WBJEE Mathematics Question Paper 2014 (Eng)

1. The number of solution(s) of the equation $\sqrt{x+1}-\sqrt{x-1}=\sqrt{4x-1}$ is/are

(A) 2 (B) 0 (C) 3 (D) 1

2. The value of |z|² + |z - 3|² + |z - i|² is minimum when z equals

(A) $2 - {2 \over 3}i$ (B) $45 + 3 i$ (C) $1+ {i \over 3}$ (D) $1- {i \over 3}$

3. If $f(x) = \left\{ {\matrix{ {2{x^2} + 1,} & {x \le 1} \cr {4{x^3} - 1,} & {x > 1} \cr } } \right.$ , then $\int^2_0 f(x)\,dx$ is

(A) 47/3 (B) 50/3 (C) 1/3 (D) 47/2

4. If $\lim \limits_{x \to 0} {{2a\sin x - \sin 2x} \over {{{\tan }^3}x}}$ exists and is equal to 1, then the value of 'a' is

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

5. The solution of the equation $log_{101} log_7(\sqrt{x+7}+\sqrt{x})=0$ is

(A) 3 (B) 7 (C) 9 (D) 49

6. The integrating factor of the differential equation

$(1+x^2)\frac{dy}{dx}+y=e^{tan^{-1}x}$ is

(A) $tan^{-1}x$ (B) 1+x² (C) $e^{tan^{-1}x}$ (D) $log_e(1+x^2)$

7. If $\sqrt{y}=cos^{-1}x$ , then it satisfies the differential equation $(1-x^2)\frac{d^2y}{dx^2}-x\frac{dy}{dx}=c$ , where c is equal to

(A) 0 (B) 3 (C) 1 (D) 2

8. The number of digits in $20^{301}$ (given $log_{10}2=0.3010$ ) is

(A) 602 (B) 301 (C) 392 (D) 391

9. The area of the region bounded by the curves y = x² and x = y² is

(A) 1/3 (B) 1/2 (C) 1/4 (D) 3

10. Let $\mathbb{R}$ be the set of all real numbers and $f:\mathbb{R}\to\mathbb{R}$ be given by ƒ(x) = 3x² + 1. Then the set $f^{-1}([1,6])$ is

(A) $\left\{ { - \sqrt {{5 \over 3},} 0,\sqrt {{5 \over 3}} } \right\}$ (B) $\left[ { - \sqrt {{5 \over 3},} \sqrt {{5 \over 3}} } \right]$ (C) $\left[ { - \sqrt {{1 \over 3},} \sqrt {{1 \over 3}} } \right]$ (D) $\left( { - \sqrt {{5 \over 3},} \sqrt {{5 \over 3}} } \right)$

11. The value of $tan\frac{\pi}{5}+2tan\frac{2{\pi}}{5}+4cot\frac{4{\pi}}{5}$ is

(A) $cot\frac{\pi}{5}$ (B) $cot\frac{2{\pi}}{5}$ (C) $cot\frac{4{\pi}}{5}$ (D) $cot\frac{3{\pi}}{5}$

12 Let ƒ(x) be a differentiable function in [2,7]. If ƒ(2) = 3 and ƒ'(x) ≤ 5 for all x in (2,7), then the maximum possible value of ƒ(x) at x = 7 is

(A) 7 (B) 15 (C) 28 (D) 14

13. Let the number of elements of the sets A and B be p and q respectively. Then the number of relations from the set A to the set B is

(A) 2^p+q (B) 2^pq (C) p + q (D) pq

14. In a △ABC, tan A and tan B are the roots of pq(x² + 1) = r²x. Then △ABC is

(A) a right angled triangle

(B) an acute angled triangle

(D) an equilateral triangle

15. If y = 4x + 3 is parallel to a tangent to the parabola y² = 12x, then its distance from the normal parallel to the given line is

(A) ${{213} \over {\sqrt {17}}}$ (B) ${{219} \over {\sqrt {17}}}$ (C) ${{211} \over {\sqrt {17}}}$ (D) ${{210} \over {\sqrt {17}}}$

16. Let the equation of an ellipse be $\frac{x^2}{144}+\frac{y^2}{25}=1$ . Then the radius of the circle with centre (0,√2) and passing through the foci of the ellipse is

(A) 9 (B) 7 (C) 11 (D) 5

17. The straight lines x + y = 0, 5x + y = 4 and x + 5y = 4 form

(A) an isosceles triangle

(B) an equilateral triangle

(D) a right angled triangle

18. If $sin^{-1}(\frac{x}{13})+cosec^{-1}(\frac{13}{12})=\frac{\pi}{2}$ , then the value of x is

(A) 5 (B) 4 (C) 12 (D) 11

19. The values of λ for which the curve (7x + 5)² + (7y + 3)² = λ²(4x + 3y - 24)² represents a parabola is

(A) $\pm\frac{6}{5}$ (B) $\pm\frac{7}{5}$ (C) $\pm\frac{1}{5}$ (D) $\pm\frac{2}{5}$

20. Let ƒ(x) = x + 1/2. The the number of real values of x for which the three unequal terms ƒ(x), ƒ(2x), ƒ(4x) are in H.P. is

(A) 1 (B) 0 (C) 3 (D) 2

21. Let ƒ(x) = 2x² + 5x + 1. If we write ƒ(x) as

ƒ(x) = a(x + 1)(x - 2) + b(x - 2)(x - 1) + c(x - 1)(x + 1)

for real numbers a, b, c then

(A) there are infinite number of choices for a, b, c

(B) only one choice for a but infinite number of choices for b and c

(C) exactly one choice for each of a, b, c

(D) more than one but finite number of choices for a, b, c

22. If α, β are the roots of ax² + bx +c = 0 (a ≠ 0) and α + h, β + h are the roots of px² + qx + r = 0 (p ≠ 0) then the ratio of the squares of their discriminants is

(A) a² : p² (B) a : p² (C) a² : p (D) a : 2p

23. Let p, q be real numbers. If α is the root of x² + 3p²x + 5q² = 0, β is a root of x² + 9p²x + 15q² = 0 and 0 < α < β, then the equation x² + 6p²x + 10q² = 0 has a root γ that always satisfies

(A) γ = α/4 + β (B) β < γ (C) γ = α/2 + β (D) α < γ < β

24. The equation of the common tangent with positive slope to the parabola y² = 8√3 x and the hyperbole 4x² - y² = 4 is

(A) y = √6 x + √2 (B) y = √6 x - √2 (C) y = √3 x + √2 (D) y = √3 x - √2

25. The point on the parabola y² = 64x, which is nearest to the line 4x + 3y + 35 = 0 has coordinates

(A) (9, -24) (B) (1, 81) (C) (4, -16) (D) (-9, -24)

26. Let z₁, z₂ be two fixed complex numbers in the Argand plane and z be an arbitrary point satisfying $|z-z_1|+|z-z_2|=2|z_1-z_2|$ . Then the locus of z will be

(A) an ellipse

(B) a straight line joining z₁ and z₂

(D) a bisector of the line segment joining z₁ and z₂

27. The function $f(x) = {{\tan \left\{ {\pi \left[ {x - {\pi \over 2}} \right]} \right\}} \over {2 + {{\left[ x \right]}^2}}}$ , where [x] denotes the greatest integer ≤ x, is

(A) continuous for all values of x

(B) discontinuous at x = π/2

(D) discontinuous at x = -2

28. The function $f(x) = a sin|x| + be^{|x|}$ is differentiable at x = 0 when

(A) 3a + b = 0 (B) 3a - b = 0 (C) a + b = 0 (D) a - b = 0

29. If the coefficient of $x^8$ in $(ax^2+\frac{1}{bx})^{13}$ is equal to the coefficient of $x^{-8}$ in $(ax-\frac{1}{bx^2})^{13}$ , then a and b will satisfy the relation

(A) ab + 1 = 0 (B) ab = 1 (C) a = 1 - b (D) a + b = -1

30. If $I=\int^2_0e^{x^4}(x- \alpha)dx=0$ , then α lies in the interval

(A) (0,2) (B) (-1,0) (C) (2,3) (D) (-2,-1)

31. The solution of the differential equation $y\frac{dy}{dx}=x\bigg[\frac{y^2}{x^2}+\frac{\varphi(\frac{y^2}{x^2})}{\varphi'(\frac{y^2}{x^2})}\bigg]$ is (where c is a constant)

(A) $\varphi(\frac{y^2}{x^2})=cx$ (B) $x\varphi(\frac{y^2}{x^2})=c$ (C) $\varphi(\frac{y^2}{x^2})=cx^2$ (D) $x^2\varphi(\frac{y^2}{x^2})=c$

32. Suppose that the equation ƒ(x) = x² + bx + c = 0 has two distinct real roots α and ß. The angle between the tangent to the curve y = ƒ(x) at the point $\big(\frac{\alpha+\beta}{2}, f(\frac{\alpha+\beta}{2})\big)$ and the positive direction of the x -axis is

(A) 0° (B) 30° (C) 60° (D)90°

33. The function ƒ(x) = x² + bx + c, where b and c real constants, describes

(A) one-to-one mapping

(B) onto mapping

(D) neither one-to-one nor onto mapping

34. Let n ≥ 2 be an integer, $A=\begin{pmatrix} cos(2\pi/n) & sin(2\pi/n) & 0\\ -sin(2\pi/n) & cos(2\pi/n) & 0\\ 0 & 0 & 1 \end{pmatrix}$ and I is the identity matrix of order 3. Then

(A) Aⁿ = I and A^n-1 ≠ I

(B) A^m ≠ I for any positive integer m

(D) A^m = 0 for positive integer m

35. Ram visiting a friend. Ram knows that his friend has 2 children and 1 of them is a boy. Assuming that a child is equally likely to be a boy or a girl, then the probability that the other child is a girl, is

(A) ½ (B) ⅓ (C) ⅔ (D) 7/10

36. The value of the sum $({^n}C_1)^2+({^n}C_2)^2+({^n}C_3)^2+......+({^n}C_n)^2$ is

(A) $(^{2n}C_n)^2$ (B) $^{2n}C_n$ (C) $^{2n}C_n+1$ (D) $^{2n}C_n-1$

37. The remainder obtained when 1! + 2! + 3! + .... + 11! is divided by 12 is

(A) 9 (B) 8 (C) 7 (D) 6

38. Out of 7 consonants and 4 vowels, the number of words (not necessarily meaningful) that can be made, each consisting of 3 consonants and 2 vowels, is

(A) 24800 (B) 25100 (C) 25200 (D) 25400

39. Let $S=\frac{2}{1} {^n}C_0+\frac{2^2}{2} {^n}C_1+\frac{2^3}{3} {^n}C_2+...+\frac{2^{n+1}}{n+1} {^n}C_n$ . Then S equals

(A) $\frac{2^{n+1}-1}{n+1}$ (B) $\frac{3^{n+1}-1}{n+1}$ (C) $\frac{3^n-1}{n}$ (D) $\frac{2^n-1}{n}$

40. Let $\mathbb{R}$ be the set of all the real numbers and $f:[-1,1]\to\mathbb{R}$ be defined by

$f(x) = \left\{ {\matrix{ {x\sin {1 \over x},} & {x \ne 0} \cr {0,} & {x = 0} \cr } } \right.$ .

Then

(A) ƒ satisfies the conditions of Rolle's theorem on [-1,1]

(B) ƒ satisfies the conditions of Lagrange's Mean Value Theorem on [-1,1]

(D) ƒ satisfies the conditions of Lagrange's Mean Value Theorem on [0,1]

41. If a, b and c are positive numbers in a G.P., then the roots of the quadratic equation

$(log_ea)x^2-(2log_eb)x+(log_ec)=0$ are

(A) -1 and $\frac{log_ec}{log_ea}$ (B) 1 and $-\frac{log_ec}{log_ea}$ (C) 1 and $log_ac$ (D) -1 and $log_ca$

42. There is a group of 265 persons who like either singing or dancing or painting. In this group 200 like singing, 110 like dancing and 55 like painting. If 60 persons like both singing and dancing, 30 like both singing and painting and 10 like all three activities, then the number of persons who like only dancing and painting is

(A) 10 (B) 20 (C) 30 (D) 40

43. The range of the function $y = 3\sin \left( {\sqrt {{{{\pi ^2}} \over {16}} - {x^2}} } \right)$ is

(A) $[0,\sqrt{3/2}]$ (B) [0,1] (C) $[0,3/\sqrt{2}]$ (D) $[0,\infty)$

44. The value of $\lim \limits_{x \to 0} {{\int_0^{{x^2}} {\cos } ({t^2})dt} \over {x\sin x}}$ is

(A) 1 (B) -1 (C) 2 (D) $log_e2$

45. Let ƒ(x) be a differentiable function and ƒ'(4) = 5. Then $\lim \limits_{x \to 2}\frac{f(4)-f(x^2)}{x-2}$ equals

(A) 0 (B) 5 (C) 20 (D) - 20

46. The sum of the series

$\sum \limits_{n=1}^{\infty} \sin ( \frac{n! \pi} {720})$ is

(A) $sin(\frac{\pi}{180})+sin(\frac{\pi}{360})+sin(\frac{\pi}{540})$

(B) $sin(\frac{\pi}{6})+sin(\frac{\pi}{30})+sin(\frac{\pi}{120})+sin(\frac{\pi}{360})$

(C) $sin(\frac{\pi}{6})+sin(\frac{\pi}{30})+sin(\frac{\pi}{120})+sin(\frac{\pi}{360})+sin(\frac{\pi}{720})$

(D) $sin(\frac{\pi}{180})+sin(\frac{\pi}{360})$

47. Let I denote the 3 X 3 identity matrix and P be a matrix obtained by rearranging the columns of I. Then

(A) there are six distinct choices for P and det(P) = 1

(B) there are six distinct choices for P and det(P) = ±1

(D) there are more than one choices for P and P^-1 = I in each choice.

48. The coefficient of x³ in the infinite series expansion of $\frac{2}{(1-x)(2-x)}$ , for |x| < 1, is

(A) -1/16 (B) 15/8 (C) -1/8 (D) 15/16

49. For every real number x, let $f(x)=\frac{x}{1!}+\frac{3}{2!}x^2+\frac{7}{3!}x^3+\frac{15}{4!}x^4+...$ .

Then the equation ƒ(x) = 0 has

(A) no real solution

(B) exactly one real solution

(D) infinite number of real solutions

50. Let S denote the sum of the infinite series $1+\frac{8}{2!}+\frac{21}{3!}+\frac{40}{4!}+\frac{65}{5!}+...$ . Then

(A) S < 8 (B) S > 12 (C) 8 < S <12 (D) S = 8

51. Let [x] denote the greatest integer less than or equal to x for any real number x. Then $\lim \limits_{n \to \infty } {{\left[ {n\sqrt 2 } \right]} \over n}$ is equal to

(A) 0 (B) 2 (C) √2 (D) 1

52. Suppose that ƒ(x) is a differentiable function such that ƒ'(x) is continuous ƒ'(0) = 1 and ƒ''(0) does not exist. Let g(x) = xƒ'(x). Then

(A) g'(0) does not exist (B) g'(0) = 0 (C) g'(0) = 1 (D) g'(0) = 2

53. Let z₁ be a fixed point on the circle of radius 1 centered at the origin in the Argand plane and z₁ ≠ ± ₁. Consider an equilateral triangle inscribed in the circle with z₁ , z₂, z₃ as the vertices taken in the counter clockwise direction. Then z₁z₂z₃ is equal to

(A) $z_1^2$ (B) $z_1^3$ (C) $z_1^4$ (D) $z_1$

54. Suppose that $z_1, z_2, z_3$ are three vertices of an equilateral triangle in the Argand plane. Let $\alpha=\frac{1}{2}(\sqrt{3}+i)$ and $\beta$ be a non-zero complex number. The points $\alpha z_1+\beta$ , $\alpha z_2+\beta$ , $\alpha z_3+\beta$ will be

(A) the vertices of an equilateral triangle

(B) the vertices of an isosceles triangle

(D) the vertices of a scalene triangel

55. The curve $y=(cosx+y)^{1/2}$ satisfies the differential equation

(A) $(2y-1)\frac{d^2y}{dx^2}+2\bigg(\frac{dy}{dx}\bigg)^2+cosx=0$

(B) $\frac{d^2y}{dx^2}-2y\bigg(\frac{dy}{dx}\bigg)^2+cosx=0$

(D) $(2y-1)\frac{d^2y}{dx^2}-\bigg(\frac{dy}{dx}\bigg)^2+cosx=0$

56. In the Argand plane, the distinct roots of $1+z+z^3+z^4=0$ (z is a complex number) represent vertices of

(A) a square (B) an equilateral triangle (C) a rhombus (D) a rectangle

57. In a ΔABC, a, b, c are the sides of the triangle opposite to the angles A, B,C respectively. Then the value of $a^3sin(B-C)+b^3sin(C-A)+c^3sin(A-B)$ is equal to

(A) 0 (B) 1 (C) 3 (D) 2

58. Let $\alpha$ , $\beta$ be the roots of $x^2-x-1=0$ and $S_n=\alpha^n+\beta^n$ , for all integers $n\ge1$ . Then for every integer $n\ge2$ ,

(A) $S_{n}+S_{n-1}=S_{n+1}$

(B) $S_{n}-S_{n-1}=S_{n+1}$

(D) $S_{n}+S_{n-1}=2S_{n+1}$

59. A fair six-faced die is rolled 12 times. The probability that each face turns up twice is equal to

(A) ${{12!} \over {6!6!{6^{12}}}}$ (B) ${{{2^{12}}} \over {{2^6}{6^{12}}}}$ (C) ${{12!} \over {{2^6}{6^{12}}}}$ (D) ${{12!} \over {{6^6}{6^{12}}}}$

60. If $\alpha$ , $\beta$ are the roots of the quadratic equation $x^2+px+q=0$ , then the values of $\alpha^3+\beta^3$ and $\alpha^4+\alpha^2\beta^2+\beta^4$ are respectively

(A) $3pq-p^3$ and $p^4-3p^2q+3q^2$

(B) $-p(3q-p^2)$ and $(p^2-q)(p^2+3q)$

(D) $3pq-p^3$ and $(p^2-q)(p^2-3q)$

61. The solution of the differential equation

$\frac{dy}{dx}+\frac{y}{x \log_ex}=\frac{1}{x}$ under the condition y = 1 when x = e is

(A) $2y= \log_ex+\frac{1}{\log_ex}$

(B) $y=\log_ex+\frac{2}{\log_ex}$

(D) $y = \log_ex+e$

62. Let ƒ(x) = max{x + |x|, x - [x]}, where [x] denotes the greatest integer ≤ x. Then the value of $\int_{-3}^3f(x)dx$ is

(A) 0 (B) 51/2 (C) 21/2 (D) 1

63. Let $X_n= \left \{ z = x + iy : |z|^2 \le \frac{1}{n} \right \}$ for all integers n ≥ 1. Then $\bigcap_{n=1}^{\infty}X_n$ is

(A) a singleton set (B) not a finite set (C) an empty set (D) a finite set with more than one elements

64. Applying Lagrange's Mean Value Theorem for a suitable function ƒ(x) in [0,h] we have ƒ(h) = ƒ(0) + hƒ'(θh), 0 < θ < 1. Then for ƒ(x) = cos x, the value of $\lim \limits_{h \to 0^{+}} \theta$ is

(A) 1 (B) 0 (C) ½ (D) ⅓

65. The equation of hyperbola whose coordinates of the foci are (±8, 0) and the length of latus rectum is 24 units, is

(A) $3x^2-y^2=48$

(B) $4x^2-y^2=48$

(D) $x^2-4y^2=48$

66. A student answers a multiple choice question with 5 alternatives, of which exactly one is correct. The probability that he knows the correct answer is p, 0 < p < 1. If he does not know the correct answer, he randomly ticks one answer. Given that he has answered the question correctly, the probability that he did not tick the answer randomly, is

(A) $\frac{3p}{4p+3}$ (B) $\frac{5p}{3p+2}$ (C) $\frac{5p}{4p+1}$ (D) $\frac{4p}{3p+1}$

67. $cos\frac{2\pi}{7}+cos\frac{4\pi}{7}+cos\frac{6\pi}{7}$

(A) is eqeal to zero

(B) lies between 0 and 3

(C) is a negative number

(D) lies between 3 and 6

68. Suppose $M=\int_0^{\pi/2}\frac{ \cos x}{x+2}dx$ and $N=\int_0^{\pi/4}\frac{ \sin x \cos x}{{x+1}^2}dx$ . Then the value of (M - N) equals

(A) $\frac{3}{\pi+2}$ (B) $\frac{2}{\pi-4}$ (C) $\frac{4}{\pi-2}$ (D) $\frac{2}{\pi+4}$

69. For any two real numbers θ and φ, we define θRφ if and only if sec²θ - tan²φ = 1. The relation R is

(A) reflexive but not transitive

(B) symmetric but not reflexive

(D) an equivalence relation

70. The minimum value of $2^{\sin x}+2^{\cos x}$ is

(A) $2^{1-1/\sqrt2}$ (B) $2^{1+1/\sqrt2}$ (C) $2^{\sqrt2}$ (D) 2

71. We define a binary relation ~ on the set of all 3 X 3 real matrices as A ~ B if and only if there exist invertible matrices P and Q such that B = PA Q^-1. The binary relation ~ is

(A) neither reflexive nor symmetric

(B) reflexive and symmetric but not transitive

(D) an equivalence relation

72. Let α, ß denote the cube roots of unity other than 1 and α ≠ ß . Let

$s=\sum \limits_{n=0}^{302}(-1)^n(\frac{\alpha}{\beta})^n$ .

Then the value of s is

(A) either - 2ω or - 2ω²

(B) either - 2ω or 2ω²

(D) either 2ω or 2ω²

73. Let $t_n$ denote the n-th term of infinite series $\frac{1}{1!}+\frac{10}{2!}+\frac{21}{3!}+\frac{34}{4!}+\frac{49}{5!}+...$ .

Then $\lim \limits_{n \to \infty} t_n$ is

(A) e (B) 0 (C) e² (D) 1

74. A particle starting from a point A and moving with a positive constant acceleration along a straight line reaches another point B in time T. Suppose that the initial velocity of the particle is u > 0 and P is the midpoint of the line AB. If the velocity of the particle at point P is $v_1$ and if the velocity at time $\frac{T}{2}$ is $v_2$ , then

(A) $v_1= v_2$ (B) $v_1 > v_2$ (C) $v_1 < v_2$ (D) $v_1 = \frac{1}{2}v_2$

75. A poker hand consists of 5 cards drawn at random from a well-shuffled pack of 52 cards. Then the probability that a poker hand consists of a pair and a triple of equal face values (for example, 2 seven and 3 kings or 2 aces and 3 queens, etc) is

(A) $\frac{6}{4165}$ (B) $\frac{23}{4165}$ (C) $\frac{1797}{4165}$ (D) $\frac{1}{4165}$

76. If u(x) and v(x) are two independent solutions of the differential equation

$\frac{d^2y}{dx^2}+b\frac{dy}{dx}+cy=0$ ,

then additional solution(s) of the given differential equation is(are)

(A) y = 5u(x) + 8v(x)

(B) y = c₁{u(x) - v(x)} + c₂v(x), c₁ and c₂ are arbitrary constants

(D) y = u(x) v(x)

77. The angle of intersection between the curves y = [|sin x| + |cos x|] and x² + y² = 10, where [x] denotes the greatest integer ≤ x, is

(A) $tan^{-1}3$ (B) $tan^{-1}(-3)$ (C) $tan^{-1}\sqrt3$ (D) $tan^{-1}(1/\sqrt3)$

78. $f(x) = \left\{ {\matrix{ {\int_0^x {\left| {1 - t} \right|} dt,} & {x > 1} \cr {x - {1 \over 2},} & {x \le 1} \cr } } \right.$

Then

(A) ƒ(x) is continuous at x = 1

(B) ƒ(x) is not continuous at x = 1

(D) ƒ(x) is not differentiable at x = 1

79. If the circle $x^2+y^2+2gx+2fy+c=0$ cuts the three circles $x^2+y^2-5=0$ , $x^2+y^2-8x-6y+10=0$ and $x^2+y^2-4x+2y-2=0$ at the extremities of their diameters, then