ANSWERS 1.3 EXERCISE. 7. 2, 1 8. (i) represents function which is surjective but not injective (ii) does not represent function.
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1 ANSWERS 87 ANSWERS. EXERCISE. (b,b), (,), (a,). [-5,5] f 5. f { ( ba, ),( db, ),( a, ),( d, ) } 4 6. f f , 8. (i) represents funtion whih is surjetive but not injetive (ii) does not represent funtion. 9. fog,5, 5,,,5. (i) f is not funtion (ii) g is funtion (iii) h is funtion (iv) k is not funtion 4., 7. Domain of R = {,,,4,... 0} and Range of R = {,,5,7,9,... 9}. R is neither refletive, nor symmetri and nor transitive.. (i) f is one-one but not onto, (ii) g is neither one-one nor onto (iii) h is bijetive, (iv) k is neither one-one nor onto.. (i) transitive (ii) symmetri (iii) refleive, symmetri and transitive (iv) transitive.. (,5) = {(,4 ),(,5 ),(,6 ),( 4,7 )(5,8),(6,9)}
2 88 MATHEMATICS 5. (i) fog 4 6 (ii) gof 6 4 (iii) fof (iv) gog (ii) & (iv) 7. (i) 8. C 9. B 0. D. B. B. A 4. C 5. C 6. B 7. D 8. A 9. B 40. B 4. A 4. A 4. C 44. B 45. D 46. A 47. B 48. R =,8, 6,6, (9,4), (,) 49. R = {(, ),(, ),(, ),(,),(,), (,), (,), (,4), (4,), (4,4), (5,5)} 50. gof = {(, ),(, ),( 4,) } and fog = {(,5 ),( 5, ),(,5) } 5. fofof 5. f ( ) = 7+ ( 4 ) 5. False 54. False 55. False 56. False 57. True 58. False 59. False 60. True 6. False 6. False. EXERCISE π 5. π 7. 0, , 4 4
3 ANSWERS tan an a + aa n 0. C. D. B. D 4. A 5. A 6. B 7. C 8. A 9. B 0. A. D. D. B 4. A 5. C 6. A 7. A 8. π 9. π φ 4. π 4. π π,π 47. y > 48. π ot 49. False 50. False 5. True 5. True 5. True 54. False 55. True. EXERCISE. 8, 8, 4 7, 7 4, 4, 4. If matri has elements then its order will be either or.. (i), (ii) 9, (iii) a y, a 0, a. (i) 9 0 (ii) 4 4. e sin e sin e sin e sin e sin e sin 5. a =, b = 6. Not possible 7. (i) 5 X+ Y= 0 (ii) 0 X Y = 0 8
4 90 MATHEMATICS 5 (iii) Z= 0 8. = 4 0., 4. A 7 5. A = [ ]. A= AB= BA = =, y = 9. 0 k k k X =,Y= 0., k k k et. where k is a real number 4. A = [ 4] 0. True when AB = BA 7. (i) 7-5 (ii) not possible 8. =, y = 4 or = 4, y =, z = 6, w = A a =, b = 4, =, d = a =, b = 0, = 44.True for all real values of α
5 ANSWERS =±, y=±, z=± 6 5. (i) (ii) inverse does not eist (iii) A 54. D 55. B 56. D 57. D 58. D 59. A 60. B 6. C 6. D 6. A 64. A 65. D 66. D 67. A 68. Null matri 69. Skew symmetri matri Retangular matri 7. Distributive 74. Symmetri matri 75. Symmetri matri 76. (i)b A (ii) ka (iii) k A B 77. Skew Symmetri matri 78. (i) Skew symmetri matri (ii) neither symmetri nor skew symmetri matri 79. Symmetri matri 80. AB = BA 8. does not eist 8. False 8. False 84. False 85. True 86. True 87. False 88. False 89. True 90. False 9. False 9. False 9. False 94. True 95. False 96. True 97. False 98. True 99. False 00. True 0. True
6 9 MATHEMATICS 4. EXERCISE. +. a (a + + y + z). y z 4. ( + y + z) (y + yz + z) 5. 6 ( + 4) 6. (a + b + ) n π = 6. θ nπ or nπ + ( ). = 0, 8. = 0, y = 5, z = 9. =, y =, z = 0. =, y =, z = 4 4. C 5. C 6. B 7. D 8. C 9. A 0. A. A. C. D 4. D 5. D 6. B 7. C 8. 7 A 9. A 40. Zero (A ) Value of the determinant 45. = y = (y z) (z ) (y + yz) 47. Zero 48. True 49. False 50. False 5. True 5. True 5. True 54. False 55. True 56. True 57. True 58. True 5. EXERCISE. Continuous at =. Disontinuous. Disontinuous 4. Continuous 5. Disontinuous 6. Continuous 7. Continuous 8. Disontinuous 9. Continuous 0. Continuous. 7 k =.. k = 4. k =± 6. a =, b = k = 7. Disontinuous at = and 5 8. Disontinuous at =, and 0. Not differentiable at =. Differentiable at = 0. Not differentiable at = 5. (log) sin os
7 ANSWERS log a log 5 log log 9. os sin 0. n n a b sin a b os a b.. sin tan se os sin sin. 4. os os sin sin.logsin sin 5. sin m n os ( ntan + mot ) a 4. a t t 45. θ -θ +θ +θ+ e θ +θ +θ 46. ot θ t 5. tan 5. sin y y os y y y os y y 55. y se y tan y se y tan y 56. y 57. y 4 4y 4y + 4 y 64. sin yos y 70. Not appliable sine f is not differentiable at =
8 94 MATHEMATICS 7., 7. (, 4) 77. 7, 4 78.,0 79. p, q 5 8. tan tan se log D 84. C 85. B 86. A 87. A 88. A 89. C 90. B 9. B 9. A 9. A 94. B 95. A 96. B False 0. True 04. True 05. True 06. False 6. EXERCISE. 8 m/s 4. ( ) v unit/se. 5. π θ = πm 8. m/s towards light, m/s litres/s, 000 litre/s. +. k = 8 4. (4, 4) (, ), (, ). (, 6), ma. slope = 6. = is the point of loal maima; loal maimum = 0 = is the point of loal minima; loal minimum = 8 = 0 is the point of infletion. 7. Rs m, m, 864 m 4 tan y= ± 8
9 ANSWERS 95. :. Rs π C 6. B 7. A 8. C 9. D 40. A 4. A 4. D 4. B 44. B 45. C 46. B 47. D 48. A 49. B 50. C 5. A 5. C 5. B 54. C 55. B 56. A 57. B 58. B 59. C 60. (, 4) 6. + y = 0 6., 6. (, ) 64. ab 7. EXERCISE. + log log sin 6. tan + C 7. tan 5 tan os sin 0. + log + +. a os + + a a. 4 /4 log sin t sin 6. 9 log 9
10 96 MATHEMATICS 7. 5 log log log log tan a a a a + sin + a. sin + log. sin sin. tan ot + 4. sin a 5. sin se ( ) e 9. tan e 0. log m 4 m. π.. 4. tan 5. log + tan a b a tan b tan a b 7. π 8. log 6 9. e tan a tan tan + + a a a a 4.
11 ANSWERS 97 e e [ sin os ] [ sin os ] 4. tan tan tan tan + log + tan tan + tan π 4 a + b a b 45. log π log 48. A 49. C 50. A 5. C 5. D 5. C 54. D 55. D 47. π log D 57. A 58. D 59. e 60. e os tan EXERCISE. sq.units. 4 p sq. units. 0 sq.units 4. 6 sq.units 5. 7 sq.units 6. 9 sq. units 7. sq. units 8. π sq.units 9. 4 sq.units sq.units. 6 sq.units. πa 4 sq. units. 6 sq. units 4. 9 sq. units 5. 9 sq.units 6. 8 π sq.units 7. 4 sq.units sq. units 9. ( ) π a + sq. units sq.units. sq. units. 8 sq.units. 5 sq.units 4. C 5. D 6. A 7. B
12 98 MATHEMATICS 8. A 9. A 0. D. A. B. A 4. C 9. EXERCISE y. k. d y d 0. e y( ) = log + k + 5. y= e. m a 6. ( a+ m) y= e + e 7. ( ) e +y + = 0 8. y= ke 9. y tan 0. y y.. d y dy ( ) 0 d = d 4. dy y y 0 d 5. y 4 6. tan y = log + 7. tan y tan y = + 8. e e tan + log y = y y 9. y ke 0.. y y ( y ) yy 0 y y e. os ysin = + + =. ( tan ) log ( ) + + y = dy 4. y 0 d sin os log 5. y os 9
13 ANSWERS 99 y 6. sin y os y sin y e 7. log tan y 8. sin os y e 9. y 0. y 0. k e y y. y. log = y 4. D 5. C 6. A 7. C 8. B 9. C 40. C 4. D 4. A 4. C 44. D 45. B 46. B 47. C 48. C 49. D 50. A 5. A 5. B 5. B 54. B 55. B 56. C 57. B 58. A 59. A 60. C 6. C 6. D 6. C 64. C 65. A 66. D 67. D 68. C 69. C 70. A 7. A 7. A 7. C 74. B 75. A 76. (i) not defined (ii) not defined (iii) (iv) dy py Q d + = (v) pdy pdy Q e = e dy+ (vi) y (vii) y 4 4 (viii) y = Ae y (i) sin os y e () = se y 77. (i) True (ii) True (iii) True (iv) True (v) False (vi) False (vii) True (viii) True (i) True () True (i) True (i) e
14 00 MATHEMATICS 0. EXERCISE. i j k. (i) i j k (ii) 6 7 j k. i j 6 k 7 4. b a = 5. k = 6. i j k 7. 6,, ;4 i,6 j, k i 4 j 4k 9. os Area of the parallelograms formed by taking any two sides represented by aband, as adjaent are equal a b b a n a b b a i j k 9. C 0. D. C. B. D 4. A 5. D 6. D 7. D 8. A 9. C 0. A. C. C. B 4. If a and b are equal vetors , k k ] [ a 4. True 4. True 4. True 44. False 45. False a b. EXERCISE. 5+5 iˆ ˆj+5kˆ. ( ) iˆ + ( y+) ˆj + ( z ) kˆ = λ( ˆj ˆj + 6 kˆ). (,, )
15 ANSWERS 0 4. os y + z = y + z = 9 9. y + 6z 7 = y z 5 = 0 y z y z. = = and = = a + by + z = a + b + 4. (, ) 5. 5 or (, 6, ) ( ) ˆj + y ˆj + (z ) kˆ = λ( iˆ + ˆj + kˆ) y + 4z = y 50z + 7 = y 4z 6 = 0 and + 4y + 4z 6 = , 7 iˆ ˆj kˆ iˆ ˆj+ 6kˆ 9. D 0. D. A. D. D 4. A y z 5. D 6. C = 4 8.,, 9. ( 5) iˆ+ ( y+ 4) ˆj+ ( z 6) kˆ=λ ( iˆ + 7 ˆj + kˆ) 40. ( ) iˆ+ ( y 4) ˆj+ ( z+ 7) kˆ= λ( iˆ 5 ˆj + kˆ) 4. + y z = 4. True 4. True 44. False 45. False 46. True 47. True 48. False 49. True. EXERCISE Minimum value = 0. Maimum = 9, minimum = 7
16 0 MATHEMATICS. Maimise Z = y, subjet to: + y 0, + y, + y 5, 0, y 0. Minimise Z y, subjet to: 5 y 0 y 5 y, 0, y 0. Maimise Z = 00 70y subjet to : y 600, 4y 800, 0, y 0 4. Maimise Z = 00 0y subjet to : y 00, y 600, y 00, 0, y 0 5. Maimise Z = y, subjet to + y 0, 8 + 5y 400, 0, y 0 6. Type A : 6, Type B : ; Maimum profit = Rs sweaters of eah type and maimum profit = Rs 48, km Model X : 5, Model Y : 0 and maimum profit = Rs 40,000. Tablet X :, Tablet Y : 6 4.Fatory I : 80 days, Fatory II : 60 days 5. Maimum :, Minimum does not eist 6. B 7. B 8. A 9. D 0. C. D. D. A 4. B 5. Linear onstraints6. Linear 7. Unbounded 8. Maimum 9. Bounded 40. Intersetion 4. Conve 4. True 4. False 44. False 45. True
17 ANSWERS 0. EXERCISE. Independent. not independent P(E) =, P(F) :,P(G) =, no pair is independent (i) 4, (ii), (iii) 4, (iv) (i) E and E our 8., 4 0 (ii) E does not our, but E ours (iii) Either E or E, or both E and E ours (iv) Either E or E ours, but not both 0. (i), (ii) 8.. Rs Epetation = Rs (i).8 (ii) (i) 8 5 4, (ii),, (iii) (appro.) X 0 P (X) (i) 50 0 (ii) 45(49) (50) (49) (iii) 0 (50) 9
18 04 MATHEMATICS X P(X) p n 6. p = not independent 4. (i) 7, (ii) (i), (ii) 9 4. (i) 0.49, (ii) 0.65, (iii) (i), (ii) 5., (iii).7 (appro.) 50. (i), (ii) (i) 4., (ii) 6.9, (iii) Mean, S.D. = Mean = 6, Variane = 56. C 57. A 58. D 59. C 60. C 6. D 6. B 6. D 64. C 65. D 66. D 67. D 68. C 69. D 70. D 7. D 7. C 7. C 74. C 75. B 76. B 77. D 78. C 79. A 80. D 8. B 8. C 8. C 84. A 85. B 86. A 87. C 88. D 89. D 90. A 9. B
19 ANSWERS D 9. D 94. False 95. True 96. False 97. False 98. True 99. True 00. True 0. True 0. False 0. True ( ) i i pii Σp Σ 08. independent
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