1. A point P lies on the circle [tex]x^2+y^2=169[/tex]. If Q = (5, 12) and R = (-12, 5), then the angle [tex]\angle QPR[/tex] is
(A) [tex]\frac{\pi}{6}[/tex] (B) [tex]\frac{\pi}{4}[/tex] (C) [tex]\frac{\pi}{3}[/tex] (D) [tex]\frac{\pi}{2}[/tex]
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) [tex]\frac{5}{2}[/tex] (B) [tex]\frac{5}{\sqrt2}[/tex] (C) 5 (D) 10
Ans: (C)
3. The locus of the midpoints of the chords of an ellipse [tex]x^2+4y^2=4[/tex] that are drawn form the positive end of the minor axis, is
(A) a circle with centre [tex](\frac{1}{2},0)[/tex] and radius 1
(B) a parabola with focus [tex](\frac{1}{2},0)[/tex] and directrix x = -1
(C) an ellipse with centre [tex](0,\frac{1} {2})[/tex], major axis 1 and minor axis [tex]\frac{1}{2}[/tex]
(D) an hyperbola with centre [tex](0,\frac{1}{2})[/tex], transverse axis 1 and conjugate axis [tex]\frac{1}{2}[/tex]
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 [tex]y-\frac{3}{2}=-3(x+\frac{1}{2})[/tex]
(B) a circle with centre [tex](-\frac{1}{2},\frac{3}{2})[/tex] and radius [tex]\frac{1}{\sqrt2}[/tex]
(C) a parabola with focus (1,2) and directrix passing through (-2,1)
(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 [tex]x-2y=t[/tex] and [tex]x+2y=\frac{1} {t}[/tex] is
(A) the straight line x=y
(B) the circle with centre at the origin and radius 1
(C) the ellipse with centre at the origin and one focus [tex](\frac{2}{\sqrt5},0)[/tex]
(D) the hyperbola with centre at the origin and one focus [tex](\frac{\sqrt5}{2},0)[/tex]
Ans: (D)
6. Let [tex]P=\begin{pmatrix} \cos \frac {\pi}{4} & - \sin \frac {\pi}{4} \\ \sin \frac {\pi}{4} & \cos \frac {\pi}{4} \end{pmatrix} [/tex] and [tex]X=\begin{pmatrix} \frac {1}{\sqrt2} \\ \frac{1}{\sqrt2} \end{pmatrix}[/tex]. Then [tex]P^3X[/tex] is equal to
(A) [tex]\begin{pmatrix} 0 \\ 1 \end{pmatrix}[/tex] (B) [tex]\begin{pmatrix} -\frac{1}{\sqrt2} \\ \frac{1} {\sqrt2} \end{pmatrix}[/tex] (C) [tex]\begin{pmatrix} -1 \\ 0 \end{pmatrix}[/tex] (D) [tex]\begin{pmatrix} -\frac{1} {\sqrt2} \\ -\frac{1}{\sqrt2} \end{pmatrix}[/tex]
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 [tex]0\le P, Q\le \frac{\pi}{2}[/tex], if [tex]sinP+cosQ=2[/tex], then the value of [tex]tan(\frac{P+Q}{2})[/tex] is equal to
(A) 1 (B) [tex]\frac{1}{\sqrt2}[/tex] (C) [tex]\frac{1}{2}[/tex] (D) [tex]\frac{\sqrt3}{2}[/tex]
Ans: (A)
9. If [tex]\alpha[/tex] and [tex]\beta[/tex] are the roots of [tex]x^2-x+1=0[/tex], then the value of [tex]\alpha^{2013}+\beta^ {2013}[/tex] is equal to
(A) 2 (B) -2 (C) -1 (D) 1
Ans: (B)
10. The value of the integral [tex]\int_{-1}^{+1} \{\frac {x^{2013}}{e^{|x|} (x^2 + \cos x)} + \frac {1}{e^{| x|}} \}dx[/tex] is equal to
(A) 0 (B) [tex]1-e^{-1}[/tex] (C) [tex]2e^{-1}[/tex] (D) [tex]2(1-e^{-1})[/tex]
Ans: (D)
11. Let
[tex]f(x)=2^{100}x+1[/tex],
[tex]g(x)=3^{100}x+1[/tex].
Then the set of real numbers x such that [tex]f(g(x))=x[/tex] is
(A) empty (B) a singleton (C) a finite set with more than one element (D) infinite
Ans: (B)
12. The limit of [tex]x \sin(e^{1/x})[/tex] as [tex]x\to0[/tex]
(A) is equal to 0 (B) is equal to 1 (C) is equal to e/2 (D) does not exist
Ans: (A)
13. Let [tex]I=\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}[/tex] and [tex]P=\begin{pmatrix} 1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & -2 \end{pmatrix}[/tex]. Then the matrix [tex]P^3+2P^2[/tex] is equal to
(A) P (B) I - P (C) 2I + P (D) 2I - P
Ans: (C)
14. If [tex]\alpha, \beta[/tex] are the roots of the quadratic equation [tex]x^2+ax+b=0, (b\ne0)[/tex]; then the quadratic equation whose roots are
[tex]\alpha-\frac{1}{\beta}[/tex], [tex]\beta-\frac{1}{\alpha}[/tex] is
(A) [tex]ax^2+a(b-1)x+(a-1)^2=0[/tex]
(B) [tex]bx^2+a(b-1)x+(b-1)^2=0[/tex]
(C) [tex]x^2+ax+b=0[/tex]
(D) [tex]abx^2+bx +a=0[/tex]
Ans: (B)
15. The value of [tex]1000[\frac{1}{1\times2}+\frac{1}{2\times3}+\frac{1}{3\times4}+...+\frac{1} {999\times1000}][/tex] is equal to
(A) 1000 (B) 999 (C) 1001 (D) 1/999
Ans: (B)
16. The value of the determinant
[tex]\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}[/tex]
is equal to
(A) 0 (B) [tex](1+a^2+b^2)[/tex] (C) [tex](1+a^2+b^2) ^2[/tex] (D) [tex](1+a^2+b^2)^3[/tex]
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) [tex]\frac{1}{4}(\sqrt5-1)[/tex] (B) [tex]\frac{1}{2}(\sqrt5+1)[/tex] (C) [tex]\frac{1}{2} (\sqrt5-1)[/tex] (D) [tex]\frac{1}{4}(\sqrt5+1)[/tex]
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 [tex]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[/tex] is equal to
(A) [tex]\frac{1}{n}[/tex] (B) [tex]\frac{n+2}{2n+1}[/tex] (C) [tex]\frac{2n-1}{n}[/tex] (D) [tex]\frac{2n-3} {3n-2}[/tex]
Ans: (A)
20. Let ƒ(θ) = (1 + sin²θ)(2 - sin²θ). Then for all values of θ
(A) ƒ(θ) > [tex]{9 \over 4}[/tex] (B) ƒ(θ) < 2 (C) ƒ(θ) > [tex]{11 \over 4}[/tex] (D) 2 ≤ ƒ(θ) ≤ [tex]{9 \over 4}[/tex]
Ans: (D)
21. Let [tex]f(x) = \left\{ {\matrix{ {{x^3} - 3x + 2} & {x < 2} \cr {{x^3} - 6{x^2} + 9x + 2} & {x \ge 2} \cr } } \right.[/tex] Then
(A) [tex]\lim \limits_{x \to 2} f(x)[/tex] does not exist
(B) ƒ is not continuous at x = 2
(C) ƒ is continuous but not differentiable at x = 2
(D) ƒ is continuous and differentiable at x = 2
Ans: (C)
22. The limit of [tex]\sum \limits_{n=1}^{1000}(-1)^nx^n[/tex] as [tex]x\to\infty[/tex]
(A) does not exist
(B) exist and equals to 0
(C) exists and approaches [tex]+\infty[/tex]
(D) exists and approaches [tex]-\infty[/tex]
Ans: (C)
23. If [tex]f(x)=e^x(x-2)^2[/tex] then
(A) ƒ is increasing in [tex](-\infty,0)[/tex] and [tex](2,\infty)[/tex] and decreasing in (0,2)
(B) ƒ is increasing in [tex](-\infty,0)[/tex] and decreasing in [tex](0,\infty)[/tex]
(C) ƒ is increasing in [tex](2, \infty)[/tex] and decreasing in [tex](-\infty,0)[/tex]
(D) ƒ is increasing in [tex](0,2)[/tex] and decreasing in [tex](-\infty,0)[/tex] and [tex](2,\infty)[/tex]
Ans: (A)
24. Let [tex]f:\mathbb{R}\to\mathbb{R}[/tex] be such that [tex]f[/tex] is injective and [tex]f(x)f(y)=f(x+y)[/tex] for all [tex]x,y \in\mathbb{R}[/tex]. If [tex]f(x), f(y), f(z)[/tex] are in G.P., then [tex]x, y, z[/tex] are in
(A) A.P. always
(B) G.P. always
(C) A.P. depending on the values of x, y, z
(D) G.P. depending on the values of x, y, z
Ans: (A)
25. The number of solutions of the equation
[tex]\frac{1}{2}log_{\sqrt3}(\frac{x+1}{x+5})+log_9(x+5) ^2=1[/tex] is
(A) 0 (B) 1 (C) 2 (D) infinite
Ans: (B)
26. The area of the region bounded by the parabola [tex]y=x^2-4x+5[/tex] and the straight line [tex]y=x+1[/tex] is
(A) 1/2 (B) 2 (C) 3 (D) 9/2
Ans: (D)
27. The value of the integral
[tex]\int_1^2e^x(log_ex+\frac{x+1}{x})dx[/tex] is
(A) [tex]e^2(1+log_e2)[/tex] (B) [tex]e^2-e[/tex] (C) [tex]e^2(1+log_e2)-e[/tex] (D) [tex]e^2-e(1+log_e2)[/tex]
Ans: (C)
28. Let [tex]P=1+\frac{1}{2\times2}+\frac{1}{3\times2^2}+......[/tex]
and [tex]Q=\frac{1}{1\times2}+\frac{1}{3\times4}+\frac{1}{5\times6}+......[/tex]
Then
(A) P = Q (B) 2P = Q (C) P = 2Q (D) P = 4Q
Ans: (C)
29. Let [tex]f(x)= \sin x+2 \cos^2 x[/tex], [tex]\frac {\pi}{4} \le x \le \frac {3 \pi}{4}[/tex]. Then ƒ attains its
(A) minimum at [tex]x=\frac{\pi}{4}[/tex]
(B) maximum at [tex]x=\frac{\pi}{2}[/tex]
(C) minimum at [tex]x=\frac{\pi}{2}[/tex]
(D) maximum at [tex]x=sin^{-1}(\frac{1}{4})[/tex]
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) [tex]{1 \over 2}[/tex] (B) [tex]{1 \over 4}[/tex] (C) [tex]{1 \over 8}[/tex] (D) [tex]{1 \over {16}}[/tex]
Ans: (B)
31. There are two coins, one unbiased with probability [tex]{1 \over 2}[/tex] of getting heads and the other one is biased with probability [tex]{3 \over 4}[/tex] 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) [tex]{2 \over 3}[/tex] (B) [tex]{3 \over 5}[/tex] (C) [tex]{1 \over 2}[/tex] (D) [tex]{2 \over 5}[/tex]
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 [tex]\frac{x^2}{4}+\frac{y^2}{9}=1[/tex]
(B) the ellipse [tex]\frac{x^2} {9}+\frac{y^2}{4}=1[/tex]
(C) the hyperbola [tex]\frac{x^2}{4}-\frac{y^2}{9}=1[/tex]
(D) the hyperbola [tex]\frac{x^2} {9}-\frac{y^2}{4}=1[/tex]
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) [tex]\frac{300}{2197}[/tex] (B) [tex]\frac{36}{85}[/tex] (C) [tex]\frac{12}{85}[/tex] (D) [tex]\frac{4}{51}[/tex]
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) [tex]\frac{36}{5}[/tex] (B) [tex]\frac{18}{5}[/tex] (C) [tex]\frac{9}{5}[/tex] (D) [tex]\frac{1}{5}[/tex]
Ans: (B)
35. The number of onto functions from the set {1, 2,.....,11} to set {1, 2,.....,10} is
(A) [tex]5 \times | \underline {11} [/tex] (B) [tex]|\underline {10} [/tex] (C) [tex]{{|\underline {11}} \over 2}[/tex] (D) [tex]10 \times |\underline {11} [/tex]
Ans: (D)
36. The limit of [tex][\frac {1}{x^2} + \frac {(2013)^x}{e^x-1} + \frac {1}{e^x-1}][/tex] as [tex]x \to 0[/tex]
(A) approaches + ∞ (B) approaches - ∞ (C) is equal to [tex]log_e (2013) [/tex] (D) does not exist
Ans: (A)
37. Let [tex]z_1=2+3i[/tex] and [tex]z_2=3+4i[/tex] be two points on the complex plane. Then the set of complex numbers [tex]z[/tex] satisfying [tex]|z-z_1|^2+|z-z_2|^2=|z_1-z_2|^2[/tex] 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) [tex]-\frac{1}{2}[/tex] (D) [tex]\frac{1}{2}[/tex]
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
(C) The girl receives at least 5 apples or the boy receives at least 8 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 [tex]\{\frac{1}{x}\sqrt{1+x}-\sqrt{1+\frac{1}{x^2}}\}[/tex] as [tex]x\to0[/tex]
(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 [tex] \cos^6 \theta + \sin^6 \theta [/tex] 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 [tex]3x^2+3y^2-9x+6y+5=0[/tex] 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 [tex]z=x+iy[/tex] where x and y are real numbers and [tex]i=\sqrt{-1}[/tex]. The points (x, y) for which [tex]\frac{z-1}{z-i}[/tex] 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
(C) pair of perpendicular straight lines
(D) hyperbola
Ans: (B)
48. The line y = x intersects the hyperbola [tex]\frac{x^2}{9}-\frac{y^2}{25}=1[/tex] at the points P and Q. The eccentricity of ellipse with PQ as major axis and minor axis of length [tex]\frac{5}{\sqrt2}[/tex] is
(A) [tex]\frac {\sqrt 5}{3}[/tex] (B) [tex]\frac {5}{\sqrt 3}[/tex] (C) [tex]\frac {5}{9}[/tex] (D)[tex]\frac{25}{9}[/tex]
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
(C) Symmetric, Transitive but not Reflexive
(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
(C) Transitive but not Reflexive
(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 [tex](1+x)^n[/tex] is 11 : 10. Then the number of terms in the expansion of [tex](1+x)^n[/tex] is
(A) 20 (B) 21 (C) 10 (D) 11
Ans: (B)
56. Let exp (x) denote the exponential function e<sup>x</sup>. If [tex]f(x) = \exp \left( {{x^{{1 \over x}}}} \right), x>0[/tex], then the minimum value of [tex]f[/tex] in the interval [2, 5] is
(A) [tex]\exp \left( {{e^{{1 \over e}}}} \right)[/tex] (B) [tex]\exp \left( {{2^{{1 \over 2}}}} \right)[/tex] (C) [tex]\exp \left( {{5^{{1 \over 5}}}} \right)[/tex] (D) [tex]\exp \left( {{3^{{1 \over 3}}}} \right)[/tex]
Ans: (C)
57. The sum of the series [tex]\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}[/tex]
(A) [tex]\frac {2^{27} - 1}{26 \times 27}[/tex] (B) [tex]\frac {2^{27}-28}{26 \times 27}[/tex] (C) [tex]\frac {1}{2}(\frac {2^{26}+1}{26 \times 27})[/tex] (D) [tex]\frac {2^{26} -1}{52}[/tex]
Ans: (B)
58. Five numbers are in A.P. With common difference ≠ 0 . If the 1<sup>st</sup>, 3<sup>rd</sup> and 4<sup>th</sup> terms are in G.P., then
(A) the 5<sup>th</sup> term is always 0
(B) the 1<sup>st</sup> term is always 0
(C) the middle term is always 0
(D) the middle term is always -2
Ans: (A)
59. The minimum value of the function [tex]f(x)=2|x-1|+|x-2|[/tex] is
(A) 0 (B) 1 (C) 2 (D) 3
Ans: (B)
60. If P, Q, R are angles of an isosceles triangle and [tex] \angle P = {\pi \over 2} [/tex], then the value of
[tex](\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) [/tex]
is equal to
(A) i (B) -i (C) 1 (D) -1
Ans: (B)
61. A line passing through the point of intersection of [tex]x+y=4[/tex] and [tex]x-y=2[/tex] makes an angle [tex]tan^ {-1}(3/4)[/tex] with the x-axis. It intersects the parabola [tex]y^2=4(x-3)[/tex] at points [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex] respectively. Then [tex]|x_1-x_2|[/tex] is equal to
(A) [tex]{16 \over 9}[/tex] (B) [tex]{32 \over 9}[/tex] (C) [tex]{40 \over 9}[/tex] (D) [tex]{80 \over 9}[/tex]
Ans: (B)
62. Let [a] denote the greatest integer which is less than or equal to a. Then the value of the integral
[tex]\int_{-{\pi \over 2}}^{\pi \over 2}[ \sin x \cos x]dx [/tex] is
(A) [tex]{\pi \over 2} [/tex] (B) [tex]\pi[/tex] (C) [tex]- \pi[/tex] (D) [tex]-{\pi \over 2}[/tex]
Ans: (D)
63. If [tex]P=\begin{pmatrix} 2 & -2 & -4\\ -1 & 3 & 4\\ 1 & -2 & -3 \end{pmatrix}[/tex] then [tex]P^5[/tex] equals
(A) P (B) 2P (C) -P (D) -2P
Ans: (A)
64. If [tex] \sin^2 \theta + 3 \cos \theta = 2 [/tex], then [tex] \cos^3 \theta + \sec^3 \theta [/tex] is
(A) 1 (B) 4 (C) 9 (D) 18
Ans: (D)
65. [tex]x = 1 + \frac {1}{2 \times |\underline 1} + \frac {1}{4 \times |\underline2} + \frac {1}{8 \times | \underline3} +......[/tex] and [tex]y = 1 + \frac {x^2}{|\underline1} + \frac {x^4}{|\underline 2} + \frac {x^6}{ | \underline 3} +......[/tex]
Then the value of [tex]log_ey[/tex] is
(A) e (B) e² (C) 1 (D) 1/e
Ans: (A)
66. The value of the infinite series
[tex]\frac {1^2 + 2^2}{|\underline 3} + \frac {1^2 + 2^2 + 3^2}{| \underline 4} + \frac {1^2 + 2^2 + 3^2 + 4^2}{|\underline 5}+.......[/tex] is
(A) e (B) 5e (C) [tex]\frac {5e}{6} - \frac {1}{2}[/tex] (D) [tex]\frac{5e}{6}[/tex]
Ans: (C)
67. The value of the integral [tex]\int_{\pi \over 6}^{\pi \over 3} \frac {(\sin x - x \cos x)}{x(x + \sin x)}dx [/tex] is equal to
(A) [tex]log_e(\frac{2(\pi+3)}{2\pi+3\sqrt3})[/tex] (B) [tex]log_e(\frac{\pi+3}{2(2\pi +3\sqrt3)})[/tex] (C) [tex]log_e(\frac{2\pi+3\sqrt3}{2(\pi+3)})[/tex] (D) [tex]log_e(\frac{2(2\pi+3\sqrt3)}{\pi +3})[/tex]
Ans: (A)
68. Let [tex]f(x) = x( \frac {1}{x-1} + \frac {1}{x} + \frac {1}{x+1}), x > 1 [/tex]. Then
(A) [tex]f(x) \le 1[/tex] (B) [tex]1 < f(x) \le 2 [/tex] (C) [tex]2 < f(x) \le 3 [/tex] (D) [tex]f(x) > 3[/tex]
Ans: (D)
69. Let [tex]F(x) = \int_0^x \frac {\cos t}{(1 + t^2)}dt [/tex], [tex]0 \le x \le 2 \pi [/tex]. Then
(A) F is increasing in [tex](\frac {\pi}{2},\frac {3\pi}{2})[/tex] and decreasing in [tex](0,\frac{\pi}{2})[/tex] and [tex](\frac{3\pi}{2},2\pi)[/tex]
(B) F is increasing in [tex](0,\pi)[/tex] and decreasing in [tex](\pi,2 \pi)[/tex]
(C) F is increasing in [tex](\pi,2\pi)[/tex] and decreasing in [tex](0,\pi)[/tex]
(D) F is increasing in [tex](0,\frac {\pi}{2})[/tex] and [tex](\frac {3 \pi}{2}, 2\pi)[/tex] and decreasing in [tex](\frac {\pi}{2},\frac {3 \pi}{2})[/tex]
Ans: (D)
70. Let [tex]f(x)=x^{2/3}, x \ge 0[/tex]. Then the area of the region enclosed by the curve [tex]y=f(x)[/tex] and three lines [tex]y=x[/tex], [tex]x=1[/tex] and [tex]x=8[/tex] is
(A) [tex]\frac{63}{2}[/tex] (B) [tex]\frac{93}{5}[/tex] (C) [tex]\frac{105}{7}[/tex] (D) [tex]\frac{129}{10}[/tex]
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 [tex]\frac {\tan \angle {PQF}}{\tan \angle {PFQ}} [/tex] 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) [tex]\frac{5}{32}[/tex] (B) [tex]\frac{3}{128}[/tex] (C) [tex]\frac{3}{256}[/tex] (D) [tex]\frac{3}{64}[/tex]
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) [tex]xy=c[/tex], c is a constant
(B) [tex]xy^2=c[/tex], c is a constant
(C) [tex]x^2y = c[/tex], c is a constant
(D) [tex]x^2y^2=c[/tex], c is a constant
Ans: (C)
74. The solution of the differential equation [tex](y^2+2x)\frac{dy}{dx}=y[/tex] satisfy x = 1, y = 1. Then the solution is
(A) [tex]x=y^2(1+log_ey)[/tex] (B) [tex]y=x^2(1+log_ex)[/tex] (C) [tex]x=y^2(1-log_ey)[/tex] (D) [tex]y=x^2(1- log_ex)[/tex]
Ans: (A)
75. The solution of the differential equation [tex]y \sin (x/y)dx = (x \sin (x/y) - y) dy [/tex] satisfying [tex]y(\pi /4)=1 [/tex] is
(A) [tex] \cos \frac{x}{y} =-log_ey + \frac {1}{\sqrt2}[/tex]
(B) [tex] \sin \frac {x}{y} = log_ey + \frac {1} {\sqrt2}[/tex]
(C) [tex] \sin \frac {x}{y} = log_ex- \frac {1}{\sqrt2}[/tex]
(D) [tex]\cos \frac {x}{y} =-log_ex - \frac {1} {\sqrt2}[/tex]
Ans: ()
76. The area of the region encloses between parabola y² = x and the line y = mx is [tex]{1 \over 48}[/tex]. Then the value of m is
(A) -2 (B) -1 (C) 1 (D) 2
Ans: (A)
77. Consider the system of equations:
[tex]x + y + z = 0 [/tex]
[tex]\alpha x + \beta y + \gamma z = 0[/tex]
[tex]\alpha^2 x + \beta^2 y + \gamma^2 z = 0[/tex]
Then the system of equations has
(A) A unique solution for all values of [tex]\alpha ,\beta ,\gamma[/tex]
(B) Infinite numbers of solutions if any two of [tex]\alpha ,\beta ,\gamma[/tex] are equal
(C) A unique solution if [tex]\alpha ,\beta , \gamma[/tex] are distinct
(D) More than one, but finite number of solutions depending on values of [tex]\alpha ,\beta ,\gamma[/tex]
Ans: (B)
78. The equations of the circles which touch both the axis and the line [tex]4x + 3y = 12 [/tex] and have centres in the first quadrant, are
(A) x² + y² - x - y + 1 = 0
(B) x² + y² - 2x - 2y + 1 = 0
(C) x² + y² - 12x - 12y + 36 = 0
(D) x² + y² - 6x - 6y + 36 = 0
Ans: (B)
79. Which of the following real valued functions is/are not even functions ?
(A) ƒ(x) = x³ sin x
(B) ƒ(x) = x² cos x
(C) ƒ(x) = exx³ sin x
(D) ƒ(x) = x-[x], where [x] denotes the greatest integer less than or equal to x
Ans: (C)
80. Let [tex]\sin \alpha [/tex], [tex]\cos \alpha [/tex] be the roots of the equation [tex]x^2 - bx + c = 0 [/tex]. Then which of the following statements is/are correct ?
(A) [tex]c \le \frac{1}{2}[/tex] (B) [tex]b \le \sqrt{2}[/tex] (C) [tex]c > \frac{1}{2}[/tex] (D) [tex]b > \sqrt{2}[/tex]
Ans: (A)
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