WBJEE Mathematics Question Paper 2012 (Eng)

               Subject: Mathematics                          

 

Duration : Two Hours                                                                                         Maximum Marks :100

Q. 1 - Q. 60 carry one mark each.

 

1.   The maximum value of |z| when the complex number z satisfies the condition$\left | z + {2 \over z}\right | = 2$ is

        (A) $\sqrt 3$      (B) $\sqrt 3 + \sqrt 2$      (C) $\sqrt 3 + 1$      (D) $\sqrt 3 - 1$

 

2.   If  $\left\( \frac {3}{2} + i \frac {\sqrt 3}{2}\right ) ^{50} = 3^{25} (x + iy)$  where x and y are real,  then the ordered pair (x, y) is

      (A) (-3, 0)        (B) (0, 3)        (C) (0, -3)      (D) $\left\( \frac {1}{2}, \frac {\sqrt 3}{2}\right )$

 

3.   If $\frac {z - 1}{z + 1}$ is purely imaginary, then

       (A) $\left | z \right | =  \frac {1}{2}$     (B) $\left | z \right | = 1$      (C) $\left | z \right | = 2$      (D) $\left | z \right | = 3$

 

4.   There are 100 students in a class. In an examination, 50 of them failed in Mathematics, 45 failed in Physics, 40 failed in Biology and 32 failed in exactly two of the three subjects. Only one student passed in all the subjects. Then the number of students failing in all the three subjects

        (A) is 12       (B) is 4        (C) is 2       (D) cannot be determined from the given information

 

5.   A vehicle registration number consists of 2 letters of English alphabet followed by 4 digits, where the first digit is not zero. Then the total number of vehicles with distinct registration numbers is

       (A) $26^2 \times 10^4$     (B) ${}^{26}{P_2} \times {}^{10}{P_4}$      (C) ${}^{26}{P_2} \times 9 \times {}^{10}{P_3}$      (D) ${26}^2 \times 9 \times {10}^3$

 

6.  The number of words that can be written using all the letters of the word 'IRRATIONAL'  is

      (A) $\frac {10 !}{{(2 !)}^3}$       (B) $\frac {10 !}{{(2 !)}^2}$      (C) $\frac {10 !}{2 !}$     (D) $10 !$

 

7.   Four speakers will address a meeting where speaker Q will always speak after speaker P .  Then the number of ways in which the order of speakers can be prepared is

       (A) 256        (B) 128       (C) 24      (D) 12

 

8.   The number of diagonals in a regular polygon of 100 sides is

       (A) 4950       (B) 4850     (C) 4750      (D) 4650

 

9.   Let the coefficients of powers of x in the 2^nd ,  3^rd  and  4^th  terms in the expansion of (1 + x)ⁿ , where n is a positive integer, be in arithmetic progression. Then the sum of the coefficients of odd powers of x in the expansion is

       (A) 32      (B) 64     (C) 128     (D) 256

 

10.   Let f(x) = ax² + bx + c,   g(x) = px² + qx + r ,  such that f (l) = g(1),     f (2) = g(2)  and  f (3) - g(3) = 2 .  Then f (4) - g(4) is

       (A) 4      (B) 5     (C) 6     (D) 7

 

11.   The sum 1 x 1! + 2 x 2! + ...... + 50 x 50! equals

       (A) 51!     (B) 51! - 1     (C) 51! + 1     (D) 2 x 51!

 

12.   Six numbers are in A.P. such that their sum is 3.  The first term is 4 times the third term.  Then the fifth term is

       (A) -15      (B) -3       (C) 9     (D) -4

 

13.   The sum of the infinite series

        $1 + {1 \over 3} + {1.3 \over 3.6} + {1.3.5 \over 3.6.9} + {1.3.5.7 \over 3.6.9.12} + ......$

is equal to

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

 

14.  The equations x² + x + a = 0  and  x² + ax + 1 = 0 have a common real root

       (A) for no value of a     (B) for exactly one value of a    (C) for exactly two values of a     (D) for exactly three values of a

 

15.  If 64, 27, 36 are the P^th,  Q^th and  R^th  terms of a G.P.,  then P + 2Q is equal to

        (A) R      (B) 2R    (C) 3R      (D) 4R

 

16.   If $(\alpha + \sqrt \beta )$ and $(\alpha - \sqrt \beta )$are the roots of the equation x² + px + q = 0 where α,  β,  p and  q are real, then the roots of the equation

       (p² - 4q)(p²x² + 4px) - 16q = 0 are

      (A) $\left ( \frac {1}{\alpha} + \frac {1}{\sqrt \beta} \right )$ and$\left ( \frac {1}{\alpha} - \frac {1}{\sqrt \beta} \right )$ 

      (B) $\left ( \frac {1}{\sqrt \alpha} + \frac {1}{\beta} \right )$ and$\left ( \frac {1}{\sqrt \alpha} - \frac {1}{\beta} \right )$

      (C) $\left ( \frac {1}{\sqrt \alpha} + \frac {1}{\sqrt \beta} \right )$ and$\left ( \frac {1}{\sqrt \alpha} - \frac {1}{\sqrt \beta} \right )$

      (D) $\left ( \sqrt \alpha + \sqrt \beta \right )$ and $\left ( \sqrt \alpha - \sqrt \beta \right )$

 

17.   The number of solutions of the equation $\log_2(x^2 + 2x - 1) = 1$ is

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

 

18.   The sum of the series

        $1 + {1 \over 2}{}^n{C_1} + {1 \over 3}{}^n{C_2} + ... + \frac {1}{n+1} {}^n{C_n}$  is equal to

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

 

19.   The value of

     $\sum ^{\infty} _{r=2}\frac {1 + 2 + ... + (r -1)}{r!}$ is equal to

        (A) $e$      (B) $2e$       (C) $\frac {e}{2}$      (D) $\frac {3e}{2}$

 

20.   If  $P = \left| {\matrix{ {{1}} & {{2}} & {{1}} \cr {{1}} & {{3}} & {{1}} \cr } } \right|$ , $Q = PP^r$,

      then the value of the determinant of  Q  is equal to

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

 

21.  The remainder obtained when$1! + 2! + ... + 95!$ is divided by 15 is

       (A) $14$      (B) $3$      (C) $1$       (D) $0$

 

22   If P, Q, R are angles of triangle PQR, then the value of

       $\left| {\matrix{ {{-1}} & {{\cos R}} & {{\cos Q}} \cr {{\cos R}} & {{-1}} & {{\cos P}} \cr {{\cos Q}} & {{\cos P}} & {{-1}} \cr } } \right|$

       is equal to

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

 

23.   The number of real values of a for which the system of equations

         $x + 3y + 5z =ax$

         $5x + y + 3z =ay$

         $3x + 5y + z =az$

         has infinite number of solutions is

        (A) $1$       (B) $2$      (C) $4$      (D) $6$

 

24.   The total number of injections (one-one into mappings) from $\left{ a_1,a_2,a_3,a_4 \right }$ to $\left{ b_1,b_2,b_3,b_4,b_5,b_6,b_7 \right }$ is

        (A) 400      (B) 420     (C) 800      (D) 840

 

25.   Let  ${(1 + x)}^{10} = \sum ^{10} _{r=0}c,x^r$ and ${(1 + x)}^{7} = \sum ^{7} _{r=0}d,x^r$ . If $P = \sum ^{5} _{r=0}c_{2r}$ and $Q = \sum ^{3} _{r=0}d_{2r + 1}$,  then

        ${P \over Q}$ is equal to

        (A) 4       (B) 8      (C) 16     (D) 32

 

26.   Two decks of playing cards are well shuffled and 26 cards are randomly distributed to a player. Then the probability that the player gets all distinct cards is

       (A) ${}^{52}{C_{26}} / {}^{104}{C_{26}}$

       (B) $2 \times {}^{52}{C_{26}} / {}^{104}{C_{26}}$

       (C) $2^{13} \times {}^{52}{C_{26}} / {}^{104}{C_{26}}$

       (D) $2^{26} \times {}^{52}{C_{26}} / {}^{104}{C_{26}}$

 

27.   An urn contains 8 red and 5 white balls. Three balls are drawn at random. Then the probability that balls of both colours are drawn is

         (A) ${49 \over 143}$      (B) ${70 \over 143}$       (C) ${3 \over 13}$      (D) ${10 \over 13}$

 

28.   Two coins are available, one fair and the other two-headed. Choose a coin and toss it once; assume that the unbiased coin is chosen with probability${3 \over 4}$.  Given that the outcome is head, the probability that the two-headed coin was chosen is

      (A) ${3 \over 5}$      (B) ${2 \over 5}$       (C) ${1 \over 5}$      (D) ${2 \over 7}$

 

29.   Let  R  be the set of real numbers and the functions f : R → R and g : R → R be defined by f(x) = x² + 2x - 3 and g(x) = x + 1.  Then the value of x for which f (g(x)) = g(f (x))  is

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

 

30.   If  a, b, c  are in arithmetic progression,  then the roots of the equation

        ax² - 2bx + c = 0 are

        (A) $1$and${c \over a}$     (B) $-{1 \over a}$and$-c$     (C) $-1$and$-{c \over a}$     (D) $-2$and$-{c \over 2a}$

 

31.  If  $\sin^{-1} x + \sin^{-1}y + \sin^{-1}z = \frac {3 \pi}{2}$ , then the value of $x^9 + y^9 + z^9 - \frac {1}{x^9 y^9 z^9}$  is equal to

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

 

32.  Let p, q, r be the sides opposite to the angles P, Q, R respectively in a triangle PQR. If r² sin P sin Q = pq, then the triangle is

       (A) equilateral      (B) acute angled but not equilateral      (C) obtuseangled       (D) right angled

 

33.  Let p, q, r  be the sides opposite to the angles P, Q, R respectively in a triangle PQR. Then $2pr \sin \left\( \frac {P-Q+R}{2}\right)$equals

      (A) p² + q² + r²     (B) p² + r² - q²      (C) q² + r² - p²     (D) p² + q² - r²

 

34.   Let P (2, -3),  Q (-2, 1) be the vertices of the triangle PQR. If the centroid of ΔPQR lies on the line 2x + 3y = 1, then the locus of R is

     (A) 2x + 3y = 9      (B) 2x - 3y = 7      (C) 3x + 2y = 5     (D) 3x - 2y = 5

 

35.   ${\lim }\limits_{x \to 0} \frac {\pi^x-1}{\sqrt{1+x}-1}$

       (A) does not exist      (B) equals log_e(π²)     (C) equals 1      (D) lies between 10 and 11

 

36.   If  f  a real-valued differentiable function such that  f(x)f '(x) < 0 for all real x, then

      (A)  f (x) must be an increasing function

      (B)  f (x) must be a decreasing function

      (C)  |f(x)| must be an increasing function

      (D)  |f(x)| must be a decreasing function

 

37.   Rolle's theorem is applicable in the interval [-2, 2] for the function

      (A) f(x) = x³       (B) f(x) = 4x⁴      (C) f(x) = 2x³ + 3      (D) f(x) = π|x|

 

38.  The solution of

     $25 \frac {d^2 y}{dx^2} - 10 \frac {dy}{dx} + y = 0$,  $y(0) = 1$,    $y(1) = 2e^{-1/5}$  is

   (A) $y = e^{5x} + e^{-5x}$     (B) $y = (1 + x)e^{5x}$      (C) $y = (1 + x)e^{x \over 5}$ (D) $y = (1 + x)e^{-{x \over 5}}$

 

39.   Let P be the midpoint of a chord joining the vertex of the parabola y² = 8x to another point on it. Then the locus of P is

      (A) $y^2 = 2x$     (B) $y^2 = 4x$      (C) $y^2 + \frac {x^2}{4} = 1$    (D) $x^2 + \frac {y^2}{4} = 1$

    

40.   The line x - 2y intersects the ellipse ${x^2 \over 4} + y = 1$  at the points P and Q. The

equation of the circle with PQ as diameter is

     (A) $x^2 + y^2 = {1 \over2}$      (B) $x^{2} + y^{2} = 1$      (C) $x^{2} + y^{2} = 2$     (D) $x^2 + y^2 = {5 \over 2}$ 

 

41.  The eccentric angle in the first quadrant of a point on the ellipse ${x^2 \over 10} + {y^2 \over 8} = 1$ at a distance 3 units from the centre of the ellipse is

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

 

42.  The transverse axis of a hyperbola is along the x-axis and its length is 2a.  The vertex of the hyperbola bisects the line segment joining the centre and the focus.  The equation of the hyperbola is

       (A) 6x² - y² = 3a²        (B) x² - 3y² = 3a²        (C) x² - 6y² = 3a²       (D) 3x² - y² = 3a²

 

43.   A point moves in such a way that the difference of its distance from two points (8, 0)  and (-8, 0) always remains 4.  Then the locus of the point is

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

 

44.   The number of integer values of m,  for which the x-coordinate of the point of intersection of the lines 3x + 4y = 9  and  y = mx + 1  is also an integer,  is

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

 

45.   If a straight line passes through the point ( α , β ) and the portion of the line intercepted between the axes is divided equally at that point, then ${x \over \alpha} + {y \over \beta}$ is

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

 

46.  The equation  y² + 4x + 4y + k = 0 represents a parabola whose latus rectum is

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

 

47.  If   the   circles   x² +y² + 2x + 2ky + 6 = 0   and   x² +y² + 2ky + k = 0   intersect orthogonally, then k is equal to

      (A) 2 or$-{3 \over 2}$     (B) -2 or$-{3 \over 2}$     (C) 2 or${3 \over 2}$    (D) -2 or${3 \over 2}$       

 

48.  If four distinct points (2k, 3k),  (2,0),  (0,3),  (0,0) lie on a circle, then

       (A) k < 0        (B) 0 < k < 1      (C) k = 1      (D) k > 1

 

49.   The line joining A(b cos a,  b sin a)  and  B(a cos β, a sin β),  where a ≠ b, is produced to the point M(x,y)  so that AM : MB = b : a.  Then $x \cos \frac {\alpha + \beta}{2} + y \sin \frac{\alpha + \beta}{2}$ is equal to

      (A) 0      (B) 1       (C) -1       (D) a² + b²

 

50.    Let the foci of the ellipse$\frac {x^2}{9} + y^2 = 1$subtend a right angle at a point P.  Then the locus of P is

      (A) x²+y² = l      (B) x²+ y² = 2       (C) x² + y² = 4      (D) x²+y² = 8

 

51.   The general solution of the differential equation

       $\frac {dy}{dx} = \frac {x + y + 1}{2x + 2y + 1}$ is

      (A) log_e |3x + 3y + 2| + 3x + 6y = c       (B) log_e |3x + 3y + 2| - 3x + 6y = c

      (C) log_e |3x + 3y + 2| - 3x - 6y = c        (D) log_e |3x + 3y + 2| + 3x - 6y = c

 

52.   The value of the integral

       $\int_{\pi /6}^{\pi /2} \left ( \frac {1 + \sin 2x + \cos 2x}{\sin x + \cos x} \right )dx$

       (A) 16       (B) 8       (C) 4      (D) 1

 

53.   The value of the integral

        $\int_{0}^{\pi \over 2} \frac {1}{1 + {(\tan x )}^{101}}dx$ 

       (A) 1     (B) $\frac {\pi}{6}$     (C) $\frac {\pi}{8}$      (D) $\frac {\pi}{4}$

 

54.   The integrating factor of the differential equation

        $3x \log_e x \frac {dy}{dx} + y = 2 \log_e x$ is given by

       (A) ${( \log_e x)}^3$      (B) ${\log_e (\log_e x)}$      (C) ${ \log_e x}$      (D) ${( \log_e x)}^{1 \over 3}$

 

55.  Number of solutions of the equation  tan x + sec x = 2 cos x,  x Ɛ [0,π] is

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

 

56.  The value of the integral

        $\int_{0}^{\pi \over 4} \frac { \sin x + \cos x}{3 + \sin 2x}dx$  is equal to

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

 

57.   Let $y = \left ({{3^x - 1} \over {3^x + 1}}\right ) \sin x + \log_e (1 + x) , x > -1$.  Then at $x = 0 , {dy \over dx}$ equals

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

 

58.   Maximum value of the function$f(x) = {x \over 8} + {2 \over x}$ on the interval [1, 6] is

        (A) 1     (B) ${1 \over 8}$     (C) ${13 \over 12}$      (D) ${17 \over 8}$

 

59.   For $-{ \pi \over 2} < x < {3 \pi \over 2}$ ,  the value of

        ${d \over dx} \left { \tan^{-1} \frac {\cos x}{1 + \sin x}\right }$ is equal to

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

 

60.   The value of the integral

        $\int_{-2}^{2} ( 1 + 2 \sin x) e^{\left | x \right |}dx$  is equal to

        (A) 0        (B) e² -1      (C) 2(e² -1)      (D) 1

Q. 61 to Q. 80 carry two marks each.

 

61.   The points representing the complex number z for which

        $arg \left ( \frac {z -2}{z + 2}\right ) = \frac {\pi}{3}$  lie on

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

 

62.   Let  a, b, c, p, q, r  be positive real numbers such that a, b, c are in G.P.  and a ^p = b^q = c ^r .  Then

        (A) p, q, r are in G.P.       (B) p, q, r are in A.P.

       (C) p, q, r are in H.P.       (D) p², q², r² are in A.P.

 

63.   Let S_k be the sum of an infinite G.P. series whose first term is k and common ratio is $\frac {k}{k + 1} \left ( k > 0 \right )$.  Then the value of

       $\sum ^{\infty} _{k=1}\frac {{(-1)}^k}{{S_{k}}$ is equal to

      (A) $\log_e 4$     (B) $\log_e 2 - 1$      (C) $1 - \log_e 2$      (D) $1 - \log_e 4$

 

64.   The quadratic equation $2x^2 - ( a^3 + 8a - 1 )x + a^2 - 4a = 0$

       possesses roots of opposite sign. Then

      (A) $a \le 0$      (B) $0  \lt a \lt 4$      (C) $4 \le a \lt 8$    (D) $a \ge 8$

 

65.  If $\log_e (x^2 - 16) \le \log_e(4x - 11)$, then

      (A) $4 \lt x \le 5$      (B) $x \lt - 4$or$x \gt 4$      (C) $-1 \le x \le 5$      (D) $x \le -1$or$x \gt 5$

 

66.  The coefficient of x¹⁰ in the expansion of

       $1 + (1 + x) + ... + (1 + x)^{20}$  is

      (A) ${}^{19}{C_{9}}$      (B) ${}^{20}{C_{10}}$      (C) ${}^{21}{C_{11}}$      (D) ${}^{22}{C_{12}}$

 

67.   The system of linear equations

        $\lambda x + y + z = 3$

        $x - y - 2z = 6$

        $- x + y + z = \mu$

        has

       (A)   infinite number of solutions for λ ≠ -1 and all  μ        (B)   infinite number of solutions for λ = -1 and  μ = 3

       (C)  no solution for λ ≠ -1

       (D)  unique solution for  λ = -1 and  μ = 3

 

68.   Let A and B be two events with P(A^c) = 0.3,  P(B) = 0.4 and$P (A \cap B^c) = 0.5$.  Then

       $P (B \left | A \cap B^c)$ is equal to

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

 

69.   Let p, q, r  be the altitudes of a triangle with area S and perimeter 2t. Then the value of

       ${1 \over p}+{1 \over q}+{1 \over r}$ is

      (A) ${s \over t}$        (B) ${t \over s}$        (C) ${s \over 2t}$      (D) ${2s \over t}$

 

70.   Let C₁ and C₂ denote the centres of the circles  x² + y² = 4  and  (x - 2)² + y² = 1 respectively and let P and Q  be their points of intersection. Then the areas of triangles C₁ PQ and C₂ PQ are in the ratio

        (A) 3 : 1      (B) 5 : 1      (C) 7 : 1       (D) 9 : 1

 

71.   A straight line through the point of intersection of the lines  x + 2y = 4  and  2x + y = 4  meets the coordinate axes at  A  and  B .  The locus of the midpoint of AB  is

       (A) 3(x + y) = 2xy      (B) 2(x + y) = 3xy     (C) 2(x + y) = xy     (D) x + y = 3xy

 

72.    Let P and Q be the points on the parabola y² = 4x so that the line segment PQ subtends right angle at the vertex.  If  PQ intersects the axis of the parabola at  R ,  then the distance of the vertex from R is

       (A) 1       (B) 2       (C) 4       (D) 6

 

73.   The incentre of an equilateral triangle is (1, 1) and the equation of one side is 3x + 4y + 3 = 0.  Then the equation of the circumcircle of the triangle is

       (A) x² + y² - 2x - 2y - 2 = 0 

       (B) x² + y² - 2x - 2y - 14 = 0

       (C) x² + y² - 2x - 2y + 2 = 0

       (D) x² + y² - 2x - 2y + 14 = 0

 

74.   The value of ${\lim }\limits_{n \to \infty} \frac {{(n!)}^{1 \over n}}{{n}}$  is

       (A) $1$      (B) ${1 \over e^2}$      (C) ${1 \over 2e}$      (D) ${1 \over e}$

 

75.   The area of the region bounded by the curves$y = x^3$, $y = {1 \over x}$,  $x = 2$  is

       (A) $4 - \log_e 2$     (B) ${1 \over 4} + \log_e 2$      (C) $3 - \log_e 2$     (D) ${15 \over 4}- \log_e 2$

76.   Let y be the solution of the differential equation

       $x \frac {dy}{dx} = \frac {y^2}{1- y \log x}$ satisfying y(l) = 1.  Then y satisfies

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

 

77.   The area of the region, bounded by the curves  y = sin ^-1 x + x(l — x)  and y = sin^-1 x - x(l - x) in the first quadrant,  is

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

 

78.  The value of the integral

       $\int^5_1 \left [ \left | x - 3 \right | + \left | 1 - x \right | \right ] dx$ is equal to

      (A) 4      (B) 8     (C) 12      (D) 16

 

79.  If   f(x)  and  g(x) are twice differentiable functions on (0, 3) satisfying f"(x) = g"(x),  f '(1) = 4, g'(1) = 6,  f(2) = 3, g(2) = 9,  then f(1) - g(1)  is

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

 

80.   Let [x] denote the greatest integer less than or equal to x, then the value of the integral

        $\int^1_{-1} \left ( \left | x \right | - 2 \left [ x \right ] \right ) dx$  is equal to

        (A) 3      (B) 2       (C) -2       (D) -3

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