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1996 Mathematics
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(1) np + (-1)n
p/12 (2) p/12 (3) np (4) (2n +1)p/2 |
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| 2. | If tan-1 x + tan-1 y+ tan-1 z = p/2 then xy + yz + zx = |
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(1) xyz
(2) 0 (3) 1 (4) x + y + z |
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| 3. | Solve: cos-1 x + sin-1 x/2 = p/6 , x = |
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(1) +1
(2) +Ö3 (3) 0 (4) 1/Ö2 |
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| 4. | The points of discontinuity of f(x) = tan ( px/x+1) other than x =-1 are: |
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(1) x = 2m +1/1 - 2m m is
any integer (2) x =2m -1/2m + 1 (3) x = 0 (4) x = p |
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| 5. | If y = log I (log x), dy/dx = |
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(1) 1/x log (log x) (2) none of these (3) 1/ log (log x) (4) 1/x log x.log (log x) |
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| 6. | 2 is a group of all points rationals under the operation * defined by a* b = ab/2. The identify of 2 is: |
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(1) 0 (2) 1/2 (3) 2 (4) 1 |
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| 7. | To be a sub-group the elements of a subset of a group must obey the axioms of: |
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(1) Closure and inverse (2) Closure and associatively (3) Closure and identify (4) Associativity and commutativity |
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| 8. | The distance between the parallel lines x2 + 2xy + y2 -6x -6y + 8 =0 is: |
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(1) 2 (2) 1/Ö2 (3) 1 (4) Ö2 |
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| 9. | The radius of the circle 3x(x-2) + 3y(y+1) =4 is: |
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(1) Ö15/4 (2) Ö31/12 (3) 2 (4) 3 |
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| 10. | If the two circles x2 + y2 + 7x + 16y -3 =0 and 2x2 -6x-4y + k =0 cut each other orthogonally , k = |
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(1) 17 (2) -37 (3) 27 (4) -47 |
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| 11. | If x = a(cos q + q sin q), y = a (sin q -q cos q cos q) d2y/dx2= |
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(1) aq
cos3q (2) sec2q/a (3) sec3q/aq (4) sec2q/q |
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| 12. | If x = t2 +t +1 and y = sin p/2 t + cos p/2t then at t = 1, dy/dx = |
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(1) p/3 (2) -p/4 (3) p/2 (4) -p/6 |
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| 13. | S = tan-1 (Ö1 + x2-1/x) and T = tan-1 x then dS/dT = |
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(1) 1/2 (2) 2 (3) 1 (4) -1 |
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| 14. | The equation to the tangent to the curve y = b e-x/a at the point where it crosses the y -axis is: |
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(1) x + y =ab (2) x/a + y/b=1 (3) ax + by =1 (4) x + y = a+ b |
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| 15. | The angle of intersection curves y2 = 2x and x2 = 16y at (0,0) is: |
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(1) p/3 (2) tan-1 (3/5) (3) p/2 (4) p/4 |
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| 16. | The length of the intercept that the circle x2 + y2 + 10x - 6y + 9 =0 makes on the x - axis is: |
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(1) 4 (2) 2 (3) 8 (4) 6 |
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| 17. | A parabola has its focus at (-4,0) and its directrix is x = 4. Its equation is: |
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(1) y2=-16x (2) x2=-8y (3) y2=8x (4) x2=9y |
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| 18. | The eccentricity of the hyperbola 36x2 - 25y2 = 900 is : |
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(1) 5 (2) Ö61/5 (3) Ö31/5 (4) 6 |
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| 19. | The straight line y = 4x + k touches the hyperbola x2/64 - y2/49 =1. Then k= |
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(1) +_ 500 (2) 56 (3) Ö251 (4) + Ö975 |
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| 20. | An ellipse has a minor axis of length 6 and the distance between its foci is 8. Its equation is: |
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(1) x2/6 + y2/9 =1 (2) x2/6 + y2/5 =1 (3) x2/25 + y2/9 =1 (4) x2/9 + y2/25 =1 |
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| 21. | òp/2o dx/1+cot x= |
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(1) 0 (2) p (3) p/4 (4) p/2 |
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| 22. | The area bounded by the curve y = 4x -x2 and the x-axis is: |
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(1) 32/3 (2) 16 (3) 32 (4) 21 1/3 |
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| 23. |
 |
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(1) 1/3 (2) p/4 (3) 1 (4) log 2 |
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| 24. | ò dx/x(xn+1) = |
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(1) n log (x/xn+1) (2) n/x log (xn+1) (3) log (xn /xn+1) (4) 1/n log (xn /xn+1) |
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| 25. | Identify the non-A belian group among the following : |
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(1) The set of all n-square,
non-singular matrices under multiplication (2) The set of all integer under addition (3) The set of all m X n matrices under addition (4) All n-square complex matrices under multiplication |
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| 26. |
 |
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(1) (2) (3) (4) |
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| 27. | For what value of
l are the vector are coplanar? |
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(1) -4 (2) -3 (3) 4 (4) 2a |
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| 28. |
 |
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(1) (2) (3) (4) |
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| 29. | The sine of the angle between the
vectors |
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(1) 3/Ö14 (2) 5/2Ö7 (3) 5/21 (4) 5/Ö7 |
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| 30. | Simplify |
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(1) 36 (2) 618 (3) 0 (4) 12 |
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| 31. |
 |
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(1) e3 (2) e4 (3) e (4) e2 |
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| 32. | If x = loga be, y = logb ca
and z = logc ab then  |
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(1) ab + bc + ca (2) abc (3) x + y+ z (4) 1 |
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| 33. |
 |
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(1) (log 9)2 (2) (log 3)2 (3) log 9 (4) 2 log 9 |
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| 34. | α, β y are the roots of x2 + px + q = 0. Then α3 + β3 + y2 = |
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(1) -pq
(2) 3pq (3) -3q (4) -p |
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| 35. | The third term of a G.P. is 4. The product of its first five terms is: |
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(1) 3.125 (2) 32 (3) 1,024 (4) 243 |
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| 36. | If x =-9 is a root of the equation =0 the other two roots are: |
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(1) 2,-7 (2) 1, 5 (3) 2,7 (4) -2,7 |
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| 37. | If a
¹ b
¹ c a root of the equation |
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(1) x =c
(2) x=0 (3) x =a (4) x= b |
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| 38. |
 |
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(1) (2) (3) (4) |
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| 39. | If w is a cube root of unity (1 -w) (1 - w2) (1 - w4) (1 - w8) = |
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(1) w
(2) 3 (3) 1 (4) 9 |
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| 40. | The real part of 1/1-cos q+i sin q is: |
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(1) tan q/2 (2) 2 (3) 1/1-cos q (4) 1/2 |
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| 41. | In a DABC if sin A/sin C = sin( A -B)/sin (B-C) then a2, b2 and c2 are in: |
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(1) H.P (2) none of these (3) A.P (4) G.P |
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| 42. | The integral part of (Ö2 + 1)6 is: |
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(1) 196 (2) 163 (3) 198 (4) 197 |
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| 43. |
 |
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(1) (2) (3) (4) |
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| 44. | From a group of 5 boys and 3 girls three persons are chosen at random. Find the probability that there are more girls than boys: |
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(1) 5/8 (2) 2/7 (3) 3/8 (4) 4/7 |
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| 45. | A and B are two independent events. The probability that both A and B occur is 1/6 and the probability that neither of them occurs is 1/3. Find the probability of A. |
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(1) 0 or 1 (2) 1/4 or 1/2 (3) 1/2 or 1/3 (4) 1/3 or 1/4 |
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| 46. | ò-0 sech x dx = |
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(1) p (2) 1 (3) p/2 +1 (4) p/2 |
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| 47. | ò-0 sin x-cos x/1 + sin x cos x dx = |
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(1) 0 (2) p/2 (3) 1 (4) p/4 |
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| 48. |
 |
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(1) p/4 (2) x= 4/p I (3) x + 1 (4) x - 4I |
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| 49. | ò Ö 1 + sin x/2 dx = |
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(1) cos x/2+ sin x/2 (2) 4 cos x/2 -4 sin x/2 (3) -4cos x/4 + 4sin x/4 (4) 4 cos x/4 +4 sin x/4 |
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| 50. | ò x/(x + 1) 2 ex dx = |
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(1) (x + 1)e x (2) none of these (3) xex (4) ex/x+1 |
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| 51. | If 15C3r = 15Cr+3, then r: |
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(1) 1/3 (2) 3/2 (3) 2 (4) 3 |
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| 52. | How many committees of 5 members can be formed from 6 gentlemen and 4 ladies? |
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(1) 252 (2) 120 (3) 10C5 (4) 10P5 |
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| 53. | How many even numbers can be formed by using all the digits 2, 3, 4, 5 , 6? |
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(1) 120 (2) 72 (3) 48 (4) 24 |
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| 54. | There are three copies of each of 4 difference books. In how many ways can they be arranged in a shelf? |
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(1) 12/3 +4 (2) 369.600 (3) 369,000 (4) 12 |
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| 55. | The equation of the base of an equilateral triangle is x + y = 2 and the vertex is (2,-1), Find the length of the side of the triangle. |
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(1) Ö2/3 (2) 4Ö3/2 (3) Ö6 (4) 2 Ö3/2 |
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| 56. | The maximum value of f (x) = log x/x, 0 < x < ¥ is: |
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(1) 1/e (2) 2/e (3) e (4) Öe |
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| 57. | For f(x) = Ö3 sin x + 3 cos x, the point x = p/6 is: |
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(1)
a point of inflection (2) none of these (3) a local minimum (4) a local maximum |
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| 58. | If y2 = ax2 + 2bx + c then y3 d2y/dx2 = |
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(1) ac -b2 (2) 4(b2 + ac) (3) b2 - 4ac (4) b2 - ac |
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| 59. | The speed v of a particle moving along a straight line is given by a + bv2 = x2, where x is its distance from the origin. The acceleration of the particle is: |
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(1) x/ab (2) x/b (3) ax (4) abx |
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| 60. | ò2-1 lxl dx = |
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(1) 3/2 (2) 1 (3) 5/2 (4) 2 |
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