GmGmwm cix¶v 2021
Aa¨vq 01 †mU I dvskb
mvaviY eüwbevÆ Pwb cÉGkv² îi 13. , A = {0}, B = {1} †mU¸‡jvi g‡a¨ †Kvb 24. s = {(x, y) : x2 + y2 + 2x 6y 6 = 0}
†mUwUi Dc‡mU msL¨v 1 n‡e? Aš^qwU eY©bvKvix e‡„ Ëi †K‡›`ªi ¯v’ bv¼
1. mvweK© †mU U Ges A †h †Kv‡bv †mU n‡j, KZ?
wb‡¤œi †Kvb m¤úKw© U mwVK? LA
MB ( 1, 3) L ( 1, 3)
K n(A) = n(U) N I A DfqB
14. hw` A = {2, 3, 5, 7} Ges B = {5, 7, 1, 4}
n(A) + n(A) = n(U) M (0, 0) N (3, 4)
M n(A) + n(U) = n(A) nq, Zvn‡j B /A = KZ? 25. S= {(x, y) : x2 + y2 = 1}, Ašq^ wUi
N n(A) + n(A) = 0 K {5, 7} {1, 4} 4 9
2. hw` U = {1, 3, 5, 7} Ges A = {3, 5, 7} 15. M {2, 5} N {1, 2, 3} †jLwP‡Îi AvKvi wb‡Pi †KvbwU?
16.
nq, Z‡e A Gi gvb KZ? A BU K e„Ë L civeË„
K {1, 5} {1} M {1, 7} N { }
3. hw` †Kv‡bv †m‡Ui Dcv`vb mmxg nq, aiv M mij‡iLv Dce„Ë
4. hvK H †m‡U n msL¨K Dcv`vb Av‡Q, 26. (x) = |x| n‡j ( 6) = KZ?
5. 27.
Zvn‡j D³ †mUwUi kw³ †m‡U KZ msL¨K C K 6 6 M 6 N
28.
Dcv`vb _vK‡e? Dc‡ii †fbwPÎvbmy v‡i wb‡Pi †KvbwU hw` (x)= 3x + 5, x R nq, Zvn‡j
2n L 2n + 1 M 2n 1N 22n QvqvKZ… As‡ki AÂjwU cKÖ vk K‡i? dvskbwU wK ai‡bi dvskb?
hw` D = {0, 1, 2, 3} Z‡e D †m‡Ui A B C L A B C K msh³y dvskb L bvj dvskb
M A B C N U C
Dc‡mU KqwU? GK-GK dvskb
S = {(2, 3), (4, 3), (5, 10), (9, 6)} Ašq^ wUi
K8 L 14 16 N 24 †iÄ †KvbwU? N †Kej Aš^q, dvskb bq
{(2, 2), (4, 2), (2, 10), (7, 7)} Aš‡^ qi
†Wv‡gb †KvbwU? {3, 10, 6} L {2, 4, 5, 9} hw` (x) = x + 7 dvskbwUi †jLwPÎ GKwU
mij †iLv nq, Z‡e †iLvwU Øviv y A‡¶i
(Ab.y 1) M {3, 10, 9, 6} N {2, 4, 5} †Q`K…Z As‡ki cwigvY KZ?
{2, 4, 7} L {2, 2, 10, 7} 17. a 0 n‡j, wØNvZ dvsk‡bi mvaviY iƒc
18.
6. M {2, 2, 10, 7} N {2, 4, 2, 5, 7} 19. wb‡Pi †KvbwU? K0 7
7.
8. S = {(x, y) : x A, y A Ges y = x2} y = ax2 + bx + c M5 N8
Ges A = {2, 1, 0, 1, 2} wb‡Pi †KvbwU S L y = ax2 + cy
Aš‡^ qi m`m¨? (Ab.y 2) 29. S = {(x, y) : y = x2} Ašq^ wUi †jLwPÎ
30. †Kgb n‡e?
K (2, 4) L (2, 4) M y = ax + b
N y = x2 + y2 + ax + by + c K mij‡iLv cive„Ë
(1, 1) N (1, 1) hw` U = {2, 3, 4, 5, 6, 7, 8, 9}, A = {x : x
M Dce„Ë N e„Ë
ev¯Íe RMZ ev wPš—v RM‡Zi e¯i‘ †h †gŠwjK msL¨v}, B = {x : x we‡Rvo msL¨v}
3x + 1 (x) + 1
†Kv‡bv mywbav© wiZ msMnÖ ‡K ejv nq|Ó nq, Z‡e A B Gi gvb KZ? hw`, (x) = 3x 1 nq Zvn‡j, (x) 1
†mU L dvskb
M Ašq^ N Dc‡mU {3, 5, 7} L {2, 5, 7} gvb KZ?
A = {1, 2, 3} Ges B = {1, 2, 3, 4} `By wU M {1, 3, 5, 7} N {4, 6, 8} 3x L 6x M 3x2 N x
hw` S = {x : x R Ges x(x 1) = x2
†mU n‡j wb‡Pi †Kvb hyw³wU mwVK? x} nq, Z‡e S = R\S wb‡Pi †KvbwU n‡e? eüc`x mgvwµ¦mPƒ K eüwbevÆ Pwb cGÉ kv² îi
9. K AB ABM ABN AB 20. KQ LR NZ 31. S = {(x, y) : x2 + y2 25 = 0 Ges x 0}
10. 32.
11. hw` A †mU, mmxg †mU B-Gi Dcv`vb nq, 21. wKQy msL¨K †jv‡Ki g‡a¨ 50 Rb evsjv, i. Ašq^ wU dvskb bq
Zvn‡j wb‡Pi †KvbwU mwVK? [hLb A B] 22. 20 Rb Bs‡iwR Ges 10 Rb evsjv I
ii. Ašq^ i †jLwPÎ GKwU Aae© „Ë
n(A) n(B) L n(A) n(B) Bs‡iwR ej‡Z cv‡i| `By wU fvlvi Aš—Zt
iii. Aš^qwUi †jLwPÎ x A‡¶i Dci
M n(A) n(B) N n(A) > n(B) GKwU fvlv KZRb ej‡Z cv‡i? Aa©Z‡j _vK‡e
A I B †mU؇qi g‡a¨ wb‡Pi †Kvb
m¤ú‡Ki© Rb¨ n(A) < n(b) †jLv hv‡e? K 50 Rb L 55 Rb Dc‡ii Z‡_¨i Av‡jv‡K wb‡Pi †KvbwU
60 Rb N 70 Rb mwVK? (Aby. 7)
K AB L AB hw` S = {x : x R Ges x2 + 1 = 0} nq,
K i I ii i I iii
M AB AB Z‡e †mU S n‡e
hw` A = {2, 3, 5} Ges B = {1, 4, 6} nq, K S = R S = M S = 2+ N S = N M ii I iii N i, ii I iii
Zvn‡j A B = wb‡Pi †KvbwU?
hw` (x) = 3x + 2 Ges g(x) = x 2 nq, †h †Kv‡bv mvwe©K †mU U Gi Rb¨
K {4} L {2, 4}
Z‡e go (2) = KZ? †hLv‡b, , g R. i. A\A =
N {2, 3, 5} K4 L 10 M 8 6 ii. A\(A\A) = A
12. hw` U = {2, 3, 4, 5, 6, 7, 8, 9} Ges A = 23. y = x2 + 6x + 3 nq Z‡e wØNvZ dvskbwUi iii. A\(A\A) =
{x : x †gŠwjK msL¨v} nq, Z‡e A = KZ? cÖwZmg A‡¶i mgxKiY †KvbwU? wb‡Pi †KvbwU mwVK?
K {3, 5, 7, 9} L {3, 5, 7} K y=5 y = 3 i Ges ii L i Ges iii
M {1, 2, 3, 5, 7} {2, 3, 5, 7} M y=1 N y=3 M ii Ges iii N i, ii Ges iii
33. hw` U = {x : x cY~ m© sL¨v, 0 < x 10}, A 36. A1 A2 Gi gvb wb‡Pi †KvbwU? (Aby. 2) 45. A B Gi mwVK Dcv`vb msL¨v KZ?
= {x : 2x > 7} Ges B = {x : 3x < 20} nq, K A1 A2 M A3 N A4 K9 L8
M5 6
Z‡e 37. wb‡Pi †KvbwU A3 A6 Gi gvb wb‡`©k
K‡i? (Ab.y 3) 46. (A B) (B A) (A B) = KZ?
i. A B ii. B A iii. A B
wb‡Pi †KvbwU mwVK? L {2, 3, 4, 5}
K A2 L A3 M A4 A6 M {2, 4} N {2, 3, 4, 5}
K i Ges ii i Ges iii 38.
M ii Ges iii N i, ii Ges iii A2 A3 Gi cwie‡Z© wb‡Pi †KvbwU wjLv wbGPi ZÅ^ nGZ 47 49 bs cÉGk²i
hvq? (Ab.y 4) Dîi `vI :
34. wb‡Pi wPÎwU j¶ Ki : †hLv‡b F : X 47.
39. K A2 L A4 M A5 A6 48. AB
40.
Y. wbGPi ZG^Åi wfwîGZ 39 I 40 bs cGÉ k²i 3x x 2x + 8
XY Dîi `vI :
3 am F = {(2, 4), (1, 1), (0, 0), (1, 1), (2, 4)} hw` n(A) = n(B) nq, Z‡e x Gi gvb
4 b
n Aš^qwUi †Wv‡gb KZ? KZ?
5c
{2, 1, 0, 1, 2} L {0, 1, 2} K7 L6
i. X Gi gvb¸‡jv‡K wb‡q MwVZ †mU‡K
Domain e‡j| M {4, 1, 0} N {2, 1, 1, 2} 8 N 10
ii. †iÄ = {a, b, c} Aš^qwUi †iÄ KZ? x = 8 n‡j, n(A B) = ?
iii. †Kv‡Wv‡gb = {a, b, c, m, n} {0, 1, 4} L {2, 1, 0, , 2} K 55 56
M {1, 4} N {2, 1, 4} M 50 N 52
wb‡Pi †KvbwU mwVK? wbGPi ZGÅ^ i AvGjvGK 41 43 bs 49. n(A B) = ?
K i I ii L ii I iii K 22 L 23
cÉGk²i Dîi `vI : M 20 24
M i I iii i, ii I iii
hw` F(x) = x 1 nq, Z‡e wbGPi ZÅ^ nGZ 50 I 51 bs cÉGk²i
35. hw` S = {(1, 4), (2, 1), (3, 0), (4, 1), 41. F(10) = KZ? (Aby. 4) Dîi `vI :
(5, 4)} nq Z‡e, 42. 50.
43. K 9 3 M 3 N 10 51. †LvkMÄ evwjKv D”P we`¨vj‡qi beg
i. S Aš^‡qi †iÄ S = {4, 1, 0, 4} F(x) = 5 n‡j x-Gi gvb KZ? (Ab.y 5)
†kÖwYi gvbweK kvLvi 50 Rb wk¶v_xi©
ii. S Aš‡^ qi wecixZ Ašq^ K5 L 24 M 25 26 g‡a¨ 29 Rb †cŠibxwZ| 24 Rb f‡‚ Mvj
dvskbwUi †Wv‡gb wb‡Pi †KvbwU? (Aby. 6)
S1 = {(4, 1), (1, 2), (0, 3), (1, 4), (4, 5)} K †Wvg F = {x R : x 1} Ges 11 Rb †cŠibxwZ I f‚‡Mvj Dfq
iii. S Ašq^ wU GKwU dvskb †Wvg F = {x R : x 1} welqB wb‡q‡Q| KZRb wk¶v_x© †cŠibxwZ
Dc‡ii Z‡_¨i Av‡jv‡K wb‡Pi †KvbwU M †Wvg F = {x R : x 1} ev f‚‡Mvj welq `By wUi †Kv‡bvwUB †bqwb?
mwVK? (Ab.y 3)
K i I ii ii I iii N †Wvg F = {x R : x > 1} Aš—Zt GKwU welq †bIqv wk¶v_x© i msL¨v
KZ?
M i I iii N i, ii I iii wbGPi ZÅ^ nGZ 44 66 bs
cÉGk²i Dîi `vI : 42 L 40
Awf®² Z^ÅwfwîK eüwbeÆvPwb cÉGk²vîi A = { 2, 3, 4} Ges B = {4, 5} n‡j M 44 N 38
wbGPi ZG^Åi AvGjvGK (3638) bs 44. A B = wb‡Pi †KvbwU? ˆKvGbv welqB ˆbqwb ‰gb wkÞv^Æxi
msLÅv KZ?
cÉGk²i Dîi `vI : K {2, 3, 4, 5} L {2, 3, 5} K5 8
cÖ‡Z¨K n N Gi Rb¨ An = {n, 2n, 3n ...... }
{4} N {3, 4, 5} M 10 N 12
Aa¨vq 02 exRMvwYwZK ivwk
mvaviY eüwbeÆvPwb cGÉ kv² îi 58. k~b¨ eûc`xi gvÎv Kx aiv nq? 64. hw` P(x) eûc`xi (x a) GKwU Drcv`K
K0 L1 nq Z‡e wb‡Pi †KvbwU mwVK?
P(x) = 4x2 3x + 2 n‡j, P(a) Gi gvb
M msÁvwqZ AmsÁvwqZ
52. wb‡Pi †KvbwU? P(a) = 0 L P(a) > 0
53.
( )54.
K a2 a + 2 L 4a2 3a 59. P(x) = 4x4 + 6x3 3x2 + x 4 n‡j, P(0) M P(a) < 0 N P(a) = a“ª eK
60. Gi gvb KZ?
4a2 3a + 2 N 4x2 3a + 2 65. hw` P(x) = 4x4 12x3 + 7x2 + 3x 2 nq Ges
K0 4 M 4 N 18
Cxpyq c‡`i gvÎv KZ? P(x) = 3x3 + 4x2 1 n‡j, P( 1) Gi gvb 1
Kp Lq 2 = 0 nq, Z‡e P(x) Gi Drcv`K
wb‡Pi †KvbwU? P
p+q N pq 0 L1 †KvbwU?
wb‡Pi †Kvb ivwkwU cwÖ Zmg? (Ab.y 1) M 1 N8 1
2
a+b+c L xy yz + zx 61. P(x, y) = 4x3 + y3 2x2 + 5xy 2 n‡j, K x L 2x 1
M x2 y2 + z2 N 2a2 5bc c2 P(1, 0) = KZ? 1
2
55. ax2 + bx + c eûc`xwUi gvÎv KZ? 0 L1 2x + 1 N 2x +
K0 L1 M 6 N 10 66. P(x) = 2x2 7x + 5 n‡j, P(2) = KZ?
2 N3 67.
5x4 + 4x3 + 6x2 + 7x 3 eûc`xwU †Kvb 62. P(x, y, z) = x3 + y3 + z3 3xyz n‡j, P(1,
63. 2, 3) Gi gvb KZ?
56. 1 L1
M2 N 2
( )57.
Pj‡Ki eûc`x? K3 18 M 36 N 54
x L x2
M x3 N x4 hw` P(x) abvZ¥K gvÎvi eûc`x nq Ges P(x) = 32x4 16x2 + 8x + 7 n‡j, P 1
P(a) = 0 nq; Z‡e P(x) Gi Drcv`K 2
6x4 2x3 + 5x + 3 eûc`xi gyL¨c` wb‡Pi wb‡Pi †KvbwU?
†KvbwU? = KZ?
K 5x L 2x3 K (x + a) (x a) K7 9
M 2x3 6x4 M P(x + a) N P(x a) M 15 N 31
68. hw` P(x) = 2x3 5x2 + 7x 8 nq Z‡e 79. 1g PjGKi ÕG© j 2q PjK, 2q PjGKi 88. i. P(x) cÖZx‡K x Gi Dci eûc`xwUi
P(x) †K (x 2) Øviv fvM Ki‡j fvM‡kl ÕG© j 3q PjK ‰es 3q PjGKi Õ©Gj 1g gv‡bi wbf©iZv wb‡`©k K‡i
KZ?
PjK emvGj hw` ivwkwU AcwiewZÆZ ii. P(x) = 3x3 4x + 10 n‡j, P(1) = 8
K0 2 ^vGK, ZGe ZvGK wK ivwk eGj?
iii. P(x) = 4x3 5x 1 n‡j, P( 1) = 0
M8 N 8 K cÖwZmg Pµ cÖwZmg
wb‡Pi †KvbwU mwVK?
69. a2 3a + 7 †K a 1 Øviv fvM Ki‡j M mggvwÎK N wZb PjK
fvM‡kl wb‡Pi †KvbwU n‡e? K i I ii L ii I iii
80. wb‡Pi †KvbwU Pµ-µwgK ivwk?
K1 5 i I iii N i, ii I iii
M8 N 11 K xy + yz zx x2y + y2z + z2x 89. i. GKwU eûc`x‡K ni Ges GKwU
70. y3 8y2 + 6y + 60 eûc`x‡K y + 2 Øviv M xy + y2z2 + z3x3 N x2y y2z z2x2 eûc`x‡K je wb‡q MwVZ fMvœ sk
fvM Ki‡j, fvM‡kl KZ n‡e? 81. a(b2 c2) + b(c2 a2) + c(a2 b2) Gi n‡”Q gj~ ` fMœvsk
K6 8 Drcv`‡K we‡kw lZ iƒc wb‡Pi †KvbwU? ii. x2 +x+1 GKwU gj~ ` fMvœ sk
(x y) (x z)
M 75 N 112 (a + b) (b + c) (c + a)
iii. 3 + 3a2 + 2 GKwU cKÖ …Z fMvœ sk
71. abv—K gvÎvi †h‡Kv‡bv eûc`xi GKwU L (a b) (b c) (c a) a+2
Drcv`K x 1 n‡e hw` I †Kej hw` M (a + b) (b + c) (c + a) wb‡Pi †KvbwU mwVK?
eûc`xwUi mnMmg‡~ ni mgwó| N 2(a2 b2 c2) i I ii L ii I iii
0 nq L 1 nq 82. wb‡Pi †KvbwU (a + b + c) (ab + bc + ca) M i I iii N i, ii I iii
abc Gi Drcv`‡K we‡kwlZ iƒc?
M fMvœ sk nq N 2 nq Awf®² Z^ÅwfwîK eüwbevÆ Pwb cÉGk²vîi
(a + b) (b + c) (c + a)
72. a2 8 b2 Gi Drcv`K †KvbwU?
3 ab L (a + b) (b + c) (c + a) P(x) = x3 6x2 + 11x 6
( )K (a 3b) a b M (a b) (b c) (c a) Dc‡ii Z‡_¨i wfwˇZ 90 92bs cÖ‡kœi
3 N 3abc DËi `vI :
( )L (a + 3b) a b 83. wb‡Pi †KvbwU cÖK…Z fMvœ sk? 90. c`Ö Ë eûc`xi aª“ec` KZ?
3
a+1 L a2 + 1 K1 L3
a2 + 1 a+1
( )M b M6 6
a + 3 (a + 3b) a2 a3 + 1
M +1 N a2 + 1 91. x = 1 n‡j, P(x) = KZ?
( ) (a 3b) a
b 0 L1
a + 3 84. wb‡Pi †KvbwU AcÖKZ… fMœvsk?
M 1 N 24
73. 2x2 3x + 1 Gi Drcv`K KZ? x+1 3x3
K x2 2 x2 1 92. c`Ö Ë eûc`xi Drcv`‡K we‡kwlZ iƒc
K (2x + 1) (x 1) (2x 1) (x 1) wb‡Pi †KvbwU?
M (x + 1) (2x 1) N (x + 1) (2x + 1) x2 1 x6
M x3 + 1 N x +4 K (x 1) (x 2) L (x + 1) (x + 2)
74. a3 a2 10a 8 eûc`xi Drcv`K wb‡Pi eüc`x mgvwµ¦mPƒ K eüwbevÆ Pwb cGÉ kv² îi (x 1) (x 2) (x 3)
†Kvb¸‡jv? N (x 1) (x + 2) (x + 3)
K (a 1) (a 2) (a 4) 85. †Kvbv Av‡jvPbvq msL¨v wb‡`k© K GKwU 3x3 + 2x2 7x + 8 GKwU eûc`x|
L (a + 1) ( a + 2) (a + 4) i. A¶icÖZxK n‡Z cv‡i Dc‡ii Z‡_¨i wfwˇZ 93 95bs cÖ‡kœi
(a + 1) ( a + 2) ( a 4) DËi `vI :
ii. PjK n‡Z cv‡i
N (a + 1) (a 2) (a 4) 93. gyL¨ mnM KZ?
94.
75. (x 1) ivwkwU wb‡Pi †Kvb eûc`xi iii. a“ª eK n‡Z cv‡i 95. 3 L2
76. Drcv`K?
77. wb‡Pi †KvbwU mwVK? M7 N8
4x3 5x2 + 3x 2
L 7x3 8x2 + 6x 36 Ki L ii 96. eûc`xwUi aª“ec` KZ?
M x4 5x3 + 7x2 4
M i I iii i, ii I iii K3 L2
N Dc‡ii me¸‡jvB 86. 3x6 5x5 + x4 + 2x 9 ivwkwU x Pj‡Ki M7 8
87. GKwU eûc`x, hvi
wb‡Pi †KvbwU 2a3 + 3a2 + 3a + 1 Gi P(x) = 3x3 + 2x2 7x + 8 n‡j, P(0) =
Drcv`‡K we‡kwlZ iƒc? i. gvÎv 6 ii. gL~ ¨c` 3x6 KZ?
K (a + 1) (a2 + a + 1) iii. aһ ec` 9 K0 L6
L (a + 1) (a2 + a + 2) wb‡Pi †KvbwU mwVK? 8 N 10
(2a + 1) (a2 + a + 1)
i I ii L i I iii eûc`x x3 + px2 x 7 Gi GKwU
N (2a 1) (a2 + a + 1) Drcv`K x + 7|
x4 5x3 + 7x2 a Gi x 3 GKwU M ii I iii N i, ii I iii GB Z‡_¨i Av‡jv‡K wb‡Pi 96 Ges 97bs
Drcv`K n‡j, a = KZ? P(x, y, z) = x3 + y3 + z3 3xyz n‡j c‡Ö kœi DËi `vI :
K1 L5 i. P(x, y, z) PµµwgK ivwk p Gi gvb KZ? (Ab.y 3)
M7 9 K7 7
78. wb‡Pi †KvbwU mggvwÎK eûc`xi D`vniY? ii. P(x, y, z) cÖwZmg ivwk M 54 N 477
7
x2 + 2xy + 5y2 iii. P(1, 2, 1) = 0
L x2 + 2x + 2y + 5y2 97. eûc`xwUi Aci Drcv`K¸‡jvi ¸Ydj
Dw³¸‡jvi †Kvb¸‡jv mZ¨? (Aby. 2) KZ? (Aby. 4)
M ax2 + by + c K i I ii L i I iii K (x 1) (x 1) L (x + 1) (x 2)
N a2 + b2 + c2 + 2a + 2bc + 2ca M ii I iii i, ii I iii M (x 1) (x + 3) (x + 1) (x 1)
Aa¨vq 03 R¨vwgwZ
mvaviY eüwbeÆvPwb cÉGk²vîi 107. DEF wÎf‡z Ri DE = DF = 6 †m. wg. I eüc`x mgvwµ¦mƒPK eüwbeÆvPwb cÉGk²vîi
98. A
EF = 6 2 n‡j, E = KZ? 119. P D
K 90 45 M 60 N 30
108. mg‡KvYx wÎfz‡Ri mg‡Kv‡Yi mwbwœ nZ
evû؇qi j¤^ Awf‡¶‡ci gvb KZ?
Q RE F
XB CY 0 L1 M2 N 1 PQR I DEF m`k„ n‡j
2 i. PQR DEF
XY †iLvs‡k AB Gi j¤^ Awf‡¶c wb‡Pi
†KvbwU? (Aby. 1) A ii. P = D, Q = E
109. PQ PR QR
DE DF EF
K AB BC M AC N XY 45 55 iii. = = = aһ eK
99. A B 375 D 375 C wb‡Pi †KvbwU mwVK?
FE ABC Gi AD ga¨gv n‡j AB Gi gvb K i I ii L i I iii ii I iiiN i, ii I iii
120. A
O wb‡Pi †KvbwU?
6.2 L 5.12 M 5.25 N 3.20
110. †Kvb mg‡KvYx wÎf‡z Ri ga¨gvÎq h_vµ‡g d, e DE
BD C
Ic‡ii wP‡Î †KvbwU j¤^ we›`?y (Ab.y 2) I f Ges AwZfRz c n‡j wb‡Pi †KvbwU mwVK? BC
KD LE MF O K d2 + e2 + f2 = c2 L 4(d2 + e2 + f2) = 5c2 ABC G BC || DE n‡j
100. GKwU mgevû wÎf‡z Ri cwÖ ZwU ga¨gvi ˆ`N¨© 3
†m.wg. n‡j cwÖ ZwU evûi ˆ`N©¨ KZ? (Aby. 3) 2(d2 + e2 + f2) = 3c2 AB AC
K 45 †m.wg. L 424 †m.wg. 111. N 3(d2 + e2 + f2) = 2c2 i. AD AE
346 †m.wg. N 259 †m.wg. 105 A ii. BEC = BDC
101. A iii. BDC I BEC Gi D”PZv mgvb
wb‡Pi †KvbwU mwVK?
B CE K i I ii L i I iii ii I iiiN i, ii I iii
60 ABC-G AB = BC n‡j ACE Gi gvb 121. P
BC wb‡Pi †KvbwU?
ABC wÎfz‡Ri AB = 6 †m. wg. I BC = 7 K 75 L 100 105 N 110
†m. wg. n‡j AC Gi gvb wb‡Pi †KvbwU?
112. ABC I DEF Gi f‚wg I D”PZv
K8 43 M 53 N 47 h_vµ‡g, 5 †m. wg. I 6 †m. wg. Ges 7
102. P QS R
†m. wg. I 8 †m. wg. n‡j ABC t DEF
60 Gi gvb wb‡Pi †KvbwU? PQR I PQS-G
13 3 K7t3 L 15 t 27
i. PQ2 < PS2 + QS2
QR M 15 t 29 15 t 28 ii. PR2 < PQ2 + QR2
113. †Kvb †iLvi Dci †Kvb we›`y †_‡K Aw¼Z iii. PQ2 < PR2 + QR2
PQR-G QR Gi gvb wb‡Pi †KvbwU?
K 2.11 L 10.12 j‡¤^i cv`we›`By H we›`yi wb‡Pi †KvbwU mwVK?
K mgvš—ivj L j¤^ K i I ii L i I iii ii I iiiN i, ii I iii
3.34 N 13.13 M Awf‡¶c j¤^ Awf‡¶c 122. D
103. a t b = c t d n‡j wb‡Pi †KvbwU mwVK? 114. †Kvb wbw`ó© †iLvs‡ki mgvši— vj †iLvs‡ki 5 35
a = c L b = c M a = cd N b = c j¤^ Awf‡¶c H †iLvs‡ki E2G F
b d a d b a d
K mgvbcy vwZK mgvb DEF Gi DG, EF Gi Dci ga¨gv|
104. D M Amgvb N e¨¯—vbycvwZK
i. DF = 274 GKK
HG 115. wc_v‡Mviv‡mi Dccv‡`¨i we¯—vi n‡Z †h
Dccv`¨wU ewYZ© n‡q‡Q †mUv Kvi Dccv`¨? ii. EF = 4 GKK
EF
K U‡jwgi L eªþv¸‡ßi iii. DEF Gi †¶Îdj = 5 eM© GKK
wP‡Î EF || HG n‡j wb‡Pi †KvbwU mwVK?
G¨v‡cv‡jvwbqv‡mi N Avi,G wdkvi wb‡Pi †KvbwU mwVK?
K DH : EH = DF : GF 116. wÎfz‡Ri ga¨gvÎq KZ Abcy v‡Z wef³ nq| i I ii L i I iiiM ii I iiiN i, ii I iii
DH : EH = DG : FG 123. P
M DE : DH = DG : DF 2:1 L3:1 M3:2 N2:3
117. `yBwU wÎfz‡Ri f‚wg mgvb n‡j Zv‡`i
N Dc‡ii me¸‡jv †¶Îdj Kx n‡e? CB
105. †Kvb wÎf‡z Ri evûÎq n2 + 1, n2 1 I 2n O
Ges n > 1 n‡j wb‡Pi †KvbwU mwVK? K e¨¯—vbcy vwZK mgvbycvwZK QAR
K wÎfRz wU mgwØevû wÎfRz wU mg‡KvYx
M wÎfzRwU welgevû N wÎfzRwU ¯z’j‡KvYx M mgvb D N Amgvb PQR wÎfz‡R PA, QB I CR wZbwU
106. D ga¨gvÎq ci¯úi O we›`‡y Z wgwjZ
118. A n‡q‡Q|
B CE F i. OA = 3 OP
2
ABC I DEF Gi Abyiƒc evû¸‡jvi
3 Abcy vZ mgvb n‡j wb‡Pi †KvbwU mwVK? ii. OQ = 2 QB
3
120
E 4F K A < D, B = F iii. CO = 1 OR
3
DEF wÎfz‡Ri DE evûi ˆ`N¨© wb‡Pi †KvbwU? L A = B, C = D
37 L 3 3 M 2 5 N 3 2 A = D, B = E Ges C = F wb‡Pi †KvbwU mwVK?
N Dc‡ii me¸‡jv K i I ii L i I iiiM ii I iii i, ii I iii
124. A D A 135. AB = AD = 3 †m.wg., BC = 2.5
FE †m.wg., CD = 3.5 †m.wg. I AM = 2
G †m.wg. n‡j, BD = KZ? [mKj †evW© Õ18]
B CE F BD C K 4.0 †m.wg. 4.5 †m.wg.
ABC I DEF m`„k n‡j D, E, F h_vµ‡g BC, AC I AB Gi M 5.5 †m.wg N 6.5 †m.wg.
i. Abiy ƒc †KvY¸‡jv mgvb n‡e ga¨we›`y n‡j
ii. Abyiƒc evû¸‡jvi Abcy vZ wb‡Pi wP‡Îi Av‡jv‡K 136 I 137bs
cÖ‡kœi DËi `vI :
mgvbycvwZK n‡e Ic‡ii wP‡Îi Av‡jv‡K 129 131 bs A
c‡Ö kœi DËi `vI : (Ab.y 4)
iii. †¶Îdj d‡ji Abcy vZ Abiy ƒc
evû؇qi e‡Mi© Abycv‡Zi mgvb n‡e 129. G we›`iy bvg wK?
wb‡Pi †KvbwU mwVK?
K i I ii L i I iii K j¤^ we›`y L Aš—t‡K›` DC B [P. †ev. Õ17]
M ii I iii i, ii I iii fi‡K›` N cwi‡K›` 136. DB Gi Dci AC Gi j¤^ Awf‡¶c
125. `By wU eûf‡z Ri †KvY¸‡jv mgvb n‡j 130. ABC Gi kxl©we›`y w`‡q Aw¼Z e„‡Ëi †KvbwU?
bvg wK? (Aby. 5) K AD DC M DB N CB
i. eûfRz Øq m`k„ ‡KvYx 137. B m~²‡KvY n‡j, AC2-Gi gvb †KvbwU?
ii. eûfzRØq m`k„ A_ev Am`„k cwie„Ë L Aš—te„Ë
K AB2 + BC2 2BC.CD
iii. eûfzRØq me`© v me©mg M ewnte„Ë N bewe›`y e„Ë AB2 + BC2 BC.BD
M AB2 + BC2 + 2AC.CD
wb‡Pi †KvbwU mwVK? 131. ABC Gi †¶‡Î wb‡Pi †KvbwU N AB2 + BC2 + 2AB.AD
i I ii L ii I iii G¨v‡cv‡jvwbqv‡mi Dccv`¨‡K mg_©b
K‡i? (Ab.y 6)
M i I iii N i, ii I iii wb‡Pi wP‡Îi Av‡jv‡K 138140bs cÖ‡kœi
K AB2 + AC2 = BC2 DËi `vI :
126. GKwU AvqZ †¶Î I GKwU eM‡© ¶Î AB2 + AC2 = 2(AD2 + BD2) [Kz. †ev. Õ16]
M AB2 + AC2 = 2(AG2 + GD2) QS = 8 †m.wg. PS = 5 P
i. ci¯úi m`k„ ‡KvY ii. ci¯úi m`„k N AB2 + AC2 = 2(BD2 + CD2)
iii. ci¯úi wem`k„ †m.wg. Ges PR = 3
wb‡Pi †KvbwU mwVK? †m.wg.| 120 R
A
QS
K i I ii i I iii 138. PS Gi j¤^ Awf‡¶c †KvbwU?
M ii I iii N i, ii I iii K PR L PQ M QS SR
139. PQ2 = KZ?
Awf®² Z^ÅwfwîK eüwbeÆvPwb cÉGk²vîi K PS2 + QS2 2 QS.SR L PS2
B DEC QS2 + 2QS.SR
P M PS2 + QS2 PS2 + QS2 +
wP‡Î ABC Gi BC †iLvi Dci AD
3 ga¨gv| 2QS.SR
35 4
Dc‡ii wP‡Îi Av‡jv‡K 132 135 bs 140. PQ Gi gvb KZ †m.wg.?
O c‡Ö kiœ DËi `vI :
QR 132. wb‡Pi †KvbwU mwVK? K 55 L 73 M 135 153
PQR wÎfz‡Ri ga¨gvÎq h_vµ‡g 4, 3 I AC2 = AB2 + BC2 + 2BD DE wb‡Pi Z‡_¨i Av‡jv‡K 141 I 142bs
35 GKK Ges Zviv ci¯úi O we›`‡y Z L AC2 = AB2 + BC2 2BD DE c‡Ö kœi DËi `vI :
†Q` K‡i| M AC2 = AB2 + BC2 + BD DE
N AC2 = AB2 + BC2 AB BC A
Dc‡ii eY©bv n‡Z 127 I 128 bs c‡Ö kiœ 133. C = 60 n‡j wb‡Pi †KvbwU mwVK?
DËi `vI :
127. OP Gi ˆ`N¨© wb‡Pi †KvbwU? D 45 B
C
AB2 = AC2 + BC2 AC BC [wm. †ev. Õ16]
K 3 GKK 3 GKK L AB2 = AC2 + BC2 + AC BC 6 ˆm.wg.
2 M AB2 = AC2 + BC2
L N AB2 > AC2 + BC2 141. BD Gi Dci AC Gi j¤^ Awf‡¶c
M 3 GKK 2 GKK 134. ACD-G AN : AG = KZ? [mKj †evW© Õ18] †KvbwU?
4 K BD CD M AB N BC
142. DC = KZ?
128. wÎf‡z Ri evû¸‡jvi e‡Mi© mgwó wb‡Pi †KvbwU? K2:1 L1:2 2 †m.wg. L 4 †m.wg.
K 4057 L 3969 3:2 N3:1 M 6 †m.wg. N 8 †m.wg.
4967 N 4129
Aa¨vq 04 R¨vwgwZK A¼b
mvaviY eüwbeÆvPwb cGÉ k²vîi 147. e„‡Ëi ewnt¯’ †Kvb we›`y †_‡K e„‡Ë KqwU 150.
¯úk©K AuvKv hvq?
143. e„‡Ëi cwiwai †Kv‡bv we›`‡y Z KqwU ¯úk©K x
AuvKv m¤¢e? 2 L3 M3 N4
148. †Kv‡bv e‡„ Ëi ewnt¯’ †Kv‡bv we›`‡y Z `By wU wPÎ
1 L2 ¯úkK© 60 †Kv‡Y wgwjZ n‡j ¯úk©K
we›`yØq †K‡›`ª KZ wWMÖx †KvY Drcbœ K‡i? x = 60 n‡j x Gi m¤ú~iK †Kv‡Yi
M 3 N AmsL¨K A‡a©‡Ki gvb KZ? (Ab.y 1)
144. 50 †Kv‡Yi m¤úi~ K †Kv‡Yi GK-cÂgvsk
wb‡Pi †KvbwU? K 30 60 M 120 N 180
K 60 L 90
151. 3.5 †m.wg., 4.5 †m.wg. Ges 5.5 †m.wg.
K 35 L 30 26 N 36 120 N 180 e¨vmva© wewkó wZbwU eË„ ci¯úi‡K
ewn¯úk© Ki‡j †K›`ªÎq Øviv Drcbœ
145. 45 †Kv‡Yi ci~ K †Kv‡Yi wظY wb‡Pi 149. †Kv‡bv e„‡Ëi ewnt¯’ †Kv‡bv we›`‡y Z `yBwU wÎf‡z Ri cwimxgv KZ †m.wg.? (Aby. 2)
†KvbwU? ¯úk©K ci¯úi 45 †KvY Drcbœ Ki‡j
¯úk©K we›`yØq †K‡›` KZ wWwMÖ †KvY K 54 L 40.5 27 N 13
K 45 L 30 90 N 60
146. mij †Kv‡Yi GK Z…Zxqvsk wb‡Pi †KvbwU? 152. mg‡KvYx wÎf‡z Ri wkit‡KvY 60 n‡j Aci
Drcbœ K‡i? †KvY KZ n‡e?
K 180 L 120 K 130 L 125 K 60 L 90 30 N 50
M 45 60 M 45 135
153. †Kv‡bv e‡„ Ëi GKwU wbw`©ó we›`‡y Z Aw¼Z 159. i. AvqZ GKwU mvgvšw— iK 163. wb‡Pi †KvbwU mwVK?
¯úk©K I H we›`y Ges †K‡›`ªi ms‡hvRK ii. eM© GKwU AvqZ
iii. i¤m^ GKwU eM© K OP = PA L OP = PB
mij‡iLv ci¯úi|
K mgvš—ivj j¤^ wb‡Pi †KvbwU mwVK? PA = PC N PA OB = AC
M mwbœwnZ †KvY N 180 Drcbœ K‡i i I ii L i I iii wb‡Pi Z‡_¨i Av‡jv‡K 164 I 165bs
M ii I iii N i, ii I iii cÖ‡kœi DËi `vI :
154. mgevû wÎf‡z Ri †h‡Kv‡bv evûi ewnt¯’
†KvY KZ n‡e? 160. PZfz ©zR Avu Kv hv‡e hw` †`Iqv _v‡K D
K 130 L 100 M 160 120 i. PviwU evû GKwU †KvY A
ii. wZbwU †KvY GKwU evû 6
6
155. ¯j‚’ ‡KvYx wÎf‡z Ri cwieË„ A¼b Ki‡i Gi E 10 C
cwi‡K›` †Kv_vq _vK‡e? iii. wZbwU evû `By wU KY© AC = AD
K AwZfRz L ewnfv© ‡M wb‡Pi †KvbwU mwVK? 164. ADC Gi gvb KZ? (Aby. 3)
Af¨š—‡i N Dc‡ii me¸‡jv K i I ii i I iii K 30 45
156. m~¶¥‡KvYx wÎf‡z Ri cwie„Ë A¼b Ki‡j M ii I iii N i, ii I iii M 60 N 75
Gi cwi‡K›` Ae¯v’ b Ki‡e? Awf®² Z^ÅwfwîK eüwbeÆvPwb cÉGk²vîi 165. ADC I AEC Gi †¶Îdj؇qi
K AwZfzR ewnfv© ‡M wb‡Pi wP‡Îi wfwˇZ 161 163 bs AbycvZ †KvbwU? (Aby. 4)
M Af¨š—‡i N Dc‡ii me¸‡jv
cÖ‡kœi DËi `vI : K 1:2 L 2:1
eüc`x mgvwµ¦mPƒ K eüwbevÆ Pwb cÉGk²vîi M 3:4 1:1
A
D 4C
157. i. mgevû wÎfz‡Ri cÖ‡Z¨K †Kv‡Yi gvb 45 O
ii. mgevû wÎf‡z Ri wZbwU evû ci¯úi P CB 3
mgvb 161. wP‡Î OCB Gi †¶‡Î wb‡Pi †KvbwU A 6 E FB
iii. evû‡f‡` wÎfRz wZb cKÖ vi mwVK?
wb‡Pi †KvbwU mwVK? Dc‡ii Z‡_¨i Av‡jv‡K 166 I 167 bs
OCB = OBC L BO = BC cÖ‡kœi DËi `vI :
K i I ii L i I iii M OB OC N Dc‡ii me¸‡jv 166. AECD †Kvb ai‡bi PZfz ©yR?
ii I iii N i, ii I iii
158. i. e‡M©i evû a n‡j cwimxgv a 2 162. wb‡Pi †KvbwU mwVK? K AvqZ‡¶Î L eM©
ii. e‡Mi© evû a n‡j KY© a 2
iii. e‡Mi© evû †`Iqv _vK‡j eM© AuvKv hvq K ACB = 1 BOC M i¤^m mvgvš—wiK
wb‡Pi †KvbwU mwVK? 2
K i I ii L i I iii ii I iiiN i, ii I iii 167. ABCD e„˯’ n‡j wb‡Pi †KvbwU mwVK?
1
BAC = 2 BOC AD = BC L DC = AF
M CD = AB N BE = BC
M AC = OA + OB N BC = AC OC
Aa¨vq 05 mgxKiY
mvaviY eüwbevÆ Pwb cÉGkv² îi 177. 5x2 2x 3 = 0 mgxKi‡Y x-Gi mnM 186. yx = 9, y2 = 3x mgxKiY †Rv‡Ui GKwU
KZ? mgvavb| (Ab.y 4)
168. ax2 + bx + c = 0 mgxKi‡Yi x Gi NvZ K5 L3 M2 2 ( )L
KZ? 178. y2 + 7y + 12 = 0 mgxKi‡Y y Gi mnM K ( 3, 3) 2 1
K1 2 M3 N4 3
KZ?
169. 2z3 z2 4z + 4 = 0 mgxKiYwU KZ K 1 7 M 12 N 19 ( ) 2 1 N ( 2, 3)
Nv‡Zi? 3
K1 L2 3 N4 179. 2x = 1 mgxKi‡Yi gj~ †KvbwU? 187. b2 4ac = 0 mgxKi‡Yi g~jØq Kxiƒc
x1 n‡e?
170. wb‡Pi †KvbwU GK PjKwewkó wÎNvZ K1 1 M2 N2 K ci¯úi mgvb
180. 5x2 + 8x = x + 2 mgxKiYwU KZ Nv‡Zi?
mgxKiY? K5 2 M8 N1 ev¯—e I ci¯úi mgvb
K 3x 3 = 3 L x3=3
M 3x2 2x 5 = 0 181. (x 5)2 = 0 mgxKi‡Y x Gi gj~ Øq wb‡Pi M Aev¯—e ci¯úi mgvb
N ev¯—e I ci¯úi Amgvb
x3 x2 + 2x 2 = 0 †KvbwU?
171. y †K PjK a‡i b3y + c = 0 mgxKiYwUi NvZ 188. b2 4ac < 0 n‡j gj~ Øq Kxiƒc n‡e?
wb‡Pi †KvbwU? K 1, 5 5, 5 M 5, 10 N 10, 25
182. (y + 1)2 (y 1)2 = 4y mgxKiYwU y Gi Aev¯—e L ev¯—e M mgvb N c~Y©eM©
K3 L0 M2 1 †Kvb gv‡bi Rb¨ wm× n‡e?
172. x2 4x 12 = 0 mgxKi‡Y x Gi g~jØq 189. b2 4ac FYvZ¥K n‡j g~jØq Kxiƒc n‡e?
K1 L4 K KvíwbK L ev¯—e
wb‡Pi †KvbwU? mKj gv‡bi Rb¨
M 100 M RwUj K I M DfqB
K 2, 6 2, 6 183. x2 x 12 = 0 mgxKiYwU‡K ax2 + bx 190. hw` x = a Ges c 0 nq Z‡e |
M 2, 6 N 2, 6 c = 0 Gi mv‡_ Zzjbv K‡i b Gi gvb x a x2 a
173. x2 + 4x + 3 = 0 mgxKi‡Y x Gi g~jØq c c c2 c2
†KvbwU? (Aby. 1) = L =
wb‡Pi †KvbwU? K0 L1 1 N3 M x = a2 N x2 = a2
c2 c c c2
K 3, 4 L 3, 4 184. 16x = 4x + 1 mgxKiYwUi mgvavb †KvbwU? 191. 2x + 7 = 4x+ 2 mgxKiYwU Kx ai‡bi
M 1, 3 1, 3 (Aby. 2) mgxKiY?
174. (z 6)2 = 0 mgxKi‡Yi gj~ KqwU? K2 L0 M4 1 K wØNvZ mgxKiY L mij mgxKiY
185. x2 x + 13 = 0 n‡j mgxKiYwUi GKwU gj~ †KvbwU?
K 1wU 2wU M 4wU N 6wU (Ab.y 3) mP~ K mgxKiY N eûNvZ mgxKiY
175. x2 x 42 = 0 mgxKi‡Yi g~j KqwU?
K 1wU L 42wU 2wU N 4wU 1 + + 51 1 51 8
K 2 L 2 192. 3x + 5 = 3x + 3 + 3 mgxKi‡Yi gj~ wb‡Pi
176. cÖ‡Z¨K exRMwYZxq m~Î GKwU| 1 + 51 N 1 + 53 †KvbwU?
K mP~ K L gvÎv A‡f` N NvZ 2 2 K2 L3 4 N5
193. 3.27x = 9x + 4 n‡j x = KZ? eüc`x mgvwµ¦mPƒ K eüwbevÆ Pwb cÉGkv² îi Awf®² Z^ÅwfwîK eüwbevÆ Pwb cÉGk²vîi
K8 L3 M4 7 208. wbðvq‡Ki †¶‡Î wb‡Pi Z‡_¨i wfwˇZ 215 I 216 bs
cÖ‡kœi DËi `vI :
194. 5x + 52x = 26 mgxKi‡Yi mgvavb wb‡Pi i. ax2 + bx + c = 0 mgxKi‡Yi (b2
†KvbwU?
2 L3 M5 N7 4ac) †K wbðvqK ejv nq `By wU abvZ¥K cY~ ©msL¨vi e‡Mi© Aši— 11
Ges ¸Ydj 30|
195. a 1 n‡j ax = am n‡e hw` †Kej hw` ii. b2 4ac > 0 n‡j mgxKiYwUi gj~ Øq 215. msL¨v `By wU Kx Kx? (Aby. 5)
wb‡Pi †KvbwU nq? ev¯—e Amgvb g~j` nq
K a=x L a=m K 1 Ges 30 L 2 Ges 15
iii. b2 4ac = 0 n‡j g~jØq mgvb nq 5 Ges 6 N 5 Ges 6
x=m N x=m wb‡Pi †KvbwU mwVK?
216. msL¨v `yBwUi e‡Mi© mgwó KZ? (Ab.y 6)
196. 3x + 9y = 12 K i I ii L i I iii
2x y = 8
M ii I iii i, ii I iii K1 L 5 M 41 N 41
Dc‡ii mgxKiY †Rv‡Ui (x, y) wb‡Pi wb‡Pi Z‡_¨i wfwˇZ 217 I 218bs
†KvbwU? 209. ax2 + bx + c = 0 mgxKi‡Yi †¶‡Î
i. a = 0 n‡j mgxKi‡Yi g~j ev¯—e n‡e c‡Ö kœi DËi `vI :
(4, 0) L (0, 4) 4x + 2 = 22x + 1 + 14 GKwU mP~ K mgxKiY|
ii. b = c = 0 n‡j, x = 0 n‡e|
M ( 4, 0) N (0, 4) 217. mgxKiYwU evgc¶‡K 2 Gi m~P‡K cKÖ vk
197. x2 xy + y2 = 21, x + y = 3 mgxKiY؇qi
mgvavb KZ? iii. a = 1 n‡j x = b b2 4c Ki‡j Kx n‡e?
2
wb‡Pi †KvbwU mwVK? K 22x + 2 22x + 4
K (1, 4) (4, 1) (4, 1), (1, 4) M 22x + 1 N 22x + 3
K i I ii L i I iii
M (1, 4) (1, 4) N (4, 1), (4, 1) 218. 22x = a a‡i mgxKiYwU‡K a Gi gva¨‡g
198. x + y = 52, x + y = 10 mgxKi‡Yi M ii I iii i, ii I iii cKÖ vk Ki|
y x
210. x2 6x + 15 x2 6x + 13 = 10 8 K 16a = 14 14a 14 = 0
mgvavb KZ? wb‡Pi Z_¨¸‡jv j¶ Ki :
(8, 2), (2, 8) L ( 8, 2), (2, 8) M 14 4a = 0 N 4a + 13 = 0
i. mgxKiYwU GKwU wØNvZ mgxKiY
M (2, 8), ( 2, 8) N †Kv‡bvwU bq wb‡Pi Z‡_¨i wfwˇZ 219 I 220 bs
ii. mgxKiYwUi gj~ ¸‡jv 1, 5 cÖ‡kœi DËi `vI :
199. `By wU abvZ¥K msL¨vi e‡M©i mgwó 41. x2 xy = 14, y2 + xy = 60
msL¨v `yBwUi ¸Ydj 20. msL¨vwU `By wU iii. mgxKiYwU GK NvZwewkó I GK
Pj‡Ki mgxKiY 219. x2 + y2 Gi gvb KZ?
KZ? K 64 L 75 74 N 84
K3I4 4I5 wb‡Pi †KvbwU mwVK?
i I ii L ii I iii 220. mgxKiY؇qi (x, y) Gi gvb wb‡Pi †KvbwU
M5I6 N7I8 n‡Z cv‡i?
200. `yBwU eM© msL¨vq e‡Mi© mgwó 13 Ges M i I iii N i, ii I iii
(7, 5) L (8, 5) M (9, 10) N (5, 7)
msL¨v `By wUi ¸Ydj 6, msL¨v `By wUi 211. 22x − 3·2x + 2 + 32 = 0 mgxKi‡Yi mgvavb
i. x = 2 wb‡Pi Z‡_¨i wfwˇZ 221 I 222bs
e‡Mi© Aš—i wbYq© Ki| cÖ‡kœi DËi `vI :
K4 5 M6 N7 ii. x = 3
1 `By wU eM‡© ¶‡Îi †¶Îd‡ji mgwó 650
201. `By wU abvZ¥K msL¨vq e‡Mi© Aš—i 11 Ges iii. x = 2 eM©wgUvi| H `yBwU eM‡© ¶Î `By evû Øviv
¸Ydj 30| msL¨v `By wU KZ? wb‡Pi †KvbwU mwVK? L ii I iii MwVZ AvqZ‡¶‡Îi †¶Îdj 323 eMw© gUvi|
i I ii
K5I4 L3I5 M i I iii N i, ii I iii 221. eM©‡¶Î `yBwUi evû h_vµ‡g x I y n‡j
6I5 N6I4
202. `yBwU abvZ¥K msL¨vq e‡M©i mgwó 337| 212. x + 2y 3 = 0, 4x y 3 = 0 †¶Îd‡ji mgwó mgxKi‡Yi gva¨‡g †`LvI|
x2 + y2 = 650 L x2 = 650 + y2
msL¨v `By wUi e‡Mi© Aš—i 175| msL¨vwU mgxKiY؇qi M 2x + 2y = 650 N x2y2 = 650
i. x Gi gvb 1
KZ? ii. y Gi gvb 1 222. †QvU eM‡© ¶ÎwUi evûi ˆ`N¨© KZ?
17 L 19 M 71 N 91
9 I 16 L 12, 15
iii. mgvavb n‡e (1, 1) wb‡Pi mgxKiY †RvU Aej¤^‡b 223 I
M 13, 14 N 15, 16 224bs c‡Ö kœi DËi `vI :
wb‡Pi †KvbwU mwVK?
203. `yB A¼ wewkó GKwU msL¨v A¼Ø‡qi mgwó K i I ii L i I iii 2x. 3y = 18
6 A¼Øq ¯v’ b wewbgq Ki‡j cÖvß msL¨vwU
g~j msL¨vi `kK ¯v’ bxq A‡¼i wZb ¸Y M ii I iii i, ii I iii 22x. 3y = 36
nq| msL¨vwU KZ? 213. x + 2y = 3 223. (x, y) Gi gvb KZ?
51 L 52 M 53 N 54 4x y = 3 mgxKiY؇q K (2, 1) (1, 2) M (4, 3) N (3, 5)
i. x Gi gvb 1
204. xy = yx Ges x = 2y mgxKiY †Rv‡Ui 224. Dc‡iv³ mgxKiY †Rv‡Ui mgvavb Ges (1, 3)
mgvavb (x, y) = KZ? ii. y Gi gvb 2 we›`y `yBwU †jLwP‡Î ¯v’ cb Ki‡j †iLvwU
K x-A‡¶i mgvš—ivj
(4, 2) L (0, 6) iii. y Gi gvb 1 y-A‡¶i mgvš—ivj
M g~j we›`y w`‡q Mgb Ki‡e
M ( 2, 4) N ( 4, 2) wb‡Pi †KvbwU mwVK? N x I y-A¶‡K †Q` Ki‡e
`yBwU abvZ¥K cY~ m© sL¨vi e‡M©i Aš—i 9
205. x2 5x + 4 = 0 mgxKi‡Yi gj~ `wy U wb‡Pi K i I ii i I iii Ges ¸Ydj 20.
†KvbwU? Dc‡ii Z‡_¨i wfwˇZ 225I 226 bs
M ïay i N ïay ii cÖ‡kœi DËi `vI :
1, 4 L 2, 4 M 3, 4 N 2, 3 225. msL¨v `yBwU Kx Kx?
214. 18yx y2x = 81, 3x = y2 mgxKiY †Rv‡Ui
206. x2 6x + 9 = 0 mgxKi‡Yi mgvavb wb‡Pi
†KvbwU? mgvavb
i. (x, y) = (2, 3)
3, 3 L 3 M 4, 3 N 2, 3
ii. (x, y) = (2, 3) K 3, 4 4, 5 M 3, 5 N 5, 6
207. x2 2x 2 = 0 mgxKi‡Yi mgvavb wb‡Pi iii. (x, y) = (0, 4)
†KvbwU? 226. msL¨v `By wUi e‡Mi© mgwó KZ?
wb‡Pi †KvbwU mwVK?
1 3 L2 3 i I ii L i I iii
M3 3 N4 3
M ii I iii N †KvbwU bq K1 L 25 41 N 61
Aa¨vq 06 AmgZv
mvaviY eüwbeÆvPwb cÉGk²vîi 241. †`vjv 14 eQ‡i eq‡m †RGmwm cix¶v Awf®² Z^ÅwfwîK eüwbevÆ Pwb cÉGk²vîi
227. 5x + 5 > 25 AmgZvwUi mgvavb †mU w`‡qwQ‡jv| 16 eQi eq‡m †m Gm.Gmwm.
†KvbwU? (Ab.y 1) cix¶v w`‡e| Zvi eZg© vb eqm x eQi wb‡“ AmgZvwU †_‡K 250 I 251 b¤^i
cÖ‡kœi DËi `vI :
S = {x R : x > 4} n‡j,|
x
L S = {x R : x < 4} K x < 14 L x > 16 x 4 + 3
M S = {x R : x 4} 14 < x < 16 N 14 > x < 16 250. AmgZvwUi mgvavb †mU †KvbwU? (Ab.y 4)
N S = {x R : x 4} 242. GKwU msL¨vi 5 ¸Y Aci GKwU msL¨vi K S = {x R : x > 4}
`By ¸Y A‡c¶v Kg| msL¨v `By wU h_vµ‡g
228. x + y = 2 mgxKiYwU‡Z x Gi †Kvb x I y n‡j, wb‡Pi †Kvb AmgZvwU mwVK? L S = {x R : x < 4}
gv‡bi Rb¨ y = 0 n‡e? (Aby. 2)
5x < 2y L 5x > 2y S = {x R : x 4}
K2 L 0 M 4 2
229. 3 < 10 AmgZvi Dfqc¶ †_‡K 2 we‡qvM M 2x < 5y N 2x > 5y N S = (x R : x 4}
Ki‡j wb‡Pi †KvbwU n‡e?
243. GK UzKiv KvM‡Ri †¶Îdj 66 eM© †m. 251. AmgZvwUi mgvavb †m‡Ui msL¨v‡iLv
wg. Zv †_‡K x †m. wg. ˆ`N¨© Ges 6 †m. †KvbwU?
K 3<8 L 1 < 10 wg. c¯Ö ’ wewkó AvqZvKvi KvMR †K‡U (Ab.y 5)
1<8 N 3 < 12 †bIqv n‡jv| Zvn‡j x Gi m¤¢ve¨ gvb| K
230. 2x > 18 Gi mgvavb wb‡Pi †KvbwU? 6 < x < 11 L 6 < x < 66 2 1 0 1 2 3 4 5 6
x>9 L x<9 M 3 < x < 22 N 6 > x > 66 L
M x > 20 N x < 16 244. †j‡Li Dci Aew¯’Z cÖ‡Z¨K we›`y P Gi 2 1 0 1 2 3 4 5 6
231. 4(y 2) < 8 Gi mgvavb †mU wb‡Pi
†KvbwU? Rb¨ |
2 1 0 1 2 3 4 5 6
N
K S = { y R : y > 4} K f (p) > 0 L f (p) < 0 2 1 0 1 2 3 4 5 6
f (p) = 0 N f (p) ≈ 0 3x 5 > 7 GKwU AmgZv|
L S = { y R : y = 4} 245. x + y = 4 mgxKiYwU‡Z x Gi †Kvb gv‡bi Dc‡ii eYb© v n‡Z 252 I 253 bs cÖ‡kœi
DËi `vI :
S = { y R : y < 4} Rb¨ y = 0 n‡e?
N S = { y R : y 4} K0 L 2 4 N 4 252. c`Ö Ë AmgZvi mgvavb wb‡Pi †KvbwU?
232. y 3 < 5 n‡j, wb‡Pi †KvbwU mwVK?
eüc`x mgvwµ¦mPƒ K eüwbeÆvPwb cGÉ k²vîi K x > 2 x>4
Ky>8 y<8 My<2 Ny>2 3
233. hw` 3(x 2) < 6 nq Z‡e wb‡Pi †KvbwU
mwVK? 246. wiZv, wgZv I exw_i eqm h_vµ‡g x, 2x I M x<4 N x < 2
3x eQi Ges Zv‡`i wZbR‡bi eq‡mi 3
Kx>4 x<4 Mx>2 Nx>6 mgwó Ab~aŸ© 60 eQi n‡j 253. wb‡Pi †KvbwU c`Ö Ë AmgZvi mgvavb
234. x < 2 n‡j, wb‡Pi †KvbwU mZ¨? i. mgm¨vwUi MvwYwZK cÖKvk x + 2x + 3x 60 †mU?
x>2 L x<2 ii. wiZvi eqm 10 eQi K S = {x R : x < 23}
iii. wgZvi eqm > 20 eQi
Mx>2 N x < 2 wb‡Pi †KvbwU mwVK? (Aby. 7)
235. 2xy + y = 3 mgxKiYwUi mwVK ¯v’ bvsK
†Kvb¸‡jv? (Aby. 3) L S = {x R : x < 4}
i, ii L i, iii S = {x R : x > 4}
K (1, 1), (2, 1) (1, 1), (1, 3) M ii, iii N i, ii I iii N S = {x R : x < 12}
M (1, 1), (2, 1) N (1, 1), (2, 1) 247. a, b I c wZbwU ev¯—e msL¨v| a > b Ges wkgy 5 UvKv `‡i x wU †cwÝj Ges 12 UvKv
236. hw` a < b nq, Z‡e c Gi abvZ¥K gv‡bi `‡i (x + 4) wU LvZv wK‡b‡Q| †gvU g~j¨
c 0 n‡j
Rb¨| i. ac > bc; hLb c > 0 Ab~aŸ© 133 UvKv n‡j, |
K a > b a < b ii. ac < bc; hLb c < 0 Dc‡ii eYb© v n‡Z 254 I 255 bs c‡Ö kœi
c c c c DËi `vI :
a bc,
M a = b N a b iii. c < hLb c > 0 254. wkgy KZ UvKv w`‡q †cwÝj wKbj?
c c c c
wb‡Pi †KvbwU mwVK? (Aby. 8) K (x + 5) UvKv L (x 5) UvKv
237. hw` x < y nq, Z‡e z Gi FYvZ¥K gv‡bi i, ii L i, iii
5
Rb¨| M ii, iii N i, ii I iii M x UvKv 5x UvKv
K x = y L x < y 248. Amgvb ivwk‡K mgvb mgvb FYvZ¥K msL¨v 255. wkgy me©vwaK KZwU †cwÝj wK‡bwQj?
z z z z
Øviv| K mev© waK 2wU L mev© waK 4wU
x > y x y i. ¸Y Ki‡j AmgZvi w`K cv‡ë hvq mev© waK 5wU N me©vwaK 12wU
z z N z z
238. 8 > 5 AmgZvi Dfq c‡¶ 3 †hvM Ki‡j ii. we‡qvM Ki‡j AmgZvi w`K cv‡ë hvq GKwU AvqZvKvi †¶‡Îi ˆ`N©¨ I cÖ‡¯i’
iii. fvM Ki‡j AmgZvi w`K cv‡ë hvq cv_K© ¨ 2 GKK Ges cÖ¯’ x GKK|
wb‡Pi †KvbwU n‡e? wb‡Pi †KvbwU mwVK? †¶Îdj 8 eM© GKK A‡c¶v eo|
K 8=8 L 8>8 K i I ii i I iii Dc‡ii eY©bv n‡Z 256 I 257 bs cÖ‡kœi
M 11 > 5 11 > 8 M ii I iii N i, ii I iii DËi `vI :
239. 2x + 5 > 11 Gi mgvavb n‡e 249. 5 < 8 AmgZvwUi |
K x>3 L x<3 i. Dfqc‡¶ 3 †hvM Ki‡j nq 8 < 11
x>3 N x3 256. mgm¨vwU AmgZvi gva¨‡g cKÖ vk Ki‡j
240. GKRb QvÎ 5 UvKv `‡i XwU Kjg Ges 4 ii. Dfqc¶‡K 2 Øviv ¸Y Ki‡j nq n‡e
UvKv `‡i (X + 4) wU LvZv wK‡b‡Q| †gvU 10 > 16 K x (x + 2) + 8 0 x (x + 2) 8
g~j¨ 124 UvKv n‡j †m me©vwaK KqwU Kjg M x (x + 2) 8 0 N 8 > x (x + 2)
iii. Dfq c¶ †_‡K 4 we‡qvM Ki‡j nq 1 > 4
wK‡b‡Q? wb‡Pi †KvbwU mwVK? 257. ˆ`N¨© cÖ‡¯’i KZ ¸Y?
K mev© waK 5wU
mev© waK 12wU L mev© waK 6wU i I ii L i I iii wظY L A‡a©K
N mev© waK 10wU
M ii I iii N i, ii I iii M GK-PZz_©vsk N `By -Z…Zxqvsk
Aa¨vq 07 Amxg aviv
mvaviY eüwbevÆ Pwb cÉGkv² îi 270. 1 1 + 1 1 + ....... avivwUi mvaviY AbycvZ KZ? 281. 0, 2, 0, 2, 0, 2, ........... AbyµgwUi
258. mgvš—i avivi n Zg c` KZ? 3 32 33 34
i. mvaviY c`, 1 + ( 1)n
K 2a + (n 1)d L a + (2n 1)d K 1 1 M3 N1
3 3 ii. r Zg c`, 2 (r we‡Rvo n‡j)
a + (n 1)d N a + (n 1)2d 271. 16 + 4 + 1+ 1 + ....... avivwUi Z…Zxq iii. AbyµgwUi 10 Zg c`, 2
259. ¸‡YvËi Amxg avivi n Zg c` KZ? 4
AvswkK mgwó KZ?
arn 1 L arn M arn 2 N arn + 1 K 20 L 19 21 N 18 wb‡Pi †KvbwU mwVK?
260. r > 1 n‡j, ¸‡YvËi Amxg avivi n c‡`i mgwó KZ? 272. 25 + 5 + 1 + 1 + ......... avivwUi 12 Zg c` KZ? K i I ii L ii I iii
5
K a (1 rn) L a (1 rn) i I iii N i, ii I iii
1r r 1
K 1 L 1 1 N 1 282. 10 + 2 + 2 + 2 + ............ GKwU Amxg ¸‡YvËi
(rn 1) (rn 1) 57 58 59 510 5 52
a r1 N a 1 r
273. 1 + 1 + 1 + 1 + ......... avivwUi c_Ö g 10wU aviv|
2 22 23 62
261. GKwU ¸‡YvËi avivi mvaviY AbycvZ 1 Ges c‡`i mgwó KZ? i. avivwUi Z…Zxq AvswkK mgwó, 5
2
AmxgZK mgwó 8 n‡j, avivwUi 1g c` KZ? 1023 511 256 1023 1
1 1 K 1024 L 512 M 512 512 ii. avivwUi mvaviY AbycvZ, 5
2 4
K2 L 4 N 274. 6 + 3 + 3 + 3 + ....... avivwUi AmxgZK mgwó KZ? iii. avivwUi beg c`, 2
2 4 56
262. 1, 3, 5, 7 Abµy gwUi 12 Zg c` †KvbwU? wb‡Pi †KvbwU mwVK?
(Aby. 1) 12 L 18 M 21 N 6
K 12 L 13 23 N 25 i I ii L ii I iii
1 275. GKwU mgvši— avivi 1g c` 9 Ges mvaviY M i I iii N i, ii I iii
n(n + Aši— 3 n‡j, avivwUi Z…Zxq c` †KvbwU?
263. †Kv‡bv Abyµ‡gi n Zg c` = 1) Gi K 12 15 M 18 N 21 3 32 33 34
7 7 7 7
3q c` †KvbwU? (Ab.y 2) 276. 1.23˙4˙ = wb‡Pi †KvbwU? 283. + + + + ........ avivwU GKwU
K 1 L 1 1 N 1 K 1234234234 ................ ¸‡YvËi Amxg aviv| 39
3 6 12 20 7
1 ( 1)n 12343434 ................. i. avivwUi Z…Zxq AvswkK mgwó,
264. †K‡bv Abyµ‡gi n Zg c` = 2
M 12342424 ................... 320
n‡j 20 Zg c` †KvbwU? (Ab.y 3) ii. avivwUi 20 Zg c`, 7
N 12342323 ................
0 L1 M1 N2 iii. avivwUi AmxgZK mgwó, 3
277. 5 5 + 5 5 + ............ avivwUi PZ_z © AvswkK mgwó KZ? 8
265. 1 + 1 + 3 + 2 + 5 + ...... avivwUi 15 Zg c` 0 L1 M1 N2
2 2 2 1 1 1 wb‡Pi †KvbwU mwVK?
†KvbwU? 278. (x + 2) + (x + 2)2 + (x + 2)3 + ....... avivwUi
i I ii L ii I iii
15 13 mvaviY AbycvZ KZ? M i I iii N i, ii I iii
K 15 2 M 2 N 13 1 1
K + L + 1)2
2 22 23 24 (x 1) (x Awf®² Z^ÅwfwîK eüwbevÆ Pwb cÉGk²vîi
266. 7 72 73 74 avivwUi mvaviY
+ + ......... (x 1 2) N (x 1
+ + 2)2
AbycvZ KZ? wb‡Pi avivwU j¶ Ki Ges 284286
eüc`x mgvwµ¦mƒPK eüwbevÆ Pwb cGÉ kv² îi b¤i^ c‡Ö kiœ DËi `vI :
2 L 2 M 7 N 7 4, 43, 49, ......
7 7 2 2 279. GKwU Abyµ‡gi n Zg c` un = 1 (1)n n‡j, GiÑ
267. GKwU ¸‡YvËi avivi 1g c` 1 Ges mvaviY i. 10 Zg c` 0
2 ii. 15 Zg c` 2 284. avivwUi 10 Zg c` †KvbwU? (Aby. 6)
Abcy vZ 7 n‡j, avivwUi AmxgZK mgwó KZ? iii. cÖ_g 12 c‡`i mgwó 12 4 4
wb‡Pi †KvbwU mwVK? (Aby. 5) K 310 39
5 7 9 7 K i, ii L i, iii M ii, iii i, ii I iii
K 7 L 5 M 7 9 M 4 N 4
280. 4 + 8 + 12 + 16 + ......... 311 312
268. mvaviY c` 2n 1 Gi Abyµg †KvbwU? i. GwU GKwU mgvš—i aviv 285. avivwUi 1g 5 c‡`i mgwó KZ? (Aby. 7)
ii. avivwUi mvaviY Aš—i 4
iii. avivwUi r Zg c` 4r
K 1, 2, 3, .... 1, 3, 5, ...... wb‡Pi †KvbwU mwVK? K 160 484
K i I ii L ii I iiiM i I iii i, ii I iii 27 81
M 2, 4, 6, ..... N 2, 3, 54, .....
M 12 N 20
9 9
269. 3 + 9 + 15 + 21 + ......... avivwUi mvaviY Aši— KZ? 286. avivwUi AmxgZK mgwó KZ? (Ab.y 8)
K3 L4 M5 6
K0 L5 6 N7
Aa¨vq 08 w·KvYwgwZ
mvaviY eüwbeÆvPwb cÉGkv² îi 289. GKwU D †Kv‡Yi eË„ xqgvb RC n‡j wb‡Pi 291. 300 †KvYwU †Kvb& PZfz ©v‡M _vK‡e? (Aby. 2)
†KvbwU mwVK? c_Ö g L wØZxq M Z…Zxq N PZz_©
287. GKwU PvKvi cwiwa 44 wgUvi n‡j PvKvi
e¨vmva© wb‡Pi †KvbwU? D R D 292. 930 †Kv‡Yi Ae¯’vb |
180 = L = R K c_Ö g PZfz ©v‡M wØZxq PZzf©v‡M
K 3.5 wgUvi 7 wgUvi 180
M Z…Zxq PZzf©v‡M N PZz_© PZzf©v‡M
M 9 wgUvi N 5 wgUvi M R = D N Dc‡ii me¸‡jv 293. †Kv‡Yi gvb wb‡Pi †KvbwU n‡j Zv Z…Zxq PZzf©v‡M Ki‡e?
288. †iwWqvb †KvY GKwU | 180
K mg‡KvY 290. sin A = 1 n‡j, sin 2A Gi gvb KZ? K 0 90 L 90 180
180 270 N 270 360
L mij‡KvY 2 294. 3.1416 †iwWqvb = KZ wWwMÖ?
(Aby. 1)
a“ª e‡KvY 1 1 K 100 (cvÖ q) L 90 (cÖvq)
N ¯’j‚ ‡KvY K 2 L 2 1 N2 M 360 (cÖvq) 180 (cÖvq)
295. Gi gvb wb‡Pi †KvbwU n‡e? 312. cos 2 = 0 n‡j, Gi gvb wb‡Pi †KvbwU? 320. Y
K 2.1416 L 3.6116 P(o, y)
3.1416 N 3.1146 K M N
2 4 3 6
296. GKwU PvKvi cwiwa 3.1416 wgUvi n‡j
e¨vm KZ n‡e? 313. C X O Q(x, o) X
Y
1 wgUvi L 3.1416 wgUvi 3
wP‡Î,
1 i. tan = x
2 y
M 2 wgUvi N wgUvi A wÎ3 fz‡R B Gi gvb wb‡Pi †KvbwU?
297. 65 42 = wb‡Pi †KvbwU? ABC y
+
K 62.8 L 66.8 65.7 N 65.9 L M N ii. cos = x2 y2
3 4 6 2
298. wÎfz‡Ri wZbwU †KvY Abcy vZ 1 t 2 t 3
n‡j ¶z`ªZi †Kv‡Yi eË„ xq gvb KZ? eüc`x mgvwµ¦mPƒ K eüwbevÆ Pwb cÉGk²vîi iii. PQ = x2 + y2
2 2 314. sin + cos = 1 n‡j Gi gvb n‡e wb‡Pi †KvbwU mwVK?
3 5
K L N i. 0 ii. 30 iii. 90 K i I ii i I iii
3 6
299. e„‡Ëi Pvc 13 †m. wg. I e¨vmva© 17 †m. wb‡Pi †KvbwU mwVK? (Aby. 3) M ii I iii N i, ii I iii
K i L ii M i I ii i I iii
wg. n‡j †K›`¯ª ’ †KvY wb‡Pi †KvbwU? 321. tan = 2 Ges sin FYvZ¥K n‡j,
3
K 40.5 L 45.23
43.81 N 53.38 315. x = 2 sin 2 Gi †¶‡Î 3
i. cos = 13
( )300. sec n‡j,
Gi gvb wb‡Pi †KvbwU? i. = 4 x = 0 ii. sin = 2
4 13
ii. = n‡j, x=0
K 2 2 M 1 N 1 2
2 2 3
2 iii. = n‡j, x = 3 iii. 2 < < 2
3 3
301. ABC-G cosec = n‡j wÎfzRwUi wb‡Pi †KvbwU mwVK?
wb‡Pi †KvbwU mwVK?
AwZfzR KZ? K i I ii L i I iii
ii I iii N i, ii I iii K i I ii i I iii
2 GKK L 7 GKK
M 1 GKK N 3 GKK M ii I iii N i, ii I iii
316.
302. ABC wÎf‡z R tan = 3 3 n‡j Gi AwZfRz x2 C 322. cot = 2 n‡j,
A 3
x
Gi gvb KZ? O B i. tan = 3
2
D
K 20 GKK L 5 GKK
2 7 GKK N 7 2 GKK 13
2 wP‡Î ii. sec = 2
5 i. DOC wÎfz‡R OD = x
303. ABC-G sin = n‡j, tan Gi gvb iii. cosec = 13
3
wb‡Pi †KvbwU? ii. AB = 2x iii. =
4 wb‡Pi †KvbwU mwVK?
K1 2 M 5 N3 wb‡Pi †KvbwU mwVK?
K i I ii L i I iii
304. sec2 tan2 Gi gvb wb‡Pi †KvbwU? K i I ii i I iii
3 3 M ii I iii N i, ii I iii M ii I iii i, ii I iii
K2 3 L3 M3 3 1 317. wPÎ Abmy v‡i A 323. C = n‡j,
4
305. sin2 cos2 Gi gvb wb‡Pi †KvbwU? 4
2 2 i. tan = 3 i. cos C = 1
4 2
K0 1 M 1 N 2 ii. sin = 5
2 3
ii. cosec C = 2 iii. cot C = 1
306. sec2 Gi gvb wb‡Pi †KvbwU? 9 B 3C wb‡Pi †KvbwU mwVK?
4 1 iii. cos2 = 25
K2 L 3 M 1 1 wb‡Pi †KvbwU mwVK? (Aby. 4) K i I ii L i I iii
2 3 M ii I iii i, ii I iii
K i I ii i I iii
M ii I iii N i, ii I iii
307. A = 6 Ges B = 6 n‡j, tan A + tan B 318. GK e¨w³ e„ËvKvi c‡_ NÈvq 6 wK. wg. Awf®² Z^ÅwfwîK eüwbeÆvPwb cÉGk²vîi
1 tan A tan B
Gi gvb wb‡Pi †KvbwU? wb‡Pi wP‡Îi Av‡jv‡K 324 325bs c‡Ö kiœ
1 †e‡M †`Š‡o 36 †m‡K‡Û †h e„ËPvc DËi `vI :
2 1 AwZµg K‡i Zv †K‡›`ª 36 †KvY Drcbœ
K1 L 3 N 3 C
( )308. tan 17 Gi gvb wb‡Pi †KvbwU? K‡i|
4 i. H e¨w³i AwZµvš— `~iZ¡ 60 wgUvi
ba
1 1 ii. e‡„ Ëi e¨vmva© 95.5 wgUvi
K1 1 M 3 N 3 iii. e„‡Ëi cwiwa 110.97 wgUvi
( )309. sec 31 + Gi gvb wb‡Pi †KvbwU? wb‡Pi †KvbwU mwVK? AB
2 3
i I ii L i I iii 324. sin B + cos C = KZ? (Ab.y 5)
2 2 3 3
3 L 3 M 2 N 2 M ii I iii N i, ii I iii 2b L 2a
a b
310. = n‡j, cos 2 Gi gvb wb‡Pi †KvbwU? 319. sin = 1 Ges tan abv—K n‡j,
6 2 M a2 + b2 N ab
ab a2 + b2
1 L 3 M 2 N 3 3 1
2 2 3 2 i. cos = 2 ii. tan = 3 325. tan B Gi gvb †KvbwU? (Ab.y 6)
311. = n‡j, 2 cos2 1 Gi gvb wb‡Pi †KvbwU? iii. cosec = 3 K a2 a b2 L a2 b b2
3
1 1 wb‡Pi †KvbwU mwVK?
K 2 2 a b
i I ii L i I iii a2 b2 a2 b2
M2 N0 M
M ii I iii N i, ii I iii
A 329. wP‡Î sin Gi gvb wb‡Pi †KvbwU? ABC wÎf‡z R tan A = 5 Ges tan A I cos
12
5 K 2x L x M x2 + 1 x2 1 A wecixZ wPý wewkó|
x2 + 1 x2 1 x2 1 x2 + 1
Dc‡ii Z‡_¨i Av‡jv‡K 334 I 335 bs
A = Ges B = c‡Ö kiœ DËi `vI :
4 3
B2C
Dc‡ii Z‡_¨i Av‡jv‡K 330 I 331 bs 334. cot A Gi gvb wb‡Pi †KvbwU?
Dc‡ii wPÎ n‡Z 326 I 327 bs cÖ‡kœi c‡Ö kœi DËi `vI : 13 12 5 12
DËi `vI : 330. tan A Gi gvb wb‡Pi †KvbwU? K 5 L 13 M 3 5
326. ABC wÎfz‡Ri AB evûi ˆ`N©¨ wb‡Pi
†KvbwU? K2 L1 1 N 3 335. ABC wÎfz‡Ri AwZfRz wb‡Pi †KvbwU?
K 13 L 12
K 3 L 3 M 3 1 331. 2 tan B Gi gvb wb‡Pi †KvbwU? 13 N 12
2 2 1 tan2 B
A
327. ABC-G cot Gi gvb wb‡Pi †KvbwU? K2 3 M0 N
K 1 L 5 2 N5 AmsÁvwqZ 10
2 2 2
C cosec = 3 2 3 Ges < < 32.
Dc‡ii Z‡_¨i Av‡jv‡K 332 I 333 bs
x2 + 1 x2 1 c‡Ö kiœ DËi `vI : BC
332. sec Gi gvb wb‡Pi †KvbwU? Dc‡ii eY©bv n‡Z 336 I 337 bs cÖ‡kiœ
DËi `vI :
A B 336. BC evûi ˆ`N©¨ wb‡Pi †KvbwU?
OD
12 †m. wg. 3 3 3 3 23 23 K 8 GKK L 4 GKK
23 23 3 3
K M 3 N 3 M 2 2 GKK 6 GKK
Dc‡ii wPÎ n‡Z 328 I 329 bs c‡Ö kœi 333. cot Gi gvb wb‡Pi †KvbwU? 337. sec Gi gvb wb‡Pi †KvbwU?
DËi `vI :
328. ODC-G OD Gi ˆ`N©¨ wb‡Pi †KvbwU? 23 L 23 10 L 10
2 2 6 2
Kx L x2 1 M x 2x M 1 N 3 M 6 N 2
2 2 2 2 6
Aa¨vq 09 m~PKxq I jMvwi`gxq dvskb
mvaviY eüwbeÆvPwb cÉGkv² îi 349. P = loga bc n‡j, 1 + p = KZ? 359. f(x) = | x | Gi †iÄ KZ?
K0 L 1 loga abc N abc x
a
a2 b2a + b 2n 2n K R L IR M {1, 1} { 1, 1}
( ) 1 350. 2n 2n + = KZ?
xa ab 1 1 360. y = ln a + x dvskbwUi †iÄ KZ?
a x
338. Gi mijgvb †KvbwU? (Aby. 1) K2 L 2n M 2n + 1 3
K0 L1 Ma x 2 R {a} L IR M R N R { a}
339. 3 (a3b5)3 = KZ? 21 361. y = | x | -GB dvsk‡bi †Wv‡gb KZ?
K a9b5 L a3b15 a3b5 N a5b3 351. hw` (64)3 + (625)2 = 3K nq Z‡e k = K ( , ) L (0, )
KZ?
340. hw` (16)x = (64)y n‡j x = KZ? K 932 L 912 R N R {0}
y
362. y = ax, a > 1 Z‡e Gi †Wv‡gb KZ?
K ( , ] L ( , 0]
K 2 3 M 4 N 3 1323 N 1225 M (0, ) ( , )
3 2 3 4
341. ax = bx ; a > 0 Ges x 0 n‡j a I b Gi 352. log2 1 -Gi gvb eüc`x mgvwµ¦mƒPK eüwbevÆ Pwb cGÉ k²vîi
b 32
m¤úK© Kxiƒc n‡e? 363. hw` a, b, p > 0 Ges a 1, b 1 nq
K 1 5 M 1 N 1
Ka>b La<b a=b Nab 25 5 5 Z‡e
a2
342. ax = 1; a > 0, x 0 n‡j a = KZ? a2 b2a + b i. loga p = logb p loga b
( )
K0 L1 1 N2 1 ab ii. loga a logb b logc c Gi gvb
343. 3x a = 5x a n‡j x = KZ? xa2
K0 a M3 N5 353. = KZ? 2
K0 L1 x Nx+1 iii. xloga y = yloga x
344. 0 < a < 1 Ges x < y n‡j †KvbwU mwVK? 354. log8 64 = ?
K ax < ay L ax = ay ax > ay N xa = ya Dc‡ii Z‡_¨i Av‡jv‡K wb‡Pi †KvbwU
K8 2 mwVK? (Ab.y 2)
1 1
345. hw` (16)x = (64) y n‡j x = KZ? M 64 N6 K i I ii L ii I iii
y 355. loga (abc) = x n‡j wb‡Pi †KvbwU mwVK?
K bx = abc ax = abc i I iii N i, ii I iii
2 L32 M43 N34 364. hw` ax = b nq, Z‡e
3 M cx = abc N ax = (abc)x = a i. loga b= x
356. hw` P = loga (bc) nq, Z‡e 1 P = ? ii. loga (ax) =
346. log5 25 + log5 625 = KZ? K loga abc L loga bc x iii. a loga b = a
wb‡Pi †KvbwU mwVK?
K 25 625 6 M 252 N ( ) loga a 1
5 bc loga (abc)
N i I ii L i I iiiM ii I iiiN i, ii I iii
5
( ) ( )347. log5 357. y = 3x †Wv‡gb KZ? 365. a > 0, n, k N, n > 1 n‡j
a 2 b 2 1
b a K ( , 0) L [0, ) i. n a = nk ak ii. = n a
+ log5 = KZ? an
0 L2 M1 N5 M (0, ) ( , ) 1 n
358. y = 3x Gi wecixZ dvsk‡bi †iÄ KZ?
348. logx 5 + logx 625 = 5 n‡j x = KZ? iii. an = a
5 L 25 M 125 N 625 wb‡Pi †KvbwU mwVK?
( , ) L ( , 0)
M (0, ) N [0, ) i I ii L i I iiiM ii I iii N i, ii I iii
366. x = loga b n‡e hw`| wb‡Pi †KvbwU mwVK? 379. x y z = KZ? L a2p
ii. a 1 iii. ax = b K i I ii L i I iii N a2p + q + 1
i. a > 0 K1
wb‡Pi †KvbwU mwVK? ii I iii N i, ii I iii a2p
K i I ii L i I iii Awf®² Z^ÅwfwîK eüwbeÆvPwb cÉGk²vîi 380. xq r . yr p . zp q = KZ?
M ii I iii i, ii I iii
367. ax = b, a > 0 Ges a 1 n‡j K0 1
372 374 bs c‡Ö kiœ DËi `vI : M a2(p2 + q2 + r2) N a2(p2 + q2 + r2)
i. loga b = x ii. loga (ax) = x hLb x, y, z 0 Ges ax = by = cz.
iii. a loga b = b 372. †KvbwU mwVK? (Aby. 3) y = ln a + x , x > 0
a x
wb‡Pi †KvbwU mwVK? yz
K i I ii L i I iii Dc‡ii eYb© v n‡Z 381 383 bs cÖ‡kœi
M ii I iii i, ii I iii K a = bz L a = cy DËi `vI :
z
368. x > 0, y > 0 Ges a 1 n‡j x = y n‡e a = cx N a b2 381. a + x > 0 Ges a x > 0 n‡j †Wv‡gb
hw` c KZ?
i. loga x > 0 ii. loga x = loga y 373. wb‡Pi †KvbwU ac Gi mgvb? (Ab.y 4)
yy yz K L (0, )
iwbii‡.Pilog†Ka yvb>wU0mwVK?
bx . bz L bx . by M ( , ) ( a, a)
K i I ii L ii yz z y 382. cÖ`Ë dvsk‡bi †Wv‡gb KZ?
M bx + y by z
M i I iii i, ii I iii N + K (0, ) R {a}
369. P = loga (bc) n‡j 1 + p = ? 374. b2 = ac n‡j wb‡Pi †KvbwU? (Aby. 5) M N ( , )
i. 1 + loga bc ii. loga a + loga bc
iwbii‡.Pilog†Ka (vabbwUc)mwVK? 1 1 2 1 1 2 383. c`Ö Ë dvsk‡bi †iÄ KZ?
x + z = y L x + y = z
K R {a} R
K i I ii L i I ii M 1 + 1 = 2 N 1 + 1 = z M (0, ) NN
M ii I iii i, ii I iii y z x x y 2
a = p, a4 = q f(x) = x + | x | hLb 2 x < 2
370. a > 0; m P Z, n , q N, n > 1, q > 1
n‡j, Dc‡ii eYb© v n‡Z 375 377 bs cÖ‡kœi Dc‡ii eYb© v n‡Z 384 386 bs cÖ‡kiœ
DËi `vI : DËi `vI :
( )i. 1 mq + nq
anq 375. a = q n‡j, q Gi NvZ KZ? 384. dvskbwU GKwU
= amq + np
m p m + p K1 4 M0 Na K jMvwi`wgK dvskb
an aq an q
ii. . = iii. amp . anq = amp + nq 376. (px)y = a2 n‡j xy = KZ? L mP~ K dvskb
wb‡Pi †KvbwU mwVK? K0 Lp 2 Nq ciggvb dvskb N wecixZ dvskb
377. pyqx = KZ? 385. cÖ`Ë dvsk‡bi †Wv‡gb KZ?
K i I ii L i I iii xy x
ii I iii N i, ii I iii
371. f(x) = ex; 2 < e < 3 Z‡e K a2 a4x + y M axy N ay K ( 2, 2) L ( 2, 2]
x = aq + r, y = ar + p, z = ap + q n‡j
i. †iLvwU (1, 0) we›`yMvgx| Dc‡ii eY©bv n‡Z 378 380 bs cÖ‡kœi M [ 2, 2] [ 2, 2]
ii. x-Gi FbvZ¥K gv‡bi Rb¨ y-Gi gvb DËi `vI : 386. c`Ö Ë dvsk‡bi †iÄ KZ?
µgvMZ n«vm cv‡e|
iii. dvskbwUi †iÄ (0, ) 378. q + r = o n‡j x = KZ? K (0, 4) L {0, 4}
1 La M0 Nq+r M (0, 4] N [0, 4]
Aa¨vq 10 wØc`x we¯w…— Z
mvaviY eüwbevÆ Pwb cGÉ k²vîi 393. (2 x) (1 + ax)5 †K x2 ch©š— we¯…Z— Ki‡j ( )401.1 + x 8
hw` 2 + 9x + cx2 cvIqv hvq, Z‡e a I c Gi 2
( )387.4 Gi we¯w—… Z‡Z x3 Gi mnM KZ?
0
= KZ? gvb (Ab.y 7) K 4 7 M 8 N 14
402. (a + x)n Gi we¯…—wZ‡Z n †Rvo n‡j KqwU ga¨c` _vK‡e?
1 L4 M0 N2 a = 1, c = 15 L a = 5, c = 15
( )388. M a = 15, c = 1 N a = 1, c = 0 K `yBwU GKwU M nwU N wZbwU
7 = KZ?
4 394. wØc`x ivwki NvZ n = 4 n‡j, c`msL¨v n‡e KZwU? 403. (a + x)n Gi we¯w—… Z‡Z n we‡Rvo n‡j we¯w…— Z‡Z
K 6wU L 4wU 5wU N 2wU
K 6! L 4!3! KZwU ga¨c` _vK‡e?
3!4! 7! 395. (1 x)8 (1 + x)7 Gi we¯w…— Z‡Z x7 Gi mnM KZ? 2wU L 3wU
M 3!3! 7! 35 L 40 M 30 N 25 M 4wU N wbYq© Kiv hvq bv
7! 4!3!
( )396. 2 8 ( )404. 1 2n
389. 120 = KZ? 1 + x x x
Gi we¯w—… Z‡Z c_Ö g c‡`i gvb KZ? Gi we¯…w— Z‡Z KqwU ga¨c` _vK‡e?
K 4! L 7! M 6! 5! K8 L 16 M 2 1 1wU L 2wU M 3wU N nwU
397. (x + 2y)5 ivwk‡Z c`msL¨v KZwU?
390. (x + y)5-Gi we¯—…wZ‡Z wØc`x mnM¸wj K5 L 4 M 10 6 ( )405.3x 1 10
n‡jv : (Ab.y 4) 2x
Gi we¯…w— Z‡Z ga¨c` KZZg?
K 5, 10, 10, 5 1, 5, 10, 10, 5, 1 398. 10C2 = KZ?
K 5-Zg 6-Zg M 10-ZgN 11-Zg
M 10, 5, 5, 10 N 1, 2, 3, 3, 2, 1 K 20 45 M 12 N 15 ( )406.
x y 10
( )391. (1 x) 1 + x 8 we¯—w… Z‡Z x Gi ( )399.x+1 6 y + x
2 x2 Gi we¯—v‡i ga¨c` n‡e
-Gi Gi we¯w—… Z‡Z x g³y c` †KvbwU?
10C 10C 10C 10C
mnM(Ab.y 5) 15 L 20 M 10 N 25 K 7 5 M 6 N 10
K 1 L 1 3 N 1 400. nC2 = †KvbwU? 407. 5C = KZ?3
2 2
n 1 n (n 1) (n 2) 5! L 5!
( )392. 4 K 2! L 3! 3!(5 5!(5
1 3)! 3)!
x2 + x2 -Gi we¯—…wZ‡Z x gy³ c` KZ? (Ab.y 6)
n (n 1) M 3! N 5!
1.2 N n (n 1) 3!(5 3)! 3!(5 1)!
K4 6 M 8 N 10
eüc`x mgvwµ¦mPƒ K eüwbeÆvPwb cÉGk²vîi Awf®² Z^ÅwfwîK eüwbeÆvPwb cÉGk²vîi 415. n = 6 n‡j ivwkwUi we¯—w… Z‡Z ga¨ c‡`i
msL¨v mnM KZ?
408. (1 + 2x + x2)3 Gi we¯—…wZ‡Z ( ) 1 n
x + x K1 L 6 M 15 20
i. c`msL¨v 4 , †hLv‡b n †Rvo msL¨v| (Aby. 2)
( ) 1 8
2x
ii. 2q c` 6x Dc‡ii Z_¨ †_‡K 411 I 412 bs cÖ‡kœi 3x2
iii. †kl c` x6 DËi `vI| Dc‡ii Z_¨vbhy vqx 416 418 bs cÖ‡kiœ
411. (r + 1) Zg c`wU x ewR©Z n‡j r Gi gvb DËi `vI :
wb‡Pi †KvbwU mwVK? (Aby. 1) KZ?
K i I ii L i I iii n 416. D³ wØc`x ivwki we¯w—… Z‡Z KqwU ga¨c`
2 _vK‡e?
ii I iii N i, ii I iii K0 L Mn 2n
409. (1 + x)5 Gi we¯—…wZ‡Z 412. n = 4 n‡j, PZz_© c` KZ? (Ab.y 3) K 8wU L 4wU
i. c`msL¨v 5wU ii. 2q c` = 5C1x1 K4 L 4x M 4 4 1wU N 5wU
iii. †kl c` = x5 x x2
417. D³ wØc`x ivwki we¯…w— Z‡Z KZZg c` ga¨c`
wb‡Pi Z‡_¨i Av‡jv‡K 413 415 bs
wb‡Pi †KvbwU mwVK? c‡Ö kœi DËi `vI : n‡e
K i I ii ii I iii ( )wØc`x ivwk 1 n K 8-Zg c` 5-Zg c`
x2
M i I iii N i, ii I iii x + G n cY~ m© sL¨v| M 4-Zg c` N 6-Zg c`
410. wb‡Pi Z_¨¸‡jv j¶ Ki : 413. ivwkwUi we¯—…wZ‡Z c‡`i msL¨v KZ? 418. D³ wØc`x we¯—w… Zi ga¨c`wUi gvb KZ?
i. 5C = 1 ii. 5C = n iii. 0! = 1
0n ( ) ( )K 8C5 3 5 3 4
Kn1 Ln n+1 N n + 1 2 8C4 2
wb‡Pi †KvbwU mwVK? 2 .x5 .x4
414. n = 6 n‡j we¯w—… Z‡Z x ewRZ© c‡`i gvb KZ? ( ) ( )M 8C2 3 2 3 4
K i I ii L ii I iii K6 15 M 20 N 30 2 N 8C4 2
i I iii N i, ii I iii .x2 .x 4
Aa¨vq 11 ¯’vbv¼ R¨vwgwZ
mvaviY eüwbeÆvPwb cGÉ kv² îi 427. y = x 3 Ges y = x + 3 Gi ( )436. 1a mg‡iL
†Q`we›`y(Aby. 7) A(a, b), B(b, a) Ges C ab
419. g~j we›`y n‡Z P(8, 6) we›`iy `~iZ¡ KZ?
10 GKK L 8 GKK K (0, 0) L (0, 3) n‡j (a + b) Gi gvb †KvbwU?
M 6 GKK N 14 GKK (3, 0) N ( 3, 3) 0 L 1 M1 N2
2
420. (1, 1) Ges (2, 2) we›`y `yBwUi ga¨eZx© `~iZ¡ KZ? 428. P we›`iy ¯v’ bv¼ (a, b) n‡j, ¯v’ bv¼wUi †KvwU wb‡Pi
†KvbwU? 437. †Kv‡bv mij‡iLv Øviv x A‡¶i abvZ¥K
K 2 2 GKK L 2 GKK w`‡Ki mv‡_ Drcbœ †KvY I Xvj m Gi
M 4 GKK 2 GKK Ka b M ab N a
b g‡a¨ m¤úK© wb‡Pi †KvbwU?
421. g~j we›`y n‡Z (sin , cos ) we›`yi `~iZ¡
wb‡Pi †KvbwU? 429. gj~ we›`y n‡Z mgZ‡j Aew¯’Z †h †Kv‡bv K m = sin L m = cos
we›`y P(x, y) Gi `i~ Z¡ KZ? M m = cot m = tan
438. A(2, 1), B(3, 5) Ges C(0, 7) we›`y
K sin + cos L sin2 K (x + y) GKK L (x2 + y2) GKK wZbwU mg‡iL n‡j, AC †iLvi Xvj KZ?
N x sin2
1 GKK x2 + y2 GKK N 1 K4 L2 M2 4
1 430. P(3, 5) we›`y n‡Z x A‡¶i ga¨eZx© `i~ Z¡ KZ?
422. {s(s a) (s b) (s G s Øviv K3 5 439. †Kv‡bv mij‡iLv A(x1, y1) Ges B(x2, y2)
c)}2
eSy vq(Ab.y 2) M 8 N 34 we›`y w`‡q AwZµg Ki‡j Gi Xvj Kx n‡e?
K wÎfz‡Ri †¶Îdj L e„‡Ëi †¶Îdj
wÎfz‡Ri Aa© cwimxgv 431. A( 3, 2), B( 5, 2) I C(2, 2) K m = x1 x2 L m = x2 x1
y1 y2 y2 y1
we›`y¸‡jv Øviv MwVZ wÎfzR‡¶‡Îi
N e„‡Ëi Aac© wiwa †¶Îdj KZ? M y1 y2 m = y2 y1
x1 x2 x2 x1
423. A 440. A(1, 2) Ges B(4, 1) we›`yØq Øviv
35 K6 L 12 14 N 28
AwZµvš— mij‡iLvi Xvj KZ?
BC 432. A(5, 6) I B ( 1, 4) we›`y `yBwUi ga¨eZx© `~iZ¡ K0 L1 1 N3
KZ? 441. A(1, 1), B(2, 2) Ges C(4, g) we›`y
wÎfRz wUi †¶Îdj(Aby. 3)
K 12 eM© GKK L 15 eM© GKK K 10 GKK L 20 GKK
6 eM© GKK N 60 eM© GKK wZbwU mg‡iLv n‡j, g Gi gvb KZ?
2 10 GKK N 4 10 GKK K4 L4 8 N8
424. A(1, 1) 433. A ( 2, 0) Ges B (1, 4) we›`y `ywUi
B(3, 3) ga¨eZx© `i~ Z¡ wb‡Pi †KvbwU? 442. y = mx + c †iLvq Xvj wb‡Pi †KvbwU?
AB †iLvi Xvj(Aby. 4)
Ky m Mx Nc
K 4 GKK L 3 GKK 443. y = 5x + 3 mij‡iLvi Xvj wb‡Pi †KvbwU?
K2 2 M0 N6
425. x 2y 10 = 0 Ges 2x + y 3 = 0
5 GKK N 6 GKK K5 5 M3 N3
444. y = 3x + 5 mij‡iLvi y A‡¶i †Q`Kvsk
†iLv؇qi Xvj؇qi ¸Ydj(Aby. 5) 434. (3, 0), (0, 4) I (0, 0) we›`y wZbwU Øviv wb‡Pi †KvbwU?
MwVZ wÎf‡z Ri †¶Îdj wb‡Pi †KvbwU?
K2 L2 M3 1 K 1 GKK L 3 GKK
x
426. y = 2 + 2 Ges 2x 10y + 20 = 0 K 5 eM© GKK 6 eM© GKK 5 GKK N 8 GKK
mgxKiYØq (Ab.y 6) M 7 eM© GKK N 8 eM© GKK 445. x A‡¶i mv‡_ 3x + 2y = 6 †iLvi †Q`
K `yBwU wfbœ †iLv wb‡`©k K‡i 435. y 2x + 3 = 0 Ges x + 2y 10 = 0 we›`y wb‡Pi †KvbwU?
M GKB †iLv wb‡`©k K‡i †iLv؇qi Xvj؇qi †hvMdj KZ?
5 3 K (3, 0) (2, 0) M (6, 0) N (2, 3)
2 2 446. y A‡¶i mv‡_ 3x + 4y = 12 †iLvi †Q`
M †iLvØq mgvš—ivj K L we›`y wb‡Pi †KvbwU?
N †iLvØq ci¯úi‡”Q`x 3 N 5 K (3, 0) (0, 3) M (4, 0) N (0, 4)
[we: `ª: mwVK DËi K Ges N] 2 2
eüc`x mgvwµ¦mPƒ K eüwbeÆvPwb cGÉ kv² îi wb‡Pi †KvbwU mwVK? Y A(9, 9)
K i L ii
447. ¯v’ bv¼ e¨e¯v’ q i I ii N i, ii I iii
B(4, 4) C(9, 4)
i. fzR I †KvwU‡K GK mv‡_ ¯’vbv¼ ejv nq 450. x + y = 2 mij‡iLvwU
ii. we›`yi ¯v’ bv¼ m~PK (x, y) GKwU X O X
i. x A¶‡K (2, 0) we›`y‡Z †Q` K‡i
µg‡Rvo eSy vq hvi cÖ_gwU fzR Ges Y
wØZxqwU †KvwU wb‡`k© K‡i ii. y A¶‡K (0, 2) we›`‡y Z †Q` K‡i wPÎ n‡Z 453455 bs cÖ‡kiœ DËi `vI :
iii. x A‡¶i Dci †KvwU k~b¨ iii. x A¶ I y A¶ †_‡K KwZ©Z As‡ki 453. A we›`y n‡Z Y A‡¶i `i~ Z¡ KZ?
mgwó 4 K 4 GKK 9 GKK
Dc‡ii Z_¨vbyhvqx wb‡Pi †KvbwU mwVK? wb‡Pi †KvbwU mwVK?
K i I ii L i I iii M 16 GKK N 81 GKK
K i, ii L i, iii 454. AC evûi `i~ Z¡ wb‡Pi †KvbwU?
M ii I iii i, ii I iii i, ii I iii
448. A( 5, 5) M ii, iii K 4 GKK L 9 GKK
B(0, 0) C(5, 5)
Awf®² Z^ÅwfwîK eüwbeÆvPwb cÉGk²vîi 5 GKK N 14 GKK
wPÎ n‡Z, 455. B we›`y n‡Z g~j we›`yi `i~ Z¡ KZ?
i. A I B we›`yi ga¨eZx© `i~ Z¡ 5 2 wb‡Pi Z‡_¨i Av‡jv‡K 451 I 452 bs
GKK cÖ‡kœi DËi `vI : K 4 GKK L 5 GKK
M 9 GKK 4 2 GKK
ii. AC †iLvi Xvj 1 x = 1, y = 1 wb‡Pi Z‡_¨i Av‡jv‡K 456 458 bs
iii. AC mij‡iLvi mgxKiY y + x = 0
Dc‡ii Z‡_¨i Av‡jv‡K wb‡Pi †KvbwU 451. †iLvØq †h we›`y‡Z †Q` K‡i Zvi cÖ‡kœi DËi `vI :
mwVK?
¯’vbv¼(Aby. 8) A(1, 3), B(2, 5) Ges C( 1, 2) wZbwU wfbœ
we›`y|
K i I ii L ii I iii K (0, 1) L (1, 0) 456. A I B Gi ga¨eZ©x `i~ Z¡ KZ?
M i I iii i, ii I iii M (0, 0) (1, 1)
449. 3x + 2y = 6 mij‡iLv 452. †iLvØq A¶Ø‡qi mv‡_ †h †¶ÎwU ˆZwi K5 5 M3 N3
i. Øviv x A‡¶i †Q`vsk 2 457. AB †iLvi Xvj KZ? N3
ii. Øviv y A‡¶i †Q`vsk 3 K‡i Zvi †¶Îdj (Ab.y 9) K0 L1 2
iii. A¶Ø‡qi mv‡_ MwVZ 5
wÎfz‡Ri K 1 eM© GKK 1 eM© GKK 458. AC †iLvi Xvj KZ? 2 2
2 5
†¶Îdj 3 13 eM© GKK M 2 eM© GKK N 4 eM© GKK K5 L 5 M
Aa¨vq 12 mgZjxq †f±i
mvaviY eüwbevÆ Pwb cGÉ kv² îi 465. b Ges BE = OB n‡j OE †f±i 470. `yBwU †f±i mgvš—ivj n‡j
OB = i. G‡`i †hv‡Mi †¶‡Î mvgvš—wiK wewa
459. u = v Ges v = w n‡j
K u w L uw †KvbwU? c‡Ö hvR¨
u=w N u<w
O bB E ii. G‡`i †hv‡Mi †¶‡Î wÎfzR wewa
460. g~j we›`iy mv‡c‡¶ Ab¨ †Kvb we›`iy Kb 2b M 3b N 4b cÖ‡hvR¨
Ae¯’vb wbY©‡qi Rb¨ †h †f±i e¨envi Kiv 466. ABCD mvgvš—wi‡Ki AC Ges BD KY©Ø‡qi iii. G‡`i ˆ`N¨© me`© v mgvb
†Q`we›`y O. Zvn‡j wb‡“i †KvbwU mwVK? Ic‡ii Dw³¸‡jvi g‡a¨ †KvbwU mwVK? (Aby. 2)
nq Zv‡K †Kvb †f±i e‡j? Ki o ii
AO = OC Ges BO = OD
K kb~ ¨ †f±i Ae¯v’ b †f±i M i I ii N i, ii I iii
L AO = OD Ges BO = OC
M GKK †f±i N mg †f±i 471. u, v, w Gi Rb¨ (u + v) + w = u + (v + w) cKÖ vk
M AO = OB Ges CO = OD
461. wÎfz‡Ri †h‡Kvb `yBwU evûi ga¨we›`Øy ‡qi K‡i
ms‡hvRK †iLvsk H wÎfz‡Ri Z…Zxq N AD = BO
i. †hvRb wewa ii. we‡qvRb wewa
evûi 467. O P A iii. mn †hvRb wewa
wb‡Pi †KvbwU mwVK?
mgvš—ivj I A‡aK© L j¤^ I A‡a©K K ïay i L ii I iii
OA
M j¤^ N A‡aK© wP‡Î, = KZ? i I iii N i, ii I iii
462. AB = CD Ges AB || CD n‡j wb‡Pi
†KvbwU mwVK? (Ab.y 3) OA = OP + PA L OA = OP PA 472. u Gi wecixZ †f±i v n‡e hw`
i. | v | = | u |
M OA = PO + PA N OA = AP + OP
AB = CD ii. v Gi aviK †iLv u Gi aviK †iLvi
L = m. †hLv‡b m > 1 468. O 3b Pa m‡½ Awfbœ ev mgvš—ivj nq
AB CD
Q iii. u Gi w`K v Gi w`‡Ki wecixZ nq
M AB + DC < O wP‡Î, = KZ? wb‡Pi †KvbwU mwVK?
OQ
N + m. = O †hLv‡b m > 1
K 3b a L a 3b K i I ii L i I iii
AB CD M a 3b 3b + a
463. U Gi aviK Ges V Gi aviK †iLvØq M ii I iii i, ii I iii
Awfbœ ev mgvš—ivj n‡j eüc`x mgvwµ¦mPƒ K eüwbeÆvPwb cÉGk²vîi 473. Dchy©³ wPÎ n‡Z C we›`y B
K U Ges V wecixZ †f±i 469. AB || DC n‡j Ae¯’vb †f±i n‡e, hw` C
m : n Abcy v‡Z wef³
L U Ges V mgvš—ivj †f±i †hLv‡b GKwU †¯‹jvi ivwk nq Z‡e,
M U Ges V GKK †f±i i. AB = m. DC, m DA
U Ges V mgvb †f±i i. na + mb ii. mb + na iii. m +n
m+n n+m a +b
ii. AB = DC iii. AB = CD
Ic‡ii Dw³¸‡jvi g‡a¨ †KvbwU mwVK?
464. m = 0 A_ev u = 0 n‡j wb‡Pi †KvbwU (Aby. 1) wb‡Pi †KvbwU mwVK?
mwVK?
K mu = 1 L mu > 1 i L ii K ïay i i Ges ii
mu = 0 N mu < 1 M ïay iii N i, ii Ges iii
M i I ii N i, ii I iii
Awf®² Z^ÅwfwîK eüwbevÆ Pwb cÉGk²vîi 477. AB †iLvsk C we›`y‡Z m t n Abycv‡Z 482. = KZ?
wb‡Pi Z‡_¨i Av‡jv‡K 474 I 475 b¤^i ewnwe©f³ n‡j AN
c‡Ö kœi DËi `vI :
AB †iLvs‡ki Dci †h‡Kv‡bv we›`y C Ges ma nb na mb K b L 3b
†Kv‡bv †f±i gj~ we›`iy mv‡c‡¶ A, B I C m n mn
we›`yi Ae¯v’ b †f±i h_vµ‡g a, b I c. c = L c = M 4b 2b
mb na nb ma wb‡Pi wPÎ Aej¤^‡b 483 I 484 bs
c = m n c = mn cÖ‡kœi DËi `vI :
M N
478. C we›`wy U AB †iLvs‡ki ga¨we›`y n‡j AD
474. AA †f±i n‡”Q 1 1 E a F
2 2 B d
i. we›`y †f±i ii. GKK †f±i c = (a + b) L c = (a b)
C
iii. k~Y¨ †f±i M c = 1 (b a) N c = 1 (b a) bc
2 2
wb‡Pi †KvbwU mwVK? (Aby. 4)
K i, ii i, iii wb‡Pi wPÎ n‡Z 479-482 bs cÖ‡kœi DËi O
`vI :
wP‡Î, AD || BC, E I F h_vµ‡g AB I
CD Gi ga¨we›`y|
M ii, iii N i, ii I iii P
475. ABC Gi †¶‡Î †KvbwU mwVK? (Aby. 5)
483. wb‡Pi †KvbwU mwVK?
K AB + BC = CA L AB + AC = BC AN K = 1 +
2
aM EF (AD CD)
M CB + BA + CA = 0 O bB Q = 1 +
2
AB + BC + CA = 0 †`Iqv Av‡Q, EF (AD BC)
wb‡Pi Z_¨ Aej¤^‡b 476478 bs c‡Ö kœi OA = a, OB = b, OA = AP, 1
DËi `vI : 2
A, B, C Gi Ae¯v’ b †f±i h_vµ‡g a, b, = Ges N, PQ Gi ga¨ we›`y| M EF = (AD BC)
c. BQ 3OB
476. AB †iLvsk C we›`y‡Z m t n Abycv‡Z 479. = KZ? N = 1
2
AB EF (BC AD)
− a + b L a − b M −a −bN a + b 484. O we›`yi mv‡c‡¶ A I B we›`iy Ae¯’vb
480. = KZ? †f±i h_vµ‡g a I b n‡j wb‡Pi †KvbwU
PQ
Aš—wef© ³ n‡j − 2a + 4b L 2a + 4b mwVK?
M 2a − 4b N − 2a − 4b
na mb na mb 1 1
K c = m+n L c = mn 481. †f±‡ii gvb KZ n‡e? = 2 (a + b) L = 2 (b a)
OE OE
ON
c = na + mb N c = ma nb a + 2b L a − 2b M = 1 (a b) N = 1 (a b)
m+n m + n M −a + 2b N a − 2b 2 2
OE OE
Aa¨vq 13 Nb R¨vwgwZ
mvaviY eüwbeÆvPwb cGÉ kv² îi 493. wcivwg‡Wi kxl© n‡Z fw‚ gi Dci Aw¼Z 500. GKwU K¨vcmy‡ji ˆ`N¨© 15 †m. wg.| Gi
j¤^‰`N©¨‡K Kx e‡j? `yB cvÖ ‡š—i Aa‡© MvjvKw… Z As‡ki c„ôZ‡ji
485. mgZj ev eµZj Øviv †ewóZ k~‡b¨i D”PZv L avi M KY© N ˆ`N©¨ †¶Îdj 36 n‡j Gi wmwjÛvi AvKw… Zi
wKQyUv ¯’vb `LjKvix e¯‡‘ K Kx e‡j? As‡ki e¨vmva© KZ?
K wcRÖ g L wcivwgW 494. mylg PZz¯’j‡Ki †KŠwYK we›`y KqwU?
K 1wU L 2wU M 3wU 4wU
Nbe¯‘ N †KvYK 495. mylg PZ¯z ’jK KqwU mgevû wÎfRz Øviv K 2 †m. wg. 3 †m. wg.
486. Nbe¯i‘ `By wU Zj †Q`Kvix †iLv‡K Kx
e‡j? †ewóZ? 4 †m. wg. 6 †m. wg.
K ˆ`N¨© L c¯Ö ’ K 2wU L 3wU 4wU N 5wU 501. †Mvj‡Ki cô„ Z‡ji †¶Îdj = ?
496. wcivwg‡Wi AvqZb wb‡Pi †KvbwU ?
avi N D”PZv 1 K 2 r2 L 4 r3
487. GKwU B‡Ui KqwU avi Av‡Q? 2 4 r2 N 2 r3
K f‚wgi cwiwa †¶Îdj
K 2wU L 4wU 502. `yBwU mgZj ci¯úi‡”Q`x Z‡e Zv‡`i
M 8wU 12wU 1 f‚wgi †¶Îdj D”PZv ga¨eZ©x wØZj †Kv‡Yi cwigvY KZ?
3
488. mvgvš—wiK Nbe¯‘‡Z KqwU mgvš—ivj M 1 fw‚ gi cwiwa D”PZv K 60 90
mgZj _v‡K? 6
M 180 N 360
K 1wU L 2wU N 1 fw‚ gi †¶Îdj D”PZv
M 4wU 6wU 12 503. 10 †m. wg. mgvb evû wewkó cÂfzR
497. wcivwg‡Wi D”PZv 8 †m. wg., fw‚ g‡¶‡Îi AvKw… Zi GKwU wcivwg‡Wi D”PZv 12 †m.
489. †h AvqZvKvi Nbe¯‘i cô„ Zj¸‡jv eM©‡¶Î, Aš—e„©‡Ëi e¨vmva© 6 †m. wg. n‡j †njv‡bv wg.| Gi †njv‡bv D”PZv KZ?
Zv‡K wK e‡j?
K †KvYK L †MvjK D”PZv KZ? K 10 †m. wg. 219 †m. wg.
K 5 †m. wg. L 8 †m. wg.
NbK N wmwjÛvi 10 †m. wg. N 12 †m. wg. M 11 †m. wg. N 14 †m. wg.
490. wb‡Pi †KvbwU‡Z ˆ`N©¨, c¯Ö ’ I †ea mgvb? 498. †KvY‡Ki e¨vmva© 6 †m. wg. Ges †njv‡bv eüc`x mgvwµ¦mPƒ K eüwbeÆvPwb cÉGkv² îi
K †KvYK L †MvjK
NbK N wmwjÊvi D”PZv 8 †m. wg. n‡j eµZ‡ji †¶Îdj 504. hw` GKwU †g‡S Kíbv Kwi Z‡e
KZ?
491. Lvov wcRÖ ‡gi cvk¦©Zj¸‡jvi AvK…wZ K 24 eM© †m. wg. 48 eM© †m. wg. i. Gi we¯—vi Amxg n‡e
†Kgb? ii. Gi Dci¯’ `By we›`iy ms‡hvRK †iLv
K eM©vKvi AvqZvKvi M 34 eM© †m. wg. N 54 eM© †m. wg. H Z‡ji Dci Aew¯’Z n‡e
499. mgeË„ f‚wgK †KvY‡Ki fw‚ gi e¨vmva© r,
M wÎfzRvKvi N i¤^vmvKvi kxl©‡KvY , D”PZv h n‡j †KvY‡Ki iii. GwU wØgvwÎK n‡e
492. wcivwg‡Wi cvk¦©Zj¸‡jvi AvKw… Z †Kgb?
K eMK© vi L i¤v^ mvKvi eµZ‡ji †¶Îdj = KZ? wb‡Pi †KvbwU mwVK?
K i I ii L i I iii
M AvqZvKvi wÎfRz vKvi r2 L r3 M r2 N r2 M ii I iii i, ii I iii
sin sin cos tan
505. GKwU Zj Abyfw‚ gK n‡e, hw` wb‡Pi †KvbwU mwVK? wb‡Pi wP‡Îi Av‡jv‡K 513 I 514 bs
K i I ii L i I iii c‡Ö kœi DËi `vI :
i. GKwU Lvov †iLv Ges HZ‡ji M ii I iii i, ii I iii
ga¨eZx© †KvY 90 nq| 4 †m. wg.
ii. ZjwU GKwU Lvov †iLvq mgvš—ivj 4 †m. wg.
Ae¯v’ b K‡i| 509. Nb‡Ki ˆewkó¨
i. c„ôZj¸‡jv eM©‡¶Î| 4 †m. wg.
iii. Zj Ges †iLvi g‡a¨ GKwU mvaviY ii. KY¸© ‡jv mgvb _v‡K| 4 †m. wg.
8 †m. wg.
we›`y _v‡K| iii. Nb‡Ki 6wU cô„ _v‡K|
wb‡Pi †KvbwU mwVK?
Ki L iii wb‡Pi †KvbwU mwVK? 4 †m. wg.
K i I ii L i I iii 4 †m. wg.
i I iii N i, ii I iii M ii I iii i, ii I iii
513. wcÖRgwUi fw‚ gi †¶Îdj KZ?
K 4 †m. wg. 16 †m. wg.
506. ˆbKZjxq PZfz ©y‡Ri M 8 †m. wg. N 64 †m. wg.
i. `ywU mwbœwnZ evû GK Z‡j Aew¯’Z| Awf®² Z^ÅwfwîK eüwbeÆvPwb cÉGk²vîi
ii. wecixZ evûØq ˆbKZjxq| 514. wcRÖ gwUi mgMÖZ‡ji †¶Îdj KZ?
wb‡Pi wP‡Îi Av‡jv‡K 510 512 bs K 128 eM© †m. wg. 160 eM© †m. wg.
iii. evû¸‡jv GKB Z‡j Aew¯’Z| c‡Ö kœi DËi `vI :
wb‡Pi †KvbwU mwVK? M 32 eM© †m. wg. N 260 eM© †m. wg.
K i i I ii
M i I iii N i, ii I iii wb‡Pi wP‡Îi Av‡jv‡K 515 I 516 bs
c‡Ö kœi DËi `vI :
507. `By wU mij‡iLv ˆbKZjxq n‡e hw`
4 †m. wg. 4 †m. wg. 2
i. †iLvØq mgvš—ivj nq| D”PZv 4 †m. wg.
ii. †iLvØq GKB Z‡ji Dci Ae¯’vb K‡i| 4 †m. wg. 4 †m. wg. (f‚wg) 12 †m. wg.
iii. Zv‡`i g‡a¨ †Kv‡bv mvaviY we›`y bv _v‡K| 510. GB Nbe¯‘i avi KqwU?
K 2wU 5wU
wb‡Pi †KvbwU mwVK? 2
Ki iii M 4wU N 8wU wPÎ : K¨vcmjy
515. K¨vcmyjwUi wmwjÛvi AvK…wZi As‡ki
M i I iii N i, ii I iii 511. Nbe¯‘wUi cvk¦Z© j¸‡jvi †¶Îdj KZ? ˆ`N¨© KZ?
508. `By wU Z‡ji ga¨eZ©x wØZj‡Kv‡Yi cwigvY
K 4 3 eM© †m. wg. L 8 3 eM© †m. wg. K 6 †m. wg. 8 †m. wg.
90| Z‡e M 2 3 eM© †m. wg. 32 eM© †m. wg. M 4 †m. wg. N 10 †m. wg.
512. Nbe¯w‘ Ui mgMZÖ ‡ji †¶Îdj KZ?
i. Zj؇qi GKwU mvaviY †iLv _vK‡e| K 16.2 eM© †m. wg. L 54.1 eM© †m. wg. 516. wmwjÛvi AvK…wZi As‡ki †¶Îdj KZ?
M 44.7 eM© †m. wg. 48 eM© †m. wg. K 16 eM© †m. wg. L 64 eM© †m. wg.
ii. Zv‡`i AmsL¨ mvaviY we›`y _vK‡e| 32 eM© †m. wg. N 90 eM© †m. wg.
iii. ZjØq ci¯úi j¤^|
Aa¨vq 14 m¤v¢ ebv
mvaviY eüwbeÆvPwb cGÉ k²vîi eüc`x mgvwµ¦mƒPK eüwbevÆ Pwb cÉGk²vîi Awf®² Z^ÅwfwîK eüwbevÆ Pwb cÉGk²vîi
517. GKmv‡_ wZbwU gy`ªv wb‡¶‡c wZbwU NUbv 522. i. †Kvb cix¶vi djvdj ev djvd‡ji wb‡Pi Z_¨ †_‡K 525 I 526 b¤^i cÖ‡kiœ
mgv‡ek‡K NUbv e‡j
GKB nIqvi m¤v¢ ebv KZ? ii. GKwU Q°v wb‡¶c Ki‡j we‡Rvo DËi `vI : (Ab.y 2)
K 1 1 msL¨v nIqvi AbKy ‚j djvdj 2wU GKwU _wj‡Z bxj ej 12wU, mv`v ej 16wU
8 4
Ges Kv‡jv ej 20wU Av‡Q| ˆ`efv‡e
M 1 N1 iii. bgby v‡¶‡Îi cÖwZwU Dcv`vb‡K
2 GKUv ej †bIqv n‡jv|
518. 1 †_‡K 200 ch©š— msL¨v¸‡jv n‡Z djvd‡ji bgybv we›`y e‡j 525. ejwU bxj nIqvi m¤v¢ ebv KZ?
wb‡Pi †KvbwU mwVK?
ˆ`ePqb c×wZ‡Z GKwU msL¨v †bIqv K i I ii L ii I iii 1 1
16 12
n‡j, msL¨vwU Nb nIqvi m¤¢vebv KZ? K L
1 L 1 i I iii N i, ii I iii M 1 1
40 200 8 4
523. i. †Kvb NUbvi m¤v¢ ebv =
5 1
M 40 N 200 D³ NUbvi AbyK‚j djvdj 526. ejwU mv`v bv nIqvi m¤v¢ ebv KZ?
519. GKwU Q°v gvi‡j 3 DVvi m¤v¢ ebv †KvbwU? mgMÖ m¤v¢ e¨ djvdj K 1 2
3 3
(Ab.y 1) ii. m¤¢vebvi gvb 0 n‡Z 1 Gi g‡a¨ _v‡K|
1 1
1 L 1 iii. GKwU gy`vª wb‡¶‡c H A_ev T Avmvi M 16 N 48
6 3
m¤¢vebv 1. wb‡“i Z_¨ †_‡K 527 I 528 bs c‡Ö kœi
2 1
M 3 N 2 wb‡Pi †KvbwU mwVK? L i I iii DËi `vI :
K i I ii
520. `By wU Q°v wb‡¶c msNwUZ bgby vwe›`y M ii I iii i, ii I iii GKwU g`y ªv‡K wZbevi wb‡¶c Kiv
KqwU? 524. i. GKwU Qv°v wb‡¶c 7 Avmvi m¤v¢ ebv k~b¨| n‡jv|(Ab.y 4)
K 6wU L 12wU ii. GKwU Q°v wb‡¶‡c we‡Rvo Avmvi m¤¢vebv 1 527. †gvU bgby vwe›`y KqwU?
3
M 18wU 36wU K0 L1
iii. hyw³ wfwËK m¤¢vebv wbY‡© q
521. `By wU g`y ªv GK mv‡_ wb‡¶c Ki‡j djvdj¸‡jv mgm¤v¢ e¨ n‡Z nq 8 N 10
`yBwU‡ZB †Uj Avmvi m¤¢vebv KZ? wb‡Pi †KvbwU mwVK? 528. wZbwU †Ubª Avmvi m¤¢vebv KZ?
1 1
K 2 4 K i I ii L ii I iii K1 1
8
3
M 4 N1 i I iii N i, ii I iii M 8 N 16
K¨v‡WU K‡jRmg~‡ni wbev© Pwb
cix¶vi cÖkœcÎ : mR„ bkxj
01 Feni Girls' Cadet College, Feni
Time : 2 Hours 35 Minute Higher Mathematics Creative Marks : 50
[N.B. : Answer total Five questions taking two questions form Algebra, two questions form Geometry & Vector, one question
form Trigonometry and Probability.]
Group-A : Algebra a. Prove that, AB + BC + CD + DE + EA = 0. 2
1 Two polynomials of variable x are P = 6x2 x a and b.
c. Prove that, PQRS is a parallelogram. 4
Q(x) = x3 5x2 2x + 24.
6 If M and N are the midpoint of UC and VD
a. If (3x + 1) is a factor of P(x), then find the value
respectively. Then prove that by vector methods
of a. 2
1
b. If Q(x) is divided by (x m) and (x n), the that UV||MN||CD and MN = 2 (UV + CD). 4
remainder is same, then prove that, A
m2 + n2 + mn 5m 5n = 2. 4
4
c. Express x3 into partial fractions. D
Q(x)
2 A = y 3x and B = 2 + 22 + 222 + 2222 + .......... B
a. Solve : b2x (b4 + b) bx 1 + b3 = 0.
2 C
b. Draw the graph when A 0. 4 a. If AP is a median of ABC, then show that
c. Find the sum of first n terms of B. 4
4AP2 = 2(AC2 + AB2) BC2. 2
3 A = 2x2 x311and B = 2x2 + xk310are two binomial b. If three median of BCD meet at G, prove that,
BC2 + CD2 + BD2 = 3(GB2 + GC2 + GD2). 4
expressions. c. Prove that, AC.BD = AB.CD + BC.AD 4
a. Find the 7th term of 1 x110. 2 Group-C : Trigonometry and Probability
7 P
b. Find the coefficient of x10 of A. 4
PRQ =
c. In the expansion of B, the coefficient x5 and x15
are equal, find the value of k. 4 QR
Group-B : Geometry and Vector a. Prove that sin2 + sin252 + sin28 + sin2 9 = 2,
2
4 If the Co-ordinates of A(1, 1), B(t, 2), C(t2, t + 3)
a. Find the slope & the intersector of y-axis of the when = 7. 2
straight line 2x 3y = 5. 2 QR PQ QR
PR PR PR
b. If A, B, C are collinear, find the admissible value b. If + = 2 then prove that,
of t. 4 cos sin = 2 sin . 4
c. The line joining the points A and B intersects the PQ QR 4
QR PQ 3
x-axis and they y-axis at the points P and Q c. + = then find the value of ,
respectively. Find the equation of PQ and the
area of POQ. 4 when 0 < < 2. 4
5 E 8 A beg contains 6 white, 8 red, and 9 black balls. A ball is
A
drawn at random.
D a. What is the probability of a ball to be black? 2
B
b. What is the probability of a ball is not to be white
or black? 4
C c. If 5 balls are drawn successively without
If P, Q, R, S, T, U and V are the midpoint of AB, BC,
CD, DE, EA, AC and AD respectively. replacement, what is the probability of all the
balls to be red? 4
K¨v‡WU K‡jRmg~‡ni wbe©vPwb cix¶vi
cÖkœcÎ I DËigvjv : eûwbev© Pwb
02
Feni Girls' Cadet College, Feni
Time : 25 Minute Higher Mathematics MCQ Marks : 25
[N.B. : Answer all the questions. Each question carries one mark. Block fully, with a ball-point pen, the circle of the letter that
stands for the correct/best answer in the "Answer Sheet" for Multiple Choice Question Type Examination.]
Candidates are asked not to leave any mark or spot on the question paper.
1. If f(x) = 2x (x 4), what is the Which one is correct? 15. sin (10 + ) = ?
x4 A i & ii B i & iii
C ii & iii D i, ii & iii A sin B sin
2. value of f1( 1)?
31 24 9. 3 + 0.3 + 0.03 + 0.003 ........ = ? C sin2 D cos2 +
3. 10. 1 10 8 1
4. A4 B3 C3 D3 A 9 B 3 C 9 D 10 16. If sin = Cosec then, = ?
11. 17. A 30 B 45 C 60 D 90
If f(x) = x 4 then answering the What is the required solution of 2 + 4 + 6 + 8 + ...... which one is
questions (2-3) xy = yx and x = 2y. 18. the 20th of the series?
Which one is the following is the A (6, 3) B (8, 4)
domain of the function? C (2, 1) D (4, 2) A 28 B 36 C 40 D 44
A Dom, f = {x R : x 4} If the value of is 520 then,
B Dom, f = {x R : x 4} x i. is a negative angle
C Dom, f = {x R : x 4} If the inequality is x 5 2 then which ii. rotate clockwise
D Dom, f = {x R : x 4} iii. The position of in 3rd quadrant
Find the value of f(a4 + 4) one is the solution set of the inequality?
where a R. A S = {x R : x 25}
A a B a2 C a3 D a4 Which one is correct?
Tossing a coin is an A i & ii B i & iii
experiment of . B S = {x R : x 25} C ii & iii D i, ii & iii
A Event B Mutually Exclusive events 19. A coin is tossed thrice. What is the
C Random experiments probability to get head and tail
D Sample space C S = {x R : x 25} respectively?
D S = {x R : x 25} 1 1 1 2
A 2 B 4 C 8 D 3
5. Suppose 5x 7 A + B 2. 12. i. In any two circles the ratios of 20. Of null the vector .
6. 1) (x x 13. the circumferences and the i. Absolute value is zero
(x 2) x 1 21. ii. Direction cannot be determined
Find the value of A and B. respective diameters are equal. 22.
ii. The centered angle produced by iii. It has non length
A A = 2, B = 3 B A = 2, B = 3 Which one is correct?
C A = 2, B = 3 D A = 2, B = 3 any arc of a circle is
i. The expression x2 + 2xy + 3y2 is proportional to its arc. A i & ii B i & iii
C ii & iii D i, ii & iii
a homogeneous iii. Radian angle is a constant angle. Which one will be zero on y-axis?
Which one is correct?
ii. Considering x, y, a, h and b the A abscissa B Ordinate
variables ax2 + 2axy + by2 is A i & iii B i & ii
homogeneous C ii & iii D i, ii & iii C Both of them D None of them
What is the distance of (6, 4)
iii. 3x2y + y2z + 8z2x 5xyz is a If A = 30, B = 45 then, which one from x-axis?
homogeneous polynomial is correct?
tanA + tanB A 4 unit B 6 unit
Which one is correct? A tan(A B) = tanA tanB
A i & iii B i & ii C 10 unit D 4 5 unit
C ii & iii D i, ii & iii 23. mu + mv = m (u + v) is true for .
7. In (1 3x)5, then what is the B tan(A B) = tanA tanB A All the value of mB All the value of u
coefficient of x2? 1 + tanA tanB C All the value of v
D All the value of u + v
A 90 B 15 C 9 D 15 C tan(A + B) = tanA tanB
8. Study the following statements : 1 + tanA tanB The height of a right circular cone
1 + tanA + tanB is 24 cm and its volume is 1232
i. The point of concurrence of the D tan(A + B) = 1 + tanA tanB cubic cm. Now answer the
medians of a triangle is called
the orthocenter. questions (24-25).
ii. The distance of the vertex from 14. i. sin23 = sin2 + 6 24. What is the formula of volume of cone?
the orthocenter of any triangle is ii. sin23= cos6 iii. sin23 =
twice the distance of the 3 A 1 rh B 1 r2hC r2h D 1 r3
opposite side of the vertex from 2 3 3 3
25. What is the radius of cone?
the circumcenter of a triangle. Which one is correct? A 4.49 cm B 7.0014 cm
iii. If two triangles are equiangular, their A i & iii B i & ii
corresponding sides are proportional. C ii & iii D i, ii & iii C 14.0014 cm D 21.0014 cm
Self test 1 ABCD 2 ABCD 3 ABCD 4 ABCD 5 ABCD 6 ABCD 7 ABCD 8 ABCD 9 ABCD
10 A B C D 11 A B C D 12 A B C D 13 A B C D 14 A B C D 15 A B C D 16 A B C D 17 A B C D 18 A B C D
19 A B C D 20 A B C D 21 A B C D 22 A B C D 23 A B C D 24 A B C D 25 A B C D
Verify your Answer :
1 D 2 D 3 B 4 B 5 D 6 D 7 A 8 C 9 B 10 D 11 A 12 D 13 B
14 D 15 A 16 D 17 C 18 D 19 B 20 D 21 A 22 A 23 A 24 B 25 B
kxl©¯’vbxq ¯‹z‡ji wbev© Pwb cix¶vi
cÖkœcÎ : mR„ bkxj
03 G. †K. nvB ¯‹zj GÛ K‡jR, `wbqv, XvKv welq ˆKvW : 1 2 6
mgq : 2 NÈv 35 wgwbU D”PZi MwYZ m„Rbkxj cÖkœ c~Yg© vb : 50
[Wvb cv‡ki msL¨v cÖ‡kœi cY~ ©gvb ÁvcK| c‡Ö Z¨K wefvM †_‡K b¨~ bZg GKwU K‡i †gvU cvu PwU cÖ‡kœi DËi w`‡Z n‡e|]
K wefvM : exRMwYZ K. wÎfz‡Ri fi‡K›` ej‡Z Kx †evS? 2
1 g(x) = x 3 Ges f(a, b, c) = a3 + b3 + c3 3abc. L. †`LvI †h,
K. g1 (x) wbY©q Ki| 2 AB2 + AC2 = 2(AD2 + BD2). 4
4 M. cÖgvY Ki †h,
L. f(a, b, c) †K Drcv`‡K we‡klY Ki|
M. f(a, b, c) = 0 n‡j cÖgvY Ki †h, AM. BC = AB. MC + AC. BM 4
a + b + c = 0 A_ev a = b = c. 4 6 ABCD PZzf©‡y Ri PviwU kxl©we›`yi ¯v’ bvsK h_vµ‡g A(0, 1),
2 22x, 2y1 = 512; 43 + 2y = 1024 B(2, 3), C(6, 7) Ges D(8, 3).
8.23x
K. mgxKiYØq †_‡K x I y PjKwewkó mij mgxKi‡Y K. wPÎmn wØZj †Kv‡Yi mÁv `vI| 2
cwiYZ Ki| 2 L. AB, BC I CD †iLvi mgxKiY wbYq© Ki| †iLv
L. mgxKiYØq mgvavb K‡i ï×Zv hvPvB Ki| 4 wZbwUi Xvj †_‡K cÖgvY Ki †h, AB || CD Ges AB I
M. x I y hw` †Kv‡bv PZzf©y‡Ri mwbœwnZ evûi ˆ`N¨© nq BC ci¯úi j¤^| 4
†hLv‡b evû؇qi Aš—f©y³ †KvY GK mg‡KvY Z‡e M. AB, BC, CD Ges DA evûi ga¨we›`y h_vµ‡g P, Q,
PZzfy©RwU AvqZ bv eM© D‡jL Ki Ges Gi †¶Îdj, R, S n‡j cÖgvY Ki †h, PQRS GKwU mvgvš—wiK|
cwimxgv Ges GKwU K‡Yi© ˆ`N©¨ wbYq© Ki| 4 (†f±‡ii mvnv‡h¨) 4
3 A = x2 + xk6 Ges B = c 4x4 M wefvM : w·KvYwgwZ I m¤¢vebv
K. c¨vm‡K‡ji wÎfz‡Ri mvnvh¨ A †K PZz_© c` ch©š— we¯—…Z 7 B
Ki| 2
x2 + 1
L. A Gi we¯—…wZ‡Z x3 Gi mnM 540 n‡j k-Gi gvb wbYq©
Ki| 4 C O x2-1 A
50 ˆm.wg. D
M. B-Gi we¯—…wZ p 8x + qx2 rx3 n‡j p, q, r Ges c Gi wP‡Î ABC GKwU e„ËKvi PvKv| PvKvi e¨vm 50 †m.wg.|
K. PvKvwU †m‡K‡Û 5 evi Niy ‡j wgwb‡U KZ `~iZ¡ AwZµg
gvb wbY©q Ki| 4
L wefvM : R¨vwgwZ I †f±i Ki‡e? 2
4 GKwU †jvnvi dvu cv †Mvj‡Ki evB‡ii e¨vm 13 †mwg. Ges L. 2cos2 = 1 + 2sin2 n‡j DÏxcK †_‡K Gi m¤v¢ e¨
†ea 2 †mwg.| gvb wbY©q Ki| †hLv‡b 0 < < . 4
K. †Mvj‡Ki duvcv As‡ki AvqZb wbY©q Ki| 2 M. BOD †_‡K cos Ges sin Gi gvb e¨envi K‡i
L. †Mvj‡K e¨enZ„ †jvnv w`‡q GKwU wb‡iU †MvjK ˆZwi Kiv cÖgvY Ki †h, tan + sec = x. 4
n‡jv| wb‡iU †Mvj‡Ki cô„ Z‡ji †¶Îdj wbY©q Ki| 4 8 GKwU Szwo‡Z 12wU jvj, 15wU mv`v Ges 8wU meyR ej Av‡Q|
M. wb‡iU †MvjKwU GKwU Nb AvK…wZi ev‡· wVKfv‡e Gu‡U ˆ`efv‡e GKwU ej †bIqv n‡jv|
hvq| ev·wUi AbwaK…Z As‡ki AvqZb wbY©q Ki| 4 K. bgby v‡¶Î ej‡Z Kx †evS? 2
5 A L. ejwU (i) meyR nIqvi m¤¢vebv Ges (ii) jvj bv nIqvi
C m¤v¢ ebv wbY©q Ki| 4
DP M. hw` cÖwZ¯’vcvb bv K‡i GKwU K‡i ci ci cuvPwU ej
B
Z‡z j †bIqv nq Z‡e me¸‡jv ej mv`v nIqvi m¤¢vebv
M
wbYq© Ki| 4
ABC Gi ga¨gv AD Ges AP BC.
04 Avigvwb‡Uvjv miKvwi D”P we`¨vjq, XvKv welq ˆKvW : 1 2 6
mgq : 2 NÈv 35 wgwbU D”PZi MwYZ m„Rbkxj cÖkœ c~Y©gvb : 50
[Wvb cv‡ki msL¨v c‡Ö kœi c~Yg© vb ÁvcK| c‡Ö Z¨K wefvM †_‡K b~¨bZg GKwU K‡i †gvU cvu PwU cÖ‡kœi DËi w`‡Z n‡e|]
K wefvM : exRMwYZ 5 A(3, 1) Ges 2x y + 4 = 0 GKwU mij‡iLvi mgxKiY|
1 : {< 1} Ges g : 21 dvskbØq K. mij‡iLvwUi Xvj wbY©q Ki| 2
2x + 2 Ges x3 L. mij‡iLvwU A¶Ø‡qi mv‡_ †h wÎfzR MVb K‡i Zvi
x1 2x + 1
(x) = g(x) = †¶Îdj wbY©q Ki| 4
K. †Wvg wbYq© Ki| 2 M. DÏxc‡Ki mij‡iLvi Dci j¤^ Ges A we›`y w`‡q hvq
L. †`LvI †h, g dvskbwU GK-GK wKš‘ AbUz bq| 4 Giƒc mij‡iLvi mgxKiY wbYq© Ki| 4
M. 1(x) wbY©q Ki Ges 31(x) = x n‡j x Gi gvb KZ? 4 6 P
2 P(x, y, z) = (x + y + z) (xy + yz + zx) ST
Ges Q = x3 + y3 + z3 3x1y1 z 1.
K. P(x, y, z) PµµwgK I cÖwZmg ivwk| 2 QR
4
4 PQR Gi PQ Gi PR Gi ga¨we›`y h_vµ‡g S Ges T.
L. Q = 0 n‡j cÖgvY Ki †h, K. PS + ST †K PR Gi gva¨‡g cÖKvk Ki| 2
x = y = z A_ev xy + yz + zx = 0 L. †f±‡ii mvnv‡h¨ cÖgvY Ki †h,
M. P(x, y, z) = xyz n‡j †`LvI †h, Ges 1 4
2
1 1 1 1 ST || QR ST = QR
y x5 y5 z5
(x + + z)5 = + + M. SQRT Gi KYØ© ‡qi ga¨we›`y h_vµ‡g M I N n‡j
3 A = 2x2 21x310 GKwU wØc`x ivwk Ges †f±‡ii mvnv‡h¨ cÖgvY Ki †h, MN || ST || QR Ges
B = (3x + 2)1 + (3x + 2)2 + (3x 2)3 + .........GKwU MN = 1 (QR ST) 4
2
¸‡YvËi aviv| M wefvM : w·KvYwgwZ I m¤¢vebv
K. A wØc`xwUi ga¨c` wbY©q Ki| 2 7 tan + sec = x n‡j,
L. A wØc`xwUi we¯—…wZ‡Z x ewR©Z c` Ges Zvi gvb wbY©q K. sec tan Gi gvb wbYq© Ki| 2
4
Ki| 4 L. †`LvI †h, cosec = x2 + 1
x2 1 4
M. x Gi Dci Kx kZ© Av‡ivc Ki‡j B avivwUi AmxgZK M. mgvavb Ki :
mgwó _vK‡e Ges †mB mgwó wbY©q Ki| 4 cot2 + cosec2 = 3; †hLv‡b 0 < < 2
L wefvM : R¨vwgwZ I †f±i 8 GKwU `By UvKvi gy`vª Pvi evi wb‡¶c Kiv n‡jv| Gi kvcjvi
4 ABC Gi AD, BE I CF ga¨gvÎq O we›`‡y Z †Q` Ki‡Q| wcV‡K L Ges wkïi wcV‡K C we‡ePbv Ki|
K. fi‡K›`ª Kx? fi‡K›` ga¨gv‡K Kx Abcy v‡Z wef³ K. GKwU Q°v wb‡¶c Kiv n‡j †Rvo A_ev †gŠwjK msL¨v
K‡i? 2 Avmvi m¤¢vebv KZ? 2
L. DÏxc‡Ki wPÎwU A¼b K‡i †`LvI †h, L. Probability tree A¼b Ki Ges bgybv‡¶ÎwU wjL| 4
AB2 + AC2 = 2(AD2 + BD2) . 4 M. bgby v‡¶Î †_‡K wb‡Pi NUbv¸‡jvi m¤v¢ ebv wbY©q Ki : 4
M. cÖgvY Ki †h, (i) Kgc‡¶ 3 L
AB2 + BC2 + AC2 = 3(AO2 + BO2 + CO2). 4 (ii) eo‡Rvi 3C
05 exYvcvwY miKvwi evwjKv D”P we`¨vjq, †MvcvjMÄ welq ˆKvW : 1 2 6
mgq : 2 NÈv 35 wgwbU D”PZi MwYZ m„Rbkxj ckÖ œ c~Yg© vb : 50
[Wvb cv‡ki msL¨v c‡Ö kœi c~Y©gvb ÁvcK| cÖ‡Z¨K wefvM †_‡K b~¨bZg GKwU K‡i †gvU cvu PwU c‡Ö kœi DËi w`‡Z n‡e|]
K wefvM : exRMwYZ 5 (i) A(3, 1) Ges 2x y + 4 = 0 GKwU mij‡iLvi mgxKiY|
1 (i) P(x) = 18x3 + 15x2 x + a eûc`xi GKwU Drcv`K (ii) ABC GKwU mgevû wÎfzR Ges AD, BE I CF Gi wZbwU
(3x 1). ga¨gv|
K. mij †iLvwUi Xvj wbYq© Ki|
(ii) F(a, b, c) = 1 + 1 + 1 3 2
a3 b3 c3 abc
K. (i) bs n‡Z a Gi gvb wbYq© Ki| 2 L. †`LvI †h AB, BE I CF mvg¨ve¯v’ q Av‡Q| 4
L. F(a, b , c) = 0 Ges ab + bc + ca 0 n‡j †`LvI †h, M. DÏxc‡Ki mij‡iLvi Dci j¤^ Ges A we›`y w`‡q hvq
a = b = c. 4 Giƒc mij‡iLvi mgxKiY wbYq© Ki| 4
M. 3x2 + 8x + 2 †K AvswkK fMœvs‡k cÖKvk Ki| 4 6 GKwU †Mvj‡Ki e¨vm 12 †m. wg. Ges GKwU AvqZvKvi
P(x)
Nbe¯i‘ gvÎv¸‡jv h_vµ‡g 3 †m. wg., 4 †m. wg. I 5 †m. wg.|
2 (i) 1 + 1 5 + (3x 1 5)2 + (3x 1 5)3 + ..... K. †MvjKwUi AvqZb wbY©q Ki| 2
3x + + +
(ii) 6 + 66 + 666 + ... `By wU aviv| L. AvqZvKvi Nbe¯‘wUi K‡Yi© mgvb aviwewkó Nb‡Ki
K. x = 1 n‡j (i) bs avivwUi mvaviY Abycv‡Zi gvb wbYq© mgMÖZ‡ji †¶Îdj wbY©q Ki| 4
Ki| 2 M. ci¯úi ewnt¯úk© K‡i Ggb GKwU e„Ë A¼b Ki hv‡`i
L. x Gi Dci Kx kZ© Av‡ivc Ki‡j (i) bs avivi e¨vmva© h_vµ‡g AvqZvKvi Nbe¯‘i gvÎv¸‡jvi mgvb
AmxgZK mgwó _vK‡e Ges †m mgwó wbY©q Ki| 4 nq| (weeiY Avek¨K) 4
M. (ii) bs avivwUi 1g n msL¨K c‡`i mgwó wbYq© Ki| 4 M wefvM : w·KvYwgwZ I m¤¢vebv
7 tan A + sin A = m, tan A sin A = n
2 2 a b
3 x2 = + 2 Ges b + a = 7. Ges sin + cos = p n‡j
33 33
K. †`LvI †h, K. †`LvI †h m2 n2 = 4 mn 2
1 1 2 L. cÖgvY Ki †h,
x = 33 3 3
L. cÖgvY Ki †h, sin4 + cos4 = 1 1 (p2 1)2 4
2
3x3 + 9x = 8. 4 M. m2 n2 = 4 mn, tan A + sin A = m n‡j †`LvI †h,
M. †`LvI †h, tan A sin A = n. 4
log a + b = 1 loga + 1 logb 4 8 A = {x : x GKwU Q°vi c„ôZ‡ji ¯^vfvweK msL¨v}
3 2 2
L wefvM : R¨vwgwZ I †f±i B = {x : x GKwU g`y ªvi wc‡Vi eY}©
4 ABC Gi j¤w^ e›`y O, cwi‡K›`ª S Ges AP GKwU ga¨gv| K. ZvwjKv c×wZ‡Z A wbYq© Ki| 2
K. cÖ`Ë Z_¨ wP‡Î †`LvI| 2 L. B Gi gy`vª i Abyiƒc wZbwU gy`ªv GK‡Î GKevi wb‡¶c
L. wÎfzRwUi fi‡K›`ª a n‡j cÖgvY Ki †h, S, G, O Ki‡j †KejgvÎ GKwU †Uj cvIqvi m¤v¢ ebv wbY©q
mg‡iL| 4 Ki| 4
M. C = 90 Ges CD AB n‡j M. GKwU g`y ªv I GKwU Q°v GK‡Î wb‡¶‡ci Probability
cÖgvY Ki †h, CD2 = AD. BD. 4 Tree A¼b K‡i Q°vq †Rvo msL¨v I g`y vª q †nW
cvIqvi m¤v¢ ebv wbYq© Ki| 4
06 nvmvb Avjx miKvwi D”P we`¨vjq, Pvu `ciy welq ˆKvW : 1 2 6
mgq : 2 NÈv 35 wgwbU D”PZi MwYZ mR„ bkxj ckÖ œ c~Yg© vb : 50
[Wvb cv‡ki msL¨v c‡Ö kœi cY~ g© vb ÁvcK| cÖ‡Z¨K wefvM †_‡K b¨~ bZg GKwU K‡i †gvU cuvPwU c‡Ö kœi DËi w`‡Z n‡e|]
K wefvM : exRMwYZ 5 PQR Gi wZbwU kxl©we›`y h_vµ‡g P(2, – 4), Q(–4, 4)
1 P(x) = – x2 + 15x + 10x3 + 9 Ges Q(x) = x3 + x3 – 6x Ges R(3, 3).
K. P(x) †K x Pj‡Ki Av`ki© ƒ‡c wj‡L Gi g~L¨ mnM wbY©q K. PQ †iLvi mgxKiY wbY©q Ki| 2
Ki| 2 L. †`LvI †h, PQR GKwU mg‡KvYx I mgwØevû wÎfzR| 4
L. p(x) †K Drcv`‡K we‡kl Y Ki| 4 M. PQR Gi †¶Îd‡ji mgvb †¶Îdj wewkó e‡„ Ëi
M. x2 +x– 1 †K AvswkK fMvœ s‡k cÖKvk Ki| 4 e¨vmva© wbY©q Ki| 4
Q(x)
2 hw` ax = by = cz nq, †hLv‡b a b c 6
K. Pp p = (p p)p n‡j, p Gi gvb wbYq© Ki| AP
2
L. hw` ab = c2 nq Z‡e cgÖ vY Ki †h, 1 + 1 = 2z. 4 DE
x y
M. abc = 1 n‡j cÖgvY Ki †h, 1 + 1 + 1 = x3yz. 4 QB RC
x3 y3 z3
ABC Gi AB I AC evûi ga¨we›`y h_vµ‡g D I E|
1 1 1
3 (i) 3x – 1 + (3x – 1)2 + (3x – 1)3 + ......... K. D`vniYmn †f±i †hv‡Mi wÎfRz wewa eY©bv Ki| 2
(ii) x – xK28 GKwU wØc`x ivwk| L. †f±i c×wZ‡Z cÖgvY Ki †h,
K. x = 1 n‡j, avivwU wbY©q K‡i mvaviY AbycvZ †ei Ki| 2 DE BC Ges DE = 1 BC. 4
2
L. x Gi Dci Kx kZ© Av‡ivc Ki‡j avivwUi AmxgZK M. P, Q, R, S h_vµ‡g BC, CE, ED Ges DB Gi
mgwó _vK‡e Zv wbYq© K‡i D³ kZ© mv‡c‡¶ avivwUi ga¨we›`y n‡j †f±i c×wZ‡Z cÖgvb Ki †h, PQRS
mgwó wbY©q Ki| 4 GKwU mvgvš—wiK| 4
M. (ii) bs Gi we¯—w… Z‡Z x2 Gi mnM 252 n‡j K Gi gvb M wefvM : w·KvYwgwZ I m¤v¢ ebv
wbY©q Ki| 4 7 sin – cos + 11,
sin + cos –
A = B = sec + tan
L wefvM : R¨vwgwZ I †f±i K. tan 10x = cot 5x n‡j x Gi gvb wbYq© Ki| 2
4 A 4
L. cÖgvY Ki †h, A = B
BPC M. B = 3 n‡j Gi gvb wYY©q Ki| †hLv‡b 0 < < 2.4
8 GKwU wbi‡c¶ gy`vª wZbevi wb‡¶c Kiv n‡jv|
ABC mgwØevû wÎfz‡R AB = AC Ges AP BC.
K. ABP Gi †¶‡Î PA †K AB I BP Gi gva¨‡g cÖKvk K. m¤v¢ e¨ NUbvi Probability tree A¼b Ki| 2
L. D‡jwLZ cix¶vq 4
Ki| 2
(i) Kgc‡¶ GKwU †nW cvIqvi m¤¢vebv KZ?
L. D, BC Gi Dci †h †Kv‡bv we›`y n‡j cÖgvY Ki †h, (ii) wZbwUB †Uj cvIqvi m¤¢vebv KZ?
AB2 – AD2 = BD. CD 4
M. ABC wÎf‡z Ri cwie¨vmva© R n‡j cÖgvY Ki †h, M. †`LvI †h, n evi gy`vª wb‡¶‡c msNwUZ NUbv 2n †K
AB2 = 2R. AP 4 mg_©b K‡i| 4
07 Av‡gbv-evKx †iwm‡WwÝqvj g‡Wj ¯‹zj GÛ K‡jR, w`bvRcyi welq ˆKvW : 1 2 6
mgq : 2 NÈv 35 wgwbU D”PZi MwYZ m„Rbkxj ckÖ œ cY~ g© vb : 50
[Wvb cv‡ki msL¨v c‡Ö kœi cY~ ©gvb ÁvcK| c‡Ö Z¨K wefvM †_‡K b~¨bZg GKwU K‡i †gvU cuvPwU c‡Ö kœi DËi w`‡Z n‡e|]
K wefvM : exRMwYZ 5 ABCD PZzf©y‡Ri kxlw© e›`y PviwU h_vµ‡g A (8, 4), B (2, 2),
1 (i) A = {x : x R Ges x2 (a + b) x + ab = 0, a, b R}, C ( 2, 2) Ges D ( 8, 4)
B = {3, 4} Ges C = {3, 4, 7, 8} K. AC K‡Y©i mgxKiY wbY©q Ki| 2
(ii) (x) = 2x + 3 L. ABCD PZfz y‡© Ri †¶Îd‡ji mgvb †¶Îdjwewkó
2x 1
GKwU eM‡© ¶‡Îi cwimxgv wbY©q Ki| 4
K. (ii) n‡Z †Wv‡gb wbY©q Ki| 2 M. ABCD UªvwcwRqv‡gi AB I CD Amgvš—ivj evû؇qi
L. (i) bs †_‡K †`LvI †h, ga¨we›`y M I N n‡j, †f±i c×wZ‡Z cÖgvY Ki †h,
P (B C) = P (B) P (C) 4 MN || AD || BC Ges MN = 1 (AD + BC) 4
2
M. (i) bs †_‡K cÖgvb Ki †h,
6 A
A (B C) = (A B) (A C) 4
l
21 h 2
2 (i) a 1 = 33 + 33, a R CO
(ii) logk (1 + x) = 2 K. †KvYK Kv‡K e‡j? D`vniY `vI|
logk x
L. DÏxc‡Ki Av‡jv‡K †`LvI †h, †KvYKwUi AvqZb,
K. jMvwi`g Kx? 2 V = 1 h3 tan2 4
3
L. (i) bs n‡Z cÖgvY Ki †h, 4
4 M. hw` r = OC = 5 †m. wg. Ges h = AO = 8 †m. wg. nq,
a3 3a2 6a 4 = 0
Z‡e †KvY‡Ki mgMÖZ‡ji †¶Îdj Ges AvqZb wbY©q
1 + 5
M. (ii) bs n‡Z †`LvI †h, x = 2 Ki| 4
3 (i) 2 + x4 5 (ii) M 4y5 `yBwU wØc`x ivwk| M wefvM : w·KvYwgwZ I m¤v¢ ebv
7 P = 1 sinA, Q = sec A tan A
K. c¨vm‡Kj mÎ~ e¨envi K‡i (1 + x)4 †K we¯—…Z Ki| 2 Ges R = 1 + sin A
L. (i) bs e¨envi K‡i (1.9975)4 Gi Avmbœ gvb wZb K. †`LvI †h, Q = P sec A 2
`kwgK ¯’vb ch©š— wbY©q Ki| 4 L. Q = ( 3)1 n‡j, A Gi gvb wbY©q Ki, †hLv‡b A
M. (ii) bs Gi we¯—…wZ‡Z M3 Gi mnM 160 n‡j y Gi gvb m~²‡KvY| 4
wbYq© Ki| 4 M. cÖgvY Ki †h, PR1 = Q2. 4
L wefvM : R¨vwgwZ I †f±i 8 GKRb †jv‡Ki XvKv †_‡K wm‡jU †U‡ª b hvIqvi m¤v¢ ebv 2 ,
4 A 9
†c‡b hvIqvi m¤v¢ ebv 91| †jvKwU wm‡jU n‡Z nweMÄ ev‡m
B PD hvIqvi m¤¢vebv 2 Ges †U‡ª b hvIqvi m¤¢vebv 37|
5
C K. ˆ`e cix¶v wK? e¨vL¨v Ki| 2
K. cÖ`Ë wP‡Îi Av‡jv‡K U‡jwgi Dccv`¨wU eYb© v Ki Ges L. Probability tree Gi mvnv‡h¨ cÖ`Ë Z_¨¸‡jv cÖKvk
MvwYwZKfv‡e †jL| 2 Ki| 4
L. cÖgvY Ki †h, AC . BD = AB . CD + BC . AD 4 M. (i) †jvKwU wm‡j‡U †U‡ª b Ges nweM‡Ä ev‡m bv hvIqvi
M. AB †K e¨vm wb‡q e¨v‡mi Dci Aw¼Z Aae© ‡„ Ëi `By wU m¤¢vebv|
R¨v AC I BD ci¯úi P we›`y‡Z †Q` Ki‡j, cÖgvY (ii) †jvKwU wm‡j‡U †U‡ª b bq Ges nweM‡Ä ev‡m
Ki †h, AB2 = AC . AP + BD. BP 4 hvIqvi m¤v¢ ebv wbYq© Ki| 4
kxl©¯v’ bxq ¯‹z‡ji wbe©vPwb cix¶vi
cÖkcœ Î I DËigvjv : eûwbev© Pwb
08 we G Gd kvnxb K‡jR, XvKv welq ˆKvW : 1 2 6
mgq : 25 wgwbU D”PZi MwYZ eûwbev© Pwb Afx¶v cY~ g© vb : 25
[we. `.ª : mieivnK…Z eûwbe©vPwb Afx¶vi DËic‡Î cÖ‡kiœ µwgK b¤^‡ii wecix‡Z cÖ`Ë eY©msewjZ e„Ëmg~n n‡Z mwVK/ m‡e©vrK…ó Dˇii e„ËwU ej c‡q›U Kjg Øviv m¤úY~ © fivU Ki|]
1. A B n‡j wb‡Pi †KvbwU mwVK? 10. GKwU wÎfz‡Ri †¶Îdj 36 eM© †m.wg. 19. 1536= KZ †iwWqvb?
Ges Gi f‚wg D”PZvi wظY| f‚wgi K 2.72
KAB=A LAB=A L 1.72
M A B = B N A B ˆ`N©¨ KZ? M 0.72 N 0.272
2. A = {a, b, c, d} Gi cKÖ Z… Dc‡mU K 6 †m.wg. L 6 2 †m.wg. 20. 1 x48 Gi we¯w…— Z‡Z x3 Gi mnM KZ?
3. KqwU?
M 12 †m.wg. N 12 2 †m.wg. 1 1 7 8
K4 L8 11. GKwU mgevû wÎfz‡Ri evûi ˆ`N©¨ 5 K 64 L 4 M 8 N 7
M 15 N 16 †m.wg. n‡j Zvi ga¨gvi ˆ`N¨© KZ? y = 3x + 3 †iLvwU P(t, 4) we›`y w`‡q
F(x) = x 1 5 dvskbwUi †Wv‡gb K 2.50 †m.wg. L 4.33 †m.wg. hvq Ges †iLvwU x I y A¶‡K
M 5 †m.wg. N 8.66 †m.wg.
†KvbwU? h_vµ‡g A I B we›`y‡Z †Q` K‡i|
12. GKwU mg‡KvYx wÎf‡z Ri ga¨gv p, q, r Ges Dc‡ii Z‡_¨i Av‡jv‡K 21 I 22 bs
K {x R : x 5} AwZf‚R d n‡j wb‡Pi †Kvb m¤úKw© U mwVK?
L {x R : x > 5} K p2 + q2 + r2 = d2 cÖ‡kiœ DËi `vI :
21. P we›`yi ¯v’ bv¼ KZ?
M {x R : x < 5} L p2 + q2 + r2 = 3d2
M 3(p2 + q2 + r2) = 4d2
N {x R : x 5} K 14 3 L 31 4
4. F(x) = 4x 9 n‡j 1(3) Gi gvb KZ? N 2(p2 + q2 + r2) = 3d2
x2
13. GKwU wÎfz‡Ri wZbwU evûi ˆ`N©¨ 6 M 31 14 1 4
K 3 L 3 M1 N3 N 3
5
†m.wg., 8 †m.wg. I 10 †m.wg.| 22. AB †iLvs‡ki ˆ`N¨© KZ?
5. y = 5x dvsk‡bi wÎfRz wUi cwie„‡Ëi e¨vmva© KZ?
i. †Wv‡gb = ( , ) K2 L5
K 3 †m.wg. L 4 †m.wg.
ii. †iÄ = (0, ) M 5 †m.wg. N 10 †m.wg. M7 N 10
23. ABC Gi AD, BE I CF ga¨gÎq
iii. wecixZ dvskb logx 5 14. x 4 = x + 12 2 mgxKi‡Yi G we›`y‡Z †Q` K‡i‡Q| G wÎfzRwUi
wb‡Pi †KvbwU mwVK?
K i I ii L i I iii exR †KvbwU? fi‡K›` Gi
M ii I iii N i, ii I iii K 5 L 7 M 13 N 15 i. AG : GD = 2 : 1
15. 2x2 7x 1 = 0 mgxKi‡Yi gj~ Øq
|x| ii. AB + AC = 2AD
6. (x) = x dvskbwUi †iÄ KZ? i. ev¯—e
7.
KR L R2 ii. Amgvb
8. iii. AD + BE + CF = 0
wb‡Pi †KvbwU mwVK?
M [1, 1] N {1, 1} iii. Ag~j` K i I ii L i I iii
wb‡Pi †KvbwU mwVK?
x3 ax2 9x 5 eûc`xwUi GKwU K i I ii L i I iii M ii I iii N i, ii I iii
Drcv`K x 5 n‡j, a Gi gvb KZ?
M ii I iii N i, ii I iii 24. 8wU NbK Mwj‡q 4096 Nb †m.wg.
K 9 L 5 AvqZ‡bi GKwU †MvjK ˆZwi Kiv
16. x2 2x 2 = 0 mgxKiYwUi wbðvqK
M 3 N3 n‡jv| Nb‡Ki c‡Ö Z¨K av‡ii ˆ`N¨©
x(x3 + 3x) KZ?
x2 eûc`xwUi aª“ec‡`i KZ?
K 4 L 8 M 12 N 16
¸Ybxq‡Ki †mU †KvbwU? K 5 †m.wg. L 6 †m.wg.
17. 2 1 + 1 1 +.....avivwUi AmxgZK M 8 †m.wg. N 10 †m.wg.
K L {1} 2 4
M {3} N {1, 3} mgwó KZ? 25. 1 †_‡K 10 ch©š— ¯^vfvweK msL¨v¸‡jvi g‡a¨
9. P(x, y) = x2 + y2 2xy n‡j, P(1, 2)
Gi gvb KZ? K 4 L 4 M 4 N4 GKwU msL¨v ˆ`efv‡e wbev© Pb Kiv n‡jv|
3 3 msL¨vwU †gŠwjK nIqvi m¤v¢ ebv KZ?
K9 L1 18. 665 †KvYwU †Kvb PZzf©v‡M Aew¯’Z? K 2 L 4 M 1 N 3
K 1g L 2q M 3q N 4_© 5 11 2 5
M 1 N 9
Self 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN 10 KLMN 11 KLMN 12 KLMN 13 KLMN
test
14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN 19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
1 L 2 M 3 K 4 N 5 K 6 N 7 N 8 N 9 K 10 M 11 L 12 N 13 M
DËigvjv 14 M 15 N 16 M 17 M 18 K 19 N 20 M 21 L 22 N 23 N 24 M 25 K
09 exi‡kôÖ gÝy x Avãyi iDd cvewjK K‡jR, XvKv welq ˆKvW : 1 2 6
mgq : 25 wgwbU D”PZi MwYZ eûwbev© Pwb Afx¶v c~Y©gvb : 25
[we. `ª. : mieivnK…Z eûwbe©vPwb Afx¶vi DËic‡Î c‡Ö kiœ µwgK b¤^‡ii wecix‡Z cÖ`Ë eYm© sewjZ e„Ëmg~n n‡Z mwVK/ m‡e©vrK…ó Dˇii e„ËwU ej c‡q›U Kjg Øviv m¤ú~Y© fivU Ki|]
1. mKvj 6:00 Uvq NÈvi KvUv I wgwb‡Ui 9. †f±i g~jwe›`ywU O n‡j Ges A I B 18. 3y+8 = 92y + 1 n‡j y-Gi gvb KZ?
KvUvi ga¨Kvi †KvY KZ †iwWqvb? we›`yi Ae¯’vb †f±i h_vµ‡g a I b K6 L 10
n‡j wb‡Pi †KvbwU mwVK? 3
K L
3 2 K AB = b a L AB = a b 7 N2
M 3
M N 2
M OA = a b N OA = b a 19. x2 2x 2 = 0 mgxKiYwUi wbðvqK
2. log 2 4 log 3 3-Gi gvb KZ? 10. e‡„ Ëi ewnt¯’ †Kv‡bv we›`y †_‡K H e‡„ Ë
K4 L6 KZwU ¯úkK© Avu Kv hvq? KZ?
M 8 N12 K1 L2 K4 L8
3. logb m.logab = KZ? M3 N4 M 12 N1+ 3
4.
K logam L logbm 11. GKwU g`y ªv‡K 3 evi wb‡¶c Kiv n‡j 20. x 9 < 3x + 1 n‡j, wb‡Pi †KvbwU
M logba N logmb bgybv we›`yi msL¨v KZ? mwVK?
(1 + y)8 Gi we¯—w… Z‡Z (r + 1) Zg K2 L3 K x>5 L x<5
c‡`i mnM †KvbwU? M6 N8 M x>5 N x<5
K 8Cr+1 L 8Cr 12. kyay GKwU evûi ˆ`N¨© †`Iqv _vK‡j 21. hw` A = {2, 3}, B = {3, 4} nq,
M 8Cr1 N 9Cr i. mgevû wÎfRz A¼b Kiv hvq
5. A(3, 2), B(6, 5) Ges C(1,4) ii. eM‡© ¶Î A¼bKiv hvq Zvn‡j
kxl©wewkó ABC †¶Îdj KZ? iii. AvqZ‡¶Î A¼b Kiv hvq i. P(A) = {{2,3}, {2}, {3}, }
ii. P(B) = {{2,4}, {2}, {4}, }
K6 eM© GKK L 9 eM© GKK wb‡Pi †KvbwU mwVK? iii. P(A B) = {{3}, }
M 18 eM© GKK N 29 eM© GKK K i I ii L i I iii
wb‡Pi wP‡Îi Av‡jv‡K 6 I 7 bs M ii I iii N i, ii I iii wb‡Pi †KvbwU mwVK?
c‡Ö kœi DËi `vI : 13. hw` n(A) = 7, n(B) = 4 Ges n(A B) = 5 K i I ii L i I iii
N i, ii I iii
f(x) = x 1 nq, Z‡e n(A B) = KZ? M ii I iii
AmgZvwUi †¶‡Î
6. f(x) = 5 n‡j, x Gi gvb KZ? K2 L6 22. x 2 1
2
K 10 L 26 M8 N 16
M 10 N 26 14. †KvbwU mggvwÎK ivwk? msL¨v‡iLv †KvbwU?
7. wb‡Pi †KvbwU mwVK? K a3 + ab2 + b4 L a3 + 3ab + b3 K 0 1 23 45 6 L 0 1 23 45 6
K †Wvg F = {x , x 1} M a3 + ab2 + b5 N a3 + a2b + b3 M 0 1 23 45 6 N 0 1 23 45 6
L †Wvg F = {x , x 1} 15. x3 + 2x2 + 2x + a Gi GKwU Drcv`K 23. 2 = KZ †iwWqvb?
M †Wvg F = {x , x 1} (x + 1) n‡j, a Gi gvb KZ? K c L c
N †Wvg F = {x , x <1} 45 90
K 5 L 1
c c
8. 3 †m. wg. D”PZvwewkó Ges 4 †m. wg. M 1 N5 M 180 N 360
fw‚ gi e¨vm wewkó mge„Ëf‚wgK 16. ABC wÎf‡z R B m~²‡KvY n‡j wb‡Pi 24. sec 2 4 Gi gvb KZ?
†KvbwU mwVK?
†KvY‡Ki K 2 L 2
K AC2 < AB2 + BC2 3
i. †njv‡bv D”PZv 13 †m. wg. L BC2 < AB2 + AC2
ii. f‚wgi †¶Îdj 16 eM© †m. wg. M AB2 > AC2 + BC2
iii. eµZ‡ji †¶Îdj 2 13 eM© N AB2 < AC2 + BC2 M 2 N2
3
†m. wg. 17. mgevû wÎf‡z Ri †h †Kv‡bv evû‡K
ewa©Z Ki‡j Drcbœ †Kv‡Yi gvb KZ? 25. sin2 () + cos2 = KZ?
wb‡Pi †KvbwU mwVK?
K i I ii L i I iii K 30 L 60 K 1 L0
M ii I iii N i, ii I iii M 90 N 120 M 1 N AmsÁvwqZ
Self 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN 10 KLMN 11 KLMN 12 KLMN 13 KLMN
test
14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN 19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
1 M 2 M 3 K 4 L 5 L 6 N 7 L 8 L 9 K 10 L 11 N 12 K 13 L
DËigvjv 14 N 15 M 16 K 17 N 18 N 19 M 20 K 21 L 22 K 23 L 24 N 25 M
10 ivYx wejvmgwY miKvwi evjK D”P we`¨vjq, †MvcvjMÄ welq ˆKvW : 1 2 6
D”PZi MwYZ eûwbev© Pwb Afx¶v
mgq : 25 wgwbU cY~ g© vb : 25
[we. `.ª : mieivnK…Z eûwbe©vPwb Afx¶vi DËic‡Î c‡Ö kiœ µwgK b¤‡^ ii wecix‡Z cÖ`Ë eYm© sewjZ e„Ëmg~n n‡Z mwVK/ m‡e©vrK…ó Dˇii eË„ wU ej c‡q›U Kjg Øviv m¤úY~ © fivU Ki|]
1. x 3 > x 2 AmgZvi mgvavb KZ? 10. f(x) = 3 x n‡j f(x) Gi †Wv‡gb 17. 2° KZ †iwWqvb?
2 3
wb‡Pi †KvbwU? c c
K x>6 L x>6 K 45 L 90
K {x R ; x < 3} L {x R ; x 3}
M x<6 N x>1 c
M {x R ; x > 3} N {x R ; x 3} M 180 N †Kv‡bvwUB bq
2. GKwU avivi 15Zg c` KZ, hvi n 11.
3. †h‡Kv‡bv a, b I c †f±‡ii Rb¨
Zg c` 1 ( 1)n ? i. a + b = b + a †f±i †hv‡Mi 18. cot = 4 Ges m~²‡KvY n‡j, cosec
1 +n 3
wewbgq wewa
K 1 L0 ii. m (b + c) = mb + mc †f±i eÈb Gi gvb KZ?
8
wewa K 3 L 2
1 1 iii. (a + b) + c = a + (b + c) †f±i 4 3
M 16 N 8
†hv‡Mi ms‡hvMwewa 5 5
(5)x + 3 = 125 Gi mgvavb KZ? M 3 N 2
K 3 L5 wb‡Pi †KvbwU mwVK? 19. 7x = y n‡j, †KvbwU mwVK?
M3 N5
K i I ii L i I iii K x = 7 log y L x = log y
7
31 M ii I iii N i, ii I iii
4. cos 3 Gi gvb KZ? M x = log y7 N x = log7 y
5. 12. †Mvj‡Ki e¨vmva© 2r GKK n‡j Gi
20. (a, 0), (0, b) Ges (1, 1) we›`Îy q
3 AvqZb KZ Nb GKK n‡e?
K1 L 2 mg‡iLv n‡j, wb‡Pi †KvbwU mwVK?
2 L 43r3
1 1 K 3 r3 K a + b = ab L a + b = ab
2 2
M N M 4 r3 32 r3 M a+b=b N a+b=1
3
1 3 N 21. GKwU Q°v I `By wU g`y vª GKmv‡_
27 2
hw` logx = nq, Z‡e x Gi gvb 13. wb‡Pi †KvbwU mwVK? wb‡¶c Kiv n‡jv| msNwUZ NUbv
KZ? K r = s L s = r KqwU?
K3 L 3 M 3 N3 M r = N s = r K 24 L 12
2 2 s M6
N 1
6. nCr Gi gvb KZ? hLb r = 0 14. 3.27x = 9x + 4 Gi mwVK mgvavb 12
K0 L1 †KvbwU? 22. (1 + 3x)5 Gi we¯—…wZ‡Z x Gi mnM
M n N Awb‡Yq© K 6 L 7 KZ?
M8 N9
7. hw` A B nq, Z‡e wb‡Pi †KvbwU K1 L5
mwVK? 15. 2y = 7x mij †iLvwU
i. gj~ we›`yMvgx M 10 N 15
KAB=A LAB=A ii. Øviv Drcbœ Kvj 7 23. xy = yx Ges y = 2x mgxKiY `By wU
M A B = B N A B iii. Øviv y A‡¶i †Q`vsk 0 GKK mgvavb wb‡Pi †KvbwU?
wb‡Pi †KvbwU mwVK?
8. hw` n (A) = 3, n (B) = 4 Ges A B K (2, 4) L (4, 2)
= nq, Z‡e n (A B) = ? K i I ii L i I iii M ( 4, 2) N (4, 2)
K3 L4 M ii I iii N i, ii I iii 24. y = 3 Ges x = y 1 mij‡iLv `By wUi
M 7 N 12 16. GKwU wbi‡c¶ gy`ªv `yBevi wb‡¶c Kiv †Q`we›`y †KvbwU?
9. bewe›`y e‡„ Ëi e¨vmva© wÎf‡z Ri n‡j, me‡P‡q †ewkevi T cvIqvi m¤v¢ ebv
K (3, 2) L (2, 3)
cwie‡„ Ëi e¨vmv‡a©i KZ¸Y? KZ? M (3, 1) N ( 1, 3)
K A‡a©K L wظY 1 1 25. sin2 ( ) + cos2 () = KZ?
4 2
K L K 1 L0
M wZb¸Y N Pvi¸Y M2 N1 M 1 N AmsÁvwqZ
Self 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN 10 KLMN 11 KLMN 12 KLMN 13 KLMN
test
14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN 19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN
------------------------------------------------------------------------------------------------------------------------------------------------------ ---------------------------
1 K 2 N 3 M 4 M 5 K 6 L 7 L 8 M 9 K 10 L 11 N 12 N 13 N
DËigvjv 14 L 15 L 16 K 17 L 18 M 19 N 20 L 21 K 22 N 23 L 24 L 25 M
11 gvwbKMÄ miKvwi D”P we`¨vjq, gvwbKMÄ welq ˆKvW : 1 2 6
mgq : 25 wgwbU D”PZi MwYZ eûwbev© Pwb Afx¶v c~Yg© vb : 25
[we. `.ª : mieivnK…Z eûwbe©vPwb Afx¶vi DËic‡Î c‡Ö kœi µwgK b¤^‡ii wecix‡Z cÖ`Ë eY©msewjZ eË„ mg~n n‡Z mwVK/ m‡e©vrK…ó Dˇii e„ËwU ej c‡q›U Kjg Øviv m¤ú~Y© fivU Ki|]
1. (A B) = n‡j, n(A B) = KZ? 11. GKwU wÎf‡z Ri cwie¨vmva© 7 †m.wg.| 19. logx x x 3 x = KZ?
2.
K n(A) n(B) L n(A) + n(B) H wÎf‡z Ri bewe›`yi e„‡Ëi e¨vmva© KZ 4 5 3 11
M n(A) + n(B) n(A B) 6 6 2 6
†m.wg.? K L M N
N n(A) + n(B) + n (A B)
K 35 L7 20. wb‡Pi wPÎwU j¶ Ki Ges mswkó
f(x) = 2x + 3 dvsk‡bi †jLwPÎ Kxiƒc
n‡e? M 14 N 49 ckÖ ¸œ ‡jvi DËi `vI :
K e˄ vKvi AB
L eµvKvi 12. wb‡Pi wP‡Î x Gi gvb KZ n‡Z cv‡i?
M mij‰iwLK N Dce„ËvKvi x+6 x + 18 P Q
3. P(x) = 4x4 12x3 + 7x2 + 3x 2 Gi D C
GKwU Drcv`K (2x + 1) n‡j, P21 = x+7 ABCD Uvª wcwRqv‡gi BD I ACK‡Yi©
ga¨we›`y h_vµ‡g P I Q n‡j, PQ =?
KZ? K 7 L 4 ( )K
1
K0 L 1 M4 N 12 M 3 N 2 2 DC AB
2
4. P = {x : x abvZ¥K cY~ © msL¨v Ges DÏxcKwU c‡o 13 I 14 bs c‡Ö kœi DËi ( )L
1 DC + AB
5x 16} n‡j, P Gi gvb †KvbwU? `vI : 2
K {0, 1, 2, 3} L {1, 2, 3} 2 + 4 8 + 16 + ....... GKwU Amxg ( )M1
2 AD + BC
M {0, 2, 3} N {0, 1, 2} aviv|
5.
( )6.
ax2 + bx + c = x2 + 2x + 1 n‡j 13. avivwUi n Zg c` = KZ? N 1 AD BC
i. a = 1 ii. b = 2 2
iii. a + b + c = 4 K 2n L 2n
wb‡Pi †KvbwU mwVK? M ( 2)n N 2n 21. (3, 8) Ges (7, p) we›`yi ms‡hvM †iLvi
Xvj 3 n‡j, p Gi gvb KZ?
K i I ii L i I iii 14. avivwUi PZ_z © AvswkK mgwó KZ?
K 4 L 1
M ii I iii N i, ii I iii K 8 L 10 M2 N4
x3 fMvœ skwUi mgvb wb‡Pi †KvbwU? M 16 N 32 DÏxcKwU c‡o 22 I 23 bs cÖ‡kœi DËi
x2 `vI :
9 15. GKwU QvÎ 3 UvKv `‡i xwU Kjg I 2 †Kvb AvqZ‡¶‡Îi ˆ`N¨© 10 †m.wg. I
K x + 9 9 L x + x 9 UvKv `‡i (x + 2)wU LvZv wK‡b‡Q|
x2 x2
†gvU g~j¨ 104 UvKvi Kg bq| †m cÖ¯’ 3 †m.wg.| G‡K e„nËi evûi
M x + 9x N x + 1
x2 9 x2 1 mew© b¤œ KZwU Kjg wKb‡Z cvi‡e? PZzw`©‡K †Nvov‡j GKwU Nbe¯‘ Drcbœ
7. Cxpyq c‡`i gvÎv KZ?
K 20 L 18 nq|
8. K C L p + qM q N pq M 12 N8 22. Nbe¯‘wUi eµZ‡ji †¶Zdj KZ eM©
9. ABC-Gi ACB = ¯j‚’ ‡KvY| AC
= 9 †m.wg.| BC = 8 †m.wg. Ges AC 16. cos = 4 Ges 0 < < n‡j, cot †m.wg.?
5 2
K 188496 L 94248
evûi j¤^ Awf‡¶c 6 †m.wg. n‡j AB = Gi gvb KZ?
KZ †m.wg.? M 282744 N 298.2744
5 5 23. Nbe¯‘wUi AvqZb KZ †m.wg.?
K 1552 L 1452 K 4 L 3
K 162832 L 194248
M 1652 N 1752 4 3
wÎfz‡Ri wZb kxl©we›`My vgx eË„ ‡K Kx ejv M 3 N 4 M 282744 N 2744298
24. m¤¢vebvi m‡e©v”P gvb KZ?
nq? 17. cosec = 2 n‡j, cos = KZ? 1 1
K kxl©eË„ L cwie„Ë K 4 L 2
K0 L1
M Aš—teË„ N ewnteË„
10. wb‡Pi wP‡Î ABC-Gi †¶Îdj KZ M 1 N 2 M0 N1
2 3 25. GKwU Q°v wb‡¶c Kiv n‡j, 4 Gi
eM© GKK?
C 18. tan2 sin2 Gi gvb †KvbwU? Kg Ges †gŠwjK msL¨v covi m¤v¢ ebv
6 6 KZ?
25 K 1 L 9 K 1 L 2
12 4 4 3
A 24 B
M 5 N 1 M 3 N 1
K 48 L 84 M 150 N 300 12 23 4 3
Self 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN 10 KLMN 11 KLMN 12 KLMN 13 KLMN
test
14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN 19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN
----------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------
1 L 2 M 3 K 4 L 5 N 6 M 7 L 8 K 9 L 10 L 11 K 12 K 13 M
DËigvjv 14 L 15 K 16 M 17 M 18 K 19 N 20 K 21 K 22 K 23 M 24 N 25 N
12 K¨v›Ub‡g›U cvewjK ¯‹zj I K‡jR, †gv‡gbkvnx welq ˆKvW : 1 2 6
mgq : 25 wgwbU D”PZi MwYZ eûwbe©vPwb Afx¶v c~Y©gvb : 25
[we. `.ª : mieivnK…Z eûwbe©vPwb Afx¶vi DËic‡Î c‡Ö kœi µwgK b¤‡^ ii wecix‡Z cÖ`Ë eYm© sewjZ e„Ëmgn~ n‡Z mwVK/ m‡e©vrK…ó Dˇii e„ËwU ej c‡q›U Kjg Øviv m¤úY~ © fivU Ki|]
1. 12 sin2 14 sin + 4 = 0 [0 < , 2] 10. GKwU Nb‡Ki K‡Yi© ˆ`N¨© 5 3 GKK f(x) = |x|
2. x
3. n‡j NbKwUi AvqZb KZ Nb GKK?
n‡j Gi gvb KZ? Z‡_¨i Av‡jv‡K 18 I 19bs cÖ‡kœi
K5 L 10
K 0 L 30 DËi `vI :
M 125 N 625
M 45 N 60 18. x < 0 n‡j f(x) Gi gvb KZ?
11. †Kvb k‡Z© (a, b) we›`ywU y A‡¶i Dci
x x x .... Amxg chš© — gvb KZ? K 1 L1
Aew¯’Z?
K x3 1 M x Nx
Mx K a b, b > 0 L a = 0, b 0
L x2 19. f(x) Gi †Wv‡gb KZ?
N x2 M a > 0, b = 0 N a = 0, b = 0
K {0} L R-{1}
2x2 x128 Gi we¯—…wZ‡Z x ewRZ© 12. a < b Ges c < 0 Gi Rb¨ wb‡Pi †KvbwU
M R-{0} NR
c`wU KZ? mwVK? 20. †Kv‡bv wÎf‡z Ri evû·qi gvb 3,4,5
88cc34..2234 88cc56..2256 a b †m. wg. n‡j ga¨gv·qi e‡Mi© mgwó
K L K ac < bc L c = c KZ?
M N M ac > bc K 6.12 eM© †m. wg.
a b L 12.5 eM© †m. wg.
4. 1 x428Gi we¯…—wZ‡Z x3 Gi mnM †KvbwU? N c > c M 37.5 eM© †m. wg.
N 150 eM© †m. wg.
K 7 L1 M0 N 7 13. 3 GKK evûwewkó mgevû wÎfz‡Ri
4 8
gag¨v·qi e‡Mi© mgwó KZ?
5. GKwU †Mvj‡Ki e¨vm 2 †m. wg. n‡j,
K 6.75 L8
†MvjKwUi cô„ Z‡ji †¶Îdj KZ eM© †m. wg.? DÏxcKwU c‡o 21 I 22bs cÖ‡kiœ DËi
M9 N 81 `vI :
K 4 L 16 M 32 N 48 4
DÏxcKwU c‡o 6 I 7 bs c‡Ö kœi DËi 14. cos 172 = KZ? A
`vI :
K0 L 1
3x + 4y 12= 0 mij‡iLvwU x I y A¶‡K
3 45
h_vµ‡g A I B we›`y‡Z †Q` K‡i‡Q| M1 N 2
D C 6 ˆm. wg. B
6. mij‡iLvwUi Xvj wb‡Pi †KvbwU? 15. log 8 x = 313 n‡j, x Gi gvb KZ?
21. BD Gi Dci AC Gi j¤^ Awf‡¶c
K 4 L 3 M 4 N3 K2 L4 †KvbwU?
3 4 3
K BD L CD M AB N BC
7. O g~jwe›`y n‡j AOB Gi †¶Îdj M 16 N 32
22. DC = KZ?
KZ eM© GKK? 16. y5 3y6 + 5y4 7 ivwkwU y Pj‡Ki
K 2 †m. wg. L 4 †m. wg.
K 3 L 4 M 6 N 12
eûc`x M 6 †m. wg. N 8 †m. wg.
8. †Kv‡bv NUbvi AbKy j‚ djvd‡ji †mU
i. gvÎv = 6 23. 4x2 3x 2 = 0 mgxKi‡Yi wbðvqK
A n‡j P(A)-Gi gvb †KvbwU?
ii. gLy ¨cv` 3y6 KZ?
K 0<P(A)<1 L 0P(A)<1
iii. aª“ec` (7) K 41 L 23 M 23 N 41
M 0P(A) 1 N -1P(A) 1
wb‡Pi †KvbwU mwVK? 24. 32x5ax+7 = 34x1, 9a1x n‡j, a =
KZ?
9. `yBwU wbi‡c¶ g`y vª wb‡¶c Ki‡j K i I ii L i I iii
K 2 L 3 M 3 N 5
i. P (gy`ªv `ywU‡Z GKB djvdj) = 1 M ii I iii N i, ii I iii
2 25. 22+22+ ..........aviwUi
i. mvaviY c` = 2(-1)n1
ii. P (Kgc‡¶ 2T) = 1 17. wb‡Pi †KvbwU 2x + 1 Gi GKwU
2 x(x 1)
ii. 15 Zg c‡`i gvb =2
iii. P (Kgc‡¶ 1T) = 3 AvswkK fMœvsk? iii. 1g 50 c‡`i mgwó = 0
4 wb‡Pi †KvbwU mwVK?
2 3 1
wb‡Pi †KvbwU mwVK? K x L x 1 x
K i I ii L i I iii M x 2 1 N 3 K i I ii L i I iii
M ii I iii N i, ii I iii x M ii I iii N i, ii I iii
Self 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN 10 KLMN 11 KLMN 12 KLMN 13 KLMN
test
14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN 19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN
------------------------------------------------------------------------------------------------------------------------ ---------------------------------------------------------
1 L 2 M 3 K 4 M 5 K 6 L 7 M 8 M 9 L 10 M 11 L 12 M 13 N
DËigvjv 14 K 15 N 16 L 17 L 18 K 19 M 20 M 21 N 22 K 23 K 24 M 25 N
13 wK‡kviMÄ miKvwi evjK D”P we`¨vjq, wK‡kviMÄ welq ˆKvW : 1 2 6
mgq : 25 wgwbU D”PZi MwYZ eûwbev© Pwb Afx¶v c~Yg© vb : 25
[we. `.ª : mieivnK…Z eûwbe©vPwb Afx¶vi DËic‡Î c‡Ö kœi µwgK b¤‡^ ii wecix‡Z cÖ`Ë eY©msewjZ eË„ mg~n n‡Z mwVK/ m‡e©vrK…ó Dˇii eË„ wU ej c‡q›U Kjg Øviv m¤úY~ © fivU Ki|]
1. f(x) = x3 5 n‡j f1 (0) = KZ? 9. x, y mgZ‡j 2x < 5 n‡j AmgZvwUi 18. A (0, 3), B (4, 2) Ges C (16, a)
†jLwPÎ wKiƒc? wZbwU we›`y| a Gi gvb KZ n‡j, we›`y
K4 L6 K x A‡¶i mgvš—ivj wZbwU mg‡iL n‡e?
M35 N1 L y - A‡¶i mgvš—ivj K0 L1 M2 N1
19. y A‡¶i Dci j¤^‡iLvi mvaviY
2. hw` a + b + c = 0 nq, Z‡e M g~jwe›`My vgx mgxKiY †KvbwU?
i. a3 + b3 + c3 = 3 abc N Aa©e„Ë
K y=b L x=b
ii. 1 = 1 = 1 10. 1 + 0.1 + 0.01 + 0.001 + avivwUi M y=a N x+y=b
a b c AmxgZK mgwó KZ?
20. u †h †Kvb †f±i Ges m †h †Kvb ev¯—
iii. (a + b)3 + 3abc = c3 K 9 L 11
10 10
wb‡Pi †KvbwU mwVK? e msL¨v n‡j, mu Øviv
10 100 i. m = 0 n‡j, mu Gi w`K u Giv
K i I ii L i I iii M 9 N 9 w`‡Ki mv‡_ mggyLx
M ii I iii N i, ii I iii 11. 1 1 1 ii. m > 0 n‡j, mu Gi w`‡K u Gi
3x + + +
3. wb‡Pi †KvbwU Øviv A‡f` eySvq? 1 + (3x 1)2 + (3x 1)3 + w`‡Ki mv‡_ mggyLx
4.
5. K f (x) = Q (x) L P (x) > Q (x) x Gi Dci wK kZ© Av‡ivc Ki‡j iii. m < 0 n‡j, mu Gi w`‡K u Gi
6. M p (x) Q (x) N p (x) Q (x) avivwUi AmxgZK mgwó _vK‡e? w`‡Ki mv‡_ wecixZgLy x
2
GKwU wÎf‡z Ri cwie¨vmva© 7 †m. wg.| K x > 3 A_ev, x < 0 wb‡Pi †KvbwU mwVK?
H wÎf‡z Ri bewe›`y e„‡Ëi e¨vmva© KZ 2 K i I ii L i I iii
3
†m. wg.? L x < A_ev, x = 0 M ii I iii N i, ii I iii
K 3.5 L7 2 A_ev, 21. kb~ ¨ †f±‡ii †¶‡Î
3
M 14 N 49 M x > x = 0 i. ciggvb kb~ ¨
hw` GKwU mgevû wÎfz‡Ri cwie¨vmva© N x < 2 A_ev, x > 0 ii. aviK †iLv †bB
3
3 †m. wg. nq Z‡e wÎf‡z Ri cwÖ ZwU 12. tan = 1, †hLv‡b, < < 2 n‡j iii. w`K wbY©q Kiv hvq
evûi ˆ`N¨© KZ? wb‡Pi †KvbwU mwVK? wb‡Pi †KvbwU mwVK?
K 6 †m.wg. L 2 3 †m.wg. K = 45° L = 225° K i I ii L ii I iii
M 3 3 †m.wg. N 4 3 †m.wg. M = 315° N 135° M i I iii N i, ii I iii
( 3 27)4 Gi mgvb wb‡Pi †KvbwU? 13. = 7 n‡j, sec2 1 Gi gvb KZ? 22. RvMwZK †Kv‡bv ¯v’ b‡K e›Ub Kiv n‡j
3 KqwU mgZj cÖ‡qvRb?
K3 L9
K 3 L3 K1 L2
M 27 N 81 M3 N 3 M3 N4
`yBwU eM©‡¶‡Îi †¶Îd‡ji mgwó 25 |x| 23. GKwU †Mvj‡Ki e¨vmva© 3 †m. wg.
eM©wgUvi Ges G‡`i `yB evû Øviv 14. f (x) = e 2 dvskbwUi †iÄ †KvbwU?
†hLv‡b 1 < x < 0 n‡j
MwVZ AvqZ‡¶‡Îi †¶Îdj 12 i. †¶Îdj 12 eM© †m. wg.
K ( 1, 0) L (1 , 0)
etwgt| ii. Aa‡© Mvj‡Ki AvqZb 4 3 Nb †m. wg.
7. eM©‡¶Î `&ywUi †¶Îd‡ji AbycvZ 1 0
KZ? M (1, e) N e iii. AvqZb 4 3
L 16 : 9
K 25 : 16 15. y = ln 5 + x Gi †iÄ wb‡Pi †KvbwU mwVK?
5 x K i I ii L i I iii
M ii I iii N i, ii I iii
M 9:4 N 4:3 K L ( 5, 5)
N + { 5 } 24. Am¤¢e NUbvi gvb me mgq KZ nq?
8. mgxKiY I AmgZvi †¶‡Î M +
16. (1 x) 1 + 2x8 Gi we¯—w… Z‡Z x Gi K1 L2
i. x2 4x + 4 > 0 AmgZvi mgvavb
mnM wb‡Pi †KvbwU? M0 N1
x=2 25. 2000 mv‡j †deª“qvwi gv‡m 5 w`b e„wó
n‡qwQj| 12 †de“ª qvwi ew„ ó nIqvi
ii. x2 + 6x + 9 = 0 mgxKi‡Yi g~jØq K 1 L 1 M3 N 1 m¤¢vebv KZ wQj?
mgvb 2 2
17. c¨vm‡K‡ji wÎfz‡Ri evg I Wvb w`‡K
iii. b2 4ac > 0 n‡j ax2 + bx + c = 0 †Kvb msL¨v _v‡K? K 5 L 5
wb‡Pi †KvbwU mwVK? 29 28
Ki L ii K4 L3 M 1 N 1
M2 N1 28 29
M ii I iii N i, ii I iii
Self 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN 10 KLMN 11 KLMN 12 KLMN 13 KLMN
test
14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN 19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN
----------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------
1 M 2 K 3 M 4 K 5 M 6 N 7 N 8 L 9 L 10 M 11 N 12 M 13 M
DËigvjv 14 M 15 K 16 M 17 M 18 L 19 K 20 M 21 K 22 N 23 L 24 M 25 K
14 AvB.B.wU miKvwi D”P we`¨vjq, bvivqYMÄ welq ˆKvW : 1 2 6
mgq : 25 wgwbU D”PZi MwYZ eûwbev© Pwb Afx¶v cY~ g© vb : 25
[we. `.ª : mieivnK…Z eûwbe©vPwb Afx¶vi DËic‡Î c‡Ö kœi µwgK b¤^‡ii wecix‡Z c`Ö Ë eYm© sewjZ eË„ mgn~ n‡Z mwVK/ m‡e©vrK…ó Dˇii e„ËwU ej c‡q›U Kjg Øviv m¤ú~Y© fivU Ki|]
1. 2x + 3 + 2x + 1 = 320 n‡j x Gi gvb 10. mgevû wÎfz‡Ri cwie¨vmva© 3 †m.wg. 17. †Kvb †iLvi j¤^ Awf‡¶‡ci ˆ`N¨©
KZ?
KZ? n‡j wÎf‡z Ri evûi ˆ`N¨© KZ?
K2 L3 K3 3 L3 K AB L AB
M5 N7 AB
M2 3 N3 M k~b¨ AB
wb‡Pi Z‡_¨i Av‡jv‡K 2 I 3 bs N
GKwU mg‡KvYx wÎf‡z Ri AwZfzR wfbœ
c‡Ö kiœ DËi `vI : 11. 18. GKwU Q°v wb‡¶‡c 2 Avmvi m¤¢vebv
4 + 4 + 4 + .......... Aci evûØq 4 †m.wg. I 6 †m.wg.| G‡K KZ? 1
3 9 e„nËi evûi PZwz `©‡K Nyiv‡j Drcbœ 2 3
3
2. Amxg c` KZ? K L
K arn 1 L Nbe¯‘ n‡e M 1 N 1
i. mge„Ëfw‚ gK †KvYK 2 6
4 ii. mge„Ëf‚wgK †ejb
M 310 N0 19. tan 265 Gi gvb KZ?
3. AmxgZ‡Ki mgwó KZ? iii. Drcbœ Nbe¯i‘ fw‚ gi †¶Îdj 36 K 1 L 1
3 3
K7 L6 eM© †m.wg.
wb‡Pi †KvbwU mwVK?
M5 N0 M 1 N1
K i L ii 2
4. 14 x9 x7 x6 Gi gvb KZ?
20. (x) = 2 x n‡j x n‡j †KvbwU
mwVK?
Ka L 1 M i I iii N ii I iii
a14 K (x) 0 L (x) 1
M 22 N a2 12. (x 2)2 + (y + 3)2 e„‡Ëi †K‡›`ªi M (x) N (x)
a14 21. (x + 3) (x 4) 0 n‡j mgvavb †mU
¯’vbv¼ KZ? †KvbwU?
5. x2 5x + 6 = 0 n‡j wbðvqK KZ?
K ( 3, 2) L (2, 3)
K1 L2 K 3x4
M4 N6 M ( 2, 3) N (3, 2) L 3<x<4
M x 3 A_ev (x 4}
6. x2 5x + 4 = 0 wK‡mi mgxKiY? 13. (0, 1) Ges (2, 3) we›`y `ywUi ga¨eZ©x `i~ Z¡
N { 3, 4}
K AwaeË„ L cive„Ë KZ?
22. GKwU wbi‡c¶ gy`vª wZbevi wb‡¶c
M Dce„Ë N e„Ë Kiv n‡jv m‡e©v”P `ywU †nW IVvi
K4 5 L2 5
7. hw` a > 1, x > 1 n‡j wb‡Pi †KvbwU M2 5 N4 5 m¤¢vebv KZ?
mZ¨ 14. m < 0 n‡j mu Gi w`K u Gi K 7 L 3
8 8
K logxa > 0 L logxa < 0
w`‡Ki M 1 N 1
M logxa < 0 N logxa = 0 8 2
K mgvb L GKgLy x
8. x x124 Gi we¯—…wZ‡Z ga¨c` KZ? 23. (2, 3) ( 4, 6) Ges (a, 12) mg‡iL
n‡j a = KZ?
M wecixZgyLx I mgvš—ivj
K9 L 11
N GKgyLx I mgvš—ivj M 10 N8
K 4x L 6 15. Amxg w`K wb‡`©kK †iLv‡K wK e‡j? 24. S = {m : x R Ges x2 + 1 = 0}
x2
K †¯‹jvi L †f±i n‡j
6
M x2 N 4x M †iLvsk N aviK K S=R L S=
M S = R+ N S=N
9. (x, y) = x3 + y3 2xy dvsk‡b ( 1, 16. wc_v‡Mvivm KZ mv‡j g„Z¨z eiY K‡i 25. 5x + 2 4x2 mgxKi‡Yi g‡~ ji cKÖ …wZ
1) we›`y‡Z Gi gvb KZ? K wLóª ce~ © 570 L wLªóc~e© 495 Kxiƒc? L Ag~j`
K RwUj
K1 L2 M 1616 N 1000 mv‡j M g~j` N mgvb
M3 N4
Self 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN 10 KLMN 11 KLMN 12 KLMN 13 KLMN
test
14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN 19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
1 M 2 L 3 L 4 K 5 K 6 L 7 K 8 L 9 L 10 N 11 K 12 L 13 M
DËigvjv 14 M 15 N 16 L 17 M 18 N 19 K 20 M 21 K 22 K 23 N 24 L 25 L
15 evª þb`x gva¨wgK evwjKv we`¨vjq, biwms`x welq ˆKvW : 1 2 6
mgq : 25 wgwbU D”PZi MwYZ eûwbev© Pwb Afx¶v cY~ ©gvb : 25
[we. `.ª : mieivnK…Z eûwbe©vPwb Afx¶vi DËic‡Î c‡Ö kiœ µwgK b¤^‡ii wecix‡Z c`Ö Ë eYm© sewjZ eË„ mgn~ n‡Z mwVK/ m‡e©vrK…ó Dˇii eË„ wU ej c‡q›U Kjg Øviv m¤úY~ © fivU Ki|]
wb‡Pi wPÎwU j¶¨ Ki Ges 1 I 2 bs 12. 3x 2y + 4 = 0 mgxKi‡Yi †¶‡Î 17. x = KZ?
c‡Ö kœi DËi `vI : 2
i. Xvj = 3 K2 3 L3 3 M4 3 N 3
A 3x x 2x + 8 B
ii. x A‡¶i abvÍK w`‡Ki mv‡_ 18. AD Gi gvb KZ?
¯yj’ ‡KvY Drcbœ K‡i
iii. x A¶‡K 34 0 we›`‡y Z †Q` K‡i K6 L2 3 M3 3 N3
1. n(A) = n(B) n‡j x Gi gvb KZ?
K 4 L 8 M 10 N 2 wb‡Pi wPÎwU j¶¨ Ki Ges 19 I 20
bs cÖ‡kiœ DËi `vI :
wb‡Pi †KvbwU mwVK? A
2. n(A B) = KZ? K i I ii L i I iii
K 8 L 16 M 24 N 32
x M ii I iii N i, ii I iii
3. (x) = n‡j 1(5) = KZ? 13. U †Kv‡bv †f±i Ges m †h‡Kv‡bv ev¯—
x 1 e msL¨v n‡j mU G †¶‡Î B CD
3 4 3 5
K 5 L 5 M 4 N 4 i. m = 0 n‡j mU = 0 n‡e wP‡Î BD = 10 cm, AD = 4 cm Ges AC = 5
cm.
4. a3 + b3 + c3 3abc = 0 n‡j ii. m < 0 n‡j mU Gi w`K U Gi w`‡K wecix‡Z 19. ABC = KZ?
i. a + b + c = 0 iii. m > 0 n‡j m-U Gi w`K U Gi
ii. a2 + b2 + c2 = ab + bc + ca avi‡Ki mv‡_ Awfbœ K 40 L 30
1 M 60
2 14. A N †KvbwU bq
iii. {(a + b + c)} 20. ACB ¯’~j‡KvY n‡j wb‡Pi †KvbwU
{(a b)2 + (b c)2 + (c a)2} = 0 FE mwVK?
wb‡Pi †KvbwU mwVK?
K AB2 = AC2 + BC + 2BC.CD
K i I ii L i I iii B C L AB2 = AC2 + BC2 = 2BC.CD
D
M AB2 = AD2 + AC2 2BC.CD
M ii I iii N i, ii I iii wP‡Îi †¶‡Î N AB2 = AD2 + CD2 2BC.CD
5. 3 4x x2 = 0 Gi †¶‡Î
i. wbðvqK = 25 ii. gj~ Øq Amgvb i. 2AC2 = 3(AD2 + BF2 + CF2) 21. mKvj 9.30 wgwb‡U Nwoi NÈv I
ii. 2AC2 = 2(AD2 + BE2 + CF2)
iii. AC2 = BC2 + AB2 wgwb‡Ui Kvu Uvi Aš—f©z³ †KvY KZ?
iii. gj~ Øq Ag~j` wb‡Pi †KvbwU mwVK?
wb‡Pi †KvbwU mwVK? K 110L 107 M 108 N 105
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 cv‡ki wPÎwU j¶¨ Ki Ges 22 I 23
M ii I iii N i, ii I iii bs cÖ‡kœi DËi `vI :
6. `yBwU abvZ¥K c~Y©msL¨vi e‡M©i Aš—i 15. wP‡Î AB = 5 cm, CD = 6 cm, AD = A
11 Ges ¸Ydj 30 n‡j msL¨v `By wUi 10 cm KY© AC = 8 cm, BD = 10 cm
e‡Mi© mgwó KZ? n‡j BC = ? x
K 41 L 61 M 36 N 61 D yC
A
7. †Kvb Abyµ‡gi n Zg c` Un = 1 Ges 22. wP‡Î < < n‡j x I y Gi g‡a¨ m¤úK©
n O 4 2
†KvbwU?
Un < 106 n‡j wb‡Pi †KvbwU mwVK? C K x>y L x<y
K n > 106 L n < 106 B
M n > 106 N 106 > n K5 M8 N9 M x=y N y = 2x
L6
8. 1 + x28 Gi we¯—…wZi cÂg c‡`i mnM KZ? wb‡Pi wPÎwU j¶¨ Ki Ges 1618 bs 23. wPÎ †_‡K y x
cÖ‡kœi DËi `vI : x x2 + y2
K 243 L 405 A i. tan = ii. cos =
M 270 N 1120 iii. sin2 + cos2 = 1
9. x + x126 Gi we¯w—… Zi x h³y c` KZ? wb‡Pi †KvbwU mwVK?
K 6 L 15 M 20 N 35 B DC K i I ii L i I iii M ii I iii
N i, ii I iii
10. A(2, 3) Ges B(3, 6) n‡j AB mij wP‡Î mgevû wÎfz‡Ri cwie„‡Ëi e¨vmva© 24. 5wU g`y vª GK‡Î wb‡¶c Ki‡j †gvU
†iLv x A‡¶i abvZ¥K w`‡Ki mv‡_ 4 †m.wg.|
KZ †KvY Drcbœ K‡i? 16. AD †K x Gi gva¨‡g cÖKvk Ki‡j bgybv we›`y KqwU n‡e?
wb‡Pi †KvbwU n‡e? K 16 L 32 M 64 N 128
K 60 L 71.56 25. wcivwg‡Wi kxl© we›`y I fw‚ gi †h‡Kv‡bv
M 45 N 90
11. Xvj 3 Ges (2, 3) we›`yMvgx mij 3 3 †KŠwYK we›`yi ms‡hvRK †iLvsk‡K wK
K 4 x L 2 x e‡j?
†iLvi mgxKiY,
K 60 L 71.56 3 3 K avi L j¤^
M 4 x2 N 2 x2 M AwZf‚R N cvkZ¦© j
M 45 N 90
Self 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN 10 KLMN 11 KLMN 12 KLMN 13 KLMN
test
14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN 19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN
------------------------------------------------------------------------------------------------------------------------------------ ---------------------------------------------
1 L 2 M 3 N 4 N 5 M 6 L 7 K 8 N 9 L 10 L 11 M 12 L 13 N
DËigvjv 14 L 15 M 16 L 17 K 18 N 19 N 20 K 21 N 22 L 23 N 24 L 25 K
16 W‡bvfvb miKvwi evwjKv D”P we`¨vjq, gv`vixciy welq ˆKvW : 1 2 6
mgq : 25 wgwbU D”PZi MwYZ eûwbev© Pwb Afx¶v cY~ ©gvb : 25
[we. `.ª : mieivnK…Z eûwbe©vPwb Afx¶vi DËic‡Î c‡Ö kœi µwgK b¤‡^ ii wecix‡Z cÖ`Ë eY©msewjZ e„Ëmgn~ n‡Z mwVK/ m‡e©vrK…ó Dˇii e„ËwU ej c‡q›U Kjg Øviv m¤ú~Y© fivU Ki|]
1. GKwU avivi 15Zg c` KZ hvi n Zg 10. sin 120° Gi gvb KZ? 17. log 2 4 log 3 3 Gi gvb KZ?
c` 1 (+n1)n? K 3 L 1 M 1 N 1 K4 L6
1 2 2 2 2 M8 N 12
K 1 L0 M 1 N 1 11. f (x) = 2x [x 4] Øviv ewYZ© 18. 2x + x16 Gi we¯—w… Z‡Z
8 16 8 x4
2. ( 5)x + 3 = 125 Gi mgvavb KZ? dvsk‡bi Rb¨ f (10) = KZ? i. c`msL¨v 7
K 3 L5 K 10 L5 M 10 N 3 ii. x g³y c` 4_© c`
M3 N5 3 10
iii. x g³y c‡`i gvb 160
DÏxcKwU c‡o 3 I 4 bs c‡Ö kœi DËi 12. 2 2 + 2 2 + avivwUi
wb‡Pi †KvbwU mwVK?
3. `vI : i. mvaviY c` = 2 ( 1) n1
K i I ii L i I iii
ii. 15 Zg c‡`i gvb = 2
hw` p (3, 5) I Q ( 4, 2) nq M ii I iii N i, ii I iii
iii. cÖ_g 50 c‡`i mgwó = 0
P I Q we›`yMvgx mij‡iLvi Xvj KZ? 19. B = {x N : 6 < 2x < 17} n‡j, P
wb‡Pi †KvbwU mwVK?
K 1 1 (B) Gi Dcv`vb msL¨v wb‡Pi †KvbwU?
L 3 K i I ii L i I iii M ii I iii
K 23 L 24
M1 N3 N i, ii I iii
M 25 N 24 + 1
4. PQ mij‡iLvi mgxKiY †KvbwU?
DÏxcKwU c‡o 13 I 14 bs cÖ‡kiœ DËi 20. (1 + 3x)5 Gi we¯…w— Zi mvnv‡h¨ x2 mnM
K x y + 2 = 0 L 3x y + 2 = 0 `vI : KZ?
M x + y + 2 = 0 N x 3y + 2 = 0 GKwU wÎfzRvKvi mylg wcÖR‡gi f‚wgi K 10 L 80
5. wb‡Pi †KvbwU cÖwZmg? cÖ‡Z¨K evûi ˆ`N©¨ 4 †m. wg. Ges D”PZv 4 M 90 N 270
K a2 + b + c L 2a2 5bc c2 †m. wg.| 21. A = {a, b, c, d} n‡j P (A) Gi
M x2 y2 + z2 N xy + yz + zx 13. wcRÖ ‡gi AvqZb KZ? Dcv`vb msL¨v KZ?
DÏxcKwU c‡o 6 I 7 bs cÖ‡kœi DËi K 16 3 Nb †m. wg. K4 L8
`vI : L 48 Nb †m. wg.
GKwU ev‡· jvj ej 16wU Ges mv`v M 16 N 32
M 48 3 Nb †m. wg.
22. tan = 3 n‡j cosec = KZ?
3
ej 24 wU| ˆ`efv‡e GKwU ej †bIqv
N 64 Nb †m. wg. K 3 L 2 M 1 N 1
n‡jv 2 3 2 2
14. wcÖR‡gi mgMÖZ‡ji †¶Îdj KZ?
6. ejwU mv`v nIqvi m¤¢vebv KZ? 23. bewe›`y e„‡Ëi e¨vmva© wÎf‡z Ri
K 48 eM© †m. wg.
3 4 1 13 cwie¨vmv‡a©i KZ ¸Y?
K 5 L 13 M 13 N 52 L 61.86 eM© †m. wg.
M 64 eM© †m. wg. 1
7. ejwU jvj nIqvi m¤¢vebv KZ? N 77.86 eM© †m. wg. K wظY L 2 ¸Y
15. †Mvj‡Ki e¨vmva© 2 GKK n‡j Gi
K 2 L 10 M 9 N 7 M 1 ¸Y N wZb ¸Y
5 13 13 13 3
8. hw` A B nq, Z‡e wb‡Pi †KvbwU AvqZb KZ Nb GKK n‡e? 24. tan 11 Gi gvb KZ?
9. mwVK? 6
K 2 r3 L 4 r3
KAB=B LAB=B 3 3 1 2
K 3 L 3
MAB=A N AB=A M 4 r3 32 r3
B N 3 3 1
2 3
240° †KvbwU †Kvb PZfz v© ‡M Ae¯v’ b 16. A M N
K‡i?
B CD Gi 25. hw` ax = 1 nq, †hLv‡b a > 0 Ges a
K c_Ö g L wØZxq ABC mgevû wÎf‡z Ri ACD 1 Zvn‡j x = KZ?
M Z…Zxq N PZz_© A‡aK© KZ wWwMÖ? K0 L1
K 30° L 60° M 90° N 120° M 1 N ±1
Self 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN 10 KLMN 11 KLMN 12 KLMN 13 KLMN
test
14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN 19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
1 N 2 M 3 K 4 M 5 N 6 K 7 K 8 L 9 L 10 K 11 M 12 M 13 K
DËigvjv 14 L 15 N 16 L 17 M 18 N 19 N 20 M 21 M 22 L 23 L 24 N 25 K
17 Kzwgjv gWv© b nvB ¯‹jz , Kwz gjv welq ˆKvW : 1 2 6
mgq : 25 wgwbU D”PZi MwYZ eûwbev© Pwb Afx¶v cY~ g© vb : 25
[we. `.ª : mieivnKZ… eûwbe©vPwb Afx¶vi DËic‡Î cÖ‡kiœ µwgK b¤^‡ii wecix‡Z c`Ö Ë eYm© sewjZ eË„ mg~n n‡Z mwVK/m‡ev© rKó… Dˇii eË„ wU ej c‡q›U Kjg Øviv m¤ú~Y© fivU Ki|]
1. P(A) Gi Dcv`vb msL¨v 64 n‡j A 10. x2 + x124 Gi we¯…—wZ‡Z x ewRZ© c` 17. Drcbœ wmwjÛviwUi D”PZv KZ †m.
Gi Dcv`vb msL¨v KZwU? wg.?
KZ? K4 L6
K4 L5 M 8 N 12
M6 N7
K1 L4 18. wmwjÛviwUi eµZ‡ji †¶Îdj KZ eM© †m.
2. F(x) = 2x 1 dvsk‡bi †Wv‡gb wb‡Pi
3. †KvbwU? M 6 N 12 wg.?
11. GKwU wÎf‡z Ri bewe›`y e‡„ Ëi e¨vmva© 5 K 12 L 24
K x R : x 12 L x R : x 21 †m. wg. n‡j, H wÎf‡z Ri cwie„‡Ëi M 36 N 42
M x R : x 12 N x R : x 12 †¶Îdj KZ eM© †m. wg.? 19. wcRÖ ‡gi f‚wgi †¶Îdj 6 eM© †m. wg.
K 6.25 L 25 Ges D”PZv 8 †m. wg. n‡j, wcRÖ gwUi
M 1002 N 100 AvqZb KZ Nb †m. wg.?
x, y, z Pj‡Ki †¶‡Î cÖwZmg ivwk
i. x + y + z 12. ABC Gi †¶‡Î K 24 L 36
M 40 N 48
ii. xy + yz + zx i. C ¯’~j‡KvY n‡j, AB2 > AC2 + BC2
iii. 2x2 5xy + z2 20. GKwU Q°v wb‡¶‡ci †¶‡Î
ii. C mg‡KvY n‡j, AB2 = AC2 + BC2 1
wb‡Pi †KvbwU mwVK? we‡Rvo msL¨v cvIqvi m¤¢vebv 2
iii. C m~²‡KvY n‡j, AB2 < AC2 + BC2 i.
K i I ii L i I iii wb‡Pi †KvbwU mwVK? ii. †gŠwjK msL¨v cvIqvi m¤v¢ ebv 1
M ii I iii N i, ii I iii 2
K i I ii L i I iii iii. 8 msL¨vwU cvIqvi m¤v¢ ebv 0
4. P(x) = 4x4 12x3 + 7x2 + 3x 31 †K 2x
5. + 1 Øviv fvM Ki‡j fvM‡kl KZ? M ii I iii N i, ii I iii wb‡Pi †KvbwU mwVK?
29 13. A (2, 3), B (5, 6) I C ( 1, 4) kxl© K i I ii L i I iii
8
K 29 L wewkó wÎfzRwUi †¶Îdj KZ eM© M ii I iii N i, ii I iii
M 29 N 29 GKK? 21. GKwU wbi‡c¶ Q°v I GKwU gy`ªv
8
K6 L7 wb‡¶‡ci †gvU bgybv we›`iy msL¨v
3y2 2y 1 = 0 mgxKiYwUi wbðvqK M 8 N 12 KZ?
KZ? 14. x 3y 12 = 0 mgxKiYwUi K 10 L 12
K 8 L 4 mij‡iLv Gi Xvj KZ? M 16 N 20
22. †Kvb NUbv A Gi Rb¨ m¤v¢ ebvi mxgv
M4 N 16 K 1 L 1 wb‡Pi †KvbwU?
ex2 5x + 5 = e1 n‡j x Gi gvb KZ? 5 4
6.
K 3, 1 L 6, 1 M 1 N 1 K 0 < P(A) 1 L 0 P(A) 1
3 2
M 2, 3 N 2, 3 M 0 P(A) < 1 N 0 < P(A) 1
7. 15. 4 Ges m~²‡KvY n‡j,
DÏxcK n‡Z 7 I 8 bs cÖ‡kœi DËi cos = 5 23. P Ges Q we›`y `By wUi Ae¯v’ b †f±i
h_vµ‡g P Ges Q n‡j PQ = KZ?
`vI : cosec Gi gvb KZ? K p q L q p
1 1 1
1 + 2 + 22 + 23 + .......... Amxg aviv| K 3 L 2 M p + q N pq
5 5
avivwUi 8g c` KZ?
1 1 M 5 N 5 24. hw` O gj~ we›`iy mv‡c‡¶ A we›`iy
32 64 3 2 Ae¯v’ b †f±i a I B we›`yi Ae¯’vb
K L
16. `ycyi 1 : 20 Uvq Nwoi NÈvi Kvu Uv I †f±i b nq Ges C we›`ywU AB
1 1
M 28 N 256 wgwb‡Ui KvUvi Aš—fz©³ †KvY KZ? †iLvsk‡K 2 : 1 Abcy v‡Z Aš—wef© ³
8. avivwUi AmxgZ‡Ki mgwó KZ n‡e? K 80 L 90 K‡i| Z‡e OC n‡e wb‡i †KvbwU?
M 100 N 110
K1 L2 K a 2b L 2a b
c`Ö Ë Z‡_¨i wfwˇZ 17 I 18 bs 2a + b a + 2b
M3 N4 M 3 N 3
9. a (x + b) < c Ges a < 0 n‡j wb‡Pi
†KvbwU mwVK? c‡Ö kiœ DËi `vI : 8x = 331 nq Z‡e x Gi gvb
K x < c b L x < c + b 6 †m. wg. e¨vmwewkó GKwU avZe 25. hw` log
a a wb‡iU †MvjK‡K Mwj‡q GKwU KZ?
M x > c b N x > c + b mge„Ëf‚wgK wmwjÛvi cÖ¯‘Z Kiv n‡jv K 32 L 16
a a M8 N4
hvi f‚wgi e¨vmva© 3 †m. wg.|
Self 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN 10 KLMN 11 KLMN 12 KLMN 13 KLMN
test
14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN 19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN
-------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------
1 M 2 L 3 K 4 K 5 N 6 M 7 M 8 L 9 M 10 M 11 N 12 N 13 K
DËigvjv 14 M 15 M 16 K 17 K 18 L 19 N 20 N 21 L 22 L 23 L 24 N 25 K
18 mv‡eiv †mvenvb miKvwi evwjKv D”P we`¨vjq, evª þYevwoqv welq ˆKvW : 1 2 6
D”PZi MwYZ eûwbev© Pwb Afx¶v
mgq : 25 wgwbU cY~ ©gvb : 25
[we. `.ª : mieivnK…Z eûwbev© Pwb Afx¶vi DËic‡Î cÖ‡kœi µwgK b¤^‡ii wecix‡Z cÖ`Ë eY©msewjZ eË„ mg~n n‡Z mwVK/m‡e©vrKó… Dˇii e„ËwU ej c‡q›U Kjg Øviv m¤ú~Y© fivU Ki|]
1. ƒ(x) = x dvsk‡bi †Wv‡gb KZ? 9. x x + 3 Gi mgvavb †KvbwU? 17. (1 + 3x)5 we¯—…wZi x2 Gi mnM KZ
|x| 2 n‡e?
K {0} L { 1, 1} K x6 L x6 K 70 L 80
M R {0} NR M x 12 N x 12 M 90 N 270
2. A B n‡j 10. 1 + 1 + 1 + 1 + ---- wmwiRwUi mvaviY 18. A ( 5, 4), B (3, 7) we›`My vgx †iLvi
2 4 8
i. A B = B Xvj
ii. B \ A = c` †KvbwU? 3 8
8 3
iii. A B = A K 1 L 1 K L
n 2n
wb‡Pi †KvbwU mwVK? M 3 N 8
8 3
K i I iii L i I ii 2 1
M n N 2n 1
M ii I iii N i, ii I iii 19. A (0, 3), B (4, 2) Ges C (16, a)
3. P(x) = x4 5x3 + 7x2 a eûc`xi 11. 2 2 + 2 2 + ----- avivwUi wZbwU we›`y 'a' Gi gvb KZ n‡j, we›`y
GKwU Drcv`K (x 2) n‡j, a = i. mvaviY c` = 2(1)n1 3wU mg‡iL n‡e?
KZ? ii. 15 Zg c‡`i gvb = 2 K0 L1
K 4 L2 iii. c_Ö g 50 c‡` mgwó = 0 M2 N3
M3 N4 wb‡Pi †KvbwU mwVK? 20. †f±i g~jwe›`y 0 n‡j wb‡Pi †KvbwU
4. ABC wÎfz‡R B m~²‡KvY n‡j wb‡Pi K i I ii L ii I iii mwVK?
5.
†KvbwU mwVK? M i I iii N i, ii I iii K OA = a b L OA + OC = AC
K AC2 < AB2 + BC2 L BC2 < 12. 6542 Gi mwVK gvb †KvbwU? M OC = c b N AB = b a
AB2 + AC2 N AB2
M AB2 > AC2 + BC2 K 65.5 L 65.6 21. 3 †m. wg. e¨vm wewkó GKwU e‡ji
>AC2 + BC2
M 65.7 N 65.8 AvqZb KZ?
x = 80 n‡j x Gi m¤úi~ K 13. mKvj 8:20 Uvq Nwoi NÈvq Kvu Uv I K 3 cm3 L 9 cm3
2
†Kv‡Yi GK-cÂgvsk KZ wWwMÖ?
wgwb‡Ui Kvu Uvi Aš—M©Z †KvY KZ M 9 cm3 N 36 cm3
K 100 L 50
n‡e? 22. GKwU Nb‡Ki evûi ˆ`N©¨ 2 †m. wg.
M 22 N 20
n‡j Zvi K‡Y©i ˆ`N©¨ KZ?
6. wÎfz‡Ri cwi‡K›`ª, fi‡K›`ª I j¤^ K 140 L 130
we›`y Øviv MwVZ wÎf‡z Ri †¶Îdj KZ M 115 N 110 K2 3 L3 2
eM© †mw›UwgUvi| 14. sin + cos = 2 n‡j, Gi gvb M4 N8
KZ?
K9 L4 L 60 23. m¤¢vebvi mxgv †KvbwU?
K 45 N 110
M0 N3 M 90 K 0<p<1 L 0p1
wb‡Pi Z‡_¨i Av‡jv‡K 7 I 8 bs M 0<p1 N 0p<1
7. cÖ‡kiœ DËi `vI : 10 24. `yBwU wbi‡c¶ gy`vª GK‡Î wb‡¶c Kiv
GKwU AvqZ‡¶‡Îi K‡Y©i ˆ`N©¨ 15. 3 y5 = 2. 3 y2 n‡j, y Gi gvb KZ? n‡jv| Dfq gy`vª q H cvIqvi m¤¢vebv
wgUvi Ges ˆ`N¨© 8 wgUvi| K1 L2 KZ?
AvqZ‡¶‡Îi cwimxgv KZ wgUvi? M 7 N 10 K 1 L 1
3 7 4 2
K 20 L 24 16. log42 + log6 6 = KZ? M 2 N 3
3 4
M 28 N 32
8. evû `By wU Øviv MwYZ eM©‡¶‡Îi K 1 L 1 25. GKwU Q°v I GKwU gy`vª wb‡¶‡ci
†¶Îd‡ji mgwó KZ eM©wgUvi? 2 3 †gvU bgby we›`yi msL¨v KZ?
K 50 L 100 M 3 N1 K 20 L 12
M 200 N 300 2 M 16 N 20
Self 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN 10 KLMN 11 KLMN 12 KLMN 13 KLMN
test
14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN 19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
1 M 2 K 3 N 4 K 5 N 6 M 7 M 8 L 9 K 10 N 11 N 12 M 13 L
DËigvjv 14 K 15 L 16 N 17 M 18 K 19 L 20 N 21 L 22 K 23 L 24 K 25 L
19 evsjv‡`k gwnjv mwgwZ evwjKv D”P we`¨vjq I K‡jR, PÆMvÖ g welq ˆKvW : 1 2 6
D”PZi MwYZ eûwbe©vPwb Afx¶v
mgq : 25 wgwbU c~Yg© vb : 25
[we. `.ª : mieivnK…Z eûwbe©vPwb Afx¶vi DËic‡Î cÖ‡kiœ µwgK b¤‡^ ii wecix‡Z cÖ`Ë eYm© sewjZ eË„ mg~n n‡Z mwVK/ m‡e©vrK…ó Dˇii e„ËwU ej c‡q›U Kjg Øviv m¤ú~Y© fivU Ki|]
1. †KvbwU mZ¨? 12. f (x) = 2 x + 5 n‡j, 19. x 4 AmgZvq
K a < b n‡j ac > bc 3 2 i. †jLwPÎwU GKwU mij‡iLv
L a > b n‡j a + c > b + c ii. mgvavb (0, 0) we›`yi Rb¨ mZ¨
M a > b n‡j ac < bc i. †iLvwUi Xvj 5 iii. mgvavb †mU S = {x : x 4}
N a < b n‡j a < b + c 2 wb‡Pi †KvbwU mwVK?
ii. †iLvwU x-A¶‡K 15 0 Ges y-
4
47 K i I ii L i I iii
2. cos2 15 + cos2 30 Gi gvb KZ? A¶‡K 0 25 we›`‡y Z †Q` K‡i M ii I iii N i, ii I iii
K3 L2 iii. †iLvwU g~jwe›`yMvgx bq 20. 7 7 + 7 7 + avivwUi 30 wU
M1 N0 wb‡Pi †KvbwU mwVK?
3. AvMvgxKvj mh~ © c~e©w`‡K DVvi m¤¢vebv c‡`i mgwó KZ?
K i I ii L i I iii
KZ? M ii I iii N i, ii I iii K 210 L0
13. D
K0 L 1 C M 30 N 210
2
2b B 21. `wy U †Kv‡Yi mgwó †iwWqvb Ges Aš—
1 3
M 6 N1 a
i †iwWqvb| e„nËi †KvYwUi gvb
4. e‡„ Ëi †K›`ª O (4, 5) eË„ wU y-A¶‡K A 6
¯úk© Ki‡j, Zvi e¨vmva© †KvbwU? wP‡Î BD Gi gvb KZ? KZ?
K1 L4 K a + 2b L 2b + c K L
2 4
M5 N9 M a+c N 2b c
5. A = {x : x N, 8 < 2x < 17} n‡j, P
(A) Gi m`m¨ msL¨v KZ? 14. = 7 n‡j, sec 2 Gi gvb KZ? M N
3 3 6
K4 L8 K 3 L3 22. 3 3 3 729 Gi gvb KZ?
M 16 N 32
6. a Gi †Kvb gv‡bi Rb¨ (a + x)5 Gi M 2 N3 12
15. wZbwU j¤^we›`y Ges kxl© n‡Z wecixZ
we¯—w… Z‡Z x3 Gi mnM 90. evûi Ici j¤w^ e›`My vgx e‡„ Ëi bvg Kx? K 39 L 39
1
K5 L4 K ewne© „Ë L bewe›`y eË„ M 33 N3
M3 N2
7. 0, 1, 0, 1, 0, 1, 0, 1 Abyµ‡gi n M Aš—eË„© N cwieË„ 23. n = 0 n‡j,
8. i. n! = 2
9. Zg c` †KvbwU? 16. mgevû wÎfz‡Ri cwie¨vmva© 3 cm
n‡j, wÎf‡z Ri GKwU evûi ˆ`N¨© KZ ii. 2cn = 1
1 ( 1)n 1 + ( 1)n n‡e? iii. ncn = 1
K 2 L 2 wb‡Pi †KvbwU mwVK?
M 1 + ( 1)n N 1 ( 1)n K3 3 L3 K i I ii L i I iii
wÎfz‡Ri wZb evûi e‡Mi© mgwó I M 2 3 N3 M ii I iii N i, ii I iii
ga¨gv·qi Dci Aw¼Z e‡Mi© mgwói 17. A
AbycvZ KZ? 24. `wy U g`y vª GK‡Î wb‡¶c Kiv n‡j, 2H
bv Avmvi m¤v¢ ebv KZ?
K 2:1 L 3:2 B OC 1 1
M 4:3 N 5:4 4 2
wP‡Î K L
3 y = 3x + 1 †iLvwU x-A‡¶i
abvZ¥K w`‡Ki mv‡_ †h †KvY Drcbœ i. OA = BC 3
1 M 4 N1
K‡i Zvi gvb KZ? ii. OA = 2 BC
25. AvqZvKvi Nbe¯i‘ ˆ`N¨© , cÖ¯’ I
K 90° L 60° iii. AOC = 2 ABC D”PZvi AbycvZ 9 : 8 : 7 Ges AvqZb
M 45° N 30°
wb‡Pi †KvbwU mwVK? 367416 Nb †m. wg. n‡j
2 K i I ii L i I iii i. D”PZv 45 †m. wg.
10. log 8 x = 3 n‡j, x Gi gvb KZ? M ii I iii N i, ii I iii
K2 L1 18. ax2 7x 1 = 0 mgxKi‡Yi ii. K‡Yi© ˆ`N©¨ 125.355 †m. wg.
M 1 N 8 wbðvq‡Ki gvb 57 n‡j, a Gi gvb iii. mgMZÖ ‡ji †¶Îdj 15471 eM© †m. wg.
11. f (x) = x 4 n‡j, f1 (2) = KZ?
K6 L5 KZ? wb‡Pi †KvbwU mwVK?
K 24 L 12 K ii L iii
M4 N2 M4 N2 M i I ii N i, ii I iii
Self 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN 10 KLMN 11 KLMN 12 KLMN 13 KLMN
test
14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN 19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN
----------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------
1 L 2 M 3 N 4 L 5 M 6 M 7 L 8 M 9 L 10 K 11 K 12 M 13 K
DËigvjv 14 M 15 L 16 L 17 M 18 N 19 L 20 L 21 L 22 L 23 M 24 M 25 K
20 PÆMÖvg miKvwi D”P we`¨vjq, PÆMvÖ g welq ˆKvW : 1 2 6
mgq : 25 wgwbU D”PZi MwYZ eûwbev© Pwb Afx¶v cY~ ©gvb : 25
[we. `.ª : mieivnKZ… eûwbev© Pwb Afx¶vi DËic‡Î cÖ‡kœi µwgK b¤^‡ii wecix‡Z cÖ`Ë eY©msewjZ e„Ëmgn~ n‡Z mwVK/m‡ev© rKó… Dˇii e„ËwU ej c‡q›U Kjg Øviv m¤úY~ © fivU Ki|]
1. 240 †KvYwU †Kvb PZzf©v‡M Ae¯v’ b 8. wØc`xwUi we¯w—… Z‡Z †gvU KZwU c` cvIqv 18. kb~ ¨ †f±‡ii ˆewkó¨ n‡”Q
K‡i? hv‡e? i. Gi ˆ`N¨© kb~ ¨
K c_Ö g L wØZxq K 3 L 6 M 7 N 12 ii. Gi Avw` we›`y I Aš—we›`y GKB
M Z…Zxq N PZz_© 1 iii. Gi †Kvb w`K †bB
2. D 9. y = x n‡j, a“ª e c`wU KZ n‡e? wb‡Pi †KvbwU mwVK?
K 20 L 1 M 4 N 20 K i I ii L i I iii
10. x x x)x n‡j, x Gi gvb KZ? M ii I iii N i, ii I iii
x = (x
E B CF K 3 L 3 M 9 N 9 19. mly g PZz¯—j‡Ki †h †Kv‡bv av‡ii ˆ`N©¨ 8
2 2 2 4
wP‡Î B, BF Gi ga¨we›`y Ges DC †m. wg.| GwUi fw‚ gi †¶Îdj KZ eM©
11. mKvj 6:00 Uvq NÈvi KuvUv I †m. wg.?
EF; wgwb‡Ui Kvu Uvi ga¨Kvi †KvY KZ
i. DE2 = DC2 + CE2
ii. DE2 + DF2 = 2 (BE2 + BD2)
iii. DF2 = BF2 + CD2 †iwWqvb? K 16 3 L 64 3
N 256 3
wb‡Pi †KvbwU mwVK? K L M N 2 M 32 3 RS
3 2 n‡j, PQ I
K i L i I ii y (y3 + 3y) 20. PQ = 3SR
M i I iii N i, ii I iii 12. y2 eûc`xi aª“eK c‡`i ci¯úi
3. mgevû wÎf‡z Ri ˆ`N©¨ 5 cm n‡j Zvi ¸Ybxq‡Ki †mU wb‡Pi †KvbwU? K mgvb L j¤^
ga¨gvi ˆ`N¨© KZ? K L {1} M mgvš—ivj I wecixZgLy x
M {3} N {1, 3} N mgvš—ivj I mggL~ x
K 2.50 cm L 4.33 cm
13. P
M 5 cm N 8.66 cm FE 21. 99 †Kv‡Yi Ae¯v’ b †Kvb PZzf©v‡M?
4. A †mUwUi Dcv`vb msL¨v 3 n‡j, Zvi
cÖKZ… Dc‡mU msL¨v KZ? K 1g L 2q
M 3q N 4_©
K3 L6 M7 N9 QR 22. †KvbwU mwVK?
D
PQR G D, E, F h_vµ‡g QR, RP I
5. wÎf‡z Ri wZbwU evûi ˆ`N¨© (GK‡K) †`Iqv PQ Gi ga¨we›`y n‡j wb‡Pi †KvbwU q
6. _vK‡j †Kvb †¶‡Î ¯’j~ ‡KvYx wÎfzR Avu Kv K s = r L r = s
mwVK?
m¤¢e? PQ PR s
K PQ + QR = RP PD + M = sr N = r
K 3, 3, 4 L 3, 4, 4 L = 2
23. sin = 12; 0 < 360 n‡j, Gi
M 3, 4, 5 N 3, 4, 6 QP OR
mgxKiY I AmgZvi †¶‡Î M QE = + N PD + QE + RF = 0 gvb
i. x2 4x + r > 0 AmgZvi mgvavb 2
14. `By wU mij‡iLv ci¯úi j¤^ n‡j
x=2 G‡`i Xvj؇qi ¸Ydj KZ? i. 45
ii. x2 + 6x + 9 = 0 mgxKi‡Yi g~jØq K1 L2 ii. 135
mgvb M3 N 1 iii. 225
iii. b2 4ac > 0 n‡j ax2 + bx + c = 15. A ( 3, 2), B ( 5, 2), C (2, 2)
wb‡Pi †KvbwU mwVK?
0 mgxKi‡Yi gj~ Øq ev¯—e I K i L i I ii
we›`y¸‡jv Øviv MwVZ wÎfz‡Ri †¶Îdj KZ? M ii I iii N i, ii I iii
Amgvb K 6 L 14
wb‡Pi †KvbwU mwVK? M 16 N 28 wb‡Pi Z_¨ Abymv‡i 24 I 25 bs
K i L ii 16. ïay cwimxgvi gva¨‡g †Kvb ai‡bi c‡Ö kœi DËi `vI :
wÎfzR A¼b m¤e¢ ? GKwU ev‡· Kv‡jv ej 20wU, bxj ej
M ii I iii N i, ii I iii K mgwØevû L mgevû
7. wb‡Pi †Kvb ivwkwU ¯^ ¯^ Pj‡Ki Rb¨ M mg‡KvYx N ¯’~j‡KvYx 12wU Ges mv`v ej 16wU Av‡Q|
cÖwZmg? 17. GKwU †Mvj‡Ki e¨vmva© 4 †m. wg. n‡j ˆ`efv‡e GKwU ej †bIqv n‡jv|
†Mvj‡Ki c„ôZ‡ji †¶Îdj KZ? 24. ejwU mv`v bv nIqvi m¤¢vebv KZ?
K 2a2 5ab + c2 L xy + yz zx K 60 eM© †m. wg. 1 1 2 1
M x2 y2 + z2 N a + b + c L 64 eM© †m. wg. K 48 L 16 M 3 N 3
M 74 eM© †m. wg.
wb‡Pi Z‡_¨i Av‡jv‡K 8 I 9 bs N 840 eM© †m. wg. 25. ejwU bxj nIqvi m¤¢vebv KZ?
cÖ‡kiœ DËi `vI : 1 1 1 1
(x y)6 GKwU wØc`x ivwk| K 4 L 8 M 12 N 16
Self 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN 10 KLMN 11 KLMN 12 KLMN 13 KLMN
test
14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN 19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
1 L 2 L 3 K 4 M 5 N 6 M 7 N 8 M 9 K 10 N 11 M 12 N 13 N
DËigvjv 14 N 15 L 16 L 17 L 18 N 19 K 20 M 21 M 22 N 23 L 24 M 25 K
21 e– evW© ¯‹zj GÛ K‡jR, wm‡jU welq ˆKvW : 1 2 6
mgq : 25 wgwbU D”PZi MwYZ eûwbe©vPwb Afx¶v c~Yg© vb : 25
[we. `ª. : mieivnK…Z eûwbev© Pwb Afx¶vi DËic‡Î c‡Ö kiœ µwgK b¤^‡ii wecix‡Z c`Ö Ë eY©msewjZ eË„ mgn~ n‡Z mwVK/m‡e©vrK…ó Dˇii eË„ wU ej c‡q›U Kjg Øviv m¤úY~ © fivU Ki|]
1. †Kv‡bv †m‡Ui m`m¨ msL¨v n n‡j 9. x = 2 n‡j ABC Gi †¶Îdj KZ? 17. 0 < < 90 n‡j wb‡Pi †KvbwU
cÖK…Z Dc‡mU msL¨v KZ? mwVK?
K 2n + 2 L 2n 2 K3 L3 3M 3 N 3
2 K sin + cos = 1
L sin + cos < 1
M 2n 1 N 2n 1 10. GKwU wÎf‡z Ri cwi‡K›`ª, fi‡K›`ª I M sin + cos > 1
2. wb‡Pi †KvbwU Abš— †mU?
j¤^we›`y †hvM Ki‡j †KvbwU MwVZ nq? N sin + cos 1
K wÎfRz L †KvYK
K {1, 2, 3, ---- 50} L {4, 5, 6} 18. sec tan = x n‡j, sec + tan = KZ?
M ¯v^ fvweK msL¨vi †mU
M AvqZ‡¶Î N mij‡iLv 1
N {x + x N, 1 < x < 10} 11. wÎfz‡Ri cwie„‡Ëi e¨vm D n‡j, x
Kx L
3. f(x) = 3 x n‡j f Gi †Wv‡gb bewe›`y e„‡Ëi e¨vmva© KZ?
4. wb‡Pi †KvbwU? 1 1 + cos
5. K D L D M 2D N 4D M cos N sin cos
4 2
K {x : x ; x < 3} 19. 300 †KvYwU †Kvb PZzf©v‡M _vK‡e?
12. 3x2 + bx + 1 = 0 mgxKi‡Yi GKwU
L {x : x ; x 3} g~j 1 n‡j b Gi gvb KZ? K cÖ_g L wØZxq
M {x : x ; x 3} M Z…Zxq N PZz_©
N {x : x ; x = 3} K 2 L 4 20. P = logabc n‡j 1 P = ?
F(x) = x x 2 Gi Rb¨ M4 N 1 i. 1 logabc
13. 2x2 7x 1 = 0 mgxKi‡Yi g~jØq
i. x = 2 Gi Rb¨ F(x) msÁvwqZ i. ev¯—e ii. logaa logabc
ii. GwU GKwU GK-GK dvskb ii. Amgvb iii. logabac
2x
iii. F1(x) = x1 iii. Agj~ ` wb‡Pi †KvbwU mwVK?
wb‡Pi †KvbwU mwVK? wb‡Pi †KvbwU mwVK? Ki L ii
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 M i I ii N i, ii I iii
M ii I iii N i, ii I iii 14. 1 avivwUi Zg wb‡Pi Z‡_¨i Av‡jv‡K 2123 bs
x6 + 3x5 2x4 5 eûc`xi g~L¨ mnM 5
25 + 5 + 1 + + ------ 12 cÖ‡kiœ DËi `vI :
†KvbwU? c` KZ? (1 + 4x + 4x2)n Gi we¯w…— Z‡Z c`msL¨v 7|
K6 L5 M3 N1 K 1 L 1 21. n Gi gvb KZ?
57 58
6. (x 5) eûc`x x3 ax2 9x 5 Gi K2 L3 M4 N5
GKwU Drcv`K| a Gi gvb KZ?
M 1 N 1 22. cÖ`Ë we¯—…wZi PZz_© c` KZ?
K3 L3 M5 N5 59 510 K 60x3 L 160x2
M 160x3 N 60x2
7. N(x) †K KLb cKÖ Z… fMœvsk ejv n‡e? 15. 1 + 0.1 + 0.01 + ------ avivwUi
D(x) AmxgZK mgwó KZ?
23. wØc`xwUi we¯—w… Z‡Z wØZxq c` 48
K N(x) Gi gvÎv = D(x) Gi gvÎv 10 9
L N(x) Gi gvÎv < D(x) Gi gvÎv K 9 L 10 n‡j, x = KZ?
M N(x) Gi gvÎ > D(x) Gi gvÎv
N N(x) Gi gvÎv D(x) Gi gvÎv 10 9 K3 L4 M5 N6
9 10
M N 24. hw` †Kv‡bv mij‡iLv x A‡¶i
wb‡Pi Z‡_¨i Av‡jv‡K 8 I 9 bs cÖ‡kiœ DËi DÏxcKwU c‡o 16 I 17 bs c‡Ö kœi DËi abvZ¥K w`‡Ki mv‡_ 60 †KvY Drcbœ
`vI : `vI :
K‡i, Z‡e Zvi Xvj KZ?
A B
K 3L 1 M 1 N 3
3 3
a
x A C 25. k~b¨ †f±‡ii †¶‡Î
b i. ciggvb kb~ ¨
B DC 16. sin + cos = KZ? ii. aviK‡iLv †bB
ABC GKwU mgevû wÎfzR| K a+b L a2 + b2 iii. w`K wbY©q Kiv hvq
8. wb‡Pi †KvbwU AD Gi gvb? a2 + b2 a+b wb‡Pi †KvbwU mwVK?
K 3 x L 3 x2 M 3x2N x2 M 2a N 2b K i I ii L ii I iii
2 4 a2 + b2 a2 + b2 M i I iii N i, ii I iii
Self 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN 10 KLMN 11 KLMN 12 KLMN 13 KLMN
test
14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN 19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
1 M 2 M 3 L 4 N 5 N 6 K 7 L 8 K 9 M 10 N 11 K 12 L 13 N
DËigvjv 14 M 15 K 16 K 17 M 18 L 19 K 20 N 21 L 22 M 23 L 24 N 25 K
22 cUqz vLvjx miKvwi evwjKv D”P we`¨vjq, cUzqvLvjx welq ˆKvW : 1 2 6
mgq : 25 wgwbU D”PZi MwYZ eûwbev© Pwb Afx¶v c~Yg© vb : 25
[we. `.ª : mieivnK…Z eûwbev© Pwb Afx¶vi DËic‡Î cÖ‡kœi µwgK b¤^‡ii wecix‡Z cÖ`Ë eYm© sewjZ e„Ëmg~n n‡Z mwVK/m‡ev© rKó… Dˇii e„ËwU ej c‡q›U Kjg Øviv m¤úY~ © fivU Ki|]
1. 5 cm, 12 cm I 13 cm evûwewkó 11. (a, b), (b, a) Ges a1 b1 we›`y¸‡jv 19. ax2 + bx + c = 0 mgxKi‡Yi wbðvqK
wÎfz‡Ri Aš—e„‡© Ëi e¨vmva© KZ?
mg‡iL n‡j wb‡Pi †KvbwU mZ¨? D = b2 4ac.
K1 L2 i. hw` D > 0 nq, Z‡e mgxKi‡Yi
M3 N4 K ab=0 L a+b=0 g~jØq ev¯—e, Amgvb
2. ƒ(x) = 4x 13 dvskbwUi †iÄ KZ? M ab = 0 N ab=0 ii. hw` D = 0 nq, Z‡e mgxKi‡Yi
3. x 5
12. GKwU m~lg PZ¯z j— ‡Ki avi‡Ki ˆ`N¨© 4 g~jØq ev¯—e, Amgvb
4. K R {1} L R {4}
†m. wg. n‡j Zvi mgMZÖ ‡ji †¶Îdj iii. hw` D < 0 nq, Z‡e mgxKi‡Yi
M R {5} N R {13}
KZ? gj~ Øq Aev¯—e
hw` g~jwe›`y †_‡K A (4, k) we›`ywUi
K 4 3 eM© †m. wg. wb‡Pi †KvbwU mwVK?
`~iZ¡ 5 GKK nq Zvn‡j k Gi abvZ¥K
K i I ii L i I iii
gvb †KvbwU? L 8 3 eM© †m. wg.
M ii I iii N i, ii I iii
K3 L4 M 12 3 eM© †m. wg.
20. H wÎf‡z Ri cwimxgv KZ hv 3.5 cm, 4.5
M5 N9 N 16 3 eM© †m. wg.
cm Ges 5.5 cm e¨vmv‡ai© wZbwU e‡„ Ëi
A I B `ywU †mU n‡j A \ B †KvbwU? 13. GKwU e„‡Ëi †K‡›`ªi ¯’vbv¼ 0 (4, 5)|
†K›`ª Øviv MwVZ hviv ci¯úi‡K
K AB L AB hw` eË„ wU y A¶‡K GKwU we›`y‡Z ewnt¯’fv‡e ¯úk© K‡i|
M A B N A B
5. xx = xx2 mgxKi‡Yi mgvavb †KvbwU? ¯úk© K‡i| eË„ wUi e¨vmva© KZ?
K 25 L 26
K0 L1 K9 L5 M 27 N 28
M2 N M4 N1 21. 3 GKwU x Pj‡Ki wØNvZ mgxKi‡Yi
6. ƒ(x) = log2x dvsk‡bi †Wv‡gb KZ? 14. (x + a)2 + (y b)2 = 9 e„ËwUi †K‡›`ªi gj~ | mgxKiYwUi g~j `wy Ui e‡Mi©
K R+ †hvMdj †KvbwU?
L R ¯v’ bv¼ KZ?
M ( , ) N †Kv‡bvwUB bq K (a, b) L ( a, b) K3 L6
7. 3x + y 5 = 0 †iLvwU x A‡¶i M ( a, b) N (a, b) M 9 N 18
8.
mv‡_ KZ †KvY Drcbœ K‡i? 15. 3x + 4y = 12 mij‡iLvwU A¶Ø‡qi 22. abvZ¥K gvÎvi †Kv‡bv eûc`xi
K 30 L 60 mv‡_ †h wÎfRz MVb K‡i Zvi †¶Îdj mnMmg‡~ ni mgwó k~b¨ n‡j eûc`xwUi
M 120 N 150 KZ? Drcv`K †KvbwU?
sin2 cos2 = cos hLb, (0 K6 L 12 K x+1 L x1
M 16 N 24 M x2 1 N x2 + 1
) n‡j Gi gvb KZ?
DÏxcKwU c‡o 23, 24 I 25 bs
K , 2 L 23, 16. wb‡Pi †KvbwU GK-GK dvskb?
3 3 cÖ‡kœi DËi `vI :
K F(x) = |x|
M , N 2, L F(x) = 1 + x2 wØc`x ivwk x + x1n G n cY~ ©msL¨v|
3 3
DÏxcKwU c‡o 9 I 10 bs cÖ‡kœi DËi M F(x) = ex 23. ivwkwUi we¯—w… Z‡Z c` msL¨v KZ?
N F(x) = x2
`vI : 17. K n1 L n+1
GKwU wbi‡c¶ gy`vª ‡K wZbevi wb‡¶c
5x +2 = x A 2 + B M n 1 N n (n 1)
2) (3x + 3x Ki‡j Kgc‡¶ `yBwU T Avmvi m¤v¢ ebv n + 1 2!
(x + 2) 2
9. A Gi gvb KZ? KZ? 24. n = 6 n‡j, we¯—w… Z‡Z x ewRZ© c‡`i gvb
K 2 L 1 K 1 L 1 KZ?
2 4
M1 N2 K 6 L 20
10. †KvbwU AvswkK fMœvsk? M 1 N 7 M 15 N 30
8 8
K 1 + 2 L 1 + 1 25. n = 6 n‡j, ivwkwUi we¯—w… Z‡Z
x + 2 3x 2 x+2 3x 2 18. b + 4c Gi mgvš—ivj †f±i †KvbwU?
ga¨c‡`i msL¨v mnM KZ?
1 2 1 1
M x + 2 3x 2 N x + 2 3x 2 K b 4c L 3b + 12c K1 L6
M 4b 4c N 4b + 4c M 16 N 20
Self 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN 10 KLMN 11 KLMN 12 KLMN 13 KLMN
test
14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN 19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN
----------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------
1 L 2 L 3 K 4 M 5 L 6 K 7 M 8 M 9 M 10 K 11 L 12 N 13 M
DËigvjv 14 M 15 K 16 M 17 K 18 L 19 L 20 M 21 L 22 L 23 L 24 L 25 N
23 SvjKvVx miKvwi D”P we`¨vjq, SvjKvVx welq ˆKvW : 1 2 6
mgq : 25 wgwbU D”PZi MwYZ eûwbe©vPwb Afx¶v cY~ ©gvb : 25
[we. `ª. : mieivnK…Z eûwbe©vPwb Afx¶vi DËic‡Î c‡Ö kœi µwgK b¤‡^ ii wecix‡Z c`Ö Ë eY©msewjZ eË„ mgn~ n‡Z mwVK/m‡e©vrKó… Dˇii e„ËwU ej c‡q›U Kjg Øviv m¤ú~Y© fivU Ki|]
1. 2x + y 3 = 0 Ges x 2y 10 = 0 10. F : R R †hLv‡b F(x) = |x 1| 18. GKwU _‡j‡Z 16wU bxj, 12wU jvj I
†iLv `By wUi Xvj؇qi ¸Ydj KZ? n‡j, dvskbwUi †Wvg ƒ KZ? 20wU mv`v ej Av‡Q| ejwU bxj bv
K2 L 2 K {1} L nIqvi m¤v¢ ebv KZ?
M 3 N 1 M + N + K 1 L 1
16 12
2. 2x + 7 = 4x + 2 n‡j x Gi gvb KZ? 11. (8, 6) we›`y n‡Z x A‡¶i `~iZ¡ KZ?
1 2
K 12 L3 K2 L6 M 4 N 3
M5 N 11
M8 N 14 19. p3 p2 10p 8 Gi Drcv`K
47
3. cos2 + cos2 30 Gi gvb KZ? 12. a > b Ges c < 0 n‡j †KvbwU?
15
K0 L1 i. ac < bc K (P + 1) (P + 2) (P 3)
M2 N3 ii. a < b L (P + 1) (P + 2) (P 4)
c c M (P + 1) (P 2) (P + 3)
4. e‡Mi© evûi I K‡Y©i ˆ`‡N©¨i Abcy vZ
iii. a + c > b + c
KZ? N (P + 1) (P + 2) (P + 4)
wb‡Pi †KvbwU mwVK?
K1: 2 L1: 3 20. 3a 2b 12 < 0 AmgZvwUi mgvavb
K i I ii L ii I iii
M 2:1 N 3:1 †KvbwU?
5. (1 2x + x2)7 Gi we¯—…wZ‡Z †gvU c` M i I iii N i, ii I iii K (4, 3) L (4, 0)
msL¨v 13. (a, 0), (0, b) Ges (1, 1) we›`y wZbwU M (4, 3) N (0, 6)
K 15 L 14 mg‡iL n‡j †KvbwU mwVK? 21. GKwU AvqZvKvi Nbe¯i‘ ˆ`N¨© 4 cm,
M8 N7
K a+b=1 L a+b=1 cÖ¯’ 3 cm, D”PZv 2 cm n‡j AvqZb
6. x Gi †Kvb gv‡bi Rb¨ 32ba4x20 = 1 M a + b = ab N a + b = ab KZ? L 24 m3
14. y = 1 2x dvskbwUi †iÄ †KvbwU? K 24 cm3
K2 L3 K ( , 1) L ( , ) M 24000 cm3 N 2400 cm3
M4 N5
7. M (0, ) N (1, ) 22. P(2, 3) Ges Q (4, 6) n‡j PQ Gi
(x a)2 + (y b)2 = 9 e„‡Ëi †K‡›`ªi `i~ Z¡ KZ?
15. P Ges Q we›`yi Ae¯’vb †f±i (b c)
¯v’ bv¼ KZ? K 13 L 117
Ges (b + c) n‡j PQ = KZ?
K (a, b) L ( a, b) M 15 N 81
M ( a, b) N (a, b) K 2a L 2c 23. 3x + y 3 = 0 †iLvi Xvj KZ?
8. 3 cm e¨vmwewkó †Mvj‡Ki AvqZb M b + c N bc K 3 L3
KZ? 16. n(n 1)! Gi gvb †KvbwU? M 1 N4
(n 2)! 3
3 5
K 2 L 2 Kn Ln1 24. RR© K¨v›Ui †Kvb †`‡ki Awaevmx?
M 7 N 9 M n(n 1) N n2 K weª‡Ub L BZvwj
2 2
17. GKwU †MvjvKvi e‡ji e¨vm 4 cm n‡j M d«vÝ N Rvgv© bx
A + B
9. ABC wÎf‡z R tan 2 = KZ? AvqZb KZ? 25. log2 2 + log5 5 = KZ?
K cot C L cot A K 4 Nb †m. wg. L 4 Nb †m. wg. K 3 L 2
2 2 3 2 3
M cot C N tan C M 2 Nb †m. wg. N 32 Nb †m. wg. M 5 N 2
3 2 3 2 2 5
Self 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN 10 KLMN 11 KLMN 12 KLMN 13 KLMN
test
14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN 19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
1 N 2 L 3 L 4 K 5 K 6 N 7 N 8 N 9 K 10 L 11 L 12 N 13 N
DËigvjv 14 K 15 L 16 M 17 N 18 N 19 L 20 K 21 K 22 K 23 K 24 N 25 K
24 cywjk jvBÝ ¯‹jz GÛ K‡jR, iscyi welq ˆKvW : 1 2 6
mgq : 25 wgwbU D”PZi MwYZ eûwbe©vPwb Afx¶v cY~ g© vb : 25
[we. `ª. : mieivnK…Z eûwbe©vPwb Afx¶vi DËic‡Î c‡Ö kœi µwgK b¤^‡ii wecix‡Z cÖ`Ë eY©msewjZ e„Ëmg~n n‡Z mwVK/ m‡e©vrK…ó Dˇii eË„ wU ej c‡q›U Kjg Øviv m¤ú~Y© fivU Ki|]
1. mvwe©K †mU U Gi †h‡Kv‡bv Dc‡mU A Gi Rb¨ 10. x = 2y n‡j, y Gi gvb KZ? 18. x3 + x1312 Gi we¯—w… Z
A/(A/A) nq, Z‡e wb‡Pi †KvbwU mwVK? K2 L3 M4 N5
K A L A M N {0} 11. 2x + 3y 7 0 AmgZvi mgvav‡bi †jLwPÎ i. ga¨c‡`i msL¨v 2wU
2. y = x2 + 6x 9 Gi †jLwPÎ †Kvb i. †h w`‡K gj~ we›`y †iLvi †m cv‡ki ii. mßg c` x ewR©Z
w`‡K †Lvjv n‡e? mKj we›`y iii. cÂg c‡`i mnM 12C4
K wb‡Pi w`‡K L Wvb w`‡K ii. (3, 3) †h w`‡K Av‡Q †iLvi †m wb‡Pi †KvbwU mwVK?
M evg w`‡K N Dc‡ii w`‡K cv‡ki mKj we›`y K i I ii L i I iii
3. S = {(x, y) : x2 + y2 25 = 0 Ges x 0} iii. mij‡iLvwU ( 1, 3) we›`yMvgx M ii I iii N i, ii I iii
n‡j wb‡Pi †KvbwU mwVK? 19. P(3, 0), Q(0, 1), R( 1, r) kxl© wewkó wÎf‡z Ri
i. Aš^qwU dvskb bq K i I ii L i I iii †¶Îdj 5 eM© GKK n‡j, r Gi gvb KZ?
ii. Aš^qwUi †jLwPÎ GKwU Aa©eË„ M ii I iii N i, ii I iii K2 L1 M0 N1
iii. Aš^qwUi †jLwPÎ x A‡¶i Dci DÏxcKwU c‡o 12 I 13 bs c‡Ö kœi DËi DÏxcKwU c‡o 20 I 21 bs cÖ‡kœi DËi
Aa©Z‡j _vK‡e `vI : `vI :
wb‡Pi †KvbwU mwVK? C
K i I ii L i I iii 1 1 + (3x 1 1)2 + (3x 1 1)3 c
3x + + +
12. x Gi Dci Kx kZ© Av‡ivc Ki‡j bB
M ii I iii N i, ii I iii aA
2x + 1 A B avivwUi AmgxZK mgwó _vK‡e? O
4. hw` x(x 1) x + x 1 nq, Z‡e A 2
5. 3 A_ev 20. AB = KZ?
6. K x > x < 0 1 1
I B Gi gvb h_vµ‡g KZ n‡e? K 2 (a b) L 2 (a + b)
K 1I 3 L 3I 1 2
M 2I 1 N 1I 2 L x < 3 A_ev x = 0 Ma+b N ba
7. 21. hw` C we›`ywU AB Gi ga¨we›`y nq,
8. ABC I C = 120, BC = 2 †m.wg. M x > 2 A_ev x > 0 Z‡e wb‡Pi †KvbwU mwVK?
Ges AC = 5 †m.wg. n‡j, AB Gi ˆ`N¨© 3
2 1 1
KZ †m.wg.? N x < 3 A_ev x > 0 K c = 2 (b + c) L c = 2 (a + b)
K 9 L 19 M 39 N 49 13. avivwUi mvaviY AbycvZ KZ? M c = 1 (b a) N c = 1 (a b)
K 3x + 1 L (3x + 1)2 2 2
ABC G CD, AC Gi j¤^
Awf‡¶c| B m~²‡KvY n‡j, AC2 M 1 N 1 DÏxcKwU c‡o 22 I 23bs c‡Ö kiœ DËi `vI :
Gi gvb †KvbwU? 3x + + 1)2 GKwU wÎfRz vKvi mylg wcÖR‡gi f‚wgi
1 (3x c‡Ö Z¨K evûi ˆ`N¨© 4 †m.wg. Ges
K AB2 + BC2 2BCCD
L AB2 + BC2 2BCBD 14. `yBwU †Kv‡Yi mgwó †iwWqvb Ges D”PZv 4 †m.wg.|
M AB2 + BC2 + 2ACCD 3
N AB2 + BC2 + 2ABAD 22. wcRÖ ‡gi AvqZb KZ?
Aš—i †iwWqvb| e„nËi †KvYwUi eË„ xq K 16 3 Nb †m.wg.
6
gvb KZ?
DÏxcKwU c‡o 6 I 7 bs cÖ‡kiœ DËi L 48 Nb †m.wg.
`vI : M 48 3 Nb †m.wg.
ABC Gi ab = 5 †m.wg. AD BC K 2 L 3 M 4 N 6 N 64 Nb †m.wg.
Ges BC = 6 †m.wg.|
ABC Gi †¶Îdj KZ eM© †m.wg.? 15. cos = 3 n‡j sin3 = KZ? 23. wcÖR‡gi mgMZÖ ‡ji †¶Îdj KZ?
2 K 48 eM© †m.wg.
K 12 L 13 M 14 N 15
K0 L 3 L 6186 eM© †m.wg.
AB I AD Gi ga¨eZx© †KvY n‡j, 2 M 64 eM© †m.wg.
1
tan = ? M 2 N1 N 7786 eM© †m.wg.
24. m¤v¢ ebvi mxgv †KvbwU?
K 3 L 2 M 1 N 1 16. log 2 4 log 33 Gi gvb KZ? K 0<p<1 L 0p1
4 3 2 3
K 4 L 6 M 8 N 12 M 0<p1 N 0p<1
DÏxcKwU c‡o 9 I 10 bs cÖ‡kœi DËi `vI :
xy = yx nq, Z‡e 17. y = 3x dvsk‡bi 25. `By wU gy`vª wb‡¶‡ci †¶‡Î
i. †Wv‡gb = ( , ) i. eo‡Rvo GKwU H cvIqvi m¤v¢ ebv = 075
9. yxxy Gi gvb †KvbwU? ii. †iÄ = (0, ) ii. Kgc‡¶ GKwU H cvIqvi m¤¢vebv 075
iii. wecixZ dvskb = logx3 iii. HH GKwU bgby v we›`y
x y wb‡Pi †KvbwU mwVK? wb‡Pi †KvbwU mwVK?
K i I ii L i I iii K i I ii L i I iii
K xy 1 L xx 1 M ii I iii N i, ii I iii M ii I iii N i, ii I iii
x y
M xy 1 N x1 x
Self 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN 10 KLMN 11 KLMN 12 KLMN 13 KLMN
test
14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN 19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
1 L 2 K 3 K 4 K 5 M 6 L 7 K 8 K 9 K 10 K 11 L 12 N 13 M
DËigvjv 14 M 15 N 16 M 17 K 18 M 19 K 20 N 21 L 22 K 23 L 24 L 25 N
NCTB KZ©K… cÖ`Ë P‚ovšÍ gvbeȇbi Av‡jv‡K
G·Kwz¬ mf g‡Wj †U÷ : m„Rbkxj
25 G·Kwz¬ mf g‡Wj †U÷ 01 welq ˆKvW : 1 2 6
mgq : 2 NÈv 35 wgwbU D”PZi MwYZ mR„ bkxj ckÖ œ c~Y©gvb : 50
[Wvb cv‡ki msL¨v cÖ‡kœi c~Y©gvb ÁvcK| c‡Ö Z¨K wefvM †_‡K b¨~ bZg GKwU K‡i †gvU cvu PwU cÖ‡kœi DËi w`‡Z n‡e|]
K wefvM : exRMwYZ 5 mij‡iLv y = 3x + 4, x A¶‡K P we›`‡y Z mij‡iLv 3x + y = 0,
1 E = {x : x R Ges x2 (a + b) x + ab = 0, a, b R},
y A¶‡K Q we›`y‡Z †Q` K‡i Ges mij‡iLvØq ci¯úi R
F = {3, 4} Ges G = {4, 5, 6}.
K. E †m‡Ui Dcv`vb wbY©q Ki| we›`‡y Z †Q` K‡i|
L. cÖgvY Ki †h,
2 K. mij‡iLv؇qi Xv‡ji ¸Ydj wbY©q Ki| 2
P(F G) = P(F) P(G). 4
4 L. R we›`My vgx Ges 4 Xvjwewkó mij‡iLvi mgxKiY wbYq©
M. †`LvI †h,
Ki| 4
E (F G) = (E F) (E G).
M. PQR Gi †¶Îdj wbY©q Ki| 4
6 A
2 f(x) = 18x3 + 15x2 x + c, g(x) = x2 4x 7 Ges h(x) = CB
x3 x2 10x 8 n‡”Q x Pj‡Ki wZbwU eûc`x| QA R
K. h(x) †K Drcv`‡K we‡kl Y Ki| 2 A, B, C h_vµ‡g QR, RP Ges PQ Gi ga¨we›`|y
L. f(x) Gi GKwU Drcv`K (3x + 2) n‡j c Gi gvb wbYq© K. PQ †f±i‡K BQ Ges CR Gi gva¨‡g cÖKvk Ki| 2
L. †`LvI †h,
Ki| 4
M. g(x) †K AvswkK fMœvs‡k cÖKvk Ki| 4 PA + QB + RC = 0. 4
h(x)
3xa= yb= zc M. †f±‡ii mvnv‡h¨ cgÖ vY Ki †h, C we›`y w`‡q Aw¼Z QR
†iLvi mgvš—ivj †iLvwU B we›`yMvgx n‡e| 4
K. a = c n‡j †`LvI †h, x = z. 2 M wefvM : w·KvYwgwZ I m¤v¢ ebv
7 N
L. hw` x = 12, y = 1 nq, Z‡e †`LvI †h,
3
ba23 ba23 1 1 O M
+ = a2 + b3. 4 P
4
wP‡Î O †K›`w ewkó GKwU e„Ë Ges OM = Pvc MN.
M. hw` abc = 1 nq, Z‡e cÖgvY Ki †h, K. †K wWwMÖ‡Z cÖKvk Ki| 2
px 1 1 + py 1 1 + pz + 1 + 1 = 1. L. cÖgvY Ki †h, GKwU aª“e †KvY| 4
+ py + pz px
M. Gi †Kvb gv‡bi Rb¨ PN OP 2 n‡e †hLv‡b,
L wefvM : R¨vwgwZ I †f±i ON + ON =
4 ABC Gi wZbwU ga¨gv AD, BE Ges CF ci¯úi G 0 < < 2. 4
we›`y‡Z †Q` K‡i| 8 GKwU c¶cvwZZ¡nxb gy`vª I GKwU Q°v GKB mv‡_ wb‡¶c
K. GD = 2 †m.wg. n‡j, AD Gi gvb wbY©q Ki| 2 Kiv n‡jv|
L. cÖgvY Ki †h, K. m¤¢ve¨ NUbvi Probability tree A¼b Ki| 2
L. gy`vª n‡Z †UBj I Q°v n‡Z we‡Rvo msL¨v cvIqvi
AB2 + BC2 = 2(AE2 + BE2). 4 m¤v¢ ebv KZ? 4
M. cÖgvY Ki †h, M. Q°v e¨ZxZ hw` ïaygvÎ gy`ªvwU‡K wZbevi wb‡¶c Kiv
3(AB2 + BC2 + AC2) = 4(AD2 + BE2 + CF2). 4 nq, Kgc‡¶ GKwU †nW cvIqvi m¤¢vebv KZ? 4
26 G·Kwz¬ mf g‡Wj †U÷ 02 welq ˆKvW : 1 2 6
mgq : 2 NÈv 35 wgwbU D”PZi MwYZ mR„ bkxj cÖkœ cY~ g© vb : 50
[Wvb cv‡ki msL¨v c‡Ö kœi c~Yg© vb ÁvcK| cÖ‡Z¨K wefvM †_‡K b¨~ bZg GKwU K‡i †gvU cuvPwU cÖ‡kœi DËi w`‡Z n‡e|]
K wefvM : exRMwYZ 5 ABC Gi BC, CA I AB evûi ga¨we›`y h_vµ‡g D, E, F|
2 2 K. AB †f±i‡K BE I CF †f±‡ii gva¨‡g cÖKvk Ki| 2
1 l = ay z, m = az x, n = ax y, A = a2 33 3 3 + 2, p 0
Ges p = loga (bc), q = logb(ca), r = logc(ab) L. cÖgvY Ki †h, AD + BE + CF = 0. 4
K. lmn = KZ?
L. A = 0 n‡j, †`LvI †h, 2 M. A I B we›`iy Ae¯v’ b †f±i a, b n‡j, C we›`y AB
3a3 + 9a = 8 4 †iLv‡K m : n Abycv‡Z Aš—we©f³ Ki‡j †`LvI †h,
M. cÖgvY Ki †h, C we›`iy Ae¯v’ b †f±i ma + nb . 4
m + n
p 1 1 + q 1 1 + r 1 1 = 1 4 6 3x + by + 1 = 0 Ges ax + 6y + 1 = 0 †iLv `ywU (5, 4)
+ + + we›`‡y Z †Q` K‡i|
2 (x) = ax + b K. Xvj Kv‡K e‡j? `By we›`yMvgx mij‡iLvi Xvj wbY©‡qi
cx + d
Ges g(x) = x2 9x 6 m~ÎwU †jL| 2
x(x 2) (x + 3)
L. †iLv؇qi cÖK…Z mgxKiY †jL| 4
K. 1x Gi gvb wbY©q Ki|
2 M. hw` cÖ_g †iLvwU x-A¶‡K A we›`‡y Z Ges wØZxq
L. a, b, c, d R n‡j, †`LvI †h, (x) dvskbwU GK-GK †iLvwU y A¶‡K B we›`‡y Z †Q` K‡i, Z‡e AB
Ges AbUz| 4 mij‡iLvi mgxKiY wbY©q Ki| 4
M. g(x) †K AvswkK fMvœ s‡k cÖKvk Ki| 4 M wefvM : w·KvYwgwZ I m¤¢vebv
3 (i) (1 + 3x)5 Ges (ii) p 3x7 `By wU wØc`x ivwk| 7 (i) tan = yx, †hLv‡b m~²‡KvY Ges x y
K. (i) †K c¨vm‡K‡ji wÎfz‡Ri mvnv‡h¨ we¯—Z… Ki| 2 (ii) tan2 A + cot2A = 2 hLb 0 < A < 2
L. (i) Gi mvnv‡h¨ (1.255)5 Gi Avmbœ gvb Pvi `kwgK K. †`LvI †h,
¯’vb ch©š— wbY©q Ki| 4 y
x2 + y2
M. (ii) Gi we¯—…wZ‡Z p3 Gi mnM 560 n‡j x Gi gvb sin = 2
wbY©q Ki| 4 L. x = 4 Ges y = 3 n‡j sin + cos Gi gvb wbYq©
sec + tan
L wefvM : R¨vwgwZ I †f±i
4 ABC wÎf‡z Ri cwi‡K›`ª O Ges AP cwie„‡Ëi GKwU e¨vm| Ki| 4
ABC wÎfz‡Ri kxl© A †_‡K wecixZ evû BC Gi Dci AD M. (ii) bs mgxKi‡Y A Gi gvb †ei Ki| 4
j¤^| 8 GKwU wek¦we`¨vj‡q 1g e‡l© 120 Rb QvÎ CSE †Z, 115 Rb
K. eþª ¸‡ßi Dccv`¨wU †jL| 2 QvÎ EEE †Z, 105 Rb QvÎ IT †Z Ges 90 Rb QvÎ MATH
L. DÏxc‡Ki Av‡jv‡K cÖgvY Ki †h, G fwZ© n‡q‡Q| GKRb Qv·K ˆ`efv‡e wbev© wPZ Kiv n‡jv|
AB.AC = AP.AD 4 K. GKRb Qv·K KZ Dcv‡q wbev© wPZ Kiv hvq? 2
M. DÏxc‡Ki wPÎ MVb K‡i B, P Ges C, P †hvM Ki‡j L. wbev© wPZ QvÎwUi MATH G bv nIqvi m¤v¢ ebv KZ? 4
ABPC e„˯’ PZzf©Rz Drcbœ nq| GB PZzf©z‡Ri KYØ© q M. wbev© wPZ QvÎwUi EEE †Z A_ev CSE †Z nIqvi
AP, BC n‡j cgÖ vY Ki †h, APBC = ABCP + BPAC. 4 m¤¢vebv KZ? 4
27 G·Kzw¬ mf g‡Wj †U÷ 03 welq ˆKvW : 1 2 6
mgq : 2 NÈv 35 wgwbU D”PZi MwYZ mR„ bkxj ckÖ œ cY~ g© vb : 50
[Wvb cv‡ki msL¨v cÖ‡kœi cY~ g© vb ÁvcK| c‡Ö Z¨K wefvM †_‡K b¨~ bZg GKwU K‡i †gvU cvu PwU c‡Ö kœi DËi w`‡Z n‡e|]
K wefvM : exRMwYZ 5 A
1 x Pj‡Ki eûc`x P(x) = 4x4 + 12x3 + bx2 3x 2 Ges
D (t 2, t E(1, 1)
Q(x) = x3 + x2 6x.
B (t, 3t) C(t2, 2t)
K. x4 5x3 + 7x2 4 eûc`x‡K x 1 Øviv fvM Ki‡j
fvM‡kl Kx n‡e Zv fvM‡kl Dccv‡`¨i mvnv‡h¨ wbYq© ABC Gi AB I AC evûi ga¨we›`y h_vµ‡g D I E|
Ki| 2
K. E we›`My vgx Ges 3 Xvjwewkó †iLvi mgxKiY wbY©q
L. P(x) eûc`xi GKwU Drcv`K x + 2 n‡j b Gi gvb
Ki| 2
L. †f±‡ii mvnv‡h¨ †`LvI †h,
wbY©q Ki Ges P(x) †K Drcv`‡K we‡klY Ki| 4 DE || BC Ges DE = 1 BC. 4
4 2
x3
M. Q(x) †K AvswkK fMvœ s‡k cÖKvk Ki| M. t Gi gvb wbYq© c~e©K BCED PZfz z©‡Ri †¶Îdj wbYq©
2 x2 + xk6 Ges x2 2 + x126 `By wU wØc`x ivwk| Ki| 4
K. log2 2 x = 3 1 n‡j x Gi gvb bY©q Ki| A
3
6
2 B OP D
L. 1g ivwkwUi we¯—w… Z‡Z x3 Gi mnM 160 n‡j k Gi gvb C
wbY©q Ki| 4 e„‡Ë Aš—wj©wLZ ABCD PZzfz‡© R AC I BD `By wU KY©|
M. 2q ivwkwUi we¯—…wZ‡Z x gy³ c` †ei Ki| 4 K. U‡jwgi Dccv`¨ weeZ„ Ki| 2
3 wb‡Pi Z_¨¸‡jv j¶ Ki : L. cÖgvY Ki †h,
AC . BD = AB . CD + BC . AD. 4
(i) 1 1 + (4x 1 1)2 + (4x 1 1)3 + … GKwU Amxg ¸‡YvËi aviv| M. AC = 12 †m.wg. n‡j, ABC †K AC Gi PZzw`‡© K
4x + + +
GKevi Nywi‡q Avb‡j †h Nbe¯‘ ˆZwi nq Zvi AvqZb
2 1
(ii) m2 +2= wbYq© Ki| Nbe¯‘wU w`‡q 4 †m.wg. `xN© I 6 †m.wg.
33 + 2
33 e¨vmwewkó KqwU wb‡iU wmwjÛvi ˆZwi Kiv hv‡e| 4
K. x = 1 n‡j (i) bs avivwU wbY©q K‡i avivi mßg c` M wefvM : w·KvYwgwZ I m¤¢vebv
†ei Ki| 2 7 A = cot + cosec 11, B = cosec + cot
cot cosec +
L. (i) bs avivwUi Dci wK kZ© Av‡ivc Ki‡j avivi
AmxgZK mgwó _vK‡e Ges †mB mgwó wbYq© Ki| 4 Ges 7P2 + 3Q2 = 4.
M. (ii) bs n‡Z †`LvI †h, K. = n‡j A Gi gvb wbY©q Ki| 2
4
3m3 + 9m 8 = 0. 4 L. P = sin Ges Q = cos n‡j tan Gi gvb wbY©q
L wefvM : R¨vwgwZ I †f±i Ki| 4
M. †`LvI †h, 4
4 PQR-Gi PX, QY Ges RZ ga¨gvÎq G we›`y‡Z †Q` K‡i
A2 B2 = 0.
Ges ABC-G AB = AC I fw‚ g BC Gi Dci S †h‡Kv‡bv
we›`y| 8 GKwU ev‡· GKwU Q°v I GKwU gy`ªv Av‡Q|
K. PQR-Gi G we›`yi bvg Kx? G we›`y PX †K Kx K. g`y vª wU `By evi wb‡¶‡ci bgby v †¶ÎwU ˆZwi K‡i
Abcy v‡Z wef³ K‡i? 2 eo‡Rvi 2T Avmvi m¤¢vebv wbY©q Ki| 2
L. PQR-G †`LvI †h, L. Q°vwU GKevi wb‡¶c Kiv n‡j †gŠwjK A_ev †Rvo
PQ2 + QR2 + PR2 = 3(GP2 + GQ2 + GR2). 4 msL¨v Avmvi m¤¢vebv wbY©q Ki| 4
M. ABC-G cÖgvY Ki †h, M. ev‡·i Q°v I g`y ªv GK‡Î GKevi wb‡¶c NUbvi
AB2 AS2 = BS . SC. 4 Probability tree ˆZwi K‡i Q°vq we‡Rvo msL¨v I
g`y vª q 4 Avmvi m¤¢vebv wbY©q Ki| 4
28 G·Kz¬wmf g‡Wj †U÷ 04 welq ˆKvW : 1 2 6
mgq : 2 NÈv 35 wgwbU D”PZi MwYZ mR„ bkxj cÖkœ c~Yg© vb : 50
[Wvb cv‡ki msL¨v c‡Ö kœi cY~ ©gvb ÁvcK| c‡Ö Z¨K wefvM †_‡K b~¨bZg GKwU K‡i †gvU cvu PwU cÖ‡kœi DËi w`‡Z n‡e|]
K wefvM : exRMwYZ
1 P(x, y, z) = (x + y + z) (xy + yz + zx) , Q = a3 + b3 + c3 3a1b1c1.
K. P(x, y, z) cÖwZmg ivwk wKbv Zv KviYmn D‡jL Ki| 2
L. Q = 0 n‡j cgÖ vY Ki †h, a = b = c A_ev ab + bc + ca = 0. 4
M. P(x, y, z) = xyz n‡j †`LvI †h, (x + 1 + z)7 = 1 + 1 + z17. 4
y x7 y7
2 A = {x : x R Ges x2 (a + b) x + ab = 0},
B = {1, 2}, C = {2, 3, 4} 2
K. A †mU‡K ZvwjKv c×wZ‡Z cÖKvk Ki|
L. †`LvI †h, P(B C) = P(B) P(C). 4
M. cÖgvY Ki †h, A (B C) = (A B) (A C). 4
3 f(x) = x2 6x + 15, g(x) = x2 6x + 13 2
K. f(x) = 10 n‡j x Gi gvb wbY©q Ki|
4
L. f(x) g(x) = 10 8 n‡j mgxKiYwU mgvavb Ki| 4
M. g(x) Gi †jLwPÎ A¼b Ki|
L wefvM : R¨vwgwZ I †f±i 2
4 ABC m~²‡KvYx wÎf‡z Ri kxlΩ q †_‡K wecixZ evû¸‡jvi Dci j¤Î^ q AD, BE I CF ci¯úi O we›`‡y Z †Q` K‡i‡Q| 4
4
K. AC = 5 †m.wg., CD = 3 †m.wg. n‡j AD Gi ˆ`N¨© wbYq© Ki|
L. cÖgvY Ki †h, AO . OD = BO . OE = CO . OF. 2
M. cÖgvY Ki †h, BC . CD = AC . CE. 4
5 A(3, 6), B( 6, 2), C( 2, 6), D(8, 4) GKB mgZ‡j Aew¯’Z PviwU we›`|y 4
K. AB Gi mgxKiY wbY©q Ki|
L. P(x, y) we›`y †_‡K x A‡¶i `~iZ¡ I A we›`iy `i~ Z¡ mgvb n‡j †`LvI †h, x2 6x + 12y + 45 = 0.
M. ABCD PZzf©z‡Ri †¶Îdj wbYq© Ki|
6 GKwU wb‡iU avZe mge„Ëf‚wgK †KvY‡Ki D”PZv 8 †m.wg. Ges fw‚ gi e¨vmva© 6 †m.wg.| D³ †KvYK‡K Mwj‡q 4 †m.wg. e¨v‡mi
K‡qKwU wb‡iU †MvjK c¯Ö ‘Z Kiv n‡jv|
K. †f±i †hv‡Mi wÎfzR wewa wPÎmn eY©bv Ki| 2
L. †KvYKwUi mgMcÖ „‡ôi †¶Îdj I AvqZb wbY©q Ki| 4
M. KqwU wb‡iU †MvjK ˆZwi Kiv hvq Zv wbY©q Ki| 4
M wefvM : w·KvYwgwZ I m¤v¢ ebv
7 P = a cos , Q = b sin .
K. P2 + Q2 Gi gvb wbY©q Ki| 2
a2 b2 4
4
L. P Q = c n‡j †`LvI †h, a sin + b cos = a2 + b2 c2
2
M. a2 = 3, b2 = 7 Ges P2 + Q2 = 4 n‡j cÖgvY Ki †h, cot = 3. 4
4
8 GKwU Q°v I GKwU g`y ªv GKmv‡_ GKevi wb‡¶c Kiv n‡jv|
K. D`vniYmn mgm¤v¢ e¨ NUbvi msÁv `vI|
L. Probability tree A¼b K‡i bgby v‡¶Î †`LvI|
M. Q°vq †gŠwjK msL¨v Ges g`y vª q T Avmvi m¤¢vebv KZ?
29 G·K¬wz mf g‡Wj †U÷ 05 welq ˆKvW : 1 2 6
mgq : 2 NÈv 35 wgwbU D”PZi MwYZ m„Rbkxj ckÖ œ c~Yg© vb : 50
[Wvb cv‡ki msL¨v cÖ‡kœi c~Y©gvb ÁvcK| cÖ‡Z¨K wefvM †_‡K b¨~ bZg GKwU K‡i †gvU cvu PwU c‡Ö kœi DËi w`‡Z n‡e|]
K wefvM : exRMwYZ 5 GKwU e‡„ Ëi PQRS GKwU Aš—wjw© LZ PZzfz©R| PR Ges QS
Gi `By wU KY© Ges QPR = SPT †hLv‡b PT †iLvsk QS
1 U = {x, y, 2, 3, 4}, A = {x, y}, B = {3, 4} Ges C = {3, 4}. †K T we›`y‡Z †Q` K‡i|
K. A \ (B C) wbY©q Ki| 2
L. (A B) C wbYq© Ki| 4 K. eY©bv Abymv‡i wPÎwU AvKu | 2
M. cÖgvY Ki †h, 4
L. cÖgvY Ki †h,
P(A) P(B) P(A B)|
PR. QS – QR. PS = PQ. RS. 4
2 P = (a + bx)6, Q = (b + ax)5 M. K Gi wP‡Îi P we›`y‡Z GKwU ¯úkK© AvuK hv ewa©Z QS
Ges R = (a + x)n n‡j, †K A we›`y‡Z †Q` K‡i Ges cÖgvY Ki †h,
K. R Gi we¯—…wZ wjL Ges m~ÎwU cÖ‡qvM K‡i P Gi we¯—w… Z AP2 = AQ. AS| 4
wbY©q Ki| 2 6 A(0, – 1), B(– 2, 3), C(6, 7) Ges D(8, 3) we›`y¸‡jv GKwU
L. hw` P Gi we¯—…wZi wØZxq I Z…Zxq c‡`i AbycvZ PZfz z©Ri PviwU kxl© we›`y|
h_vµ‡g Q Gi we¯—w… Zi wØZxq I ZZ… xq c‡`i K. PZzfz©‡Ri KYØ© ‡qi ˆ`N©¨ wbYq© Ki| 2
Abycv‡Zi mgvb nq Z‡e †`LvI †h, a : b = 5 : 2 4 L. †`LvI †h, PZfz ©zRwU GKwU AvqZ| 4
M. †`LvI †h, Q Gi we¯—…wZi †Rvo ¯’vbxq cig aª“eK¸wji M. AB, BC, CD, DA evû¸‡jvi ga¨we›`y h_vµ‡g P, Q,
†hvMdj we‡Rvo ¯’vbxq cig aª“eK¸wji †hvMd‡ji R, S| †f±i c×wZ‡Z †`LvI †h, PQRS GKwU
mgvb| Ggb GKwU wØc`x ivwk D‡jL Ki hvi †¶‡ÎI mvgvš—wiK| 4
DcwiD³ welqwU mZ¨ n‡e| 4 M wefvM : w·KvYwgwZ I m¤¢vebv
3 x2 – 5x + 4 = 0 GKwU wØNvZ mgxKiY| 7
K. cÖ`Ë mgxKiYwUi †jLwPÎ A¼b Ki‡j, x Gi †Kvb A
gvb mgxKiYwUi mgvavb wb‡`©k Ki‡e? 2 a
L. †jLwPÎ A¼b K‡i mgxKiYwUi mgvavb Ki| 4
M. ax2 + bx + c = 0 Gi mv‡_ Zzjbv K‡i mgxKiYwUi O bB
g~‡ji cKÖ …wZ wbY©q Ki Ges gj~ ¸‡jv wbY©q Ki| 4 K. wPÎ Abyhvqx, cot Gi gvb KZ? 2
L wefvM : R¨vwgwZ I †f±i L. †`LvI †h,
P a sin – b cos = 1 – a22+b2b2. 4
a sin + b cos
4
M. a b2 + b b2 = 2 n‡j, Gi gvb wbYq© Ki| 4
a2 + a2 +
60
8 GKwU Q°v I `yBwU wbi‡c¶ gy`ªv wb‡¶c Kiv n‡jv|
R QT S
K. `yBwU gy`ªv wb‡¶‡ci bgybv‡¶ÎwU ˆZwi K‡i eo‡Rvi 2T
wP‡Î PRS Gi T, RS Gi ga¨we›`y|
PRS = 60, PQ = 4 †m.wg., RS = 6 †m.wg.|
K. PR Gi ˆ`N©¨ wbY©q Ki| 2 Avmvi m¤¢vebv wbYq© Ki| 2
L. R¨vwgwZK c×wZ‡Z cÖgvY Ki †h, L. Q°vwU GKevi wb‡¶c Kiv n‡j †Rvo msL¨v A_ev 3
PR2 + PS2 = 2(PT2 + RT2) 4 Øviv wefvR¨ msL¨v DVvi m¤¢vebv wbYq© Ki| 4
M. A¼‡bi weeiYmn Ggb GKwU wÎfzR A¼b Ki hvi fw‚ g M. GKwU Q°v I GKwU g`y ªv wb‡¶c NUbvi Probability
RS, wki:†KvY R Gi mgvb Ges Aci `yB evûi Aš—i Tree ˆZwi K‡i Q°vq we‡Rvo msL¨v I gy`vª q H Avmvi
2 †m.wg.| 4 m¤¢vebv wbYq© Ki| 4
30 G·K¬wz mf g‡Wj †U÷ 06 welq ˆKvW : 1 2 6
mgq : 2 NÈv 35 wgwbU D”PZi MwYZ mR„ bkxj cÖkœ c~Y©gvb : 50
[Wvb cv‡ki msL¨v c‡Ö kœi cY~ g© vb ÁvcK| c‡Ö Z¨K wefvM †_‡K b~¨bZg GKwU K‡i †gvU cuvPwU c‡Ö kœi DËi w`‡Z n‡e|]
K wefvM : exRMwYZ 5 ABCD PZzf‡©z Ri A(6, 4), B(2, 2), C( 2, 2), D( 6, 4)
1 P I Q `yBwU mvš— †mU kxl©we›`ymgn~ Nwoi Kvu Uvi wecixZ w`‡K AvewZ©Z|
Ges f(y) = y3 2y2 + 31. K. CD Gi ˆ`N©¨ KZ? 2
y2 2y
L. ABCD PZzfz©‡Ri †¶Îd‡ji mgvb †¶Îdjwewkó
K. f(y) = 3y + 2 Gi †Wv‡gb KZ? 2 GKwU eM‡© ¶‡Îi K‡Y©i ˆ`N©¨ wbY©q Ki|
4
L. f(y) †K AvswkK fMœvs‡k cwiYZ Ki| 4 M. ABCD GKwU Uvª wcwRqvg Ges P I Q h_vµ‡g AB I
M. †`LvI †h, CD Gi ga¨we›`y n‡j †f±‡ii mvnv‡h¨ cÖgvY Ki †h,
n(P Q) = n(P) + n(Q) n(P Q). 4 PQ || AD || BC Ges PQ = 1 (AD + BC). 4
2
2 x + xy + xy2 + …… GKwU ¸‡YvËi aviv
K. avivwUi `kg c` wbYq© Ki| 2 6 4 †m.wg. e¨v‡mi GKwU †jŠn †MvjK‡K wcwU‡q 2 †m.wg. cyi“
3
L. x=1 Ges y = 1 n‡j, avivwUi AmxgZK mgwó hw` GKwU e„ËvKvi †jŠncvZ cÖ¯Z‘ Kiv n‡jv|
2
K. †jŠn †Mvj‡Ki c„ôZ‡ji †¶Îdj KZ? 2
_v‡K Z‡e Zv wbY©q Ki| 4
L. H †jŠncv‡Zi e¨vmva© wbY©q Ki| 4
M. x-Gi ¯‡’ j 6, xy Gi ¯‡’ j 66 Ges xy2 Gi ¯‡’ j 666
M. †Mvj‡Ki c„ôZ‡ji †¶Îdj, 6 †m.wg. e¨vmva© wewkó
emv‡j, †h aviv cvIqv hvq Zvi cÖ_g n msL¨K c‡`i
wmwjÛv‡ii eµZ‡ji †¶Îd‡ji mgvb n‡j wmwjÛv‡ii
mgwó wbY©q Ki| 4
mgMÖc„‡ôi †¶Îdj I AvqZb wbY©q Ki| 4
3 P = 1 2x7 M wefvM : w·KvYwgwZ I m¤v¢ ebv
Q = K x27 7 B
R = (1 x) (1 + ax)6 C x2 + y2 x2 y2
A
OD
K. c¨vm‡Kj wÎfzR m~‡Î P †K cÂg chš© — we¯—Z… Ki| 2 wP‡Î ABC GKwU e„ËvKvi PvKv Ges PvKvwUi AB Pv‡ci
L. Q Gi k3 Gi mnM 560 n‡j x Gi gvb wbY©q Ki| 4 ˆ`N©¨ 44 †m.wg. Ges †Kv‡Yi e„Ëxq cwigvc 1.
M. R = 1 + bx2 n‡j a I b Gi gvb KZ? 4 K. PvKvwU 1 evi Ny‡i KZ wgUvi `i~ Z¡ AwZµg Ki‡e? 2
L wefvM : R¨vwgwZ I †f±i L. ABC PvKvwU cÖwZ †m‡K‡Û 5 evi AvewZ©Z n‡j PvKvwU
4 1 wK.wg. c_ AwZµg Ki‡Z KZ mgq jvM‡e? 4
PS M. wPÎ n‡Z cÖgvY Ki †h,
QM O tan + sec = xy. 4
R
K. PQ Ges PS Gi j¤^ Awf‡¶c KZ? 2 8 GKwU Q°v I `wy U gy`ªv GK‡Î wb‡¶c Kiv n‡jv|
L. cÖgvY Ki †h,
K. bgby v †¶Î wK? 2
PQ2 + PS2 = 2(PQ2 + QO2). 4 L. Probability tree Gu‡K bgybv‡¶Î wjL| 4
M. †f±‡ii mvnv‡h¨ cÖgvY Ki †h, M. P(we‡Rvo I 2H) Ges P (†gŠwjK msL¨v Ges 2T)
PO = OR Ges QO = OS. 4 wbY©q K‡i G‡`i †hvMdj KZ? 4
31 G·Kz¬wmf g‡Wj †U÷ 07 welq ˆKvW : 1 2 6
mgq : 2 NÈv 35 wgwbU D”PZi MwYZ mR„ bkxj ckÖ œ c~Yg© vb : 50
[Wvb cv‡ki msL¨v cÖ‡kœi c~Y©gvb ÁvcK| c‡Ö Z¨K wefvM †_‡K b~¨bZg GKwU K‡i †gvU cuvPwU c‡Ö kœi DËi w`‡Z n‡e|]
K wefvM : exRMwYZ 5 A( 1, 3) Ges B(5, 15) we›`Øy ‡qi ms‡hvM †iLv x-A¶ Ges
1 P(x, y, z) = (x + y + z) (xy + yz + zx) xyz y-A¶‡K h_vµ‡g C I D we›`y‡Z †Q` K‡i| A¶Ø‡qi
Ges Q(x) = x3 1 1. †Q`we›`y O|
K. QK KvM‡R wPÎ Gu‡K C I D we›`y‡Z ¯’vbv¼ wbY©q Ki| 2
2
K. †`LvI †h, P(x, y, z) PµµwgK I cÖwZmg| L. CD †iLvi mgxKiY Ges COD Gi †¶Îdj wbY©q
L. P(x, y, z) = 0 n‡j, †`LvI †h, Ki| 4
4
(x + y + z)3 = x3 + y3 + z3.
M. †KvYK AvKv‡ii GKwU Zuveiy D”PZv BD Gi ˆ`‡N¨© i
M. Q(x) †K AvswkK fMvœ s‡k cÖKvk Ki| 4
2 ax = by = cz ; †hLv‡b a b c mgvb (wgUv‡i) n‡j Zvu eywU Øviv 2000 eMw© gUvi Rwg
wNi‡Z cÖ‡qvRbxq K¨vbfv‡mi cwigvY wbY©q Ki| 4
Ges 92p = 3p+1.
6 GKwU mg‡KvYx wÎfz‡Ri AwZfzR a = 5 †m.wg. Ges Aci
K. wØZxq DÏxcK n‡Z p Gi gvb wbY©q Ki| 2
L. hw` x = 2 Ges y = 3 nq, Z‡e †`LvI †h, evû؇qi Aš—i d = 1 †m.wg.|
ba32 + ba32 = a+ 1 . K. wÎfzRwUi Aci evû؇qi ˆ`N¨© wbY©q Ki| 2
3b
4 L. A¼‡bi wPý I weeiYmn wÎfzRwU A¼b Ki| 4
M. abc = 1 n‡j, †`LvI †h, M. AwZfz‡Ri mgvb e¨vm wewkó GKwU e„Ë A¼b Ki hv
1 1 1 `yBwU wbw`ó© we›`y w`‡q hvq (A¼‡bi wPý I weeiY
x y z
+ + = 0 Ges x3 + y3 + z3 = 3(xyz)1. 4 Avek¨K)| 4
3 †Kv‡bv avivi n Zg c` Un = (2x + 1)n2 M wefvM : w·KvYwgwZ I m¤¢vebv
Ges GKwU wØc`x ivwk x2 + xk6. 7 tan = xy, x y.
K. avivwU wbYq© Ki| 2 K. cÖgvY Ki †h,
L. x Gi Dci Kx kZ© Av‡ivc Ki‡j avivwUi AmxgZK cos = y y2. 2
x2 +
mgwó _vK‡e Ges †mB mgwó wbYq© Ki| 4
x sin + y cos
M. wØc`x ivwkwUi we¯—w… Z‡Z x3- Gi mnM 160 n‡j k Gi L. DÏxc‡Ki Av‡jv‡K x sin y cos Gi gvb wbYq©
gvb wbYq© Ki| 4 Ki| 4
L wefvM : R¨vwgwZ I †f±i M. mgvavb Ki :
4 ABC G AD GKwU ga¨gv Ges PQRS UªvwcwRqv‡g sin + cos = 2, hLb 0 < 2. 4
PS || QR. 8 GKwU wbi‡c¶ gy`ªv I GKwU Q°v wb‡¶c Kiv n‡jv|
K. U‡jwgi Dccv`¨wU wjL Ges wPÎmn e¨vL¨v Ki| 2 K. m¤v¢ e¨ NUbvi Probability tree A¼b Ki| 2
L. cÖgvY Ki †h, AB2 + AC2 = 2(AD2 + BD2) 4 L. bgby v‡¶Î wj‡L gy`vª q †Uj I Q°vq we‡Rvo msL¨v
M. PQ Ges SR evûi ga¨we›`y h_vµ‡g M I N n‡j cvIqvi m¤¢vebv wbY©q Ki| 4
†f±‡ii mvnv‡h¨ cÖgvY Ki †h, M. ïagy vÎ gy`ªvwU‡K hw` 3 evi wb‡¶c Kiv nq Z‡e
MN = 1 (PS + QP) Ges MN || PS || QR. 4 Kgc‡¶ GKwU †nW cvIqvi m¤¢vebv wbY©q Ki|
2
(bgybv‡¶Î †`Lv‡Z n‡e) 4
32 G·Kz¬wmf g‡Wj †U÷ 08 welq ˆKvW : 1 2 6
mgq : 2 NÈv 35 wgwbU D”PZi MwYZ mR„ bkxj cÖkœ cY~ g© vb : 50
[Wvb cv‡ki msL¨v c‡Ö kœi c~Y©gvb ÁvcK| cÖ‡Z¨K wefvM †_‡K b~¨bZg GKwU K‡i †gvU cuvPwU c‡Ö kœi DËi w`‡Z n‡e|]
K wefvM : exRMwYZ 5 ABC Gi cwi‡K›`ª P, j¤^we›`y Q Ges AD GKwU ga¨gv|
1 P(x) = x3 + 5x2 + 6x + 8 PQ, AD ga¨gv‡K O we›`‡y Z †Q` K‡i| ABC Gi cwie‡„ Ëi
Q(x) = (x 1)(x x3 2)(x 3) e¨vm 4 †mwg.|
K. msÁv wjL| Ae¯’vb †f±i (wPÎmn) 2
K. f(x) = 2x 1 n‡j, f 31 Gi gvb wbY©q Ki| 2 L. cÖgvY Ki †h, O, P Ges Q GKB mij‡iLvq Aew¯’Z| 4
2x + 3
L. hw` P(x) †K (x a) Ges (x b) Øviv fvM Ki‡j M. cwie‡„ Ëi e¨vm‡K A¶ a‡i PZwz `©‡K Niy v‡j †h Nbe¯‘
GKB fvM‡kl _v‡K †hLv‡b a b Z‡e, cÖgvY Ki †h, ˆZwi nq Zv GKwU NbvK…wZi ev‡· wVKfv‡e G‡u U hvq|
a2 + b2 + ab + 5a + 5b + 6 = 0 4 ev·wUi AbwaK…Z As‡ki AvqZb wbY©q Ki| 4
M. Q(x) †K AvswkK fMvœ s‡k cÖKvk Ki| 4 6 A
2 (i) 2 + x45 FE
(ii) k 4y5 `By wU wØc`x ivwk| BDC
K. c¨vm‡K‡ji wÎfzR e¨envi K‡i (1 + 3y)4 †K we¯—…Z ABC ‰ AD, BE I CF wZbwU gaÅgv 2
K. AB †K BE I CF Gi gva¨‡g cÖKvk Ki|
Ki| 2
L. cÖgvY Ki †h,
L. (i) bs e¨envi K‡i (1.9975)5 Gi Avmbœ gvb wZb
`kwgK ¯v’ b chš© — wbY©q Ki| 4 AD + BE + CF = 0 4
M. (ii) Gi we¯—w… Z‡Z k3-Gi mnM 160 n‡j y Gi gvb wbY©q M. A(3, 4), B(4, 2), C(6, 1) Ges P(k, 3) we›`y PviwU
Ki| 4 Nwoi Kvu Uvi wecixZ w`‡K AvewZ©Z n‡j Ges ABCP
3 1 1) + (4x 1 1)2 + (4x 1 1)3 + ..... GKwU Abš— ¸‡YvËi aviv| PZzf©y‡Ri †¶Îdj ABC Gi †¶Îd‡ji wZb¸Y n‡j
(4x
k Gi gvb wbY©q Ki| 4
K. x = 1 n‡j avivwUi mvaviY AYcy vZ wbYq© Ki| 2
M wefvM : w·KvYwgwZ I m¤¢vebv
L. x = 5 , n‡j avivwUi c_Ö g c` Ges c_Ö g 10wU c‡`i
4 5
7 (i) tan = 12 Ges cos FYvZ¥K
mgwó wbY©q Ki| 4
M. x-Gi Dci Kx kZ© Av‡ivc Ki‡j cÖ`Ë avivwUi (ii) A = cot + cosec 1 Ges B = cot + cosec
cot cosec + 1
AmxgZK mgwó _vK‡e Ges †mB mgwó wbYq© Ki| 4
K. wPÎmn †iwWqvb †Kv‡Yi msÁv wjL| 2
L wefvM : R¨vwgwZ I †f±i L. (i) bs e¨envi K‡i cgÖ vY Ki †h,
4 P
sin + cos () = 51 4
sec() + tan 26
Q
O M. (ii) bs e¨envi K‡i cgÖ vY Ki †h,
S A2 B2 = 0 4
R 8 GKwU Q°v I GKwU gy`vª GK‡Î wb‡¶c Kiv n‡jv|
O ˆK±`wÊ ewkÓ¡ PQRS ‰KwU eî† | 2 K. msÁv wjL : NUbv, ci¯úi wew”Qbœ NUbvewj| 2
K. wPÎmn †iLvs‡ki j¤^ Awf‡¶c eY©bv Ki|
L. cÖgvY Ki †h, PQ. RS + PS. QR = PR. QS 4 L. Probability tree ˆZwi K‡i bgybv‡¶ÎwU wjL| AZtci
M. PQ †K e¨vm a‡i Aw¼Z Aa©e‡„ Ëi `yBwU R¨v PC I QD Q°vq 5 Ges gy`ªvq H Avmvi m¤¢vebv wbY©q Ki| 4
ci¯úi M we›`y‡Z †Q` K‡i| M. gy`vª q Kgc‡¶ 1wU T Ges Q°vq 2 I 3 Gi ¸wYZK
cÖgvY Ki †h, PQ2 = PC.PM + QD.QM. 4 Avmvi m¤v¢ ebv wbY©q Ki| 4
33 G·Kzw¬ mf g‡Wj †U÷ 09 welq ˆKvW : 1 2 6
mgq : 2 NÈv 35 wgwbU D”PZi MwYZ mR„ bkxj ckÖ œ c~Yg© vb : 50
[Wvb cv‡ki msL¨v c‡Ö kœi c~Y©gvb ÁvcK| cÖ‡Z¨K wefvM †_‡K b~¨bZg GKwU K‡i †gvU cvu PwU c‡Ö kœi DËi w`‡Z n‡e|]
K wefvM : exRMwYZ 2
1 F(x) = 2xx–+51; x R 4
4
K. †Wvg F wbYq© Ki|
L. F GKwU GK GK dvskb wKbv wbav© iY Ki|
M. F–1(x) = x – 5 n‡j x Gi m¤v¢ e¨ gvb wbY©q Ki|
2 Q(x) = x3 + 6x2 + 11x + 6
K. †`LvI †h, x + 2, Q(x) Gi GKwU Drcv`K| 2
L. Q(x) †K (x – 2a) Ges (x – 2b) Øviv fvM Ki‡j hw` GKB fvM‡kl _v‡K †hLv‡b a b, Z‡e †`LvI †h, 4
4
4a2 + 4ab + 4b2 + 12a + 12b + 11 = 0
M. x3 †K AvswkK fMœvs‡k cÖKvk Ki|
Q(x)
3 C = x2 + xP26, D = 1 – x26 2
4
K. D †K c¨vm‡K‡ji wÎf‡z Ri mvnv‡h¨ ZZ… xq c` ch©š— we¯—Z… Ki| 4
L. C Gi we¯—…wZ‡Z x ewR©Z c`wUi gvb 14580 n‡j P Gi gvb wbY©q Ki|
M. x Gi Nv‡Zi DaŸ© µgvbymv‡i (3 + x)D †K x4 ch©š— we¯—…Z K‡i 3.1 (0.95)6 Gi gvb wbYq© Ki|
L wefvM : R¨vwgwZ I †f±i
4 DEF Gi DA, EB I FC ga¨gvÎq ci¯úi‡K G we›`y‡Z †Q` K‡i‡Q|
K. GA = 3cm n‡j, AD Gi ˆ`N¨© wbY©q Ki| 2
L. cÖgvY Ki †h, DE2 + EF2 = 2(BD2 + BE2) 4
M. cÖgvY Ki †h, 13(DE2 + EF2 + DF2) = GD2 + GE2 + GF2. 4
5 A(–3, 3), B(–3, –3), C(4 , –3) I D(4, 3) GKwU PZfz z©‡Ri PviwU kxl© we›`y|
K. BD †iLvi mgxKiY wbY©q Ki| 2
L. PZfz ©zRwU AvqZ bv mvgvš—wiK Zv wbYq© Ki| 4
M. hw` E, F, G I H h_vµ‡g AB, BC, CD I DA Gi ga¨we›`y nq Z‡e †f±i c×wZ‡Z cgÖ vY Ki †h, EFGH GKwU mvgvš—wiK| 4
6 5x + 3y = 30 †iLvwU x A¶‡K A we›`‡y Z, y A¶‡K B we›`y‡Z †Q` K‡i‡Q|
K. AB †iLvi Xvj wbYq© Ki| 2
L. gj~ we›`y O Ges OA I OB †iLvs‡ki ga¨we›`y h_vµ‡g C I D | †f±‡ii mvnv‡h¨ cÖgvY Ki †h, CD AB Ges CD = 21AB 4
M. COD †K OD evûi PZwz `©‡K GKevi †Nviv‡j †h Nbe¯‘ Drcbœ nq †mwU A¼b K‡i Gi AvqZb wbY©q Ki| 4
M wefvM : w·KvYwgwZ I m¤v¢ ebv
7 a = sin, b = cos
K. GKwU †Kv‡Yi gvb lvUg~jK c×wZ‡Z D Ges e„Ëxq c×wZ‡Z RC n‡j †`LvI †h, D = R 2
180
4
L. 7a2 + 3b2 – 4 = 0 n‡j cÖgvY Ki †h, b cosec = 3. 4
M. 3ab–1 + 3ba–1 – 4 = 0 n‡j, Gi m¤¢ve¨ gvb wbY©q Ki, hLb 0 < < 2.
8 GKwU wbi‡c¶ gy`vª I GKwU Q°v G‡K‡Î wb‡¶c Kiv n‡jv| 2
K. ïay Q°vq †gŠwjK msL¨v cvIqvi m¤¢vebv wbY©q Ki| 4
L. DÏxc‡Ki Av‡jv‡K Probability tree A¼b K‡i bgybv‡¶Î n‡Z gy`vª q †nW I Q°vq †Rvo msL¨v Avmvi m¤v¢ ebv wbY©q Ki| 4
M. hw` gy`vª ev‡` Q°vwU `yBevi wb‡¶c Kiv nq Z‡e bgybv‡¶Î n‡Z GKB djvdj bv cvIqvi m¤¢vebv wbY©q Ki|
34 G·Kzw¬ mf g‡Wj †U÷ 10 welq ˆKvW : 1 2 6
mgq : 2 NÈv 35 wgwbU D”PZi MwYZ m„Rbkxj ckÖ œ c~Y©gvb : 50
[Wvb cv‡ki msL¨v cÖ‡kœi c~Yg© vb ÁvcK| c‡Ö Z¨K wefvM †_‡K b~¨bZg GKwU K‡i †gvU cvu PwU c‡Ö kœi DËi w`‡Z n‡e|]
K wefvM : exRMwYZ 5 6 †m. wg. evûwewkó mylg lof‡y Ri Dci Aew¯’Z GKwU
1 E, F I G wZbwU mvš— †mU †hLv‡b,
wcivwg‡Wi D”PZv 8 †m.wg.|
E = {x : x 1R
K. GKwU †jvnvi duvcv †Mvj‡Ki evB‡ii e¨vm 13 †m. wg. Ges
Ges x2 +( + ) x + = 0}
F = {a, b, c} Ges G = {b, c, d} †ea 2 †m. wg. Gi duvcv As‡ki AvqZb wbY©q Ki| 2
K. E †mUwU‡K ZvwjKv c×wZ‡Z cÖKvk Ki|
L. F\G wbYq© Ki Ges †`LvI †h, L. wcivwg‡Wi mgMZÖ ‡ji †¶Îdj I AvqZb wbY©q Ki| 4
P(F G) = P(F) P(G). 2 M. lofzRwUi evûi ˆ`N©¨ hw` GKwU mylg PZz¯—j‡Ki
4
M. cÖgvY Ki †h, 4 av‡ii ˆ`N¨© nq Z‡e PZz¯—jKwUi AvqZb wbY©q Ki| 4
E (F G) (E F) (E G). 6 A
MN
2 (3a 2)1 + (3a2)2 + (3a 2)3 + ........GKwU aviv| BC
ABC-Gi AB I AC evûi ga¨we›`y M I N
K. mgwó wbY©q Ki : (hw` _v‡K) : 2 + 4 + 8 + 16 + ....... 2 K. (AM + MN ) †f±i‡K AC Gi gva¨‡g cKÖ vk Ki| 2
L. a = 4 n‡j avivwUi lô c` Ges avivwUi cÖ_g AvUwU L. †f±‡ii gva¨‡g cÖgvY Ki
3
MN || BC Ges MN = 1 BC. 4
2
c‡`i mgwó wbY©q Ki| 4
M. DÏxc‡Ki Uvª wcwRqv‡gi KYØ© ‡qi ga¨we›`y P I Q n‡j
M. a-Gi Dci Kx kZ© Av‡ivc Ki‡j avivwUi AmxgZK
†f±‡ii mvnv‡h¨ cÖgvY Ki †h,
mgwó _vK‡e Ges †mB mgwó wbY©q Ki| 4
PQ || MN || BC Ges PQ = 1 (BC MN) 4
2
m x
3 g(x) = 1n m + x M wefvM : w·KvYwgwZ I m¤¢vebv
7 cot + cosec = m
Ges F(p, q, r) = p2 + q3 + r3 3pqr
K. x2 2x 2 = 0 mgxKi‡Yi g~j؇qi cÖKw… Z wbY©q Ki| 2 Ges cosecA + cotA = P
L. g(x) Gi †Wv‡gb I †iÄ wbY©q Ki| 4 K. 32 15 18 †K †iwWqv‡b cÖKvk Ki| 2
M. p = y + z x, q = z + x y Ges r = x + y + z n‡j L. cÖgvY Ki †h,
†`LvI †h, F(x, y, z) : F(p, q, r) = 1 : 4. 4 cos = m2 1 4
m2 + 1
L wefvM : R¨vwgwZ I †f±i M. P = 3 n‡j A Gi gvb wbY©q Ki;
4 PQR Gi cwi‡K›`ª S j¤^we›`y O Ges fi †K›`ª G| †hLv‡b 0 A 2. 4
K. PQR Gi GKwU ga¨gv 12 †m.wg. n‡j PG wbY©q Ki| 2 8 `yBwU gy`vª I GKwU Q°v wb‡¶c Kiv n‡jv|
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