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# AMU - Mathematics - 2003

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• ### AMU - Mathematics - 2003

1. 1. AMU –PAST PAPERSMATHEMATICS - UNSOLVED PAPER - 2003
2. 2. SECTION – I CRITICAL REASONING SKILLS
3. 3. 01 Problem If var (x) = 8.25, var (y) = 33.96 and cov (x, y) = 10.2, then the correlation coefficient is : a. 0.89 b. - 0.98 c. 0.61 d. - 0.16
4. 4. 02 Problem If the standard deviation of a variable x is then the standard deviation of another variable ax b is : c ax b a. c a b. c c. d. none of these
5. 5. 03 Problem If x 15, y 36, xy 110, x 5 then cov(x,y) equals : a. 1/5 b. -1/5 c. 2/5 d. -2/5
6. 6. 04 Problem The two lines of regression are given by 3x + 2y = 26 and 6x + y = 31. the coefficient of correlation between x and y = is : a. - 1/3 b. 1/3 c. - 1/2 d. 1/2
7. 7. 05 Problem The A.M of 4, 7, y and 9 is 7. Then the value of y is : a. 8 b. 10 c. 28 d. 0
8. 8. 06 Problem The median from the table Value 7 8 10 9 11 12 13 Frequency 2 1 4 5 6 1 3 Is : a. 100 b. 10 c. 110 d. 1110
9. 9. 07 Problem The mode of the following series 3, 4, 2, 1, 7, 6, 7, 6, 8, 9, 5 is : a. 5 b. 6 c. 7 d. 8
10. 10. 08 Problem The mean-deviation and coefficient of mean deviation from the data. Weight (in kg) 54, 50, 40, 40, 42, 51, 45, 47, 55, 57 is : a. 0.0900 b. 0.0956 c. 0.0056 d. 0.0946
11. 11. 09 Problem The S.D, coefficient of S.D, and coefficient of variation from the given data Class interval 0-10 10-20 20-30 30-40 40-50 Frequency 2 10 8 4 6 is a. 50 b. 51.9 c. 47.9 d. 51.8
12. 12. 10 Problem There are n letters and n addressed envelopes, the probability that all the letters are not kept in the right envelope, is : 1 a. n! 1 b. 1 - n ! 1 c. 1 - n d. n!
13. 13. 11 Problem If A and B are two events such that A P( A) 0 and P(B) 1, then P B a. A 1 P B A 1 P b. B 1 P( A B) c. P(B) P( A) d. P(B)
14. 14. 12 Problem A coin is tossed 3 times, the probability of getting exactly two heads is : a. 3/8 b. 1/2 c. 1/4 d. none of these
15. 15. 13 Problem Six boys and six girls sit in a row. What is the probability that the boys and girls sit alternatively : a. 1/462 b. 1/924 c. 1/2 d. none of these
16. 16. 14 Problem 3 8 If A and B are two independent events such that P( A B ) and P( A B) 25 25 then P(A) is equal to : 1 a. 5 3 b. 8 2 c. 5 4 5 d.
17. 17. 15 Problem In a college of 300 students every student read 5 newspapers and every newspaper is read by 60 students. The number of newspaper is : a. At least 30 b. At most 20 c. Exactly 25 d. None of these
18. 18. 16 Problem 1 x is equal to : dx 1 x a. 1 1 sin x 1 x2 c 2 b. sin 1 x 1 1 x 2 c 2 c. sin 1 x 1 x2 c d. sin 1 x 1 x2 c
19. 19. 17 Problem x 1 is equal to : 3 ex d x (x 1) ex a. c (x 1)2 ex b. c (x 1)2 ex c c. (x 1)3 ex c d. (x 1)3
20. 20. 18 Problem 1 dx is equal to : x 2 (x 4 1)3 / 4 (x 4 1)1 / 4 a. c x (x 4 1)1 / 4 b. - c x 3 (x 4 1)3 / 4 c c. y x 4 (x 4 1)3 / 4 d. 3 c x
21. 21. 19 Problem If y = cos (sin x2) then at x = dy is equal to : , 2 dx a. - 2 b. 2 c. - 2 2 d. 0
22. 22. 20 Problem dy If y log x log x log x log x ... then dx is equal to : x a. 2y 1 x b. 2y 1 1 c. x(2y 1) 1 d. x(1 2y )
23. 23. 21 Problem If f(x + y) = f(x).f(y) for all x and y and f (5) = 2, f’(0) = 3 then f’(5) will be : a. 2 b. 4 c. 6 d. 8
24. 24. 22 Problem On the interval [0, 1] the function x25 (1 - x)75 takes its maximum value at the point : a. 9 b. 1/2 c. 1/3 d. 1/4
25. 25. 23 Problem If y = x log x 3 d2y is equal to : then x a bx dx 2 dy a. x dx –y 2 b. dy x y dx dy c. y -x dx 2 dy d. y x dx
26. 26. 24 Problem The first derivative of the function cos 1 sin 1 x xx with respect to x at x = 2 1 is : a. 3/4 b. 0 c. 1/2 d. - 1/2
27. 27. 25 Problem Function f(x) = 2x3 – 9x2 + 12x + 99 is monotonically decreasing when : a. x < 2 b. x > 2 c. x > 1 d. 1 < x < 2
28. 28. 26 Problem x x If f (x) and g(x) , where 0 < x 1, then in this interval : sin x tan x a. Both f(x) and g(x) and increasing function b. Both f(x) and g(x) and decreasing function c. f(x) is an increasing function d. g(x) is an increasing function
29. 29. 27 Problem 1 If x sin x 0 is equal to : f (x) x then lim f (x) x 0 0 x 0 a. 1 b. 0 c. - 1 d. none
30. 30. 28 Problem log(1 ax ) log(1 bx ) The function f (x) is not5 defined at x = 0, the value x of which should be assigned to f at x = 0, so that it is continuous at x = 0, is : a. a – b b. a + b c. log a + log b d. log a – log b
31. 31. 29 Problem Domain of the function f ( x) 2 2x x 2 is : a. 3 x 3 b. 1 3 x 1 3 c. 2 x 2 d. 2 3 x 2 3
32. 32. 30 Problem The function f(x) = [x] cos 2x 1 where [.] denotes the greatest integer 2 function, is discontinuous at : a. All x b. No x c. All integer points d. x which is not an integer
33. 33. 31 Problem x a2 2 sin x 0 x /4 If the function , is continuous in the f (x) x cot x b /4 x /2 b sin2x a cos 2x /2 x interval [0, ] then the value of (a, b) are : a. (-1, -1) b. (0, 0) c. (-1, 1) d. (1)
34. 34. 32 Problem a.{(b + c) x (a + b+ c)} is equal to : a. 0 b. [abc] c. [abc] + [bca] d. none of these
35. 35. 33 Problem The number of vectors of unit length perpendicular to vectors a = (1, 1, 0) and b = (0, 1, 1) is : a. Three b. One c. Two d.
36. 36. 34 Problem The points with position vectors 60i + 3j, 40i – 8j, ai – 52j are collinear, if a is equal to : a. - 40 b. 40 c. 20 d. -20
37. 37. 35 Problem If a = 4i + 6j and b = 3j + 4k, then the component of along b is : 18 a. (3 j 4k ) 10 3 b. 18 (3 j 4k ) 35 c. 18 (3 j 4k ) 3 d. 3j + 4k
38. 38. 36 Problem The unit vector perpendicular to the vectors 6i +2j + 3k and 3i – 6j – 2k is : 2i 3j 6k a. 7 2i 3j 6k b. 7 2i 3j 6k c. 7 2i 3j 6k d. 7
39. 39. 37 Problem A unit vector a makes an angle 4 with z-axis, if a + i + j as a unit vector, then a is equal to : i j k a. 2 2 2 i j k b. 2 2 2 c. i j k 2 2 2 i j k d. 2 2 2
40. 40. 38 Problem     If AO OB BO OC , then A, B, C from : a. Equilateral triangle b. Right angled triangle c. Isosceles triangle d. Line
41. 41. 39 Problem The area of a parallelogram whose adjacent sides are i – 2j + 3k and 2i + j – 4k, is : a. 5 3 b. 105 3 c. 10 6 d. 5 6
42. 42. 40 Problem The sum of the first n terms of the series 1 3 7 15 ... is : 2 4 8 16 a. 2n – n – 1 b. 1 – 2-n c. n + 2-n – 1 d. 2n – 1
43. 43. 41 Problem If the first and (2n -1)th terms of an A.P, G.P. and H.P. are equal and their nth terms are respectively a, b and c, then : a. a b c b. a c=b c. ac – b2 = 0 d. a and c both
44. 44. 42 Problem If a1, a2, a3 … an are in H.P., then a1a2 + a2a3 + ….an-1an will be equal to : a. a1an b. na1an c. (n -1)a1an d. none of these
45. 45. 43 Problem The sum of n terms of the following series 1 + (1 + x) + (1 + x + x2) + .. will be : 1 xn a. 1 x (x(1 x n ) b. 1 x n(1 x) x(1 xn) c. (1 x)2 d. none of these
46. 46. 44 Problem If Sn = nP + n (n -1)8, where Sn denotes the sum of the first n terms of an A.P. then the common difference is : a. P + Q b. 2P + 3Q c. 2Q d. Q
47. 47. 45 Problem For any odd integer n 1 n3 – (n -1)3 + ….+ (-1)n -1, 13 is equal to : 1 a. (n - 1)2 (2n -1) 2 b. 1 (n - 1)2 (2n -1) 4 c. 1 (n + 1)2 (2n -1) 2 1 d. 4 (n + 1)2 (2n -1)
48. 48. 46 Problem If P = (1, 0), Q = (1, 0) and R = (2, 0) are three given points then the locus of a point S satisfying the relation SQ2 + SR2 = 2SP2 is : a. A straight line parallel to x-axis b. A circle through origin c. A circle with centre at the origin d. A straight line parallel to y-axis
49. 49. 47 Problem If the vertices of a triangle be (2, 1), (5, 2) and (3, 4) then its circumcentre is : a. 13 9 , 2 2 13 9 b. , 4 4 9 13 c. , 4 4 d. None of this
50. 50. 48 Problem The line x + y = 4 divides the line joining the points (-1, 1) and (5, 7) in the ratio : a. 2 : 1 b. 1 : 2 c. 1 : 2 externally d. none of these
51. 51. 49 Problem Given the points A (0, 4) and B (0, - 4) then the equation of the locus of the point P (x, y) such that, | AP – BP | = 6, is : x2 y2 a. 1 7 9 x2 y2 b. 1 9 7 x2 y2 1 c. 7 9 y2 x2 1 d. 9 7
52. 52. 50 Problem If P (1, 2), Q(4, 6), R(5, 7) and S(a, b) are the vertices of a parallelogram PQRS, then : a. a = 2, b = 4 b. a = b, b = 4 c. a = 2, b = 3 d. a = 3, b = 5
53. 53. 51 Problem The incentre of the triangle formed by (0, 0), (5, 12), (16, 12) is : a. (7, 9) b. (9, 7) c. (-9, 7) d. (-7, 9)
54. 54. 52 Problem 1 4 20 The roots of the equation are : 1 2 5 0 1 2x 5x 2 a. - 1, - 2 b. - 1, 2 c. 1, -2 d. 1, 2
55. 55. 53 Problem x + ky – z = 0, 3x – ky – z = 0 and x – 3y + z = 0 has non-zero solution for k is equal to : a. - 1 b. 0 c. 1 d. 2
56. 56. 54 Problem x y z 9 If x y 5 then the value of (x, y, z) is : y z 7 a. (4, 3, 2) b. (3, 2, 4) c. (2, 3, 4) d. none
57. 57. 55 Problem If A is n x n matrix, then adj(adj A) = a. | A |n – 1 A b. | A |n – 2 A c. | A | n d. none
58. 58. 56 Problem a2 a 1 The value of determinant cos(nx ) cos(n 1)x cos(n 2)x is independent of sin(nx ) sin(n 1)x sin(n 2)x : a. n b. a c. x d. none of these
59. 59. 57 Problem 3 1 5 1 If X then X is equal to : 4 1 2 3 a. 3 4 14 13 3 4 b. 14 13 3 4 c. 14 13 3 4 d. 14 13
60. 60. 58 Problem 1 0 1 If matrix A = and its inverse is denoted by a11 a12 a13 then the 3 4 5 1 0 6 7 A a21 a22 a23 value of a23 is equal to : a31 a32 a33 a. 21/20 b. 1/5 c. -2/5 d. 2/5
61. 61. 59 Problem 1 2 5 The rank of the matrix 2 4 a 4 is : 1 2 a 1 a. 1 if a = 6 b. 2 if a = 1 c. 3 if a = 2 d. 4 if a = - 6
62. 62. 60 Problem cos sin + sin cos is equal to : cos sin sin cos cos cos 0 0 a. 0 0 1 0 b. 0 0 0 1 c. 1 0 1 0 d. 0 1
63. 63. 61 Problem The solution of differential equation (sin x + cos x)dy + (cos x – sinx)dx = 0 is : a. ex (sin x + cos x) + c = 0 b. ey (sin x + cos x) = c c. ey (cos x + sin x) = c d. ex (sin x - cos x) = c
64. 64. 62 Problem ab z is a group of all positive rationals under the operation* defined by a * b = 2 the identity of z is : a. 2 b. 1 c. 0 1 d. 2
65. 65. 63 Problem The average of 5 quantities is 6, the average of the them is 4, then the average of remaining two numbers is : a. 9 b. 6 c. 10 d. 5
66. 66. 64 Problem A man can swim down stream at 8 km/h and up stream at 2 km/h, then the man’s rate in still water and the speed of current is : a. (5, 3) b. (5, 4) c. (3, 5) d. (3, 3)
67. 67. 65 Problem Equation of curve through point (1, 0) which satisfies the differential equation (1 + y2)dx – xydy = 0 is : a. x2 + y2 = 4 b. x2 - y2 = 1 c. 2x2 + y2 = 4 d. none of these
68. 68. 66 Problem The order of the differential equation whose general solution is given by y (c1 c2 ) cos(x c3 ) c 4e x c5 , where, c1, c2, c3, c4 and c5 are arbitrary constants, is : a. 5 b. 6 c. 3 d. 2
69. 69. 67 Problem The equation of the base of an equilateral triangle is x + y = 2 and the vertex is (2, -1) the length of the side of the triangle is : a. 3 /2 / 2 /3 b. 2 c. 2 /3 3 /2 d.
70. 70. 68 Problem The distance between the line 3x + 4y = 9 and 6x + 8y = 15 is : a. 3/2 b. 3/10 c. 6 d. 0
71. 71. 69 Problem The area enclosed with in the curve | x | | + |y| = 1 is : a. 2 b. 1 c. 3 d. 2
72. 72. 70 Problem Co-ordinates of the foot of the perpendicular drawn from (0, 0) to the line joining (a cos , a sin ) and (a cos , a sin ) are : a b a. 2 2 a a b. (cos cos ), (sin sin ) 2 2 cos , sin c. 2 2 b 0, d. 2
73. 73. 71 Problem The lines a1x + b1y + c1 = 0 and z2x + b2y + c2 = 0 are perpendicular to each other if : a. a1b2 – b1a2 = 0 b. a1b2 + b1a2 = 0 2 2 c. a1 b2 b1 a2 0 d. a1b1 – b2a2 = 0
74. 74. 72 Problem The angle between the pair of straight lines. y 2 sin2 xy sin2 x 2 (cos2 1) 1, is : a. /3 b. /4 c. 2 /3 d. none of these
75. 75. 73 Problem If two circles (x - 1)2 + (y - 3)2 = r2 and x2 + y2 – 8x +2y + 8 = 0 intersect in two distinct point, then : a. 2 < r < 8 b. r = 2 c. r < 2 d. r > 2
76. 76. 74 Problem The equation of circle passing through the points (0, 0) (0, b) and (a, b) is : a. x2 + y2 + ax + by = 0 b. x2 + y2 - ax + by = 0 c. x2 + y2 - ax - by = 0 d. x2 + y2 + ax - by = 0
77. 77. 75 Problem Two circles S1 = x2 + y2 + 2g1x + 2f1y + c1 = 0 and S1 = x2 + y2 + 2g2x + 2f2y + c2 = 0 each other orthogonally, then : a. 2g1g2 + 2f1f2 = c1 + c2 b. 2g1g2 - 2f1f2 = c1 + c2 c. 2g1g2 + 2f1f2 = c1 - c2 d. 2g1g2 - 2f1f2 = c1 - c2
78. 78. 76 Problem The equation 13[(x -1)2 + (y - 2)2] = 3 (2x + 3y - 2)2 represents : a. Parabola b. Ellipse c. Hyperbola d. None of these
79. 79. 77 Problem Vertex of the parabola 9x2 – 6x + 36y + 9 = 0 is : a. (1/3, - 2/9) b. (-1/3, - 1/2) c. (- 1/3, 1/2) d. (1/3, 1/2)
80. 80. 78 Problem If the foci and vertices of an ellipse be ( 1,0) and ( 2,0) then the minor axis of the ellipse is : a. 2 5 b. 2 c. 4 d. 2 3
81. 81. 79 Problem the one which does not represent a hyperbola is : a. xy = 1 b. x2 – y2 = 5 c. (x - 1) (y – 3) = 0 d. x2 – y2 = 0
82. 82. 80 Problem Eccentricity of the conic 16x2 + 7y2 = 112 is : a. 3/ 7 b. 7/16 c. 3/4 d. 4/3
83. 83. 81 Problem If f and g are continuous function on [0, a] satisfying f(x) = f(a - x) and g(x) + g(a - a x) = 2, then f (x) g (x) = 0 a a. f (x) dx 0 0 b. f (x) dx a a c. 2 f (x) dx 0 d. none
84. 84. 82 Problem a The value of which satisfying sin x dx sin2 , [ (0, 2 )] /2 equal to : a. /2 b. 3 /2 c. 7 /6 d. all the above
85. 85. 83 Problem n The value of | sin x | dx is : 0 a. 2n + 1 + cos b. 2n +1 – cos c. 2n +1 d. 2n + cos
86. 86. 84 Problem 2 The value of | 2 sin x | dx [.] represents greatest integer function, is : a. - b. - 2 c. -5 /3 d. 5 /3
87. 87. 85 Problem By Simpsons rule, the value of is : a. 1.358 b. 1.958 c. 1.625 d. 1.458
88. 88. 86 Problem The value of x0 (the initial value of x) to get the solution in interval (0.5, 0.75) of the equation x3 – 5x + 3 = 0 by Newton Raphson method is : a. 0.5 b. 0.75 c. 0.625 d. 0.60
89. 89. 87 Problem By bisection method, the real root of the equation x3 – 9x + 1 = 0 lying between x = 2 and x = 4 is nearer to : a. 2.2 b. 2.75 c. 5.5 d. 4.0
90. 90. 88 Problem By false-poisitioning, the second approximation of a root of equation f(x) = 0 is (where, x0, x1 are initial and first approximations resp.) f (x0 ) a. x0 = f (x1 ) f (x0 ) b. x0 f (x1 ) x1f (x0 ) f (x1 ) f (x0 ) c. x0 f (x0 ) x1f (x1 ) f (x1 ) f (x0 ) x0 f (x1 ) x1f (x0 ) d. f (x1 ) f (x0 )
91. 91. 89 Problem Let f(0) = 1, f(1) = 2.72, then the Trapezoidal rule gives approximation value of 1 f (x) dx is : 0 a. 3.72 b. 1.86 c. 1.72 d. 0.86
92. 92. 90 Problem If and are imaginary cube roots of unity, then 4 4 1 a. 3 b. 0 c. 1 d. 2
93. 93. 91 Problem z 5i the complex number z = x + iy, which satisfy the equation 1 lies on : z 5i a. real axes b. the line y = 5 c. a circle passing through the origin d. none of these
94. 94. 92 Problem m If 1 i 1 , then the least integeral value of m is : 1 i a. 2 b. 4 c. 8 d. none of these
95. 95. 93 Problem If 1 , , 2 are the cube roots of unity, then (a + b +c 2)3 + (a + b 2 + c )3 is equal to a + b + c = 0 a. 27 abc b. 0 c. 3 abc d. none of these
96. 96. 94 Problem If is an imaginary cube root of unity, then (1 + w – w2)7 equals : a. 128 b. - 128 c. 128 2 d. - 128 2
97. 97. 95 Problem 13 The value of the (i n i n 1 ) where i 1, equals : n 1 a. i b. i – 1 c. - i d. 0
98. 98. 96 Problem The number of ways in which 6 rings can be worm on four fingers of one hand is : a. 46 b. 6c4 c. 64 d. 24
99. 99. 97 Problem Number greater than 1000 but not greater than 4000 which can be formed with the digits 1, 2, 3, 4, are : a. 350 b. 357 c. 450 d. 576
100. 100. 98 Problem How many can be formed from the letters of the word DOGMATIC if all the vowels remain together: a. 4140 b. 4320 c. 432 d. 43
101. 101. 99 Problem If A = sin2 + cos4 , then for all real values of : a. 1 A 2 3 b. A 1 4 13 A 1 c. 16 3 13 A d. 4 16
102. 102. 100 Problem In triangle ABC, if A = 450, B = 750, then a c 2 = a. 0 b. 1 c. b d. 2b
103. 103. FOR SOLUTIONS VISIT WWW.VASISTA.NET