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# AMU - Mathematics - 2007

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• ### AMU - Mathematics - 2007

1. 1. AMU –PAST PAPERSMATHEMATICS - UNSOLVED PAPER - 2007
2. 2. SECTION – I CRITICAL REASONING SKILLS
3. 3. 01 Problem The function f : R f :R R defined by f(x) = (x - 1) (x - 2) (x - 3) is a. One-one but not onto b. Onto but not one-one c. Both one-one and onto d. Neither one-one nor onto
4. 4. 02 Problem If R is an equivalence relation on a set A, then R-1 is a. Reflexive only b. Symmetric but not transitive c. Equivalence d. None of the above
5. 5. 03 Problem If the complex numbers z1,z2,z3 are in AP, then they lie on a a. A circle b. A parabola c. Line d. Ellipse
6. 6. 04 Problem Let a, b, c be in AP and |a| < 1, |b| < 1, |c| < 1. If x = 1 + a + a2 + ….. To , y = 1 + b + b2 + …...to , z = 1 + c + c2 + …… to , then x, y, z are in a. AP b. GP c. HP d. None of these
7. 7. 05 Problem a b 1 If loge 2 2 (loge a + loge b), then a. a = b b b. a = 2 c. 2a = b d. a = b/3
8. 8. 06 Problem 9 The number of real solutions the equation 10 = -3 + x – x2 is a. 0 b. 1 c. 2 d. none of these
9. 9. 07 Problem If f(x) = ax + b and g (x) = cx + d, then f{g(x)} = g{(x)} is equivalent to a. f(a) = g(c) b. f(b) = g(b) c. f(d) = g(b) d. f(c) = g(a)
10. 10. 08 Problem (1+ i)8 + (1 - i)8 equal to a. 28 b. 25 c. 24 cos 4 d. 28 cos 8
11. 11. 09 Problem The value of 3 cosec 200 – sec 200 is a. 2 b. 4 c. - 4 d. none of these
12. 12. 10 Problem If x, y, z are in HP, then log (x + z) + log (x – 2y + z) is equal to a. log (x - z) b. 2 log (x - z) c. 3 log (x - z) d. 4 log (x - z)
13. 13. 11 Problem The lines 2x – 3y – 5 = 0 and 3x – 4y = 7 are diameters of circle of area 154 sq unit, then the equation of the circle is a. x2 + y2 + 2x – 2y – 62 = 0 b. x2 + y2 + 2x –2y – 47 = 0 c. x2 + y2 - 2x + 2y – 47 = 0 d. x2 + y2 - 2x + 2y – 62 = 0
14. 14. 12 Problem Which of the following is a point on the common chord of the circle x2 + y2 + 2x – 3y + 6 = 0 ? x2 + y2 + x – 8y – 13 = 0 ? a. (1, -2) b. (1, 4) c. (1, 2) d. (1, -4)
15. 15. 13 Problem The angle of depressions of the top and the foot of a chimney as seen from the top of a second chimney, which is 150 m high and standing on the same level as the first are and respectively, then the distance between their tops when 4 5 is tan and tan 3 2 150 a. 3 M b. 100 3m c. 150 m d. 100 m
16. 16. 14 Problem If one root is square of the other root of the equation x2 + px + q = 0, then the relation between p and q is a. p3 – (3p - 1)q + q2 = 0 b. p3 – (3p + 1)q + q2 = 0 c. p3 + (3p - 1)q + q2 = 0 d. p3 + (3p + 1)q + q2 = 0
17. 17. 15 Problem 100 100 Cm (x - 3)100 – m. 2m is m 0 a. 100C 47 b. 100C 53 c. -100C53 d. -100C100
18. 18. 16 Problem If (-3,2) lies on the circle x2 + y2 + 2gx + 2fy + c = 0 which is concentric with the circle x2 + y2 + 6x + 8y – 5 = 0, then c is equal to a. 11 b. - 11 c. 24 d. 100
19. 19. 17 Problem    If a ˆ i ˆ j ˆ, b k ˆ i 3ˆ j ˆ 5k and c 7ˆ i j ˆ 9ˆ 11k , then the area of parallelogram having diagonals is a. 4 6 sq unit 1 b. 21 sq unit 2 c. 6 sq unit 2 d. 6 sq unit
20. 20. 18 Problem  ˆ  ˆ The centre of the circle given by r .(ˆ 2ˆ 2k ) i j 15 and | r ( ˆ 2k ) | j 4 is a. (0, 1, 2) b. (1, 3, 4) c. (-1, 3, 4) d. none of these
21. 21. 19 Problem 1 5 7 If A = 0 7 9 , then trace of matrix A is 11 8 9 a. 17 b. 25 c. 3 d. 12
22. 22. 20 Problem The value of the determinant cos sin 1 is sin cos 1 cos( ) sin( ) 1 a. Independent of b. Independent of c. Independent of and d. None of the above
23. 23. 21 Problem A committee of five is to be chosen from a group of 9 people. The probability that a certain married couple will either serve together or not at all, is a. 1 2 b. 5 9 4 c. 9 d. 2 3
24. 24. 22 Problem The maximum value of 4 sin2 x – 12 sin x + 7 is a. 25 b. 4 c. does not exit d. none of these
25. 25. 23 Problem If a point P(4, 3) is shifted by a distance unit parallel to the line y = x, then coordinates of P in new position are a. (5, 4) b. (5 + 2 ,4+ 2 ) c. (5 - 2 ,4- 2) d. none of the above
26. 26. 24 Problem A straight line through the point A (3, 4) is such that its intercept between the axis is bisected at A. Its equation is a. 3x – 4y + 7 = 0 b. 4x + 3y = 24 c. 3x + 4y = 25 d. x + y = 7
27. 27. 25 Problem If (- 4, 5)is one vertex and 7 x – y + 8 = 0 is one diagonal of a square, then the equation of second diagonal is a. x + 3y = 21 b. 2x – 3y = 7 c. x + 7y = 31 d. 2x + 3y = 21
28. 28. 26 Problem The equation 2x2 – 24xy + 11y2 = 0 represents a. Two parallel lines b. Two perpendicular lines c. Two lines passing through the origin d. A circle
29. 29. 27 Problem The tangent at (1, 7) to the curve x2 = y – 6 touches the circle x2 + y2 + 16x + 12y + c = 0 at a. (6, 7) b. (-6, 7) c. (6, -7) d. (-6, - 7)
30. 30. 28 Problem The equation of straight line through the intersection of the lines x – 2y = 1 and x + 3y = 2 and parallel to 3x + 4y = 0 is a. 3x + 4y + 5 = 0 b. 3x + 4y – 10 = 0 c. 3x + 4y – 5 = 0 d. 3x + 4y + 6 = 0
31. 31. 29 Problem dx equals sin x cos x 2 1 x tan c a. 2 2 8 1 x tan c b. 2 2 8 1 x c. cot c 2 2 8 1 x cot c d. 2 2 8
32. 32. 30 Problem 2x 2 3 x 1 1 x If dx a log b tan c , then value of a and b are (x 2 1)(x 2 4) x 1 2 a. (1, -1) b. (-1, 1) 1 1 , c. 2 2 1 1 , d. 2 2
33. 33. 31 Problem cosec4 x dx is equal to cot3 x a. cot x + 3 +c tan3 x b. tan x + c 3 cot3 x c. - cot x - 3 +c tan3 x d. - tan x - c 3
34. 34. 32 Problem The value of integral 1 1 x is dx 0 1 x a. 2 +1 b. -1 2 c. - 1 d. 1
35. 35. 33 Problem 1 1 The value of I x x dx is 0 2 1 a. 3 1 b. 4 1 c. 8 d. none of these
36. 36. 34 Problem The slope of tangents drawn from a point (4, 10) to the parabola y2 = 9x are 1 3 a. , 4 4 1 9 b. , 4 4 c. 1 1 , 4 3 d. none of these
37. 37. 35 Problem x2 y2 The line x = at2 meets the ellipse 1 in the real points, iff a2 b2 a. | t | < 2 b. | t | 1 c. | t | > 1 d. none of these
38. 38. 36 Problem x y The eccentricity of the ellipse which meets the straight line 1on the 7 2 x y axes of x and the straight line 1 on the axis of y and whose axes lie 3 5 along the axes of coordinates, is 3 2 a. 7 2 6 b. 7 c. 3 7 d. none of these
39. 39. 37 Problem 2 If x y2 (a > b) and x2 – y2 = c2 cut at right angles, then 2 1 a b2 a. a2 + b2 = 2c2 b. b2 - a2 = 2c2 c. a2 - b2 = 2c2 d. a2b2 = 2c2
40. 40. 38 Problem The equation of the conic with focus at (1, -1) directrix along x – y +1 = 0 and with eccentricity is a. x2 – y2 = 1 b. xy = 1 c. 2xy – 4x + 4y + 1 = 0 d. 2xy + 4x – 4y – 1 = 0
41. 41. 39 Problem The sum of all five digit numbers that can be formed using the digits 1, 2, 3, 4, 5 when repetition of digits is not allowed, is a. 366000 b. 660000 c. 360000 d. 3999960
42. 42. 40 Problem There are 5 letters and 5 different envelopes. The number of ways in which all the letters can be put in wrong envelope, is a. 119 b. 44 c. 59 d. 40
43. 43. 41 Problem 12 22 12 22 32 12 22 32 42 The sum of the series 1 .... is 2! 3! 4! a. 3e 17 b. 6 e 13 c. e 6 19 d. e 6
44. 44. 42 Problem The coefficient of xn in the expansion of loga(1 + x) is ( 1)n 1 a. n b. ( 1)n 1 log e a n n 1 c. ( 1) loge a n d. ( 1)a log e a n
45. 45. 43 Problem 46 n If the mean of n observation 12, 22, 32, …, n2 is , then n is equal to 11 a. 11 b. 12 c. 23 d. 22
46. 46. 44 Problem If a plane meets the coordinate axes at A, B and C in such a way that the centroid of ABC is at the point (1, 2, 3) the equation of the plane is x y z 1 a. 1 2 3 x y z b. 1 3 6 9 x y z 1 c. 1 2 3 3 d. none of these
47. 47. 45 Problem The projections of a directed line segment on the coordinate axes are 12, 4, 3, The DC’s of the line are a. 12 4 3 , , 13 13 13 12 4 3 b. , , 13 13 13 12 4 3 c. , , 13 13 13 d. None of these
48. 48. 46 Problem       The value of a (b c ) x (a b c) is   a. 2 [abc ]   b. [abc ] c. 0 d. none of these
49. 49. 47 Problem     Let a 2ˆ i ˆ j ˆ k, b ˆ i 2ˆ j ˆ k and a unit vector c be coplanar. If c is   perpendicular to a , then c is equal to 1 ˆ a. ( ˆ j k) 2 1 b. i j ˆ ( ˆ ˆ k) 3 1 ˆ c. (i 2ˆ) j 5 1 ˆ d. (i j ˆ ˆ k) 3
50. 50. 48 Problem    If a, b, c are the position vectors of the vertices of an equilateral triangle whose orthocenter is at the origin, then    a. a b c  0    b. a2 b2 c2    c. a b c d. none of these
51. 51. 49 Problem The points with position vectors 60ˆ i 3ˆ 40ˆ j, i 8 ˆ ai j, ˆ 52 ˆ j are collinear, if a. a = - 40 b. a = 40 c. a = 20 d. none of these
52. 52. 50 Problem Area lying in the first quadrant 3y and bounded by the circle x2 + y2 = 4, the line x = and x-axis is a. sq unit b. 2 sq unit c. 3 sq unit d. none of these
53. 53. 51 Problem 1/ x The value of lim tan 1 x is x 2 a. 0 b. 1 c. - 1 d. e
54. 54. 52 Problem If f(x) = mx 1, x is continuous at x = , then 2 2 sin x n, x 2 a. m = l, n = 0 n b. m = 1 2 c. n = m 2 d. m = n = 2
55. 55. 53 Problem The domain of the function f ( x ) 4 x2 is sin 1 (2 x) a. [0, 2] b. [0, 2) c. [1, 2) d. [1, 2]
56. 56. 54 Problem The general solution of the differential equation (1 + y2)dx + (1 + x2)dy = 0 is a. x – y = c (1 - xy) b. x – y = c (1 + xy) c. x + y = c (1 - xy) d. x + y = c (1 + xy)
57. 57. 55 Problem 3/2 2 The order and degree of the differential equation dy are 1 dx respectively d2y dx 2 a. 2, 2 b. 2, 3 c. 2, 1 d. none of these
58. 58. 56 Problem 1 3 1 1 The matrix A satisfying the equation 0 1 A 0 1 is 1 4 a. 1 0 1 4 b. 1 0 1 4 c. 0 1 d. none of these
59. 59. 57 Problem The relation R defined on the set of natural numbers as {(a, b) : a differs from b by 3} is given a. {(1, 4), (2, 5), (3, 6), ….} b. {(4, 1), (5, 2), (6, 3), … } c. {(1, 3), (2, 6), (3, 9), ….} d. none of the above
60. 60. 58 Problem dy dx h The solution of dx by k represents a parabola when a. a = 0, b = 0 b. a = 1, b = 2 c. a = 0, b 0 d. a = 2, b = 1
61. 61. 59 Problem dy 2yx 1 The solution of the differential equation is dx 1 x2 (1 x 2 )2 a. y (1 + x2) = c + tan-1 x y b. c + tan-1 x 1 x2 c. y log (1+ x2) = c + tan-1 x d. y (1+ x2) = c + sin-1 x
62. 62. 60 Problem If x, y, z are all distinct and x x2 1 x3 = 0, then the value of xyz is 2 3 y y 1 y 2 z z 1 z3 a. - 2 b. - 1 c. - 3 d. none of these
63. 63. 61 Problem The probability that at least one of the events A and B occurs is 0.6. If A and B occur simultaneously with probability 0.2, then P( A) P(B) is a. 0.4 b. 0.8 c. 1.2 d. 1.4
64. 64. 62 Problem If A and B are two events such that P(A) > 0 and P(B) 1, then P( A / B) is equal to a. 1- P (A/ B ) b. 1- P( A /B) 1 P( A B) c. P(B) P( A) d. P(B)
65. 65. 63 Problem A letter is taken out at random from ‘ASSISTANT’ and another is taken out from ‘STATISTICS’. The probability that they are the same letters, is 1 a. 45 13 b. 90 19 c. 90 d. none of these
66. 66. 64 Problem If 3p and 4p are resultant of a force 5p, then angle between 3p and 5p is 1 3 a. sin 5 b. 1 4 sin 5 c. 900 d. none of these
67. 67. 65 Problem Resultant velocity of two velocities 30 km/h and 60 km/h making an angle 600 with each other is a. 90 km/h b. 30 km/h c. 30 7 km/h d. none of these
68. 68. 66 Problem A ball falls of from rest from top of a tower. If the ball reaches the foot of the tower is 3s, then height of tower is (take g = 10 m/s2) a. 45 m b. 50 m c. 40 m d. none of these
69. 69. 67 Problem Two trains A and B 100 km apart are traveling towards each other with starting speeds of 50 km/h. The train A is accelerating at 18 km/h2 and B deaccelerating at 18 km/h2. The distance where the engines cross each other from the initial position of A is a. 50 km b. 68 km c. 32 km d. 59 km
70. 70. 68 Problem If 2 tan-1 (cos x) = tan-1 (2 cosec x), then the value of x is 3 a. 4 b. 4 c. 3 d. none of these
71. 71. 69 Problem Let a be any element in a Boolean algebra B. If a + x = 1 and ax = 0, then a. x = 1 b. x = 0 c. x = a d. x = a’
72. 72. 70 Problem Dual of (x + y) . (x + 1) = x + x . y + y is a. (x .y) + (x . 0) = x . (x + y) .y b. (x .y) + (x .1) = x . (x + y) .y c. (x .y) (x .0) = x . (x + y) .y d. none of these
73. 73. FOR SOLUTIONS VISIT WWW.VASISTA.NET