WBJEE Mathematics Question Paper 2011 (Eng)

WBJEE - 2011 -  Mathematics

1.   The eccentricity of the hyperbola [tex]4{x^2} - 9{y^2} = 36[/tex] is

(A) [tex]{{\sqrt {11} } \over 3}[/tex]       (B) [tex]{{\sqrt {15} } \over 3}[/tex]      (C)  [tex]{{\sqrt {13} } \over 3}[/tex]      (D) [tex]{{\sqrt {14} } \over 3}[/tex]

Ans : (C)

2.   The length of the latus rectum of the ellipse [tex]16{x^2} + 25{y^2} = 400[/tex] 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) [tex]\left( {{3 \over 2},1} \right)[/tex]       (D) [tex]\left( { - {7 \over 2},1} \right)[/tex]

Ans : (B)

4.   The coordinates of a moving point p are [tex](2{t^2} + 4,4t + 6)[/tex]. 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 [tex]{x^2} + {y^2} - 2x = 0[/tex] is AB.  Equation of the circle with AB as  diameter is

(A) [tex]{x^2} + {y^2} = 1[/tex]         (B) [tex]x(x - 1) + y(y - 1) = 0[/tex]

(C) [tex]{x^2} + {y^2} = 2[/tex]         (D) [tex](x - 1)(x - 2) + (y - 1) + (y - 2) = 0[/tex]

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) [tex]5x + y - 11 = 0[/tex]        (B) [tex]5x + 13y + 5 = 0[/tex]       (C) [tex]5x - 13y + 5 = 0[/tex]        (D) [tex]13x - 5y + 5 = 0[/tex]

Ans : (A)

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

(A) [tex]{{1 - {t^2}} \over {1 + {t^2}}}[/tex]      (B) [tex]{{{t^2} - 1} \over {1 + {t^2}}}[/tex]      (C) [tex]{{ - \left| {1 - {t^2}} \right|} \over {1 + {t^2}}}[/tex]     (D) [tex]{{1 + {t^2}} \over {\left| {1 - {t^2}} \right|}}[/tex]

Ans : (C)

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

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

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 [tex]\tan \alpha = {a \over {a + 1}}[/tex] and [tex]\tan \beta = {1 \over {2a + 1}}[/tex] then [tex]\alpha + \beta [/tex] is

(A) [tex]{\pi \over 4}[/tex]        (B) [tex]{\pi \over 3}[/tex]       (C) [tex]{\pi \over 2}[/tex]       (D) [tex]\pi [/tex]

Ans : (A)

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

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

Ans : (B)

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

(A) [tex]{a^2} + {b^2} + 2ac = 0[/tex]        (B) [tex]{a^2} - {b^2} + 2ac = 0[/tex]

(C) [tex]{a^2} + {c^2} + 2ab = 0[/tex]       (D) [tex]{a^2} - {b^2} - 2ac = 0[/tex]

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 [tex]A = \left( {\matrix{ 3 & {x - 1} \cr {2x + 3} & {x + 2} \cr } } \right)[/tex] is a symmetric matrix, then the value of x  is

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

Ans : (C)

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

(A) z is purely real      (B) z is purely imaginary     (C) [tex]z + \bar z = 0[/tex]      (D) [tex](z - \bar z)i[/tex] is purely imaginary

Ans : (A)

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

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

Ans : (D)

22.   If [tex]\sin \theta + \cos \theta = 0[/tex] and [tex]0 < \theta < \pi[/tex],  then [tex]\theta [/tex]

(A) 0       (B) [tex]{\pi \over 4}[/tex]     (C) [tex]{\pi \over 2}[/tex]    (D)  [tex]{{3\pi} \over 4}[/tex]

Ans : (D)

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

(A) 0     (B) [tex]{1 \over {\sqrt 2 }}[/tex]      (C) [tex] - {1 \over {\sqrt 2 }}[/tex]      (D) [tex]{1 \over {2\sqrt 2 }}[/tex]

Ans : (B)

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

(A)[tex]\pi [/tex]       (B)[tex]{\pi \over 2}[/tex]       (C) [tex]{\pi \over 3}[/tex]      (D) [tex]{\pi \over 4}[/tex]

Ans : (A)

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

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

Ans : (B)

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

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

Ans : (C)

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

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

Ans : (D)

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

(A) applicable when [tex]{\pi \over 2} \le x \le {{3\pi } \over 2}[/tex]        (B) applicable when [tex]0 \le x \le {\pi \over 2}[/tex]

(C) applicable when [tex]0 \le x \le \pi [/tex]         (D) applicable when [tex]{\pi \over 4} \le x \le {\pi \over 2}[/tex]

Ans : (A)

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

(A) [tex](A + Bx){e^{5x}}[/tex]        (B) [tex](A + Bx){e^{4x}}[/tex]       (C) [tex](A + B{x^2}){e^{4x}}[/tex]      (D) [tex](A + B{x^4}){e^{4x}}[/tex]

Ans : (B)

30.   If [tex]{x^2} + {y^2} = 4[/tex], then [tex]y{{dy} \over {dx}} + x = [/tex]

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

Ans : (B)

31.  [tex]\int {{{{x^3}dy} \over {1 + {x^8}}} = } [/tex]

(A) [tex]4{\tan ^{ - 1}}{x^3} + c[/tex]      (B) [tex]{1 \over 2}{\tan ^{ - 1}}{x^4} + c[/tex]      

(C) [tex]x + 4{\tan ^{ - 1}}{x^4} + c[/tex]       (D) [tex]{x^2} + {1 \over 4}{\tan ^{ - 1}}{x^4} + c[/tex]

Ans : (B)

32.  [tex]\int\limits_\pi ^{16\pi } {\left| {\sin x} \right|dx} = [/tex]

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

Ans : (C)

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

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

Ans: (C)

34.  [tex]f(x) = \left\{ {\matrix{0 \cr {x - 3} \cr } } \right.\matrix{, \cr, \cr } \matrix{{x = 0} \cr {x > 0} \cr} [/tex] 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.    [tex]\int {{{\cos 2x} \over {\cos x}}dx} = [/tex]

(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.   [tex]\int {{{{{\sin }^8}x - {{\cos }^8}x} \over {1 - 2{{\sin }^2}x{{\cos }^2}x}}dx} [/tex]

(A)  [tex] - {1 \over 2}\sin 2x + C[/tex]       (B) [tex]{1 \over 2}\sin 2x + C[/tex]      (C) [tex]{1 \over 2}\sin x + C[/tex]      (D) [tex] - {1 \over 2}\sin x + C[/tex]

Ans : (A)

38.   The general solution of the differential equation [tex]{\log _e}\left( {{{dy} \over {dx}}} \right) = x + y[/tex] 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 [tex]y = {A \over x} + B{x^2}[/tex],  then [tex]{x^2}{{{d^2}y} \over {d{x^2}}} = [/tex]

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

Ans: (A)

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

(A) [tex]\omega [/tex]      (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) [tex]{1 \over 2}[/tex]      (B) [tex]{1 \over 4}[/tex]       (C)  [tex]{1 \over 3}[/tex]      (D) [tex]{1 \over 6}[/tex]

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) [tex]{1 \over 16}[/tex]      (B) [tex]{1 \over 2}[/tex]       (C)  [tex]{1 \over 8}[/tex]      (D) [tex]{1 \over 4}[/tex]

Ans : (B)

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

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

Ans :(D)

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

(A)  [tex]2{\log _e}2 + 1[/tex]      (B) [tex]2{\log _e}2[/tex]       (C) [tex]2{\log _e}2 - 1[/tex]      (D) [tex]{\log _e}2 - 1[/tex]

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 [tex]x \ne 0[/tex], 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 [tex]{{ab} \over {{{(a + b)}^2}}} = [/tex]

(A) [tex]{{{p^2}} \over {qr}}[/tex]       (B) [tex]{{pr} \over {{q^2}}}[/tex]      (C) [tex]{{{p^2}} \over {pr}}[/tex]      (D) [tex]{{pq} \over {{r^2}}}[/tex]

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 [tex]z = (1 + {t^2}) + i\sqrt {1 + {t^2}} [/tex] in the complex plane is

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

Ans : (B)

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

(A) [tex]2\cos n\theta [/tex]       (B) [tex]2\sin n\theta [/tex]      (C) [tex]2i \cos n\theta [/tex]      (D) [tex]2i \sin n\theta [/tex]

Ans : (A)

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

(A) [tex]{{3n} \over {\omega - 1}}[/tex]       (B) [tex]3n(\omega - 1)[/tex]      (C) [tex]{{\omega - 1} \over {3n}}[/tex]      (D) 0

Ans : (A)

57.  If [tex]{\log _3}x + {\log _3}y = 2 + {\log _3}2[/tex] and [tex]{\log _3}(x + y) = 2[/tex], 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 [tex]{\log _7}2 = \lambda [/tex] then value of [tex]{\log _{49}}(28)[/tex] is

(A) [tex](2\lambda +1)[/tex]       (B) [tex](2\lambda + 3)[/tex]      (C) [tex]{1 \over 2}(2\lambda + 1)[/tex]     (D) [tex]2(2\lambda + 1)[/tex]

Ans : (C)

59.  The sequence [tex]\log a,\log {{{a^2}} \over b},\log {{{a^3}} \over {{b^2}}}, \cdots \cdots [/tex] 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.   [tex]\left| {\matrix{{a - b} & {b - c} & {c - a}  \cr {b - c} & {c - a} & {a - b}  \cr {c - a} & {a - b} & {b - c}  \cr }} \right| = [/tex]

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

Ans : (A)

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

(A) [tex]{2 \over 3}[/tex] sq. units     (B) [tex]{1 \over 2}[/tex] units       (C) [tex]{1 \over 3}[/tex] units      (D) [tex]{1 \over 6}[/tex] units

Ans:  (D)

63.   Let [tex]f(x) = {x^3}{e^{ - 3x}},x > 0[/tex]. 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) [tex] {{20} \over 3}[/tex] sq. unit      (B) [tex] {{16} \over 3}[/tex] sq. unit      (C) [tex] {{14} \over 3}[/tex] sq. unit     (D)[tex] {{10} \over 3}[/tex] 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) [tex]{{\sqrt 2 - 1} \over 2}[/tex] sec       (B) [tex]{{\sqrt 2 + 1} \over 2}[/tex] sec      (C) [tex](1 + \sqrt 2 )[/tex] sec      (D) [tex](\sqrt 2 - 1)[/tex] sec

Ans : (A)

66.  The solution of [tex]{{dy} \over {dx}} = {y \over x} + \tan {y \over x}[/tex] 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 [tex]{{dy} \over {dx}} - {{3{x^2}y} \over {1 + {x^3}}} = {{{{\sin }^2}(x)} \over {1 + x}}[/tex] is

(A) [tex]{e^{1 + {x^3}}}[/tex]       (B) [tex]\log (1 + {x^3})[/tex]        (C) [tex]1 + {x^3}[/tex]       (D) [tex]{1 \over {1 + {x^3}}}[/tex]

Ans : (D)

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

(A) [tex]y{y_1} = y_2^2[/tex]       (B) [tex]y{y_2} = y_1^2[/tex]      (C) [tex]yy_1^2 = {y_2}[/tex]      (D) [tex]yy_2^2 = {y_1}[/tex]

Ans : (B)

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

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

Ans: (C)

70.  The value of [tex]\int\limits_0^\pi{{{\sin }^{50}}x} {\cos^{49}}xdx[/tex] is

(A) 0      (B) [tex]{\pi \over 4} [/tex]       (C) [tex]{\pi \over 2} [/tex]      (D) 1

Ans: (A)

71.  [tex]\int {{2^x}(f'(x) + f(x)\log 2)dx} [/tex] is

(A) [tex]{2^x}f'(x) + C[/tex]     (B) [tex]{2^x}f(x) + C[/tex]     (C) [tex]{2^x}(\log 2)f(x) + C[/tex]     (D) [tex](\log 2)f(x) + C[/tex]

Ans:(B)

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

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

Ans: (B)

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

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

Ans : (A)

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

(A) n      (B) [tex]{{n+1} \over 2} [/tex]       (C)[tex]{{n(n+1)} \over 2} [/tex]      (D)[tex]{{n(n-1)} \over 2} [/tex]

Ans: (C)

75.   [tex]\lim \limits_{x \to 0} {{\sin (\pi {{\sin }^2}x)} \over {{x^2}}}[/tex] =

(A) [tex]{\pi ^2}[/tex]       (B) [tex]3\pi [/tex]      (C) [tex]2\pi [/tex]      (D) [tex]\pi [/tex]

Ans: (D)

76.   If the function [tex]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 } [/tex] is continuous at x = 2, then

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

Ans : (A)

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

       If ƒ(x) is continuous at x = 2, the value of [tex]\lambda [/tex] will be

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

Ans : (A)

78.   The even function of the following is

(A) [tex]f(x) = {{{a^x} + {a^{- x}}} \over {{a^x} - {a^{- x}}}}[/tex]        (B) [tex]f(x) = {{{a^x} + 1} \over {{a^x} - 1}}[/tex]

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

Ans : (C)

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

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

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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