WB-JEE - 2009-Mathematics
1. If C is the reflecton of A (2, 4) in x-axis and B is the reflection of C in y-axis, then |AB| is
(A) 20 (B) 2√5 (C) 4√5 (D) 4
Ans : (C)
2. The value of [tex]\cos {15^ \circ }\cos 7{{{1^ \circ }} \over 2}\sin 7{{{1^ \circ }} \over 2}[/tex] is
(A) [tex]{1 \over 2}[/tex] (B) [tex]{1 \over 8}[/tex] (C) [tex]{1 \over 4}[/tex] (D) [tex]{1 \over {16}}[/tex]
Ans : (B)
3. The value of integral [tex]\int\limits_{ - 1}^1 {{{\left| {x + 2} \right|} \over {x + 2}}} dx[/tex] is
(A) 1 (B) 2 (C) 0 (D) –1
Ans : (B)
4. The line [tex]y = 2{t^2}[/tex] intersects the ellipse [tex]{{{x^2}} \over 9} + {{{y^2}} \over 4} = 1[/tex] in real points if
(A) | t | ≤ 1 (B) | t | < 1 (C) | t | > 1 (D) | t | ≥ 1
Ans : (A)
5. General solution of [tex]\sin x + \cos x = {\min }\limits_{a \in IR} \left\{ {1,{a^2} - 4a + 6} \right\}[/tex] is
(A) [tex]{{n\pi } \over 2} + {( - 1)^n}{\pi \over 4}[/tex] (B) [tex]2n\pi + {( - 1)^n}{\pi \over 4}[/tex] (C) [tex]n\pi + {( - 1)^{n + 1}}{\pi \over 4}[/tex] (D) [tex]n\pi + {( - 1)^n}{\pi \over 4} - {\pi \over 4}[/tex]
Ans : (D)
6. If A and B square matrices of the same order and AB = 3I, then A–1 is equal to
(A) 3B (B) [tex]{1 \over 3}B [/tex] (C) [tex]3{B^{ - 1}}[/tex] (D) [tex]{1 \over 3}{B^{ - 1}}[/tex]
Ans : (B)
7. The co-ordinates of the focus of the parabola described parametrically by [tex]x = 5{t^2} + 2[/tex] , [tex]y= 10t + 4[/tex] are
(A) (7, 4) (B) (3, 4) (C) (3, –4) (D) (–7, 4)
Ans : (A)
8. For any two sets A and B, A – (A – B) equals
(A) B (B) A – B (C) A ∩ B (D) Ac ∩ Bc
Ans : (C)
9. If a = 2√2 , b = 6 , A = 45°, then
(A) no triangle is possible (B) one triangle is possible
(C) two triangle are possible (D) either no triangle or two triangles are possible
Ans : (A)
10. A Mapping from IN to IN is defined as follows :f : IN → INf(n) = (n + 5)2 , n ∈ IN(IN is the set of natural numbers). Then
(A) f is not one-to-one (B) f is onto (C) f is both one-to-one and onto (D) f is one-to-one but not onto
Ans : (D)
11. In a triangle ABC if [tex]\sin A\sin B = {{ab} \over {{c^2}}}[/tex] then the triangle is
(A) equilateral (B) isosceles (C) right angled (D) obtuse angled
Ans : (C)
12. [tex]\int {{{dx} \over {\sin x + \sqrt 3 \cos x}}} [/tex] equals
(A) [tex]{1 \over 2}1n\left| {\tan \left( {{x \over 2} - {\pi \over 6}} \right)} \right| + c[/tex] (B) [tex]{1 \over 2}1n\left| {\tan \left( {{x \over 4} - {\pi \over 6}} \right)} \right| + c[/tex]
(C) [tex]{1 \over 2}1n\left| {\tan \left( {{x \over 2} + {\pi \over 6}} \right)} \right| + c[/tex] (D) [tex]{1 \over 2}1n\left| {\tan \left( {{x \over 4} + {\pi \over 3}} \right)} \right| + c[/tex]
where c is an arbitrary constant
Ans : (C)
13. The value of [tex]\left( {1 + \cos {\pi \over 6}} \right)\left( {1 + \cos {\pi \over 3}} \right)\left( {1 + \cos {{2\pi} \over 3}} \right)\left( {1 + \cos {{7\pi } \over 6}} \right)[/tex] is
(A) [tex]{3 \over {16}}[/tex] (B) [tex]{3 \over 8}[/tex] (C) [tex]{3 \over 4}[/tex] (D) [tex]{1 \over 2}[/tex]
Ans : (A)
14. If [tex]P = {1 \over 2}{\sin ^2}\theta + {1 \over 3}{\cos ^2}\theta [/tex] then
(A) [tex]{1 \over 3} \le P \le {1 \over 2}[/tex] (B) [tex]P \ge {1 \over 2}[/tex] (C) [tex]2 \le P \le 3[/tex] (D) [tex] - {{\sqrt {13} } \over 6} \le P \le {{\sqrt {13} } \over 6}[/tex]
Ans : (A)
15. A positive acute angle is divided into two parts whose tangents are [tex]{1 \over 2}[/tex] and [tex]{1 \over 3}[/tex]. Then the angle is
(A) [tex]{\pi \over 4}[/tex] (B) [tex]{\pi \over 5}[/tex] (C) [tex]{\pi \over 3}[/tex] (D) [tex]{\pi \over6}[/tex]
Ans : (A)
16. If [tex]f(x) = f(a - x)[/tex] then [tex]\int\limits_0^a x f(x)dx[/tex] is equal to
(A) [tex]\int\limits_0^a f (x)dx[/tex] (B) [tex]{{{a^2}} \over 2}\int\limits_0^a f (x)dx[/tex] (C) [tex]{a \over2}\int\limits_0^a f (x)dx[/tex] (D) [tex] - {a \over 2}\int\limits_0^a f (x)dx[/tex]
Ans : (C)
17. The value of [tex]\int\limits_0^\infty {{{dx} \over {({x^2} + 4)({x^2} + 9)}}} [/tex] is
(A) [tex]{\pi \over {60}}[/tex] (B) [tex]{\pi \over {20}}[/tex] (C) [tex]{\pi \over {40}}[/tex] (D) [tex]{\pi \over {80}}[/tex]
Ans : (A)
18. If [tex]{I_1} = \int\limits_0^{\pi /4} {{{\sin }^2}xdx} [/tex] and [tex]{I_1} = \int\limits_0^{\pi /4} {{{\cos }^2}xdx} [/tex] , then,
(A) [tex]{I_1} = {I_2}[/tex] (B) [tex]{I_1} < {I_2}[/tex] (C) [tex]{I_1} > {I_2}[/tex] (D) [tex]{I_2} = {I_1}+ \pi /4[/tex]
Ans : (B)
19. The second order derivative of a sin3t with respect to a cos3t at [tex]t = {\pi \over 4}[/tex] is
(A) 2 (B) [tex]{1 \over {12a}}[/tex] (C) [tex]{{4\sqrt 2 } \over {3a}}[/tex] (D) [tex]{{3a} \over {4\sqrt 2}}[/tex]
Ans : (C)
20. The smallest value of 5 cos θ + 12 is
(A) 5 (B) 12 (C) 7 (D) 17
Ans : (C)
21. The general solution of the differential equation [tex]{{dy} \over {dx}} = {e^{y + x}} + {e^{y - x}}[/tex] is
(A) e–y = ex – e–x + c (B) e–y = e-x – ex + c (C) e–y = ex + e–x + c (D) ey = ex + e–x + c
where c is an arbitrary constant
Ans : (B)
22. Product of any r consecutive natural numbers is always divisible by
(A) r ! (B) (r + 4) ! (C) (r + 1) ! (D) (r + 2) !
Ans : (A)
23. The integrating factor of the differential equation [tex]x\log x{{dy} \over {dx}} + y = 2\log x[/tex] is given by
(A) ex (B) log x (C) log (log x) (D) x
Ans : (B)
24. If x² + y² = 1 then
(A) yy′′ − (2y′)² + 1 = 0 (B) yy′′ + ( y′)² +1 = 0 (C) yy′′ − (y′)² −1 = 0 (D) yy′′ + (2y′)² + 1 = 0
Ans : (B)
25. If c0, c1, c2, ..................., cn denote the co-efficients in the expansion of (1 + x)ⁿ
then the value of c1 + 2c2 + 3c3 + ..... + ncn is
(A) n.2n-1 (B) (n + 1)n-1 (C) (n + 1)2n (D) (n + 2) 2n-1
Ans. (A)
26. A polygon has 44 diagonals. The number of its sides is
(A) 10 (B) 11 (C) 12 (D) 13
Ans : (B)
27. If α, β be the roots of x² – a(x – 1) + b = 0, then the value of [tex]{1 \over {{\alpha ^2} - a\alpha }} + {1 \over{{\beta ^2} - a\beta }} + {2 \over {a + b}}[/tex]
(A) [tex]{4 \over {a + b}}[/tex] (B) [tex]{1 \over {a + b}}[/tex] (C) 0 (D) –1
Ans : (C)
28. The angle between the lines joining the foci of an ellipse to one particular extremity of the minor axis is 90° . The eccentricity of the ellipse is
(A) [tex]{1 \over 8}[/tex] (B) [tex]{1 \over {\sqrt 3 }}[/tex] (C) [tex]\sqrt {{2 \over 3}} [/tex] (D) [tex]\sqrt{{1 \over 2}} [/tex]
Ans : (D)
29. The order of the differential equation [tex]{{{d^2}y} \over {d{x^2}}} = \sqrt {1 - {{\left( {{{dy} \over {dx}}}\right)}^2}} [/tex] is
(A) 3 (B) 2 (C) 1 (D) 4
Ans : (B)
30. The sum of all real roots of the equation |x – 2|² + |x – 2| – 2 = 0
(A) 7 (B) 4 (C) 1 (D) 5
Ans : (B)
31. If [tex]\int\limits_{ - 1}^4 {f(x)dx = 4} [/tex] and [tex]\int\limits_2^4 {\left\{ {3 - f(x)} \right\}dx = 7} [/tex] then the value of [tex]\int\limits_{ - 1}^2 {f(x)dx} [/tex]
(A) –2 (B) 3 (C) 4 (D) 5
Ans : (D)
32. For each n∈ N, 23n – 1 is divisible by
(A) 7 (B) 8 (C) 6 (D) 16
where N is a set of natural numbers
Ans : (A)
33. The Rolle’s theorem is applicable in the interval – 1 ≤ x ≤ 1 for the function
(A) ƒ(x) = x (B) ƒ(x) = x² (C) ƒ(x) = 2x3 + 3 (D) ƒ(x) = |x|
Ans : (B)
34. The distance covered by a particle in t seconds is given by x = 3 + 8t – 4t² . After 1 second velocity will be
(A) 0 unit/second (B) 3 units/second (C) 4 units/second (D) 7 units/second
Ans : (A)
35. If the co-efficients of x² and x³ in the expansion of (3 + ax)9 be same, then the value of ‘a’ is
(A) [tex]{3 \over 7}[/tex] (B) [tex]{7 \over 3}[/tex] (C) [tex]{7 \over 9}[/tex] (D) [tex]{9 \over 7}[/tex]
Ans : (D)
36. The value of [tex]\left( {{1 \over {{{\log }_3}12}} + {1 \over {{{\log }_4}12}}} \right)[/tex] is
(A) 0 (B) [tex]{1 \over 2}[/tex] (C) 1 (D) 2
Ans : (C)
37. If x = loga bc, y = logb ca, z = log<sub>c</sub> ab, then the value of [tex]{1\over {1 + x}} + {1 \over {1 + y}} + {1 \over {1 + z}}[/tex] will be
(A) x + y + z (B) 1 (C) ab + bc + ca (D) abc
Ans : (B)
38. Using binomial theorem, the value of (0.999)³ correct to 3 decimal places is
(A) 0.999 (B) 0.998 (C) 0.997 (D) 0.995
Ans : (C)
39. If the rate of increase of the radius of a circle is 5 cm/.sec., then the rate of increase of its area, when the radius is 20 cm, will be
(A) 10π (B) 20π (C) 200π (D) 400π
Ans : (C)
40. The quadratic equation whose roots are three times the roots of 3ax² + 3bx + c = 0 is
(A) ax² + 3bx + 3c = 0 (B) ax² + 3bx + c = 0 (C) 9ax² + 9bx + c = 0 (D) ax² + bx + 3c = 0
Ans : (A)
41. Angle between y² = x and x² = y at the origin is
(A) [tex]2{\tan ^{ - 1}}\left( {{3 \over 4}} \right)[/tex] (B) [tex]{\tan ^{ - 1}}\left( {{4 \over 3}} \right)[/tex] (C)[tex]{\pi \over 2}[/tex] (D) [tex]{\pi \over 4}[/tex]
Ans : (C)
42. In triangle ABC, a = 2, b = 3 and [tex]\sin A = {2 \over 3}[/tex] , then B is equal to
(A) 30° (B) 60° (C) 90° (D) 120°
Ans : (C)
43. [tex]\int\limits_0^{1000} {{e^{x - [x]}}} [/tex] is equal to
(A) [tex]{{{e^{1000}} - 1} \over {e - 1}}[/tex] (B) [tex]{{{e^{1000}} - 1} \over {1000}}[/tex] (C) [tex]{{e - 1}\over {1000}}[/tex] (D) 1000 (e – 1)
Ans : (D)
44. The coefficient of xⁿ, where n is any positive integer, in the expansion of (1 + 2x + 3x² + ......... ∞)½ is
(A) 1 (B) [tex]{{n + 1} \over 2}[/tex] (C) 2n + 1 (D) n + 1
Ans : (A)
45. The circles x² + y² – 10x + 16 = 0 and x² + y² = a² intersect at two distinct points if
(A) a < 2 (B) 2 < a < 8 (C) a > 8 (D) a = 2
Ans. (B)
46. [tex]\int {{{{{\sin }^{ - 1}}x} \over {\sqrt {1 - {x^2}} }}dx} [/tex] is equal to
(A) [tex]\log ({\sin ^{ - 1}}x) + c[/tex] (B) [tex]{1 \over 2}{({\sin ^{ - 1}}x)^2} + c[/tex] (C) [tex]\log \left({\sqrt {1 - {x^2}} } \right) + c[/tex] (D) [tex]\sin \left( {{{\cos }^{ - 1}}x} \right) + c[/tex]
where c is an arbitrary constant
Ans : (B)
47. The number of points on the line x + y = 4 which are unit distance apart from the line 2x + 2y = 5 is
(A) 0 (B) 1 (C) 2 (D) Infinity
Ans : (A)
48. Simplest form of [tex]{2 \over {\sqrt {2 + \sqrt {2 + \sqrt {2 + 2\cos 4x} } } }}[/tex] is
(A) [tex]\sec {x \over 2}[/tex] (B) sec x (C) cosec x (D) 1
Ans : (A)
49. If [tex]y = {\tan ^{- 1}}\sqrt {{{1 - \sin x} \over {1 + \sin x}}} [/tex] , then the value of [tex]{{dy} \over {dx}}[/tex] at [tex]x = {\pi \over 6}[/tex]
(A) [tex] - {1 \over 2}[/tex] (B) [tex] {1 \over 2}[/tex] (C) 1 (D) –1
Ans : (A)
50. If three positive real numbers a , b , c are in A.P. and abc = 4 then minimum possible value of b is
(A) 23/2 (B) 22/3 (C) 21/3 (D) 25/2
Ans : (B)
51. If [tex]5\cos 2\theta + 2{\cos ^2} {{\theta} \over 2} + 1 = 0[/tex] , when (0 < θ < π), then the values of θ are :
(A) [tex]{\pi \over 3} \pm \pi [/tex] (B) [tex]{\pi \over 3},{\cos ^{ - 1}}\left ( {{3 \over 5}} \right)[/tex] (C) [tex] {\cos ^{ - 1}}\left ( {{3 \over 5}} \right) \pm \pi [/tex] (D) [tex]{\pi \over 3},\pi - {\cos ^{ - 1}}\left ( {{3 \over5}} \right)[/tex]
Ans : (D)
52. For any complex number z, the minimum value of |z| + |z – 1| is
(A) 0 (B) 1 (C) 2 (D) –1
Ans : (B)
53. For the two circles x² + y² = 16 and x² + y² – 2y = 0 there is / are
(A) one pair of common tangents (B) only one common tangent
(C) three common tangents (D) no common tangent
Ans : (D)
54. If C is a point on the line segment joining A (–3, 4) and B (2, 1) such that AC = 2BC , then the coordinateof C is
(A) [tex]\left( {{1 \over 3},2} \right)[/tex] (B) [tex]\left( {2,{1 \over 3}} \right)[/tex] (C) (2, 7) (D) (7, 2)
Ans : (A)
55. If a , b , c are real, then both the roots of the equation (x – b) (x – c) + (x – c) (x – a) + (x – a) (x – b) = 0 are always
(A) positive (B) negative (C) real (D) imaginary
Ans : (C)
56. The sum of the infinite series [tex]1 + {1 \over {2!}} + {{1.3} \over {4!}} + {{1.3.5} \over {6!}} + \cdots \cdots [/tex]
(A) e (B) e² (C) √e (D) [tex]{1 \over e}[/tex]
Ans : (C)
57. The point (–4, 5) is the vertex of a square and one of its diagonals is 7x – y + 8 = 0. The equation of the other diagonal is
(A) 7x – y + 23 = 0 (B) 7y + x = 30 (C) 7y + x = 31 (D) x – 7y = 30
Ans : (C)
58. The domain of definition of the function [tex]f(x) = \sqrt {1 + {{\log }_e}(1 - x)} [/tex] is
(A) [tex] - \infty < x \le 0[/tex] (B) [tex] - \infty < x \le {{e - 1} \over e}[/tex] (C) [tex] - \infty < x \le 1[/tex] (D) [tex]x \ge 1 - e[/tex]
Ans : (B)
59. For what value of m, [tex]{{{a^{m + 1}} + {b^{m + 1}}} \over {{a^m} + {b^m}}}[/tex] is the arithmetic meanof ‘a’ and ‘b’ ?
(A) 1 (B) 0 (C) 2 (D) None
Ans : (B)
60. The value of the limit [tex] {\lim }\limits_{x \to 1} {{\sin ({e^{x - 1}} - 1)} \over {\log x}}[/tex] is
(A) 0 (B) e (C) [tex]{1 \over e}[/tex] (D) 1
Ans : (D)
61. Let [tex]f(x) = {{\sqrt {x + 3} } \over {x + 1}}[/tex] then the value of [tex] {Lt}\limits_{x \to - 3 - 0} f(x)[/tex] is
(A) 0 (B) does not exist (C) [tex]{1 \over 2}[/tex] (D) [tex]-{1 \over 2}[/tex]
Ans : (B)
62. ƒ(x) = x + | x | is continuous for
(A) x∈(−∞,∞) (B) x∈(−∞,∞) −{0} (C) only x > 0 (D) no value of x
Ans : (A)
63. [tex]\tan \left[ {{\pi \over 4} + {1 \over 2}{{\cos }^{ - 1}}\left( {{a \over b}} \right)} \right] + \tan \left[ {{\pi \over4} - {1 \over 2}{{\cos }^{ - 1}}\left( {{a \over b}} \right)} \right][/tex] is equal to
(A) [tex]{{2a} \over b}[/tex] (B) [tex]{{2b} \over a}[/tex] (C) [tex]{a \over b}[/tex] (D) [tex]{b \over a}[/tex]
Ans : (B)
64. If [tex]i = \sqrt { - 1} [/tex] and n is a positive integer, then [tex]{i^n} + {i^{n + 1}} + {i^{n + 2}} + {i^{n + 3}}[/tex] is euqal to
(A) 1 (B) i (C) iⁿ (D) 0
Ans : (D)
65. [tex]\int {{{dx} \over {x(x + 1)}}} [/tex] equals
(A) [tex]ln\left| {{{x + 1} \over x}} \right| + c[/tex] (B) [tex]ln\left| {{x \over {x + 1}}} \right| + c[/tex] (C)[tex] ln\left| {{{x - 1} \over x}} \right| + c[/tex] (D) [tex]ln\left| {{{x - 1} \over {x + 1}}} \right| + c[/tex]
where c is an arbitrary constant.
Ans : (B)
66. If a, b, c are in G.P. (a > 1, b > 1, c > 1), then for any real number x (with x > 0, x ≠ 1), loga x , logb x, logc x are in
(A) G..P. (B) A.P. (C) H.P. (D) G..P. but not in H.P.
Ans : (C)
67. A line through the point A (2, 0) which makes an angle of 30° with the positive direction of x-axis is rotated about A in clockwise direction through an angle 15°. Then the equation of the straight line in the new position is
(A) (2 - √3)x + y - 4 + 2√3 = 0 (B) (2 - √3)x - y - 4 + 2√3 = 0
(C) (2 - √3)x - y + 4 + 2√3 = 0 (D) (2 - √3)x + y + 4 + 2√3 = 0
Ans : (B)
68. The equation [tex]\sqrt 3 \sin x + \cos x = 4[/tex] has
(A) only one solution (B) two solutions (C) infinitely many solutions (D) no solution
Ans : (D)
69. The slope at any point of a curve y = ƒ(x) is given by [tex]{{dy} \over {dx}} = 3{x^2}[/tex] and it passes through (–1 , 1). The equation of the curve is
(A) y = x³ + 2 (B) y = – x³ – 2 (C) y = 3x³ + 4 (D) y = – x³ + 2
Ans : (A)
70. The modulus of [tex]{{1 - i} \over {3 + i}} + {{4i} \over 5}[/tex] is
(A) [tex]\sqrt 5 [/tex] unit (B) [tex]{{\sqrt {11} } \over 5}[/tex] unit (C) [tex]{{\sqrt 5 } \over 5}[/tex] unit (D) [tex]{{\sqrt {12} } \over 5}[/tex] unit
Ans : (C)
71. The equation of the tangent to the conic x² – y² – 8x + 2y + 11 = 0 at (2, 1) is
(A) x + 2 = 0 (B) 2x + 1 = 0 (C) x + y + 1 = 0 (D) x – 2 = 0
Ans : (D)
72. A and B are two independent events such that P(A∪B') = 0.8 and P(A) = 0.3. The P(B) is
(A) [tex]{2 \over 7}[/tex] (B) [tex]{2 \over 3}[/tex] (C) [tex]{3 \over 8}[/tex] (D) [tex]{1 \over 8}[/tex]
Ans : (A)
73. The total number of tangents through the point (3, 5) that can be drawn to the ellipses 3x² + 5y² = 32 and 25x² + 9y² = 450 is
(A) 0 (B) 2 (C) 3 (D) 4
Ans : (C)
74. The value of [tex] {\lim }\limits_{n \to \infty } \left[ {{n \over {{n^2} + {1^2}}} + {n \over {{n^2} + {2^2}}} + \cdots \cdots {n \over {{n^2} + {n^2}}}} \right][/tex] is
(A) [tex]{\pi \over 4}[/tex] (B) log 2 (C) zero (D)1
Ans : (A)
75. A particle is moving in a straight line. At time t, the distance between the particle from its starting pointis given by x = t – 6t² + t³. Its acceleration will be zero at
(A) t = 1 unit time (B) t = 2 unit time (C) t = 3 unit time (D) t = 4 unit time
Ans : (B)
76. Three numbers are chosen at random from 1 to 20. The probability that they are consecutive is
(A) [tex]{1 \over {190}}[/tex] (B) [tex]{1 \over {120}}[/tex] (C) [tex]{3 \over {190}}[/tex] (D) [tex]{5\over {190}}[/tex]
Ans : (C)
77. The co-ordinates of the foot of the perpendicular from (0, 0) upon the line x + y = 2 are
(A) (2, –1) (B) (–2, 1) (C) (1, 1) (D) (1, 2)
Ans : (C)
78. If A is a square matrix then,
(A) A + AT is symmetric (B) AAT is skew - symmetric (C) AT + A is skew-symmetric (D) ATA is skew symmetric
Ans : (A)
79. The equation of the chord of the circle x² + y² – 4x = 0 whose mid point is (1, 0) is
(A) y = 2 (B) y = 1 (C) x = 2 (D) x = 1
Ans : (D)
80. If A² – A + I = 0, then the inverse of the matrix A is
(A) A – I (B) I – A (C) A + I (D) A
Ans : (B)
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