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# AMU - Mathematics - 2005

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• ### AMU - Mathematics - 2005

1. 1. AMU –PAST PAPERSMATHEMATICS - UNSOLVED PAPER - 2005
2. 2. SECTION – I CRITICAL REASONING SKILLS
3. 3. 01 Problem ~p q is logically equivalent to : a. P q b. q p c. ~ (p q) d. ~ (q p)
4. 4. 02 Problem The angle of elevation of the top of an incomplete vertical pillar at a horizontal distance of 100 m from its base is 450. If the angle of elevation of the top of the complete pillar at the same point is to be 600, then the height of the incomplete pillar is to be increased by : a. 50 2 b. 100 m c. 100 ( 3 -1)m d. 100 ( 3 + 1)m
5. 5. 03 Problem 1 2 3 8 What must be the matrix X, if 2X 3 4 7 2 ? 1 3 a. 2 1 1 3 b. 2 1 2 6 c. 4 2 2 6 d. 4 2
6. 6. 04 Problem The value of 1 1 1 is : bc ca ab b c c a a b a. 1 b. 0 c. (a - b) (b - c) (c - a) d. (a + b) (b + c) (c + a)
7. 7. 05 Problem 441 441 443 The value of 445 446 447 is : 449 450 451 a. 441 x 446 x 4510 b. 0 c. - 1 d. 1
8. 8. 06 Problem    ˆ ˆ (a ˆ)ˆ i i (a ˆ)ˆ j j (a k )k is equal to :  a. a  b. 2 a  c. 3 a  d. 0
9. 9. 07 Problem cos 2 sin 2 Inverse of the matrix is : sin 2 cos 2 a. cos 2 sin 2 sin 2 cos 2 b. cos 2 sin2 sin2 cos 2 cos 2 sin2 c. sin2 cos 2 cos 2 sin 2 sin 2 cos 2 d.
10. 10. 08 Problem   a 3, b 4,   If then a value of for which a b is perpendicular to   a b is : 9 a. 16 3 b. 4 3 c. 2 4 d. 3
11. 11. 09 Problem   The projection a 2i j ˆ ˆ 3ˆ 2k on b i j ˆ ˆ 2ˆ 3k is : 1 a. 14 2 b. 14 c. 14 2 d. 14
12. 12. 10 Problem Let A = {x : x is a multiple of 3} and B = {x : x is a multiple of 5}. Then A B is given by : a. {3, 6, 9, ………… } b. {5, 10, 15, 20, ………..} c. {15, 30, 45, …………….} d. none of the above
13. 13. 11 Problem The maximum of the function 3 cos x – 4 sin x is : a. 2 b. 3 c. 4 d. 5
14. 14. 12 Problem If the distance ‘s’ metres traversed by a particle in t seconds is given by s = t3 – 3 t2, then the velocity of the particle when the acceleration is zero, (in m/s) is : a. 3 b. - 2 c. - 3 d. 2
15. 15. 13 Problem For the curve yn = an-1 x if the subnormal at any point is a constant, then n is equal to : a. 1 b. 2 c. -2 d. -1
16. 16. 14 Problem d2 x If x = A cos 4t + B sin 4t, then is equal to : dt 2 a. - 16 x b. 16 x c. x d. - x
17. 17. 15 Problem If tangent to the curve x = at2, y = 2at is perpendicular to x-axis, then its point of contact is : a. (a, a) b. (0, a) c. (0, 0) d. (a, 0)
18. 18. 16 Problem dy 1 cos 2y The general solution of the differential equation 0 is dx 1 cos 2x given by : a. tan y + cot x = c b. tan y – cot x = c c. tan x – cot y = c d. tan x + cot y = c
19. 19. 17 Problem 3/4 1/3 2 The degree of the differential equation : dy d2y is : 1 dx dx 2 a. 1/3 b. 4 c. 9 d. 3/4
20. 20. 18 Problem The area enclosed between the curves y = x3 and y = is , (in sq unit) : a. 5 3 5 b. 4 5 c. 12 12 d. 5
21. 21. 19 Problem /8 cos3 4 d is equal to : 0 5 a. 3 5 b. 4 1 c. 3 1 d. 6
22. 22. 20 Problem /2 cos x sin x dx is equal to : 0 1 cos x sin x a. 0 b. 2 c. 4 d. 6
23. 23. 21 Problem If ax2 – y2 + 4x – y = 0 represents a pair of lines, then a is equal to : a. -16 b. 16 c. 4 d. -4
24. 24. 22 Problem What is the equation of the locus of a point which moves such that 4 times its distance from the x-axis is the square of its distance from the origin ? a. x2 – y2 – 4y = 0 b. x2 + y2 – 4|y| = 0 c. x2 + y2 – 4x = 0 d. x2 + y2 – 4|x| = 0
25. 25. 23 Problem Equation of the straight line making equal intercepts on the axes and passing through the point (2, 4) is : a. 4x – y – 4 = 0 b. 2x + y – 8 = 0 c. x + y – 6 = 0 d. x + 2y – 10 = 0
26. 26. 24 Problem If the area of the triangle with vertices (x, 0) (1, 1) and (0, 2) is 4 square unit, then the value of x is : a. - 2 b. - 4 c. - 6 d. 8
27. 27. 25 Problem is equal to : lim 2 cot 2 a. 0 b. - 1 c. 1 d.
28. 28. 26 Problem The co-axial system of circles given by x2 + y2 + 2gx + c = 0 for c < 0 represents : a. Intersecting circles b. Non-intersecting circles c. Touching circles d. Touching or non-intersecting circles
29. 29. 27 Problem The radius of the circle passing through the point (6, 2) and two of whose diameters are x + y = 6 and x + 2y = 4 is : a. 4 b. 6 c. 20 d. 20
30. 30. 28 Problem If (0, 6) and (0, 3) are respectively the vertex and focus of a parabola, then its equation is a. x2 + 12y = 72 b. x2 - 12y = 72 c. x2 – 12x = 72 d. y2 + 12x = 72
31. 31. 29 Problem For the ellipse 24x2 + 9y2 – 120x –90y + 225 = 0, the eccentricity is equal to : a. 2 5 b. 3 5 15 c. 24 1 d. 5
32. 32. 30 Problem x2 y2 x2 y2 1 If the foci of the ellipse 1 and the hyperbola 16 b2 144 81 25 coincide, then the value of b2 is : a. 1 b. 7 c. 5 d. 9
33. 33. 31 Problem The differential coefficient of f(sin x) with respect to x where f(x) = log x is : a. tan x b. cot x c. f(cos x) 1 d. x
34. 34. 32 Problem 1 cos x If f(x) = , x 0 is continuous at x = 0, then the value of k is : x k, x 0 a. 0 1 b. 2 1 c. 4 1 d. - 2
35. 35. 33 Problem If 1 3i 2 is : then (3 3 )4 2 a. 16 b. - 16 c. 16 d. 16 2
36. 36. 34 Problem If y = tan-1 (sec x – tan x), then is equal to : a. 2 b. -2 1 c. 2 1 d. - 2
37. 37. 35 Problem 1 1 If x 2 cos then x n is equal to : 2 xn a. 2n cos b. 2n cos n c. 2i sin n d. 2 cos n
38. 38. 36 Problem 1 |1 x | dx is equal to : 1 a. - 2 b. 0 c. 2 d. 4
39. 39. 37 Problem dx is equal to : 7 x(x 1) a. x7 log c x7 1 b. 1 x7 log 7 c 7 x 1 x7 1 c. log c x7 1 x7 1 log c d. 7 x7
40. 40. 38 Problem xe x dx is equal to : a. 2 x e x 4 xe x c b. x (2 x 4 x 4)e c x c. (2x 4 x 4)e c x (1 4 x )e c d.
41. 41. 39 Problem dx is equal to : 2 x 2x 2 a. sin-1 (x + 1) + c b. sin-1 (x + 1) + c c. tan-1 (x - 1) + c d. tan-1 (x + 1) + c
42. 42. 40 Problem If a tangent to the curve y = 6x – x2 is parallel to the line 4x – 2y – 1 = 0, then point of tangency on the curve is : a. (2, 8) b. (8, 2) c. (6, 1) d. (4, 2)
43. 43. 41 Problem 0.5737373……… is equal to : a. 284 497 284 b. 495 568 c. 999 567 d. 990
44. 44. 42 Problem The number of solutions for the equation x2 – 5 |x| + 6 = 0 is : a. 4 b. 3 c. 2 d. 1
45. 45. 43 Problem How many numbers of 6 digits can be formed from the digits of the number 112233 ? a. 30 b. 60 c. 90 d. 120
46. 46. 44 Problem The last digit in 7300 a. 7 b. 9 c. 1 d. 3
47. 47. 45 Problem log x log y log z If then xyz is equal to : a b b c c a a. 0 b. 1 c. - 1 d. 2
48. 48. 46 Problem The smallest positive integer n for which (1 + i)2n = (1 - i)2n is : a. 1 b. 2 c. 3 d. 4
49. 49. 47 Problem If cos-1 p + cos-1 q + cos-1 r = then p2 + q2 + r2 + 2ppr is equal to : a. 3 b. 1 c. 2 d. - 1
50. 50. 48 Problem x 5 If sin 1 cosec-1 , then x is equal to : 5 4 2 a. 1 b. 4 c. 3 d. 5
51. 51. 49 Problem If 0 x 2 and 81sin x 2 81cos x 30, then x is equal to : a. 6 b. 2 c. 4 3 d. 4
52. 52. 50 Problem x2 y2 The equation of the director circle of the hyperbola 1 is given by : 16 4 a. x2 + y2 =16 b. x2 + y2 = 4 c. x2 + y2 = 20 d. x2 + y2 = 12
53. 53. 51 Problem The normals at the extremities of the latus rectum of parabola intersects the axis at an angle of : a. Less than 900 b. Greater than 900 c. 900 d. none of the above
54. 54. 52 Problem The circle x2 + y2 – 8x + 4y + 4 = 0 touches : a. x-axis b. y-axis c. both axes d. neither x-axis nor y-axis
55. 55. 53 Problem If A = {1, 2, 3} and B = {3, 8}, then (A B) x ( A B) is : a. {(3, 1), (3, 3), (3, 8)} b. {(1, 3), (2, 3), (3, 3), (8, 3)} c. {(1, 2), (2, 2), (3, 3), (8, 8)} d. {(8, 3), (8, 2), (8, 1), (8, 8)}
56. 56. 54 Problem The condition that one root of the equation ax2 + bx + c = 0 may be double of the other, is a. b2 = 9ac b. 2b2 = 9ac c. 2b2 = ac d. b2 = ac
57. 57. 55 Problem The value of k so that x2 + y2 + kx + 4y + 2 = 0 and 2 (x2 + y2) – 4x – 3y + k = 0 cut orthogonally is : a. 10 3 8 b. 3 10 c. - 3 8 d. 3
58. 58. 56 Problem (3 x 1) 4 is equal to : lim 1 x X 1 a. e12 b. e-12 c. e4 d. e3
59. 59. 57 Problem If A + B + C = 1800, then A B is equal to : tan tan 2 2 a. 0 b. 1 c. 2 d. 3
60. 60. 58 Problem In a triangle ABC, If b = 2, B = 300 then the area of the circumcircle of triangle ABC in square unit is : a. b. 2 c. 4 d. 6
61. 61. 59 Problem If sin x + sin2 x = 1, then cos12 x + 3 cos10 x + 3 cos8 x + cos6 x is equal to : a. 1 b. 2 c. 3 d. 0
62. 62. 60 Problem If R denotes the set of all real number, then the function f :R R defined f(x) = |x| is : a. One-one only b. Onto only c. Both one-one and onto d. Neither one-one nor onto
63. 63. 61 Problem If f(x) = 2x3 + mx2 – 13x + n and 2,3 are roots of the equation f(x) = 0, then the values of m and n are a. - 5, - 30 b. - 5, 30 c. 5, 30 d. none of these
64. 64. 62 Problem If p1, p2, p3 are respectively the perpendiculars from the vertices of a triangle to the opposite sides, then p1p2p3 is equal to : a. a2b2c2 b. 2 a2b2c2 4a2 b2c 3 c. R2 a2 b2 c 2 d. 8R2
65. 65. 63 Problem If 5 cos 2 cos2 1 0, , then is equal to : 2 a. 3 1 3 , cos b. 3 5 1 3 c. cos 5 1 3 , cos d. 3 5
66. 66. 64 Problem The two forces acting at a point, the maximum effect is obtained when their resultant is 4N If they act at right angles, then their resultant is 3N. Then the forces are : a. 1 1 (2 3)N and (2 3)N 2 2 b. (2 3)N and (2 3)N 1 1 c. (2 2)N and (2 2)N 2 2 (2 2)N and (2 2)N d.
67. 67. 65 Problem The resultant R of two forces P and Q act at right angles to P. Then the angle between the forces is : 1 P a. cos Q 1 P cos b. Q 1 P sin c. Q 1 P d. sin Q
68. 68. 66 Problem A body starts from rest and moves with a uniform acceleration. The ratio of the distance covered in nth sec to the distance covered in n second is : 2 1 a. n n2 1 1 b. n2 n 2 1 c. n2 n 2 1 d. n n2
69. 69. 67 Problem Two points move in the same straight line starting at the same moment from the same point in the same direction. The first moves with constant velocity u and the second starts from rest with constant acceleration f, then distance between the two points will be maximum at time : 2u u a. t C. t f 2f u u2 t t b. f D. f
70. 70. 68 Problem The equation of the plane containing the line x 1 y 3 z 2 and 3 2 1 the point (0, 7, -7) is : a. x + y + z = 1 b. x + y + z = 2 c. x + y + z = 0 d. none of these
71. 71. 69 Problem A plane passes through a fixed point (p, q) and cut the axes in A, B, C. Then the locus of the centre of the sphere OABCi. : p q r a. 2 x y z b. p q r 1 x y z c. p q r 3 x y z d. None of these
72. 72. 70 Problem The value of 12.C1 + 32 . C3 + 52 . C5 + … is : a. n (n -1)n-2 + n. 2n-1 b. n(n - 1)2n-2 c. n(n -1) . 2n-3 d. none of the above
73. 73. FOR SOLUTIONS VISIT WWW.VASISTA.NET
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