼q
6. ‰KwU eGMÆi Š`NÆÅ 2 ˆm.wg. KGYÆi Š`NÆÅ KZ? ii. a : b = c : d ‰i mij AbycvZ b : a = d : c
K 2 2 ˆm.wg. L 3 2 ˆm.wg.
h^vKÌGg r1 I r2 nGj
iii. 19, 91 mnGgŒwjK
M 2 ˆm.wg. N 2 3 ˆm.wg. i. e†î«¼q ewn:Õ·kÆ KiGj PQ = r1 + r2 nGe
wbGPi ˆKvbwU mwVK?
7. x, y, z evÕ¦e msLÅv ‰es x < y nGj xz > yz nGe K i I ii L ii I iii ii. e†î«¼q A¯¦:Õ·kÆ KiGj PQ = r1 ~ r2
hLb z M i, ii I iii N i I iii iii. e†î«¼q ciÕ·iGK ˆQ` KiGe hw`
K FYvñK L abvñK 24. i. 6 ‰i ˆgŒwjK àYbxqK 2wU = r1 + r2 < PQ < r1  r2 nq|
M 0 N –1 ii. wòGKvYwgwZK AbycvZ 6wU wbGPi ˆKvbwU mwVK?
8. gaÅK wbYÆGq Fc «¼viv wK eySvq? iii. mKj mƒòB AGf` K i, ii I iii L ii I iii
K ˆhvwRZ MYmsLÅv L wbÁ²mxgv wbGPi ˆKvbwU mwVK? M i I iii N i I ii
M ˆkÉwYmxgv N MYmsLÅv K i, ii I iii L ii I iii wbGPi ZG^Åi wfwîGZ (32-34) bs cÉGk²i Dîi `vI :
1
9. a2  3a + 1 = 0 nGj, a2 + a2 ‰i gvb KZ? M i I iii N i I ii 2  4 + 8  16 + ...........
K 6 L7 25. i. f„wg ˆiLv nGœQ f„wgZGj AewÕ©Z ˆh ˆKvb 32. avivwU ˆKvb cÉK‡wZi?
M 9 N 10 mij ˆiLv K mgv¯¦i L àGYvîi
10. a + b = 2, a2 + b2 = 4 nGj, 3ab ‰i gvb KZ? 7
ii. x = nGj,
x+1
=3 M mmxg N Amxg
K 0 L 2 M 1 N 3 3 x1
iii. mij ˆiLvi gvòv `yB
33. avivwUi 6Ó¤ c` KZ?
11. 4 + 7 + 10 + ........ avivwUi KZ Zg c` 301? K 32 L 128
K 98 L 129 wbGPi ˆKvbwU mwVK?
M 64 N 64
M 100 N 101 K i, ii I iii L ii I iii
12. PZzfzÆGRi KqwU Dcvî Rvbv ^vKGj PZzfzÆRwU M i N i I iii
34. avivwUi 1g 7wU cG`i mgwÓ¡ KZ?
AuvKv hvq? K 96 L 96
26. (a + 2)2 = 0 nGj,
M 86 N 86
K wZbwU L PviwU
2
i. a + 2a + 1 = 0
M cuvPwU N QqwU ii. a2 = 4a  4 wbGPi ZG^Åi wfwîGZ (35-37) bs cÉGk²i Dîi `vI :
13. mgGKvYx wòfzGRi cwie†î AsKb KiGj Zvi
iii. a = 2 `yB AâwewkÓ¡ ‰KwU msLÅvi `kK Õ©vbxq Aâ
wbGPi ˆKvbwU mwVK? ‰KK Õ©vbxq AGâi w«¼àY ‰es AsK«¼Gqi àYdj 32.
ˆK±`Ê ˆKv^vq ^vGK? K ii I iii L i, ii I iii
K jGÁ¼i Dci L AwZfzGRi Dci 35. ‰KK Õ©vbxq Aâ x nGj msLÅvwU KZ?
M i I iii N i I ii K 2x L 3x M 12x N 21x
M f„wgi Dci N wòfzGRi Dci
27. i. we±`y e†Gîi ˆKvb eÅvmvaÆ ˆbB| 36. A⫼q Õ©vb wewbgq KiGj msLÅwU KZ nGe?
14. `yBwU evü I ZvG`i ‰KwUi wecixZ ˆKvY ˆ`Iqv
ii. `yBwU e†î ciÕ·iGK `yBfvGe Õ·kÆ KiGZ K 3x L 4x M 12x N 21x
^vKGj, KqwU wòfzR Aâb Kiv mÁ¿e? cvGi|
K 1 L3 37. x = 1 nGj, gƒj msLÅvi mvG^ Õ©vb wewbgqK‡Z
iii. wZbwU we±`y «¼viv MwVZ wòfzGRi ˆÞòdj
M 4 N2 msLÅvi cv^ÆKÅ KZ?
15. ‰KwU AvqZGÞGòi `yBwU mw®²wnZ evüi Š`NÆÅ 3 kƒbÅ nGe hw` we±`y wZbwU ‰KB ˆiLvq K 9 L 9 M 19 N 19
I 4 nGj ‰i KGYÆi Š`NÆÅ nGe AeÕ©vb KGi| 38. Z^ÅvbymvGi cÉvµ¦ msLÅv ˆKvbwU?
K 5 L3 M3 2 N 5
wbGPi ˆKvbwU mwVK? K 84 L 48 M 42 N 24
K i, ii I iii L ii I iii
16. ˆZvgvi MwYZ eBGqi gvòv KZ? wbGPi ZG^Åi wfwîGZ (39-40) bs cÉGk²i Dîi `vI :
K 0 L2 M1 N3 M i I iii N i I ii
Iwb cÉwZ WRb Kjv 60 UvKv `Gi 8 WRb ‰es 50
17. (x  2)3 = 0 mgxKiGY x ‰i NvZ KZ? 28. i. mgevü mgGKvYx wòfzGRi mgGKvY evG`
Aci `yBwU ˆKvY ciÕ·i mgvb UvKv `Gi 12 WRb Kjv KÌq KGi|
K 1 L2 M3 N4
18. (1, 5) we±`ywU ˆKvb PZzfÆvGM cGoGQ? ii. mƒßGKvYx wòfzGRi `yBwU ˆKvY mƒßGKvY 39. ˆgvU 20 WRb Kjvi `vg KZ?
K cÉ^g L w«¼Zxq iii. ‰K PjKwewkÓ¡ mgxKiY, x2 + x  2 = 0 K 1200 UvKv L 1160 UvKv
M Z‡Zxq N PZz^Æ wbGPi ˆKvbwU mwVK? M 1100 UvKv N 1080 UvKv
19. (2x  6) + 5 = 5 ‰i mgvavb wbGPi ˆKvbwU? K i, ii I iii L iii 40. WRb cÉwZ 6 UvKv jvf nGj, ˆgvU KZ UvKv jvf
K 3 L5 M 6 N 1 M i I ii N i I iii nq?
20. mx2 + nx + a = 0 ‰KwU w«¼NvZ mgxKiY nGe bv, 29. i. MwYZ eB `yB gvwòK K 54 UvKv L 62 UvKv
hLb m ‰i gvb ii. wòfzGRi ˆh ˆKvb `yB evüi gaÅwe±`yi M 88 UvKv N 96 UvKv
K 0 L 1 M 2 N 3 msGhvMKvix ˆiLv ‰i Z‡Zxq evüi mgv¯¦ivj
1 M 2 M 3 M 4 M 5 N 6 M 7 K 8 K 9 L 10 K 11 M 12 M 13 L 14 K 15 K 16 N 17 M 18 L 19 K 20 K
Dîi

21 N 22 N 23 N 24 K 25 M 26 K 27 K 28 N 29 L 30 L 31 N 32 L 33 M 34 N 35 N 36 M 37 K 38 K 39 N 40 L
MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkc
œ ò I DËi : eûwbe©vPwb 17

221. KÅv´ŸbGg´Ÿ cvewjK Õ•zj I KGjR, ˆgvGgbkvnx welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1. ‰KwU eGMÆi evüi Š`NÆÅ w«¼àY nGj, Zvi ˆÞòdj 15. ˆkÉwY eÅeavb 10 nGj, ˆkÉwY msLÅv KZ? wbGPi ˆKvbwU mwVK?
KZàY e†w«¬ cvGe? K 9 L8 K i I ii L ii I iii
M 7 N6
K 2 àY L 4 àY M i I iii N i, ii I iii
16. sec230  cosec290 ‰i gvb KZ? 29. a + b = 3 ‰es ab = 1 nGj, a3 + b3 + (a  b)2
M 8 àY N 6 àY
4 2
2. A = {1, 2, 3, 4, 5} nGj, P(A) ‰i Dcv`vb msLÅv K
3
L ‰i gvb KZ?
3
KZ? K 23 L 31
1 1
K 16 L 20 M N M 41 N 49
2 3
M 32 N 25 17. 2x + y = 5
30. (x1 + y1)−1 ‰i gvb KZ?
3. hw` x, y, z KÌwgK mgvbycvwZK nq ZGe, ˆKvbwU x+y 1
3x  2y = 11 mgxKiY ˆRvGU (x, y) ‰i gvb ˆKvbwU? K
y
L
x+y
mwVK? K (3, 1) L (3, 1) xy
K y2 = zx L x2 = yz M (2, 1) N (5, 2) M x+y N
x+y
M z2 = xy N x2 = y2z 18. S = {(3, 1), (3, 2), (4, 3), (5, 4)} A®¼Gqi ˆWvGgb 31. wZbwU evüi Š`NÆÅ ˆm.wg. ‰ ˆ`Iqv nj| ˆKvb
4. ABC I DEF m`†k ‰es Abyi…c evü«¼Gqi KZ? ˆÞGò wòfzR AsKb Kiv mÁ¿e?
AbycvZ 3:2| ZvG`i ˆÞòdj«¼Gqi AbycvZ KZ? K {3, 3, 4, 5} L {1, 2, 3, 4} K 3, 5, 6 L 4, 5, 9
K 4:5 L 7:3 M {2, 3, 4, 5} N {3, 4, 5} M 5, 6, 12 N 6, 7, 16
M 8:3 N 9:4 19. (3, 1) ‰es (3, 1)
we±`y `yBwUi AeÕ©vb QK 32. 5
nGj, sec  tan = KZ?
tan + sec =
5. iÁ¼Gmi NƒYÆb ˆKvY KZ wWMÉx? KvMGRi ˆKvb PZzfÆvGM? 2
K 60 L 90 2 3
K 2q I 3q L 2q I 4^Æ K L
M 180 N 120 5 5
M 1g I 4^Æ N 1g I 3q 6 4
wbGPi ZG^Åi wfwîGZ 6−7 bs cÉGk²i Dîi `vI: 20. gƒj` msLÅv ˆKvbwU? M
5
N
5
ABC ‰KwU mgGKvYx wòfzR| B = 90 ‰es AC ‰i K 13 L 14 33. m8 + m4  2 ‰i Drcv`K ˆKvbwU?
gaÅwe±`y D M 15 N 16 K m4  2 L m3  1
6. ˆKvb mÁ·KÆwU mwVK? 21. (x) = x2 + 3x + 2 nGj (1) ‰i gvb KZ? M m2 + 2 N m+1
K AB2 + BC2 = AC2 L BC2 = AB2  AC2 K 2 L0 34. mvwKe I ZvwgGgi ivGbi AbycvZ 9 : 7| ZvG`i
M AC2 + BC2 = AB2 N AB2 + AC2 = BC2 M 1 N6 ˆgvU ivb 2000 nGj mvwKGei ivb KZ?
7. BD = KZ? 22. mKj gƒj` I Agƒj` msLÅvGK wK eGj? K 250 L 875
1 1 M 1125 N 1600
K AB L AC K Õ¼vfvweK L ˆgŒwjK msLÅv
2 2
M cƒYÆ msLÅv N evÕ¦e msLÅv
35. jvf ÞwZi ˆÞGò-
1
M 2AC N BC i. KÌqgƒjÅ-weKÌqgƒjÅ = ÞwZ
2 23. hw` A = {a, b, c, d} nq ZGe A ‰i cÉK‡Z
ii. weKÌqgƒjÅ - KÌqgƒjÅ = jvf
8. mgGKvYx wòfzGRi mƒßGKvY«¼q mgvb nGj, DcGmU KqwU?
iii. jvf ev ÞwZ KÌqgƒGjÅi Dci wbfÆi KGi
ˆKvYàGjvi AbycvZ wbGPi ˆKvbwU? K 16 L 15
K 2:2:3 L 1:1:2 M 14 N 8 wbGPi ˆKvbwU mwVK?
M 2:3:4 N 1:1:4 24. 1  1 + 2  ...... avivwUi AÓ¡g c` KZ? K i I ii L i I iii
. 2 M ii I iii N i, ii I iii
9. 0.13 = KZ? K 16 L 8
1 3
36. log9 + log27 + log81 + ... avivi mvaviY A¯¦i
K L M 8 N 32
6 15 KZ?
2 5
25. K log6 L log3
M N C
M 2 log3 N log3
15 12
10. kZKiv evwlÆK 4.50 UvKv nvi gybvdvq 450 UvKvi 37. GKvbwU AmsævwqZ?
4 eQGii gybvdv KZ UvKv? 1 K sec90 L cos90
M cosec90 N tan0
K 72 UvKv L 79 UvKv
M 81 UvKv N 85 UvKv
38. 0.4305 ‰i jGMi cƒYÆK KZ?
K 2 L4
11. log 216 ‰i gvb KZ? A
1 B
M 1 N3
K 2 L3 39. ‰KwU mgw«¼evü wòfzGRi f„wg 60cm ‰es mgvb
M 4 N8 cosB = KZ?
1 mgvb evüi Š`NÆÅ 50 ˆm.wg. nGj, wòfzRwUi
12. mgevü wòfzGRi evüi Š`NÆÅ 8 ˆm.wg. nGj ‰i K 1 L
2 ˆÞòdj KZ?
DœPZv KZ? K 500cm2 L 820cm2
M 2 N2
K 2 3 L4 3 M 936cm2 N 1200cm2
26. loga200 = 2 nGj, a ‰i gvb KZ?
M 16 3 N 32 3 40.
K 10 2 L5 2
13. ‰KwU NbGKi cÉwZwU avi 3 ˆm.wg. nGj NbGKi i. 1 + 2 + 3 + ..... + n =
n(n + 1)
M 5 3 N 10 5 2
KGYÆi Š`NÆÅ KZ ˆm.wg.?
27. DcvGîi msLÅv n weGRvo nGj gaÅK nGe- ii. 12 + 22 + 32 + ...... + n2 =
n(n + 1)(2n  1)
K 2 3 L3 2 6
n n1
M 3 3 N9 K
2
Zg c` L
2
Zg c` iii. 1gc` a ‰es mvaviY A¯¦i d nGj mgv¯¦i
10 Rb kÉwgGKi gãjevGii cvwikÉwgK wbÁ²i…c: n+1 n+2 avivi n Zg c` = a + (n  1)d.
150, 130, 145, 140, 190, 180, 165, 175, 200
M
2
Zg c` N
2
Zg c`
wbGPi ˆKvbwU mwVK?
DcGii ZG^Åi AvGjvGK 14-15 bs cÉGk²i Dîi `vI: 28. AbycvZ I mgvbycvGZ agÆ AbymvGi
K i I ii L ii I iii
14. DcvîàGjvi cwimi KZ? i. a t b = b t a
M i I iii N i, ii I iii
K 69 L 70 ii. a t b = c t d nGj ad = bc
M 71 N 75 iii. a t b = 5 t 3 nGj a t 5 = b t 3
1 L 2 M 3 K 4 N 5 M 6 K 7 L 8 L 9 M 10 M 11 N 12 L 13 M 14 M 15 L 16 N 17 K 18 N 19 L 20 N
Dîi

21 L 22 N 23 L 24 L 25 L 26 K 27 M 28 M 29 K 30 N 31 K 32 K 33 N 34 M 35 N 36 L 37 K 38 M 39 N 40 M
18 MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkœcò I DËi : eûwbe©vPwb

222. KÅv´ŸbGg´Ÿ ˆevWÆ DœP we`Åvjq, ˆgvGgbkvnx welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
a b 17. ‰KwU PvKvi eÅvm 4.5 wgUvi| PvKvwU 360 wgUvi DcGii wPGò O we±`yGZ P we±`yi D®²wZ ˆKvY
1. a+b+c=0 nGj ( ) ‰i gvb KZ?
+
c c c^ AwZKÌg KiGZ KZ evi NyiGe? ˆKvbwU?
K 1 L0 K cÉvq 25 evi L cÉvq 18 evi K BOQ L AOP
M 1 N2 M cÉvq 19 evi N cÉvq 20 evi M POB N POQ
2. (x) = x  6x + 5 eüc`xi ‰KwU Drcv`K nGjv 18. mgGKvYx wòfzGRi mƒßGKvY«¼Gqi ‰KwU  nGj wbGPi Z^Å ˆ^GK 29 I 30 bs cÉGk²i Dîi `vI:
2

K x3 L x2 ABC ‰ ABC = 90, ACB = , AB = 3 ‰es BC

M x1 N x+1
AciwU KZ?
K 90 +  L 90   = 4 ˆm. wg.
3. 10% ÞwZGZ 120 UvKvi ‰KwU wRwbGmi weKÌq M   90 N 180   29. tanA ‰i gvb KZ?
gƒjÅ KZ? 19. x log2 = log4 nGj x ‰i gvb KZ? 3 4 4 3
K 12 L 98 K L M N
K 1 L2 5 5 3 4
M 108 N 112
..
M 3 N4 30. sec2  tan2 = KZ?
4. 1. 3 3 ‰i mvgvbÅ f™²vsk wbGPi ˆKvbwU? wbGPi Z^Å ˆ^GK 20 I 21 bs cÉGk²i Dîi `vI: 5 4
4 43 K L
D 4 5
K L
3 330 3
133 132 M N1
4
M N
990 990 31. e†îÕ© mvg¯¦wiK ‰KwU
5. wbGPi MvwYwZK evKÅàGjv jÞÅ Ki: A C K AvqZGÞò L eMÆGÞò
i. 4x = (x + 1)2  (x  1)2
2
M iÁ¼m N UÇvwcwRqvg
x+1 2 32.
ii. x =
2( ) ( )

x1
2
B
A
x+y 2 2
iii. xy =
2( ) ( )+
xy
2 20. BCD + BAD = KZ?
K 90 L 120
D E
wbGPi ˆKvbwU mwVK? M 180 N 360
K i I ii L i I iii C
21. A = 83 nGj C = KZ? B
M ii I iii N i, ii I iii
K 90 L 97 D I E h^vKÌGg AB I AC evüi gaÅwe±`y nGj 
6. 7 ‰i 7 wfwîK log KZ? M 180 N 360 i. DE || BC
K
3
L
2 22. ABC ‰ PQ || BC nGj wbGPi ˆKvbwU mwVK? 1
2 3 K AP: PB = AQ : QC ii. DE = BC
2
M 1 N3 L AB : PQ = AC : PQ 1
1 1 M AB : AC = PQ : BC iii. AD = AB
7. x:y=3:7 nGj, x2 : y2 = KZ? N PQ : BC = BP : BQ
2
K 49:9 L 9 : 49
wbGPi ˆKvbwU mwVK?
M 10:21
23. wbGPi evKÅàGjv jÞ Ki
N 21:10 K i I ii L ii I iii
i. ‰KwU ˆiLv `yBwU mijGiLvGK ˆQ` KiGj M i I iii N i, ii I iii
wbGPi Z^Å ˆ^GK 8 I 9 bs cÉGk²i Dîi `vI:
A = {a, b, c}, B = {b, c, d} nGj,
AvUwU ˆKvY Drc®² nq 33. ˆKw±`Êq cÉeYZvi cwigvc 
8. n(A  B) = KZ? ii. ‰K mijGKvY = 180 K 1wU L 2wU
K {a, b, c, d} L2 iii. ˆiLvi Š`NÆÅ, cÉÕ© I DœPZv AvGQ M 3wU N 4wU
M 4 N {b, c} wbGPi ˆKvbwU mwVK? 34. ‰KwU MvGQi Š`NÆÅ I Qvqvi Š`GNÆÅi AbycvZ
9. A  B ‰i Dcv`vb msLÅv KZ? K i I ii L i I iii 3 : 3 nGj D®²wZ ˆKvY KZ wWMÉx?
K 9 L 16 M ii I iii N i, ii I iii K 30 L 45 M 60 N 90
M 27 N 30 24. ‰KwU UÇvwcwRqvGgi DœPZv 8 ˆm.wg. ‰es 35. ‰KwU NbGKi KYÆ 6 3 wgUvi nGj, ‰i avi KZ
10. ‰KwU eGMÆi KGYÆi Š`NÆÅ I cwimxgvi AbycvZ mgv¯¦ivj evü«¼Gqi Š`NÆÅ h^vKÌGg 9 ˆm.wg. ‰es wgUvi?
ˆKvbwU nGe? 7 ˆm.wg. nGj ‰i ˆÞòdj KZ eMÆ ˆm.wg.? K 4 L5
K 1:2 2 L 1: 2 K 24 L 64 M 6 N7
M 1:2 N 1:4 M 96 N 504 36. ‰KwU mgGKvYx wòfzGRi AwZfzR 3 wgUvi ‰i
m
11. m<n nGj xxn = KZ? wbGPi wPGò ABC mgevü wòfzR AD  BC ‰es AB = f„wg msj™² ˆKvY 30 nGj jGÁ¼i Š`NÆÅ KZ wgUvi?
mn 2 ZG^Åi wfwîGZ 25 ‰es 26 bs cÉGk²i Dîi `vI: 1 3
K x L xnm K L
1 1 A 2
3
M mn N nm
x x 1
M N1
12. 3 ˆm.wg. eÅvmvaÆwewkÓ¡ e†Gîi ˆK±`Ê ˆ^GK 2 2
ˆm.wg. `ƒGii RÅv ‰i Š`NÆÅ KZ? 37. 4x+1 = 2x2 nGj
K 5 L2 5 i. x =  4
ii. x + 4 = 0
M 3 5 N5 5 B C iii. x =  3
D
13. x : y = 4 : 5 ‰es y : z = 4 : 5 nGj wbGPi ˆKvbwU mwVK?
i. x, y, z KÌwgK mgvbycvwZ
25. BD = KZ?
K 1 L 2 K i I ii L ii I iii
ii. x : z = 16:25
iii. x = z M 2 N4 M i I iii N i, ii I iii
wbGPi ˆKvbwU mwVK? 26. wòfzRwUi DœPZv KZ? 38. DcvGî eÅen†Z msLÅvmgƒn nGjv
K i I ii L i I iii 4 K PjK
K ‰KK L 3 ‰KK
M ii I iii N i, ii I iii 3 L MYmsLÅv
2 M KÌgGhvwRZ MYmsLÅv
14.
1
,  1, 2 avivwUi ˆKvb c` 8 2? M ‰KK N2 3 ‰KK
2 3 N a˂eK
K 8Zg c` L 9Zg c` 27. cÉGZÅK e†Gîi cwiwa eÅvGmi KZàY? 39. sin + cos = 1 nGj  ‰i gvb KZ?
M 6 Zg c` N 5Zg c` K

L
 K 30 L 45 M 60 N 90
15. 32x = 81 nGj x ‰i gvb KZ? 2 3 40. i. e†Gîi ˆKvb we±`yGZ ‰KvwaK Õ·kÆ AvKv hvq
M  N 2 ii. e†Gîi ˆK±`Ê cÉGZÅK eÅvGmi gaÅwe±`y
3 1
K 3 L2 M
2
N
3 28. iii. ˆKvb Pvc AaÆe†î nGj Zvi wWMÉx cwigvY 180
Q P
16. ‰KwU wòfzGRi wZbwU ˆKvGYi AbycvZ 3 : 4 : 5 wbGPi ˆKvbwU mwVK?
nGj ˆKvY wZbwUi cwigvc K i I ii L ii I iii
B O A
K 90, 30, 60 L 45, 60, 75 M i I iii N i, ii I iii
M 80, 10, 90 N 50, 60, 70
1 K 2 M 3 M 4 N 5 K 6 M 7 K 8 M 9 K 10 K 11 K 12 L 13 K 14 L 15 L 16 L 17 K 18 L 19 L 20 M
Dîi

21 L 22 K 23 K 24 L 25 K 26 L 27 M 28 L 29 M 30 N 31 K 32 N 33 M 34 M 35 M 36 L 37 K 38 L 39 N 40 L
MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkc
œ ò I DËi : eûwbe©vPwb 19

223. cywjk jvBb DœP we`Åvjq, gqgbwmsn welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1. mKj gƒj` I Agƒj` msLÅv wbGq evÕ¦e msLÅvi 14. ACB ‰i gvb KZ? 27. eMÆGÞò `yBwUi cwimxgvi mgwÓ¡ KZ?
ˆmU wbGPi ˆKvbwU «¼viv MwVZ? K 55 L 60 K 4(M + N) L 4M+N
K I LX M 65 N 75 M 16(M+N) N 4M  4N
M R NA 28. KvMGRi cƒeÆ gƒjÅ t eZÆgvb gƒjÅ = 2 t 3| cƒGeÆi
15. ACB + BAC = KZ?
2.  (cvB) Kx RvZxq msLÅv? K 105 L 115 Zzjbvq kZKiv KZ e†w«¬ ˆcGqGQ?
K K‡wòg L Õ¼vfvweK M 125 N 135 K 30% L 25%
M gƒj` N Agƒj` 16. mgGKvYx mgw«¼evü wòfzGRi mgvb mgvb ˆKvY«¼Gqi M 75% N 50%
3. A = (3, 6, 9) nGj n(A) ‰i gvb KZ? cÉGZÅKwUi gvb KZ? 29. x + y = 6 ‰es x  y = 2 nGj mgxKiGYi mgvavb
K 2 L3 M4 N8
K 60 L 30
KZ?
4. A = {1, 2, 3, 4, 5, 6} nGj K (4, 2) L (4, 2)
M 90 N 45
i. 8  A M (3, 3) N (4, 2)
17. ÷ay cwimxgv ˆ`Iqv ^vKGj ˆKvb PZzfzÆRwU AuvKv
ii. 4  A 30. 4x + 3y = 7 ‰es 8x  6y = 2 mgxKiY ˆRvU
iii. O  A hvq?
i. mãwZcƒYÆ
wbGPi ˆKvbwU mwVK? K iÁ¼m L eMÆGÞò
ii. AmsLÅ mgvavb AvGQ
K i I ii L i I iii M AvqZGÞò N mvgv¯¦wiK
iii. mgvavb AbbÅ
M ii I iii N i, ii I iii 18. e†Gîi eÅvm r nGj, ZLb e†Gîi cwiwa ˆKvbwU? wbGPi ˆKvbwU mwVK?
5. x2  2 6  5 = 0 nGj, x  1x ‰i gvb KZ? K r L 2r
K i I ii L i I iii
M 3r N 4r
K 2 2 L2 3 M ii I iii N i, ii I iii
19.
M 2 N3 2 31. (2, 6) we±`yGZ fzR I ˆKvwU KZ?
i. e†Gîi eÅvm e†îGK ˆQ` KGi bv
ewYÆZ ZG^Åi AvGjvGK 6 I 7 bs cÉGk²i Dîi `vI : K fzR 2 ˆKvwU 4 L fzR 2 ˆKvwU 6
ii. e†Gîi Š`NÆÅGKB cwiwa ejv nq
1
x− =5 nGj M fzR 3 ˆKvwU 6 N fzR 2 ˆKvwU 2
x iii. mgvb mgvb e†Gîi eÅvmvaÆ mgvb 32. 4  11  18 avivwUi mvaviY A¯¦i KZ?
1
6. x2 + ‰i gvb wbGPi ˆKvbwU? wbGPi ˆKvbwU mwVK? K +8 L 8
x2
K 23 L 24 M 27 N 29 K i I ii L i I iii M +7 N 7
12 M ii I iii N i, ii I iii 33. 1  1 + 2 avivwUi mvaviY AbycvZ KZ?
7. (x + ) ‰i gvb wbGPi ˆKvbwU?
x 2 2
K 23 L 24 20. ABC ‰i sin = nGj tan ‰i gvb wbGPi 1
5 K L 1
M 27 N 29
ˆKvbwU? 2
1
8. 2
x  4x + 1 = 0 nGj x  ‰i gvb
x
KZ? M 2 N 2
K 5 L3
K 3 3 L2 3 M 1 N2 34. mvgv¯¦wiGKi `yBwU mw®²wnZ ˆKvGYi mgwÓ¡ KZ
M 4 3 N5 3 21. A = 30 nGj cot2A  1 = KZ? wWMÉx?
9. (21 + 51)1 ‰i gvb KZ? K 1 L2 K 90 L 180
K 0 L1 M 3 N4 M 270 N 360

M
10
N
7 22. sin30  cos60 ‰i gvb KZ? 35. ABC ‰ AC2 = AB2 + BC2 nGj ˆKvbwU
mZÅ?
7 10 mgGKvY
K A = 1 L B = 1 mgGKvY
1
10. wbGPi Z^Å jÞÅ Ki- K 1 L
2 M C = 1 mgGKvY N A+B = 1 mgGKvY
1
i. a2 =
a2 M
1
N
1 36. ‰KwU mgw«¼evü wòfzGRi f„wg M ‰es mgvb
3 4
ii. 20 = 1 mgvb evüi Š`NÆÅ N nGj, ˆÞòdj KZ nGe?
ap 23. 48 wgUvi jÁ¼v ‰KwU LuywU ˆfGã wewœQ®² bv nGq M
iii. n = anp K 4N2  M2 L 4N2  M2
a f„wgi mvG^ 30 ˆKvY Drc®² KGi| MvQwU KZ 4
wbGPi ˆKvbwU mwVK? DœPZvq ˆfGãwQj? M
M
4M2  N2 N
3M
4N2  M2
4 4
K i I ii Li I iii K 48 wgUvi L 24 wgUvi
37. 400 ˆmw´ŸwgUvi eMÆvKvi ‰KwU cv^Gii ˆÞòdj
M ii I iii N i, ii I iii M 16 wgUvi N 32 wgUvi
KZ eMÆ wgUvi?
24. ‰KwU euvGki ˆMvov nBGZ 60 wgUvi `ƒGi f„wgi
11. log5
3
5( )( 5 ‰i gvb wbGPi ˆKvbwU?
) mvG^ ‰KwU we±`yGZ euvkwUi D®²wZ ˆKvY 45 |
K 0.16 eMÆ wgUvi L 16 eMÆ wgUvi
6 5 7 2 M 160 eMÆ wgUvi N 1600 eMÆ wgUvi
K L M N euvkwUi DœPZv KZ?
5 6 5 5 38. ‰KwU e†Gîi cwiwa 220 wgUvi| e†îwUi eÅvmvaÆ
12. x+2
= 2 mgxKiGYi exR KZ? K 60 wgUvi L 60 3 wgUvi KZ?
2x + 1
4 3 M 30 3 wgUvi N 20 3 wgUvi K 21.028 wgUvi L 35.014 wgUvi
K L 25. a t b = c t d nGj, wbGPi ˆKvbwU ‰Kv¯¦KiGYi
3 4 M 45.036 wgUvi N 70.028 wgUvi
M 
4
N0 D`vniY? 39. ‰KwU mge†îf„wgK ˆKvYGKi DœPZv 16 ˆm.wg.
3
13. (x  1) (x  2) = 0 nGj x ‰i gvb KZ? K =
c a b d
L = ‰es eÅvmvaÆ 12 ˆm. wg. nGj, ˆnjvb D®²wZ KZ?
d b c a
K (1, 2) L (1, 2) K 17 ˆm.wg. L 18 ˆm.wg.
a b a c
M (1, 2) N (0, 2) M = N = M 19 ˆm.wg. N 20 ˆm.wg.
c d b d
wbGPi wPòwUi wfwîGZ 14 I 15 bs cÉGk²i Dîi `vI: wbGPi DóxcGKi AvGjvGK 26 I 27 bs cÉGk²i Dîi 40. ˆKvb cwimsLÅvb 360 ‰i Ask wnGmGe DcwÕ©Z
A `vI: nGj, Zv ˆKvb aiGbi wPò?
`yB eGMÆi evüi Š`NÆÅ h^vKÌGg M I N K AvqZGjL L ˆiLwPò

26. `yBwU eMÆGÞGòi ˆÞòdGji mgwÓ¡ KZ? M cvBwPò N AwRf ˆiLv

115
K M2  N2 L M2 + N2
B C D
M M2  N N M2 + N

1 M 2 N 3 L 4 K 5 K 6 M 7 N 8 L 9 M 10 K 11 L 12 N 13 K 14 M 15 L 16 N 17 L 18 K 19 M 20 N
Dîi

21 L 22 N 23 M 24 K 25 M 26 L 27 K 28 N 29 K 30 L 31 L 32 N 33 N 34 L 35 L 36 K 37 L 38 L 39 N 40 M
20 MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkœcò I DËi : eûwbe©vPwb

224. K‡wl wek¼we`Åvjq nvB Õ•zj, gqgbwmsn welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1. kƒbÅ ‰KwU Õ¼vfvweK msLÅv
i. 17. wbGPi Z^ÅàGjv jÞÅ Ki: 27. ‰KwU eMÆGÞò I ‰i KGYÆi Dci AwâZ
mKj Õ¼vfvweK msLÅv cƒYÆmsLÅv
ii. i. tan90 ‰i gvb 0 eMÆGÞGòi ˆÞòdGji AbycvZ KZ?
`yBwU Õ¼vfvweK msLÅvi ˆhvMdj Õ¼vfvweK msLÅv
iii. cosA K 1:5 L 1:4
ii. cotA = M 1:3 N 1:2
wbGPi ˆKvbwU mwVK? sinA
K i I ii L i I iii iii.  1 cos  1 28. gaÅK KÌgvbymvGi mvRvGbv DcvîGK KZfvGM
M ii I iii N i, ii I iii wbGPi ˆKvbwU mwVK? wefÚ KGi?
K i I ii L i I iii K 5 L4
2. mKj gƒj` I Agƒj` msLÅv wbGq Kx MwVZ nq? M 3 N2
K Õ¼vfvweK msLÅv L ˆgŒwjK msLÅv M ii I iii N i, ii I iii
wbGPi wPGòi AvGjvGK 29-31 bs cÉGk²i Dîi `vI:
M gƒj` msLÅv N evÕ¦e msLÅv wbGPi wPò Abyhvqx 18 I 19 bs cÉGk²i Dîi `vI:
A
3. A = {x : x2  2x = 0} nGj ˆKvbwU mZÅ? A
K {0} L {0, 1}
M {0, 2} N {}
4. n Dcv`vb wewkÓ¡ ‰KwU ˆmGUi Dcv`vb msLÅv KZ? C
K 2n L n (n  1)
M n2 N 2n1 C
B 5 †m.wg.
1 1
5. x + = 3 nGj x  = KZ?
x x 29. ABC ‰i ˆÞòdj KZ?
K 2 5 L 5
60
K 2 21 eMÆ ˆm.wg. L 21 eMÆ ˆm.wg.
M 3 5 N5 5 D B M 2 12 eMÆ ˆm.wg. N 12 eMÆ ˆm.wg.
wbGPi ZG^Åi wfwîGZ (6-8) bs cÉGk²i Dîi `vI: wgUvi 8 30. ABC ‰i DœPZv KZ ˆm.wg.?
(x) = x3 + 6x2y + 11xy2 + 6y3 ‰es 18. BCD ‰i cwigvY KZ? K 7 L 13
4 2
g(x) = 2x  3x  3x  2 K 30 L 45 M 17 N 21
6. (x) = x3 + 6x2y + 11xy2 + 6y3 ‰ (y) ‰i gvb M 60 N 90 31. A ‰i gvb KZ?
KZ? 19. i. AB ‰i Š`NÆÅ 24 3 wgUvi K cÉvq 30.51 L cÉvq 21.51
3 3
K 14y L 20y CD ‰i Š`NÆÅ 16 wgUvi
M 18y3 N 24y3
ii. M cÉvq 20.51 N cÉvq 47.15
7. x = KZ emvGj (x) = 0 nGe? iii. BC ‰i Š`NÆÅ 8 3 wgUvi wbGPi ZG^Åi wfwîGZ (32-34) bs cÉGk²i Dîi `vI:
K x L y wbGPi ˆKvbwU mwVK? ˆKvGbv ‰K kxGZi ˆgŒmyGg wmGjGUi Rvbyqvix gvGm 13
M xy N xy K i I ii L i I iii w`Gbi Zvcgvòv wbGP ˆ`Iqv nGjv:
8. wbGPi ˆKvbwU (x) ‰i Drcv`K? M ii I iii N i, ii I iii 14, 14, 14, 13, 10, 10, 11, 12, 11, 14, 9, 8, 6
K x+y L x  2y 20. `yBwU msLÅvi AbycvZ 3 : 5 ‰es j.mv.à 60 | 32. Mo Zvcgvòv KZ?
M x  3y N xy K 9.53 L 10.53
‰G`i M.mv.à KZ? M 11.23 N 12.53
9. a > 0, b > 0, x  0 kGZÆ ax = bx nGj ˆKvbwU K 1 L2
mwVK? M 3 N4
33. gaÅK Zvcgvòv KZ?
K a=b L a>b>0 K 8 L 9
x y 4=0
M ax = bx
10. Õ¼vfvweK msLÅv N ‰i
N ab = x
ŠeævwbK i…c a  10n nGj
21. 3x  3y  10 = 0 } mgxKiY ˆRvUwUi ŠewkÓ¡Å M 10
34. cÉPziK Zvcgvòv KZ?
N 11

logN ‰i gvb KZ?

ˆKvbwU? K 14 L 12
K a + loga L a + logn K AmvgémÅ L mvgémÅ M 11 N 10
M n + loga N n + logn M wbfÆikxj N mgvavb AbÅbÅ 35. ‰KwU mgGKvYx wòfzGRi evüàGjvi AbycvZ 3 : 4 : 5
11. `yBwU ivwk a I b ‰i àYdj ab = 0 nGj 22. wcZvi eZÆgvb eqm cyGòi eZÆgvb eqGmi 5 àY| ‰i cwimxgv 48 wgUvi nGj AwZfzR KZ?
i. a = 0 nGj b = 0 nGe 15 eQi ci wcZvi eqm cyGòi eqGmi w«¼àY K 20 wgUvi L 22wgUvi
ii. a = 1 nGj b = 0 nGe nGe| wcZvi eZÆgvb eqm KZ? M 26wgUvi N 85wgUvi
iii. b = 1 nGj a = 0 nGe K 40 eQi L 35 eQi 36. wPGò
wbGPi ˆKvbwU mwVK? M 30 eQi N 25 eQi D
A
K i I ii L i I iii wbGPi ZG^Åi AvGjvGK (23-24)bs cÉGk²i Dîi `vI:
M ii I iii N i, ii I iii ‰KwU mgv¯¦i avivi `kg c` 37 I ‰Kwesk c` 81
12. `yBwU Õ¼vfvweK msLÅvi cv^ÆKÅ 2 ‰es ZvG`i 23. avivwUi cÉ^g c` KZ?
àYdj 15 nGj e†nîg msLÅvwU KZ? K 1 L2 B C
K 5 L3 M 3 N4
M 2 N1 i. DAB AebwZ ˆKvY
24. i. avivwUi cÉ^g 8wU cG`i mgwÓ¡ 120 ii. ABC D®²wZ ˆKvY
13. F
ii. avivwUi mvaviY A¯¦i 4 iii. DAB = ABC
P 30 iii. avivwUi cç`k c` 61 wbGPi ˆKvbwU mwVK?
A B wbGPi ˆKvbwU mwVK? K i I ii L i I iii
K i I ii L i I iii M ii I iii N i, ii I iii
C 37. a, b, c evÕ¦e msLÅv ‰es a < b I c > 0 nGj
Q D M ii I iii N i, ii I iii
E wbGPi wPGòi AvGjvGK (25-26) bs cÉGk²i Dîi `vI: wbGPi ˆKvbwU mwVK?
D C K ac > bc L ac < bc
wPGò CQE = KZ? M ab > ac N ac  bc
K 60 L 30 wbGPi AbyGœQ` nGZ 38 I 39 bs cÉGk²i Dîi `vI:
M 45 N 35
5% nvi gybvdvq 1000 UvKv 2 eQGii RbÅ wewbGqvM
14. ABC mgGKvYx wòfzGR D, AwZfzR AC ‰i
Kiv nGjv|
gaÅwe±`y, AD = 3 ˆm.wg. ‰es AB = 2 ˆm.wg. 38. mij gybvdv KZ?
nGj BC ‰i Š`NÆÅ KZ? K 50 UvKv L 100 UvKv
A B
K 3 2 ˆm.wg. L 4 2 ˆm.wg.
M 120 UvKv N 150 UvKv
M 5 2 ˆm.wg. N 6 2 ˆm.wg. 25. wPòwUi cÉwZmvgÅ ˆiLv KqwU? 39. mij gybvdv I PKÌe†w«¬ gybvdvi cv^ÆKÅ KZ?
15. wbw`ÆÓ¡ AvqZ AuvKGZ KgcGÞ KZwU Dcvî cÉGqvRb? K 4wU L 3wU K 2.5 UvKv L 5 UvKv
K 5wU L 4wU M 2wU N 1wU M 10 UvKv N 20 UvKv
M 3wU N 2wU 26. A ˆKw±`ÊK PvcwU gyGQ w`Gj KqwU cÉwZmvgÅ ˆiLv 40. ‰KwU mylg AÓ¡fzGRi kxlÆGKvGYi cwigvY KZ?
16. wòfzGRi evüàGjvi jÁ¼w«¼L´£K AsKb KGi wbGPi AewkÓ¡ ^vKGe? K 135 L 120
ˆKvbwU AuvKv hvq? K 1wU L 2wU M 90 N 60
K iÁ¼m L AvqZGÞò M 3wU N 4wU
M A¯¦e†î N cwie†î
1 M 2 N 3 M 4 K 5 L 6 N 7 L 8 K 9 K 10 M 11 M 12 K 13 L 14 L 15 N 16 N 17 M 18 K 19 M 20 N
Dîi

21 K 22 N 23 K 24 K 25 K 26 K 27 N 28 N 29 K 30 N 31 N 32 M 33 N 34 K 35 K 36 N 37 L 38 L 39 K 40 K
MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkc
œ ò I DËi : eûwbe©vPwb 21

225. Rvgvjcyi wRjv Õ•zj, Rvgvjcyi welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1. wbGPi ˆKvbwU gƒj` msLÅv? wbGPi ˆKvbwU mwVK? sin = KZ?
K 2 L 13 K i I ii L i I iii 4 12
K L
M 156 N 256 M ii I iii N i, ii I iii 13 13
2. 3 I 5 ‰i gvGS cƒYÆ msLÅv KZwU? 20. GKvb kGZÆ a0 = 1? M
13
N
13
K a=0 L a0 12 5
K 1 L2
M 3 N0 M a>0 N a1 31. 30 bs ‰i wPò ˆ^GK Dîi `vI:
22 21. N = 10N nGj, logN ‰i cƒYÆK KZ? cot2 = KZ?
3. 7
, 9 , 25 BZÅvw` ˆKvb aiGYi msLÅv?
K 10 Ln 144 25
K L
K cƒYÆ L Õ¼vfvweK M 1 N 100 169 169
M gƒj` N ˆhŒwMK 22. ‰KwU msLÅvGK a  10 AvKvGi ˆjLvi
n
RbÅ kZÆ M
25
N
144
4. B =  nGj P(B) wbGPi ˆKvbwU? ˆKvbwU? 144 25
K {} L0 K 1 < a < 10 L 1  a  10 wPGòi AvGjvGK (32-34) bs cÉGk²i Dîi `vI:
M 1  a < 10 N 1 < a  10 A
M  N {1}
5. Y = {x  N : x < 1} ˆmUwUi Dcv`vb msLÅv 23. C
KZ? 4
K 0 L1
A B 
M 2 N AmsLÅ O C B
Error!
6. A  B = A ‰es Bc = A nGj wbGPi ˆKvbwU wPò «¼viv wbG`ÆwkZ AOC I BOC ciÕ·i
mwVK? wK ˆKvY wbG`Æk KGi? 32. wPGò AB = 4, BC = 3 . AB,  ‰i gvb KZ
K B= L A= K mgGKvY L mijGKvY wWwMÉ?
M A B N AB=A M mÁ·ƒiK ˆKvY N cƒiK ˆKvY K 30 L 45
7. (x + y, 0) = (1, x  y) nGj, (x, y) = KZ? M 60 N 90
K (0, 0) L (1, 1) C
33. AC-‰i Š`NÆÅ KZ ‰KK?
1 1
M (2, 2) N ( )
2 2
A
O
90
B
K 5
M 8
L6
N 10
8. a + b = a  b nGj, b ‰i gvb KZ? 34.  = 60 nGj, BC ‰i Š`NÆÅ KZ?
K 1 L0 4
1 D K 3AB L
M
2
N2 24. BOD ‰i gvb KZ wWwMÉ? 3
1 K 30 L 45 M 8AB N 9AB
9. a2 + 1 = 2a nGj, a + = KZ?
a M 60 N 90 35. ‰KwU eMÆGÞGò ˆÞòdj 169 eMÆ ˆm.wg. nGj
K 1 L0 25. ABC ‰ AB = AC ‰es B = 25 nGj A cwimxgv KZ ˆm.wg.?
M 2 N4 ‰i gvb KZ? K 13 L 26
p + q = 5 , p  q = 3 nGj, K 30 L 60 M 52 N 65
10. pq ‰i gvb KZ? M 65 N 130 36. ‰KwU iÁ¼Gmi KYÆ«¼Gqi AGaÆK h^vKÌGg 5 ˆm.wg.
K
1
L1
26. wPGò ABC-‰ C = 2A nGj A ‰i gvb ‰es 7 ˆm.wg.| iÁ¼mwUi ˆÞòdj KZ eMÆ
2 KZ? ˆm.wg.?
M 2 N8 K 10 L 30 K 12 L 24
11. 10pq(p2 + q2) ‰i gvb KZ? M 45 N 60 M 35 N 70
K 15 L 5 15 27. D 37. P + 25 + Q + 625 + ........... ‰KwU àGYvîi aviv
M 20 N 25
1 1
A
Y nGj
12. a + = 2 nGj, a  = KZ?
a i. mvaviY A¯¦i = 5
a
K 0 L1 40 ii. p = 5
30
M 2 N4 B C iii. mvaviYc` = 5n
x y
+ = 2 nGj,
wPGò y ‰i gvb KZ wWwMÉ? wbGPi ˆKvbwU mwVK?
y x K 70 L 60 K i I ii L i I iii
13. (x  y)2 = KZ? M 80 N 90 A M ii I iii N i, ii I iii
K 0 L2 28. wPGò O ˆK±`Ê wewkÓ¡ 3
M 3 N4 38. ‰KwU àGYvîi avivi cÉ^g c` ‰es mvaviY
e†Gî ABC mgevü|
14. (x  y) + 3xy(x  y) ‰i gvb KZ?
3
AOB ‰i gvb KZ O
2
K 0 L3 AbycvZ
2
nGj, avivwUi Z‡Zxq c` wbGPi
M 4 N9 wWwMÉ? 3
B C
15. (x + y) ‰i gvb wbGPi ˆKvbwU?
3
ˆKvbwU?
K 2x3 L 8x3
K 300 L 240 K 3 L 2
M 0 N 4x
M 180 N 120 1 1
16. (x) = x2  5x + 6 ‰es (x) = 0 nGj, x = KZ? M N
1 2 3
K 2, 3 L 5, 1 29. sin = nGj, tan = KZ?
M 2, 3 N 1,  5
2 wbGÁ²i Z^Å ˆ^GK 39 I 40 bs cÉGk²i Dîi `vI:
1 1
3 K L ‰KwU we`ÅvjGqi bZzb feb wbgÆvGYi KvGR wbGqvwRZ
17. 512 + 23 ‰i mijdj ˆKvbwU? 2 3
10 Rb kÉwgGKi Š`wbK gRyix (UvKvq) ˆ`Iqv nGjv:
K 0 L 16 3 150, 130, 145, 170, 200, 180, 190, 140, 165, 175
M 3 N
M 64 N 75.88 2
39. DcvGîi cwimi KZ?
18. log10 x =  2, x = KZ? 30. A K 50 L 71
K 0.02 L 0.1 
M 85 N 90
M 0.001 N 0.01 13
40. 10 ˆkÉwY eÅeavb wbGq ˆkÉwY msLÅv KZ?
19. 10 wfwîK log ‰i ˆÞGò K 8 L6
i. log101 = 0 M 4 N2
ii. log 0 = AmsævwqZ B C
iii. log10(100) = 2 12

1 N 2 K 3 M 4 K 5 K 6 K 7 N 8 L 9 M 10 K 11 M 12 K 13 K 14 K 15 L 16 K 17 L 18 N 19 N 20 L
Dîi

21 L 22 M 23 M 24 N 25 N 26 L 27 K 28 N 29 L 30 L 31 L 32 K 33 M 34 L 35 M 36 N 37 M 38 N 39 L 40 K
22 MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkœcò I DËi : eûwbe©vPwb

226. ‰m.wf. miKvwi evwjKv DœP we`Åvjq, wKGkviMé welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1. mKj gƒj` ‰es Agƒj` msLÅvGK wK msLÅv ejv 15. cosec (90  ) = 2 nGj cos = KZ? 28. `yBwU KÌwgK abvñK msLÅvi eGMÆi A¯¦i 9|
nq? 3 1 1 msLÅv `yBwU KZ?
K 2 L M N K 6, 7 L 4, 5 M 3, 4 N 5, 6
K Õ¼vfvweK L cƒYÆ 2 2 2
16. 29. ‰KwU AvqZvKvi NGii Š`NÆÅ cÉÕ© AGcÞv 2
M evÕ¦e N ˆgŒwjK A
   wgUvi ˆewk ‰es ‰KwU KYÆ 34 wgUvi| NiwUi
2. 2.4 × 0. 81 ‰i gvb KZ? wPGò, cÉÕ© KZ wgUvi?
 
K 2 L 0. 2 M 0.12 N 1. 2 i.   K 5 L8 M3 N 10
B C
3. A = {a, b, c} ‰es B = {b, c} nGj P(A\B) ‰i AC 30. mgv¯¦i avivi 1g c` a, mvaviY A¯¦i d ‰es
ii. sec (90  ) =
Dcv`vb msLÅv KqwU?
BC c`msLÅv n nGj
AC
K 8 L4 M2 N1 iii. cosec (90  ) = i. avivwUi n Zg c` = a + nd  d
AB
4. A = {1, 2, 3} nGj, A × A wbYÆq KiGj KqwU ii. avivwUi 12Zg c` = a + 12d
wbGPi ˆKvbwU mwVK? n
KÌgGRvo cvIqv hvGe? K i I ii L i I iii iii. avivwUi mgwÓ¡ = {2a + nd  d}
2
K 5 L7 M8 N9 M ii I iii N i, ii I iii wbGPi ˆKvbwU mwVK?
3 17. ‰KwU gB f„wgi mvG^ 60 ˆKvY Drc®² KGi 30 K i I ii L i I iii
5. (x + y)2 = 27 ‰es xy = 0 wgUvi DuPz ˆ`qvj Õ·kÆ KGi| gBGqi Š`NÆÅ KZ? M ii I iii N i, ii I iii
i. x + y = 3 K 20 3 wg. L 60 3 wg. 31. wòfzGRi wZbwU evüi Š`NÆÅ ˆ`qv AvGQ; wbGPi
ii. x3y + xy3 = 3 30
ˆKvb ˆÞGò mgGKvYx wòfzR AsKb mÁ¿e bq?
M wg. N 60 wg.
iii. x3 + y3 = 3 3 3 K 3cm, 4cm, 5cm L 6cm, 8cm, 10cm
wbGPi ˆKvbwU mwVK? 18. P M 5cm, 7cm, 9cm N 5cm, 12cm, 13cm
K i I ii L i I iii 32. i. cÉGZÅK mxgve«¬
mgZj ˆÞGòi wbw`ÆÓ¡
M ii I iii N i, ii I iii A B ˆÞòdj iGqGQ|
O ii. `yBwU wòfzR ˆÞGòi ˆÞòdj mgvb nGjB
x= 3 + 2 nGj 6 I 7bs cÉGk²i Dîi `vI:
wPòvbymvGi, Q wòfzR meÆmg|
1
6. x+
x
‰i gvb KZ? i. P we±`yi D®²wZ ˆKvY POB iii. `yBwU wòfzR meÆmg nGj ZvG`i ˆÞòdj
x= ii. O we±`yGZ Q we±`yi AebwZ ˆKvY QOA mgvb|
K 2 3 L3 2 M2 2 N4 6 iii. POQ nGjv f„-ˆiLv wbGPi ˆKvbwU mwVK?
1 wbGPi ˆKvbwU mwVK? K i I ii L i I iii
7. x3  3 ‰i gvb KZ?
x K i I ii L i I iii M ii I iii N i, ii I iii
K 2 2 L 18 3 M 22 2 N 30 3 M ii I iii N i, ii I iii 33. ‰KwU wòfzGRi `yBwU evüi Š`NÆÅ h^vKÌGg 9cm I
8. 7% nvi mij gybvdvq 650 UvKvi KZ eQGii 19. e†Gîi eÅvmvaÆ 12 àY KiGj ˆÞòdj KZ àY nGe? 8cm ‰es A¯¦fƒÆÚ ˆKvY 30 nGj ˆÞòdj KZ?
gybvdv 273 UvKv? K 9 eMÆ ˆm.wg. L 18 eMÆ ˆm.wg.
1
K 2 L4 M6 N8 K
4
L4 M 16 N 32 M 36 eMÆ ˆm.wg. N 72 eMÆ ˆm.wg.
9. 22x6 = 32x6 nGj, x ‰i gvb KZ? 20. avGb Pvj I ZzGli AbycvZ 7 t 3 nGj ‰GZ 34. ‰KwU mgw«¼evü wòfzGRi cwimxgv 16 ˆm.wg. ‰es
K 0 L1 M2 N3 kZKiv wK cwigvY Pvj AvGQ? f„wg 6 ˆm.wg. nGj
10. ŠeævwbK msLÅvi Av`kÆi…c a × 10n wbGPi K 50 L 60 M 70 N 80 i. mgvb mgvb evüi Š`NÆÅ 5 ˆm.wg.
ii. ˆÞòdj 12 eMÆ ˆm.wg.
ˆKvbwU a ‰i gvGbi mwVK mxgv? 21. PviRbGK 2500 UvKv 12 t 14 t 16 t 18 AbycvGZ fvM
iii. DœPZv 8 ˆm.wg.
K 10 < a < 10 L 1  a  10 KGi ˆ`Iqv nGjv
M 1  a < 10 N 10  a  10
wbGPi ˆKvbwU mwVK?
1g I 2q Rb ‰i UvKvi AbycvZ wbGPi ˆKvbwU? K i I ii L i I iii
11. mgxKiGYi
x2  x  12 = 0 K 2t1 L 1t2 M ii I iii N i, ii I iii
i. ‰KwU PjK x M 3t4 N 4t3
35. ‰KwU eMÆGÞGòi cwimxgv 20 nGj ‰i ˆÞòdj
ii. NvZ 2 22. a1x + b1y = c1 ‰es a2x + b2y = c2 mgxKiY
KZ?
iii. x ‰i exR (4, 3) «¼Gqi mgvavb ˆbB KviY K 4 L5 M 16 N 25
wbGPi ˆKvbwU mwVK? K
a1 b1 c1
= =
a1 b1 c1
L =  36. e†îKjvi ˆÞòdj KZ eMÆ ‰KK?
b2 b2 c2 a2 b2 c2 q  2pr q  r2
K i I ii L i I iii a1 b1 c1 a1 b1 c1 K L
M   N   360 360
M ii I iii N i, ii I iii a2 b2 c2 a2 b2 c2
360 360 2
12. AB mijGiLv ‰KwU ˆmU nGj ‰i Dcv`vb wbGPi 23. 3x  4y = 10 ‰es 6x  8y = 18 ‰i mgvavb M
2pr
. N
q
.r 

ˆKvbwU? mÁ·GKÆ wK ejv hvq? 37. n weGRvo nGj, gaÅK wbYÆGqi mƒò ˆKvbwU?
K ˆbB L AbbÅ M AmsLÅ N `yBwU n1 n+1
K we±`y L ˆiLvsk M Zj N RMZ K Zg c` L Zg c`
24. 6x  y = 1 nGj x ‰i gvb y ‰i gvaÅGg cÉKvk 2 2
13. wPGò, KiGj  n n
A
C K
y+1
L 6y + 1
M Zg c`
2
N ( ) Zg c`
2
+1

P Q 6 wbGPi ZG^Åi AvGjvGK 38-40 bÁ¼i cÉGk²i Dîi `vI :

M
y1
N 6y  1
kxZKvGj evsjvG`Gki ˆKvGbv ‰KwU AçGji 10 w`Gbi
D
R S
6 Zvcgvòv nj : 14, 14, 14, 13, 12, 13, 10, 10,
x y
25. ‰es x3 + y2 = 1 nGj x I y ‰i mÁ·KÆ 11, 12|
+ =1
2 3
B ˆKvbwU? 38. DcGii Dcvîàwji cÉPziK KZ?
K 11 L 12 M 13 N 14
K x<y L x>y
wbGPi ˆKvb ˆRvov ˆQ`GKi ‰KB cvGki A¯¦Õ© ˆKvY? M xy N x=y 39. DcvGîi Mo Zvcgvòv KZ?
K CDR, CDS L QCD, RDS 26. wbGPi ˆKvb we±`ywU y AÞGiLvi Dci AewÕ©Z? K 12.1 L 12.2 M 12.3 N 12.4
M DCQ, CDS N SDC, PCD K (0, 4) L (4, 0) 40. DcvGîi gaÅg KZ?
M (8, 0) N (4, 4) K 11.5 L 12 M 12.5 N 13
14. e†Gîi wfZGi AewÕ©Z ˆKvGbv we±`y ˆ^GK e†Gî
KqwU Õ·kÆK AuvKv hvGe? 27. (3, 0) I (5, 0) we±`y«¼Gqi `ƒiZ½ KZ ‰KK?
K 2 L4 M8 N 16
K 0 L1 M2 N3

1 M 2 K 3 M 4 N 5 L 6 K 7 M 8 M 9 N 10 M 11 K 12 K 13 M 14 K 15 M 16 N 17 K 18 K 19 K 20 M
Dîi

21 K 22 L 23 K 24 K 25 N 26 K 27 M 28 L 29 M 30 L 31 M 32 L 33 L 34 K 35 N 36 L 37 L 38 N 39 M 40 M
MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkc
œ ò I DËi : eûwbe©vPwb 23

227. ‰.wf.ˆR.‰g. miKvwi evwjKv DœP we`Åvjq, gy´ÖxMé welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1. a, b, c evÕ¦e msLÅv ‰es a> b I c < 0 nGj 18. x  x  12 = 0 mgxKiGYi
2
wbGPi ˆKvbwU mwVK?
wbGPi ˆKvbwU mwVK? i. ‰KwU PjK x ii. NvZ 2 K i I ii L ii I iii
K ac > bc L ac < bc iii. x ‰i exR (4, 3) M i I iii N i, ii I iii
M ab > bc N ac  bc wbGPi ˆKvbwU mwVK? 30. tan = 8 nGj sec = KZ?
2. wbGPi ˆKvbwUi gvb ˆgŒwjK msLÅv? K i I ii L i I iii K 8 L 5 M 3 N3
20 16 81 27 M ii I iii N i, ii I iii 3
K L M N 31. sinA =
4
nGj tanA ‰i gvb KZ?
4 3 5
2 19. ABC = ACB nGj AB : AC = ?
3. wbGPi evKÅàGjv jÞ Ki: K 2:1 L 1:2 M 1:1 N 2:3 K
3
L
7
M
3
N
3
3 7 7
i. 0 ‰KwU Õ¼vfvweK msLÅv 20. 7
A A
ii. mKj Õ¼vfvweK msLÅv cƒYÆ msLÅv 32. A = 60 nGj cot ‰i gvb KZ?
2
iii. 7 ‰KwU gƒj` msLÅv 3 2 1
wbGPi ˆKvbwU mwVK? K 3 L M N
2 3 3
K i L ii 33. sinA = cosAnGj A ‰i gvb KZ?
M i I ii N ii I iii 60
K 30 L 45 M 60 N 90
4. N  Z  Q  R nGj mvweÆK ˆmU wbGPi ˆKvbwU? B C
34.
K N LZ MQ NR
wPGò BAC = 2ACB nGj, ACB = KZ? A
5. (x) = 3x  9 ‰ x ‰i ˆKvb gvGbi RbÅ (x) = 6 K 30 L 40 M 60 N 80 2
nGe? 21. wòfzGRi wZbwU ˆKvGYi AbycvZ 2 : 3 : 4 nGj
K 9 L5 M1 N0 
e†nîg ˆKvYwU KZ? B C
6. S = {(x, y) : x  A, y  A ‰es y = x2} ˆhLvGb,
K 20 L 30 M 40 N 80 3
A = {3, 2, 1, 0, 1, 2} ˆmUwUi ˆWvGgb KZ? 22. ‰KwU mgGKvYx wòfzGRi mƒßGKvY«¼q ‰KwU wPGò cosec = KZ?
K {1, 0, 1} L {0, 1}
M {0, 1, 2} N {0, 1, 4}
AciwUi w«¼àY nGj, Þz`Ë mƒßGKvY KZ? K
3
L
2
M
1
N2
K 30 L 45 M 60 N 90 2 3 2
7. a + b = 8, ab = 15 nGj (a  b) ‰i gvb KZ?
K 2 L2 M 4 N 4
wbGPi wPGòi AvGjvGK 23 I 24bs cÉGk²i Dîi `vI: 35. ‰KwU MvGQi Qvqvi Š`NÆÅ 10 3 ‰es kxlÆwe±`y
1 x Qvqvi ˆkl cÉvG¯¦i mvG^ 30 D®²wZ ˆKvY ŠZwi
8. x + = 5 nGj, 2 = KZ? A
x x  3x + 1 KGi| MvQwUi DœPZv KZ?
1 1 K 10 wgUvi L 20 wgUvi
K L M1 N 1
2 2 M 25 wgUvi N 30 wgUvi
9. 1
p + 2 ‰i mvG^ KZ ˆhvM KiGj ˆhvMdj
2
30° 36. ˆKvb mgxKiY ˆRvGUi wbfÆikxjZvi kZÆ ˆKvbwU?
p a1 b1 a1 c1
O B K  L 
cƒYÆeMÆ msLÅv nGe? C a2 b2 a2 c2
K 0 L1 M2 N4 a1 b1 c1 a1 b1 c1
M = = N  
10. (x) ˆK (2x  b) «¼viv fvM KiGj fvMGkl nGe- 23. AOB ‰i cƒiK ˆKvGYi cwigvY wbGPi ˆKvbwU? a2 b2 c2 a2 b2 c2
K (b) L (b) K 30 L 60 M 150 N 180 37. 2, 4, 6 ..... AbyKÌgwUi mvaviY c` ˆKvbwU?
b b 24. wPGò cÉ`wkÆZ cÉe†«¬ ˆKvY ‰es AOB ‰i K n
1
Ln M 2n N 3n
M  ()
2
N ( )

2 mÁ·ƒiK ˆKvGYi mgwÓ¡ KZ?
38.
2
11. ‰KwU Kjg 11 UvKvq weKÌq KiGj 10% jvf K 480 L 330 M 210 N 180
E
nq| KjgwUi KÌqgƒjÅ KZ? 25. ‰KwU AvqZGÞGòi `yBwU mw®²wnZ evüi Š`NÆÅ 2
K 9 L 10 M 11 N 12 I 1 nGj ‰i KGYÆi Š`NÆÅ KZ?
12. a2  3a + 1 = 0 nGj K 5 L3 M3 2 N 5
A D
1
i. a + = 3 26. ˆKvb PZzfzÆGRi KYÆ«¼q mgvb I ciÕ·iGK
a mgGKvGY mgw«¼Lw´£Z KGi|
1
ii. a2 + 2 = 1 K UÇvwcwRqvg L iÁ¼m
a B C
1 M eMÆGÞò N mvgv¯¦wiK
iii. a3 + 3 = 0
a 27. ˆK±`ÊMvgx e†Gîi RÅv ‰i Š`NÆÅ 6 ˆm.wg. nGj wPGò, ABCD ‰KwU eMÆ, hvi cÉGZÅK evüi Š`NÆÅ
wbGPi ˆKvbwU mwVK? e†Gîi eÅvm KZ ˆm.wg.? 22 wgUvi ‰es AEB ˆÞòwU ‰KwU AaÆe†î|
K i I ii L i I iii K 12 L6 M3 N2 mÁ·ƒYÆ ˆÞòwUi ˆÞòdj KZ?
M ii I iii N i, ii I iii 28. C K 380.13 L 484.38
M 664.31 N 674.07
13. 51x = 625 nGj x ‰i gvb KZ? 39. mylg lofzGRi cÉwZwU ˆKvGYi gvb KZ?
K 4 L 3 M3 N4 K 60 L 90
A B M 108 N 120
4
14. x2 = 81 nGj x ‰i gvb KZ? O 40. MYmsLÅv wbGekb mviwY nGZ-
K 9 L 4 M  3 N 2 n h

15.
1
log2 ‰i gvb KZ?
i. gaÅK = L + 2 c
f
f1
( ) 
fm
8
ii. cÉPziK = L + h
K 3 L 2 M
1
N3 O ˆK±`ÊwewkÓ¡ e†Gî AB eÅvm ‰es BAC = 35 f1 + f2
n+1
2 nGj ACB ‰i gvb KZ? iii. MvwYwZK Mo =
16. 0.000000772 ‰i ŠeævwbK i…c wbGPi ˆKvbwU? K 55 L 45 M 35 N 90
2
K 7.72107 L 7.72106 wbGPi ˆKvbwU mwVK?
29. ˆKvGbv wòfzGR- K i I ii L i I iii
M 7.72106 N 7.72107
i. ‰KwU A¯¦e†Æî AuvKv hvq M ii I iii N i, ii I iii
17. y2 = 2 y ‰i mgvavb ˆmU KZ?
ii. ‰KwU cwie†î AuvKv hvq
K { 2} L ( 2)
iii. ‰KwU ewneƆî AuvKv hvq
M {0, 2 } N (0, 2 )
1 L 2 M 3 L 4 N 5 L 6 K 7 K 8 K 9 M 10 M 11 L 12 N 13 N 14 M 15 K 16 N 17 M 18 K 19 M 20 L
Dîi

21 N 22 K 23 L 24 K 25 N 26 M 27 L 28 K 29 N 30 N 31 K 32 K 33 L 34 N 35 K 36 M 37 M 38 N 39 N 40 K
24 MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkœcò I DËi : eûwbe©vPwb

228. AvB. B. wU miKvwi DœP we`Åvjq, bvivqYMé welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1. (x  1) = 0 mgxKiGYi gƒj KqwU?
2
14. ˆKvb wòfzGRi KqwU ewnte†î AuvKv hvq? 27. ˆejbwUi f„wgi ˆÞòdj KZ eMÆ ˆm. wg.?
K 3 L2 K 1wU L 2wU K 9 L 15
M 1 N4 M 3wU N 4wU M 30 N 48
2. x2  (a + b)x + ab = 0 mgxKiGYi mgvavb ˆmU 15. mgevü wòfzGRi evü I gaÅgvi Š`GNÆÅi AbycvZ 28. ‰KwU MvGQi cv`G`k ˆ^GK 60 wgUvi `ƒGi ˆKvb
ˆKvbwU? KZ? we±`yGZ MvGQi kxGlÆi D®²wZ ˆKvY 60 | MvGQi
K {a, b} L {a,  b}
K 3t2 L 2t 3 DœPZv KZ wgUvi?
M { a, b} N { a,  b}
M 2t3 N 1t2 K 20 3 L 60 3
3. nGj x t z = KZ?
x t y = 7 t 5, y t z = 5 t 7 wbGPi ZG^Åi AvGjvGK 16 I 17 bs cÉGk²i Dîi `vI : 20
M N 30 3
K 35 t 49 L 35 t 35 A 3
M 25 t 49 N 49 t 25 29. ABC mgGKvYx wòfzGRi C mgGKvY, B =
4. wbGPi ˆKvb kGZÆ ax + by + c = 0 I px + qy + r 40 nGj@
30
= 0 mgxKiY ˆRvU mvgémÅcƒYÆ I wbfÆikxj? 60 K sinB + cosB = 1 L sin2B + cos2C = 1
a b a b c D C M sin2B + cos2B = 1 N cosec2B + cos2B = 1
K  L = =
p q p q r B 30. e†Gîi KqwU cÉwZmvgÅ ˆiLv iGqGQ?
a b c a b K 4 L8
M = 
p q r
N =
p q wPGò, AB ‰KwU MvQ
16. MvQwUi DœPZv AB = KZ ˆm. wg.? M 16 AmsLÅv
N
5. wbGPi Z^ÅàGjv jÞÅ Ki : 31. 5 eQi cƒGeÆ wcZv I cyGòi eqm wQj 35 eQi I
K 34.64 L 44.64
i. 2x  y = 0 I x  2y = 0 mgxKiY«¼q ciÕ·i M 33.64 N 34.146 5 eQi| ZvnGj 5 eQi ci ZvG`i eqGmi
wbfÆikxj 17. MvQwUi cv`G`k ˆ^GK f„ZjÕ© C we±`yi `ƒiZ½ AbycvZ KZ?
ii. x  2y + 3 = 0 mgxKiGYi ˆjLwPò ( 3, 0) BC = KZ ˆm. wg.? K 2:1 L 3:1
we±`yMvgx K 30 L 40 M 4:1 N 17 : 11
iii. 3x + 4y = 1 mgxKiGYi ˆjLwPò ‰KwU M 20 N 60 32. ‰KwU mgv¯¦i avivi 18 Zg c`  14 Zg c` = 32
mijGiLv 18. ‰KwU AvqZGÞGòi `yBwU mw®²wnZ evüi Š`NÆÅ avivwUi mvaviY A¯¦i KZ?
wbGPi ˆKvbwU mwVK? h^vKÌGg 7 ˆm. wg. ‰es 5 ˆm. wg. nGj ‰i K 2 L 4 M 6 N 8
K i I ii L ii I iii cwimxgvi w«¼àY KZ ˆm. wg.? 33. O ˆK±`ÊwewkÓ¡ e†Gî
RÅv AB = RÅv AC, OM 
K 12 L 24 AB ‰es ON  AC nGj@
M i I iii N i, ii I iii
M 36 N 48
6. p, q, r, s mgv¯¦i avivi PviwU KÌwgK c` nGj@ 19. wbGPi ˆKvbwU AwewœQ®² PjK bq? i. AOM = AON
r+p q+r ii. AB = 2AN
K q=
2
L p=
2 K eqm L DœPZv we±`y PviwU mge†î
iii. A, M, O, N
p+s p+r M IRb N RbmsLÅv wbGPi ˆKvbwU mwVK?
M r= N s=
2 2 20. ˆKvbwU wbYÆGq KÌgGhvwRZ MYmsLÅvi cÉGqvRb nq? K i I ii L i I iii
wbGPi avivwUi wfwîGZ 7 I 8bs cÉGk²i Dîi `vI : K AwRf ˆiLv L AvqZGjL
M ii I iii N i, ii I iii
log2 + log4 + log8 + ......... M cÉPziK N Mo
34. mygGbi evwlÆK 4% nvGi 800 UvKvi 4 eQGii
7. avivwUi mvaviY A¯¦i ˆKvbwU? 21. ‰KwU NbGKi avi 6 ˆm. wg. nGj, ‰i c†Ó¤ZGji
K 2 L4 gybvdv KZ?
KGYÆi Š`NÆÅ KZ?
M log2 N 2log2 K 130 UvKv L 127 UvKv
K 6 2 ˆm. wg. L 4 2 ˆm. wg.
8. avivwUi mµ¦g c` KZ? M 128 UvKv N 228 UvKv
M 3 3 ˆm. wg. N 2 6 ˆm. wg.
K log32 L log64 35. ‰KwU QvMj p UvKvq weKÌq KiGj x% jvf nq,
22. mylg lofzGRi KqwU cÉwZmvgÅ ˆiLv AvGQ?
M log128 N log256 QvMjwUi KÌqgƒjÅ KZ?
9. ABC ‰es DEF m`†k ‰es Abyi…c evü«¼Gqi K 4wU L 10wU 100x 100p
M 6wU N 5wU K L
AbycvZ 3 t 2 nGj, ZvG`i ˆÞòdGji AbycvZ 100 + p 100 + x
KZ? 23. ‰KwU mylg cçfzGRi cÉwZ evüi Š`NÆÅ 4 ˆm. wg. M
100
N
100 + p

K 5t1 L 3t2
nGj ‰i ˆÞòdj KZ eMÆ ˆm. wg.? 100 + p 100x
K 5.506 L 5.606 36. mvaviY jMvwi`g wbGPi ˆKvbwU?
M 8t2 N 9t4 K logex L lnx
M 5.056 N 27.528
10. e†Gîi eÅvmvaÆ 2 ˆm. wg. ‰es e†Gîi ˆKG±`Ê Drc®² n d M log102 N log p
ˆKvY 180 nGj, e†îKjvi ˆÞòdj KZ eMÆ ˆm. 24. i. gaÅK = L + ( ) 2
 Fc 
m 37. ˆKvb aiGbi msLÅv?
e = 2.71828 ......
wg.? ii. Mo = a + hu f™²vsk
K L FYvZÄK
K  L 2 1 M Agƒj` N gƒj`
M 3 N 4 iii. cÉPziK = L +  h
1 + 2 38. log2717 ‰i AskK KZ?
wbGPi Z^ÅvbymvGi (11-13) bs cÉGk²i Dîi `vI: wbGPi ˆKvbwU mwVK? K 0.43408 L 4.308
ABCD iÁ¼mwUi ˆÞòdj 192 eMÆ ˆm. wg. ‰es ‰KwU K i I ii L i I iii M 0.48408 N 1.43408
KGYÆi Š`NÆÅ BD = 24 ˆm. wg.| AE iÁ¼mwUi DœPZv| M ii I iii N i, ii I iii wbGPi ZG^Åi AvGjvGK 39 I 40 bs cÉGk²i Dîi `vI :
11. AC KGYÆi Š`NÆÅ KZ? 25. eMÆGÞGòi cwimxgv 20 wgUvi nGj, ˆÞòdj KZ 3.6  105 ŠeævwbK i…Gc cÉKvwkZ msLÅv|
K 96 ˆm. wg. L 48 ˆm. wg. eMÆ. wg.? 39. cÉ`î msLÅvwUi mvaviY `kwgKi…c ˆKvbwU?
M 20 ˆm. wg. N 16 ˆm. wg. K 9 L 16 M 25 N 36 K 0.00036 L 0.0000036
12. BC evüi Š`NÆÅ KZ ˆm. wg.? wbGPi ZG^Åi AvGjvGK 26 I 27 bs cÉGk²i Dîi `vI : M 0.36 N 0.000036
K 14.42 L 16 mge†îf„wgK ˆejGbi f„wgi eÅvmvaÆ 3 ˆm. wg. I DœPZv 40. msLÅvwUi mvaviY jGMi cƒYÆK ˆKvbwU?
M 16.24 N 20.42 5 ˆm. wg.| K 3 L 2
M 5 N5
13. AE ‰i Š`NÆÅ KZ ˆm. wg.? 26. ˆejbwUi AvqZb KZ Nb ˆm.wg.?
K 14.4 L 13.31 K 15 L 30
M 12.2 N 11.3 M 45 N 75
1 L 2 K 3 L 4 L 5 L 6 K 7 M 8 M 9 N 10 L 11 N 12 K 13 L 14 M 15 L 16 K 17 M 18 N 19 N 20 K
Dîi

21 K 22 M 23 N 24 N 25 M 26 M 27 K 28 L 29 M 30 N 31 L 32 N 33 N 34 M 35 L 36 M 37 M 38 K 39 N 40 M
MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkc
œ ò I DËi : eûwbe©vPwb 25

229. biwms`x miKvwi evwjKv DœP we`Åvjq, biwms`x welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1. cÉ^g n msLÅK weGRvo Õ¼vfvweK msLÅvi mgwÓ¡ 14. ‰KwU mgGKvYx wòfzGRi mgGKvY msj™² `yB evü 29. `yBwU mgv¯¦ivj mijGiLvGK Aci ‰KwU
nGjv@ 4 wgUvi ‰es 3 wgUvi| ‰i cwimxgv KZ wgUvi? mijGiLv ˆQ` KiGj A¯¦tÕ© ˆKvYmgƒGni mgwÓ¡
K
n(n + 1)
L
n(n – 1) K 10 L 12 KZ?
2 2 M 15 N 20 K 360 L 270
M n2 N 2n + 1
2. a1x + b1y + c1 = 0
15. ˆKvbwU wewœQ®² PjK?
‰es a2x + b2y + c2 = 0
M 90 N 180
K Zvcgvòv L eqm 30. ˆKvGbv mgGKvYx wòfzGRi mƒßGKvY«¼Gqi cv^ÆKÅ
mgxKiY ˆRvUwUi mgvavb bv ^vKGj wbGPi
M DœPZv N RbmsLÅv 6 nGj, e†nîg ˆKvGYi gvb@
ˆKvbwU mwVK?
K 38 L 41
K
a1 b1
= L ≠
a1 b1 ‰KwU eMÆGÞGòi ˆÞòdj 432 eMÆwgUvi| M 92 N 48
a2 b2 a2 b2 DcGii ZG^Åi AvGjvGK 16 I 17bs cÉGk²i Dîi `vI: 31. ˆKvGbv e†Gî ‰KwU mgevü wòfzR A¯¦wjÆwLZ nGj,
a1 b1 c1 a1 b1 c1
M = =
a2 b2 c2
N = ≠
a2 b2 c2 16. eMÆGÞòwUi evüi Š`NÆÅ KZ wgUvi? wòfzRwUi kxlÆwe±`yGZ Õ·kÆKàGjv ˆh wòfzR MVb
3. y = 2x dvskGbi ˆjLwPò ˆKgb nGe? K 8 3 L 10 3 KGi Zv nGe@
K mijGiLv L eKÌGiLv M 11 3 N 12 3 K mgGKvYx L mgevü
M e†î N cive†î 17. eMÆGÞòwUi KGYÆi Š`NÆÅ KZ wgUvi? M welgevü N Õ©ƒjGKvYx
1
4. a2 – 5 a + 1 = 0 nGj a + a = KZ? K 8 6 L 10 6 32. mviwY fzÚKiGYi RbÅ cÉ^Gg ˆKvbwU wbYÆq KiGZ
M 11 6 N 12 6 nGe?
K 2 L 3
18. AvqZGjL ˆ^GK ˆKvb ˆjLwPò Au vKv hvq? K ˆkÉwYmsLÅv L ˆkÉwY eÅeavb
M 2+ 3 N 5
5. a + b + c = 0 nGj, a3 + b3 + c3 = KZ? K cvBwPò L `íwPò M cwimi N MYmsLÅv
K 0 L3 M AwRf ˆiLv N MYmsLÅv eüfzR 33. A
1 19. ˆh ˆKvGbv RÅvwgwZK wPGòi KgcGÞ KZ gvòvi
M 3abc N
3abc
NƒYÆb cÉwZmgZv AvGQ? D E
6. 1 + 13 + 19 + ……….. avivwUi 6Ó¤ I 7g cG`i K 1 L2
ˆhvMdj KZ? M 3 N4 wPGò BC 7 DE B C

K
1
L 729
20. mgevü wòfz G Ri cÉ w ZmvgÅ ˆiLv KqwU? wbGPi ˆKvbwU mwVK?
729 K 1 L2 AB AC AB AC
1 4 K = L =
M N M 3 N4 AD AE BD AE
243 729
21. sin( – 90) = KZ? M
AB AC
N
AB AC
7. a : b = c : d ˆK ‰Kv¯¦KiY KiGj Kx nGe? K –cos L cos
=
BC BC
=
DE BC
K bc L a:c=b:d 34. ‰KwU AvqZGÞGòi Š`NÆÅ I cÉÕ© w«¼àY KiGj
a+b c+d M cosec N –cosec
M a:d=b:c N
a
=
d 22. A ˆÞòdj KZàY evoGe?
f(x) = x3 + x2 + x + 1  K 2 àY L 3 àY
DcGii ZG^Åi wfwîGZ 8 I 9 bs cÉGk²i Dîi `vI : M 4 àY N 6 àY
8. f(x) ‰i ‰KwU Drcv`K nGjv@ 35. sin + cos = 1 nGj, sin.cos = KZ?
K x2 + 1 L x3 + 1 K 0 L –1
M x–1 N x2 – 1 B C
1
1 M 1 N
9. f(x) ˆK (x + 3) «¼viv fvM KiGj fvMGkl KZ tan = nGj, AB : BC = KZ? 2
nGe? 3 36. 0.0176 ‰i ŠeævwbK i…c ˆKvbwU?
K 20 L 10 K 1:2 L 1: 2 K 1.76 × 10–2 L 17.6 × 10–3
M –20 N –10 M 3:1 N 1: 3 M 176 × 10–4 N .176 × 10–2
10. a > 0 ‰es a ≠ 1 nGj@
23. wewkÓ¡ AaÆe†Gîi ˆÞòdj KZ eMÆ 37. NbeÕ§i gvòv KqwU?
r ‰KK eÅvmvaÆ
logbM K 1 L2
i. logaM =
logb a ‰KK?
r
ii. logaM = rlogaM K r2 L 2r2 M 3 N AmsLÅ
iii. logaM + logaN = loga (M × N) 2r r2 38. n weGRvo msLÅv nGj, gaÅK nGe@
M N
wbGPi ˆKvbwU mwVK? 2 2 K
n+2
Zg c` L
n–2
Zg c`
2 2
K i I ii L i I iii 24. ‰KwU `í I ‰i Qvqvi Š`GNÆÅi AbycvZ 1 : 3 n+1 n
M ii I iii N i, ii I iii nGj, mƒGhÆi D®²wZ ˆKvY KZ? M
2
Zg c` N Zg c`
2
11. mgevü wòfzGRi ‰K evüi Š`NÆÅ 6 ˆm.wg. nGj, K 30 L 45 A
‰i ˆÞòdj KZ eMÆ ˆm.wg.? M 60 N 90
K 12 3 L9 3 25. wbGPi ˆKvb mijGiLvwU gƒj we±`yMvgx? B D
M 6 3 N3 3 K x+y=5 L x– y=7
12. 2r eÅvmvaÆ wewkÓ¡ e†Gîi@ M 5x – 3y = 0 N 5x – 3y = 4
i. e†nîg RÅv 4r 26. ˆKvGbv e†Gîi AwaPvGc A¯¦wjÆwLZ ˆKvY ˆKvbwU? C
ii. cwiwa 4r K mƒßGKvY L Õ©ƒjGKvY DcGii ZG^Åi AvGjvGK 39 ‰es 40 bs cÉGk²i Dîi
iii. ˆÞòdj 4r2 M mgGKvY N cÉe†«¬GKvY `vI|
wbGPi ˆKvbwU mwVK? 27. msLÅv mƒPK Z^Åvewj «¼viv cwimsLÅvGbi Kx 39. ABC I  ADC ‰i gGaÅ mÁ·KÆ Kx?
K i I ii L ii I iii K mgGKvY L cƒiKGKvY
wbG`Æk KGi?
M i I iii N i, ii I iii M mÁ·ƒiK ˆKvY N cÉe†«¬GKvY
9 K djvdj L Dcvî
13. ‰KwU mgevü wòfzGRi cwimxgv 2 cm nGj, ‰i M Abym®¬vb N weGkÏlY
40. i. ABC Õ©ƒjGKvY nGj ADC mƒßGKvY
ˆÞòdj@ ii. BAD = 45 nGj, BCD = 45
28. ‰KwU eMÆGÞò r eÅvmvaÆ wewkÓ¡ ‰KwU e†Gî iii. ABC +  ADC = 180
K
2 3
L
3 A¯¦ÆwjwLZ nGj e†î I eGMÆi ˆÞòdGji AbycvZ wbGPi ˆKvbwU mwVK?
4 2
KZ? K i I ii L ii I iii
9 3 3 3
M N K 4: L :4 M i I iii N i, ii I iii
16 16
M 2: N :2
1 M 2 N 3 K 4 N 5 M 6 N 7 L 8 K 9 M 10 N 11 L 12 N 13 M 14 L 15 N 16 N 17 N 18 N 19 K 20 M
Dîi

21 K 22 M 23 N 24 K 25 M 26 K 27 L 28 N 29 K 30 N 31 L 32 M 33 K 34 M 35 K 36 K 37 M 38 M 39 M 40 M
26 MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkœcò I DËi : eûwbe©vPwb

230. dwi`cyi wRjv Õ•zj, dwi`cyi welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1. {x  ô : 6 < x < 7 ‰es x ˆgŒwjK msLÅv} ˆK 15. ACD gvb KZ wWwMÉ? 28. BCDE eMÆGÞGòi BD KGYÆi Š`NÆÅ KZ?
ZvwjKv c«¬wZGZ cÉKvk KiGj ˆKvbwU nq? K 60 L 90 K 3 2 L4 2
K {} L {0} M {} N {6} M 120 N 100 M 5 2 N2 2
2.
1 1
x + = 2 nGj, x2 + 2 ‰i gvb KZ?
16. 4.5 ˆm.wg. eÅvmvaÆ wewkÓ¡ e†Gîi ˆK±`ÊMvgx RÅv 29. ABCD UÇvwcwRqvGgi
ˆÞòdj KZ eMÆ ˆm.wg.?
x x ‰i Š`NÆÅ KZ ˆm.wg.? K 23 L 22
K 0 L 2 M2 N4 M 21 N 20
K 7 ˆm.wg. L 9 ˆm.wg.
3. a > 0, b > 0 ‰es a, b  R nGj 30. mge†îf„wgK ˆejGbi f„wgi eÅvmvaÆ 3 ˆm.wg. ‰es
M 10 ˆm.wg. N 12 ˆm.wg.
i. logaa  loga b = 1
17. sin3 = cos3 nGe hLb  KZ? DœPZv 4 ˆm.wg. nGj-
ii. logbb logc c = 1
K 15 L 30 i. f„wgi ˆÞòdj 9 eMÆ ˆm.wg.
3 5
iii. loga ( a a)= M 20 N 45 ii. AvqZb 70 Nb ˆm.wg.
6
iii. mgMÉZGji ˆÞòdj 42 eMÆ ˆm.wg.
wbGPi ˆKvbwU mwVK? 18. cosecA = ab nGj, tan A = KZ?
K i I ii L ii I iii
wbGPi ˆKvbwU mwVK?
b a K i I ii L i I iii
M i I iii N i, ii I iii K L
a2 + b2 a2  b2 M ii I iii N i, ii I iii
4. 5 5 ‰i 5 wfwîK log ‰i gvb KZ? b a
3 2 3 4 M N 2 2
31. mge†îf„wgK ˆKvYGKi ˆnjvGbv D®²wZ 5 ˆm.wg.
2 2
K L M N a b a +b ‰es DœPZv 4 ˆm.wg. nGj-
2 3 4 3
5. ‰KwU AvqZvKvi NGii Š`NÆÅ cÉGÕ©i wZbàY| 19. ‰KwU wgbvGii DœPZv 20 3 wgUvi ‰es Qvqvi i. ˆKvYGKi eÅvmvaÆ 3 ˆm.wg.
‰i ˆÞòdj 147 eMÆwgUvi nGj, cwimxgv KZ Š`NÆÅ 20 wgUvi nGj ‰i AebwZ ˆKvY KZ wWwMÉ? ii. ˆKvYGKi f„wgi ˆÞòdj 9 eMÆ ˆm.wg.
wgUvi? K 30 L 45 iii. ˆKvYGKi eÅvmvaÆ 4.5 ˆm.wg.
K 65 L 56 M 64 N 46 M 60 N 90 wbGPi ˆKvbwU mwVK?
6. (x) = x5 + 5x  3 nGj, (1) ‰i gvb KZ? 20. `ywU e†Gîi eÅvmvGaÆi AbycvZ 1 : 3 nGj, ˆÞòdGji K i I ii L i I iii
K 7 L 8 M 9 N 10 AbycvZ KZ? M ii I iii N i, ii I iii
7. ˆKvb wòfzGRi ˆKvYàGjvi AbycvZ 1 : 2 : 3 ‰es K 1:3 L 2:3
32. AwRf ˆiLvi ˆÞGò wbGPi ˆKvbwU mwVK?
Þz`ËZi evüi Š`NÆÅ 10 ˆm.wg. nGj, e†nîg evüi M 1:9 N 1:5
K wbÁ²Mvgx L Da»ÆMvgx
Š`NÆÅ KZ? wbGPi DóxcKwU cGov ‰es (21 I 22) bs cÉGk²i Dîi
M DjÁ¼ N mgv¯¦ivj
K 10 3 L 20 M 20 3 N 30 `vI:
33. mviwYfzÚKiGYi ˆÞGò cÉ^Gg ˆKvbwU wbYÆq
8. wbGPi ˆKvbwU Agƒj` msLÅv? 6 + a + b + 162 àGYvîi avivfzÚ|
KiGZ nq?
3 3 1 21. avivwUi mvaviY AbycvZ KZ? K ˆkÉwY msLÅv L cwimi
K 81 L 9 M N K 2 L3
5 2 M MYmsLÅv N ˆkÉwY eÅeavb
9. (x) = x2  3x + 2 nGj ‰i Drcv`K- M 4 N5
22. (b  a) ‰i gvb ˆKvbwU? wbGPi ZG^Åi AvGjvGK (34 I 35) bs cÉGk²i Dîi
i. x + 3
ii. x  2 K 18 L 36 `vI:
iii. x  1 M 54 N 60 dwi`cyi wRjv Õ•zGji ‰KwU bZzb feb wbgÆvGYi KvGR
wbGPi ˆKvbwU mwVK? 23. ABC mgevü wòfzGRi B ‰i mgw«¼L´£K AC wbGqvwRZ 10 Rb kÉwgGKi Š`wbK gRyix (UvKvq)
K i I ii L ii I iii evüGK D we±`yGZ ˆQ` KiGj CD : DA = KZ? ˆ`Iqv nGjv| 175, 165, 140, 190, 180, 200, 170, 145,
M i I iii N i, ii I iii K 1:2 L 1:3 130, 150|
10. ‹eævwbK msLÅvi Av`kÆi…c a  10n| wbGPi M 1:1 N 1:4 34. DcvGîi cwimi KZ?
ˆKvbwU a ‰i gvGbi mwVK mxgv? 24. ‰KwU mgevü wòfzGRi NƒYÆb cÉwZmvgÅZvi gvòv K 50 L 71
K 10 </ a < 10 L 1  a < 10 KZ? M 85 N 90
M 1 < a </ 10 N a > 10 K 1 L3 35. 10 ˆkÉwY eÅeavb wbGq ˆkÉwYmsLÅv KZ nGe?
11. ‰KwU mgGKvYx wòfzGRi mƒßGKvY«¼Gqi ‰KwU M 2 N4 K 8 L6
AciwUi w«¼àY, mƒßGKvY«¼Gqi cwigvGYi AbycvZ 25. ‰KwU mgGKvYx wòfzGRi f„wg 12 wgUvi ‰es M 4 N2
KZ? AwZfzR 13 wgUvi nGj, ‰i ˆÞòdj KZ eMÆ 36. wbGPi Z^ÅàGjv jÞÅ Ki-
K 2:3 L 1:1 i. mKj exRMwYZxq mƒòB AGf`
wgUvi?
M 2:1 N 3:2 ii. mKj AGf`B mgxKiY
K 30 L 40
12. ÷ay cwimxgv RvbGjB- M 20 N 25 iii. mKj mgxKiYB AGf`
i. eMÆ AuvKv mÁ¿e 26. ‰KwU mgevü wòfzGRi evüi Š`NÆÅ 2 wgUvi nGj, wbGPi ˆKvbwU mwVK?
ii. mgevü wòfzR AuvKv mÁ¿e K i I ii L i I iii
DœPZv KZ wgUvi nGe?
iii. AvqZGÞò AuvKv mÁ¿e M ii I iii N i, ii I iii
3 3
wbGPi ˆKvbwU mwVK? K
2
L
4 37. ‰KwU eGMÆi evüi Š`NÆÅ cÉGÕ©i w«¼àY nGj ‰i
K i I ii L ii I iii 4 ˆÞòdj KZ àY nGe?
M i I iii N i, ii I iii M 3 N
3 K 3 àY L 2 àY
13. mgevü ABC ‰i AD  BC ‰es AB = 2 27. ‰KwU iÁ¼Gmi KYÆ«¼Gqi AGaÆK h^vKÌGg 5 ˆm.wg. M 4 àY N 5 àY
ˆm.wg. nGj, AD = KZ? ‰es 7 ˆm.wg.| iÁ¼Gmi ˆÞòdj KZ eMÆ ˆm.wg.? 38. ‰KwU AvqZvKvi NGii Š`NÆÅ cÉGÕ©i w«¼àY ‰es
K 2 L 3 M3 N 5 K 70 L 35 ˆÞòdj 512 eMÆ wgUvi nGj, cwimxgv KZ?
wbGPi ZG^Åi AvGjvGK (14 I 15) bs cÉGk²i Dîi `vI: M 12 N 24 K 96 L 48
mgevü ABC ‰i BC ˆK D chƯ¦ ewaÆZ Kiv wbGPi wPGòi AvGjvGK (28 I 29) bs cÉGk²i Dîi `vI: M 52 N 56
nGjv| 3 ˆm.wg. E
39. e†Gîi cwiwa I eÅvmvGaÆi AbycvZ?
A K 7 : 22 L 2 : 1
14. ABC ‰i gvb KZ wWwMÉ? D
A M 1: N :2
ˆm.wg.

40. ‰KwU e†Gîi ˆh ˆKvb Õ©ƒjGKvY «¼viv Drc®² Pvc-

K AwaPvc L DcPvc
5

C M Awae†î N AaÆe†î
C B B
K 30 L 60 M 45 N 15
1 K 2 K 3 L 4 K 5 L 6 M 7 L 8 L 9 L 10 L 11 M 12 K 13 L 14 L 15 M 16 L 17 K 18 M 19 M 20 M
Dîi

21 L 22 L 23 M 24 L 25 K 26 M 27 K 28 L 29 L 30 L 31 K 32 L 33 L 34 L 35 K 36 K 37 M 38 K 39 L 40 L
MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkœcò I DËi : eûwbe©vPwb 27

231. cvjs Zzjvmvi ài‚`vm miKvwi DœP we`Åvjq, kixqZcyi welq ˆKvW : 1 0 9
mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1. (2x + 3y, 4) = (10, 3x  5y) nGj, (x, y) ‰i 15. 4 + x + y + 32 àGYvîi aviv nGj (x, y) = ?
gvb KZ? K (4, 8) ABCD ‰KwU iÁ¼Gmi KYÆ«¼q ciÕ·i O we±`yGZ ˆQ`
L (6, 12)
K (2, 1) L (1, 1) M (8, 16) N (9, 12)
KGi| AC = 8 ˆm.wg. ‰es BD = 6 ˆm.wg.
M (2, 2) N (4, 2) 16. AvqZGjL ‰i DœPZv ˆKvbwU? 28. iÁ¼mwUi evüi Š`NÆÅ KZ?
2. p2  3p + 1 = 0 nGj, 1
p2 + 2 ‰i gvb KZ? K MYmsLÅv K 5 ˆm.wg. L 4 ˆm.wg.
p
K 5 L7 L KÌGgvGhvwRZ MYmsLÅv M 3 ˆm.wg. N 2 ˆm.wg.
M 11 N 13 M ˆkÉwY gaÅgvb 29. OBC ‰i ˆÞòdj KZ?
3. S = {(2, 4), (1, 1), (0, 0), (1, 1), (2, 4)} nGj, N ˆkÉwY eÅeavb K 8 ˆm.wg. 2 L 6 ˆm.wg.2
ˆKvb dvskbwU S A®¼qGK cÉKvk KGi? 17. wbGPi ˆKvbwU ˆK±`Êxq cÉeYZv bq? M 4 ˆm.wg. 2
N 3 ˆm.wg.2
K y = x2 L x = y2
K Mo L gaÅK 30. iÁ¼m ˆÞòwUi ˆÞòdj KZ?
M y = x2 + 1 N x = y2 + 1
M cÉPziK N GkÉwY eÅeavb K 48 ˆm.wg.2 L 24 ˆm.wg.2
4. a : b = 3 : 4, b : c = 6 : 7 nGj, a : b : c = KZ?
K 3:4:1 L 3:9:7 18. 3x  4y = 0 ‰es 2x  3y = 1 mgxKiY ˆRvGUi M 12 ˆm.wg. 2
N 6 ˆm.wg.2
M 9 : 12 : 14 N 12 : 6 : 7 mgvavb ˆKvbwU? 31. a + b = 2, a + b = 2 nGj, ab = KZ?
2 2

5. ˆKvb àGYvîi avivi 5g c` 16 ‰es 8g c` 128 K (1, 2) L (2, 1) K 2 L1

nGj, avivwUi 1g 4 cG`i mgwÓ¡ KZ? M (3, 4) N (4, 3) 1
M 0 N
K 14 L 15 19. ABC ‰ C = 1 mgGKvY ‰es B = 2A 2
M 16 N 17 nGj, wbGPi ˆKvbwU mwVK? 32. ‰KwU wmwj´£vGii f„wgi eÅvmvaÆ 3 ˆm.wg. nGj
6. 7 7 ‰i 7 wfwîK jM wbGPi ˆKvbwU? K AB = 2BC L AC = 2BC Dnvi f„wgi ˆÞòdj KZ?
2 3 M AB = AC + BC N AC = BC
K L K 9 eMÆ ˆm.wg. L 2 eMÆ ˆm.wg.
3 2
20. e†Gîi cwiwa I eÅvGmi AbycvZ ˆKvbwU? M 4 eMÆ ˆm.wg. N 4 eMÆ ˆm.wg.
49
M 49 N K :1 L 2 : 1
7 33. wbGPi ˆKvbwU Q = {x  ô : x2
> 15 ‰es x3 < 36}
M 2 : 1 N 3 : 1
1 1 ˆmU ‰i ZvwjKv c«¬wZ cÉ K vk Ki|
( )
7. (x  a  b) a + b = 0 mgxKiGY x ‰i gvb 21. 0.0245 ‰i ŠeævwbK i…c ˆKvbwU?
K Q = {2} L Q = {3}
KZ? K 2.45  101 L 2.45  102
M {4} N
M 2.45  103 N 2.45  104
1 1 1 1 34. ˆMvjGKi AvqZb wbYÆGqi mƒòwU ˆKvbwU?
K ( )
+
a b
L (a + b), ( )+
22. cixÞvi bÁ¼i I RbmsLÅv ˆKvb aiGbi PjK?
a b
4
M  (a + b) N (a + b) K wewœP®² PjK L AwewœQ®² PjK K r2h L r3
3
wbGPi ZG^Åi AvGjvGK 8 I 9 bs cÉGk²i Dîi `vI : M evÕ¦e PjK N AevÕ¦e PjK M 2rh N r3
‰KwU AvqZvKvi NbeÕ§i Š`NÆÅ, cÉÕ© I DœPZv 23. mgMÉ ZGji ˆÞòdj = 6A2 eMÆ ‰KK ˆKvbwUi 35. avGb Pvj I ZzGli AbycvZ 7 : 3 nGj, ‰GZ
h^vKÌGg 10 ˆm.wg., 5 ˆm.wg. I 4 ˆm.wg.| mvG^ mÁ·KÆhyÚ? kZKiv wK cwigvY Pvj AvGQ?
8. AvqZvKvi NbeÕ§wUi mgMÉZGji ˆÞòdj KZ? K NbK L AvqZGÞò K 70% L 60%
K 110 eMÆ ˆm.wg. L 200 eMÆ ˆm.wg. M 40% N 30%
M eMÆGÞò N UÇvwcwRqvg
M 220 eMÆ ˆm.wg. N 440 eMÆ ˆm.wg. 36. ˆmU C nGZ ˆmU B ‰ ‰KwU mÁ·KÆ Ñ nGj,
24. A = 30 nGj, 2tan2A + cot2A ‰i gvb KZ?
9. NbeÕ§wUi KGYÆi Š`NÆÅ KZ ˆm.wg.? 11 11
wbGPi ˆKvbwU mwVK?
K 11.874 L 23.749 K L K RC L RB
6 3
M 35.622 N 47.497 3 1 M RCB N CBR
M N
10. Õ·kÆ we±`yGZ Õ·kÆGKi Dci jÁ¼ wbGPi ˆKvbwU 11 37. 22x+1 = 128 nGj, x ‰i gvb ˆKvbwU?
3
w`Gq hvq? 25. 60 wgUvi `xNÆ ‰KwU gB ˆ`IqvGji mwnZ 45 K 5 L4
K RÅv L ˆK±`Ê ˆKvY Drc®² KGi| ˆ`IqvjwUi DœPZv KZ nGe? M 3 N2
M ˆQ`K N Õ·kÆ ˆiLvsk K 42.43 wgUvi L 30 wgUvi 38. kx3  6x2  4x + 3 ‰i ‰KwU Drcv`K (2x + 1)
11. ‰KwU eGMÆ A¯¦wjÆwLZ e†Gî KqwU Õ·kÆK AvGQ? M 60 wgUvi N 120 wgUvi nGj, k = ?
K 1 L2 K 28 L 14
26. 2x + 2x ‰i Drcv`K ˆKvbwU?
2
M 3 N4 M 14 N 28
K 2 L 2x
12. sec2 1  sin2 = KZ?  
M 2x N 2x 39. 4.3  5.7 = KZ?
K sin L cos
M tan N sec 27. wbGPi ˆKvbwU gƒj` msLÅv bq? ..
K 20.21
..
L 24.5 1
13. ‰KwU eGMÆi evüi ˆ`Lv wZbàY e†w«¬ ˆcGj K 0.4 L 9 . .
M 25. 037 N 25.30
ˆÞòdj KZàY e†w«¬ cvGe? M 5.639 N 11
x a+2
K 2 àY L 3 àY wbGPi wPGòi AvGjvGK (28-30) bs cÉGk²i Dîi `vI : 40. =
y a2
nGj, (x + y) : (x  y) = KZ?

M 4 àY N 9 àY A K 2:a L 4:a
B
14. e†îKjvi ˆÞòdj ˆKvbwU? M a:2 N 2a : 1
O
  2r r2
K L
360 360
360 360r2 D C
M N
2r 

1 M 2 L 3 K 4 M 5 L 6 L 7 N 8 M 9 K 10 L 11 N 12 N 13 N 14 L 15 M 16 K 17 N 18 N 19 K 20 K
Dîi

21 L 22 K 23 K 24 L 25 K 26 L 27 N 28 K 29 L 30 L 31 L 32 K 33 N 34 L 35 K 36 M 37 M 38 N 39 M 40 M
28 MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkœcò I DËi : eûwbe©vPwb

232. Kvw`ivev` KÅv´ŸbGg´Ÿ cvewjK Õ•zj, bvGUvi welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1. logx9 = 2, x ‰i gvb wbGPi ˆKvbwU? 14. wPGò ABC mgevü ‰es cÉGZÅK evüi Š`NÆÅ 4 27. cƒYÆ eMÆ bq ‰i…c abvñK cƒYÆ msLÅvi eMÆgƒj
K 2 L 3 wgUvi nGj A ˆKvb aiGbi msLÅv?
M  N 18 K Õ¼vfvweK L cƒYÆ
2.
1
= 81 nGj x ‰i gvb wbGPi ˆKvbwU? i. BD = 2 wgUvi B M gƒj` N Agƒj`
C
3 x ii. AD = 2 5 D A = {1, 2, 3} ‰es B = {2, 3, 4}
K 4 L3
iii. ABC ‰i ˆÞòdj = 4 3 eMÆwgUvi DcGii ˆmU«¼Gqi AvGjvGK 28 I 29bs cÉGk²i Dîi
M 3 N 4
3. wbGPi ˆKvbwU mwVK? `vI:
7 7 ‰i 7 wfwîK jM KZ?
1 3 K i I ii L ii I iii 28. A ˆmGUi cÉK‡Z DcGmGUi msLÅv KZ?
K L M i I iii N i, ii I iii K 3 L6
2 2
M 7 N8
M 2 N3 15. wMÉK kõ Metron ‰i A^Æ Kx?
4. ‰KwU eGMÆi evüi Š`NÆÅ 40 wgUvi| ‰i cÉGZÅK 29. A  B wbGPi ˆKvbwU?
K cwimxgv L cwiwgwZ
K {1} L {4}
evüi Š`NÆÅ 10% evoGj M cwigvc N avi M {2, 3} N {1, 2, 3, 4}
i. cÉGZÅK evüi Š`NÆÅ nGe 44 wgUvi 16. 3 log102 + 2 log102 ‰i gvb ˆKvbwU? 30. eÅvm «¼viv e†î mgw«¼LwíZ nq-‰i hywÚgƒjK
ii. ‰i ˆÞòdj evoGe 21% K log1032 L log1016 cÉgvY ˆK ˆ`b?
iii. ‰i KGYÆi Š`NÆÅ evoGe 10 2 M log108 N log104
K cx^vGMvivm L BDwKÑW
wbGPi ˆKvbwU mwVK? 17. (x1 + y1)1 ‰i gvb ˆKvbwU? M ˆ^wjm N ‰wiÕ¡Uj
x+y 1
K i I ii L ii I iii K L 31. 3 + a + b + 81 àGYvîi avivfzÚ nGj b ‰i gvb
xy x+y
M i I iii N i, ii I iii xy KZ?
5. `yBwU msLÅvi AbycvZ 5 : 6 ‰G`i j.mv.à. 150 M x+y N K 9 L 12
x +y
nGj, M.mv.à. KZ? 1 M 18 N 27
K 2 L3
18. 2
{(x + y)2 + (x  y)2} ‰i gvb KZ? 32. ‰KwU NbGKi KGYÆi Š`NÆÅ 6 3 wgUvi nGj, ‰i
M 5
A
N 11 K 2(x2 + y2) L 2xy AvqZb KZ NbwgUvi?
M x2 + y2 N 4xy K 36 L 144
19. tanA =
1
nGj, cot2A ‰i gvb KZ? M 216 N 512
3 33. K t L = 3 t 4, L t M = 5 t 4 nGj,
KtLtM
O 1 1 = KZ?
K L
C 3 3 K 15 t 20 t 16 L 16 t 15 t 20
B
M 3 N3 M 20 t 16 t 15 N 20 t 15 t 16
wbGPi wPGòi AvGjvGK 20 I 21bs cÉGk²i Dîi `vI: 1  tan2
34. hw` A = 45 nq ZGe 1 + tan2 = ?
A
6. wPGò A = 60 nGj, OBC = KZ? B K 1 L0
K 30 L 45 1
M 60 N 120 M N2
2
wbGPi cÉ`î ZG^Åi AvGjvGK 7 I 8bs cÉGk²i Dîi 35. wòfzGRi wZbwU ˆKvGYi AbycvZ 2 t3t 4 nGj
C
`vI: D e†nîg ˆKvYwU KZ?
`yB AsK wewkÓ¡ ‰KwU msLÅvi `kK Õ©vbxq AsK K 40 L 80
‰KK Õ©vbxq AsGKi w«¼àY ‰es AsK«¼Gqi àYdj wPGò ABCD ‰KwU iÁ¼m, hvi KYÆ AD = 12 ˆm.wg., M 60 N 100
32| BC = 16 ˆm.wg. 36. wbGPi ˆKvbwU wewœQ®² PjK?
7. ‰KK Õ©vbxq AsK x nGj, msLÅv ˆKvbwU? 20. CD evüi Š`NÆÅ KZ? K eqm L cÉvµ¦ bÁ¼i
K 3x L 12x K 10 L 14 M DœPZv N IRb
M 21x N 30x M 28 N 100 37. Q ‰KwU duuvKv ˆmU nGj P(Q) ‰i m`mÅ msLÅv
8. Z^ÅvbymvGi cÉvµ¦ msLÅv ˆKvbwU? 21. CD evüi mgvb eÅvmvaÆ wewkÓ¡ e†Gîi cwiwa KZ? KZ?
K 84 L 48 K 31.416 ˆm.wg. L 62.832 ˆm.wg. K 0 L1
M 42 N 24
M 314.16 ˆm.wg. N 628.32 ˆm.wg. M 2 N3
9. e†îKjvi ˆÞòdj wbGPi ˆKvbwU? 22. ‰KwU e†Gîi ˆK±`Ê nGZ mg`ƒieZÆx `ywU RÅv AB I
D
  C
K  r2 L  r3 CD.
O
9 9 K AB > CD L CD > AB
  M AB  CD = 0 N AB  CD > 0 A
M r3 N r2
200 360 23. wbGPi Z^ÅàGjv jÞ Ki:
10. f(a) = a2  3a + 2 nGj, a ‰i ˆKvb gvGbi RbÅ i. cÉGZÅK dvskb ‰K-‰KwU A®¼q B
f(a) = 0 nGe?
ii. f(x) = x2 + 1 nGj, f(1) = 2 DcGii wPGòi AvGjvGK 38 I 39bs cÉGk²i Dîi `vI:
K 0 L2
iii. f(x) = x2 + 2 nGj, f(2) = 6 38. ABC I ADC ‰i mÁ·KÆ wKi…c?
M (1, 2) N (1,  2)
11. 2n  5 + 3 = 2 ‰i mwVK mgvavb ˆmU wbGPi ˆKvbwU mwVK? K mgGKvY L Õ©ƒjGKvY
K i I ii L ii I iii M mÁ·ƒiK ˆKvY N cÉe†«¬ ˆKvY
ˆKvbwUi?
M i I iii N i, ii I iii 39. ABCD e†îÕ© PZzfƃGRi
K {3} L { 3}
M { 3} N 24. 2sec2 + tan2 ‰i gvb KZ? i. ABC Õ©ƒjGKvY nGj, ADC mƒþGKvY
K 1 L2 ii. BAD = 45 nGj, BCD = 45 nGe
12. NbGKi avGii Š`NÆÅ 3 nGj, cÉwZ ZGji
M 1 N0
ˆÞòdj KZ? iii. ABC + ADC = 180
25. AvqZGÞGòi Š`NÆÅ 12 wgUvi ‰es KYÆ 13 wg. wbGPi ˆKvbwU mwVK?
K 3 eMÆwgUvi L 3 eMÆwgUvi
nGj cÉÕ© KZ? K i I ii L ii I iii
M 3 3 eMÆwgUvi N 18 eMÆwgUvi
K 25 wgUvi L 13 wgUvi M i I iii N i, ii I iii
sinA
13. A = 30 nGj 1  sin2A ‰i gvb KZ? M 12 wgUvi N 5 wgUvi 40. msM†nxZ DcvîGK cwimsLÅvGbi Kx eGj?
2 1 1
26. x + x =2 nGj, x - x = KZ? K Drcv`K L mviwY
K L2 3
3 M KuvPvgvj N WvUv
K 0 L1
M 4 3 N6 3
M 2 N4
Dîigvjv

1 M 2 K 3 L 4 K 5 M 6 M 7 M 8 K 9 N 10 M 11 N 12 L 13 K 14 M 15 M 16 K 17 N 18 M 19 N 20 K
21 L 22 M 23 N 24 K 25 N 26 K 27 N 28 M 29 K 30 M 31 N 32 M 33 K 34 L 35 L 36 L 37 L 38 M 39 M 40 M
MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkc
œ ò I DËi : eûwbe©vPwb 29

233. AviwmwmAvB cvewjK Õ•zj AÅv´£ KGjR, iscyi welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1. cÉGZÅK AaÆe†Gîi wWwMÉ cwigvc KZ? wbGPi ˆKvbwU mwVK? wbGPi ˆKvbwU mwVK?
K 210 L 360 K i I ii
M 180 N 90 L i I iii K i I ii L i I iii
2. wbGPi Z^ÅàGjv jÞ Ki M ii I iii N i, ii I iii M ii I iii N i, ii I iii
i. cÉGZÅK mxgve«¬ mgZj ˆÞGòi wbw`ÆÓ¡ 14. a, b, c KÌwgK mgvbycvZx nGj wbGPi ˆKvbwU 30. 2.329 ‰i jGMi cƒYÆK KZ?
ˆÞòdj iGqGQ mwVK? K 1 L0 M2 N3

ii. `yBwU wòfzRGÞGòi ˆÞòdj mgvb nGjB K a:b=c:b L b2 = ac
M a:c=b:c N c2 = ab
wbGPi MvwYwZK evKÅ nGZ (31 I 32) bs cÉGk²i Dîi
wòfzR `yBwU meÆmg `vI:
a1x + b1y = c1
iii. `ywU wòfzR meÆmg nGj ZvG`i ˆÞòdj mgvb 15.
wbGPi ˆKvbwU mwVK?
}
a2x + b2y = c2 mgxKiY ˆRvGUi AmsLÅ mgvavb (x + y, 0) = (1, x  y) nGj
AvGQ ˆKvb kGZÆ @ 31. x ‰i gvb KZ?
K i I ii L i I iii a1 b1 a1 b1 c1 1 1
K  L =  K 0 L M– N1
M ii I iii N i, ii I iii a2 b2 a2 b2 c2 2 2
3. NƒYÆb ˆKvY 90 nGj NƒYÆb cÉwZmgZvi gvòv KZ? a1 b 1 a1 b1 c1 32. x I y «¼ viv MwVZ ˆmU ˆKvbwU?
M = N = =
a2 b2 a2 b2 c2 1 1
4.
K 4 L2 M1
wbGPi wPGò cÉwZmvgÅ ˆiLvi msLÅv KZ?
N3
16. ˆKvGbv avivi n Zg c` 2n – 4 nGj avivwUi 3wU K { } 2
0 L { } – 1
2
cG`i mgwÓ¡ KZ? 11
K 0 L2 M4 N –2
M { } 22
 N 

wbGPi ZG^Åi AvGjvGK 17 I 18 bs cÉGk²i Dîi `vI : wbGPi ZG^Åi AvGjvGK 33 I 34bs cÉGk²i Dîi `vI :
1 A
128 + 64 + 32 + .......... + ‰KwU àGYvîi aviv|
2
60
K 2 L4 M3 N5 17. avivwUGZ KqwU c` AvGQ? O
5. RÅvwgwZK Dccv`Å cÉgvGY mvaviYZ KqwU avc K 10 L 11 M 14 N 8
^vGK? 18. avivwUi 7wU cG`i mgwÓ¡ KZ? C
K 255.5 L 63.5 B
K 2 L4 M3 N5 M 254 N 93
wPGò AB = AC
cÉ`î Z^Å AejÁ¼Gb (6 I 7) bs cÉGk²i Dîi `vI : 33. BOC ‰i gvb KZ?
19. 5 + 8 + 11 + 14 + ........ avivwUi ˆKvb c` 302? K 60 L 120 M 100 N 80
1 1
x + = 2 3 ‰es a  = 2 2 K 110 L 120
34. AB = AC = 3 ˆm.wg. nGj BC = KZ?
x a M 100 N 101
6. x ‰i gvb KZ? 20. ˆKw±`Ëq cÉeYZvi cwigvc KqwU? K 2 ˆm.wg. L 4 ˆm.wg.
K 3+ 2 L 2 3–2 2 K 4 L 2 M 1 ˆm.wg. N 3 ˆm.wg.
M 2 3– 2 N 2– 3 M 3 wU N 7 wU 35. wPGò PA I PB `yBwU Õ·kÆK nGj
7. 1
a2 + 2 ‰i gvb KZ? 21. àYevPK bq ‰gb msLÅv mƒPK Z^ÅvejxGK Kx A
a
eGj?
K 9 L8 M 10 N7
K ˆkÉwY eÅeavb L Dcvî
8. wbGPi ˆKvbwU gƒj` msLÅv @ P O
M cwimi N ˆkÉwY gaÅgv
3 3 6 16
K
4
L M
2
N
13
22. mgevü wòfzGRi cÉwZmvgÅ ˆiLv KqwU?
49 B
K 3 wU L 2 wU
1 i. OA = OB
9. sin = nGj sec = KZ? M 4 wU N 1 wU
2 ii. OAP = OBP = 1 mgGKvY
2 1 3
23. 5% nvGi PKÌe†w«¬ gybvdvq 1000 UvKvi 2 eQGii iii. PA = PB
K 2 L
3
M
2
N
2 PKÌe†w«¬ gybvdv KZ? wbGPi ˆKvbwU mwVK?
K 1350 UvKv L 350 UvKv K i I ii L i I iii
10. wPòwU jÞ Ki : A
M 1102.50 UvKv N 102.50 UvKv M ii I iii N i, ii I iii
24.  
0.35  0.08 = KZ? 36. 3 3 ‰i 3 wfwîK log ‰i gvb KZ?
4 3 2 3 4
K 4 L7 M9 N6 K L M N
2 3 4 3
90 25. A = {x  ô : x2 > 15 ‰es x3 < 60} ‰i ZvwjKv
C

B c«¬wZGZ cÉKvk @ wbGPi MYmsLÅv wbGekb mviwY ˆ^GK 36 I 37 bs
4
K {4} L {} cÉGk²i Dîi `vI :
i.  = 60 ˆkÉwY 21-30 31-40 41-50 51-60
M {1, 2, 3} N {3, 4}
ii. AC = 4 2 7x + 2 MYmsLÅv 15 25 35 45
26. f (x) = nGj f(–1) ‰i gvb KZ?
iii. sin + cos = 2 5x – 3 37. gaÅK ˆkÉwY wbGPi ˆKvbwU?
wbGPi ˆKvbwU mwVK? 8 7 5 K 21-30 L 41-50
K L M N0
K i I ii L i I iii 5 5 8 M 51-60 N 31-40
M ii I iii N i, ii I iii 27. wbGPi ˆKvbwU AwewœQ®² PjGKi D`vniY? 38. cÉPziK wbYÆGqi ˆÞGò f2 ‰i gvb KZ?
11. mƒGhÆi D®²wZ ˆKvY 60 nGj ˆKvGbv ‰KwU 6 K cixÞvi bÁ¼i L RbmsLÅv K 0 L 45 M 10 N 35

wgUvi Š`GNÆÅi LuywUi Qvqvi Š`NÆÅ KZ wgUvi? M eqm N Zvcgvòv 39. x + y = 3 ‰es x – y = 2 nGj xy = KZ?
x x 1 1 1
K 3 3 L5 M6 3 N2 3 28. a
+ a = + b mgxKiGY x = KZ?
b
K
3
L5 M
4
N
5
12 K (a – b) L ab
12. nGj sin = KZ? 1 3 1
cos =
12
13
5 13 13
M a N b
40. x – = nGj 8 x  3 = KZ?
x 2
3
x ( )
K
13
L
13
M
12
N
5
29. i. a0 = 1
K 63 L
63
M8 N1
1 8
13. wbGPi Z^ÅàGjv jÞÅ Ki : n
i. sin2 = 1 + cos2 ii. a = an
an
ii. sec2 = 1 + tan2 iii. = an – m
am
iii. coses2 – cot2 = 1
1 M 2 L 3 K 4 M 5 L 6 K 7 M 8 L 9 K 10 M 11 N 12 L 13 M 14 L 15 N 16 K 17 N 18 M 19 M 20 M
Dîi

21 L 22 K 23 N 24 K 25 L 26 M 27 M 28 L 29 N 30 L 31 L 32 M 33 L 34 N 35 N 36 K 37 L 38 L 39 M 40 K
30 MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkœcò I DËi : eûwbe©vPwb

234. kvnxb ‰KvGWgx Õ•zj ‰´£ KGjR, ˆdbx welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1. 3 I 4 ‰i
gvGS cƒYÆ msLÅv KqwU? wbGPi Z^Å ˆ^GK 16 I 17bs cÉGk²i Dîi `vI: 28. e†Gîi eÅvmvaÆ 2 ˆm.wg. ‰es e†Gîi ˆKG±`Ê Drc®²
K 1 wU L 2 wU 19, 20, 12, 13, 14, 15, 16, 17 ˆKvY 180 nGj, e†îKjvi ˆÞòdj KZ e.
M 3 wU N 4 wU
16. cÉ`î ZG^Åi MvwYwZK Mo KZ? ˆm.wg.?
K 15.75 L 14.75
2. dvskGbi wPòi…cGK eGj M 14.5 N 13.5 K  L 2
K ˆjLwPò L AwRf ˆiLv 17. gaÅK ˆKvbwU? M 3 N 4
M eüfzR N e†î K 14.5 L 17.5 wbGPi wPòwU jÞÅ Ki ‰es 2931 bs cÉGk²i Dîi `vI :
M 15.5 N 14.75
3. P(A) ‰i Dcv`vb msLÅv 32 nGj, A ‰i Dcv`vb
18. cÉPziK wbGPi ˆKvbwU?
msLÅv KZ? K 14 L 15
K 4 L5 O
M 19 ˆbB N cÉPziK
M 6 N7
19. ‰KwU eGMÆi KGYÆi Š`NÆÅI cwimxgvi AbycvZ
4. e†îÕ© PZzfzÆGRi wecixZ ˆKvYàGjvi mgwÓ¡ KZ? KZ?
K 45 L 90 A C B
M 180 N 360 K 1:2 2 L 1: 2
5. hw` (x) ‰i gvòv abvñK ‰es a  0 nq ZGe M 1:4 N 2 2:1 29. OC  AB, OA = 4 ˆm.wg., OC = 3 ˆm.wg. nGj,
(x) ˆK ax + b «¼viv fvM KiGj fvMGkl KZ 20. 5 Rb QvGòi BsGiRxi bÁ¼i nGjv t 45, 40, 48, AB = KZ?
nGe? 42, 46 | cÉ`î DcvîàGjv K 25 ˆm.wg. L 7 ˆm.wg.
i. wewœQ®² PjGKi D`vniY M 7 ˆm.wg. N 2 7 ˆm.wg.
1 a
K  ()a
L () b ii. ˆKvGbv cwimsLÅvb bq
iii. gaÅK nGe 48
30. OB = 4 nGj e†Gîi eÅvm KZ?
b a K 8 ˆm.wg. L 5 ˆm.wg.
M   ( )
a
N ( )
b wbGPi ˆKvbwU mwVK?
K i L ii I iii
M 25 ˆm.wg. N 16 ˆm.wg.
6. x2  2x + 1 = 0 nGj, 31. OAB = 50 nGj AOB ‰i gvb KZ?
1 M i I iii N i, ii I iii K 80 L 90
i. x + = 2
x 21. ABC I DEF m`†k ‰es Abyi…c evü«¼Gqi M 100 N 120
1 AbycvZ 3 : 2 nGj, ZvG`i ˆÞòdGji AbycvZ 32. ˆKvb wòfzGRi KqwU ewne†Æî AuvKv hvq?
ii. x2 + 2 = 4
x KZ nGe? K 1wU L 2wU
1 K 5t1 L 3t2 M 8t2 N 9t4 M 3wU N 4wU
iii. x  =0
x wbGPi Z^Å ˆ^GK 22-24 bs cÉGk²i Dîi `vI : 33. mgevü wòfzGRi evü I gaÅgvi Š`GNÆÅi AbycvZ
A D
wbGPi ˆKvbwU mwVK? KZ?
K i I ii L i I iii K 3t2 L 2t 3
M ii I iii N i, ii I iii M 2t5 N 5t2
7. 144 ‰i 2 3 wfwîK jM KZ? 34. `yBwU wòfzGRi gGaÅ wbGPi ˆKvbwU mgvb nIqv
K 2 L2 3 mGZ½I wòfzR«¼q meÆmg nq bv?
M 3 N4 K wZb ˆKvY
B E C
8. Z^ÅàGjv jÞÅ Ki : L wZb evü
ABCD iÁ¼mwUi ˆÞòdj 192 e.ˆm.wg. ‰es ‰KwU
i. mvaviY jGMi AskK abvñK msLÅv M `yB evü I A¯¦fƒÆÚ ˆKvY
KGYÆi Š`NÆÅ BD = 24 ˆm.wg.| AE iÁ¼mwUi DœPZv|
ii. AskK gƒjZ ‰KwU Agƒj` msLÅv N `yB ˆKvY ‰K evü
22. AC KGYÆi Š`NÆÅ KZ?
iii. log5 + log4  log2 = log1 K 96 ˆm.wg. L 48 ˆm.wg.
wbGPi Z^Å ˆ^GK 35 I 36bs cÉGk²i Dîi `vI :
wbGPi ˆKvbwU mwVK? M 20 ˆm.wg. N 16 ˆm.wg.
A
K i I ii L i I iii 23. BC evüi Š`NÆÅ KZ?
M ii I iii N i, ii I iii K 14.42 ˆm.wg. L 16 ˆm.wg. 60
9. kZKiv evwlÆK 7 UvKv nvi mij gybvdvq 650 M 20 ˆm.wg. N 48 ˆm.wg.
UvKvi KZ eQGii gybvdv 273 UvKv nGe? 24. AE ‰i Š`NÆÅ KZ? 60
K 1 L2 K 14.4 ˆm.wg. L 13.31 ˆm.wg. D C
20 wg. B
M 3 N6 M 12.2 ˆm.wg. N 11.3 ˆm.wg.
p q 2 2
wbGPi Z^Å ˆ^GK 25, 26 I 27bs cÉGk²i Dîi `vI : DcGii wPGò AB ‰KwU MvQ, BD = 20 wg.
10. + =2
q p
nGj, pq2 + qp2 = ? 35. MvQwUi DœPZv, AB = KZ wg.?
A
K 0 L2 K 34.641 wg. L 44.641 wg.
M 4 N6 M 33.641 wg. N 34.146 wg.
11. `yB msLÅvi AbycvZ 2 t 3 ‰es M.mv.à 4 nGj 36. BC = KZ wgUvi?
e†nîg msLÅv KZ? K 30 wg. L 40 wg.
K 8 L 10
M 50 wg. N 60 wg.
M 12 N 14 B
12. Avwgb I mvwKGei eqGmi AbycvZ 5 t 3|
D C 37. sin + cos = 1 nGj,  ‰i gvb KZ?
wPGò, ABC mgevü wòfzR| AD  BC ‰es K 30 L 45
AvwgGbi eqm 20 eQi nGj KZ eQi ci ZvG`i AB = 2 ‰KK| M 60 N 90
eqGmi AbycvZ 7 t 5 nGe? 25. BD = KZ? cos A 1
K 5 L6 38. sin A = nGj, A ‰i gvb KZ?
K 1 ‰KK L 2 ‰KK 3
M 8 N 10
M 2 ‰KK N 4 ‰KK K 30 L 45
13. wbGPi ˆKvb we±`ywU Y AÞGiLvi Dci AewÕ©Z? 26. wòfzRwUi DœPZv KZ? M 60 N 90
K (1, 4) L (4, 0)
M (0, 4) N (4, 1) K
4
‰KK L 3 ‰KK 39. sin A = 23 sec A = 2 nGj, tanA ‰i gvb KZ?
wbGPi Z^Å ˆ^GK 14 I 15bs cÉGk²i Dîi `vI : 3
2 K 3 3 L6 2
log3 + log9 + log27 + ..... M ‰KK N 2 3 ‰KK
M 9 2 N 3
14. avivwUi mµ¦g c` KZ? 3
K log 729 L log 2187 27. wòfzRwUi ˆÞòdj KZ? 40. e†Gîi ˆh ˆKvGbv `yBwU we±`y ˆhvM KiGj ˆhvRK
M log 243 N log 81 3 3 ˆiLvwUGK eGj
K e. ‰KK L e. ‰KK K RÅv L eÅvmvaÆ
15. avivwUi 1g 10wU cG`i mgwÓ¡ KZ? 4 2
K 110 log3 L 10 log3 M
2
e. ‰KK N 3 e. ‰KK M eÅvm N cwiwa
M 44 log 3 N 55 log3 3
1 L 2 K 3 L 4 M 5 M 6 L 7 N 8 K 9 N 10 L 11 M 12 M 13 M 14 L 15 N 16 K 17 M 18 N 19 K 20 K
Dîi

21 N 22 N 23 K 24 L 25 K 26 L 27 N 28 L 29 N 30 K 31 K 32 M 33 L 34 K 35 K 36 N 37 N 38 M 39 N 40 K
MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkc
œ ò I DËi : eûwbe©vPwb 31

235. ˆMvwe±`cyi miKvwi DœP we`Åvjq, nweMé welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1. P, Q I R we±`y wZbwU ‰KB mijGiLvq AewÕ©Z 14. AvGqkvi wbKU 1wU Kgjv ^vKGj `yBRGbi ˆgvU iii. cÉPziK
bv nGj KqwU mgZj cvIqv hvGe? KqwU Kgjv AvGQ? wbGPi ˆKvbwU mwVK?
K 0 L1 K 1wU L 2wU K i I ii L i I iii
M 3 N AmsLÅ M 3wU N 4wU M ii I iii N i, ii I iii
2. wòfzGRi ‰KwU ˆKvY 95 nGj ZvGK Kx wòfzR eGj? 15. 3 + 10 + 17 + 24 + ......... avivi AÓ¡g c` KZ nGe? wbGPi ZG^Åi wfwîGZ 29 I 30 bs cÉGk²i Dîi `vI :
K mƒßGKvYx L Õ©ƒjGKvYx K 35 L 42
wfKvi‚bwbmv bƒb Õ•zj ‰´£ KGjGRi ‰KwU bZzb feb
M mgevü N mgGKvYx M 52 N 62
wbgÆvGYi KvGR wbGqvwRZ 10 Rb kÉwgGKi Š`wbK gRywi
3. wòfzGRi `yBwU ˆKvY 60 I 75 nGj Aci 16. x + y + z + .......... àGYvîi avivi mvaviY
ˆKvGYi gvb KZ wWwMÉ? AbycvZ 3 nGj wbGPi ˆKvbwU mwVK? (UvKvq) ˆ`Iqv nGjv :
K 90 L 60 M 45 N 30 K x = 3y L y = 3x 150, 130, 145, 170, 200, 180, 190, 140, 165, 175|
4. ‰KwU iÁ¼Gmi cwimxgv a ‰es ‰KwU ˆKvY x M x=
3
N xy =
1 29. DcvGîi cwimi KZ?
y 3 K 50 L 71
ˆ`Iqv ^vKGj wbGPi ˆKvb kGZÆ ‰wU AuvKv mÁ¿e?
K x = 0 L x = 120
17. ABC mgevü wòfzGRi BC evüi gaÅwe±`y D nGj M 85 N 90
M x = 180 N x = 200 BAD = KZ wWwMÉ? 30. 10 ˆkÉwY eÅeavb wbGq ˆkÉwYmsLÅv KZ nGe?
K 20 L 30 K 8 L6
wbGPi wPGòi AvGjvGK 5 I 6 bs cÉGk²i Dîi `vI :
M 40 N 60 M 4 N2
B
C 18. ‰KRb gvbyGli wPGò KqwU cÉwZmvgÅ ˆiLv we`Ågvb? 31. 3 I 4 ‰i gvGS Agƒj` msLÅv ˆKvbwU?
4
K 1wU L 2wU K 4 L 2
3
O M 3wU N 4wU
M 5 N 18
A
19. ‰KwU mylg cçfzGRi NƒYÆbGK±`Ê ˆKvbwU? 32. wbGPi ˆKvbàGjv m`†k Ave†Z `kwgK msLÅv?
K fiGK±`Ê L cÉ^g evüi gaÅwe±`y .. . . ..
O ˆK±`Ê wewkÓ¡ e†Gî AB eÅvm| M cÉ^g evüi kxlÆwe±`y N ˆkl evüi kxlÆwe±`y K 6.435, 2.893 L 12.45, 6.32
5. O ˆK±`ÊwewkÓ¡ e†Gî ACB Kxi…c ˆKvY? 20. AvqZGÞGòi Š`NÆÅ cÉGÕ©i w«¼àY ‰es cwimxgv .. . . . .
M 0.345, 7.457 N 9.453, 125.897
K e†îÕ© L AaÆe†îÕ© 60 wgUvi nGj cÉÕ© KZ?
M ˆK±`ÊÕ© N cÉe†«¬
33. P = {x  ô : 2 < x  6} ‰es Q = {x  ô : x
K 5 L 10
6. AOB = KZ wWMÉx? M 12 N 20 ˆRvo msLÅv ‰es x  8} nGj P  Q = KZ?
K 0 L 45 M 90 N 180 21. ˆKvGbv AvqZGÞGòi Š`NÆÅ ‰i cÉGÕ©i w«¼àY| K {2, 4, 6, 8} L {4, 8}
M {4, 6} N {2, 4, 6}
5
7. sec + tan = 2 nGj sec  tan ‰i gvb Š`NÆ Å 8 ˆm.wg. nGj ˆÞòdj KZ eMÆ ˆm.wg.?
K 128 L 48 34. f(x) = x2 nGj f(x) +1
= KZ?
wbGPi ˆKvbwU? M 32 N 16 2
f(x)  1
x + 1 (x + 1)2
K
5
L
2 22. ˆKvGbv eMÆGÞGòi ˆÞòdj 25 eMÆ ˆm.wg. nGj K 2
x 1
L
(x  1) 2
2 5
5 3
DÚ eGMÆ i ˆh ˆKvb KGYÆ i Dci AwâZ x 2
+ x x2 + 2x + 1
M N eMÆ G ÞGòi ˆÞòdj KZ eMÆ ˆm.wg.? M 2 N 2
3 5 x  x x  2x + 1
K 25 L 50 cÉ`î ZG^Åi wfwîGZ wbGPi 35 I 36 bs cÉGk²i Dîi
8. sinA = tanA nGj A = ? M 100 N 625
K 0 L 30 `vI :
M 45 N 90
23. ‰KwU wòfzGRi `yBwU evüi Š`NÆÅ h^vKÌGg 9
ˆm.wg. I 8 ˆm.wg. ‰es A¯¦fzÆÚ ˆKvY 30 nGj, f(x) = x  x + x  x + x  x ‰KwU x PjGKi eüc`x|
6 5 4 3 2

9. ˆKvGbv mgGKvYx wòfzGR 60 ˆKvY AâGbi 35. wbGPi ˆKvbwU eüc`xi ‰KwU Drcv`K?
ˆÞGò f„wg jGÁ¼i wKi…c nGe? ˆÞòdj KZ?
K x+1 Lx
K eo L mgvb K 9 eMÆ ˆm.wg. L 18 eMÆ ˆm.wg. M x2 + 1 N x2  1
M GQvU N w«¼àY M 36 eMÆ ˆm.wg. N 72 eMÆ ˆm.wg.
36. (x  1) «¼viv f(x) ˆK fvM KiGj fvMGkGli gvb
10. ‰KwU cZvKvi LyuwU ˆfGã, fvãv Ask f„wgi 24. ‰KwU eMÆGÞGòi ˆÞòdj 169 eMÆ ˆm.wg. nGj KZ?
mvG^ 30 ˆKvY Drc®² KGi| LyuwUi fvãv AsGki ‰i cwimxgv KZ ˆm.wg.? K 1 L x+1
K 13 L 26
Š`NÆÅ 16 wgUvi nGj, `´£vqgvb AsGki Š`NÆÅ KZ M 52 N 65 M 0 N 1
wgUvi? 25. ‰KwU e†Gîi eÅvmvaÆ 14 ˆm.wg. ‰es ‰KwU PvGci 37. 5 5 ‰i 5 wfwîK jM KZ?
K 8 L8 3 Š`NÆÅ 32 ˆm.wg. nGj, e†îKjvi ˆÞòdj KZ eMÆ 3
K L 5
M 16 N 16 3 2
2 2
ˆm.wg.? 5
11. a, b, c KÌwgK mgvbycvZx nGj, ab2 ++ bc2 = KZ? K 224 L 333 M
2
N5
M 444 N 555
a 38. 9.813  10 ‰i mvaviY `kwgK i…c KZ?
3
K ac L
c 26. wmwj´£vGii eÅvmvaÆ 2 ˆm.wg. nGj K 9813 L 981.3
c i. `yB cÉvG¯¦i ˆÞòdj 8 eMÆ ˆm.wg. M 0.9813 N 0.009813
M N a+c
a ii. ‰i f„wgi ˆÞòdj 8 eMÆ ˆm.wg. 2
12. `ywU msLÅvi AbycvZ 3 : 4 ‰es ZvG`i j.mv.à iii. eÅvm 4 ˆm.wg.
39. ‰KwU msLÅv hv x ‰i 3 àY| msLÅvwU wbGPi
180 nGj, cÉ^g msLÅvwU KZ? wbGPi ˆKvbwU mwVK? ˆKvbwU?
K 15 L 20 K i I ii L i I iii 2x 3x
M 45 N 60 K L
M ii I iii N i, ii I iii 3 2
wbGPi ZG^Åi wfwîGZ 13 I 14 bs cÉGk²i Dîi `vI : 27. ˆKvbwU wbYÆq KiGZ KÌgGhvwRZ MYmsLÅvi M x+
2
N
3+2
AwbK I AvGqkvi wbKU h^vKÌGg x I y msLÅK Kgjv cÉGqvRb nq?
3 x
AvGQ| AwbGKi AvGqkv AGcÞv 2wU Kgjv ˆewk AvGQ| 40. x2  2x + 1 = 0 mgxKiY ax2 + bx + c = 0
K AvqZGjL L ˆkÉwYmsLÅv
13. DcGii eYÆbv mvGcGÞ wbGPi ˆKvbwU mwVK mgxKiGYi mvG^ Zzjbv KiGj a ‰i gvb Kx?
M AwRfGiLv N cwimi
mgxKiY? K 2 L0
28. ˆKw±`Êq cÉeYZvi cwigvc M 1 N2
K x+2=y L x=y+2
i. MvwYwZK Mo
M x+y=2 N x+y+2=0
ii. gaÅK
Dîigvjv

1 L 2 L 3 M 4 L 5 L 6 N 7 L 8 K 9 M 10 K 11 L 12 M 13 L 14 N 15 M 16 L 17 L 18 K 19 K 20 L

21 M 22 L 23 L 24 M 25 K 26 L 27 M 28 N 29 L 30 K 31 M 32 N 33 M 34 K 35 L 36 M 37 K 38 N 39 K 40 M
32 MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkœcò I DËi : eûwbe©vPwb

236. AvjgWvãv cvBjU gvaÅwgK evwjKv we`Åvjq, PzqvWvãv welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1. wbGPi ˆKvbwU Agƒj` msLÅv 15. 27. sec = KZ?
A
36 5 1 2
K L K L
2 2 2 3
120 O
M 25 N 49 3
M N 3
2. wbGPi evKÅàGjv jÞÅ Ki : 2
i. kƒbÅ ‰KwU Õ¼vfvweK msLÅv
C D 28. 6 ˆm.wg. jÁ¼v ‰KwU gB ˆ`IqvGji Qv` eivei
ii. 8 ‰wU Agƒj` msLÅv wPGòi COD = KZ? ˆVm w`Gq f„wgi mvG^ 45 ˆKvY Drc®² KGi|
iii. mKj Õ¼vfvweK msLÅv gƒj` msLÅv K 60 L 120 ˆ`IqvjwUi DœPZv KZ?
wbGPi ˆKvbwU mwVK? M 85 N 90 K 3 2 L2 3
K i I ii L i I iii 16. e†Gîi ˆK±`Ê ‰es eÅvGmi KqwU mvaviY we±`y M 4 3 N6 2
M ii I iii N i, ii I iii ^vGK? 29. D A
1 1 K 4wU L 3wU
3. p
p
= 3 nGj p2 + 2 ‰i gvb KZ?
p M 2wU N 1wU
60

K 11 L5 A C 90
M 7 N 13 B C
4. x3 + 6x2 + 11x + 6 ivwkGZ x ‰i gvb KZ nGj, A we±`yGZ B we±`yi AebwZ ˆKvGYi cwigvY KZ?
E F
ivwkwUi gvb kƒbÅ nGe? O K 90 L 60
K 6 L1
M 45 N 30
M 2 N 2 B D
30. e†Gîi ˆKG±`Ê ˆgvU Drc®² ˆKvY = KZ?
5. log5x = 3 nGj x ‰i gvb KZ?
K 180 L 90
K 100 L 15 DcGii wPò AbymvGi 17 ˆ^GK 20 bs cÉGk²i Dîi `vI :
M 360 N 280
M 125 N 10 17. OEA ‰i gvb KZ?
6. 0.000435 ‰i jGMi cƒYÆK KZ? K 60
31. ‰KwU e†Gîi cwiwa 10 ˆm.wg. nGj, e†Gîi
L 90
 M 120 N 45 eÅvmvaÆ KZ ˆm.wg.?
K 4 L4
18. OE = 3 ˆm.wg.
‰es K 3 ˆm.wg.
AE = 4 ˆm.wg.
nGj OC ‰i L 6 ˆm.wg.
M 3

N3 gvb KZ? M 4 ˆm.wg. N 5 ˆm.wg.
7. x2  2x + 1 = 0 nGj K 5 ˆm.wg. L 6 ˆm.wg. 32. wbGPi Z^ÅàGjv jÞÅ Ki
1 M 7 ˆm.wg. N 9 ˆm.wg. i. eMÆGÞGòi KGYÆi Š`NÆÅ = 4a ‰KK
i. x + = 2
x
19. AE = 4 ˆm.wg. nGj CD ‰i Dci AswKZ eGMÆi ii. AvqZGÞGòi KYÆ d = a2 + b2 ‰KK
1
ii. x2 + 2 = 4 ˆÞòdj KZ? iii. NbGKi KGYÆi Š`NÆÅ = 3a ‰KK
x
1 K 16 ˆm.wg. L 36 ˆm.wg. wbGPi ˆKvbwU mwVK?
iii. x  =0 K i I ii L i I iii
x M 20 ˆm.wg. N 64 ˆm.wg.
wbGPi ˆKvbwU mwVK? 20. 60 ˆKvGYi ŠiwLK mÁ·ƒiK ˆKvY KZ wWMÉx? M ii I iii N i, ii I iii
K i I ii L i I iii K 90 L 30 33. e†Gîi cwiwa I eÅvmvGaÆi AbycvZ KZ?
M ii I iii N i, ii I iii M 180 N 120 K 2 : 1 L 1:
64 + 32 + 16 + 8 + ...... 21. wòfzGRi wZbwU ˆKvY ˆ`Iqv ^vKGj wewf®² M 22 : 8 N :1
DcGii ZG^Åi wfwîGZ 8 I 9 bs cÉGk²i Dîi `vI : ˆÞòdGji KZàGjv wòfzR AuvKv mÁ¿e? 34. wbGPi ˆKvbwU ˆKw±`Êq cÉeYZv bq?
8. avivwUi mvaviY AbycvZ KZ nGe? K 3wU L 4wU K Mo L eÅeavb
K 2 L1 M AmsLÅ N 5wU M cÉPziK N gaÅK

M
1
N0 22. ˆMvjGKi gvòv KqwU? 35. ˆKvGbv ˆkÉwYi eÅvwµ¦ 18-22 ‰es avc wePzÅwZ 0
2 K 3 L2 nGj AvbygvwbK Mo KZ?
9. avivwUi AÓ¡g c` KZ? M 1 N0 K 19 L 18
K 4 L2 23. wbGPi ˆKvb mÁ·KÆwU mwVK? M 22 N 20
M 1 N
1
K cosec2 + cot2 = 1 L tan2 = sec2  1 wbGPi ZG^Åi AvGjvGK 36-38 bs cÉGk²i Dîi `vI :
2
M sec2 + tan2 = 1 N cot2 = cosec2 + 1 ˆkÉwY 21-30 31-40 41-50 51-60
10. ‰KwU eGMÆi evüi Š`NÆÅ w«¼àY nGj Dnvi ˆÞòdj
24. sin cosec2q  1 = KZ? MYmsLÅv 4 12 8 10
KZ àY e†w«¬ cvGe?
K 2 àY L 4 àY
K 1 L sin 36. DóxcGKi gaÅK ˆkÉwYi Ea»Æmxgv KZ?
M cos N sin tan K 40 L 41
M 8 àY N 6 àY M 31 N 50
wbGPi ZG^Åi wfwîGZ 25-27 bs cÉGk²i Dîi `vI :
11. ax  by = ab = bx  ay cÉ`î mgxKiY«¼Gqi 37. DcGivÚ ZG^Åi cÉPziK KZ (cÉvq)?
A
mgvavbt K 34.33 L 41.83
2a M 37.67 N 47.67
K (ab

ab
a+b a+b ) L (ab,  ab) a
38. DcGivÚ ZG^Åi KÌgGhvwRZ MYmsLÅvi Ea»Æmxgv
M a, b N (a + b, a  b) KZ?
12. ‰KwU eGMÆi KqwU cÉwZmvgÅ ˆiLv AvGQ? K 20 L 12
O B
K 3wU L 2wU M 5wU N 4wU 3a M 34 N 30
13. A = {2, 3, 4} ‰es B = {2, 3, a} nGj 25. sin = KZ? 39. x2  x  12 = 0 mgxKiGYi gƒj«¼q wbGPi ˆKvbwU?
A  B = KZ? 3 K 3 L 4, 3
K L2 M 3, 4 N 3, 4
K {4, a} L {2, 3, 4, a} 2
M {2, 3} N {2, 4}
M
2
N
1 40. log2 + log4 + log8 + ...... avivwUi 7 Zg c`
14. ÷ay cwimxgv Rvbv ^vKGj wbGPi ˆKvbwU AuvKv 3 2 KZ?
mÁ¿e? 26. tan = KZ? K log 32 L log 64
M log 256 N log 128
K AvqZb L mvgv¯¦wiK 2 1 3
K L M 3 N
M eMÆ N UÇvwcwRqvg 3 3 2

1 K 2 M 3 K 4 N 5 M 6 L 7 L 8 M 9 N 10 L 11 K 12 N 13 M 14 M 15 K 16 N 17 L 18 K 19 N 20 N
Dîi

21 M 22 K 23 L 24 M 25 N 26 L 27 L 28 K 29 N 30 M 31 N 32 M 33 K 34 L 35 N 36 N 37 M 38 M 39 L 40 N
MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkc
œ ò I DËi : eûwbe©vPwb 33

237. ewikvj we.‰g.Õ•zj, ewikvj welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1. wbGPi ˆKvbwU Agƒj` msLÅv? wbGPi ˆKvbwU mwVK? cÉ`î ZG^Åi AvGjvGK 27-29 bs cÉGk²i Dîi
8 35 K i I ii L i I iii `vI:
K L D C
2 5 M ii I iii N i, ii I iii

M
50
N
16 15. cos = 1 nGj  ‰i gvb KZ?
2 F
2 4 K 0 L 45
2. Ave†Z `kwgK f™²vsk msLÅv ˆKvb aiGbi msLÅv? M 60 N 90 E
K gƒj` msLÅv L Agƒj` msLÅv 16. wbGPi Z^ÅàGjv jÞÅ Ki: A B
M cƒYÆ msLÅv N Õ¼vfvweK msLÅv i. tan90 ‰i gvb 0 wPGò AE = 2AB, AE = 10 ˆm.wg., DE = 5 5
3. A  B = A nGj wbGPi ˆKvbwU mwVK? cosA ˆm.wg.
ii. cotA =
K A, B ‰i DcGmU L B, A ‰i DcGmU sinA 27. AD = KZ ˆm.wg.?
M A ‰es B ‰i m`mÅ msLÅv mgvb iii. 1  cos  1 K 5 ˆm.wg. L 8 ˆm.wg.
N B AekÅB A ‰i cÉK‡Z DcGmU wbGPi ˆKvbwU mwVK? M 10 ˆm.wg. N 12 ˆm.wg.
4. ˆh ˆKvb ˆmU A ‰i RbÅ K i I ii L i I iii 28. ABCD ˆÞòwUi ˆÞòdj KZ eMÆ ˆm.wg.?
i. A  Ac =  M ii I iii N i, ii I iii K 20 eMÆ ˆm.wg. L 25 eMÆ ˆm.wg.
ii. B =  nGj A  B = A 17. f„wg I jGÁ¼i gGaÅ ˆKvb mÁ·GKÆi RbÅ f„wg I M 30 eMÆ ˆm.wg. N 35 eMÆ ˆm.wg.
iii. B =  nGj A  B = A AwZfzGRi gaÅeZÆx ˆKvY 45 nq| 29. ADE ˆÞòwUi ˆÞòdj mgvb
wbGPi ˆKvbwU mwVK? K f„wg = jÁ¼ L f„wg  jÁ¼ i. ABCD ˆÞòwUi ˆÞòdj
K i I ii L ii I iii M f„wg  jÁ¼ N f„wg  jÁ¼ ii. 2(ABC ‰i ˆÞòdj)
M i I iii N i, ii I iii 18. A we±`yGZ B ‰i AebwZ ˆKvY 60 | B we±`yGZ A iii. 3(ADF ‰i ˆÞòdj)
5. x3  y3 = 9, x2 + xy + y2 = 3 nGj x  y = ? ‰i D®²wZ ˆKvGYi cwigvY KZ? wbGPi ˆKvbwU mwVK?
K 1 L3 K 90 L 60 K i I ii L i I iii
M 6 N 27 M 45 N 30 M ii I iii N i, ii I iii
6. ‰KwU Kjg 11 UvKvq weKÌq KiGj 10% jvf 19. cçfzGRi ˆKvYàGjvi AbycvZ 2 t 3 t 4 t 5 t 6 30. e†îÕ© eGMÆi ˆÞòdj 8 eMÆ ˆm.wg. nGj e†îwUi
nq| KjgwUi KÌqgƒjÅ KZ? e†nîg ˆKvYwUi gvb KZ? ˆÞòdj KZ eMÆ ˆm.wg.?
K 8 UvKv L 9 UvKv K 180 L 162 K 12.57 cÉvq L 15.27 cÉvq
M 10 UvKv N 12 UvKv M 150 N 75 M 17.25 cÉvq N 25.17 cÉvq
7. wbGPi ˆKvbwUi RbÅ b = n a mwVK? 20. 1  1  x =  1 nGj x ‰i gvb KZ? wbGPi wPGòi AvGjvGK 31-33 bs cÉGk²i Dîi `vI:
1+ 1x 2 A
K b  0. bn = a L b  0. bn = a
M a  0. an = b N a  0. an = b K 2 L 4
M 6 N 8
8. wbGPi Z^ÅàGjv jÞÅ Ki 5 cm 5 cm
21. wbGPi ˆKvbwUi ˆjLwPò ‰KwU mijGiLv?
i. e ‰KwU Agƒj` msLÅv K x2 + y2 = 1 L 2x + y = 8
ii. logex ˆK lnx AvKvGi cÉKvk Kiv hvq M y2 = 4x N x2  4y
B
4 cm
C
iii. logex ˆK ˆbwcqvi jMvwi`g ejv nq| 22. ax  cy = 0, cx  ay = c2  a2 mgxKiY ˆRvUwU
wbGPi ˆKvbwU mwVK? 31. ABC ‰i ˆÞòdj KZ?
i. mgém K 2 21 eMÆ ˆm.wg. L 21 eMÆ ˆm.wg.
K i I ii L i I iii ii. wbfÆikxj M 2 12 eMÆ ˆm.wg. N 12 eMÆ ˆm.wg.
M ii I iii N i, ii I iii iii. AbbÅ mgvavb wewkÓ¡
9. ‰KwU mgxKiGYi ŠewkÓ¡Å wbGPi ˆKvbwU? 32. ABC ‰i DœPZv KZ ˆm.wg.?
wbGPi ˆKvbwU mwVK? K 7 ˆm.wg. L 13 ˆm.wg.
K mKj mgxKiYB AGf` K i I ii L i I iii M 17 ˆm.wg. N 21 ˆm.wg.
L mgxKiGYi Dfq cGÞ eüc`xi gvòv mgvb ^vGK
M ii I iii N i, ii I iii 33. A ‰i gvb KZ?
M mgxKiY PjGKi mKj gvGbi RbÅ mZÅ nq 23. wbGPi Z^ÅàGjv jÞÅ Ki: K cÉvq 30.51 L cÉvq 21.51
N mgxKiGYi `yB cGÞ `yBwU eüc`x ^vKGZ cvGi
i.  5n  1  ‰KwU mgv¯¦i aviv M cÉvq 20.51 N cÉvq 47.16
10. y(y  5) =  6 mgxKiGYi KqwU gƒj AvGQ? ii.  6n  ‰KwU àGYvîi aviv
K 1wU L 2wU
34. AvbygvwbK Mo 17 nGj 15-19 ˆkÉwYi avc wePzÅwZ
iii.  4n1  avivwUi cÉ^g n msLÅK KZ?
M 3wU N bvB
11. RÅvwgwZK `ƒiGZ½i aviYvi Õ¼xKvhÆ f ˆK wK ejv cG`i mgwÓ¡ 1 (4n  1) K 3 L 2
3 M 1 N0
nq? wbGPi ˆKvbwU mwVK? 35. Dcvî mgƒGni cÉwZwbwaZ½ KGi ˆKvbwU?
K `ƒiZ½ Õ¼xKvhÆ L i‚jvi Õ¼xKvhÆ K i I ii L i I iii K ˆkÉwY L MYmsLÅv
M i‚jvi Õ©vcb Õ¼xKvhÆ N AvcZb Õ¼xKvhÆ M ii I iii N i, ii I iii M ˆKw±`Êq gvb N cwimi
12. mvgv¯¦wiK AâGbi RbÅ cÉGhvRÅ 24. x + 3x + 5x +  + 149x avivwUi KZwU c` 36. wbGPi ˆKvbwU x3  3xy2 + 2y3 ‰i ‰KwU Drcv`K?
i. `yBwU msj™² evü ‰es ZvG`i A¯¦fzÆÚ ‰KwU ˆKvY iGqGQ? K x+y L x + 2y
ii. `yBwU KYÆ I ZvG`i A¯¦fzÆÚ ‰KwU ˆKvY K 75 L 77 M x  2y N x  3y
iii. `yBwU KYÆ I ‰KwU evü M 78 N 79
37. (x) = 3x + 1 nGj f 1 ‰i gvb KZ?
wbGPi ˆKvbwU mwVK? 3x  1 2 ()
K i I ii L i I iii 25. wPòwUi NƒYÆb ˆKvY KZ wWwMÉ? K
5
L2
2
M ii I iii N i, ii I iii M 5 N3
13. e†Gîi ˆKG±`Êi wbKUZi ˆKvbwU? K 30 L 45
38. cos  sin = 2 sin nGj cos + sin = KZ?
K Þz`ËZi RÅv L e†nîi RÅv M 60 N 90
K 2 tan L 2 sec
M Õ·kÆK N Õ·kÆ we±`y 26. cÉwZmvgÅ ˆiLv AvGQ
M 2 cot N 2cos
14. i. ˆKvb PZzfzÆGRi `yBwU wecixZ ˆKvY cƒiK i. gyGLvGmi wPGò
39. 3x = 9 3 nGj x ‰i gvb KZ?
nGj kxlÆwe±`y PviwU mge†î ii. gmwRG`i wPGò
K 2.5 L 3.52
ii. mgGiL bq ‰gb wZbwU we±`y w`Gq ‰KwU gvò iii. wmwjsdÅvGbi wPGò M 4.52 N 5.52
e†î AuvKv hvq wbGPi ˆKvbwU mwVK? 40. y = 1 nGj y + y + 1 ‰i gvb KZ?
3 2

iii. `yBwU e†Gîi e†nîg RÅv ‰i Š`NÆÅ mgvb nGj K i I ii L i I iii K 0 L1

ZvG`i eÅvmvaÆ mgvb| M ii I iii N i, ii I iii M 1 N3

1 L 2 K 3 L 4 K 5 L 6 M 7 K 8 N 9 N 10 L 11 K 12 N 13 L 14 M 15 M 16 M 17 K 18 L 19 L 20 N
Dîi

21 L 22 L 23 L 24 K 25 N 26 N 27 K 28 L 29 K 30 K 31 K 32 N 33 N 34 N 35 L 36 L 37 M 38 N 39 K 40 N
34 MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkœcò I DËi : eûwbe©vPwb

238. nvwjgv LvZzb evwjKv gvaÅwgK we`Åvjq, ewikvj welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1. wbGPi ˆKvbwU mmxg `kwgK f™²vsk? wbGPi ZG^Åi AvGjvGK 17 I 18 bs cÉGk²i Dîi `vI : 28. a + b + c + d + ... mgv¯¦i avivi A¯¦fzÆÚ nGj
K 2.11 L 5.234 wbGPi ˆKvbwU mwVK?
M 2.1914... N 1.243 K a=b L db=a
2. 9.6372... msLÅvwUi `yB `kwgK Õ©vb chƯ¦ Avm®² O M a+b=d N ba=dc
A B
gvb wbGPi ˆKvbwU? 29. 3  6 + 12  24 + ....avivwUi mvaviY AbycvZ
K 9.63 L 9.64 wbGPi ˆKvbwU?
M .96 N 96.37 C K 3 L2
3. ˆfbwPGò DcGmU ˆevSvGZ wK wPn× eÅen†Z nq? M 2 N 3
17. O ˆK±`ÊwewkÓ¡ e†Gî AB eÅvm ‰es ACB ‰KwU
K e†îvKvi ev wòfzRvKvi ˆÞò 30. A
AaÆe†îÕ© ˆKvY| ACB ‰i gvb KZ?
L 'U' A^ev 'C' wPn× K 45 L 60
M 'C' A^ev 'C' wPn× M 90 N 120 h
N AvqZvKvi ev eMÆvKvi ˆÞGò 18. O ˆK±`ÊwewkÓ¡ e†Gî AB eÅvm ‰es ACB ‰KwU
4. A  X ‰i A^Æ Kx? e†îÕ© ˆKvY nGj, AOB ‰i gvb KZ? B
D a
C
K A, a ‰i mgvb L A, X ‰i mgvb K 90 L 45
M X, A ‰i DcGmU N A, X ‰i DcGmU M 120 N 180 ABC ‰i ˆÞòdj Kx nGe?
1
5. wbGPi evKÅàGjv jÞÅ Ki 19. sin A = 34 nGj, cosec A ‰i wòGKvYwgwZK AbycvZ K ah eMÆ ‰KK L ah eMÆ ‰KK
2
i. KÌgGRvo nGœQ `yB Dcv`vbwewkÓ¡ ˆmU
KZ? M 2ah eMÆ ‰KK N ah ‰KK
1
ii. ˆKvb ˆmGUi Dcv`vb msLÅv n nGj P(A) ‰i 4 1 2
Dcv`vb msLÅv 2n K
3
L
3 31. i. `yBwU m`†kGKvYx wòfzGRi Abyi…c evüàGjvi
iii. KvGZÆmxq àYR KÌgGRvo mgƒGni ˆmU 1 3 AbycvZ aË‚eK
M N ii. `yBwU wPò m`†kÅ nGj ˆmàGjv meÆ`v meÆmg
wbGPi ˆKvbwU mwVK? 4 4
K i L iii 20. wbGPi Z^ÅàGjv jÞÅ Ki nGe
M i I iii N i, ii I iii i. sin 30 + cos 60 = 1 iii. `yBwU wòfzR m`†kGKvYx nGj ZvG`i Abyi…c
6. x + y =12, x y = 2 nGj, xy ‰i gvb KZ nGe? ii. sec260  tan260 = 0 evüàGjv mgvbycvwZK
iii. sin 90 + cos 90 = 1
K 74 L 35 wbGPi ˆKvbwU mwVK?
M 53 N 36 wbGPi ˆKvbwU mwVK? K i I ii L ii I iii
7. (x  a), (x) ‰i Drcv`K nGe, hw` ‰es ˆKej hw` K i I ii L i I iii
M i I iii N i, ii I iii
K (x) = 0 nq L (x  a) = 0 nq M ii I iii N i, ii I iii
32. ˆiLv cÉwZmgZvGK Kx ejv nq?
M (a) = 0 nq N (a) = 0 nq wbGPi ZG^Åi AvGjvGK 21 I 22 bs cÉGk²i Dîi `vI : K cÉwZdjb cÉwZmgZv L cÉwZmiY cÉwZmgZv
C
8. ‰KwU LvZv 11 UvKvq weKÌq KiGj 10% jvf B
M Avqbv cÉwZmgZv N ˆj´Ö cÉwZmgZv
nq| ‰i KÌqgƒjÅ KZ wQj? 33. ‰KB f„wgi Dci ‰es ‰KB mgv¯¦ivj ˆiLvhyMGji
K 10 UvKv L 9 UvKv gGaÅ AewÕ©Z mvgv¯¦wiK ˆÞòmgƒGni ˆÞòdj
M 8 UvKv N 10.50 UvKv ˆKgb?
K f™²vsk L wecixZ
3 3 D
A
9. 4  4 ‰i gvb KZ?
M mgvb N Amgvb
K 1 L0 DcGii wPGòi ABC I ACD wbG`Æk Kiv nGqGQ|
M 2 N 2 34. A I B ciÕ·i mÁ·ƒiK ˆKvY| A = 115
21. ABC ‰i RbÅ C ‰i wecixZ evü ˆKvbwU?
10. 0.032 ‰i cƒYÆK KZ? K AB L BC nGj, B = KZ?
K 0 L2 M AC N BD K 65 L 75
M 1 N3 22. ABC ‰i RbÅ mw®²wnZ evü ˆKvbwU? M 85 N 90
wbGPi ZG^Åi AvGjvGK 11 I 12 bs cÉGk²i Dîi `vI : K AB L CD 35. ˆKvbwU mvgv¯¦wiGKi ˆÞòdj wbYÆGqi mƒò?
1
‰KwU mgxKiY
x2 + 2x  35 = 0 M BC N AC K (Š`NÆÅ + DœPZv) L (Š`NÆÅ  cÉÕ©)
2
11. cÉ`î mgxKiGY PjK wbGPi ˆKvbwU? 23. a : b = c : d nGj,
a+b c+d
ˆK Kx cÉwKÌqv
b
=
d M 2 (Š`NÆÅ  cÉÕ©) N f„wg  DœPZv
K 2 Lx
M x2 N0 eGj? wbGPi ZG^Åi AvGjvGK 36 I 37 bs cÉGk²i Dîi `vI :
12. cÉ`î mgxKiYwU ˆKvb aiGbi mgxKiY? K ˆhvRb L weGqvRb ‰KwU mge†îf„wgK ˆejGbi DœPZv 10 ˆm.wg. ‰es
K ‰K PjKwewkÓ¡ ‰KNvZ mgxKiY M ˆhvRb-weGqvRb N DcGii meàGjv f„wgi eÅvm 14 ˆm.wg.
L `yB PjKwewkÓ¡ ‰KNvZ mgxKiY 24. ˆKvGbv e†Gîi ˆÞòdj 16 àY e†w«¬ KiGj ‰i 36. mge†îf„wgK ˆejGbi eÅvmvaÆ KZ?
M ‰K PjKwewkÓ¡ w«¼NvZ mgxKiY eÅvmvaÆ KZàY e†w«¬ cvGe? K 7 ˆm.wg. L 14 ˆm.wg.
N `yB PjKwewkÓ¡ w«¼NvZ mgxKiY K 8 àY L 16 àY M 24 ˆm.wg. N 26 ˆm.wg.
13. mgZGj `ywU iwkÄi cÉvµ¦ we±`y ‰KB nGj Kx M 2 àY N 4 àY 37. mge†îf„wgK ˆejGbi AvqZb wbGPi ˆKvbwU?
Drc®² nq? 25. wZbRb ˆRGj 690 wU gvQ aGiGQ| ZvG`i AsGki K 219.91 Nb ˆm.wg. (cÉvq)
2 4 5 L 1539.38 Nb ˆm.wg. (cÉvq)
K ˆiLv L ˆKvY AbycvZ 3 : 5 : 6 nGj
M 3078.77 Nb ˆm.wg. (cÉvq)
M wòfzR N Zj i. 1g ˆRGj cvq 200wU N 6157.54 Nb ˆm.wg. (cÉvq)
14. ABC ‰ A ‰i mgw«¼L´£K AD ‰es AB = ii. 2q ˆRGj cvq 240 wU
AC nGj
38. wbGPi ˆKvbwU wbYÆq KiGZ KÌGgvGhvwRZ MYmsLÅvi
A iii. 3q ˆRGj cvq 300wU cÉGqvRb nq?
i. BD = DC
ii. AD  BC
wbGPi ˆKvbwU mwVK? K AwRfGiLv L AvqZGjL
iii. ABD = BAD K i I iii L i I ii M cÉPziK N Mo
wbGPi ˆKvbwU mwVK? B C M i, ii I iii N ˆKvbwUB bq 39. 27, 29, 30 ˆKvGbv DcvGîi MYmsLÅv nGj Zvi
K i I ii L i I iii
D 26. ‰KwU wòfzGRi wZb evüi AbycvZ 2 : 3 : 4 ‰es w«¼Zxq ˆkÉwYi KÌgGhvwRZ MYmsLÅv wbGPi ˆKvbwU?
M ii I iii N i, ii I iii cwimxgv 36 ˆm.wg. nGj Þz`ËZi evüi Š`NÆÅ wbGPi K 27 L 29
15. wòfzGRi wkitGKvGYi mgw«¼Lw´£Z ˆiLv f„wgGK ˆKvbwU? M 56 N 86
mgw«¼Lw´£Z KiGj wòfzRwU Kx? K 8 ˆm.wg. L 12 ˆm.wg. 40. ˆKvb ˆkÉwYi eÅvwµ¦ 18  22 ‰es avc wePzÅwZ 0
K Õ©ƒjGKvYx L mgGKvYx M 16 ˆm.wg. N 20 ˆm.wg. nGj AvbygvwbK Mo KZ?
M mgevü N mgw«¼evü 27. evMvGbi Š`NÆÅ I cÉÕ© h^vKÌGg x I y wgUvi nGj K 20 L 19
Zvi KGYÆi Š`NÆÅ KZ wgUvi? M 18 N 22
16. ABCD mvgv¯¦wiGKi B = 100 nGj, C = KZ?
K 100 L 90 K x2 + y2 L x2 + y2
M 80 N 60 M xy N x+y

1 K 2 L 3 K 4 N 5 N 6 L 7 M 8 K 9 K 10 L 11 L 12 M 13 L 14 K 15 N 16 M 17 M 18 N 19 K 20 L
Dîi

21 K 22 M 23 K 24 N 25 L 26 K 27 L 28 N 29 M 30 L 31 M 32 K 33 M 34 K 35 N 36 K 37 L 38 K 39 M 40 K
MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkc
œ ò I DËi : eûwbe©vPwb 35

239. evDdj Av`kÆ evwjKv gvaÅwgK we`Åvjq, cUzqvLvjx welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
àY nGj 14. ‰i gvb KZ? 28. BC-‰i Š`NÆÅ KZ wgUvi?
2 1 1 1
(2a + 3b )
1. ‰KwU msLÅv Aci ‰KwU msLÅvi 5 a ab K 10 L 12
K L
msLÅv«¼Gqi ˆhvMdj 3a + 2b 3a + 2b M 14 N 16
2x 5x ab ab n+1
K L M
2a + 3b
N
2a + 2b
29. i. 1 + 2 + 3+......... + n =
2
5 2
M x+
2x
N x+
5x 15.  nGj sec2  tan2 = ? ii. 12 + 22 + 32 + ...... + n2 =
n(n + 1) (2n + 1)
5 2 K 1 L2 6
2. msLÅv `yBwUi mgwÓ¡ 98 nGj eo msLÅvwU KZ? M 3 N4 n (n + 1)2
2
iii. 13 + 23 + 33 + ..... n3 =
K 50 L 60 16. ‰KwU eMÆGÞGòi evüi Š`NÆÅ 4 2 nGj ‰i KGYÆi 4
M 70 N 80 Š`NÆÅ KZ? wbGPi ˆKvbwU mwVK?
3. ABCD PZzfzÆRwU e†Gî A¯¦ÆwjwLZ ‰es ABC = K 4 L6 K i I ii L i I iii
120 ZvnGj ADC = ? M 8 N 10 M ii I iii N i, ii I iii
K 120 L 90
17. 128 + 64 + 32 + 16 + ...... avivwUi ˆKvb c` 2
1 30. ‰KwU eGMÆi KqwU cÉwZmvgÅ ˆiLv AvGQ?
M 60 N 30 K 4wU L 6wU
4. i. e†Gîi mKj mgvb RÅv ˆK±`Ê nGZ mg`ƒieZxÆ K 7g L 8g M 8wU N 10wU
ii. e†Gîi DciÕ© ˆKvb we±`yGZ ‰KwU gvò M 9g N 10g wbGPi ZG^Åi AvGjvGK 31 I 32bs cÉGk²i Dîi `vI:
Õ·kÆK AuvKv hvq| wbGPi wPGòi AvGjvGK 18 I 19 bs cÉGk²i Dîi `vI : ‰KwU AvqZvKvi evMvGbi cÉGÕ©i w«¼àY Š`NÆÅ AGcÞv
iii. mgGKvYx wòfzGRi AwZfzRGK eÅvm aGi e†î A
10 wgUvi ˆewk|
AsKb KiGj Zv mgGKŒwYK we±`y w`Gq hvGe| 2a 31. evMvbwUi Š`NÆÅ x wgUvi, cÉÕ© y wgUvi nGj
wbGPi ˆKvbwU mwVK? mgxKiYwU wK nGe?
K i I ii L i I iii  K x = 2y + 10 L 2x = y  10
M ii I iii N i, ii I iii C M 5x = 2y  10 N 2y = x + 10
B
a 32. evMvbwUi cwimxgv 100 wgUvi nGj evMvbwUi
5. tanA = 43 nGj cosA-‰i gvb KZ? 18. AB = KZ? Š`NÆÅ KZ wgUvi?
3 5 a K 30 wg. L 40 wg.
K L K L a
5 3 2
4 5 M 50 wg. N 60 wg.
M N M 2a N3 33. (x) = x + 5x  3 nGj, (1) ‰i gvb KZ?
4
5 4
f1
19. KZ? K 3 L 5
6. cÉPziK = L + f + f  h ‰LvGb h = ? K 30 L 45 M 7 N 9
1 2
M 60 N 90 34. i. 2(a2 + b2) = (a + b)2  (a  b)2
K wbÁ²mxgv L DœP mxgv
M ˆkÉwY eÅeavb N MYmsLÅv
20. a, b, c KÌwgK mgvbycvZx nGj wbGPi ˆKvbwU ii. 2(ab + bc + ca) = (a + b + c)2  (a2 + b2 + c2)
7. B = {3, 4} nGj mwVK? iii. a3 + b3 = (a + b) (a2  ab + b2)
K a2 = bc L ab = bc wbGPi ˆKvbwU mwVK?
i. , B ˆmGUi DcGmU M a=b=c N b2 = ac K i I ii L i I iii
ii. B- ˆmGUi DcGmU msLÅv 4 21. i. ˆejGbi f„wgi ˆÞòdj r  2
M ii I iii N i, ii I iii
iii. B ˆmUwU eYÆbvgƒjK c«¬wZGZ wjLv nGqGQ ii. ˆMvjGKi c†Ó¤ZGji ˆÞòdj 4r  2
35. ‰KwU e†Gîi eÅvmvaÆ 14 ˆm.wg.| ‰KwU eGMÆi
wbGPi ˆKvbwU mwVK? iii. NbGKi mgMÉ ZGji ˆÞòdj a3. ˆÞòdj DÚ e†Gîi ˆÞòdGji mgvb| eMÆGÞòwUi
K i I ii L i I iii wbGPi ˆKvbwU mwVK? Š`NÆÅ KZ?
M ii I iii N i, ii I iii K i I ii L i I iii K 24.21 L 24.41
8. (x  3) -‰i gƒj KqwU?
2
M ii I iii N i, ii I iii M 24.61 N 24.81
K 2 L3 wbGPi Z^Å ˆ^GK 22 I 23 bs cÉGk²i Dîi `vI : wbGPi ZG^Åi AvGjvGK 36 I 37bs cÉGk²i Dîi `vI :
M 4 N6
log3 + log9 + log27 + ....
wbGPi ZG^Åi AvGjvGK 09 I 10 bs cÉGk²i Dîi `vI : 22. avivwUi mvaviY A¯¦i KZ? cÉvµ¦ bÁ¼i 1-10 11-20 21-30 31-40 41-50
m2 + 1 = 2m nGj
MYmsLÅv 7 10 16 18 9
K log3 L log9
1 M 2 log3 N 3 log3
36. Dcvî mgƒ n GK KqwU ˆkÉ w YGZ webÅÕ¦ Kiv nGqGQ?
9. m2 + m2 = ? K 4 L5
23. avivwUi 10Zg c` KZ? M 6 N7
K 1 L 2 M 3 N 4 K log 1000 L log 8000
1 37. mviwYGZ DcwÕ©Z DcvGîi ˆkÉwY eÅvwµ¦ KZ?
10. m4 + m4 = ? M log 59049 N log 72900
K 10 L9
K 1 L 2 M 3 N 4
24. `y BwU abvñK msLÅvi cv^Æ K Å 4| ˆQvU msLÅvwUi M 8 N7
11. ABC mgevü wòfzGRi B I C-‰i mgw«¼LíK eMÆ eo msLÅvwUi w«¼àGYi mgvb| eowUi gvb KZ? 38. ax = 0, a2x + b2y = b3 mgxKiY ˆRvGUi mgvavb
K 2 L4
O we±`yGZ ˆQ` KGiGQ? M 6 N8
KZ?
K (a, b) L (0, b3)
A 25. x : y ‰i eÅvÕ¦vbycvZ nGe M (0, b) N (a2, b2)
K x:y L y:z
1 1
39. ABCD mvgv¯¦wiK nGj wbGPi ˆKvbwU mwVK?
M : N x: y A
x y D
O
26. `yBwU msLÅvi AbycvZ 3 : 4 ‰es ZvG`i j.mv.à
O
180| msLÅv `ywU KZ?
B C K 60, 45 L 45, 60
BOC -‰i gvb wbGPi ˆKvbwU? M 60, 80 N 90, 120
B C
K 90 L 105 wbGPi wPGòi AvGjvGK 27 I 28 bs cÉGk²i Dîi `vI :
M 120 N 135 K AC = BC L AD = AC
12. AaÆe†îÕ© ˆKvGYi cwigvY KZ? D B M AO = OB N OA = OC
K 150 L 120 M 90 N 60 30 40. ˆKvb mgGKvYx wòfzGRi mƒß ˆKvY«¼Gqi cv^ÆKÅ
13. (x + y, 0) = (1, x  y) nGj (x, y) = ? 7 wg. 8 nGj e†nîg ˆKvGYi gvb
1 1 1 1 K 38 L 41
K ( )
4 4
L ( )

3 3 27.
C
ABC = KZ?
A M 42 N 49
1 1 2 2 K 30 L 60
M ( )
2 2
N ( )

3 3 M 75 N 90
1 M 2 M 3 M 4 N 5 K 6 M 7 K 8 K 9 L 10 L 11 M 12 M 13 M 14 L 15 K 16 M 17 M 18 L 19 L 20 N
Dîi

21 K 22 K 23 M 24 N 25 M 26 L 27 L 28 M 29 M 30 K 31 N 32 K 33 M 34 M 35 N 36 L 37 K 38 M 39 N 40 N
36 MwYZ (Avewk¨K)  wbe©vPwb cix¶vi cÖkœcò I DËi : eûwbe©vPwb

240. Õ¼i…cKvVx KGjwRGqU ‰KvGWgx, wcGivRcyi welq ˆKvW : 1 0 9

mgq  40 wgwbU MwYZ (AvewkÅK) cƒYÆgvb  40
1. 7 ˆKvb aiGbi msLÅv? wbGPi ZG^Åi AvGjvGK 17-18 bs cÉGk²i Dîi `vI : i. sin30 =
BC
AC
K Õ¼vfvweK msLÅv L gƒj` msLÅv ‰KwU msLÅv Aci msLÅvi 5 àY|
2
BC
M Agƒj` msLÅv N cƒYÆmsLÅv ii. cos A =
AC
17. ‰KwU msLÅv x nGj AciwU KZ?
2. 0.2  0.02  0.002  2 mwVK gvb KZ?
2x 5x iii. AC = 40 wg.
K 0.000016 L 0.0000016 K L
5 2 wbGPi ˆKvbwU mwVK?
M 0.00016 N 0.0016 2x 5x
M x+ N x+ K i I ii Li I iii
3. A = {x, y, z}ˆKvb aiGbi ˆmU? 5 2
K Amxg ˆmU L mmxg ˆmU M ii I iii N i, ii I iii
18. msLÅv `yBwUi mgwÓ¡ 98 nGj eo msLÅvwU KZ?
M msGhvM ˆmU N ˆQ` ˆmU K 50 L 60 31. sin2 45 + cos2 45 = ?
M 70 N 80 1
4. (x) = x2  7x + 6 ‰es (x) = 0 nGj x ‰i gvb K L 1
KZ? 19. ‰KwU eGMÆi evüi Š`NÆÅ 2 àY nGj, ˆÞòdj 2

K 3, 4 L 6, 1 KZàY nGe? M 2 N2 2
M  6, 1 N 6, 1 K wZbàY L PviàY 32. cot =
5
nGj, sin ‰i gvb KZ?
12
5.
1 1
a + = 3 nGj a3 + 3 ‰i gvb KZ? M cuvPàY N QqàY
a a 12 12
20. wòfzGRi f„wg 4 ˆm.wg. ‰es ˆÞòdj 16 e. ˆm. wg. K L
13 5
K 2 L0
M 4 N6
nGj DœPZv KZ? 13 5
M N
K 2 L4 12 12
6. log 7 = ?
7 M 6 N8 wbGPi ZG^Åi AvGjvGK 33-35 bs cÉGk²i Dîi `vI :
1
K L2 21. ‰KwU eGMÆi evüi Š`NÆÅ 6 ˆm.wg. nGj ‰i ABC mgevn wòfzGR AD  BC ‰es AB = 2 ‰KK
2
cwimxgv KZ? 33. BD gvb KZ ‰KK?
M 2 N 7
K 12 L 18 1 1
7. (x  3)3 = 0 mgxKiGYi KqwU gƒj AvGQ? M 24 N 30 K L
2 2
K 0 L1
22. cos ‰i mGeÆvœP gvb KZ?
M 2 N3 M 1 N 2
K 1 L1
8. log5 x = 3 nGj x = ? M 0 N2
34. wòfzRwUi DœPZv KZ ‰KK?
K 5 L 15 2
23. ‰KwU e†Gîi eÅvm 2 ˆm.wg.| ‰i ˆÞòdj KZ K L 3
M 125 N 625 3
9. x  2x + 1 = 0 nGj
2 eMÆ ˆm.wg.? 4
K  L 2 M N2 3
1 3
i. x + = 2
x M 3 N 4
35. ABC wòfzRwUi ˆÞòdj KZ e: ‰KK?
1 1  tan2A
2
ii. x + =4 24. hw` A = 45 nq ZGe 1 + tan2A
=? K 3 L2 3
x
1 K 1 L0 M 4 3 N6
iii. x  =0
x M
1
N2 36. ABC wòfzGRi evüàGjvi AbycvZ 3 : 4 : 5 nGj
2
wbGPi ˆKvbwU mwVK? ABC wòfzRwU nGe
25. e†Gîi cwiwa I eÅvGmi AbycvZ KZ?
K i I ii L i I iii i. Õ©ƒjGKvYx
K r:1 L 22 : 7
M ii I iii N i, ii I iii M 1:r N 2 : 1 ii. mgGKvYx
10. a + b = 2, ‰es ab = 1 nGj, a3 + b3 = ? 26. ˆMvjGKi AvqZGbi mƒò ˆKvbwU? iii. welgevü
K 0 L1
M 2 N3 K
4 3
r N. ‰KK L 4r3 N. ‰KK wbGPi ˆKvbwU mwVK?
3
11. `yBwU msLÅvi AbycvZ 2 : 3 ‰es M.mv.à 4 nGj, 4 K i I ii L i I iii
M 3r3 N. ‰KK N r3 N. ‰KK M ii I iii N i, ii I iii
e†nîg msLÅv KZ? 3
K 10 L 12 27. 5 : 8 = 15 : x nGj x ‰i gvb KZ? wbGPi ZG^Åi AvGjvGK 37-38bs cÉGk²i Dîi `vI :
M 14 N 16 K 24 L 40 cÉvµ¦ bÁ¼i 60-69 70-79 80-89 90-100
12. wbGPi ˆKvbwU y AGÞi Dci AewÕ©Z? M 75 N 120
28. cƒYÆ msLÅvi ˆmUGK Kx «¼viv cÉKvk Kiv nq? MYmsLÅv 5 10 20 15
K (0, 4) L (4, 0)
M (8, 0) N (4, 4) K Z LQ 37. gaÅK ˆkÉwYi DœPmxgv KZ?
M N NR K 89 L 79
wbGPi ZG^Åi AvGjvGK 1315 bs cÉGk²i Dîi `vI :
29. PZzfzÆR AvKvi RbÅ KqwU Dcvî Rvbv cÉGqvRb? M 69 N 99
log 3 + log 9 + log 27 + ........
13. avivwUi mvaviY A¯¦i KZ? K 3wU L 4wU 38. DcvGîi ‰i 2 gvb KZ?
K log 3 L log 9 M 5wU N 6wU K 5 L 10
M log 6 N log 12 M 15 N 20
30.
14. avivwUi 5g c` KZ? 39. 2, 4, 6, 9, 8, 15 DcvîàGjvi cÉPziK KZ?
A K 6 L7
K log 81 L log 243
M log 729 N log2187 M 15 N cÉPziK ˆbB
15. avivwUi 1g 10wU cG`i mgwÓ¡ KZ? 40. ˆKvbwU AwewœQ®² PjK?
K 5 log 3 L 10 log 3
K ˆjvK msLÅv L mgq
M 55 log 3 N 110 log 3
M cÉvµ¦ bÁ¼i N kÉwgGKi ˆeZb
16. a, b, c KÌwgK mgvbycvZx nGj ˆKvbwU mwVK? 60
B
K a2 = bc L ab = bc C 20 wg.
M a=b=c N b2 = ac

1 M 2 K 3 L 4 L 5 L 6 L 7 N 8 M 9 L 10 M 11 L 12 K 13 K 14 L 15 M 16 N 17 K 18 M 19 L 20 N
Dîi

21 M 22 L 23 K 24 L 25 L 26 K 27 K 28 K 29 M 30 L 31 L 32 K 33 M 34 L 35 K 36 M 37 K 38 M 39 N 40 L