1. UPSEEPAST PAPERSMATHEMATICS - UNSOLVED PAPER 2007

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 If is perpendicular to b and c,| a | 2,| b | 3,| c | 4 and the angle between 2 b and c, is , then [a bc] is equal to 3 a. 43 b. 63 c. 12 3 3 d. 18 4. 02 Problem The general solution y2dx + (x2 xy + y2)dy = 0 is x a. tan-1y + log y + c = 0 x b. 2 tan-1+ log x + c = 0 y c. log (y + x2 y 2 ) + log y + c = 0 x d. sinh-1 y + log y + c = 0 5. 03 Problem The vector ixj 3k is rotated through an angle and doubled in magnitude, then it (4x 2) 4ibecomes j 2k.The value of x is 2 a.,2 31 b. 3,22 c. 3,0 d. {2, 7} 6. 04 Problem a ball of mass 3 kg moving with a velocity of 3 m/s collides with another ball of mass 1 kg moving with velocity u in the opposite direction. If the first ball2 comes to rest after the impact and e = , 7 then u is in m/s, is a. 13 3 b. 17 319 c.3 23 d. 3 7. 05 Problem Three forces of magnitude 30, 60 and P acting at a point are in equilibrium. If the angle between the first two is 600, the value of P is a. 30 7 b. 30 3 c. 206 d. 25 2 8. |a b| 06 ProblemLet be two equal vectors inclined at an angle , then a sin is equal to 2 |a b|a. 2 b.|a b| 2 c.|a b| d.|a b| 9. 07 Problem 2 The solution of the equation d ye 2x is 2dx 1 2x cx a. yed 42 b. y = e-2x + cx + d c. y = e-2x + cx2 + d d. y = e-2x + cx3 + d 10. 08 Problem dx is equal to2x4x 13 a. log (x2 + 4x + 13) + c1 1 x 2 b. tan c3 3 c. log (2x + 4) + c 2x 4 d.2c(x4 x 13)2 11. 09 Problem The value of 3x 1is dx2 x 2 ( x 1)161 a. log96161 b. log 961 c. 2 log 2-6 d. log 4 13 6 12. 10 Problem/4 5 /4 /4 (cos x sin x)dx(sin x cos x)dx(cos x sin x)dx is equal to0/4 2 a.2 -2 b. 2 2 - 2 c. 3 2 - 2 d. 4 2 - 2 13. 11 Problem The length of the chord of the parabola x2 = 4y passing through the vertex and having slope cot is a. 4 cos cosec2 b. 4 tan sec c. 4 sin sec2 d. none of these 14. 12 Problem x2 y2 Equation of tangents to the ellipse 1 , which are perpendicular to the 94 line 3x + 4y = 7, are a. 4x3y 20 b. 4x3y12 c.4x 3y 2 d.4x3y1 15. 13 Problem From the point P(16,7) tangents PQ and PR are drawn to the circle x2 + y2 2x 4y 20 = 0. If c be the centre of the circle, then area of quadrilateral PQCR is a. 450 sq unit b. 15 sq unit c. 50 sq unit d. 75 sq unit 16. 14 Problem The distance between the lines 3x + 4y = 9 and 6x + 8y = 15 is3 a. 2 3 b. 10 c. 6 d. none of these 17. 15 Problemb If tan x = , then the value a cos 2x + b sin 2x isa a. a b. a - b c. a + b d. b 18. 16 Problem The maximum value of is a. 3 b. 4 c. 5 d. none of these 19. 17 Problem In a triangle ABC, right angled at C, the value of cot A + cot B is a. c2ab b. a + ba2 c. bcb2 d. ac 20. 18 Problem The records of a hospital show that 10% of the cases of a certain disease are fatal. If 6 patients are suffering from the disease, then the probability that only three will die, is a. 8748 x 10-5 b. 1458 x 10-5 c. 1458 x 10-6 d. 41 x 10-6 21. 19 Problem Out of 40 consecutive natural numbers, two are chosen at random, Probability the that sum of the number is odd, is14 a. 2920 b. 391 c.2 d. none of these 22. 20 Problem If arg(z) = , then arg is equal to (z ) a. b. c. d. - 23. 21 Problemz 1 If z is a complex number such that z 1is purely imaginary, then |z| is equal to a. 0 b. 1 c. 2 d. none of these 24. 22 Problem If , are the roots of the equation tx2 + mx + n = 0, then the equation 1whose 1 tantan 3 3 roots are and , is a. l4x2 nl (m2 2nl)x + n4 = 0 b. l4x2 + nl (m2 2nl)x + n4 = 0 c. l4x2 + nl (m2 2nl)x - n4 = 0 d. l4x2 nl (m2 + 2nl)x + n4 = 0 25. 23 Problem 11 If the 7th term of a HP isand the 12th term is ,then the 20th term is1025 a. 1411 b. 451 c. 49 1 d.37 26. 24 Problem The value of 21/4. 41/8. 81/16 . 161/323 a. 25 b. 2 c. 2 d. 1 27. 25 Problem Out of 6 boys and 4 girls, a group of 7 is to be formed .In how many ways can this be done, if the group is to have a majority of boys ? a. 120 b. 80 c. 90 d. 100 28. 26 Problem1 1[2 1 -1] is equal to2 21 a. 2 2 1 1 21 1 b.4 22 c. [-1] d. not defined 29. 27 Problem The domain of the function 1x2 issin log2 2 a. [-1, 2] {0} b. [-2, 2] (-1, 1) c. [-2, 2] {0} d. [1, 2] 30. 28 Problem (2x 3)(3x 4) is equal to lim x (4x 5)(5x 6) 1 a. 10 b.01 c. 5 3 d. 10 31. 29 Problemx 1, x2 function is f (x)a continuous function 2 x 3,x2 a. for x = 2 only b. for all real values of x such that x 2 c. for all real value of x d. for all integral values of x only 32. 30 Problem Differential coefficient of sec x is 1 a. sec x sin x4 x 1 b.(sec x )3 / 2.sin x 4 x1 c. x sec x sin x21 d. x (sec x )3 / 2 sin x2 33. 31 Problem The function x5 5x4 + 5x3 - 1 is a. Neither maximum nor minimum at x = 0 b. Maximum at x = 0 c. Maximum at x = 1 and minimum at x = 3 d. Minimum at x = 0 34. 32 Problem dy If 1y2 , then dxis equal to a. X 1 y2 b. 1 2y 2 c.1 y21 2y 2 d. 0 35. 33 Problem If the planes x + 2y + kz = 0 and 2x + y 2z = 0, are at right angles, then the value of k is a. 2 b. -21 c. 21 d. - 2 36. 34 Problem The ratio in which the line joining (2, 4, 5), (3, 5, -4) is divided by the yz-plane is a. 2 : 3 b. 3 : 2 c. - 2 : 3 d. 4 : - 3 37. 35 Problem The radical centre of the circles x2 + y2 16x + 60 = 0, x2 + y2 12x + 27 = 0 and x2 + y2 12y + 8 = 0 is3313, a.433 b., 13 4 c. 33 ,13 4 d. none of these 38. 36 Problem If the lines 3x + 4y + 1 = 0, 5x + y + 3 = 0 and 2x + y 1 = 0 are concurrent, then is equal to a. - 8 b. 8 c. 4 d. -4 39. 37 Problem ax / 2 dx is equal toxa ax1 a. sin 1 axclog a1 b. tan 1 axclog a c. 2 a x ax c d. log (ax - 1) + c 40. 38 Problem The value of1 x4 1isdx 0 x2 11 a. 6 (3 - 4 )1 b. 6 (3 + 4)1 c. 6 (3 + 4 )1 d. 6 (3 - 4) 41. 39 Problem dy The solution of the differential equation dx= y tan x 2 sin x, is a. y sin x = c + sin 2x1 b. y cos x = c + 2sin 2x c. y cos x = c sin 2x1 d. y cos x = c + cos 2x2 42. 40 Problem If a 1 1 and b ,( 2, m) are two collinear vectors, then m is equal to a. 2 b. 4 c. 3 d. 0 43. 41 Problem A ball falls from rest from top of a tower. If the ball reaches the foot of the tower in 3 s, then height of tower is (take g = 10 m/s2) a. 45 m b. 50 m c. 40 m d. none of these 44. 42 Problem A particle is thrown vertically upwards with a velocity of 490 cm/s. It will return to this position after a. 1 s b. 0.5 s c. 2 s d. none of these 45. 43 Problem The resultant of two forces P and 2P . 3 If 1st forces is doubled and reversed. The resultant of forces is a. 2P 3 b. P3 c. 4P d. none of these 46. 44 Problem2 The value of 1 log 2 + (log2)(log2)3 + is2! 3! a. log 3 b. log 2 c. 1/2 d. None of these 47. 45 Problemx The maximum f ( x)on [-1, 1] is4 x x2 a. - 131 b. - 41 c. 5 d. 16 48. 46 Problemex dx is equal to(2 e x )(e x1) ex 1 a. log+c ex 2 ex 2 log b.ex 1 +cex1 c. ex2 +cex2 d. ex1+c 49. 47 Problem If the radius of a circle be increasing at a uniform rate of 2 cm/s. The area of increasing of area of circle, at the instant when the radius is 20 cm, is a. 70cm2/s b. 70 cm2/s c. 80cm2/s d. 80 cm2/s 50. 48 Problem 5 boys and 5 girls are sitting in a row randomly. The probability that boys and girls sit alternatively, is a.51261 b. 42 4 c. 1261 d.126 51. 49 Problem1 If P(A) = P(B) = x and P(A B) = P(A B) = , then x is equal to3 1 a. 21 b. 3 1 c. 41 d. 6 52. 50 Problem The focus of parabola y2 x 2y + 2 = 0 is1 a. ,04 b. (1, 2)5,1 c. 43 5 , d. 4 2 53. 51 Problem the equation of normal at the point (0, 3) of the ellipse 9x2 + 5y2 = 45 is a. x-axis b. y-axis c. y + 3 = 0 d. y 3 = 0 54. 52 Problem x2 y2 The equation of the tangent parallel to y x +5 = 0 drawn to 1 is 32 a. x y + 1 = 0 b. x y + 2 = 0 c. x + y - 1 = 0 d. x + y + 2 = 0 55. 53 Problem Let the functions f, g, h are defined from the set of real numbers R to R such that f(x) = x2 1, g (x) =(x2 1) and0, if x 0 h(x) = x, if x 0, then ho(fog)(x) is defined by a. x b. x2 c. 0 d. none of these 56. 54 Problem The angle of elevation of the sun, if the length of the shadow of a tower is times the height of the pole, is a. 1500 b. 300 c. 600 d. 450 57. 55 Problem If sin A = n sin B, then n 1 A B is equal totann 12 A B a. sin2A B b. tan 2 c. cot A B 2 d. none of these 58. 56 Problem Both the roots of the given equation (x - a) (x - b) + (x - b) (x - c) + (x - c) (x - a) = 0 are always a. Positive b. Negative c. Real d. Imaginary 59. 57 Problem 3 tan-1 a is equal to3a a3 a. tan-1 1 3a23a a3 b. tan-1 1 3a23a a3 c. tan-1 1 3a23a a3 d. tan-1 1 3a2 60. 58 Problem 1 2i In which quadrant of the complex plane, the pointlies ? 1 i a. Fourth b. First c. Second d. Third 61. 59 Problem The argument of the complex number 13 4 5i 9iis a.3 b.4 c.5 d. 6 62. 60 Problem If sin and cos are the roots of the equation px2 + qx + r = 0, then a. p2 + q2 2pr = 0 b. p2 - q2 + 2pr = 0 c. p2 - q2 2pr = 0 d. p2 + q2 + 2pr = 0 63. 61 Problem if a, b, c are in GP, then the equations ax2 + 2bx + c = 0 and dx2 + 2ex + f = 0 have a common root, ifd e fare in , ,a b c a. AP b. GP c. HP d. None of these 64. 62 Problem The sum of 24 terms of the following series 2 8 18 32 .... is a. 300 b. 300 2 c. 200 2 d. none of these 65. 63 Problem x = 0, the function f(x) = |x| is a. continuous but not differentiable b. discontinuous and differentiable c. discontinuous and not differentiable d. continuous and differentiable 66. 64 Problem12 In the expansion of2 1, the term independent of x is 2xx a. 8th b. 7th c. 9th d. 10th 67. 65 Problem The general value of in the equation cos 1 is, tan 12 a. 2n ,nI 6 7 b. 2n,n I4 c. n ( 1)n ,n I 3 d. n( 1)n ,nI 4 68. 66 Problem In a ABC, if r1 = 2r2 = 3r3, thena 4 a.b 5 b. a5b4 c. a + b 2c = 0 d. 2a = b + c 69. 67 Problem1 2 If A = , then A-1 is equal to3 552 a. 315 / 11 2 / 11 b. 3 / 111 / 115 / 11 2 / 11 c. 3 / 11 1 / 115 2 d. 31 70. 68 Problem The range of f(x) = cos x sin x is a. [-1, 1] b. (-1, 2) c. , 2 2 d. [- 2, 2] 71. 69 Problem the value of x2 bx 4 islimx x2 ax 5b a. a b. 0 c. 14 d. 5 72. 70 Problemsin x Letx 0 if f(x) is continuous at x = 0, then k is equalf (x) 5x,k,x 0 to a.5 b.5 c. 1 d. 0 73. 71 Problem If be the angle between the vectors a 2i 2 j k and b 6 3ij 2k , then a. 4cos 213 b. cos 19 2 c. cos195 d. cos21 74. 72 Problem Let a, b and c be vectors with magnitudes 3, 4 and 5 respectively and a b c 0 then ,the values of a b b c c a is a. 47 b. 25 c. 50 d. -25 75. 73 Problem A particle is moving in a straight line with uniform velocity. Its acceleration is a. Positive b. Negative c. Zero d. None of these 76. 74 Problem If a particle is projected with a velocity 49 m/s making an angle 600 with the horizontal, its time of flight is a. 10 3 b. 53 s c.3 s d. none of these 77. 75 Problem If two equal components (P) of a force (F) are acting at an angle 1200, then F is a. P b. P 2 c. 2PP d.2 78. FOR SOLUTIONS VISIT WWW.VASISTA.NET