WBJEE - 2010 - Mathematics
1. The value of [tex]{{\cot x - \tan x} \over {\cot 2x}}[/tex] 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 [tex]A = \left[ {\matrix{ 2 & 1 & 3 \cr 4 & 1 & 0 \cr } } \right][/tex] and [tex]B = \left[ {\matrix{ 1& { - 1} \cr 0 & 2 \cr 5 & 0 \cr } } \right][/tex], then AB will be
(A) [tex]\left[ {\matrix {{17} & 0 \cr 4 & {- 2} \cr } } \right][/tex] (B) [tex]\left[ {\matrix{ 4 & 0 \cr 0 & 4 \cr } } \right][/tex] (C) [tex]\left[ {\matrix{ {17} & 4 \cr 0 & { - 2} \cr } } \right][/tex] (D) [tex]\left[ {\matrix{ 0 & 0 \cr 0 & 0 \cr } } \right][/tex]
Ans : (A)
6. ω is an imaginary cube root of unity and [tex]\left[ {\matrix{ {x + {\omega ^2}} & \omega & 1 \cr \omega & {{\omega ^2}} & {1 + x} \cr 1 & {x + \omega } & {{\omega ^2}} \cr } } \right] = 0[/tex] then one of the values of x is
(A) 1 (B) 0 (C) –1 (D) 2
Ans : (B)
7. If [tex]A = \left[{\matrix{1 & 2 \cr {-4} & {- 1} \cr } } \right][/tex] then A–1 is
(A) [tex]{1 \over 7}\left[{\matrix{ { - 1} & { - 2} \cr 4 & 1 \cr } } \right][/tex] (B) [tex]{1 \over 7}\left[ {\matrix{1 & 2 \cr { - 4} & { - 1} \cr } } \right][/tex] (C) [tex]{1 \over 7}\left[ {\matrix{ { - 1} & { - 2} \cr 4 & 1 \cr } } \right][/tex] (D) Does not exist
Ans : Both (A) & (C)
8. The value of [tex]{2 \over {3!}} + {4 \over {5!}} + {6 \over {7!}} + \cdots \cdots [/tex] is
(A) [tex]{e^{{1 \over 2}}}[/tex] (B) [tex]{e^{ - 1}}[/tex] (C) e (D) [tex]{e^{{1 \over 3}}}[/tex]
Ans : (B)
9. If sum of an infinite geometric series is [tex]{4 \over 5}[/tex] and its 1st term is [tex]{3 \over 4}[/tex], then its common ratio is
(A) [tex]{7 \over {16}}[/tex] (B) [tex]{9 \over {16}}[/tex] (C) [tex]{1 \over 9}[/tex] (D) [tex]{7 \over 9}[/tex]
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 [tex]{}^{n - 1}{C_3} + {}^{n - 1}{C_4} > {}^n{C_3}[/tex], then n is just greater than integer
(A) 5 (B) 6 (C) 4 (D) 7
Ans : (D)
12. If in the expansion of [tex]{(a - 2b)^n}[/tex], the sum of the 5th and 6th term is zero, then the value of [tex]{a \over b}[/tex] is
(A) [tex]{{n - 4} \over 5}[/tex] (B) [tex]{{2(n - 4)} \over 5}[/tex] (C) [tex]{5 \over {n - 4}}[/tex] (D) [tex]{5 \over {2(n - 4)}}[/tex]
Ans : (B)
13. [tex]({2^{3n}} - 1)[/tex] 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 [tex]{(1 + x)^{59}}[/tex] , when expanded in ascending powers of x is
(A) [tex]{2^{59}}[/tex] (B) [tex]{2^{58}}[/tex] (C) [tex]{2^{30}}[/tex] (D) [tex]{2^{29}}[/tex]
Ans : (B)
15. If [tex]{\left( {1 - x + {x^2}} \right)^n} = {a_0} + {a_1}x + \cdots \cdots + {a_{2n}}{x^{2n}}[/tex], then the value of [tex]{a_0} + {a_2} + {a_4} + \cdots \cdots + {a_{2n}}[/tex] is
(A)[tex]{3^n} + {1 \over 2}[/tex] (B) [tex]{3^n} - {1 \over 2}[/tex] (C) [tex]{{{3^n} - 1} \over 2}[/tex] (D) [tex]{{{3^n} + 1} \over 2}[/tex]
Ans : (D)
16. If α, β be the roots of the quadratic equation x² + x + 1 = 0 then the equation whose roots are [tex]{\alpha ^{19}},{\beta ^7}[/tex] 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 [tex]{x^2} + 15\left| x \right| + 14 > 0[/tex] has
(A) only positive solutions (B) only negative solutions (C) no solution (D) both positive and negative solution
Ans : (C)
19. If [tex]z = {4 \over {1 - i}}[/tex], then [tex]\overline z [/tex] is (where [tex]\overline z [/tex] is complexconjugate of z )
(A) [tex]2(1 + i)[/tex] (B) [tex](1 + i)[/tex] (C) [tex]{2 \over {1 - i}}[/tex] (D) [tex]{4 \over {1 + i}}[/tex]
Ans : (D)
20. If [tex] - \pi < \arg (z) < {\pi \over 2}[/tex] then [tex]\arg \overline z - \arg ( - \overline z )[/tex] is
(A) [tex]\pi [/tex] (B) [tex] - \pi [/tex] (C) [tex]{\pi \over 2}[/tex] (D) [tex] - {\pi \over 2}[/tex]
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) [tex]{1 \over {36}}[/tex] (B) [tex]{3 \over {36}}[/tex] (C) [tex]{{11} \over {36}}[/tex] (D) [tex]{{20}\over {36}}[/tex]
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 [tex]{{{{\log }_3}5 \times {{\log }_{25}}27 \times {{\log }_{49}}7} \over {{{\log }_{81}}3}}[/tex] is
(A) 1 (B) 6 (C) [tex]{2 \over 3}[/tex] (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 [tex]({\log _{c + b}}a + {\log _{c - b}}a)/(2{\log _{c + b}}a \times {\log _{c - b}}a)[/tex] will be
(A) 2 (B) –1 (C) [tex]{1 \over 2}[/tex] (D) 1
Ans : (D)
25. Sum of n terms of the following series [tex]{1^3} + {3^3} + {5^3} + {7^3} + \cdots \cdots [/tex] is
(A) [tex]{n^2}(2{n^2} - 1)[/tex] (B) [tex]{n^3}(n - 1)[/tex] (C) [tex]{n^3} + 8n + 4[/tex] (D) [tex]2{n^4} + 3{n^2}[/tex]
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 [tex]{{{x^{n + 1}} + {y^{n + 1}}} \over {{x^n} + {y^n}}}[/tex] is the geometric mean of x and y is
(A) [tex]n = - {1 \over 2}[/tex] (B) [tex]n = {1 \over 2}[/tex] (C) [tex]n = 1 [/tex] (D) [tex]n = - 1[/tex]
Ans : (A)
28. If angles A, B and C are in A.P., then [tex]{{a + c} \over b}[/tex] is equal to
(A) [tex]2\sin {{A - C} \over 2}[/tex] (B) [tex]2\cos {{A - C} \over 2}[/tex] (C) [tex]\cos {{A - C} \over 2}[/tex] (D) [tex]\sin {{A - C} \over 2}[/tex]
Ans : (B)
29. If [tex]{{\cos A} \over 3} = {{\cos B} \over 4} = {1 \over 5},{\pi \over 2} < A < 0, - {\pi \over 2} < B < 0[/tex] then value of [tex]2\sin A + 4\sin B[/tex] is
(A) 4 (B) –2 (C) –4 (D) 0
Ans : (C)
30. The value of [tex]{{\cot {{54}^ \circ }} \over {\tan {{36}^ \circ }}} + {{\tan {{20}^ \circ }} \over {\cot {{70}^ \circ}}}[/tex] is
(A) 0 (B) 2 (C) 3 (D) 1
Ans : (B)
31. If [tex]sin6\theta + sin4\theta + sin2\theta = 0[/tex] then the general value of θ is
(A) [tex]{{n\pi} \over 4},n\pi \pm {\pi \over 3}[/tex] (B) [tex]{{n\pi } \over 4},n\pi \pm {\pi \over 6}[/tex] (C) [tex]{{n\pi } \over 4}, 2n\pi \pm {\pi \over 3}[/tex] (D) [tex]{{n\pi} \over 4}, 2n\pi \pm {\pi \over 6}[/tex]
Ans : (A)
32. In a Δ ABC, [tex]2ac\sin {{A - B + C} \over 2}[/tex] is equal to
(A) [tex]{a^2} + {b^2} - {c^2}[/tex] (B) [tex]{c^2} + {a^2} - {b^2}[/tex] (C) [tex]{b^2} - {a^2} - {c^2}[/tex] (D) [tex]{c^2} - {a^2} - {b^2}[/tex]
Ans : (B)
33. Value of [tex]{\tan ^{ - 1}}\left( {{{\sin 2 - 1} \over {\cos 2}}} \right)[/tex] is
(A) [tex]{\pi \over 2} - 1[/tex] (B) [tex]1 - {\pi \over 4}[/tex] (C) [tex]2 - {\pi \over 2}[/tex] (D) [tex]{\pi \over 4} - 1[/tex]
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 [tex]{x^2} + {y^2} - 6x - 8y = 0[/tex] 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 [tex]{t_1}[/tex] and [tex]{t_2}[/tex] be the parameters of the end points of a focal chord for the parabola y² = 4ax, then which one is true ?
(A) [tex]{t_1}{t_2} = 1[/tex] (B) [tex]{{{t_1}} \over {{t_2}}} = 1[/tex] (C) [tex]{t_1}{t_2} = - 1[/tex] (D) [tex]{t_1} + {t_2} = - 1[/tex]
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) [tex]{1 \over 4}[/tex] (B) [tex]{1 \over 3}[/tex] (C) [tex]{1 \over 2}[/tex] (D) [tex]{2 \over 3}[/tex]
Ans : (C)
39. For different values of α , the locus of the point of intersection of the two straight lines [tex]\sqrt 3 x - y -4\sqrt 3 \alpha = 0[/tex] and [tex]\sqrt 3 \alpha x + \alpha y - 4\sqrt 3 = 0[/tex] is
(A) a hyperbola with eccentricity 2
(B) an ellipse with eccentricity [tex]\sqrt {{2 \over 3}} [/tex]
(C) a hyperbola with eccentricity [tex]\sqrt {{{19} \over {16}}} [/tex]
(D) an ellipse with eccentricity [tex]{3 \over 4}[/tex]
Ans : (A)
40. The area of the region bounded by y² = x and y = |x| is
(A) [tex]{1 \over 3}[/tex] sq.unit (B) [tex]{1 \over 6}[/tex] sq.unit (C) [tex]{2 \over 3}[/tex] 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) [tex]f = {v^3}{{{d^2}t} \over {d{x^2}}}[/tex] (B) [tex]f = - {v^3}{{{d^2}t} \over {d{x^2}}}[/tex] (C) [tex]f = {v^2}{{{d^2}t} \over {d{x^2}}}[/tex] (D) [tex]f = - {v^2}{{{d^2}t} \over {d{x^2}}}[/tex]
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 [tex]f(x) = {x^4} - 4{x^3} + 4{x^2} + 40[/tex] 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 [tex]x = {t^4} - k{t^3}[/tex] . 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) = ex sin x has maximum slope, is
(A) [tex]{\pi \over 4}[/tex] (B) [tex]{\pi \over 2}[/tex] (C) [tex]\pi [/tex] (D) [tex]{{3\pi } \over 2}[/tex]
Ans : (B)
46. The minimum value of [tex]f(x) = {e^{\left( {{x^4} - {x^3} + {x^2}} \right)}}[/tex] is
(A) e (B) – e (C) 1 (D) –1
Ans : (C)
47. [tex]\int {{{\log \sqrt x } \over {3x}}} dx[/tex] is equal to
(A) [tex]{1 \over 3}{\left( {\log \sqrt x } \right)^2} + C[/tex] (B) [tex]{2 \over 3}{\left( {\log \sqrt x } \right)^2} + C[/tex] (C) [tex]{2 \over 3}{\left( {\log x} \right)^2} + C[/tex] (D) [tex]{1 \over 3}{\left( {\log x} \right)^2} + C[/tex]
Ans : (A)
48. [tex]\int {{e^x}} \left( {{2 \over x} - {2 \over {{x^2}}}} \right)dx[/tex] is equal to
(A) [tex]{{{e^x}} \over x} + C[/tex] (B) [tex]{{{e^x}} \over {2{x^2}}} + C[/tex] (C) [tex]{{2{e^x}} \over x} + C[/tex] (D) [tex]{{2{e^x}} \over {{x^2}}} + C[/tex]
Ans : (C)
49. The value of the integral [tex]\int {{{dx} \over {{{\left( {{e^x} + {e^{ - x}}} \right)}^2}}}} [/tex] is
(A) [tex]{1 \over 2}\left( {{e^{2x}} + 1} \right) + C[/tex] (B) [tex]{1 \over 2}\left( {{e^{ - 2x}} + 1} \right) + C[/tex] (C) [tex] - {1 \over 2}{\left( {{e^{2x}} + 1} \right)^{ - 1}} + C[/tex] (D) [tex]{1 \over 4}\left( {{e^{2x}} - 1} \right) + C[/tex]
Ans : (C)
50. The value of [tex] {Lt}\limits_{x \to 0} {{{{\sin }^2}x + \cos x - 1} \over {{x^2}}}[/tex] is
(A) 1 (B) [tex]{1 \over 2}[/tex] (C) [tex] - {1 \over 2}[/tex] (D) 0
Ans : (B)
51. The value of [tex] {Lt}\limits_{x \to 0} {\left( {{{1 + 5{x^2}} \over {1 + 3{x^2}}}} \right)^{{1 \over {{x^2}}}}}[/tex] is
(A) [tex]{e^2}[/tex] (B) e (C) [tex]{1 \over e}[/tex] (D) [tex]{1 \over {e^2}}[/tex]
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) [tex]f(x) = 1 + {\left( {x - 2} \right)^{{2 \over 3}}}[/tex] 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 [tex] {Lt}\limits_{x \to 5} {{xf(5) - 5f(x)} \over {x - 5}}[/tex] is given by
(A) 35 (B) – 35 (C) 28 (D) – 28
Ans : (D)
54. If [tex]y = (1 + x)(1 + {x^2})(1 + {x^4}) \cdots (1 + {x^{2n}})[/tex] then the value of [tex]{\left( {{{dy} \over{dx}}} \right)_{x = 0}}[/tex] is
(A) 0 (B) –1 (C) 1 (D) 2
Ans : (C)
55. The value of ƒ(0) so that the function [tex]f(x) = {{1 - \cos (1 - \cos x)} \over {{x^4}}}[/tex] is continuous everywhere is
(A) [tex]{1 \over 2}[/tex] (B) [tex]{1 \over 4}[/tex] (C) [tex]{1 \over 6}[/tex] (D) [tex]{1 \over 8}[/tex]
Ans : (D)
56. [tex]\int {\sqrt {1 + \cos x} dx} [/tex] is equal to
(A) [tex]2\sqrt 2 \cos {x \over 2} + C[/tex] (B) [tex]2\sqrt 2 \sin {x \over 2} + C[/tex] (C) [tex]\sqrt 2 \cos {x \over 2} + C[/tex] (D) [tex]\sqrt 2 \sin {x \over 2} + C[/tex]
Ans : (B)
57. The function [tex]f(x) = \sec \left[ {\log \left( {x + \sqrt {1 + {x^2}} } \right)} \right][/tex] is
(A) odd (B) even (C) neither odd nor even (D) constant
Ans : (B)
58. [tex] {\lim }\limits_{x \to 0} {{\sin \left| x \right|} \over x}[/tex] 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 [tex]y = {x^2} - 3x + 2[/tex] 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 [tex]f(x) = \sqrt {{{\cos }^{ - 1}}\left( {{{1 - \left| x \right|} \over 2}} \right)} [/tex] 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) [tex]-{3 \over 4}[/tex] (D) [tex]{3 \over 4}[/tex]
Ans : (A)
63. The general solution of the different equation [tex]100{{{d^2}y} \over {d{x^2}}} - 20{{dy} \over {dx}} + y =0[/tex] is
(A) [tex]y = ({c_1} + {c_2}x){e^x}[/tex] (B) [tex]y = ({c_1} + {c_2}x){e^{ - x}}[/tex] (C) [tex]y = ({c_1} + {c_2}x){e^{{x \over {10}}}}[/tex] (D) [tex]y = {c_1}{e^x} + {c_2}{e^{ - x}}[/tex]
Ans : (C)
64. If y′′ – 3y′ + 2y = 0 where y(0) = 1, y′(0) = 0, then the value of y at x = loge2 is
(A) 1 (B) –1 (C) 2 (D) 0
Ans : (D)
65. The degree of the differential equation [tex]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 [/tex]
(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(ex + x −1) (B) y = 2(ex − x −1) (C) y = 2(ex − x +1) (D)y = 2(ex + 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 [tex]\int\limits_0^{\pi /2} {{{\sin }^5}xdx} [/tex] is
(A) [tex]{4 \over {15}}[/tex] (B) [tex]{8 \over 5}[/tex] (C) [tex]{8 \over {15}}[/tex] (D) [tex]{4 \over 5}[/tex]
Ans : (C)
69. If [tex]{d \over {dx}}\left\{ {f(x)} \right\} = g(x)[/tex], then [tex]\int\limits_a^b {f(x)g(x)dx} [/tex] is equal to
(A) [tex]{1 \over 2}\left[ {{f^2}(b) - {f^2}(a)} \right][/tex] (B) [tex]{1 \over 2}\left[ {{g^2}(b) - {g^2}(a)} \right][/tex] (C) [tex]f(b) - f(a)[/tex] (D) [tex]{1 \over 2}\left[ {g({b^2}) - g({a^2})} \right][/tex]
Ans : (A)
70. If [tex]{I_1} = \int\limits_0^{3\pi } {f({{\cos }^2}x)dx} [/tex] and [tex]{I_2} = \int\limits_0^\pi {f({{\cos }^2}x)dx} [/tex] then
(A) [tex]{I_1} = {I_2}[/tex] (B) [tex]3{I_1} = {I_2}[/tex] (C) [tex]{I_1} = 3{I_2}[/tex] (D) [tex]{I_1} = 5{I_2}[/tex]
Ans : (C)
71. The value of [tex]I = \int\limits_{ - \pi /2}^{\pi /2} {\left| {\sin x} \right|dx} [/tex] is
(A) 0 (B) 2 (C) – 2 (D) – 2 < I < 2
Ans : (B)
72. If [tex]I = \int\limits_0^I {{{dx} \over {1 + {x^{\pi /2}}}}} [/tex], then
(A) [tex]{\log _e}2 < 1 < \pi /4[/tex] (B) [tex]{\log _e}2 > 1[/tex] (C) [tex]I = \pi /4[/tex] (D) [tex]I = {\log_e}2 [/tex]
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) [tex]13{1 \over 2}[/tex] sq. units (D) 14 sq. units
Ans : (D)
74. The area bounded by the parabolas y = 4x², [tex]y = {{{x^2}} \over 9}[/tex] and the line y = 2 is
(A) [tex]{{5\sqrt 2 } \over 3}[/tex] sq. units (B) [tex]{{10\sqrt 2 } \over 3}[/tex] sq. units (C) [tex]{{15\sqrt 2 }\over 3}[/tex] sq. units (D) [tex]{{20\sqrt 2 } \over 3}[/tex] 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) [tex]{1 \over p} + {1 \over q} = 1[/tex] (B) [tex]{1 \over p} + {1 \over q} = 2[/tex] (C) [tex]{1 \over p} + {1 \over q} = 3[/tex] (D) [tex]{1 \over p} + {3 \over q} = 1[/tex]
Ans : (C)
77. The equations [tex]y = \pm \sqrt {3x} [/tex], 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, [tex]\angle R = \pi /2[/tex] . If tan [tex]\left( {{p \over 2}} \right)[/tex] and tan [tex]\left({{Q \over 2}} \right)[/tex] 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 [tex]{{\sin {{55}^ \circ } - \cos {{55}^ \circ }} \over {\sin {{10}^ \circ }}}[/tex] is
(A) [tex]{1 \over {\sqrt 2 }}[/tex] (B) 2 (C) 1 (D) [tex]\sqrt 2 [/tex]
Ans : (D)
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