WBJEE Mathematics Question Paper 2013 (Eng)

administratorThu, 10/07/2014 - 9:48pm

1. A point P lies on the circle $x^2+y^2=169$ . If Q = (5, 12) and R = (-12, 5), then the angle $\angle QPR$ is

(A) $\frac{\pi}{6}$ (B) $\frac{\pi}{4}$ (C) $\frac{\pi}{3}$ (D) $\frac{\pi}{2}$

Ans: (B)

2. A circle passing through (0,0), (2,6), (6,2) cuts the x-axis at the point P ≠ (0,0). Then the length of OP, where O is origin, is

(A) $\frac{5}{2}$ (B) $\frac{5}{\sqrt2}$ (C) 5 (D) 10

Ans: (C)

3. The locus of the midpoints of the chords of an ellipse that are drawn form the positive end of the minor axis, is

(A) a circle with centre $(\frac{1}{2},0)$ and radius 1

(B) a parabola with focus $(\frac{1}{2},0)$ and directrix x = -1

(D) an hyperbola with centre $(0,\frac{1}{2})$ , transverse axis 1 and conjugate axis $\frac{1}{2}$

Ans: (No Option is correct)

4. A point moves so that the sum of squares of its distances from the points (1,2) and (-2,1) is always 6. Then its locus is

(A) the straight line $y-\frac{3}{2}=-3(x+\frac{1}{2})$

(B) a circle with centre and radius $\frac{1}{\sqrt2}$

(D) an ellipse with foci (1,2) and (-2,1)

Ans: (B)

5. For the variable t, the locus of the points of intersection of lines $x-2y=t$ and is

(A) the straight line x=y

(B) the circle with centre at the origin and radius 1

(D) the hyperbola with centre at the origin and one focus $(\frac{\sqrt5}{2},0)$

Ans: (D)

6. Let $P=\begin{pmatrix} \cos \frac {\pi}{4} & - \sin \frac {\pi}{4} \\ \sin \frac {\pi}{4} & \cos \frac {\pi}{4} \end{pmatrix}$ and $X=\begin{pmatrix} \frac {1}{\sqrt2} \\ \frac{1}{\sqrt2} \end{pmatrix}$ . Then $P^3X$ is equal to

(A) $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ (B) $\begin{pmatrix} -\frac{1}{\sqrt2} \\ \frac{1} {\sqrt2} \end{pmatrix}$ (C) $\begin{pmatrix} -1 \\ 0 \end{pmatrix}$ (D)

Ans: (C)

7. The number of solutions of the equation x+y+z = 10 in positive integers x, y, z, is equal to

(A) 36 (B) 55 (C) 72 (D) 45

Ans: (A)

8. For $0\le P, Q\le \frac{\pi}{2}$ , if $sinP+cosQ=2$ , then the value of is equal to

(A) 1 (B) $\frac{1}{\sqrt2}$ (C) $\frac{1}{2}$ (D) $\frac{\sqrt3}{2}$

Ans: (A)

9. If and $\beta$ are the roots of , then the value of is equal to

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

Ans: (B)

10. The value of the integral $\int_{-1}^{+1} \{\frac {x^{2013}}{e^{|x|} (x^2 + \cos x)} + \frac {1}{e^{| x|}} \}dx$ is equal to

(A) 0 (B) (C) $2e^{-1}$ (D) $2(1-e^{-1})$

Ans: (D)

11. Let

$f(x)=2^{100}x+1$ ,

Then the set of real numbers x such that is

(A) empty (B) a singleton (C) a finite set with more than one element (D) infinite

Ans: (B)

12. The limit of $x \sin(e^{1/x})$ as $x\to0$

(A) is equal to 0 (B) is equal to 1 (C) is equal to e/2 (D) does not exist

Ans: (A)

13. Let and $P=\begin{pmatrix} 1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & -2 \end{pmatrix}$ . Then the matrix is equal to

(A) P (B) I - P (C) 2I + P (D) 2I - P

Ans: (C)

14. If are the roots of the quadratic equation ; then the quadratic equation whose roots are

, $\beta-\frac{1}{\alpha}$ is

(A) $ax^2+a(b-1)x+(a-1)^2=0$

(B) $bx^2+a(b-1)x+(b-1)^2=0$

(D) $abx^2+bx +a=0$

Ans: (B)

15. The value of $1000[\frac{1}{1\times2}+\frac{1}{2\times3}+\frac{1}{3\times4}+...+\frac{1} {999\times1000}]$ is equal to

(A) 1000 (B) 999 (C) 1001 (D) 1/999

Ans: (B)

16. The value of the determinant

$\begin{vmatrix} 1+a^2-b^2 & 2ab & -2b\\ 2ab & 1-a^2+b^2 & 2a \\ 2b & -2a & 1-a^2-b^2 \end{vmatrix}$

is equal to

(A) 0 (B) (C) (D)

Ans: (D)

17. If the distance between the foci of an ellipse is equal to the length of the latus rectum, then its eccentricity is

(A) $\frac{1}{4}(\sqrt5-1)$ (B) $\frac{1}{2}(\sqrt5+1)$ (C) $\frac{1}{2} (\sqrt5-1)$ (D) $\frac{1}{4}(\sqrt5+1)$

Ans: (C)

18. For the curve x² + 4xy + 8y² = 64 the tangents are parallel to the x-axis only at the points

(A) (0,2√2) and (0,-2√2) (B) (8,-4) and (-8,4) (C) (8√2, -2√2) and (-8√2, 2√2) (D) (8,0) and (-8,0)

Ans: (B)

19. The value of $I = \int_0^{\pi \over 4}( \tan^{n+1}x)dx + \frac {1}{2} \int_0^{\pi \over 2} \tan^{n-1} (x/2)dx$ is equal to

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

Ans: (A)

20. Let ƒ(θ) = (1 + sin²θ)(2 - sin²θ). Then for all values of θ

(A) ƒ(θ) > ${9 \over 4}$ (B) ƒ(θ) < 2 (C) ƒ(θ) > ${11 \over 4}$ (D) 2 ≤ ƒ(θ) ≤ ${9 \over 4}$

Ans: (D)

21. Let $f(x) = \left\{ {\matrix{ {{x^3} - 3x + 2} & {x < 2} \cr {{x^3} - 6{x^2} + 9x + 2} & {x \ge 2} \cr } } \right.$ Then

(A) $\lim \limits_{x \to 2} f(x)$ does not exist

(B) ƒ is not continuous at x = 2

(D) ƒ is continuous and differentiable at x = 2

Ans: (C)

22. The limit of $\sum \limits_{n=1}^{1000}(-1)^nx^n$ as

(A) does not exist

(B) exist and equals to 0

(D) exists and approaches

Ans: (C)

23. If $f(x)=e^x(x-2)^2$ then

(A) ƒ is increasing in and $(2,\infty)$ and decreasing in (0,2)

(B) ƒ is increasing in and decreasing in $(0,\infty)$

(D) ƒ is increasing in and decreasing in and $(2,\infty)$

Ans: (A)

24. Let $f:\mathbb{R}\to\mathbb{R}$ be such that is injective and for all . If are in G.P., then are in

(A) A.P. always

(B) G.P. always

(D) G.P. depending on the values of x, y, z

Ans: (A)

25. The number of solutions of the equation

$\frac{1}{2}log_{\sqrt3}(\frac{x+1}{x+5})+log_9(x+5) ^2=1$ is

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

Ans: (B)

26. The area of the region bounded by the parabola $y=x^2-4x+5$ and the straight line is

(A) 1/2 (B) 2 (C) 3 (D) 9/2

Ans: (D)

27. The value of the integral

$\int_1^2e^x(log_ex+\frac{x+1}{x})dx$ is

(A) $e^2(1+log_e2)$ (B) $e^2-e$ (C) $e^2(1+log_e2)-e$ (D) $e^2-e(1+log_e2)$

Ans: (C)

28. Let $P=1+\frac{1}{2\times2}+\frac{1}{3\times2^2}+......$

and $Q=\frac{1}{1\times2}+\frac{1}{3\times4}+\frac{1}{5\times6}+......$

Then

(A) P = Q (B) 2P = Q (C) P = 2Q (D) P = 4Q

Ans: (C)

29. Let , $\frac {\pi}{4} \le x \le \frac {3 \pi}{4}$ . Then ƒ attains its

(A) minimum at

(B) maximum at $x=\frac{\pi}{2}$

(D) maximum at

Ans: (C)

30. Each of a and b can take values 1 or 2 with equal probability. The probability that the equation ax² + bx + 1 = 0 has real roots, is equal to

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

Ans: (B)

31. There are two coins, one unbiased with probability ${1 \over 2}$ of getting heads and the other one is biased with probability ${3 \over 4}$ of getting heads. A coin is selected at random and tossed. It shows heads up. Then the probability that the unbiased coin was selected is

(A) ${2 \over 3}$ (B) ${3 \over 5}$ (C) ${1 \over 2}$ (D) ${2 \over 5}$

Ans: (D)

32. For the variable t, the locus of the point of intersection of the lines 3tx - 2y + 6t = 0 and 3x + 2ty - 6 = 0 is

(A) the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$

(B) the ellipse $\frac{x^2} {9}+\frac{y^2}{4}=1$

(D) the hyperbola $\frac{x^2} {9}-\frac{y^2}{4}=1$

Ans: (A)

33. Cards are drawn one-by-one without replacement from a well shuffled pack of 52 cards. Then the probability that a face card (Jack, Queen or King) will appear for the first time on the third turn is equal to

(A) $\frac{300}{2197}$ (B) $\frac{36}{85}$ (C) $\frac{12}{85}$ (D) $\frac{4}{51}$

Ans: (C)

34. Lines x + y = 1 and 3y = x + 3 intersect the ellipse x² + 9y² = 9 at the points P,Q,R. The area of the triangle PQR is

(A) $\frac{36}{5}$ (B) $\frac{18}{5}$ (C) $\frac{9}{5}$ (D) $\frac{1}{5}$

Ans: (B)

35. The number of onto functions from the set {1, 2,…..,11} to set {1, 2,…..,10} is

(A) $5 \times | \underline {11}$ (B) $|\underline {10}$ (C) ${{|\underline {11}} \over 2}$ (D) $10 \times |\underline {11}$

Ans: (D)

36. The limit of $[\frac {1}{x^2} + \frac {(2013)^x}{e^x-1} + \frac {1}{e^x-1}]$ as

(A) approaches + ∞ (B) approaches - ∞ (C) is equal to $log_e (2013)$ (D) does not exist

Ans: (A)

37. Let $z_1=2+3i$ and $z_2=3+4i$ be two points on the complex plane. Then the set of complex numbers $z$ satisfying $|z-z_1|^2+|z-z_2|^2=|z_1-z_2|^2$ represents

(A) a straight line (B) a point (C) a circle (D) a pair of straight line

Ans: (C)

38. Let p(x) be a quadratic polynomial with constant term 1. Suppose p(x) when divided by x-1 leaves remainder 2 and when divided by x+1 leaves remainder 4. Then the sum of the roots of p(x) = 0 is

(A) -1 (B) 1 (C) $-\frac{1}{2}$ (D) $\frac{1}{2}$

Ans: (D)

39. Eleven apples are distributed among a girl and a boy. Then which one of the following statements is true ?

(A) At least one of them will receive 7 apples

(B) The girl receives at least 4 apples or the boy receives at least 9 apples

(D) The girl receives at least 4 apples or the boy receives at least 8 apples

Ans: ()

40. Five numbers are in H.P. The middle term is 1 and the ratio of the second and the fourth terms is 2 : 1. Then the sum of the first three terms is

(A) 11/2 (B) 5 (C) 2 (C) 14/3

Ans: (A)

41. The limit of $\{\frac{1}{x}\sqrt{1+x}-\sqrt{1+\frac{1}{x^2}}\}$ as $x\to0$

(A) does not exist (B) is equal to 1/2 (C) is equal to 0 (D) is equal to 1

Ans: (A)

42. The maximum and minimum values of $\cos^6 \theta + \sin^6 \theta$ are respectively

(A) 1 and 1/4 (B) 1 and 0 (C) 2 and 0 (D) 1 and 1/2

Ans: (A)

43. If a, b, c are in A.P., then the straight line ax + 2by + c = 0 will always pass through a fixed point whose co-ordinates are

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

Ans: (A)

44. If one end of a diameter of the circle is (1, 2), then the other end is

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

Ans: (C)

45. The value of cos²75° + cos²45° + cos²15° - cos²30° - cos²60° is

(A) 0 (B) 1 (C) 1/2 (D) 1/4

Ans: (C)

46. Suppose $z=x+iy$ where x and y are real numbers and . The points (x, y) for which is real, lie on

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

Ans: (D)

47. The equation 2x² + 5xy - 12y² = 0 represents a

(A) circle

(B) pair of non-perpendicular intersecting straight lines

(D) hyperbola

Ans: (B)

48. The line y = x intersects the hyperbola at the points P and Q. The eccentricity of ellipse with PQ as major axis and minor axis of length $\frac{5}{\sqrt2}$ is

(A) $\frac {\sqrt 5}{3}$ (B) $\frac {5}{\sqrt 3}$ (C) $\frac {5}{9}$ (D)

Ans: ()

49. The equation of the circle passing through the point (1, 1) and the points of intersection of x² + y² - 6x - 8 = 0 and x² + y² - 6 = 0 is

(A) x² + y² + 3x - 5 = 0 (B) x² + y² - 4x + 2 = 0 (C) x² + y² + 6x - 4 = 0 (D) x² + y² - 4y -2 = 0

Ans: (A)

50. Six positive numbers are in G.P., such that their product is 1000. If the fourth term is 1, then the last term is

(A) 1000 (B) 100 (C) 1/100 (D) 1/1000

Ans: (C)

51. In the set of all 3 x 3 real matrices a relation is defined as follows. A matrix A is related to a matrix B if and only if there is a non-singular 3x3 matrix P such that B = P-1AP. This relation is

(A) Reflexive, Symmetric but not Transitive

(B) Reflexive, Transitive but not Symmetric

(D) an Equivalence relation

Ans: (D)

52. The number of lines which pass through the point (2, -3) and are at a distance 8 from the point (-1, 2) is

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

Ans: (D)

53. If α, β are the roots of the quadratic equation ax² + bx + c = 0 and 3b² = 16ac then

(A) α = 4β or ß = 4α (B) α = -4β or β = -4α (C) α = 3β or β = 3α (D) α = -3β or β = -3α

Ans: (C)

54. For any two real numbers a and b, we define a R b if and only if sin² a + cos² b = 1. The relation R is

(A) Reflexive but not Symmetric

(B) Symmetric but not Transitive

(D) an Equivalence relation

Ans: (D)

55. Let n be a positive even integer. The ratio of the largest coefficient and the 2nd largest coefficient in the expansion of is 11 : 10. Then the number of terms in the expansion of is

(A) 20 (B) 21 (C) 10 (D) 11

Ans: (B)

56. Let exp (x) denote the exponential function ex. If , then the minimum value of in the interval [2, 5] is

(A) (B) (C) (D)

Ans: (C)

57. The sum of the series $\frac {1}{1 \times 2}{^{25}}C_0 + \frac {1}{2 \times 3}{^{25}}C_1 + \frac {1}{3 \times 4}{^{25}}C_2 +...+ \frac {1}{26 \times 27}{^{25}}C_{25}$

(A) $\frac {2^{27} - 1}{26 \times 27}$ (B) $\frac {2^{27}-28}{26 \times 27}$ (C) $\frac {1}{2}(\frac {2^{26}+1}{26 \times 27})$ (D) $\frac {2^{26} -1}{52}$

Ans: (B)

58. Five numbers are in A.P. With common difference ≠ 0 . If the 1st, 3rd and 4th terms are in G.P., then

(A) the 5th term is always 0

(B) the 1st term is always 0

(D) the middle term is always -2

Ans: (A)

59. The minimum value of the function is

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

Ans: (B)

60. If P, Q, R are angles of an isosceles triangle and $\angle P = {\pi \over 2}$ , then the value of

$(\cos \frac {P}{3} - i \sin {P \over 3})^3 + ( \cos Q + i \sin Q)( \cos R -i \sin R ) + ( \cos P -i \sin P )( \cos Q -i \sin Q)( \cos R -i \sin R)$

is equal to

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

Ans: (B)

61. A line passing through the point of intersection of $x+y=4$ and makes an angle $tan^ {-1}(3/4)$ with the x-axis. It intersects the parabola at points and $(x_2,y_2)$ respectively. Then $|x_1-x_2|$ is equal to

(A) ${16 \over 9}$ (B) ${32 \over 9}$ (C) ${40 \over 9}$ (D) ${80 \over 9}$

Ans: (B)

62. Let [a] denote the greatest integer which is less than or equal to a. Then the value of the integral

$\int_{-{\pi \over 2}}^{\pi \over 2}[ \sin x \cos x]dx$ is

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

Ans: (D)

63. If $P=\begin{pmatrix} 2 & -2 & -4\\ -1 & 3 & 4\\ 1 & -2 & -3 \end{pmatrix}$ then $P^5$ equals

(A) P (B) 2P (C) -P (D) -2P

Ans: (A)

64. If $\sin^2 \theta + 3 \cos \theta = 2$ , then $\cos^3 \theta + \sec^3 \theta$ is

(A) 1 (B) 4 (C) 9 (D) 18

Ans: (D)

65. $x = 1 + \frac {1}{2 \times |\underline 1} + \frac {1}{4 \times |\underline2} + \frac {1}{8 \times | \underline3} +......$ and $y = 1 + \frac {x^2}{|\underline1} + \frac {x^4}{|\underline 2} + \frac {x^6}{ | \underline 3} +......$

Then the value of $log_ey$ is

(A) e (B) e² (C) 1 (D) 1/e

Ans: (A)

66. The value of the infinite series

(A) e (B) 5e (C) $\frac {5e}{6} - \frac {1}{2}$ (D) $\frac{5e}{6}$

Ans: (C)

67. The value of the integral $\int_{\pi \over 6}^{\pi \over 3} \frac {(\sin x - x \cos x)}{x(x + \sin x)}dx$ is equal to

(A) $log_e(\frac{2(\pi+3)}{2\pi+3\sqrt3})$ (B) $log_e(\frac{\pi+3}{2(2\pi +3\sqrt3)})$ (C) $log_e(\frac{2\pi+3\sqrt3}{2(\pi+3)})$ (D) $log_e(\frac{2(2\pi+3\sqrt3)}{\pi +3})$

Ans: (A)

68. Let . Then

(A) (B) $1 < f(x) \le 2$ (C) $2 < f(x) \le 3$ (D)

Ans: (D)

69. Let , $0 \le x \le 2 \pi$ . Then

(A) F is increasing in $(\frac {\pi}{2},\frac {3\pi}{2})$ and decreasing in $(0,\frac{\pi}{2})$ and $(\frac{3\pi}{2},2\pi)$

(B) F is increasing in $(0,\pi)$ and decreasing in $(\pi,2 \pi)$

(D) F is increasing in $(0,\frac {\pi}{2})$ and $(\frac {3 \pi}{2}, 2\pi)$ and decreasing in

Ans: (D)

70. Let . Then the area of the region enclosed by the curve and three lines , $x=1$ and is

(A) $\frac{63}{2}$ (B) $\frac{93}{5}$ (C) $\frac{105}{7}$ (D) $\frac{129}{10}$

Ans: (D)

71. Let P be a point on the parabola y² = 4ax with focus F.

Let Q denote the foot of the perpendicular from P onto the directrix. Then $\frac {\tan \angle {PQF}}{\tan \angle {PFQ}}$ is

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

Ans: (A)

72. An objective type test paper has 5 questions. Out of these 5 questions, 3 questions have four options each (A, B, C, D) with one option being the correct answer. The other 2 questions have two options each, namely True and False. A candidate randomly ticks the options. Then the probability that he/she will tick the correct option in at least four questions, is

(A) (B) $\frac{3}{128}$ (C) (D) $\frac{3}{64}$

Ans: (D)

73. A family of curves is such that the length intercepted on the y-axis between the origin and the tangent at a point is three the ordinate of the point of contact. The family of curves is

(A) , c is a constant

(B) $xy^2=c$ , c is a constant