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1997 Mathematics
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(1) 4abc (2) (c) 0 (4) abc |
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| 2. |
 |
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(1) (2) (3) (4) |
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| 3. |
 |
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(1)
(2) (3) (4) |
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| 4. |
 is equal to: |
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(1) 0 (2) 2 (3) 1 (4) -1 |
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| 5. |
 |
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(1) Ö3 (2) Ö5 (3) 3 (4) 2 |
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| 6. | The smallest positive integer n for which (l+i/I -i)n =1 is: |
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(1) 3
(2) 4 (3) 0 (4) 2 |
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| 7. | If w is a cube root of unity, and n is a multiple of 3, then I + wn +2wn is: |
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(1) 3 (2) 2 (3) 1 (4) 0 |
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| 8. | If A = [ 1, 2, 3, 4, 5] then the number of proper subsets of A is: |
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(1) 30
(2) 120 (3) 32 (4) 31 |
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| 9. | The negative of q n ~ (p Ù r) is: |
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(1) ~q
Ù (p
Ù r) (2) ~q Ù ~ (p Ù r) (3) ~q v (p Ù r) (4) -q v ~ (p Ù r) |
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| 10. | Which of the following is false? |
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(1) In an abelian group (ab)2 = a2b2
" a,b,
Î G (2) In an abelian group (ab)-1 = a-1b-1 " a,b, Î G (3) In a group of even order there exists an element other than identity which is its own inverse (4) In an abelian group, every element is its own inverse. |
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| 11. | If 4 sin-1 (x) + cos-1 (x) = p then x = |
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(1) 1/3 (2) 2/3 (3) 2 (4) 1/2 |
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| 12. | If logp x = α,and logq x = β then the value of logp/q x= |
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(1) αβ/
β - α (2) β - α/αβ (3) αβ/α - β (4) α - β/αβ |
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| 13. | If the slope of one line the pair ax2 + 4xy + y2 = 0 is 3 times the other then a is: |
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(1) -1 (2) 1 (3) -3 (4) 3 |
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| 14. | If a = 5, b =13, c =12 in D ABC, then tan (B/4) is: |
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(1) Ö2 +1 (2) Ö2 -1 (3) Ö3 -1 (4) Ö3 +1 |
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| 15. | ò elog(tanx) dx = |
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(1) e(tanx)+c (2) tan x + c (3) log (sec x) +e (4) log (tan x) +c |
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| 16. | òp/20 esin xcos x.dx = |
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(1) e (2) e + 2 (3) e -1 (4) e + 1 |
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| 17. | òp/20 1/1+ (cot x)101 dx = |
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(1) p/4 (2) 0 (3) p/2 (4) p/3 |
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| 18. | ò dx/(1 + ex ) ( 1 + e-x) = |
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(1) 1/ex
+ c (2) 1/(1 + e)2 +c (3) 1/(1 + ex) + c (4) 1/(1 + ex) +c |
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| 19. |
 |
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(1) p/Öa2
- b2 (2) -1/Öa2 - b2 (3) 1/Öa2 - b2 (4) 2/Öa2 - b2 |
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| 20. | The maximum value of sin x + cos x is: |
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(1) Ö3
(2) 2 (3) Ö2 (4) - Ö2 |
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| 21. |
 |
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(1) 2/3 (2) 1/4 (3) 3/2 (4) 1/2 |
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| 22. | If the function f(x) = { x2 - (A + 2) x +A/x-2 for x ¹ 2 is continuous x = 2 then: |
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(1) A =-1 (2) A = 3 (3) A = 1 (4) A = 0 |
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| 23. |
 |
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(1) p/6 (2) p/3 (3) log 2 (4) p/4 |
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| 24. |
 |
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(1) 2
(2) 1/2 (3) e2 (4) e |
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| 25. |
 |
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(1) p/2 (2) 1 (3) 0 (4) 2/p |
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| 26. | In the group G = {0, 1, 2, 3, 4, 5} under addition module 6, a subgroup is: |
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(1) ( 0, 3, 5)
(2) ( 0, 4, 5) (3) ( 0, 1, 3) (4) ( 0, 2 ,4) |
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| 27. | In a group G = { 2, 4, 6, 8 } under X mod 10 the identity element is: |
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(1) 8 (2) 6 (3) 2 (4) 4 |
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| 28. | If 
then its characteristic roots are: |
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(1) 1,5 (2) 6,1 (3) 1, -5 (4) -1,5 |
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| 29. | If in a square matrix A = aij we find a ij "ij then A is: |
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(1) Transpose matrix (2) Skew symmetric matrix (3) Diagonal matrix (4) Triangular matrix |
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| 30. | The value of x for which the matix
has no inverse is: |
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(1) 3 (2) 0 (3) 2 (4) -2 |
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| 31. | If p is a prime number and p is the product of all prime numbers less than or equal to p then: |
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(1) p + 1 is not a prime number (2) p - 1 is a prime number (3) p + 1 is a composite number (4) p + 1 is a prime number |
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| 32. | The unit digit in 1337 is: |
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(1) 6 (2) 3 (3) 5 (4) 2 |
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| 33. | The equation of the tangent to the circle x2 + y2 + 2gx + 2fy + c = 0 at the origin is |
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(1) x =0 (2) y =0 (3) fx + gy =0 (4) gx + fy =o |
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| 34. | The point of intersection of two perpendicular tangents to x2/a2 + y2/b2 =1 lies on |
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(1) x2 + y2 = a2 + b2 (2) x2 + y2 = b2 (3) x2 + y2 = a2 (4) x2/a2 - y2/b2 =1 |
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| 35. |
 |
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(1) 2100 (2) 250 (3) (10050) (4) (50)50 |
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| 36. | The degree of the differential
equation  |
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(1) 6 (2) 4 (3) 1 (4) 2 |
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| 37. | The solution of the differential equation cos x cos dx + sin y dy = 0 is |
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(1) tan x = c
(2) sec x - sec y = c (3) sin x = cos y (4) cos x = c sin y |
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| 38. | The area bounded by the curve y = sin x between x = 0 and x = 2p is given by |
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(1) 3 sq. units (2) 4 sq. units (3) 0 (4) 2 sq units |
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| 39. | The proposition (p ® - p) Ù (- p ® p) is |
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(1) Neither tautology nor
contradiction (2) Tautology and contradiction (3) Tautology (4) Contradiction |
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| 40. | If two tangents drawn from a point P to the parabola y2 = 4x are at right angles, then the locus of P is |
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(1) x + 1 = 0
(2) 2x -1 = 0 (3) x -1 = 0 (4) 2x + 1 = 0 |
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| 41. | In an ellipse x2/a2 + y2/b2 =1, If the distance between the directrices is 3 times the distance between the foci, then the eccentricity is: |
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(1) 1/2 (2) 1/3 (3) 1/Ö2 (4) 1/Ö3 |
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| 42. | The value of cos-2 (-1) - sin-1(1) is |
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(1) -3p/2 (2) 3p/2 (3) p/2 (4) p |
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| 43. | If 3 sin-1 (2x/1 + x2) -4 cos-1 (1 - x2)/1 + x2) + 2 tan-1 ( 2x/1 -x2) = p/3 then x = |
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(1) 2
(2) 1 (3) 1/Ö2 (4) 1/Ö3 |
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| 44. | If sin A = sin B and cos A = cos B, then |
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(1) A = 2 np+
B (2) A = 2np- B(n Î Z) (3) A = np + B (4) A = np - B |
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| 45. | (I + i)6 + ( I - i)6= |
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(1)
32 (2) 8 (3) 0 (4) 1 |
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| 46. | If a, b, c are three unequal numbers such that a, b, c are in A.P., and b - a, c - b, a are in G.P. then a : b : c = |
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(1) 2 : 3 : 5 (2) 1 : 2 : 4 (3) 1 : 3 : 5 (4) 1 : 2: 3 |
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| 47. | If the coefficients of (2r + 4)th and (r-2)th terms in the expansion of (1 + x)18 are equal then the value of r is |
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(1) 6 (2) 8 (3) 3 (4) 5 |
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| 48. | The value of tan 10 tan 20 ....tan 890 is |
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(1) 1/Ö2 (2) 2 (3) ¥ (4) 1 |
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| 49. | If A + B + C = p and cos A = cos B cos C then the value of cot B cot C = |
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(1) 1
(2) 1/3 (3) 1/2 (4) 2 |
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| 50. | From the top of a lighthouse, the angles of depression of two stations on opposite sides of it a distance a part are α and β. The height of the lighthouse is: |
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(1) a/cot
α + cot β
(2) a / cot α cot β (3) a tan α tan β/tan α + tan β (4) a cot α cotβ/cot α + cos β |
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| 51. | If y = Ösin x+ Ö sin x + Ösin x +..... to infinity then dy/dx = |
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(1) 2y -1/cos x (2) y2/cos x (3) cos x/2y -1 (4) sin x/2y-1 |
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| 52. | The derivative of tan-1 [ Ö1 + x2 -1/x] with reference to tan-1 (x): |
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(1) 1 (2) 1/2 (3) 1/3 (4) 1/4 |
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| 53. | If f(x) = cos-1 [1-( log x)2/1 + ( log x)2] then f' (e) is |
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(1) 1/e (2) Does not exist (3) 1 (4) 2/e |
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| 54. | If y = tan-1 [Ö1 - cos x/1 + cos x] then d2y/dx2 is |
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(1) 1/2 (2) 1/cos x (3) 0 (4) 1 |
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| 55. | If f(m, n) = òp/20 cosm cos nx dx and f(m,n) = m/m + n f(m -1, n-1) then f(n,n) = |
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(1) p/2n (2) p/4 (3) 1/2 f(m,n) (4) p/2n+1 |
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| 56. | The equation x2 + y2 - 2xy -1 = 0 represents: |
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(1) Two perpendicular straight lines (2) Hyperbola (3) A circle (4) Two parallel straight lines |
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| 57. | The given two circles x2 + y2 - 2x + 6y + 6 =0 and x2 + y2 - 5x + 6y + 15 = 0 are |
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(1) Concentric circles
(2) Intersecting type of circles (3) Touch each other externally (4) Touch each other internally |
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| 58. | 3x - 5y - 8 = 0 is the radical axis of a coaxial system of circles. If x2 + y2 + 2x - 4y + 7 = 0 is a number of the system, the member of the system through (1, 0) is given by: |
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(1) x2 + y2 +
2x - 4y + 5 =0 (2) x2 + y2 + 5x - 9y - 1 =0 (3) x2 + y2 + 8x - 14y - 9 =0 (4) x2 + y2 - 8x - 14y + 9 =0 |
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| 59. | If P is any point on the hyperbola (x-1)2/9 - (y+1)2/16 =1, and S1 and S2 are its foci, then lS1P- S2Pl = |
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(1) 6 (2) 8 (3) 3 (4) 4 |
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| 60. | If y = 2x + k is tangent to y2 = 8x, then k = |
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(1) 3 (2) 1 (3) 2 (4) 1/2 |
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