WBJEE Mathematics Question Paper 2010 (Eng)

WBJEE - 2010 - Mathematics

1.  The value of ${{\cot x - \tan x} \over {\cot 2x}}$ is

(A) 1    (B) 2     (C) –1      (D) 4

Ans : (B)

2.  The number of points of intersection of 2y = 1 and y = sin x, in  −2π ≤ x ≤ 2π is

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

Ans : (D)

3.  Let R be the set of real numbers and the mapping ƒ : R → R and g : R → R be defined by ƒ(x) = 5 – x²  and g(x) = 3x – 4, then the value of (fog)(–1) is

(A) –44       (B) –54       (C) –32      (D) –64

Ans : (A)

4.  A = {1, 2, 3, 4},  B = {1, 2, 3, 4, 5, 6} are two sets, and function ƒ : A → B is defined by ƒ(x) = x + 2∀x ∈A , then the function ƒ is

(A) bijective        (B) onto        (C) one–one        (D) many–one

Ans : (C)

5.  If the matrices $A = \left[ {\matrix{ 2 & 1 & 3 \cr 4 & 1 & 0 \cr } } \right]$ and $B = \left[ {\matrix{ 1& { - 1} \cr 0 & 2 \cr 5 & 0 \cr } } \right]$, then AB will be

(A) $\left[ {\matrix {{17} & 0 \cr 4 & {- 2} \cr } } \right]$      (B) $\left[ {\matrix{ 4 & 0 \cr 0 & 4 \cr } } \right]$      (C) $\left[ {\matrix{ {17} & 4 \cr 0 & { - 2} \cr } } \right]$       (D)  $\left[ {\matrix{ 0 & 0 \cr 0 & 0 \cr } } \right]$

Ans : (A)

6.   ω is an imaginary cube root of unity and $\left[ {\matrix{ {x + {\omega ^2}} & \omega & 1 \cr \omega & {{\omega ^2}} & {1 + x} \cr 1 & {x + \omega } & {{\omega ^2}} \cr } } \right] = 0$ then one of the values of x is

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

Ans : (B)

7.  If $A = \left[{\matrix{1 & 2 \cr {-4} & {- 1} \cr } } \right]$ then A^–1 is

(A) ${1 \over 7}\left[{\matrix{ { - 1} & { - 2} \cr 4 & 1 \cr } } \right]$      (B) ${1 \over 7}\left[ {\matrix{1 & 2 \cr { - 4} & { - 1} \cr } } \right]$       (C) ${1 \over 7}\left[ {\matrix{ { - 1} & { - 2} \cr 4 & 1 \cr } } \right]$       (D) Does not exist

Ans : Both (A) & (C)

8.  The value of ${2 \over {3!}} + {4 \over {5!}} + {6 \over {7!}} + \cdots \cdots$ is

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

Ans : (B)

9.  If sum of an infinite geometric series is ${4 \over 5}$ and its 1st term is ${3 \over 4}$, then its  common ratio is

(A) ${7 \over {16}}$       (B) ${9 \over {16}}$       (C) ${1 \over 9}$      (D) ${7 \over 9}$

Ans : (A)

10.  The number of permutations by taking all letters and keeping the vowels of the word COMBINE in the odd places is

(A) 96     (B) 144      (C) 512      (D) 576

Ans : (D)

11.  If ${}^{n - 1}{C_3} + {}^{n - 1}{C_4} > {}^n{C_3}$, then n is just greater than integer

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

Ans : (D)

12.  If in the expansion of ${(a - 2b)^n}$, the sum of the 5th and 6th term is zero, then the value of ${a \over b}$ is

(A) ${{n - 4} \over 5}$       (B)  ${{2(n - 4)} \over 5}$       (C) ${5 \over {n - 4}}$      (D) ${5 \over {2(n - 4)}}$

Ans : (B)

13.  $({2^{3n}} - 1)$ will be divisible by ( ∀n ∈ N )

(A) 25      (B) 8       (C) 7      (D) 3

Ans : (C)

14.  Sum of the last 30 coefficients in the expansion of ${(1 + x)^{59}}$ , when expanded in ascending powers of x is

(A) ${2^{59}}$       (B) ${2^{58}}$        (C) ${2^{30}}$      (D) ${2^{29}}$

Ans : (B)

15.  If ${\left( {1 - x + {x^2}} \right)^n} = {a_0} + {a_1}x + \cdots \cdots + {a_{2n}}{x^{2n}}$, then the value of ${a_0} + {a_2} + {a_4} + \cdots \cdots + {a_{2n}}$ is

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

Ans : (D)

16.   If α,  β be the roots of the quadratic equation x² + x + 1 = 0 then the equation whose roots are ${\alpha ^{19}},{\beta ^7}$ is

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

Ans : (D)

17.  The roots of the quadratic equation x² - 2√3x - 22 = 0 are :

(A) imaginry       (B) real, rational and equal      (C) real, irrational and unequal       (D) real, rational and unequal

Ans : (C)

18.   The qudratic equation ${x^2} + 15\left| x \right| + 14 > 0$ has

(A) only positive solutions      (B) only negative solutions       (C) no solution       (D) both positive and negative solution

Ans : (C)

19.  If $z = {4 \over {1 - i}}$, then $\overline z$ is (where $\overline z$ is complexconjugate of z )

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

Ans : (D)

20.  If $- \pi < \arg (z) < {\pi \over 2}$ then $\arg \overline z - \arg ( - \overline z )$ is

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

Ans : (A)

21.  Two dice are tossed once. The probability of getting an even number at the first die or a total of 8 is

(A) ${1 \over {36}}$      (B) ${3 \over {36}}$      (C)  ${{11} \over {36}}$     (D) ${{20}\over {36}}$

Ans : (D)

22.  The probability that at least one of A and B occurs is 0.6 . If A and B occur simultaneously with probability 0.3, then P(A′) + P(B′) is

(A) 0.9     (B) 0.15       (C) 1.1     (D) 1.2

Ans : (C)

23.  The value of ${{{{\log }_3}5 \times {{\log }_{25}}27 \times {{\log }_{49}}7} \over {{{\log }_{81}}3}}$ is

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

Ans : (D)

24.   In a right-angled triangle, the sides are a, b and c, with c as hypotenuse, and c − b ≠ 1, c + b ≠ 1 . Then the value of $({\log _{c + b}}a + {\log _{c - b}}a)/(2{\log _{c + b}}a \times {\log _{c - b}}a)$ will be

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

Ans : (D)

25.   Sum of n terms of the following series ${1^3} + {3^3} + {5^3} + {7^3} + \cdots \cdots$ is

(A) ${n^2}(2{n^2} - 1)$       (B) ${n^3}(n - 1)$       (C) ${n^3} + 8n + 4$      (D) $2{n^4} + 3{n^2}$

Ans : (A)

26.   G.. M. and H. M. of two numbers are 10 and 8 respectively. The numbers are :

(A) 5,  20        (B) 4,  25       (C) 2, 50        (D) 1, 100

Ans : (A)

27.  The value of n for which ${{{x^{n + 1}} + {y^{n + 1}}} \over {{x^n} + {y^n}}}$ is the geometric mean of x and y is

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

Ans : (A)

28.  If angles A, B and C are in A.P., then ${{a + c} \over b}$ is equal to

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

Ans : (B)

29.  If ${{\cos A} \over 3} = {{\cos B} \over 4} = {1 \over 5},{\pi \over 2} < A < 0, - {\pi \over 2} < B < 0$ then value of $2\sin A + 4\sin B$ is

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

Ans : (C)

30.  The value of ${{\cot {{54}^ \circ }} \over {\tan {{36}^ \circ }}} + {{\tan {{20}^ \circ }} \over {\cot {{70}^ \circ}}}$ is

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

Ans : (B)

31.   If $sin6\theta + sin4\theta + sin2\theta = 0$ then the general value of θ is

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

Ans : (A)

32.   In a Δ ABC, $2ac\sin {{A - B + C} \over 2}$ is equal to

(A) ${a^2} + {b^2} - {c^2}$       (B) ${c^2} + {a^2} - {b^2}$       (C) ${b^2} - {a^2} - {c^2}$    (D) ${c^2} - {a^2} - {b^2}$

Ans : (B)

33.   Value of ${\tan ^{ - 1}}\left( {{{\sin 2 - 1} \over {\cos 2}}} \right)$ is

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

Ans : (D)

34.  The straight line 3x + y = 9 divides the line segment joining the points (1,3) and (2,7) in the ratio

(A) 3 : 4 externally       (B) 3 : 4 internally      (C) 4 : 5 internally       (D) 5 : 6 externally

Ans : (B)

35.  If the sum of distances from a point P on two mutually perpendicular straight lines is 1 unit, then thelocus of P is

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

Ans : (D)

36.  The straight line x + y – 1 = 0 meets the circle ${x^2} + {y^2} - 6x - 8y = 0$ at A and B. Then the equation of the circle of which AB is a diameter is

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

Ans : (A)

37.   If ${t_1}$ and ${t_2}$ be the parameters of the end points of a focal chord for the parabola y² = 4ax, then which one is true ?

(A) ${t_1}{t_2} = 1$      (B) ${{{t_1}} \over {{t_2}}} = 1$      (C) ${t_1}{t_2} = - 1$     (D) ${t_1} + {t_2} = - 1$

Ans : (C)

38.  S and T are the foci of an ellipse and B is end point of the minor axis. If STB is an equilateral triangle, the eccentricity of the ellipse is

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

Ans : (C)

39.   For different values of α , the locus of the point of intersection of the two straight lines $\sqrt 3 x - y -4\sqrt 3 \alpha = 0$ and $\sqrt 3 \alpha x + \alpha y - 4\sqrt 3 = 0$ is

(A) a hyperbola with eccentricity 2

(B) an ellipse with eccentricity $\sqrt {{2 \over 3}}$    

(C) a hyperbola with eccentricity $\sqrt {{{19} \over {16}}}$

(D) an ellipse with eccentricity ${3 \over 4}$

Ans : (A)

40.  The area of the region bounded by y² = x and y = |x| is

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

Ans : (B)

41.  If the displacement, velocity and acceleration of a particle at time, t be x, v and f respectively, then which one is true ?

(A) $f = {v^3}{{{d^2}t} \over {d{x^2}}}$       (B) $f = - {v^3}{{{d^2}t} \over {d{x^2}}}$       (C) $f = {v^2}{{{d^2}t} \over {d{x^2}}}$       (D) $f = - {v^2}{{{d^2}t} \over {d{x^2}}}$

Ans : (B)

42.  The displacement x of a particle at time t is given by x = At² + Bt + C where A, B, C are constants and vis velocity of a particle, then the value of 4Ax – v² is

(A) 4AC + B²     (B) 4AC – B²      (C) 2AC – B²       (D) 2AC + B²

Ans : (B)

43.   For what values of x, the function $f(x) = {x^4} - 4{x^3} + 4{x^2} + 40$ is monotone decreasing ?

(A) 0 < x < 1      (B) 1 < x < 2       (C) 2 < x < 3       (D) 4 < x < 5

Ans : (B)

44.  The displacement of a particle at time t is x, where $x = {t^4} - k{t^3}$ .  If the velocity of the particle at time t = 2 is minimum, then

(A) k = 4      (B) k = – 4       (C) k = 8       (D) k = – 8

Ans : (A)

45.  The point in the interval [0,2π], where ƒ(x) = e^x sin x has maximum slope, is

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

Ans : (B)

46.  The minimum value of $f(x) = {e^{\left( {{x^4} - {x^3} + {x^2}} \right)}}$ is

(A) e       (B) – e      (C) 1       (D) –1

Ans : (C)

47.  $\int {{{\log \sqrt x } \over {3x}}} dx$ is equal to

(A) ${1 \over 3}{\left( {\log \sqrt x } \right)^2} + C$       (B) ${2 \over 3}{\left( {\log \sqrt x } \right)^2} + C$       (C) ${2 \over 3}{\left( {\log x} \right)^2} + C$      (D) ${1 \over 3}{\left( {\log x} \right)^2} + C$

Ans : (A)

48.  $\int {{e^x}} \left( {{2 \over x} - {2 \over {{x^2}}}} \right)dx$ is equal to

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

Ans : (C)

49.  The value of the integral $\int {{{dx} \over {{{\left( {{e^x} + {e^{ - x}}} \right)}^2}}}}$ is

(A) ${1 \over 2}\left( {{e^{2x}} + 1} \right) + C$        (B) ${1 \over 2}\left( {{e^{ - 2x}} + 1} \right) + C$       (C) $- {1 \over 2}{\left( {{e^{2x}} + 1} \right)^{ - 1}} + C$        (D) ${1 \over 4}\left( {{e^{2x}} - 1} \right) + C$

Ans : (C)

50.   The value of ${Lt}\limits_{x \to 0} {{{{\sin }^2}x + \cos x - 1} \over {{x^2}}}$ is

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

Ans : (B)

51.  The value of ${Lt}\limits_{x \to 0} {\left( {{{1 + 5{x^2}} \over {1 + 3{x^2}}}} \right)^{{1 \over {{x^2}}}}}$ is

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

Ans : (A)

52.  In which of the following functions, Rolle’s theorem is applicable ?

(A) ƒ(x) =|x| in − 2 ≤ x ≤ 2         (B) ƒ(x) = tan x in 0 ≤ x ≤ π      

(C) $f(x) = 1 + {\left( {x - 2} \right)^{{2 \over 3}}}$ in 1 ≤ x ≤ 3       (D) ƒ(x) = x (x − 2)² in 0 ≤ x ≤ 2

Ans : (D)

53.  If ƒ(5) = 7 and ƒ′(5) = 7 then ${Lt}\limits_{x \to 5} {{xf(5) - 5f(x)} \over {x - 5}}$ is given by

(A) 35      (B) – 35     (C) 28      (D) – 28

Ans : (D)

54.  If $y = (1 + x)(1 + {x^2})(1 + {x^4}) \cdots (1 + {x^{2n}})$ then the value of ${\left( {{{dy} \over{dx}}} \right)_{x = 0}}$ is

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

Ans : (C)

55.  The value of ƒ(0) so that the function $f(x) = {{1 - \cos (1 - \cos x)} \over {{x^4}}}$ is continuous everywhere is

(A) ${1 \over 2}$       (B) ${1 \over 4}$      (C) ${1 \over 6}$      (D) ${1 \over 8}$

Ans : (D)

56.  $\int {\sqrt {1 + \cos x} dx}$ is equal to

(A) $2\sqrt 2 \cos {x \over 2} + C$       (B) $2\sqrt 2 \sin {x \over 2} + C$       (C) $\sqrt 2 \cos {x \over 2} + C$      (D) $\sqrt 2 \sin {x \over 2} + C$

Ans : (B)

57.  The function $f(x) = \sec \left[ {\log \left( {x + \sqrt {1 + {x^2}} } \right)} \right]$ is

(A) odd       (B) even       (C) neither odd nor even       (D) constant

Ans : (B)

58.  ${\lim }\limits_{x \to 0} {{\sin \left| x \right|} \over x}$ is equal to

(A) 1      (B) 0      (C) positive infinity       (D) does not exist

Ans : (D)

59.  The co-ordinates of the point on the curve $y = {x^2} - 3x + 2$ where the tangent is perpendicular to the straight line y = x are

(A) (0, 2)      (B) (1, 0)      (C) (–1, 6)      (D) (2, –2)

Ans : (B)

60.  The domain of the function $f(x) = \sqrt {{{\cos }^{ - 1}}\left( {{{1 - \left| x \right|} \over 2}} \right)}$ is

(A) (–3, 3)       (B) [–3, –3]      (C) (−∞,−3)U(3,∞)      (D) (−∞,−3]U[3,∞)

Ans : (B)

61.  If the line ax + by + c = 0 is a tangent to the curve xy = 4, then

(A) a < 0,  b > 0       (B) a ≤ 0,  b > 0       (C) a < 0,  b < 0       (D) a ≤ 0,  b < 0

Ans : (C)

62.  If the normal to the curve y = ƒ(x) at the point (3, 4) make an angle 3π/4 with the positive x-axis, then ƒ′(3) is

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

Ans : (A)

63.  The general solution of the different equation $100{{{d^2}y} \over {d{x^2}}} - 20{{dy} \over {dx}} + y =0$ is

(A) $y = ({c_1} + {c_2}x){e^x}$      (B) $y = ({c_1} + {c_2}x){e^{ - x}}$        (C) $y = ({c_1} + {c_2}x){e^{{x \over {10}}}}$       (D) $y = {c_1}{e^x} + {c_2}{e^{ - x}}$

Ans : (C)

64.  If y′′ – 3y′ + 2y = 0 where y(0) = 1, y′(0) = 0, then the value of y at x = log_e2 is

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

Ans : (D)

65.  The degree of the differential equation $x = 1 + \left( {{{dy} \over {dx}}} \right) + {1 \over {2!}}{\left({{{dy} \over {dx}}} \right)^2} + {1 \over {3!}}{\left( {{{dy} \over {dx}}} \right)^3} + \cdots \cdots$

(A) 3     (B) 2      (C) 1     (D) not defined

Ans : (C)

66.  The equation of one of the curves whose slope at any point is equal to y + 2x is

(A) y = 2(e^x + x −1)       (B) y = 2(e^x − x −1)       (C) y = 2(e^x − x +1)     (D)y = 2(e^x + x +1)

Ans : (B)

67.   Solution of the differential equation xdy – ydx = 0 represents a

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

Ans : (D)

68.   The value of the integral $\int\limits_0^{\pi /2} {{{\sin }^5}xdx}$ is

(A) ${4 \over {15}}$       (B) ${8 \over 5}$       (C)  ${8 \over {15}}$       (D) ${4 \over 5}$

Ans : (C)

69.   If ${d \over {dx}}\left\{ {f(x)} \right\} = g(x)$, then $\int\limits_a^b {f(x)g(x)dx}$ is equal to

(A) ${1 \over 2}\left[ {{f^2}(b) - {f^2}(a)} \right]$       (B) ${1 \over 2}\left[ {{g^2}(b) - {g^2}(a)} \right]$       (C) $f(b) - f(a)$       (D) ${1 \over 2}\left[ {g({b^2}) - g({a^2})} \right]$

Ans : (A)

70.   If ${I_1} = \int\limits_0^{3\pi } {f({{\cos }^2}x)dx}$ and ${I_2} = \int\limits_0^\pi {f({{\cos }^2}x)dx}$ then

(A) ${I_1} = {I_2}$       (B) $3{I_1} = {I_2}$        (C) ${I_1} = 3{I_2}$        (D) ${I_1} = 5{I_2}$

Ans : (C)

71.   The value of $I = \int\limits_{ - \pi /2}^{\pi /2} {\left| {\sin x} \right|dx}$ is

(A) 0       (B) 2      (C) – 2      (D) – 2 < I < 2

Ans : (B)

72.   If $I = \int\limits_0^I {{{dx} \over {1 + {x^{\pi /2}}}}}$, then

(A) ${\log _e}2 < 1 < \pi /4$       (B) ${\log _e}2 > 1$       (C) $I = \pi /4$     (D) $I = {\log_e}2$

Ans : (A)

73.   The area enclosed by y = 3x – 5,  y = 0,  x = 3 and x = 5 is

(A) 12 sq. units      (B) 13 sq. units      (C) $13{1 \over 2}$ sq. units     (D) 14 sq. units

Ans : (D)

74.  The area bounded by the parabolas y = 4x²,  $y = {{{x^2}} \over 9}$ and the line y = 2 is

(A) ${{5\sqrt 2 } \over 3}$ sq. units       (B) ${{10\sqrt 2 } \over 3}$ sq. units       (C) ${{15\sqrt 2 }\over 3}$ sq. units       (D) ${{20\sqrt 2 } \over 3}$ sq. units

Ans : (B)

75.  The equation of normal of x² + y² – 2x + 4y – 5 = 0 at (2, 1) is

(A) y = 3x – 5      (B) 2y = 3x – 4      (C) y = 3x + 4      (D) y = x + 1

Ans : (A)

76.   If the three points (3q, 0), (0, 3p) and (1, 1) are collinear then which one is true ?

(A) ${1 \over p} + {1 \over q} = 1$       (B) ${1 \over p} + {1 \over q} = 2$        (C) ${1 \over p} + {1 \over q} = 3$      (D) ${1 \over p} + {3 \over q} = 1$

Ans : (C)

77.   The equations $y = \pm \sqrt {3x}$, y = 1 are the sides of

(A) an equilateral triangle       (B) a right angled triangle        (C) an isosceles triangle        (D) an obtuse angled triangle

Ans : (A)

78.  The equations of the lines through (1, 1) and making angles of 45° with the line x + y = 0 are

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

Ans : (D)

79.   In a triangle PQR, $\angle R = \pi /2$ . If tan $\left( {{p \over 2}} \right)$  and tan $\left({{Q \over 2}} \right)$ are roots of ax² + bx + c = 0, where a ≠ 0, then which one is true ?

(A) c = a + b      (B) a = b + c       (C) b = a + c       (D) b = c

Ans : (A)

80.  The value of ${{\sin {{55}^ \circ } - \cos {{55}^ \circ }} \over {\sin {{10}^ \circ }}}$ is

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

Ans : (D)

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