1. AMU PAST PAPERSMATHEMATICS - UNSOLVED PAPER - 2007

2. SECTION I CRITICAL REASONING SKILLS 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. 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. 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. 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. 05Problema b 1 If loge2 2(loge a + loge b), then a. a = b b b. a =2 c. 2a = b d. a = b/3 8. 06Problem9 The number of real solutions the equation 10= -3 + x x2 is a. 0 b. 1 c. 2 d. none of these 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. 08 Problem (1+ i)8 + (1 - i)8 equal to a. 28 b. 25 c. 24 cos4 d. 28 cos8 11. 09 Problem The value of 3 cosec 200 sec 200 is a. 2 b. 4 c. - 4 d. none of these 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. 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. 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. 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 areandrespectively, then the distance between their tops when4 5 istan and tan3 2150 a. 3M b. 1003m c. 150 m d. 100 m 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. 15 Problem100100Cm (x - 3)100 m. 2m ism 0 a. 100C 47 b. 100C 53 c. -100C53 d. -100C100 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. 17 Problem If a i j, bk i 3j5k and c 7ij 9 11k , then the area of parallelogram having diagonals is a. 4 6sq unit1 b. 21sq unit2 c. 6 sq unit2 d.6sq unit 20. 18 Problem The centre of the circle given by r .( 2 2k ) ij15 and | r ( 2k ) |j4 is a. (0, 1, 2) b. (1, 3, 4) c. (-1, 3, 4) d. none of these 21. 19 Problem15 7 If A = 07 9 , then trace of matrix A is11 8 9 a. 17 b. 25 c. 3 d. 12 22. 20 Problem The value of the determinant cos sin1issincos 1cos() sin() 1 a. Independent of b. Independent of c. Independent ofand d. None of the above 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. 12 b. 594 c.9 d. 23 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. 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. 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. 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. 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. 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. 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. 29 Problemdx equalssin x cos x21x tan c a. 22 81 xtanc b. 2 281x c. cotc 2 2 81x cot c d. 22 8 32. 30 Problem2x 2 3x 1 1 x Ifdx 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 21 1 , d. 2 2 33. 31 Problemcosec4 x dx is equal tocot3 x a. cot x + 3+c tan3 x b. tan x + c 3cot3 x c. - cot x - 3 +c tan3 x d. - tan x - c 3 34. 32 Problem The value of integral 1 1 xis dx 0 1 x a. 2 +1 b. -12 c. - 1 d. 1 35. 33 Problem1 1 The value of I x x dx is0 21 a.31 b. 41 c.8 d. none of these 36. 34 Problem The slope of tangents drawn from a point (4, 10) to the parabola y2 = 9x are1 3 a.,4 41 9 b.,4 4 c. 1 1 ,4 3 d. none of these 37. 35 Problemx2 y2 The line x = at2 meets the ellipse 1 in the real points, iffa2 b2 a. | t | < 2 b. | t | 1 c. | t | > 1 d. none of these 38. 36 Problemx y The eccentricity of the ellipse which meets the straight line1onthe7 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, is3 2 a.72 6 b. 7 c.3 7 d. none of these 39. 37 Problem2 If x y2 (a > b) and x2 y2 = c2 cut at right angles, then 2 1a b2 a. a2 + b2 = 2c2 b. b2 - a2 = 2c2 c. a2 - b2 = 2c2 d. a2b2 = 2c2 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. 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. 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. 41 Problem 12 22 12 22 32 12 22 32 42 The sum of the series 1.... is2!3!4! a. 3e17 b. 6 e13 c.e 619 d.e 6 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 nn 1 c. ( 1) loge an d. ( 1)a log e an 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. 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 isx y z1 a. 1 2 3x y z b. 13 6 9x y z 1 c. 1 2 3 3 d. none of these 47. 45 Problem The projections of a directed line segment on the coordinate axes are 12, 4, 3, The DCs of the line are a. 124 3 , ,13 13 13124 3 b.,,13 13 1312 4 3 c. ,,13 13 13 d. None of these 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. 47 Problem Let a 2i jk, b i 2j k and a unit vector c be coplanar. If c is perpendicular to a , then c is equal to 1 a.( j k) 2 1 b.i j ( k) 3 1 c.(i 2) j 5 1 d.(i j k) 3 50. 48 Problem If a, b, care the position vectors of the vertices of an equilateral triangle whose orthocenter is at the origin, then a. a b c 0 b. a2b2 c2 c. a bc d. none of these 51. 49 Problem The points with position vectors 60i 3 40 j,i 8 ai j, 52 j are collinear, if a. a = - 40 b. a = 40 c. a = 20 d. none of these 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. 3sq unit d. none of these 53. 51 Problem1/ x The value of lim tan 1xis x2 a. 0 b. 1 c. - 1 d. e 54. 52 Problem If f(x) = mx1, x is continuous at x = , then22 sin x n, x2 a. m = l, n = 0 n b. m =1 2 c. n = m 2 d. m = n = 2 55. 53 Problem The domain of the function f ( x )4 x2 issin 1 (2 x) a. [0, 2] b. [0, 2) c. [1, 2) d. [1, 2] 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. 55 Problem3/22 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. 56 Problem1 3 1 1 The matrix A satisfying the equation 0 1A0 1is1 4 a. 1 01 4 b. 1014 c. 0 1 d. none of these 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. 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. 59 Problem dy2yx1 The solution of the differential equation is dx 1 x2 (1 x 2 )2 a. y (1 + x2) = c + tan-1 xy b.c + tan-1 x1x2 c. y log (1+ x2) = c + tan-1 x d. y (1+ x2) = c + sin-1 x 62. 60 Problem If x, y, z are all distinct and x x21 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. 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. 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. 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, is1 a. 4513 b. 9019 c.90 d. none of these 66. 64 Problem If 3p and 4p are resultant of a force 5p, then angle between 3p and 5p is1 3 a. sin5 b. 1 4sin5 c. 900 d. none of these 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. 307 km/h d. none of these 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. 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. 68 Problem If 2 tan-1 (cos x) = tan-1 (2 cosec x), then the value of x is3 a.4 b. 4 c. 3 d. none of these 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. 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. FOR SOLUTIONS VISIT WWW.VASISTA.NET