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# AMU - Mathematics - 2002

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### AMU - Mathematics - 2002

1. 1. AMU –PAST PAPERSMATHEMATICS- UNSOLVED PAPER - 2002
2. 2. SECTION – I Single Correct Answer Type  There are five parts in this question. Four choices are given for each part and one of them is correct. Indicate you choice of the correct answer for each part in your answer-book by writing the letter (a), (b), (c) or (d) whichever is appropriate
3. 3. 01 Problem The area of a parallelogram whose adjacent sides are ˆ i 2ˆ j 3k and 2ˆ i ˆ j ˆ 4k is : a. 10 3 b. 5 3 c. 5 6 d. 10 6
4. 4. 02 Problem   If a j ˆ iˆ 2 ˆ 3k and b 3ˆ i ˆ j ˆ 2k , then the unit vector perpendicular   to a and b is : i j ˆ ˆ ˆ k a. 3 ˆ i ˆ j ˆ k b. 3 ˆ i ˆ j ˆ k c. 3 ˆ i ˆ j ˆ k d. 3
5. 5. 03 Problem If the 10th term of a geometric progression is a and the 4th term it is 4, then its 7th term is : a. 9/4 b. 4/9 c. 36 d. 6
6. 6. 04 Problem a a the harmonic mean of and is : 1 ab 1 ab a. 1 1 a2b2 a b. 1 a2b2 c. a a d. (1 a2 b2 )
7. 7. 05 Problem The general value of obtained from the equation cos 2 sin is : a. 2n 2 b. n ( 1)n 2 c. n 2 2 d. 2 2
8. 8. 06 Problem 1 5 The principal value of sin is : 3 a. 5 3 b. 5 3 c. 3 d. 4 3
9. 9. 07 Problem The equation of the plane passes through (2,3,4) and parallel to the plane x + 2y + 4z = 5 is : a. x + 2y + 4z = 10 b. x + 2y + 4z = 3 c. x + 2y + 4z = 24 d. x + y + 2z = 2
10. 10. 08 Problem The distance of the point (2, 3, 4) from the plane 3x – 5y + 2z + 11 = 0 is : a. 2 b. 9 c. 10 d. 1
11. 11. 09 Problem For the function f(x) = x2 – 6x + 8 2 x 4, the value of x for which f’(x) vanishes has : a. 9/4 b. 5/2 c. 3 d. 7/2
12. 12. 10 Problem 1 From Mean value theoren f(b) – f(a) = (b - af)’(x1) a < x1 < b is : f(x) = x then x equal to 2ab a. a b b a b. b a c. ab a b d. 2
13. 13. 11 Problem If 1 2 1 0 then A and B 3 0 2 3 a. AB = BA b. B2 = B c. AB BA d. A2 = A
14. 14. 12 Problem 1 If for real values of x, cos x then : x a. is an acute angle b. is a right angle c. is an obtuse angle d. no value is possible
15. 15. 13 Problem One of the equations of the lines passing through the point (3, -2) and inclined at 600 at the line 3x y 1: a. x – y = 3 b. x + 2 = 0 c. x + y = 0 d. y + 2 = 0
16. 16. 14 Problem The lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 are perpendicular to each other : a. a1b1 – b1a2 = 0 b. a12b2 – b12a2 = 0 c. a1b1 – b2a2 = 0 d. a1a2 + b1b2 = 0
17. 17. 15 Problem /2 dx is equal to : 0 1 tan x a. b. /2 c. 3 d. /4
18. 18. 16 Problem cos3 x dx is equal to : 0 a. b. 1 c. 0 d. - 1
19. 19. 17 Problem The angle between the vectors (2ˆ i 6ˆ j ˆ 3k ) and (12ˆ i 4ˆ j ˆ 3k ) is : a. 1 1 cos 9 b. 1 9 cos 11 1 9 c. cos 91 1 1 d. cos 10
20. 20. 18 Problem If the vectors ˆ ai 2ˆ j ˆ 3k and ˆ i 5ˆ j ˆ 9k are perpendicular to each other then a is equal to : a. 5 b. -6 c. - 5 d. 6
21. 21. 19 Problem 200 If i2 = -1 the value of i n is : n 1 a. 0 b. 50 c. - 50 d. 100
22. 22. 20 Problem If c i a ib , where a, b, c ae real, then a2 + b2 is equal to : c i a. 7 b. 1 c. c2 d. - c2
23. 23. 21 Problem The direction cosines to the normal plane x + 2y – 3z + 9 = 0 are : 1 2 3 a. , 14 14 14 1 2 3 b. , , 10 10 14 1 2 1 c. , , 10 14 14 1 2 3 , , d. 14 14 14
24. 24. 22 Problem The equation of a plane which cuts equal intercepts of unit lengths on the axes, is : a. x + y + z = 0 b. x + y + z = 1 c. x + y – z = 1 x y z d. 1 a b a
25. 25. 23 Problem Performing 3 iterations of bisection method of smallest positive approximate roots of equation x3 – 5x + 1 = 0 is : a. 0.25 b. 0.125 c. 0.05 d. 0.1875
26. 26. 24 Problem For the smallest positive root of transcendental equation x – e-x = 0 interval is : a. (1, 2) b. (0, 1) c. (2, 3) d. (-1, 0)
27. 27. 25 Problem The equation of the sides of a triangle are x + y – 5 = 0, x – y + 1 = 0 and y – 1 = 0 then the coordinate of the circumcentre are : a. (2, 1) b. (1, 2) c. (2, -2) d. (1, -2)
28. 28. 26 Problem A stick of length ‘l’ rests against the floor and a wall of a room. If the stick begins to slide on the floor, then the locus of its middle point is : a. A straight line b. Circle c. Parabola d. Ellipse
29. 29. 27 Problem The probability, that a leap year has 53 Sundays is : a. 2/7 b. 3/7 c. 4/7 d. 1/7
30. 30. 28 Problem In tossing 10 coins the probability of getting 5 heads is : a. 1 2 63 b. 256 c. 193 250 9 d. 128
31. 31. 29 Problem After second iteration of Newton-Raphson method the positive root of equation x2 = 3 is, (taking initial approximation 3 ) 2 a. 7 4 97 b. 56 3 c. 2 d. 347 200
32. 32. 30 Problem By false positioning the second approximation of a root of equation f(x) = 0 is (where x0, x1 are initial and first approximations respectively) : a. x1 - f (x0 ) f (x1 ) f (x0 ) f (x0 ) b. x0 – f (x1 ) f (x0 ) x0 f (x 1 ) x1f (x0 ) c. f (x1 ) f (x0 ) d. x0 f (x 0 ) x1f (x1 ) f (x1 ) f (x0 )
33. 33. 31 Problem 13 16 19 14 17 20 is equal to : 15 18 21 a. 57 b. - 39 c. 96 d. 0
34. 34. 32 Problem Which one of the following statements is true ? a. If | A | 0, then b. | A adj A | = | A | (n - 1) where A = | an | n x n c. If A’ = A, then A is a square matrix d. Determination of non-square matrix is zero e. Non-singular square matrix does not have a unique inverse
35. 35. 33 Problem 1 3 dx is equal to : x x 1 x2 a. log c 2 (1 x 2 ) b. log x (1 – x2) + c (1 x) c. log c x(1 x) 1 (1 x 2 ) d. log c 2 x2
36. 36. 34 Problem x 1 2 e x dxis equal to : x 1 a. e x c x x b. e c x x c. e ce x 2 x 1 d. log x x c
37. 37. 35 Problem dy Solution of differential equation dx + ay = emx is : a. (a + m)y = emx + c b. y = emx + ce-ax c. (a + m) = emx + c d. (a + m) y = emx + ce-ax
38. 38. 36 Problem Integrating factor of differential equation cos x + y sin x = 1 is : a. sec x b. sin x c. cos x d. tan x
39. 39. 37 Problem Solution of differential equation dy – sin x sin y dx = 0 is : a. cos x tan y = c b. cos x sin y = c y c. ecos x tan =c 2 d. ecos x tan y = c
40. 40. 38 Problem Solution of differential equation x dy – y dx = 0 represents : a. Rectangular hyperbala b. Parabola whose vertex is at origin c. Circle whose centre is at origin d. Straight line passing through origin
41. 41. 39 Problem With the help of Trapezoidal rule for numerical integration and the following table : x 0 0.25 0.50 0.75 1 f(x) 0 0.0625 0.2500 0.5625 1 1 The value of f (x) dx is : 0 a. 0.33334 b. 0.34375 c. 0.34457 d. 0.35342
42. 42. 40 Problem By the application of Simpson’s one-third rule for numerical integrations, with 1 dx two sub-intervals, the value of 0 1 x is : a. 17 36 17 b. 25 25 c. 36 17 d. 24
43. 43. 41 Problem 1 1 If matrix A = 1 1 , then : 1 1 a. A’ = 1 1 1 1 b. A-1 = 1 1 1 1 c. Adj A = 1 1 d. A= 1 1 Where is a non zero scalar
44. 44. 42 Problem 9 3 1/2 1/2 If for AX = B, B = 52 and A-1 = 4 3/4 5/4 then X is equal to : 0 2 3/4 3/4 3 a. 3/4 3/4 1/2 b. 1/2 2 c. 4 2 3 d. 1 3 5
45. 45. 43 Problem The sum of first n terms of the series 1 3 7 15 is : ... 2 4 8 16 a. n + 2 - n – 1 b. n2 – n – 1 c. 1 – 2- n d. 2n – 1
46. 46. 44 Problem 1 The coefficient of x in the expansion of 1 x2 x in ascending powers of x, when | x | < 1, is : a. 0 1 b. 2 1 c. 2 d. 1
47. 47. 45 Problem The angle between two lines is : 1 1 a. cos 9 1 4 b. cos 9 1 2 c. cos 9 1 3 cos d. 9
48. 48. 46 Problem   a 2ˆ i ˆ j ˆ 8k and b ˆ i 3ˆ j ˆ 4k , If then the magnitude of is equal to : a. 13 13 b. 3 3 c. 13 4 d. 13
49. 49. 47 Problem The line y = 2x + c is tangent to the parabola y2 = 4x then c is equal to : 1 a. 2 1 b. 2 1 c. 3 d. 4
50. 50. 48 Problem The eccentricity of the ellipse, 4x2 + 9y2 + 8x + 36y + 4 = 0 is : 3 a. 5 b. 5 3 5 c. 6 2 d. 3
51. 51. 49 Problem sin2 x cos2 x dx is equal to : sin2 x cos2 x a. tan x + cot x + c b. cosec x + sec x + c c. tan x + sec x + c d. tan x + cosec x + c
52. 52. 50 Problem If sum of two numbers is 3, the maximum value of the product of first and the square of second is : a. 4 b. 3 c. 2 d. 1
53. 53. 51 Problem Three lines ex – y = 2, 5x + ay = 3 and 2x + y = 3 are concurrent, then a is equal to : a. 2 b. 3 c. - 2 d. 1
54. 54. 52 Problem j i j ˆ The pair of straight line joining the origin to the points ˆ intersection2of the line y iof 2 ˆ 3k and ˆ ˆ 4k = 2 x + c and the circles x2 + y2 = 2 are at right angles, if : a. c2 = 9 = 0 b. c2 = 10 = 0 c. c2 – 4 = 0 d. c2 – 8 = c
55. 55. 53 Problem The direction ratio of the diagonals of a cube which joins the origin to the opposite corner are (when the three con current edges of the cube are co- ordinate axes) : a. 1, 2, 3 b. 2, -2, 1 c. 1, 1, 1 2 2 2 d. , , 3 3 3
56. 56. 54 Problem The cosine of the angle between any two diagonals of a cube is : 1 a. 3 1 b. 3 1 c. 2 2 d. 3
57. 57. 55 Problem If x f (a) is equal to : f (x) then x 1 f (a 1) a. f(a2) 1 b. f a c. f(-a) a f d. a 1
58. 58. 56 Problem If the domain of the function f(x) = x2 – 6x + 7 is (- , ), then the range of function is, a. (-2, ) b. (- , ) c. (- 2, 1) d. (- , -2)
59. 59. 57 Problem    a (a x b) is equal to : a. 0 b. a2 c. a2 + ab   d. a, b
60. 60. 58 Problem    If a 3ˆ 7ˆ 5k, b i j 3ˆ 3ˆ 3k and c i j i j ˆ 7ˆ 5ˆ 3k are the three coterminous edges of a parallelepiped, then its volume is : a. 210 b. 108 c. 308 d. 272
61. 61. 59 Problem (2x 3)(3x 4) is equal to : lim x (4x 5)(5x 6) a. 1 10 b. 0 1 c. 5 3 d. 10
62. 62. 60 Problem loge x lim is equal to : x 1 x 1 a. 1 b. 2 1 c. 2 d. 0
63. 63. 61 Problem Differential coefficient of sec x is : 1 a. sec x sin x 4 x 1 b. (sec x )3 / 2 sin x 4 x 1 c. x . sec x sin x 2 1 x (sec x )3 / 2 sin x d. 2
64. 64. 62 Problem If y e1 log e x then the value dy is equal to : dx a. 2 b. 1 c. 0 d. loge xe
65. 65. 63 Problem The co-ordinate of a point P are (3, 12, 4) with respect to origin O. Then the direction cosines of OP are : 3 1 2 a. , , 13 13 13 3 12 4 b. , , 13 13 13 1 1 1 c. , , 4 3 2 d. 2, 12, 4
66. 66. 64 Problem 2 The equation of the normal to the hyperbola x y2 at the point (8, 3 3) is : 1 16 9 a. 3 x + 2y = 25 b. 2x + 3 y = 25 c. y + 2x = 25 d. x + y = 25
67. 67. 65 Problem Differential equation for y A cos x sin x, where a and B are arbitary constants, is 2 a. d y2 y 0 dx d2y 2 y 0 b. dx 2 d2y 2 c. y 0 dx 2 d2y y 0 d. dx 2
68. 68. 66 Problem 1/4 2 2 Order and degree of differential equation d y y dy are : dx 2 dx a. 4 and 2 b. 2 and 4 c. 1 and 2 d. 1 and 4
69. 69. 67 Problem The function which is continuous for all real values of x and differentiable at x = 0 is :. a. x1/2 b. | x | c. log x d. sin x
70. 70. 68 Problem x 1 x 2 Function f (x) is an continuous function : 2x 3 x 2 a. For x = 2 only b. For all real value of x such that x 2 c. For all real value of x d. For all integral value of x only
71. 71. 69 Problem The area of the curve x2 + y2 = 2ax is : a. 4 a2 b. a2 1 2 c. a 2 d. 2 a2
72. 72. 70 Problem By graphical method, the solution of linear programming problem maxmize z = 3x1 + 5x2 subjecdt to 3x1 2x2 18, x1 4, x2 6, x1 0, x2 0 is : a. x1 = 4, x2 = 6, z = 4.2 b. x1 = 4, x2 = 3, z = 2.7 c. x1 = 2, x2 = 6, z = 36 d. x1 = 2, x2 = 0, z = 6
73. 73. 71 Problem Maximum value of f(x) = sin x + cos x is : a. 2 b. 2 c. 1 d. 1 2
74. 74. 72 Problem For all real values of x, increasing function f(x) is : a. x-1 b. x3 c. x2 d. x4
75. 75. 73 Problem 2x2 + 7xy + 3y2 + 8x + 14y + = 0 will represent a pair of straight lines when is equal to : a. 8 b. 6 c. 4 d. 2
76. 76. 74 Problem The gradient of one of the lines x2 + hxy + 2y2 = 0 is twice that of the other, then h is equal to : a. 2 3 b. 2 c. 3 d. 1
77. 77. 75 Problem A coin is tossed three times is succession. If E is the event that three are at least two heads and F is the event in which first throw is a head, then P(E/F) equal to : a. 3 4 3 b. 8 1 c. 2 1 d. 8
78. 78. 76 Problem In a box there are 2 rad, 3 black and 4 white balls. Out of these three balls are drawn together. The probability of these being of same colour is : 5 a. 84 1 b. 21 1 c. 84 d. none of these
79. 79. 77 Problem If 7th term of a Harmonic progression is 8 and the 8th term is 7, then its 5th term is 56 a. 15 b. 14 27 c. 14 d. 16
80. 80. 78 Problem If the sum of the first n terms of a series be 5n2 + 2n, then its second term is : a. 42 b. 17 c. 24 d. 7
81. 81. 79 Problem If the conjugate (x + iy) (1 – 2i) be 1 + i, then : 3 a. y = 5 1 i b. x iy 1 2i 1 i c. x iy 1 2i 1 x d. 5
82. 82. 80 Problem If a is an imaginary cube root of unity, then for the value of 3n 1 3n 3 3n 5 a. -1 b. 0 c. 3 d. 1
83. 83. 81 Problem The value of 15C 2 2 15C1 2 15C2 2 ... 15C15 is : 0 a. 0 b. - 15 c. 15 d. 51
84. 84. 82 Problem (loge n)2 (loge n)4 1 ... is equal to : 2! 4! 1 a. 2 (n + n-1) b. 1 n c. n 1 d. (en + e-n) 2
85. 85. 83 Problem ax da is equal to : ax 1 c a. x 1 ax 1 b. e loge a e c. ax loge a + c d. none of these
86. 86. 84 Problem Two numbers with in the bracket denote the ranks of 10 students of a class in two subjects – (1, 10), (2, 9), (3, 8), (4, 7), (5, 6), (6, 5)(7, 4), (8, 3), (9, 2), (10, 1) then rank correlation coefficient is : a. - 1 b. 0 c. 0.5 d. 1
87. 87. 85 Problem There are 5 roads leading to a town from a village. The number of different ways in which a villager can go to the town return backs is : a. 20 b. 25 c. 5 d. 10
88. 88. 86 Problem If (1 + x)n = C0 + C1x + C2x2 + ….Cnxn then the value of Cn = 2C1 + 3C2 + ….+ (n + 1) Cn will be : a. (n + 2)2n b. (n + 1)2n – 1 c. (n + 1)2n d. (n + 2)2n - 1
89. 89. 87 Problem In a triangle ABC, 2 cos A = sin B cosec C then a. 2a = bc b. a = b c. b = c d. c = a
90. 90. 88 Problem If tan x b then the value of a cos 2x + b sin 2x is : a a. a b. a – b c. a + b d. b
91. 91. 89 Problem In angle of a triangle are in the ratio of 2 : 3 : 7, then the sides are in the ratio of : a. 2 : ( 3+ 1) : 2 b. 2 : 2 : ( 3+ 1) c. 2 : 2 : ( 3+ 1) d. 2 : ( 3 + 1) : 2
92. 92. 90 Problem 1 1 ac 1 bc is equal to : 1 1 ad 1 bc 1 1 ac 1 bc a. a + b + c b. 1 c. 0 d. 3
93. 93. 91 Problem The angle of elevation of the sum if the length of the shadow of a tower is 3 times the height of the pole is : a. 1500 b. 300 c. 600 d. 450
94. 94. 92 Problem If the sides of triangle are 3, 5, 7 then : a. Triangle is right-angled b. One angle is obtuse c. All its angles are acute d. None of these
95. 95. 93 Problem Area bounded by lines y = 2 + x, y = 2- x and x = 2 is : a. 16 b. 8 c. 4 d. 3
96. 96. 94 Problem Area bounded by parabola y2 = x and straight line 2y = x is : a. 4 3 b. 1 2 c. 3 1 d. 3
97. 97. 95 Problem The latusrectum of parabola y2 = 5x + 4y + 1 is : a. 10 b. 5 5 c. 4 5 d. 2
98. 98. 96 Problem x = 7 touches the circles x2 + y2 – 4x – 6y – 12 = 0 then the co-ordinates of the point of contact : a. (7, 4) b. (7, 3) c. (7, 2) d. (7, 8)
99. 99. 97 Problem If the roots of the equation, (a2 + b2) t2 – 2(ac + bd) t + (c2 + d2) = 0 are equal then : a. ad + bc = 0 a c b. b d c. ab = dc d. ac = bc
100. 100. 98 Problem 2 If the roots a, of the equation x bx 1 are such that a 0 , ax c 1 then the value of is : 1 a. c b. 0 a b c. a b a b d. a b
101. 101. 99 Problem The equation of the tangents of the ellipse 9x2 + 16y2 = 144 from the point (2, 3) are : a. y = 3, x + y = 5 b. y = 3, x = 2 c. y = 2, x = 3 d. y = 3, x = 5
102. 102. 100 Problem The latus rectum of the hyperbola 9x2 – 16y2 – 18x – 32y – 151 = 0 9 a. 4 3 b. 2 c. 9 9 d. 2
103. 103. FOR SOLUTIONS VISIT WWW.VASISTA.NET