1. UPSEEPAST PAPERSMATHEMATICS - UNSOLVED PAPER 2001

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 A man swims at a speed of 5 km/h. He wants to cross a canal of 120 m wide, in a direction perpendicular to the direction of flow. If the canal flows at 4 km/h, the direction and the time taken by the man to cross the canal are :1 1tan , 24 min2 a.1 3tan,144 s 4 b.1 1 tan ,100 s 2 c. d. none of these 4. 02 Problem y ey ey ..... If x ey e , thendydx is equal to :1 a.x1 x b.x c. x1 x d. None of these 5. 03 Problem The acceleration of a particle moving in a straight line, a time t is (2t + 1) m/s2. If 4 m/s is the initial velocity of the particle, then its velocity after 2 s is : a. 4 m/s b. 8 m/s c. 10 m/s d. none of these 6. 04 Problema bc 2a 2a If a + b + c = 0, then determinant 2b b ca 2b is equal to :2c 2c c ab a. 0 b. 1 c. 2 d. 3 7. 05 Problem A body of weight 40 kg rests on a rough horizontal plane, whose coefficient of friction is 0.25. The least force which is acting horizontally would move the body of : a. 40 kg wt b. 20 kg wt c. 35 kg wt d. 10 kg wt 8. 06 Problem If log alog log c then aabbcc is equal to :b c c aa b a. -1 b. 1 c. 2 d. none of these 9. 07 Problem The co-ordinates of a point on the parabola y2 = 8x whose focal distance is 4, is : a. (2, 4) b. (4, 2) c. (4, -2) d. (2, 4) 10. 08 Problem Three letters are written to different persons and addresses on three envelopes are also written. Without looking at the addresses, the probability that the letters to into the right envelope is :2 a.31 b.281 c. 271 d. 9 11. 09 Problem The subtangent, ordinate and subnormal to the parabola y2 = 4ax at a point different from the origin are in : a. GP b. AP c. HP d. None of these 12. 10Problem If a 3 i j 2k and b 2 i j k , then a x (a b) is equal to a. 3 a b. 0 c. 3 14 d. none of these 13. 11 Problemtan x Evaluate dx :sin x cos x a. cot x c b. 2 cot x c c. tan x c d. 2 tan x c 14. 12 Problem Two cars start off to race with velocities u, u and move with uniform acceleration f, f; the result being a dead heat. The time taken by cars is :5 ff a. ufuf u u b. 2ff 5 ff c.uf uf u u d. 2ff 15. 13 Problem The difference between the greatest and least values of the functionx xt 1 dt on [2, 3] is : 0 a. 3 b. 27 c. 211 d. 2 16. 14 Problem If the circle x2 + y2 = a2 intersects the hyperbola xy = c2 in four points P(x1, y1), Q(x2, y2), R(x3, y3), S(x4, y4) then : a. x1 + x2 + x3 + x4 = 1 b. y1 + y2 + y3 + y4 = 0 c. x1x2x3x4 = c3 d. y1y2y3y4 = c3 17. 15 Problem The vector moment about the point i 2j 3kof the resultant of the force i 2j5k and 3j4k acting at the point is : 2i 3 j k5 i j 4k a. b. 3 i j 4k c. 5 i j 4k d. none of these 18. 16 Problem x dx is equal to :1 x4 a. tan-1 x2 + c b. log (1 + x4) + c1 c. 2tan-1 x2 + c d. none of these 19. 17 Problem If the intercept made on the line y = mx by lines y = 2 and y = 6 is less than by 5, then the range of the value of m is : 4 4, a.3 3 44 b.,, 33 c.3 3, 4 4 d. none of the above 20. 18 Problem The equation of the tangent to the curve y = e-|x| at the point where the curve cuts the line x = 1 is : a. e(x + y) = 1 b. y + 2x = 1 c. y + x = e d. none of these 21. 19 Problem Let z = 1 t + i t2 t 2, where t is a real parameter. The locus of z in the argand plane is : a. an ellipse b. a hyperbola c. a straight line d. none of these 22. 20 Problem Let z = 1- t + i t2 t 2, where t is a real parameter. The locus of z in the argand plane is : a. An ellipse b. A hyperbola c. A straight line d. None of these 23. 21 Problem Differential equation for y A cos x B sin x where A and B are arbitrary constants, is : a. d2y 2y0dx 2 b. d2y y 0dx 2d2y c. y0dx 2d2y2 d.y 0dx 2 24. 22 Problem A billiard ball collides directly with another ball of same mass having in rest. If the coefficient of restitution is e, then ratio of their velocities will be : a. 2 e : 2 + e b. 1 e : 1 + e c. 1 e2 : 1 + e2e e d. :1 e 1 e 25. 23 Problem From the gun cartridge of mass M, a fire arm of mass m with velocity u relative to gun cartridge is fired. The real velocities of fire arms and gun cartridge will be respectively : Mm mu, a. M m M mM m M m b. , Mu Mu c. u Mm u M m , M M d. none of the above 26. 24 Problem Is the equation (ab + ca + bc) sin = 2(a2 + b2 + c2) possible for real values of a, b, c? a. Possible b. Not possible c. Insufficient data d. None of these 27. 25 Problem The equation 3 sin2 x + 10 cos x 6 = 0 is satisfied, if : 1 1 a. x ncos 3 1 1 b. x 2n cos 3 1 1 c. x ncos 3 1 1 d. x 2n cos 6 28. 26 Problem A train whose mass is 6 metric tons, moves at the rate of 72 km/h. After applying brakes at stops at a distance of 500 m. What is the force exerted by brakes, obtaining it to be uniform ? a. 800 N b. 1600 N c. 3200 N d. 6400 N 29. 27 Problem Six girls are entering in a dance room with 10 boys to form a circle so that every girl is in between two boys, then the probability of doing so, such that two specified boy remains together, is : a.415 b.715 c.215 d. none of these 30. 28 Problem 2 n 1 If 1, , 2 ,......, n 1are the n roots of unity, then : 1 1 ..... 1 equals : a. 0 b. 1 c. n d. n2 31. 29 Problem The number of common tangent to the circles (x + 1)2 + (y + 4)2 = 40 and (x - 2)2 + (y- 5)2 = 10 are : a. 1 b. 2 c. 3 d. 4 32. 30 Problem If f(x) = xx, then f(x) is decreasing in interval : a. ] 0, e[1 b. ]0, [e c. ]0, 1[ d. none of these 33. 31 Problem The angle between the vectors 2i 3 j k and 2i j k is : a. 2 b.4 c.3 d. 0 34. 32 Problem A man falls vertically under gravity with a box of mass m on his head then the reaction force is : a. mg b. 2 mg c. zero d. 1.5 mg 35. 33 Problem100 The value of [ x ] dx is equal to : (where [.] is the greatest integer)0 a. 400 b. 600 c. 415 d. 615 36. 34 Problem 5 9 13 .....n terms 17 If , then n is equal to :7 9 11 .....(n 1)terms 16 a. 7 b. 12 c. 8 d. none of these 37. 35 Problem1 aa2 The value of the determinant cos n 1 x cos nx cos n1 xis zero, if :sin n 1 x sin nx sin n1 x a. sin x = 0 b. cos x = 0 c. a = 01a2 d. cos x2a 38. 36 Problem If in a triangle ABC, sin A, sin B, sin C are in AP, then : a. The altitudes are in AP b. The altitudes are in HP c. The altitudes are in GP d. None of the above 39. 37 Problem A variable chord is drawn through the origin to the circle x2 + y2 2ax = 0. The locus of the centre of the circle drawn on this chordas diameter is : a. x2 + y2 + ax = 0 b. x2 + y2 + ay = 0 c. x2 + y2 - ax = 0 d. x2 + y2 - ay = 0 40. 38 Problem The straight lines x + y = 0, 3x + y 4 = 0 and x + 3y 4 = 0 from a triangle which is : a. Isosceles b. Right angled c. Equilateral d. None of these 41. 39 Problem For three vectors u, v, w which of the following expression is not equal to any of the remaining three ? a. u (v x w) b. (v x w) u c. v (u x w) d. (u x v) u 42. 40 Problem[f ( x )g ( x ) f ( x )g( x )]dx is equal to : f (x) a.g (x) b. f(x) g(x) f(x)g(x) c. f(x)g(x) f(x)g(x) d. f(x)g(x) + f(x)g(x) 43. 41 Problemxe xis equal to : 2 dx1 x ex a.cx 1 b. ex(x + 1) + cex c c. (x 1)2ex d.c 1 x2 44. 42 Problem The top of a hill observed from the top and bottom of a building of height h is at angles of elevation p and q respectively. The height of the hill is :h cos q a. cot q cot ph cot p b. cot p cot q h tan p c. tan p tan q d. none of these 45. 43 Problem The probability that in a random arrangement of the letters of the word UNIVERSITY, the two Is do not come together is :4 a.5 b. 15 1 c.10 9 d. 10 46. 44 Problem If a and bare two vectors such that , a x b a b 0 then : a. A is equal to zero b. B is equal to zero c. Either a or b is zero d. Both a and b are necessarily zero 47. 45 Problem2 2 If (1 + i) (1 - 2i) (1 - 3i). (1 - ni) = i then equals : a. 1 . 2 . 3 ..n b. 12. 22 . 32 . n2 c. 12 + 22 + 32 + n2 d. 2 .5 . 10 .. (n2 + 1) 48. 46 Problem If f(x) = ax + b and g(x) = cx + d, then f[g(x)] = f[f(x)] is equivalent to : a. f(a) = g(c) b. f(b) = g(b) c. f(d) = g (b) d. f(c) = g (a) 49. 47 Problem A rough plane is inclined at an angle to the horizon. A body is just to slide due to its own weight. The angle of friction would be : a. tan-1 b. c. tan d. 2 50. 48 Problem If a, then f(x) has maximum value at x = 3, then :f (x) x2x a. a < -27 b. a > -27 c. a > 27 d. a < 27 51. 49 Problem If a + b + c = 0, |a| = 3, |b| = 5, |c| = 7 then the angle between a and b is equal to : a.62 b.35 c.3 d. 3 52. 50 Problem A bag contains 4 red, 6 white and 5 black balls. 2 balls are drawn at random. Find the probability of getting one red and one white ball is :2 a. 3 b.435 c. 1510 8 d.35 53. 51 Problem 5 53 In two events P( A B) 6 P( A) 6 ,P B2then A and B are : a. Independent b. Mutually exclusive c. Mutually exhaustive d. Dependent 54. 52 Problem If P(not A) = 0.7, P(B) = 0.7 and P(B/A) = 0.5, then P(A/B) equals : 3 a.13 b.314 c.112 d. none of these 55. 53 Problemp q y r zp q r If = 0, then the value of x y z is :p x q r zp x q y r a. 0 b. 1 c. 2 d. 4pqr 56. 54 Problemx m Let f :RR be a function defined by f x , where m n, then :x n a. f is one-one onto b. f is one-one into c. f is many-one onto d. f is many-one into 57. 55 Problem 1.3 1.3.5 The sum of the series 1 ... is :6 6.8 a. 1 b. 0 c. d. 4 58. 56 Problem The locus of the pole of normal chords of an ellipse is given by :a6b6 2 a. a2b2x2y2a3 b3 2 a2b2 b. x2 y2 a6b6 2 c. a2b2 x2y2 a3b3 2a2b2 d.x2y2 59. 57 Problem A body is projected through an angle from vertical so that its range is half of maximum range. Value of is : a. 600 b. 750 c. 300 d. 22.40 60. 58 Problem The sun of the magnitudes of two forces acting at a point is 18 and magnitudes of their resultant is 12. If the resultant is at 900 with the force of smaller magnitude, then their magnitudes are : a. 3, 15 b. 4, 14 c. 5, 13 d. 6, 12 61. 59 Problem To be semigroup the elements of a subset of a group must obey the axioms of : a. Associativity and commutativity b. Closure and identity c. Closure and associativity d. Closure and inverse 62. 60 Problem Let A and B be two events such that, 51 1 then :P A B ,P A B and P A63 2 a. P(B) P (A) b. P(A) = P(B) c. A and B are independent d. A and B are mutually exclusive 63. 61 Problem2 If z = z3i 5 , then the locus of z is a : a. Circle b. Hyperbola c. Parabola d. None of these 64. 62 Problem Let a > 0, b > 0 and c > 0. Then both the roots of the equation ax2 + bx + c = 0 a. Are real and negative b. Have negative real parts c. Are rational numbers d. None of the above 65. 63 Problem If 1 11+ . upto n terms, thenytan 1tan 1tan 1 1 x x2 x 23x 3 x 2 5x 7 y(0) is equal to : 1 a. 1 n2n2 b.1 n2 n c. 1 n2 d. none of these 66. 64 Problem Equation of the tangent to the hyperbola 2x2 3y2 = 6 which is parallel to the line y = 3x + 4 is : a. y = 3x + 5 b. y = 3x 5 c. y = 3x + 5 and y = 3x 5 d. none of the above 67. 65 Problem 12x12x Differential coefficient of tan 1 x2with respect to sin1 x2 will be : a. 1 b. -1 c. -1/2 d. x 68. 66 Problem A particle is moving in a straight line with constant acceleration a. If x is the space described in t seconds and x is the space described during next t seconds, then a is equal to : 2xx a.t t tt 2xx b.t t tt 2xx c.t t tt 2 x x d.tt t t 69. 67 Problem The numbers P, Q and R for which the function f(x) = Pe2x + Qex + Rx satisfies the conditions f(0) = -1, f (log 2) = 31 and log 4 [f (x ) Rx]dx 39 are given by : 02 a. P = 2, Q = -3, R = 4 b. P = -5, Q = 2, R = 3 c. P = 5, Q = -2, R = 3 d. P = 5, Q = -6, R = 3 70. 68 Problem 2excos x is equal to :limx 0x23 a. 2 b. 12 c. 23 d. none of these 71. 69 Problem(x 2 x 6)2 lim is : x2(x 2)2 a. 6 b. 25 c. 9 d. 16 72. 70 Problem If in a triangle ABC, B600 , then : a. (a - b)2 = c2 ab b. (b - c)2 = a2 bc c. (c - a)2 = b2 ac d. a2 + b2 + c2 = 2b2 ac 73. 71 Problem 10 The coefficient of the term independent of x in the expansion of x3is3 2x 2 :5 a. 47 b. 4 c. 94 d. none of these 74. 72 Problem A set contains (2n +1) elements. The number of subsets of the set which contain at most n element, is : a. 2n b. 2n+1 c. 2n-1 d. 22n 75. 73 Problem A unit vector perpendicular to the vector 4i j3k and 2i j2k is :1 a. (i 2j2k )3 1 b.( i2 j2k ) 31 c. (2i2j2k)31 (2i2j 2k ) d. 3 76. 74 Problem The radius of the incircle of a triangle whose sides are 18, 24 and 30 cms, is : a. 2 cm b. 4 cm c. 6 cm d. 9 cm 77. 75 Problem The area in the first quadrant bound by y = 4x2, x = 0, y = 1 and y = 4 is :7 a. sq unit34 b. sq unit5 c. 3 sq unit4 d. none of these 78. 76 Problem A particle is projected vertically upwards at a height h after t1 seconds and again after t2 seconds from the start. Then h is equal to : a. 1 g(t t2) 12 1 b.g(t1 + t2) 2 c. 1 Gt1t22 d. None of these 79. 77 Problem If sin + cosec =2, then sin2 + cosec2 is equal to : a. 1 b. 4 c. 2 d. none of these 80. 78 Problem/2 sin x The value ofdx , is :0 sin x cos x a.2 b.4 c.8 d. 6 81. 79 Problemsin2 y 1 cos ysin y The value of expression 1 is equal to : 1 cos y sin y 1 cos y a. 0 b. 1 c. - sin y d. cos y 82. 80 Problema1 0 If f(x) = ax a1 , then f(2x) f(x) equal to : ax 2 ax a a. a (2a + 3x) b. ax (2x + 3a) c. ax (2a + 3x) d. x (2a + 3x) 83. 81 Problem 2 2 1 2 3 2 3 3 If is a non-real cube root of unity, then 2 2 is equal 2 3 3 3 2 to : a. -2 b. 2 c. - d. 0 84. 82 Problem ab If in a ABC , cos Acos B then : a. sin2 A + sin2 B = sin2 C b. 2 sin A cos B = sin C c. 2 sin A sin B sin C = 1 d. none of the above 85. 83 Problem The graph of the function y = f(x) has a unique tangent at the point (a, 0)loge {1 6f (x)} through which the graph passes, Then lim is :x a 3f (x) a. 0 b. 1 c. 2 d. none of these 86. 84 Problemna is equal to : lim 1sinn n a. ea b. e c. e2a d. 0 87. 85 Problem3c If the equation ax2 + 2bx 3c = 0 has no real roots and4 < a + b, then : a. c < 0 b. c > 0 c. c0 d. c = 0 88. 86 Problem The line 3x 4y = touches the circle x2 + y2 4x 8y 5 = 0 if the value of is : a. - 35 b. 5 c. 20 d. 31 89. 87 Problem If OA i 2j 3k, OB 3i j2k, OC 2i 3 j k. Then AB AC is equal to : a. 0 b. 17 c. 15 d. none of these 90. 88 Problem The value of tan2 (sec-1 2) + cot2 (cosec-1 3) is a. 15 b. 13 c. 11 d. 10 91. 89 Problem The sum of all proper divisor of 9900 is : a. 29351 b. 23951 c. 33851 d. none of these 92. 90 Problem The combined equation of the pair of lines through the point (1, 0) and parallel to the lines represented 2x2 xy y2 = 0 is : a. 2x2 xy y2 4x y = 0 b. 2x2 xy y2 4x + y + 2 = 0 c. 2x2 + xy + y2 2x + y = 0 d. none of the above 93. 91 Problem a 1 2 If a, b, c are in AP, then , ,are in :bc c b a. AP b. GP c. HP d. None of these 94. 92 Problem A particle is in equilibrium when the forces , u u F1 10k, F2 (4 i 12 j 3k), F2(4i 12j3k)13 13v F3( 4i j 12 3k) and F4 (cos sin ) act on it, then : i j 1365 v 65 cot a. 3 b. u = 65 (1 3 cot ) c. w = 65 cosec d. none of the above 95. 93 Problem There are 10 points in a plane out of these 6 are collinear. The number of triangles formed by joining these point is : a. 100 b. 120 c. 150 d. none of these 96. 94 Problem Ifx and yare two unit vectors and is the angle between them, then1 |x y| is equal to :2 a. 0 b.2 sin c.2 cos d.2 97. 95 Problem a b c a b x a c If a, b and c are three non-coplanar vectors, thenis equal to : a. 0 b. [a b c ] c. 2 [a b c ] d. - [a b c ] 98. 96 Problem The coefficient of x5 in the expansion of (1 + x2)5 (1+ x)4 is : a. 30 b. 60 c. 40 d. none of these 99. 97 Problem The function f(x) = x3 3x is : a. Increasing on (- , -1)(1, ) and decreasing on (-1, 1) b. Decreasing on (- , -1)(1, ) and increasing on (-1, 1) c. Increasing on (0, ) and decreasing on (- , 0) d. decreasing on (0, ) and increasing on (- , 0) 100. 98 Problem A man in a balloon rising vertically with an acceleration of 4.9 m/s2, releases a ball 2 s after the balloon is let go from the ground. The greatest height above the ground reached by the ball, is : a. 19.6 m b. 14.7 m c. 9.8 m d. 24.5 m 101. 99 Problem A bag contain n + 1 coins. It is known that one of these coins shows heads on both sides, whereas the other coins are fair. One coin is selected at random and 7 tossed. If the probability that toss results in heads is, then the value of n is :12 a. 3 b. 4 c. 5 d. none of these 102. 100 Problem xIf (x)sin t 2dt , then(1) is equal to : 1/ xa. sin 1b. 2 sin 1 3c. 2 sin 1d. none of these 103. FOR SOLUTIONS VISIT WWW.VASISTA.NET