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1998 Mathematics
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(1) H.P.
(2) both G.P and H.P (3) A.P (4) G.P |
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| 2. | If x = cos q + i sin q, then x2n-1/x2n+1= | |
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(1) i tan (nq)
(2) tan (nq) (3) tan q (4) i tan q |
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| 3. | If y = tan-1 (sin x/ 1 + cos x) then dy/dx = | |
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(1) 1/3 (2) 2 (3) 1/2 (4) 3/2 |
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| 4. | If 2x2 + 4xy + 3y2 = 0 then d2y/dx2= | |
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(1) 4/3 (2) 1/(2x + y)2 (3) 1/2 (4) 0 |
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| 5. | Which of the following in not true? | |
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(1) If lim f(x) exists then f(x) is
continuous at x = a x®a (2) The identity function f(x) is continuous for all real x (3) A differentiable function is a continuous function (4) The function f(x) = [x] is continuous at the origin |
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| 6. | Which of the following is always true? | |
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(1) ~ (p
Ù q) º (~p
Ù ~q) (2) ~ (p ® q) º (p v ~q) (3) ~(p v~q) º (~p v~q) (4) (p ® q) º (~ q ® ~P) |
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| 7. | The characteristic
roots of the matrix  |
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(1) a/x, b/y, c/z (2) ax, by, cz (3) a, b, c (4) x, y, z |
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| 8. |
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(1) p/3 (2) p/6 (3) p/2 (4) p/4 |
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| 9. |
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(1) -3/2 (2) 1/2 (3) 2/3 (4) -2/3 |
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| 10. | The area of the parallelogram two of
whose adjacent sides are |
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(1) 10 sq. units (2) 5 sq. units (3) 10 Ö3 sq. units (4) 5Ö3 sq. units |
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| 11. | The ellipse x2/a2 + y2/b2 = 1 cuts X -axis at A and A' and Y-axis at B and B'. The line joining the focus S and B makes an angle 3p/4 with the X - axis. This eccentricity of the ellipse is: | |
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(1) 1/3 (2) 1/4 (3) 1/Ö2 (4) 1/2 |
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| 12. | The point of intersection of two perpendicular tangents to x2/a2 - y2/b2 =1 lies on the circle: | |
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(1) x2 + y2 = a2
- b2 (2) x2 + y2 = a2 + b2 (3) x2 + y2 = a2 (4) x2 + y2 = b2 |
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| 13. | The equation of the normal to the hyperbola x2/16 - y2/9 =1 at (-4, 0) is: | |
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(1) x = 1 (2) Y = 0 (3) 2X - 3Y =1 (4) X = 0 |
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| 14. | If sin-1 (x) + sin-1 (y) + sin-1 (z) = 3p/2 the value of (x100 + y100 + z100) - (x101 + y101 + z101) = | |
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(1) 2 (2) 4 (3) 0 (4) 1 |
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| 15. | The value of sin [tan-1 (1 - x2/2x) + cos-1 (1 - x2/1 + x2) ] = | |
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(1) 1 (2) 1/3 (3) 0 (4) 1/2 |
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| 16. | The condition that the line x/a + y/b = 1 be a tangent to the curve x2/3 + y2/3 =1 is | |
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(1) a2 + b2 + 1 (2) 1/a2 + 1/b2 =1 (3) a2 + b2 = 2 (4) a2 + b2 =-2 |
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| 17. | Let x1, x2,x3... be positive Integers in A.P, such that x1+ x2+ x3 =12 and x4 + x6 =14 then x5 is: | |
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(1)
9 (2) 8 (3) 7 (4) 10 |
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| 18. |
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(1) 5 (2) 7 (3) 3 (4) 2 |
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| 19. | A circle and a square have the same perimeter, Then: | |
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(1) Area of the square is larger (2) The area of the circle is p times the area of the square (3) Their areas are equal (4) Area of the circle is larger |
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| 20. | The equation x2 + y2 - 2x - 4y + 5 = 0 represents: | |
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(1) an ellipse (2) a point (3) a circle (4) a pair of straight lines |
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| 21. | Which of the following is not correct? | |
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(1) If A is any square matrix the A (adj A) =A) A=lAl (2) A square matrix A is invertible if f lAl ¹ 0. (3) If A is any square matrix and B is the transpose of the matrix of cofactors of corresponding elements of A, then B is called the adjoint of A. (4) Every non-singular matrix has a unique inverse |
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| 22. | The solution of system of equation 1 + x + x2 +.....+ x23 = 0 and 1 + x + x2 + .... + x19 = 0, x ¹ 0 is | |
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(1) x =-1 (2) x =- 4Ö1 (3) x = Ö1 (4) x = 3Ö1 |
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| 23. | Let f(x+y) = f(x). f(x). f(y) V x and y. Suppose f(5) =2, f'(0) =3, then f'(5) is: | |
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(1) 6 (2) 8 (3) 3 (4) 4 |
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| 24. |
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(1) (2) (3) (4) |
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| 25. | If log7 (log5 (Öx +5 + Öx) = 0 then the value of x is: | |
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(1) 3 (2) 5 (3) 1 (4) 4 |
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| 26. |
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(1) -p/2 (2) -p/4 (3) p/2 (4) p/4 |
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| 27. | The value of |
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(1) 6 abcd (2) 4 abcd (3) abcd (4) 8 abcd |
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| 28. | The combined equation through the origin perpendicular to the lines represented by ax2 + 2hxy + by2 = 0 is given by: | |
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(1) bx2 -2hxy - ay2
=0 (2) ax2 -2hxy - by2 =0 (3) bx2 -2hxy + ay2 =0 (4) bx2 + 2hxy +ay2 =0 |
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| 29. | When 730 is divided by 5 the remainder is : | |
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(1) 3 (2) 2 (3) 1 (4) 4 |
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| 30. | If a and b are natural such that a2 - b2 is a prime number, then: | |
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(1) a2 - b2 = 1 (2) a2 - b2 = a + b (3) a2 - b2 = a - b (4) a2 - b2 = 2 |
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| 31. | If Um = òp/40 tann x dx for n ³ 2, then Um + Un-2= | |
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(1) 1/n +1 (2) 1/n + 1/n -1 (3) 1/n -1 (4) 1/n |
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| 32. | The equation of the parabola whose vertex is (1,-2) and focus (1, -1) is; | |
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(1) x2 = 4 (y + 2) (2) (x+1)2 = 4(y + 2) (3) (x + 1)2= 4(y - 2) (4) (x - 1)2 = 4(y + 2) |
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| 33. | If the normals to the parabola y2 = 4ax at (at21) and (at22, at2) are mutually perpendicular, then : | |
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(1) t1t2=-1 (2) t1-t2=-1 (3) t1+t2=-1 (4) t1t2=1 |
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| 34. |
If (I+i) ( I+ 2i) (I + 3i)...(I + ni) = x + iy then the value of 2, 5, 10....(I + n2) = |
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(1) x2 - y2
(2) x2 + y2 (3) Ö x + ÖY/Öx - Öy (4) Öx/2 + /Öy/3 |
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| 35. | A circle is drawn on the major axis of the ellipse 9x2 + 16y2 = 144 as diameter. The equation of the circle is: | |
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(1) x2 + y2 =3 (2) x2 + y2 =4 (3) x2 + y2 =9 (4) x2 + y2 =16 |
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| 36. | In the expansion (I + x)50 the sum of the coefficients of odd powers of x is: | |
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(1) 250 (2) 251 (3) 0 (4) 249 |
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| 37. | If sin is q, cos q and tan q are in geometric progression, then cot6 q -cot2 q= | |
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(1) 2 (2) 3 (3) 1 (4) 1/2 |
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| 38. | If A and B are acute angles such that sin that sin A= sin 2 B, and 2 cos2 A=3 cos2 B then A is: | |
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(1) p/3 (2) p/8 (3) p/4 (4) p/6 |
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| 39. | The value of f (0) so that the function f(x) = 2x-sin-1(x)/2x+sin-1(x) is continuous at each point domain is equal to | |
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(1) -1/3 (2) 1/3 (3) 2 (4) 2/3 |
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| 40. |
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(1) 2 (2) 1/2 (3) 0 (4) 1 |
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| 41. |
® ®
® ® If a and b and unit vectors and q is the angle between them then (a + b) = |
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(1) 2 cos q/2 (2) 2 cos q (3) 2 units (4) 2 sin q/2 |
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| 42. | In the group G = [1, 3, 7, 9] under addition modulo 10, (3 x 7-1)-1 is equal to: | |
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(1) 3 (2) 7 (3) 5 (4) 9 |
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| 43. | Which of the following statements is not true? | |
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(1) The cube roots of unity are 1, 1-iÖ3/2 , 1+iÖ3/2
, (2) In an abelian group, (ab)2 = a2b2 for all a, b Î G (3) The identity element in a group is unique (4) In a group of even order, there exists an element a ¹ c such that a-1 = a |
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| 44. | The circle x2 + y2 - 2x + 6y + 6 = 0 and x2 + y2- 5x + 6y + 15 =0 | |
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(1) Cut each other orthogonally (2) Concentric circle (3) Touch each other internally (4) Touch each other externally |
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| 45. | The limiting points of the coaxial system of circles x2 + y2 + 2gx + c =0 is | |
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(1) (+
Öc,0) (2) (+c,0) (3) (+ Ög,0) (4) (+ Ö-c,0) |
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| 46. | The value of tan [ cos-1(4/5) + tan-1 ( 2/3) ] is: | |
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(1) 7/6 (2) 16/7 (3) 6/17 (4) 17/6 |
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| 47. | The equation of the circle circumscribing the triangle formed by the points ( 0,0). (1,0) and (0,1) is: | |
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(1) x2 + y2 -
x - y =0 (2) x2 + y2 + x - y - 2 =0 (3) x2 + y2 + x + y =0 (4) x2 + y2 + x - y + 2 =0 |
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| 48. | If f(x) =1-x/1+x then f(f( cos x) equals: | |
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(1) tan2x/2 (2) cos x (3) x (4) 1-cos x/1+cos x |
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| 49. | In a triangle ABC, if sin A = 2 sin C cos B then: | |
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(1) a = b (2) a = b/2 (3) b = c (4) c = a |
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| 50. | If the area of an expanding circular region increases at a constant rate with respect to time, then the rate of increase of the perimeter with respect to time: | |
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(1) varies directly is square of the
radius (2) remains constant (3) varies directly as radius (4) varies inversely as radius |
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| 51. | ò esin2 xsin 2x dx = | |
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(1) sin2 x + c
(2) sin 2x + c (3) esin2x + c (4) esin2 x + c |
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| 52. | òp/20 log (tan x) dx = | |
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(1) 0 (2) p2/4 (3) p/4 (4) p/2 |
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| 53. | In rule method the null set is represented by: | |
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(1) [x l x ¹x] (2) [x l x =x] (3) f (4) [ ] |
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| 54 |
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(1) 7/6 (2) 6/7 (3) 49/36 (4) 36/49 |
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| 55. | The correct match of the following table is given by: | |
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(1)
Leibnitz (2) Euler (3) Cayley Hamilton (4) George Boole (5) De Moivre |
eiq Mathematical logic Calculus (eiq)n= ei(nq) Theory of matrices |
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(1)
1 = (d), 2 = (a), 3= (c), 4 = (b), 5 = (a) (2) 1 = (b), 2 = (d), 3= (c), 4 = (c), 5 = (e) (3) 1 = (c), 2 = (a), 3= (d), 4 = (b), 5 = (e) (4) 1 = (c), 2 = (a), 3= (c), 4 = (b), 5 = (d) |
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| 56. |
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(1) e (4) -e (3) 1/e (4) 1/e |
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| 57. | ò-aa lxl dx is equal to: | |
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(1) a2 (2) a2/4 (3) 0 (4) a2/2 |
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| 58. | The elimination of A and B from the equation y2 = Ax + B gives the differential equation of order: | |
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(1) Third (2) Zero (3) First (4) Second |
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| 59. | If p and q are the order and degree of the differential equation y2(d2y/dx2)2 + 3x ( dy/dx) + x2y2 = sin x then: | |
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(1) p >q
(2) p/q=1/2 (3) p = q (4) p < q |
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| 60. | "The diagonals of a rhombus are perpendicular". The contrapositive of the above statement is | |
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(1) If the diagonals are not
perpendicular then the figure is not rhombus (2) If the diagonals are not perpendicular then the figure is a rhombus (3) If the diagonals are perpendicular, then the figure is a rhombus (4) If the figure is not a rhombus, then its diagonals are not perpendicular. |
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