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# AMU - Mathematics - 1996

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• ### AMU - Mathematics - 1996

1. 1. AMU –PAST PAPERSMATHEMATICS - UNSOLVED PAPER - 1996
2. 2. SECTION – I CRITICAL REASONING SKILLS
3. 3. 01 Problem 2 The differential equation of all ‘Simple Harmonic Motions’ of given period n is : x = a cos (nt + b) d2 x a. dt 2 + nx = 0 d2 x b. + n 2x = 0 dt 2 d2 x c. - n2 x = 0 dt 2 d2 x 1 d. + x=0 dt 2 n2
4. 4. 02 Problem The elimination of constants A, B and C from y = A + Bx – Ce-x leads the differential equation : a. y’ + y’’ = 0 b. y’’ – y’’’ = 0 c. y’ + ex = 0 d. y’’ + ex = 0
5. 5. 03 Problem The solution of the differential equation xdy + ydx = 0 is : a. xy = c b. x2 + y2 = 0 c. xy logx = 1 d. log xy = c
6. 6. 04 Problem dy The solution of the equation +2y tan x = sin x, is : dx a. y sec2 x = sec x + c b. y sec x = tan x + c c. y = sec x + c sec2 x d. none of these
7. 7. 05 Problem x a is equal to: dx x a x a. c log a b. 2a x c log a x c. 2a .log a  c d. None of these
8. 8. 06 Problem 1/2 dx  1/4 xx 2 is equal to :  a. 6  b. 3  c. 4 d. 0
9. 9. 07 Problem  /2 1 The value of  0 3 1  tan x dx is : a. 0 b. 1 c.  /4 d. 2 
10. 10. 08 Problem Q dx The value of  P x , (P  Q) is : a. log (Q - P) b. log P c. log Q d. log (P Q)
11. 11. 09 Problem For a biased dice the probabilities of different faces to turn up are given below : Face: 1 2 3 4 5 6 Probabili 0.1 0.32 0.21 0.15 0.05 0.17 ty The dice is tossed and you are told that either the face 1 or 2 has turned up. The probability that it is face is : a. 1 7 4 b. 7 5 c. 21 4 d. 21
12. 12. 10 Problem 1 x /2 If A  1 tan x dx and B   x / 2 cot x dx then A + B is equals : a. 0 b. 1 c. 2  d. 4
13. 13. 11 Problem If f’(x) = (x - a)2n (x - b)2p+1 where x and p are positive integers, then : a. x = a is a point of minimum b. x = a is a point of maximum c. x is not a point of maximum or minimum d. none of these
14. 14. 12 Problem the inverse of a symmetric matrix is : a. diagnol matrix b. skew symmetric c. square matrix d. a symmetric matrix
15. 15. 13 Problem  a  p The matrix product     equals :  b  x[xyz ]x q  c    r    pqr  abc a. xyz b. xyz  pqr abc c. pqr  abc xyz d. none of these
16. 16. 14 Problem In a group of 52 persons, 16 drink tea but not coffee, while 33 drink tea. How many persons drink coffee but not tea : a. 17 b. 36 c. 23 d. 19
17. 17. 15 Problem If the length of major axis of an ellipse is three times the length of minor axis, then the eccentricity is : 1 a. 3 1 b. 2 1 c. 3 2 2 d. 3
18. 18. 16 Problem A conic section with centricity e is a parabola if : a. e = 0 b. e < 1 c. e > 1 d. e = 1
19. 19. 17 Problem For the ellipse 3x2 + 4y2 = 12 length of the latusrectum is : a. 3 b. 4 3 c. 5 2 d. 5
20. 20. 18 Problem If (2, 0) is the vertex and y-axis is the direction of a parabola, then its focus is a. (2, 0) b. (-2, 0) c. (4, 0) d. (-4, 0)
21. 21. 19 Problem The length of tangent from the point (6,-7) to the circle : 3x2 + 3y2 – 7x – 6y = 12 is a. 6 b. 9 c. 7 d. 13
22. 22. 20 Problem Which of the following lines is fathest from the origin ? a. x – y + 1 = 0 b. 2x – y + 3 = 0 c. x + 2y – 2 = 0 d. x + y – 2 = 0
23. 23. 21 Problem The normal to a given curve is parallel to x-axis if : dy a. dx =0 dy b. dx =1 c. dx =0 dy dx d. =1 dy
24. 24. 22 Problem The distance between the lines 3x + 4y = 9 and 6x + 8y = 15 is : 3 a. 2 3 b. 10 c. 6 d. 9 4
25. 25. 23 Problem a line passes through (2, 2) and is perpendicular to the line 3x + y = 3. Its y- intercept is : 1 a. 3 b. 2 3 c. 1 4 d. 3
26. 26. 24 Problem If the volume in cm3 and surface and surface area is cm2 of a sphere are numerically equal Then the radius of the sphere in cm is : a. 2 b. 3 c. 4 d. 5
27. 27. 25 Problem the locus of a point represented by x  a  t  1  y  a  t  1  is : 2 t    2 t    a. an ellipse b. a circle c. a pair of straight lines d. none of these
28. 28. 26 Problem The equation of a tangent parallel to y = x drawn x2 y 2 is :  1 3 2 a. x – y + 1 = 0 b. x – y + 2 = 0 c. x + y – 1 = 0 d. x – y + 2 = 0
29. 29. 27 Problem Which of the following is not correct : a. A function f such that f(x) = x – 1 for every integer x is binary operation b. The binary operation positive on IR is commutative c. The set{(a, b)/a I and b = - a} is a binary operation on I d. The division on IR – [0] is not as associative operation
30. 30. 28 Problem  If cos   - 0.6 and 1800 <  < 2700, then tan 4 is equal to : a. 5 2 (1  5) b. 2 c. (1  5) 2 ( 5  1) d. 2
31. 31. 29 Problem z1 z Let z1 and z2 be two complex number such that  2  1. Then : z2 z1 a. z1, z2 are collinear b. z1, z2 and the origin form a right angled triangle c. z1, z2 and the origin form an equilateral triangle d. none of these
32. 32. 30 Problem 1i 3 The amplitude of is : 3 1  a. 6  b. 4  c. 3  d. 2
33. 33. 31 Problem In a G.P. if (m + n)th terms is p and (m - 1)th term is q then mth term is : p a. q q b. p c. Pq d. pq
34. 34. 32 Problem In a G.P. of positive terms, any term is equal to the sum of the next two terms. Then the common ratio of the G.P. is a. 2 cos 180 b. sin 18 c. cos 18 d. 2 sin 18
35. 35. 33 Problem If n+2C8 n-2P4 = 57 : 16, then the value of x is : a. 20 b. 19 c. 18 d. 17
36. 36. 34 Problem A library has a copies of one book, b copies of each of two books, c copies of each of three books, and single copies of d books. The total number of ways in which these books can be distributed is : (a  b  c  d)! a. a! b ! c ! (a  2b  3c  d)! b. a !(b !)2 (c !)3 (a  2b  3c  d)! c. a! b ! c ! d. None of these
37. 37. 35 Problem Three identical dice are rolled. The probability that the same number will appear on each of them is : 1 a. 6 1 b. 36 1 c. 18 3 d. 28
38. 38. 36 Problem If a, b, c are real numbers a  0. If , is a root of a2x2 + bx + c = 0, is a root of a2x2 – bx + c = 0 and 0 < a < b, then the equation a2x2 + 2bx + 2c = 0 has a root that always satisfied : a.      2 b.      2 c.   d.     
39. 39. 37 Problem The solution of log7log5 ( x 5  x ) = 0 is : a. 4 b. 5 c. 6 d. 9
40. 40. 38 Problem 1 The most general value of  , satisfying the two equations, cos  = , tan   1 2 is 5 a. 2n  4  b. 2n  4 5 c. n  4  d. (2n  1)  4
41. 41. 39 Problem Three faces of a fair dice are yellow, two faces red and only one is blue. The dice is tosses three times. The probability that the colour yellow, red and blue appears in the first, second and the third tosses respectively is : a. 1 30 b. 1 25 1 c. 36 1 d. 32
42. 42. 40 Problem If a2 + b2 + c2 = 1, then ab + bc + ca lies in the interval : a. 1   2 ,2   b. [-1, 2]  1  c.   2 ,1    1 d.  1, 2   
43. 43. 41 Problem The common tangent to the parabola y2 = 4ax and x2 = 4ay is : a. x + y + a = 0 b. x + y - a = 0 c. x - y + a = 0 d. x - y - a = 0
44. 44. 42 Problem The solution of the equation cos2  + sin  + 1 = 0, lies in the interval :    a.  4 , 4     3  b.  4 , 4     3 5  c.  4 , 4     5 7  d.  4 , 4   
45. 45. 43 Problem If cosec  - cot = q, then the value of cot  is : 1  q2 a. q 1  q2 b. q q c. 1  q2 q d. 1  q2
46. 46. 44 Problem The length of the intercept on the normal at the point (at2, 2at) of the parabola y2 = 4ax made by the circle on the line joining the focus and p as diameter is : a. a 1  t2 b. a 1  t2 c. a 1t d. a 1t
47. 47. 45 Problem From the top of a light house 60 metres high with its base at the sea level, the angle of depression of a boat is 150. The distance of the boat from the foot of the light house is :  3  1   3  1  a.   60 metres  3  1   3  1  b.   60 metres  3  1 c.   3  1  metres   d. none of these
48. 48. 46 Problem The expression 1  sin  1  sin   equal to :  for   1  sin  1  sin  2 2 cos  a. cos2  2 sin  b. sin2  2 cos  c. sin2  d. none of these
49. 49. 47 Problem x2 y2 If m1 and m2 are the slopes of the tangent to the parabola  1 , which 15 16 passes through the point (6, 2) the value of (m1 + m2) is : a. 14 11 4 b. 11 11 c. 4 24 d. 11
50. 50. 48 Problem If f’(x) = sin (log x) and y = f  2 x  3  , then dy equals :  3  2x    dx 1 a. sin (log x) . x log x 12   2x  3   b. 2 sin log   (3  2x)   3  2x     2x  3   c. sin log     3  2x   d. none of these
51. 51. 49 Problem The derivatives of tan-1  2x  w.r.t. sin-1 2x  is :  2  1  x 2  1  x    a. 2 b. 4 c. 1 d. 2
52. 52. 50 Problem x  m equals to : lim 1  x   x   a. em b. e-m c. m-e d. me
53. 53. 51 Problem x 2 If y  f (x )  then : x 1 a. f(1) = 3 b. 2 = f(4) c. f is rational function of x d. y increases with x for x < 1
54. 54. 52 Problem x The co-ordinates of the points on the curve, f (x)  1  x where the tangent to 2 the curve has greatest slope is : a. (0, 2) b. (0, 0) c. (0, 1) d. (1, 1)
55. 55. 53 Problem the maximum value of xy subjected to x + y = 8 is : a. 8 b. 16 c. 20 d. 24
56. 56. 54 Problem df If f = xy then is equal to : dx a. x log f y b. xy . x c. xy log x d. yx
57. 57. 55 Problem The curve y - exy + x = 0 has a vertical tangent at the point : a. (1, 1) b. at no point c. (0, 1) d. (1, 0)
58. 58. 56 Problem The sum of the series 1 1 : 1   ... (1  2) (1  2  3) a. 1 2 b. 1 c. 2 3 d. 2
59. 59. 57 Problem Let S be a finite set containing n elements. Then the total number of binary operations on S is : 2 a. nn b. nn c. n2 2 d. 2n
60. 60. 58 Problem The ratio of the volume of a cube to that of the sphere which will fit inside the cube : a. 5 :  b. 7 :  c. 6 :  d. 3 : 
61. 61. 59 Problem The function f(x) = cos (log (x + x2  1 )) is : a. Even b. Constant c. Odd d. None of these
62. 62. 60 Problem The perimeter of a given rectangle is x, its area will maximum of its sides are : a. x x , 2 2 x x b. , 3 3 x x c. , 6 3 x x d. , 4 4
63. 63. 61 Problem d 3y dy Which of the following functions hold the property of 3  =0? dx dx a. y = ex b. y = cos x c. y = tan x d. y = sin x
64. 64. 62 Problem 0.5 the domain of definition of the function : x.1  2( x  4) + (x + 4)0.5 + 2  ( x  4)0.5 4(x + 4)0.5 is : a. R b. (- 4, 4) c. R+ d. (-4, 0)  (0, )
65. 65. 63 Problem If a, b, c, d are in H.P. then : a. ab > cd b. ad > bc c. ac > bd d. none of these
66. 66. 64 Problem If  1 x  then the value of cos   x is : tan  tan , 2 1 x 2 1  x cos  a. sin x b. cos x c. cos  d. sin 
67. 67. 65 Problem A round balloon of radius r subtends an angle at the eye of the observer while the angle of elevation of its centre is  , then the height of the centre of balloon is : a. r tan  sin    b. r cosec  2    c. r tan  and    d. r sin 2   cosec , 
68. 68. 66 Problem 1 a 1 1 If a-1 + b-1 + c-1 = 0 such that 1 1 b 1  then the value of  1 1 1 c is : a. 0 b. - abc c. abc d. none of these
69. 69. 67 Problem 4 5 2  If A =   then adj. (A) equals : 5 4 2  2  2 8  36 36 18    a. 36 36 18 18 18  9    36 36 18  b.    36 36 18  18  18 9   0 0 0 c.   0 0 0 0 0 0   d. none of these
70. 70. 68 Problem The statement of k = 1 10 by 2, do remit in : a. 3 cycles b. 2 cycles c. 8 cycles d. 9 cycles
71. 71. 69 Problem The Newton’s method converges first of f’(x) is : a. Small b. mid c. large d. zero
72. 72. 70 Problem  are unit vectors and  is the angle between them then    If a and b a  b 2 is :  a. sin 2 b. sin  c. 2 sin  d. sin 2 
73. 73. 71 Problem The area of ABC with vertices A (1, -1, 2), B (2, 1, -1) and C (3, -1, 2) is : a. 13 2 sq. units b. 3 6 sq. units c. 13 sq. units d. 15 3 sq. units
74. 74. 72 Problem If p, q, r be three non-zero vectors, then equation p . q = p . r implies : a. q = r b. p is orthogonal to both q and r c. p is orthogonal to q – r d. either q = r or p is orthogonal to q – r.
75. 75. 73 Problem   j ˆ i j ˆ  i j ˆ ˆ  a2 ˆ  a3 k, b  b1ˆ  b2 ˆ  b3 k, c  c1ˆ  c2 ˆ  c3 k, be three non- Let a  a1i   zero vectors such that is a unit vector perpendicular to both a and b . If the    a1 a2 a3 a and b is angle between 6 then b1 b2 b3 is equal to : c1 c2 c3 a. 0 b. 1 1  2  2 | a||b| c. 4 3  2  2 d. | a||b| 4
76. 76. 74 Problem       The scalar a.{(b  c ) x (a  b  c )} equals : a. 0    b. 2 [a b c ]    c. [a b c ] d. none of these
77. 77. 75 Problem If f(x) = x1/x-1 for all positive x = 1 and if f is continuous at 1, then equals : a. 0 1 b. e c. e d. e2
78. 78. 76 Problem the point equidistant from the four points (0, 0, 0), (3/2, 0, 0), (0,5/2, 0) and (0, 0, 7/2) is :  2 1 2 a.  3 3 5     3 b.  3,2, 5    3 5 7 c. 4, 4, 4   1   2 , 0, 1  d.  
79. 79. 77 Problem The area bounded by the curve y = x sin x and the x-axis between x = 0 and x = 2  is : a. 2  b. 3  c. 4  d. 
80. 80. 78 Problem The value of (i)I where i2 = -1 is : a. Whole imaginary number b. Whole real number c. Zero d. E  /2
81. 81. 79 Problem The equations of the sphere with A (2, 3, 5) and B (4, 9, -3) as the ends of a diameter is : a. x2 + y2 + z2 – 6x – 12y – 2z + 20 = 0 b. 2x2 + 2y2 + 2z2 – x - y – z + 1 = 0 c. 3x2 + 3y2 + 3z2 – 2x – 2y –2z –1 = 0 d. none of these
82. 82. 80 Problem If (2, 0) is the vertex and y-axis the direction of a parabola, then its focus is : a. (2, 0) b. (-2, 0) c. (4, 0) d. (-4, 0)
83. 83. 81 Problem If f(x) = log (x + a), for x  a , then the value of f’(x) is equal to : 1 a. x a 1 b. | x  a| 1 c. | x  a| 1 d. | x  a|
84. 84. 82 Problem In a collection of 6 mathematics books and 4 physics books, the probability that 3 particular mathematics books will be together is : 1 a. 8 1 b. 10 1 c. 15 1 d. 20
85. 85. 83 Problem Find the mode from the data given below : Marks 0-5 10-15 15-20 15-20 20-25 25-30 Obtained 18 20 25 30 16 14 a. 16.1 b. 16.2 c. 16.7 d. 16.3
86. 86. 84 Problem If the mean of four observations is 20 and when a constant c is added to each observation, the mean becomes 22. The value of c is : a. - 2 b. 2 c. 4 d. 6
87. 87. 85 Problem The area of surface generated by rotating the circle x = b =  cos y = a + b sin  , 0    2 abut x-axis is : a.  ab sq. units b. ab sq. units x c. 2 loge x c d. ( x )a loge x c
88. 88. 86 Problem The eigin values of the matrix 5 4 are : A    1 2 a. 2, 4 b. 1, 6 c. 3, -3 d. 1, -2
89. 89. 87 Problem x y 2 z 3 the length of perpendicular from the point P(3, -1, 11) to the line   2 3 4 is : a. 2 13 b. 53 c. 2 14 d. 8
90. 90. 88 Problem the angle A, B, C of a triangle are in the ratio of 3 : 5 : 4, then a + i 2 is equal to : a. 3b b. 2b c. 2b d. 4b
91. 91. 89 Problem If the angle between the lines joining the foci of an ellipse to an extremity of the minor axis is 900, the eccentricity of the ellipse is : 1 a. 8 11 b. 4 c. 3 2 1 d. 2
92. 92. 90 Problem If a man running at the rate of 15 km per hour crosses a bridge in 5 minutes, the length of the bridge in metres is : a. 7500 b. 1250 c. 1000 1 d. 1333 3
93. 93. 91 Problem If  x   y  15,  x 2   y 2  49,  xy  44 and n = 5, then byx is equal to : 1 a. - 3 2 b. - 3 c. - 1 4 1 d. - 2
94. 94. 92 Problem If (x + 1) is a factor of x4 + (p -3)x3 – (3p - 5)x2 + (2p -10) x + 5 then the value of p is : a. 2 b. 1 c. 3 d. 4
95. 95. 93 Problem Two finite sets have m and n elements. The total number of subsets of the first set is 56 more than the total number of subsets of the second set. The values of m and n are : a. 7, 6 b. 6, 3 c. 5, 1 d. 8, 7
96. 96. 94 Problem The circle x2 = y2 – 2ax – 2ay + a2 = 0 touches : a. x-axis only b. y-axis only c. both axis d. none of these
97. 97. 95 Problem If |A| = 2 and A is a 2 x 2 matrix, what is the value of det, {adj. {adj. (adj. A2)}} is equal to a. 4 b. 16 c. 64 d. 128
98. 98. 96 Problem If y = (4x - 5) is a tangent to the curve y2 = px3 + q at (2, 3), then : a. p = -2, q = -7 b. p = -2, q = 7 c. p = 2, q = -7 d. p = 2, q = 7
99. 99. 97 Problem given that x > 0, y > 0, x > y and z 0. the inequality which is not always correct is : a. x + z > y + z b. x – z > x – z c. xz > y > z x y  2 d. z 2 z
100. 100. 98 Problem Given that log10343 = 2.5353. The least n such that 7n > 105 is : a. 4 b. 3 c. n d. 6
101. 101. 99 Problem the smallest angle of the triangle whose sides are 6 12, 48, 24 is :  a. 4  b. 6 c.  3 d. none of these
102. 102. 100 Problem 1 p   If f’(x) = (1  cos x) for 0  x  2 and that f(0) = 3, then f 2   lies in the interval : a. 1   2 ,1      b. 4 , 2      c. 3, ,3    4  2    d. 3  ,3    4 2
103. 103. FOR SOLUTIONS VISIT WWW.VASISTA.NET