WBJEE Mathematics Question Paper 2011 (Eng)

WBJEE - 2011 -  Mathematics

1.   The eccentricity of the hyperbola $4{x^2} - 9{y^2} = 36$ is

(A) ${{\sqrt {11} } \over 3}$       (B) ${{\sqrt {15} } \over 3}$      (C)  ${{\sqrt {13} } \over 3}$      (D) ${{\sqrt {14} } \over 3}$

Ans : (C)

2.   The length of the latus rectum of the ellipse $16{x^2} + 25{y^2} = 400$ is

(A) 5/16 unit         (B) 32/5 unit          (C) 16/5 unit         (D) 5/32 unit

Ans : (B)

3.   The vertex of the parabola y² + 6x - 2y + 13 = 0 is

(A)  (1, - 1)       (B) (-2,  1)       (C) $\left( {{3 \over 2},1} \right)$       (D) $\left( { - {7 \over 2},1} \right)$

Ans : (B)

4.   The coordinates of a moving point p are $(2{t^2} + 4,4t + 6)$. Then its locus will be a

(A) circle         (B) straight line          (C) parabola         (D) ellipse

Ans : (C)

5.   The equation 8x² + 12y² - 4x + 4y - 1 = 0 represents

(A) an ellipse        (B) a hyperbola       (C) a parabola        (D) a circle

Ans : (A)

6.  If the straight line y = mx lies outside of the circle x² + y² - 20y + 90 = 0, then the value of m will satisfy

(A) m < 3        (B) |m| < 3         (C) m > 3         (D) |m| > 3

Ans : (B)

7.  The locus of the centre of a circle which passes through two variable points (a, 0), (–a, 0) is

(A) x = 1       (B) x + y = a      (C) x + y = 2a        (D) x = 0

Ans : (D)

8.   The coordinates of the two points lying on x + y = 4 and at a unit distance from the straight line 4x + 3y = 10 are

(A) (–3, 1), (7, 11)      (B) (3, 1), (–7, 11)       (C) (3, 1), (7, 11)       (D) (5, 3), (–1, 2)

Ans : (B)

9.   The intercept on the line y = x by the circle ${x^2} + {y^2} - 2x = 0$ is AB.  Equation of the circle with AB as  diameter is

(A) ${x^2} + {y^2} = 1$         (B) $x(x - 1) + y(y - 1) = 0$

(C) ${x^2} + {y^2} = 2$         (D) $(x - 1)(x - 2) + (y - 1) + (y - 2) = 0$

Ans : (B)

 10.   If the coordinates of one end of a diameter of the circle x² + y² + 4x - 8y + 5 = 0, is (2,1), the coordinates of the other end is

(A) (–6, –7)     (B) (6, 7)       (C) (–6, 7)      (D) (7, –6)

Ans : (C)

11.  If the three points A(1,6),  B(3, –4) and C(x, y) are collinear then the equation satisfying by x and y is

(A) $5x + y - 11 = 0$        (B) $5x + 13y + 5 = 0$       (C) $5x - 13y + 5 = 0$        (D) $13x - 5y + 5 = 0$

Ans : (A)

12.   If $\sin \theta = {{2t} \over {1 + {t^2}}}$ and $\theta$ lies in the second quadrant, then $\cos \theta$ is equal to

(A) ${{1 - {t^2}} \over {1 + {t^2}}}$      (B) ${{{t^2} - 1} \over {1 + {t^2}}}$      (C) ${{ - \left| {1 - {t^2}} \right|} \over {1 + {t^2}}}$     (D) ${{1 + {t^2}} \over {\left| {1 - {t^2}} \right|}}$

Ans : (C)

13.    The solutions set of inequation cos^–1x < sin^–1x is

(A)  $\left[ { - 1,1} \right]$       (B) $\left[ {{1 \over {\sqrt 2 }},1} \right]$      (C) $\left[ {0,1} \right]$      (D) $\left( {{1 \over {\sqrt 2 }},1} \right]$

Ans : (D)

14.   The number of solutions of 2sin x + cos x = 3 is

(A) 1        (B) 2       (C) infinite         (D) No solution

Ans : (D)

15.  Let $\tan \alpha = {a \over {a + 1}}$ and $\tan \beta = {1 \over {2a + 1}}$ then $\alpha + \beta$ is

(A) ${\pi \over 4}$        (B) ${\pi \over 3}$       (C) ${\pi \over 2}$       (D) $\pi$

Ans : (A)

16. If $\theta + \phi = {\pi \over 4}$, then  $(1 + \tan \theta )(1 + \tan \phi )$ is equal to

(A) 1       (B) 2        (C) 5/2        (D) 1/3

Ans : (B)

17.   If $\sin \theta$  and $\cos \theta$ are the roots of the equation $a{x^2} - bx + c = 0$, then a, b and c satisfy the relation

(A) ${a^2} + {b^2} + 2ac = 0$        (B) ${a^2} - {b^2} + 2ac = 0$

(C) ${a^2} + {c^2} + 2ab = 0$       (D) ${a^2} - {b^2} - 2ac = 0$

Ans : (B)

18.   If A and B are two matrices such that A + B and AB are both defined, then

(A) A and B can be any matrices          (B) A, B are square matrices not necessarily of the same order

(C) A, B are square matrices of the same order        (D) Number of columns of A = number of rows of B

Ans : (C)

19.   If $A = \left( {\matrix{ 3 & {x - 1} \cr {2x + 3} & {x + 2} \cr } } \right)$ is a symmetric matrix, then the value of x  is

(A) 4      (B) 3     (C) –4     (D) –3

Ans : (C)

20.  If $Z = \left( {\matrix{1 & {1 + 2i} & { - 5i}\cr {1 - 2i} & { - 3} & {5 + 3i}\cr {5i} & {5 - 3i} & 7 \cr } } \right)$ then  $\left( {i = \sqrt { - 1} } \right)$

(A) z is purely real      (B) z is purely imaginary     (C) $z + \bar z = 0$      (D) $(z - \bar z)i$ is purely imaginary

Ans : (A)

21.  The equation of the locus of the point of intersection of the straight lines $x\sin \theta + (1 - \cos \theta )y = a\sin \theta$ and $x\sin \theta - (1 + \cos \theta)y + a\sin \theta = 0$ is

(A) $y = \pm ax$        (B) $x = \pm ay$        (C) ${y^2} = 4x$      (D) ${x^2} + {y^2} = {a^2}$

Ans : (D)

22.   If $\sin \theta + \cos \theta = 0$ and $0 < \theta < \pi$,  then $\theta$

(A) 0       (B) ${\pi \over 4}$     (C) ${\pi \over 2}$    (D)  ${{3\pi} \over 4}$

Ans : (D)

23.   The value of cos 15° – sin 15° is

(A) 0     (B) ${1 \over {\sqrt 2 }}$      (C) $- {1 \over {\sqrt 2 }}$      (D) ${1 \over {2\sqrt 2 }}$

Ans : (B)

24.   The period of the function ƒ(x)= cos 4x + tan 3x is

(A)$\pi$       (B)${\pi \over 2}$       (C) ${\pi \over 3}$      (D) ${\pi \over 4}$

Ans : (A)

25.   If $y = 2{x^3} - 2{x^2} + 3x - 5$, then for x = 2 and $\triangle x = 0.1$ value of is $\triangle y$ is

(A) 2.002        (B) 1.9       (C) 0        (D) 0.9

Ans : (B)

26. The approximate value of $\sqrt[5] {33}$ correct to 4 decimal places is

(A) 2.0000        (B) 2.1001       (C) 2.0125       (D) 2.0500

Ans : (C)

27.   The value of $\int\limits_{ - 2}^2 {(x\cos x + \sin x + 1)dx}$ is

(A) 2       (B) 0        (C) – 2        (D) 4

Ans : (D)

28.  For the function $f(x) = {e^{\cos x}}$, Rolle’s theorem is

(A) applicable when ${\pi \over 2} \le x \le {{3\pi } \over 2}$        (B) applicable when $0 \le x \le {\pi \over 2}$

(C) applicable when $0 \le x \le \pi$         (D) applicable when ${\pi \over 4} \le x \le {\pi \over 2}$

Ans : (A)

29.  The general solution of the differential equation ${{{d^2}y} \over {d{x^2}}} + 8{{dy} \over {dx}} + 16y = 0$ is

(A) $(A + Bx){e^{5x}}$        (B) $(A + Bx){e^{4x}}$       (C) $(A + B{x^2}){e^{4x}}$      (D) $(A + B{x^4}){e^{4x}}$

Ans : (B)

30.   If ${x^2} + {y^2} = 4$, then $y{{dy} \over {dx}} + x =$

(A) 4      (B) 0       (C) 1     (D) -1

Ans : (B)

31.  $\int {{{{x^3}dy} \over {1 + {x^8}}} = }$

(A) $4{\tan ^{ - 1}}{x^3} + c$      (B) ${1 \over 2}{\tan ^{ - 1}}{x^4} + c$      

(C) $x + 4{\tan ^{ - 1}}{x^4} + c$       (D) ${x^2} + {1 \over 4}{\tan ^{ - 1}}{x^4} + c$

Ans : (B)

32.  $\int\limits_\pi ^{16\pi } {\left| {\sin x} \right|dx} =$

(A) 0      (B) 32      (C) 30      (D) 28

Ans : (C)

33.  The degree and order of the differential equation $y = x{\left( {{{dy} \over {dx}}} \right)^2} + {\left( {{{dx} \over {dy}}} \right)^2}$ are respectively

(A) 1, 1      (B) 2, 1       (C) 4, 1      (D) 1, 4

Ans: (C)

34.  $f(x) = \left\{ {\matrix{0 \cr {x - 3} \cr } } \right.\matrix{, \cr, \cr } \matrix{{x = 0} \cr {x > 0} \cr}$ The function ƒ(x) is

(A) increasing when x ≥ 0         (B) strictly increasing when x > 0

(C) Strictly increasing at x = 0       (D) not continuous at x = 0 and so it is not increasing when x > 0

Ans :(B)

35.   The function ƒ(x) = ax + b is strictly increasing for all real x if

(A) a > 0        (B) a < 0        (C) a = 0      (D) a ≤ 0

Ans : (A)

36.    $\int {{{\cos 2x} \over {\cos x}}dx} =$

(A) 2 sin x + log | sec x + tan x | + C       (B) 2 sin x – log |sec x – tan x| + c

(C) 2 sin x – log |sec x + tan x| + C          (D) 2 sin x + log |sec x – tan x| + C

Ans:  (C)

37.   $\int {{{{{\sin }^8}x - {{\cos }^8}x} \over {1 - 2{{\sin }^2}x{{\cos }^2}x}}dx}$

(A)  $- {1 \over 2}\sin 2x + C$       (B) ${1 \over 2}\sin 2x + C$      (C) ${1 \over 2}\sin x + C$      (D) $- {1 \over 2}\sin x + C$

Ans : (A)

38.   The general solution of the differential equation ${\log _e}\left( {{{dy} \over {dx}}} \right) = x + y$ is

(A) e^x + e^–y = C       (B) e^x + e^y = C      (C) e^y + e^-x = C     (D) e^-x + e^-y = C

Ans : (A)

39.   If $y = {A \over x} + B{x^2}$,  then ${x^2}{{{d^2}y} \over {d{x^2}}} =$

(A) 2y      (B) y²        (C) y³       (D) y⁴

Ans: (A)

40.  If one of the cube roots of 1 be $\omega$, then $\left| {\matrix{1 & {1 + {\omega ^2}} & {{\omega ^2}} \cr {1 - i} & { - 1} & {{\omega ^2} - 1}  \cr    { - i} & { - 1 + \omega } & { - 1}  \cr } } \right|$ =

(A) $\omega$      (B) i      (C) 1      (D) 0

Ans: (D)

41.  4 boys and 2 girls occupy seats in a row at random. Then the probability that the two girls occupy seats side by side is

(A) ${1 \over 2}$      (B) ${1 \over 4}$       (C)  ${1 \over 3}$      (D) ${1 \over 6}$

Ans : (C)

42.  A coin is tossed again and again.  If tail appears on first three tosses, then the chance that head appears on fourth toss is

(A) ${1 \over 16}$      (B) ${1 \over 2}$       (C)  ${1 \over 8}$      (D) ${1 \over 4}$

Ans : (B)

43.   The coefficient of Xⁿ in the expansion of ${e^{7x} + {e^x}} \over {e^{3x}}$ is

(A) ${{{4^{n - 1}} - {{( - 2)}^{n - 1}}} \over {\left| {n\limits_ - }}$      (B) ${{{4^{n - 1}} - {2^{n - 1}}} \over {\left|{n\limits_-}}$      (C) ${{{4^n} - {2^n}}\over {\left|{ n\limits_ - }}$      (D)${{{4^n}+{{(-2)}^n}}\over {\left| {n\limits_ - }}$

Ans :(D)

44.  The sum of the series ${1 \over {1.2}} - {1 \over {2.3}} + {1 \over {3.4}} - \cdots\cdots \infty$ is

(A)  $2{\log _e}2 + 1$      (B) $2{\log _e}2$       (C) $2{\log _e}2 - 1$      (D) ${\log _e}2 - 1$

Ans : (C)

45.  The number (101)¹⁰⁰ – 1 is divisible by

(A) 10⁴       (B) 10⁶       (C) 10⁸      (D) 10¹²

Ans : (A)

46.  If A and B are coefficients of xⁿ in the expansions of (1+ x)²ⁿ and (1+x)^{2n – 1} respectively, then A/B is equal to

(A) 4        (B) 2       (C) 9      (D) 6

Ans : (B)

47.    If n > 1 is an integer and $x \ne 0$, then (1 + x)ⁿ – nx – 1 is divisible by

(A) nx³        (B) n³x         (C) x         (D) nx

Ans : (C)

48.   If ⁿC₄, ⁿC₅ and ⁿC₆ are in A.P., then n is

(A) 7 or 14        (B) 7       (C) 14        (D) 14 or 21

Ans : (A)

49.  The number of diagonals in a polygon is 20. The number of sides of the polygon is

(A) 5       (B) 6       (C) 8       (D) 10

Ans : (C)

50.  ¹⁵C₃ + ¹⁵C₅ + ......... + ¹⁵C₁₅ =

(A) 2¹⁴       (B) 2¹⁴ – 15      (C) 2¹⁴ + 15       (D) 2¹⁴ – 1

Ans : (B)

51.  Let a, b, c be three real numbers such that a + 2b + 4c = 0. Then the equation ax² + bx + c = 0

(A) has both the roots complex          (B) has its roots lying within – 1 < x < 0

(C) has one of roots equal to ½          (D) has its roots lying within 2 < x < 6

Ans : (C)

52.  If the ratio of the roots of the equation px² + qx + r = 0 is a : b, then ${{ab} \over {{{(a + b)}^2}}} =$

(A) ${{{p^2}} \over {qr}}$       (B) ${{pr} \over {{q^2}}}$      (C) ${{{p^2}} \over {pr}}$      (D) ${{pq} \over {{r^2}}}$

Ans : (B)

53.  If α and ß are the roots of the equation x² + x + 1 = 0, then the equation whose roots are α¹⁹ and ß⁷ is

(A) x² – x – 1 = 0       (B) x² – x + 1 = 0       (C) x² + x – 1= 0     (D) x² + x + 1 = 0

Ans : (D)

54.   For the real parameter t, the locus of the complex number $z = (1 + {t^2}) + i\sqrt {1 + {t^2}}$ in the complex plane is

(A) an ellipse        (B) a parabola        (C) a circle        (D) a hyperbola

Ans : (B)

55.  If $x + {1 \over x} = 2\cos \theta$, then for any integer n, ${x^n} + {1 \over {{x^n}}} =$

(A) $2\cos n\theta$       (B) $2\sin n\theta$      (C) $2i \cos n\theta$      (D) $2i \sin n\theta$

Ans : (A)

56.  If $\omega \ne 1$ is a cube root of unity, then the sum of the series $S = 1 + 2\omega + 3{\omega^2} + \cdots \cdots + 3n{\omega^{3n - 1}}$ is

(A) ${{3n} \over {\omega - 1}}$       (B) $3n(\omega - 1)$      (C) ${{\omega - 1} \over {3n}}$      (D) 0

Ans : (A)

57.  If ${\log _3}x + {\log _3}y = 2 + {\log _3}2$ and ${\log _3}(x + y) = 2$, then

(A) x = 1,   y = 8       (B) x = 8,  y = 1       (C) x = 3,   y = 6        (D) x = 9,  y = 3

Ans : (C)

58.  If ${\log _7}2 = \lambda$ then value of ${\log _{49}}(28)$ is

(A) $(2\lambda +1)$       (B) $(2\lambda + 3)$      (C) ${1 \over 2}(2\lambda + 1)$     (D) $2(2\lambda + 1)$

Ans : (C)

59.  The sequence $\log a,\log {{{a^2}} \over b},\log {{{a^3}} \over {{b^2}}}, \cdots \cdots$ is

(A) a G.P.      (B) an A.P.       (C) a H.P.     (D) both a G.P. and a H.P

Ans : (B)

60.  If in a triangle ABC, sin A, sin B, sin C are in A.P., then

(A) the altitudes are in A.P.          (B) the altitudes are in H.P.

(C) the angles are in A.P.             (D) the angles are in H.P.

Ans : (B)

61.   $\left| {\matrix{{a - b} & {b - c} & {c - a}  \cr {b - c} & {c - a} & {a - b}  \cr {c - a} & {a - b} & {b - c}  \cr }} \right| =$

(A) 0       (B) – 1         (C) 1        (D) 2

Ans : (A)

62.  The area enclosed between y² = x and y = x is

(A) ${2 \over 3}$ sq. units     (B) ${1 \over 2}$ units       (C) ${1 \over 3}$ units      (D) ${1 \over 6}$ units

Ans:  (D)

63.   Let $f(x) = {x^3}{e^{ - 3x}},x > 0$. Then the maximum value of ƒ(x) is

(A) e^-3      (B) 3e^-3       (C) 27e^-9      (D) ∞

Ans : (A)

64.   The area bounded by y² = 4x and x² = 4y is

(A) ${{20} \over 3}$ sq. unit      (B) ${{16} \over 3}$ sq. unit      (C) ${{14} \over 3}$ sq. unit     (D)${{10} \over 3}$ sq. unit

Ans: (B)

65.   The acceleration of a particle starting from rest moving in a straight line with uniform acceleration is 8m/sec². The time taken by the particle to move the second metre is

(A) ${{\sqrt 2 - 1} \over 2}$ sec       (B) ${{\sqrt 2 + 1} \over 2}$ sec      (C) $(1 + \sqrt 2 )$ sec      (D) $(\sqrt 2 - 1)$ sec

Ans : (A)

66.  The solution of ${{dy} \over {dx}} = {y \over x} + \tan {y \over x}$ is

(A) x = c sin(y/x)        (B) x = c sin(xy)       (C) y = c sin(y/x)       (D) xy = c sin (x/y)

Ans : (A)

67.   Integrating Factor (I.F.) of the defferential equation ${{dy} \over {dx}} - {{3{x^2}y} \over {1 + {x^3}}} = {{{{\sin }^2}(x)} \over {1 + x}}$ is

(A) ${e^{1 + {x^3}}}$       (B) $\log (1 + {x^3})$        (C) $1 + {x^3}$       (D) ${1 \over {1 + {x^3}}}$

Ans : (D)

68. The differential equation of y = ae^bx (a & b are parameters) is

(A) $y{y_1} = y_2^2$       (B) $y{y_2} = y_1^2$      (C) $yy_1^2 = {y_2}$      (D) $yy_2^2 = {y_1}$

Ans : (B)

69.  The value of $\lim\limits_{n \to \infty}\sum\limits_{r = 1}^n {{{{r^3}} \over {{r^4} + {n^4}}}}$ is

(A) ${1 \over 2}{\log _e}(1/2)$       (B) ${1 \over 4}{\log _e}(1/2)$      (C)${1 \over 4}{\log _e}2$     (D) ${1 \over 2}{\log _e}2$

Ans: (C)

70.  The value of $\int\limits_0^\pi{{{\sin }^{50}}x} {\cos^{49}}xdx$ is

(A) 0      (B) ${\pi \over 4}$       (C) ${\pi \over 2}$      (D) 1

Ans: (A)

71.  $\int {{2^x}(f'(x) + f(x)\log 2)dx}$ is

(A) ${2^x}f'(x) + C$     (B) ${2^x}f(x) + C$     (C) ${2^x}(\log 2)f(x) + C$     (D) $(\log 2)f(x) + C$

Ans:(B)

72.  Let $f(x) = {\tan ^{ - 1}}x$. Then $f'(x) + f''(x)$ is =0, when x is equal to

(A) 0      (B) +1      (C) i      (D) -i

Ans: (B)

73.  If $y={\tan^{- 1}}{{\sqrt {1+{x^2}}- 1}\over x}$, then y'(1)=

(A) 1/4      (B) 1/2      (C) -1/4      (D) -1/2

Ans : (A)

74.  The value of $\lim\limits_{x \to 1}{{x+{x^2}+ \cdots + {x^n}- n} \over {x-1}}$ is

(A) n      (B) ${{n+1} \over 2}$       (C)${{n(n+1)} \over 2}$      (D)${{n(n-1)} \over 2}$

Ans: (C)

75.   $\lim \limits_{x \to 0} {{\sin (\pi {{\sin }^2}x)} \over {{x^2}}}$ =

(A) ${\pi ^2}$       (B) $3\pi$      (C) $2\pi$      (D) $\pi$

Ans: (D)

76.   If the function $f(x) = \left\{ {\matrix{ {{{{x^2} - (A + 2)x + A} \over {x - 2}}} \cr 2 \cr }} \right.\matrix{ {for} & {x \ne 2} \cr {for} & {x = 2} \cr }$ is continuous at x = 2, then

(A) A = 0       (B) A = 1     (C) A = – 1      (D) A = 2

Ans : (A)

77.  $f(x)= \left\{{\matrix{{\left[x \right] + \left[{-x} \right],}\cr \lambda \cr}}\right.\matrix{{when} &{x \ne 2} \cr {when} &{x = 2} \cr}$

       If ƒ(x) is continuous at x = 2, the value of $\lambda$ will be

(A) – 1       (B) 1        (C) 0       (D) 2

Ans : (A)

78.   The even function of the following is

(A) $f(x) = {{{a^x} + {a^{- x}}} \over {{a^x} - {a^{- x}}}}$        (B) $f(x) = {{{a^x} + 1} \over {{a^x} - 1}}$

(C) $f(x) = x.{{{a^x} - 1} \over {{a^x} + 1}}$       (D) $f(x) = {\log _2}\left( {x + \sqrt {{x^2} + 1} } \right)$

Ans : (C)

79.  If ƒ(x + 2y, x – 2y) = xy, then ƒ(x, y) is equal to

(A) ${1\over 4}xy$       (B) ${1 \over 4}({x^2} - {y^2})$       (C) ${1 \over 8}({x^2} - {y^2})$      (D) ${1 \over 2}({x^2} + {y^2})$

Ans: (C)

80.   The locus of the middle points of all chords of the parabola y² = 4ax passing through the vertex is

(A) a straight line        (B) an ellipse         (C) a parabola         (D) a circle

Ans : (C)

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