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MATHEMATICS 1. Solve: tan + tan 4tan tantan 4tan 7. = (1) nn/12 (2) /12 (3) n (4) n/2 2. If tan-1 x + tan-1 y+ tan-1 z =/2 then xy + yz + zx = (1) xyz (2) 0 (3) 1 (4) x + y + z 3. Solve: cos-1 x + sin-1 x/2 =/6 , x = (1) +1 (2) +3 (3) 0 (4) 1/2 4. The points of discontinuity of f(x) = tan ( x/x+1) other than x =-1 are: (1) x = 2m +1/1 - 2m m is any integer (2) x =2m -1/2m + 1 (3) x = 0 (4) x = 5. If y = log I (log x), dy/dx = (1) 1/x log (log x) (2) none of these (3) 1/ log (log x) (4) 1/x log x.log (log x) 6. 2 is a group of all points rationals under the operation * defined by a* b = ab/2. The identify of 2 is: (1) 0 (2) 1/2 (3) 2 (4) 1 7. To be a sub-group the elements of a subset of a group must obey the axioms of: (1) Closure and inverse (2) Closure and associatively (3) Closure and identify (4) Associativity and commutativity 8. The distance between the parallel lines x2 + 2xy + y2 -6x -6y + 8 =0 is: (1) 2 (2) 1/2 (3) 1 (4) 2 9. The radius of the circle 3x(x-2) + 3y(y+1) =4 is: (1) 15/4 (2) 31/12 (3) 2 (4) 3 10. If the two circles x2 + y2 + 7x + 16y -3 =0 and 2x2 -6x-4y + k =0 cut each other orthogonally , k = (1) 17 (2) -37 (3) 27 (4) -47 11. If x = a(cos sin ), y = a (sin cos cos ) d2y/dx2= cos (1) a 3 (2) sec2/a (3)sec3/a (4) sec2/ 12. If x = t2 +t +1 and y = sin/2 t + cos /2t then at t = 1, dy/dx = /3 (2) /4 (3) /2 (4) -/6 13. S = tan-1 (1 + x2-1/x) and T = tan-1 x then dS/dT = (1) 1/2 (2) 2 (3) 1 (4) -1 14. The equation to the tangent to the curve y = b e-x/a at the point where it crosses the y -axis is: (1) x + y =ab (2) x/a + y/b=1 (3) ax + by =1 (4) x + y = a+ b 15. The angle of intersection curves y2 = 2x and x2 = 16y at (0,0) is: (1) /3 (2) tan-1 (3/5) (3) /2 (4) /4 16. The length of the intercept that the circle x2 + y2 + 10x - 6y + 9 =0 makes on the x - axis is: (1) 4 (2) 2 (3) 8 (4) 6 17. A parabola has its focus at (-4,0) and its directrix is x = 4. Its equation is: (1) y2=-16x (2) x2=-8y (3) y2=8x (4) x2=9y 18. The eccentricity of the hyperbola 36x2 - 25y2 = 900 is : (1) 5 (2) 61/5 (3) 31/5 (4) 6 19. The straight line y = 4x + k touches the hyperbola x2/64 - y2/49 =1. Then k= (1) +_ 500 (2) 56 (3) 251 (4) + 975 20. An ellipse has a minor axis of length 6 and the distance between its foci is 8. Its equation is: (1) x2/6 + y2/9 =1 (2) x2/6 + y2/5 =1 (3) x2/25 + y2/9 =1 (4) x2/9 + y2/25 =1 21. dx/1+cot x= (1) 0 (2) (3)/4 (4) /2 22. The area bounded by the curve y = 4x -x2 and the x-axis is: (1) 32/3 (2) 16 (3) 32 (4) 21 1/3 23. (1) 1/3 (2) /4 (3) 1 (4) log 2 24. dx/x(xn+1) = (1) n log (x/xn+1) (2) n/x log (xn+1) (3) log (xn /xn+1) (4) 1/n log (xn /xn+1) 25. Identify the non-A belian group among the following : (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 26. (1) (2) (3) (4) 27. For what value of are the vector are coplanar? (1) -4 (2) -3 (3) 4 (4) 2a 28. (1) (2) (3) (4) 29. The sine of the angle between the vectors (1) 3/14 (2) 5/7 (3) 5/21 (4) 5/7 30. Simplify (1) 36 (2) 618 (3) 0 (4) 12 31. (1) e3 (2) e4 (3) e (4) e2 32. If x = loga be, y = logb ca and z = logc ab then (1) ab + bc + ca (2) abc (3) x + y+ z (1) (log 9)2 (4) 1 33. (2) (log 3)2 (3) log 9 (4) 2 log 9 34. α, β y are the roots of x2 + px + q = 0. Then α3 + β3 + y2 = (1) -pq (2) 3pq (3) -3q (4) -p 35. The third term of a G.P. is 4. The product of its first five terms is: 3.125 (1) (2) 32 (3) 1,024 (4) 243 36. If x =-9 is a root of the equation =0 the other two roots are: (1) 2,-7 (2) 1, 5 (3) 2,7 (4) -2,7 37. If a b c a root of the equation (1) x =c (2) x=0 (3) x =a (4) x= b 38. (1) (2) (3) (4) 39. If w is a cube root of unity (1 -w) (1 - w2) (1 - w4) (1 - w8) = (2) 3 (3) 1 (4) 9 40. The real part of 1/1-cos +i sin is: (2) 2 (1) tan /2 (1) (3) 1/1-cos (4) 1/2 41. In a ABC if sin A/sin C = sin( A -B)/sin (B-C) then a2, b2 and c2 are in: (1) H.P (2) none of these (3) A.P (4) G.P 42. The integral part of (is: (1) 196 (2) 163 (3) 198 (4) 197 43. (1) (2) (3) (4) 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: (1) 5/8 (2) 2/7 (3) 3/8 (4) 4/7 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. (1) 0 or 1 (2) 1/4 or 1/2 (3) 1/2 or 1/3 (4) 1/3 or 1/4 46. sech x dx = (1) (2) 1 (3) /2 +1 (4) /2 47. sin x-cos x/1 + sin x cos x dx = (1) 0 (2) /2 (3) 1 (4) /4 48. (1) /4 (2) x= 4/I (3) x + 1 (4) x - 4I 49. sin x/2 dx = (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 50. x/(x + 1) ex dx = (1) (x + 1)e x (2) none of these (3) xex (4) ex/x+1 51. If 15C3r = 15Cr+3, then r: (1) 1/3 (2) 3/2 (3) 2 (4) 3 52. How many committees of 5 members can be formed from 6 gentlemen and 4 ladies? (1) 252 (2) 120 (3) 10C5 (4) 10P5 53. How many even numbers can be formed by using all the digits 2, 3, 4, 5 , 6? (1) 120 (2) 72 (3) 48 (4) 24 54. There are three copies of each of 4 difference books. In how many ways can they be arranged in a shelf? (1) 12/3 +4 (2) 369.600 (3) 369,000 (4) 12 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. (1) 2/3 (2) 43/2 (3) 6 (4) 2 3/2 56. The maximum value of f (x) = log x/x, 0 < x < is (1) 1/e (2) 2/e (3) e (4) e 57. For f(x) = 3 sin x + 3 cos x, the point x = /6 is: (1) a point of inflection (2) none of these (3) a local minimum (4) a local maximum 58. If y2 = ax2 + 2bx + c then y3 d2y/dx2 = (1) ac -b2 (2) 4(b2 + ac) (3) b2 - 4ac (4) b2 - ac 59. The speed v of a particle moving along a straight line is given by a + bv = x2, where x is its distance from the origin. The acceleration of the particle is: 2 (1) x/ab (2) x/b (3) ax (4) abx 60. 2-1 lxl dx = (2) 1 (3) 5/2 (4) 2 (1) 3/2