1. UPSEEPAST PAPERMATHEMATICS- UNSOLVED PAPER - 2004

2. SECTION I Single Correct Answer Type There are five parts in this question. Four choices are given for each part and one of them iscorrect. Indicate you choice of the correct answer for each part in your answer-book bywriting the letter (a), (b), (c) or (d) whichever is appropriate 3. 01 Problem 200 j The coefficient of x100 in the expansion of 1 x is : j 0200 a. 100201 b. 102200 c. 101201 d. 100 4. 02 Problem The circles x2 + y2 10x + 16 = 0 and x2 + y2 = r2 intersect each other at two distinct points, if : a. r < 2 b. r > 8 c. 2 < r < 8 d. 2 r 8 5. 03 Problem Three numbers are in AP such that their sum is 18 and sum of their equares is 158. The greatest number among them is : a. 10 b. 11 c. 12 d. none of these 6. 04 Problem Let be three vectors. Then, scalar triple product [a b c] is equal to :a, b and c a. [ b a c] b. [a c b] c. [ c b a] d. [b c a] 7. 05 Problem The roots of the equation x4 2x3 + x = 380 are : a. 5, - 4,1 5 321 53 b. -5, 4, -2 1 5 3 c. 5, 4,21 53 d. -5, -4, 2 8. 06 Problem Let y x ....dy is equal to :xx, then dx a. yxy-1y2 b. x 1 y log xy c.x 1 y log x d. none of these 9. 07 Problem31tan xis equal to :dx1 x2 a. 3 (tan-1 x)2 + c4 b.tan 1 x c4 c. (tan-1 x)4 + c d. none of these 10. 08 Problem X speaks truth in 60% and Y in 50% of the cases. The probability that they contradict each other narrating the same incident is : a. b. 1/3 c. d. 2/3 11. 09 Problem A set contains 2n + 1 elements. The number of subsets of this set containing more than n elements is equal to : a. 2n-1 b. 2n c. 2n+1 d. 22n 12. 1310 Problem The area between the parabola y = x2 and the line y = x is :1 a. 6sq unit1 b. sq unit31 c. 2sq unit d. none of these 13. 11 Problem The eccentricity of the hyperbola 5x2 4y2 + 20 x + 8y = 4 is : a. 23 b.2 c. 2 d. 3 14. 12 Problem e x esin x is equal to : lim x 0 x sin x a. -1 b. 0 c. 1 d. none of these 15. 13 Problemx dx is equal to : 0 1 sin x a. - b. /2 c. d. none of these 16. 14 Problem A man of mass 80 kg is traveling in a lift. The reaction between the floor of the lift and the man when the lift is accelerating upwards at 4 m/s2 and the acceleration due to gravity g = 9.81 m/s2, is equal to : a. 884.8 N b. 784.8 N c. 464 N d. 1104.8 N 17. 15 Problem The argument of 1 i 3 / 1 i 3 is : a. 600 b. 1200 c. 2100 d. 2400 18. 16 Problem The points z1, z2, z3, z4 in a complex plane are vertices of a parallelogram taken in order, then : a. z1 + z4 = z2 + z3 b. z1 + z3 = z2 + z4 c. z1 + z2 = z3 + z4 d. none of these 19. 17 Problem 2 The harmonic mean between two numbers is 14 5 and the geometric mean is 24. The greater number between them is : a. 72 b. 54 c. 36 d. none of these 20. 18 Problem The angle between two forces each equal to P when their resultant is also equal to P is : a. 2/3 b. /3 c. d. /2 21. 19 Problem The solution of the differential equation sec2 x and y dx + sec2y tan x dy = 0 is : a. tan y tan x = ctan y b. tan xctan2 x c.c tan y d. none of these 22. 20 Problem The real roots of the equation x2/3 + x1/3 2 = 0 are : a. 1, 8 b. - 1, -8 c. - 1, 8 d. 1, -8 23. 21 Problem Let f (x) = g(x) = ex. Then, (gof)(0) is : a. 1 b. -1 c. 0 d. none of these 24. 22 Problem Cosine of the angle between two diagonals of a cube is equal to :2 a. 6 b. 131 c.2 d. none of these 25. 23 Problem In a certain population 10% of the people are rich, 5% are famous and 3% are rich and famous. The probability that a person picked at random from the population is either famous or rich but not both, is equal to : a. 0.7 b. 0.08 c. 0.09 d. 0.12 26. 24 Problem Three numbers are in GP such that their sum is 38 and their product is 1728. The greatest number among them is : a. 18 b. 16 c. 14 d. none of these 27. 25 Problem The equation of the circle touching x = 0, y = 0 and x = 4 is : a. x2 + y2 4x 4y + 16 = 0 b. x2 + y2 8x 8y + 16 = 0 c. x2 + y2 + 4x + 4y - 4 = 0 d. x2 + y2 4x 4y + 4 = 0 28. 26 Problem1, when x is rational Let f(x) = f x then lim f (x ) is :0, when x is irraitonal x 0 a. 0 b. 1 c. 1/2 d. none of these 29. 27 Problem | a | 4,| b | 4,|c | 2 and a, b and c are three vectors with magnitude such that a is perpendicular a b c ,b is perpendicular to (c a) and c is perpendicular to (a b) . It follows that |ab c | is equal to : a. 9 b. 6 c. 5 d. 4 30. 28 Problem Let z1 and z2 be complex numbers, then |z1 + z2|2 + |z1 z2|2 is equal to : a. |z1|2 + |z2|2 b. 2 (|z1|2 + |z2|2) c. 2(z12 + z22) d. 4z1z2 31. 29 Problem2 If tan tan3tan 3 3, then : tan 2 = 1 tan 3 = 1 tan2 = 1 tan3 = 1 32. 30 Problem d2 y Let y = t10 + 1 and x = t8 + 1, then 2 is equal to : dx5 a. 2 t b. 20t8 5 c.16t 6 d. none of these 33. 31 Problem The vectorsABij 3 5 4k and ACij 5 5 2k are the side of a triangle ABC. The length of the median through A is : a. 13 unit b. 2 5 unit c. 5 unit d. 10 unit 34. 32 Problem If a, b, c are three non-coplanar vectors, then (a b c).[(a b) x(a c)] is a. 0 b. 2 [a b c] c. [a b c] d. [a b c] 35. 33 Problemdxis equal to :x( x 5 1)15 5 a. 5 log x (x 1) c1 x5 1logc b. 5x51 x5 c. log 5c5 x1 d. none of the above 36. 34 Problem A function f on R into itself is continuous at a point a in R, iff for each > 0, there exists,> 0 such that : a. | f (x) f (a) ||xa| b. | f (x) f (a) | |xa| c. | xa| | f (x) f (a) | d. | xa| | f (x)f (a) | 37. 35 Problemx sin xis equal to :dx1 cos xx a. x tan +c2x b. x sec2 2 + cx c. log cos2 d. none of these 38. 36 Problem A straight line through the point (1, 1) meets the x-axis at A and the y-axis at B. The locus of the mid point of AB is : a. 2xy + x + y = 0 b. x + y 2xy = 0 c. x + y + 2 = 0 d. x + y 2 = 0 39. 37 Problem2 4 5 If A 4 810, then rank of A is equal to :6 1215 a. 0 b. 1 c. 2 d. 3 40. 38 Problem A bag contains 8 red and 7 black balls. Two balls are drawn at random. The probability that both the balls are of the same colour, is :14 a.1511 b. 15 c.715 4 d.15 41. 39 Problem2 x2y2 1 If sin, then x must be : 2x a. - 3 b. - 2 c. 1 d. none of these 42. 40 Problem The solution of equation cos2 + sin + 1 = 0 lies in the interval : a. , 4 4 b. 3 ,4 4 c. 3 , 5445 7 d. , 4 4 43. 41 Problem Coefficient of x19 in the polynomial (x 1) (x - 2) ..(x - 20) is equal to : a. 210 b. - 210 c. 20! d. None of these 44. 42 Problem Two pillars of equal height stand on either side of a road way which is 60 m wide. The a point in the road way between the pillars, the elevation of the top of pillars are 600 and 300. The height of the pillars is : a. 15 3m15 b.3 m c. 15 m d. 20 m 45. 43 Problem A light string passing over a light smooth pulley carries masses of 3 kg and 5kg at its ends. If the string is allowed to move from the rest, the acceleration of the motion is equal to : a. (g/2)m/s2 b. (g/4)m/s2 c. 2g m/s2 d. 4g m/s2 46. 44 Problem The equation of the directrix of the parabola x2 + 8y 2x = 7 is : a. y = 3 b. y = -3 c. y = 2 d. y = 0 47. 45 Problem If iz4 + 1 = 0, the z can take the value :1 i a. 2 b. cosi sin 8 81 c. 4i d. i 48. 46 Problem If a i j k, bij 2 3 k, and c i j are coplanar vectors, the value of a is : a. - 433 b. 44 c. 3 d. 2 49. 47 Problem The equation of the tangent parallel to y x + 5 = 0 drawn to a. x y 1= 0 b. x y + 2 = 0 c. x + y 1 = 0 d. x + y + 2 = 0 50. 48 Problem The equation y2 x2 + 2x 1 = 0 represents : a. A hyperbola b. An ellipse c. A pair of straight lines d. A rectangular hyperbola 51. 49 Problem The minimum value of 3 sin + 4 cos is : a. 5 b. 1 c. 3 d. - 5 52. 50 Problem A man in swimming with the uniform velocity of 6 km/h straight across a river which is flowing at the rate of 2 km/h. If the breadth of the river is 300 m, the distance between the point and the man is initially directed to and the point it will reach on the opposite bank of the river is equal to : a. 100 m b. 200 m c. 300 m d. 400 m 53. 51 Problem A ball is thrown vertically upwards from the ground with velocity 15 m/s and rebounds from the ground with one-third of its striking velocity. The ratio of its greatest heights before and after striking the ground is equal to : a. 4 : 1 b. 9 : 1 c. 5 : 1 d. 3 : 1 54. 52 Problem If the position vectors of the vertices A, B, C of a triangle ABC arej i j 7 10k, 6 6k and ij4 9 6krespectively, the triangle is : a. Equilateral b. Isosceles c. Scalene d. Right angled and isosceles also 55. 53 Problem The number of solutions of the equation 2 cos (ex) = 5x + 5-x, are : a. No solution b. One solution c. Two solution d. Infinitely many solutions 56. 54 Problem Probability of throwing 16 in one throw with three dice is ; a.136 1 b.18 c. 1721 d.9 57. 55 Problem The differential equation of all straight lines passing through origin is : dy a. yx dxdy b. dx =y+xdy c. dx =yx d. Nome of these 58. 56 Problem Ifa, b, c are three unit vectors such that a b c 0 where 0 is null vector, thenb c c aa b is : a. - 3 b. - 23 c. - 2 d. 0 59. 57 Problem The expression equal to : a. -1 b. 0 c. 1 d. none of these 60. 58 Problem is equal to : a.4 b.6 c. 32 d. 3 61. 59 Problem If f(x) = (a - xn)1/n, where a > 0 and n N, then f0f(x) is equal to : a. a b. x c. xn d. an 62. 60 Problem The number of reflexive relations of a set with four elements is equal to : a. 216 b. 212 c. 28 d. 24 63. 61 Problem The maximum horizontal range of a ball projected with a velocity of 39.2 m/s is (take g = 9.8 m/s2) a. 100 m b. 127 m c. 157 m d. 177 m 64. 62 Problem Maximum value of sin x cos x is equal to : a. 2 b. 1 c. 0 d. none of these 65. 63 Problem The equation of the bisector of the acute angles between the lines 3x 4y + 7 = 0 and 12x + 5y -2 = 0 is : a. 99x 27y 81 = 0 b. 11x -3y + 9 = 0 c. 21x + 77y 101 = 0 d. 21 x + 77y + 101 = 0 66. 64 Problem To reduce the differential equation + P(x)y = Q(x).yn to linear form, the subsitution is a. v = b. v = c. v = yn d. v = yn 1 67. 65 Problem A particle possess two velocities simultaneously at an angle of tan-1 to each other. Their resultant is 15 m/s. If one velocity is 13 m/s, then the other will be : a. 5 m/s b. 4 m/s c. 12 m/s d. 13 m/s 68. 66 Problem If in the expansion of (1 + x)21, the coefficients of xr and xr + 1 be equal, then r is equal to : a. 9 b. 10 c. 11 d. 12 69. 67 Problem A train is running at 5 m/s and a man jumps out of it with a velocity 10 m/s in a direction making an angle of 600 with the direction of the train. The velocity of the man relative to the ground is equal to : a. 12.24 m/s b. 11.25 m/s c. 14.23 m/s d. 13.23 m/s 70. 68 Problem A ball is projected vertically upward with a velocity 112 m/s. The time taken by it to return to the point of projection is (g = 10 m/s2) : a. 11 s b. 33 s c. 5.5 s d. 22 s 71. 69 Problem If the sides of triangle are 4, 5 and 6 cm, then area of the triangle is equal to :15 a.4 cm215 b. 7 cm2 4 4 c.7 cm215 d. none of these 72. 70 Problem The volume of a spherical cap of height h cut off from a sphere of radius a is equal to : a. 3 h2 (3a - h) b. (a - h)(2a2 h2 - ah)4 c. h33 d. none of the above 73. 71 Problem The eccentricity of the hyperbola conjugate to x2 3y2 = 2x + 8 is :2 a. 3 b. 3 c. 2 d. none of these 74. 72 Problem The area of the parallelogram whose adjacent sides are is : a. 2 b. 4 c. 17 d. 2 13 75. 73 Problem Integrating factor of the differential equation dyis :P(x)y Q(x) dx a. P dx b.Q dx c.P dxe d. eQ dx 76. 74 Problem Angle of intersection of the curves r = sin + cos and r = 2 sin is equal to : a. 2 b. 3 c.4 d. none of these 77. 75 Problem1 Define f on R into itself by x sin , when x 0, then :f (x) x 0, when x 0 a. f is continuous at 0 but not differentiable at 0 b. f is both continuous and differentiable at 0 c. f is differentiable but not continuous at 0 d. none of the above 78. FOR SOLUTIONS VISIT WWW.VASISTA.NET