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1995 Mathematics
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(1) a parabola (2) an ellipse (3) a circle (4) a hyperbola |
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| 2. | In n is a positive integer then n(n2 -1) (n2 -4) is divisible by: |
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(1) 4.5.6 (2) 5.6.7 (3) 2.4.6 (4) 3.4.5 |
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| 3. | The equation to the hyperbola with eccentricity 2 and foci at (+ 2, 0) is : |
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(1) 3x2 -y2 =3 (2) x2 - y2 = 4 (3) x2/4 -y2/9 =1 (4) x2/1 - y2/4 =1 |
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| 4. | If y = mx + 6 is a tangent to the hyperbola x2/100 - y2/49 =1, then m = |
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(1) Ö20/17 (2) Ö17/20 (3) Ö3/20 (4) 70 |
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| 5. | What is the locus of the centre of a circle which touches externally two given circles? |
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(1) A parabola (2) An ellipse (3) A hyperbola (4) A circle |
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| 6. | If 100 = x(mod 7), then the least positive value of x is: |
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(1) 1 (2) 3 (3) 4 (4) 2 |
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| 7. | (sin q + i cos q)4 = |
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(1) sin 4q +
cos 4q (2) sin 4q - i cos 4q (3) cos 4q + i sin 4q (4) cos 4q - i sin 4 q |
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| 8. | Which of the following assertion is not true? |
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(1) The cross product of vectors is
not commutative (2) If a set has n element its power set has 2n elements (3) In a box product the dot and the cross cannot be interchanged (4) Matrix multiplication is associative |
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| 9. | In the polynomial (x-1) (x - 2) ..... (x - 100) the coefficient of x99 is: |
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(1) -3000 (2) -5050 (3) -Ð100 (4) -9090 |
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| 10. |
 |
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(1) 0
(2) 1 (3) -1 (4) 3 |
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| 11. |
 |
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(1) 4 abc (2) 2(Öab + Öbc + Öca (3) -1 (4) 2Öabc |
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| 12. | If 2np3 = 2. nP4, then n = |
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(1) 4 (2) 6 (3) 8 (4) 10 |
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| 13. | The area bounded by the curve y = log x, the x-axis and the line x = 2 is : |
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(1) log 2 (2) 2 (3) 3/2 (4) 2 log 2-1 |
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| 14. | The angle between the curves x2 - y2 =1 xy = Ö2 at (Ö2,1) is: |
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(1) 900 (2) 600 (3) tan-1(3/Ö2) (4) 450 |
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| 15. | If x2 + xy + y2 = 0, d2y/dx2= |
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(1) x/y (2) 0 (3) Ö1+x2 (4) 2 |
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| 16. | What is a singleton? |
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(1) It is a single set (2) It is a singular matrix (3) It is a set with a single element (4) It is a null set |
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| 17. | Let A = [2 ,4, 6, 8] and define R = [(2, 4) (4,2) (4, 6), (6,4)]. Then R is: |
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(1) reflexive (2) symmetric (3) transitive (4) anti-symmetric |
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| 18. |
 |
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(1)
1 (2) 0 (3) x-a + x-b +x-c (4) x |
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| 19. | Below is a list of relations among people, which of them is reflexive and symmetric? |
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(1) is taller than (2) lives within 30 miles of (3) recognizes (4) is married to. |
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| 20. | If all red, boiled crabs are dead and all red, dead crabs are boiled does it follow that all dead, boiled crabs are red? |
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(1) yes (2) no (3) insufficient data (4) inconsistent data |
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| 21. | The length of the tangent drawn from any point on the circle x2 + y2 2gx + 2fy + c1= 0 to the circle x2 + y2 + 2gx + 2fy + c2= 0 is : |
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(1) c1 + c2 (2) c2 - c1 (3) Öc12 + c22 (4) Öc2 - c1, where c2- c1 > 0 |
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| 22. | The vertex of the parabola y2 = 5x + 4y + 1 is: |
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(1) (-1,2)
(2) (1,-2) (3) (0, 3) (4) (2, -2) |
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| 23. | The axis of the parabola 2x2 + 5y - 3x + 4 = 0 is: |
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(1)
x = 3/4 (2) y = 3/4 (3) x =-1/2 (4) x - 3y = 5 |
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| 24. | The tangents drawn at the extremities of a focal chord of a parabola: |
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(1) intersect on the tangent at the
vertex (2) intersect on the directrix (3) intersect at an angle of 450 (4) are parallel |
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| 25. | If the angle between the lines joining the foci of an ellipse to an extremity of the minor axis is 900 the eccentricity of the ellipse is: |
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(1)
1/2 (2) 1/Ö2 (3) 3/4 (4) 2/Ö5 |
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| 26 | Define a function f : A ® B as follows: f(x) = x-2/x-3 Then f-1 (x) = |
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(1) x - 3/x-2 (2) 2x -1/3x - 2 (3) x + 2/x + 3 (4) 2 - 3x/1 - x |
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| 27. |
 |
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(1) 5 (2) 5 + Ö5 (3) 1 + Ö2I/2 (4) Ö5-1/2 |
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| 28. |
 |
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(1) 41 (2) 51 (3) 31 (4) 26 |
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| 29. |
 |
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(1) (2) (3) (4) |
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| 30. | Find the non-zero values of x is if |
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(1) +1 (2) +Ö2 (3) +Ö3 (4) Ö2,Ö3 |
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| 31. | How many diagonals can be drawn in a polygon of n sides? |
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(1) n(n-1)/2 (2) n(n+1)/2 (3) n(n-3)/2 (4) n(n+3)/2 |
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| 32. | The conjugate and the reciprocal of a complex number z = x + iy are equal. Then |
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(1) x = 1, y = 0 (2) x = 0, y = 1 (2) x = y = 1 (4) X2 + Y2 = 1 |
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| 33. | The combined equation of the y-axis and the line x = 1 is: |
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(1) x2 = x (2) xy = 1 (3) x2 = 1 (4) (x - 1)y = 0 |
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| 34. | The angle between the pair of straight lines 2x2 - 5xy + 2y2 + 7x -11y + 5 = 0 is |
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(1) tan-11/Ö2 (2) tan-13/4 (3) tan-14/3 (4) 600 |
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| 35. |
 |
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(1) (2) (3) (4) |
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| 36. | Two events A and B occur with probabilities 0.25 and 0.50. The probability that they occur simultaneously is 0.14. What is the probability that neither of them occurs? |
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(1) 0.39 (2) 0.61 (3) 0.72 (4) 0.28 |
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| 37. | The probability that an event A happens in a trial is 0.4. Three Independent trials are made. The probability that A happens at least once is: |
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(1) 0.216 (2) 0.784 (3) 0.064 (4) 0.936 |
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| 38. |
 |
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(1) Ö2 (2) 1 (3) 0 (4) p/4 |
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| 39. | The letters of the word TREASON are arranged in a row in all possible ways. How many of them begin with T and end with N? |
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(1) 60 (2) 120 (3) 240 (4) 720 |
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| 40. | If y = sin-1 (cos x) dy/dx = |
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(1) 1/sin x (2) cos-1 x (3) -1 (4) 1/2 |
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| 41. | If
w is a cube root of unity  |
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(1) 1 (2) w (3) 0 (4) w2 |
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| 42. | 1 1 1......1 (91 times) is: |
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(1) a composite number
(2) a prime number (3) a surd (4) irrational |
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| 43. | Factorize |
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(1) (x-α) (x-β) (x-y) (2) xαβylmn (3) (x+α) (x+β) (x+y) (4) (x-l) (x-m) (x-n) |
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| 44. | sin-1(1/Ö5) + cos-1 3= |
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(1) p/3 (2) p/2 (3) p/6 (4) p/4 |
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| 45. | Solve sin x - cos x = Ö2 .x = |
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(1) 2np (2) np (3) (2n + 1)p (4) 2np + 3p/4, (n is any integer) |
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| 46. | If y2(2a -x) = x3, then dy/dx at (a, a) is: |
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(1) 1/a (2) 1/2a (3) 1 (4) 2 |
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| 47. |
 |
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(1) 3/2 (2) log (3/2) (3) 1 (4) 0 |
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| 48. |
 |
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(1) 9/14 (2) 3/7 (3) -9/14 (4) -1 |
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| 49. |
 |
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(1) 0 (2) ¥ (3) 1 (4) -1/2 |
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| 50. | If xp occurs in the expansion of
(x+1/x)n its coefficient must be: |
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(1) (2) (3) (4) |
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| 51. | òp/20 log tan x dx = |
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(1) 1 (2) p/4 (3) p/2 (4) 0 |
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| 52. | ò cos Öx dx = |
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(1) Öx
sin Öx (2) 2[Öx sin Öx + cosÖx] (3) 2[x sin Öx +cos Öx) (4) x sin Öx /2 |
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| 53. | Six boys and six girls are seated in
a row randomly. What is the probability that the boys and girls sit
alternately? |
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(1) (2) (3) (4) |
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| 54. | ò 1 / (ex-1) dx = |
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(1) ex-1 (2) log(1-e-x) (3) log(ex-1) (4) tan-1(e-x) |
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| 55. | ò log x dx= |
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(1) x log x (2) log (x+1) (3) x/log x (4) x log x-x |
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| 56. | In the group C= {0,1,3,4,5} under addition modulo 6, subgroup is: |
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(1)
{0,1,3} (2) {0,4,5} (3) {0,2,4} (4) {0,3,5} |
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| 57. | The set of positive integers including zero is not a group under addition because: |
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(1) closure fails (2) associative law does not hold (3) there is no identify (4) there is no inverse |
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| 58. | The set of all integers under the operation * defined by a*b = a + b + 1 is a group. Its identify is: |
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(1) -1 (2) -2 (3) 1 (4) 0 |
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| 59. | log tan 10 + log tan 20 +....+ log tan 890 = |
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(1) 1 (2) 0 (3) tan 10 (4) tan 890 |
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| 60 | The least positive integer to which 89 x 111 x 135 is congruent modulo 11 is: |
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(1) 3 (2) 2 (3) 1 (4) 7 |
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| 61. | If U = cos-1 (1-x2/1+x2) and V = tan-1 (2x/x-x2), then dU/dV = |
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(1) 0 (2) 1 (3) 1/2 (4) 2 |
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| 62. | The equation to the tangent to the curve (x/a)n + (y/b)n = 2 at (a, b) is: |
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(1) x/a + y/b =2 (2) x/a + y/b =1/2 (3) x/b + y/a=2 (4) ax + by =2 |
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| 63. | The maximum value of 4 sin2 x + 3 cos2 x is: |
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(1) 3 (2) 4 (3) 5 (4) 7 |
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| 64. | ò20 lI -xl dx = |
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(1) 0 (2) 1 (3) 2 (4) 3/2 |
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| 65. | ò x2-1/x2+1 dx = |
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(1) tan -1 x2 (2) x + tan -1 x (3) x - 2 tan -1 x (4) log (1 + x4) |
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| 66. |
 are coplanar then |
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(1) a + b+ c =0 (2) abc =-1 (3) a + b + c = abc +2 (4) ab + bc + ca =0 |
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| 67. |  |
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(1) 3 (2) 4 (3) 5 (4) 6 |
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| 68. | A real, non-zero solution to the system of equation 1+ x + x2 +.....+ x20=0 1 + x + x2 +....+ x16 =0 is |
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(1) -2 (2) 1 (3) 2 (4) -1 |
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| 69. | The adjacent sides of a
parallelogram are Its area is |
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(1) 3Ö5 (2) 5Ö3 (3) Ö15 (4) 5Ö3/2 |
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| 70. | How should we define f(x) = log(1 + ax) - log (1-bx)/x at x = 0 so as to make it continuous there f(0) = |
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(1) a-b (2) log a + log b (3) a + b (4) log (a/b) |
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| 71. | If is α, β y are roots of the equation x3 + qx + r = 0, then (1+α) (1+β) (1+y)= |
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(1) qr (2) 1+q-r (3) -q/r (4) 1+qr |
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| 72. | p.q,r and in A.P. and each is numerically less than unity. |
| let x = 1 + p + p2 +....¥ | |
| y = 1 + q + q2+.....¥ | |
| and z = 1 + r + r2 +.....¥ Then x, y, z are in | |
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(1) A.P (2) G.P (3) H.P (4) None of these |
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| 73 | If z4 = i then z = |
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(1) 1 (2) i (3) cis p/4 (4) cis p/8 |
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| 74. | If (1+i/1-i)m = 1, then m = |
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(1) 4 (2) 5 (3) 6 (4) 7 |
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| 75. | 4 cos 360 + cot 7 10/2 = |
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(1) -Ö1 +
Ö 2 + Ö 3 + Ö 4 + Ö 5+ Ö 6 (2) Ö1 + Ö 2 + Ö 3 + Ö 4 + Ö 5+ Ö 6 (3) Ö5 + 1 (4) Ö5 + Ö3 |
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