# WBJEE Mathematics Question Paper 2013 (Eng)

1.  A point P lies on the circle $$x^2+y^2=169$$. If Q = (5, 12) and R = (-12, 5), then the angle $$\angle QPR$$ is

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

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) $$\frac{5}{2}$$         (B) $$\frac{5}{\sqrt2}$$       (C) 5        (D) 10

Ans: (C)

3. The locus of the midpoints of the chords of an ellipse $$x^2+4y^2=4$$ that are drawn form the positive end of the minor axis, is

(A) a circle with centre $$(\frac{1}{2},0)$$ and radius 1

(B) a parabola with focus $$(\frac{1}{2},0)$$ and directrix x = -1

(C) an ellipse with centre $$(0,\frac{1} {2})$$, major axis 1 and minor axis $$\frac{1}{2}$$

(D) an hyperbola with centre $$(0,\frac{1}{2})$$, transverse axis 1 and conjugate axis $$\frac{1}{2}$$

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 $$y-\frac{3}{2}=-3(x+\frac{1}{2})$$

(B) a circle with centre $$(-\frac{1}{2},\frac{3}{2})$$ and radius $$\frac{1}{\sqrt2}$$

(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 $$x-2y=t$$ and $$x+2y=\frac{1} {t}$$ 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 $$(\frac{2}{\sqrt5},0)$$

(D) the hyperbola with centre at the origin and one focus $$(\frac{\sqrt5}{2},0)$$

Ans: (D)

6.  Let $$P=\begin{pmatrix} \cos \frac {\pi}{4} & - \sin \frac {\pi}{4} \\ \sin \frac {\pi}{4} & \cos \frac {\pi}{4} \end{pmatrix}$$ and $$X=\begin{pmatrix} \frac {1}{\sqrt2} \\ \frac{1}{\sqrt2} \end{pmatrix}$$. Then $$P^3X$$ is equal to

(A) $$\begin{pmatrix} 0 \\ 1 \end{pmatrix}$$        (B) $$\begin{pmatrix} -\frac{1}{\sqrt2} \\ \frac{1} {\sqrt2} \end{pmatrix}$$        (C) $$\begin{pmatrix} -1 \\ 0 \end{pmatrix}$$        (D) $$\begin{pmatrix} -\frac{1} {\sqrt2} \\ -\frac{1}{\sqrt2} \end{pmatrix}$$

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 $$0\le P, Q\le \frac{\pi}{2}$$, if $$sinP+cosQ=2$$, then the value of $$tan(\frac{P+Q}{2})$$ is equal to

(A) 1       (B) $$\frac{1}{\sqrt2}$$         (C) $$\frac{1}{2}$$        (D) $$\frac{\sqrt3}{2}$$

Ans: (A)

9.  If $$\alpha$$ and $$\beta$$ are the roots of $$x^2-x+1=0$$, then the value of $$\alpha^{2013}+\beta^ {2013}$$ is equal to

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

Ans: (B)

10.  The value of the integral $$\int_{-1}^{+1} \{\frac {x^{2013}}{e^{|x|} (x^2 + \cos x)} + \frac {1}{e^{| x|}} \}dx$$ is equal to

(A) 0       (B) $$1-e^{-1}$$        (C) $$2e^{-1}$$        (D) $$2(1-e^{-1})$$

Ans: (D)

11.   Let

$$f(x)=2^{100}x+1$$,

$$g(x)=3^{100}x+1$$.

Then the set of real numbers x such that $$f(g(x))=x$$ is

(A) empty         (B)  a singleton        (C) a finite set with more than one element         (D) infinite

Ans: (B)

12.  The limit of $$x \sin(e^{1/x})$$ as $$x\to0$$

(A) is equal to 0      (B) is equal to 1       (C) is equal to e/2        (D) does not exist

Ans: (A)

13. Let $$I=\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}$$ and $$P=\begin{pmatrix} 1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & -2 \end{pmatrix}$$. Then the matrix $$P^3+2P^2$$ is equal to

(A) P       (B) I - P       (C) 2I + P        (D) 2I - P

Ans: (C)

14.  If $$\alpha, \beta$$ are the roots of the quadratic equation $$x^2+ax+b=0, (b\ne0)$$; then the quadratic equation whose roots are

$$\alpha-\frac{1}{\beta}$$, $$\beta-\frac{1}{\alpha}$$ is

(A) $$ax^2+a(b-1)x+(a-1)^2=0$$

(B) $$bx^2+a(b-1)x+(b-1)^2=0$$

(C) $$x^2+ax+b=0$$

(D) $$abx^2+bx +a=0$$

Ans: (B)

15.  The value of $$1000[\frac{1}{1\times2}+\frac{1}{2\times3}+\frac{1}{3\times4}+...+\frac{1} {999\times1000}]$$ is equal to

(A) 1000        (B) 999        (C) 1001         (D) 1/999

Ans: (B)

16.  The value of the determinant

$$\begin{vmatrix} 1+a^2-b^2 & 2ab & -2b\\ 2ab & 1-a^2+b^2 & 2a \\ 2b & -2a & 1-a^2-b^2 \end{vmatrix}$$

is equal to

(A) 0        (B) $$(1+a^2+b^2)$$         (C) $$(1+a^2+b^2) ^2$$         (D) $$(1+a^2+b^2)^3$$

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) $$\frac{1}{4}(\sqrt5-1)$$         (B) $$\frac{1}{2}(\sqrt5+1)$$         (C) $$\frac{1}{2} (\sqrt5-1)$$        (D) $$\frac{1}{4}(\sqrt5+1)$$

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 $$I = \int_0^{\pi \over 4}( \tan^{n+1}x)dx + \frac {1}{2} \int_0^{\pi \over 2} \tan^{n-1} (x/2)dx$$ is equal to

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

Ans: (A)

20.  Let ƒ(θ) = (1 + sin²θ)(2 - sin²θ). Then for all values of θ

(A) ƒ(θ) > $${9 \over 4}$$        (B) ƒ(θ) < 2        (C) ƒ(θ) > $${11 \over 4}$$       (D) 2 ≤ ƒ(θ) ≤ $${9 \over 4}$$

Ans: (D)

21.   Let $$f(x) = \left\{ {\matrix{ {{x^3} - 3x + 2} & {x < 2} \cr {{x^3} - 6{x^2} + 9x + 2} & {x \ge 2} \cr } } \right.$$ Then

(A) $$\lim \limits_{x \to 2} f(x)$$ 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 $$\sum \limits_{n=1}^{1000}(-1)^nx^n$$ as $$x\to\infty$$

(A) does not exist

(B) exist and equals to 0

(C) exists and approaches $$+\infty$$

(D) exists and approaches $$-\infty$$

Ans: (C)

23.   If $$f(x)=e^x(x-2)^2$$ then

(A) ƒ is increasing in $$(-\infty,0)$$ and $$(2,\infty)$$ and decreasing in (0,2)

(B) ƒ is increasing in $$(-\infty,0)$$ and decreasing in $$(0,\infty)$$

(C) ƒ is increasing in $$(2, \infty)$$ and decreasing in $$(-\infty,0)$$

(D) ƒ is increasing in $$(0,2)$$ and decreasing in $$(-\infty,0)$$ and $$(2,\infty)$$

Ans: (A)

24.  Let $$f:\mathbb{R}\to\mathbb{R}$$ be such that $$f$$ is injective and $$f(x)f(y)=f(x+y)$$ for all $$x,y \in\mathbb{R}$$. If $$f(x), f(y), f(z)$$ are in G.P., then $$x, y, z$$ 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

$$\frac{1}{2}log_{\sqrt3}(\frac{x+1}{x+5})+log_9(x+5) ^2=1$$ is

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

Ans: (B)

26.  The area of the region bounded by the parabola $$y=x^2-4x+5$$ and the straight line $$y=x+1$$ is

(A) 1/2        (B) 2         (C) 3        (D) 9/2

Ans: (D)

27.  The value of the integral

$$\int_1^2e^x(log_ex+\frac{x+1}{x})dx$$ is

(A) $$e^2(1+log_e2)$$       (B) $$e^2-e$$        (C) $$e^2(1+log_e2)-e$$         (D) $$e^2-e(1+log_e2)$$

Ans: (C)

28.   Let $$P=1+\frac{1}{2\times2}+\frac{1}{3\times2^2}+......$$

and $$Q=\frac{1}{1\times2}+\frac{1}{3\times4}+\frac{1}{5\times6}+......$$

Then

(A) P = Q       (B)  2P = Q        (C) P = 2Q          (D) P = 4Q

Ans: (C)

29.  Let $$f(x)= \sin x+2 \cos^2 x$$, $$\frac {\pi}{4} \le x \le \frac {3 \pi}{4}$$. Then ƒ attains its

(A) minimum at $$x=\frac{\pi}{4}$$

(B) maximum at $$x=\frac{\pi}{2}$$

(C) minimum at $$x=\frac{\pi}{2}$$

(D) maximum at $$x=sin^{-1}(\frac{1}{4})$$

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) $${1 \over 2}$$        (B) $${1 \over 4}$$         (C) $${1 \over 8}$$         (D) $${1 \over {16}}$$

Ans: (B)

31.  There are two coins, one unbiased with probability $${1 \over 2}$$ of getting heads and the other one is biased with probability $${3 \over 4}$$ 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) $${2 \over 3}$$         (B) $${3 \over 5}$$         (C) $${1 \over 2}$$        (D) $${2 \over 5}$$

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 $$\frac{x^2}{4}+\frac{y^2}{9}=1$$

(B) the ellipse $$\frac{x^2} {9}+\frac{y^2}{4}=1$$

(C) the hyperbola $$\frac{x^2}{4}-\frac{y^2}{9}=1$$

(D) the hyperbola $$\frac{x^2} {9}-\frac{y^2}{4}=1$$

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) $$\frac{300}{2197}$$        (B) $$\frac{36}{85}$$        (C) $$\frac{12}{85}$$        (D) $$\frac{4}{51}$$

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) $$\frac{36}{5}$$         (B) $$\frac{18}{5}$$          (C) $$\frac{9}{5}$$         (D) $$\frac{1}{5}$$

Ans: (B)

35.  The number of onto functions from the set {1, 2,.....,11} to set {1, 2,.....,10} is

(A) $$5 \times | \underline {11}$$         (B)  $$|\underline {10}$$         (C) $${{|\underline {11}} \over 2}$$       (D) $$10 \times |\underline {11}$$

Ans: (D)

36.  The limit of $$[\frac {1}{x^2} + \frac {(2013)^x}{e^x-1} + \frac {1}{e^x-1}]$$ as $$x \to 0$$

(A) approaches + ∞       (B) approaches - ∞        (C) is equal to $$log_e (2013)$$        (D) does not exist

Ans: (A)

37.  Let $$z_1=2+3i$$ and $$z_2=3+4i$$ be two points on the complex plane. Then the set of complex numbers $$z$$ satisfying $$|z-z_1|^2+|z-z_2|^2=|z_1-z_2|^2$$ 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) $$-\frac{1}{2}$$        (D) $$\frac{1}{2}$$

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 $$\{\frac{1}{x}\sqrt{1+x}-\sqrt{1+\frac{1}{x^2}}\}$$ as $$x\to0$$

(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 $$\cos^6 \theta + \sin^6 \theta$$ 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 $$3x^2+3y^2-9x+6y+5=0$$ 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 $$z=x+iy$$ where x and y are real numbers and $$i=\sqrt{-1}$$. The points (x, y) for which $$\frac{z-1}{z-i}$$ 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 $$\frac{x^2}{9}-\frac{y^2}{25}=1$$ at the points P and Q. The eccentricity of ellipse with PQ as major axis and minor axis of length $$\frac{5}{\sqrt2}$$ is

(A) $$\frac {\sqrt 5}{3}$$        (B) $$\frac {5}{\sqrt 3}$$         (C) $$\frac {5}{9}$$        (D)$$\frac{25}{9}$$

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 $$(1+x)^n$$ is 11 : 10. Then the number of terms in the expansion of $$(1+x)^n$$ is

(A) 20       (B) 21       (C) 10       (D) 11

Ans: (B)

56.  Let exp (x) denote the exponential function e<sup>x</sup>. If $$f(x) = \exp \left( {{x^{{1 \over x}}}} \right), x>0$$, then the minimum value of $$f$$ in the interval [2, 5] is

(A) $$\exp \left( {{e^{{1 \over e}}}} \right)$$        (B) $$\exp \left( {{2^{{1 \over 2}}}} \right)$$        (C) $$\exp \left( {{5^{{1 \over 5}}}} \right)$$        (D) $$\exp \left( {{3^{{1 \over 3}}}} \right)$$

Ans: (C)

57.  The sum of the series $$\frac {1}{1 \times 2}{^{25}}C_0 + \frac {1}{2 \times 3}{^{25}}C_1 + \frac {1}{3 \times 4}{^{25}}C_2 +...+ \frac {1}{26 \times 27}{^{25}}C_{25}$$

(A) $$\frac {2^{27} - 1}{26 \times 27}$$       (B) $$\frac {2^{27}-28}{26 \times 27}$$       (C) $$\frac {1}{2}(\frac {2^{26}+1}{26 \times 27})$$       (D) $$\frac {2^{26} -1}{52}$$

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 $$f(x)=2|x-1|+|x-2|$$ is

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

Ans: (B)

60.  If P, Q, R are angles of an isosceles triangle and $$\angle P = {\pi \over 2}$$, then the value of

$$(\cos \frac {P}{3} - i \sin {P \over 3})^3 + ( \cos Q + i \sin Q)( \cos R -i \sin R ) + ( \cos P -i \sin P )( \cos Q -i \sin Q)( \cos R -i \sin R)$$

is equal to

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

Ans: (B)

61.  A line passing through the point of intersection of $$x+y=4$$ and $$x-y=2$$ makes an angle $$tan^ {-1}(3/4)$$ with the x-axis. It intersects the parabola $$y^2=4(x-3)$$ at points $$(x_1,y_1)$$ and $$(x_2,y_2)$$ respectively. Then $$|x_1-x_2|$$ is equal to

(A) $${16 \over 9}$$         (B) $${32 \over 9}$$         (C) $${40 \over 9}$$         (D) $${80 \over 9}$$

Ans: (B)

62.  Let [a] denote the greatest integer which is less than or equal to a. Then the value of the integral

$$\int_{-{\pi \over 2}}^{\pi \over 2}[ \sin x \cos x]dx$$ is

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

Ans: (D)

63.  If $$P=\begin{pmatrix} 2 & -2 & -4\\ -1 & 3 & 4\\ 1 & -2 & -3 \end{pmatrix}$$ then $$P^5$$ equals

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

Ans: (A)

64.  If $$\sin^2 \theta + 3 \cos \theta = 2$$, then $$\cos^3 \theta + \sec^3 \theta$$ is

(A) 1         (B) 4         (C) 9         (D) 18

Ans: (D)

65.  $$x = 1 + \frac {1}{2 \times |\underline 1} + \frac {1}{4 \times |\underline2} + \frac {1}{8 \times | \underline3} +......$$ and $$y = 1 + \frac {x^2}{|\underline1} + \frac {x^4}{|\underline 2} + \frac {x^6}{ | \underline 3} +......$$

Then the value of $$log_ey$$ is

(A) e         (B) e²          (C) 1         (D) 1/e

Ans: (A)

66.  The value of the infinite series

$$\frac {1^2 + 2^2}{|\underline 3} + \frac {1^2 + 2^2 + 3^2}{| \underline 4} + \frac {1^2 + 2^2 + 3^2 + 4^2}{|\underline 5}+.......$$ is

(A) e         (B) 5e          (C) $$\frac {5e}{6} - \frac {1}{2}$$         (D) $$\frac{5e}{6}$$

Ans: (C)

67.  The value of the integral $$\int_{\pi \over 6}^{\pi \over 3} \frac {(\sin x - x \cos x)}{x(x + \sin x)}dx$$ is equal to

(A) $$log_e(\frac{2(\pi+3)}{2\pi+3\sqrt3})$$         (B) $$log_e(\frac{\pi+3}{2(2\pi +3\sqrt3)})$$        (C) $$log_e(\frac{2\pi+3\sqrt3}{2(\pi+3)})$$         (D) $$log_e(\frac{2(2\pi+3\sqrt3)}{\pi +3})$$

Ans: (A)

68.  Let $$f(x) = x( \frac {1}{x-1} + \frac {1}{x} + \frac {1}{x+1}), x > 1$$. Then

(A) $$f(x) \le 1$$          (B) $$1 < f(x) \le 2$$         (C) $$2 < f(x) \le 3$$         (D) $$f(x) > 3$$

Ans: (D)

69.  Let $$F(x) = \int_0^x \frac {\cos t}{(1 + t^2)}dt$$, $$0 \le x \le 2 \pi$$. Then

(A) F is increasing in $$(\frac {\pi}{2},\frac {3\pi}{2})$$ and decreasing in $$(0,\frac{\pi}{2})$$ and $$(\frac{3\pi}{2},2\pi)$$

(B) F is increasing in $$(0,\pi)$$ and decreasing in $$(\pi,2 \pi)$$

(C) F is increasing in $$(\pi,2\pi)$$ and decreasing in $$(0,\pi)$$

(D) F is increasing in $$(0,\frac {\pi}{2})$$ and $$(\frac {3 \pi}{2}, 2\pi)$$ and decreasing in $$(\frac {\pi}{2},\frac {3 \pi}{2})$$

Ans: (D)

70.  Let $$f(x)=x^{2/3}, x \ge 0$$. Then the area of the region enclosed by the curve $$y=f(x)$$ and three lines $$y=x$$, $$x=1$$ and $$x=8$$ is

(A) $$\frac{63}{2}$$         (B) $$\frac{93}{5}$$         (C) $$\frac{105}{7}$$         (D) $$\frac{129}{10}$$

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 $$\frac {\tan \angle {PQF}}{\tan \angle {PFQ}}$$ 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) $$\frac{5}{32}$$         (B) $$\frac{3}{128}$$         (C) $$\frac{3}{256}$$        (D) $$\frac{3}{64}$$

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) $$xy=c$$, c is a constant

(B) $$xy^2=c$$, c is a constant

(C) $$x^2y = c$$, c is a constant

(D) $$x^2y^2=c$$, c is a constant

Ans: (C)

74.  The solution of the differential equation $$(y^2+2x)\frac{dy}{dx}=y$$ satisfy x = 1, y = 1. Then the solution is

(A) $$x=y^2(1+log_ey)$$        (B) $$y=x^2(1+log_ex)$$         (C) $$x=y^2(1-log_ey)$$        (D) $$y=x^2(1- log_ex)$$

Ans: (A)

75.  The solution of the differential equation $$y \sin (x/y)dx = (x \sin (x/y) - y) dy$$ satisfying $$y(\pi /4)=1$$ is

(A) $$\cos \frac{x}{y} =-log_ey + \frac {1}{\sqrt2}$$

(B) $$\sin \frac {x}{y} = log_ey + \frac {1} {\sqrt2}$$

(C) $$\sin \frac {x}{y} = log_ex- \frac {1}{\sqrt2}$$

(D) $$\cos \frac {x}{y} =-log_ex - \frac {1} {\sqrt2}$$

Ans: ()

76. The area of the region encloses between parabola y² = x and the line y = mx is $${1 \over 48}$$. Then the value of m is

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

Ans: (A)

77.  Consider the system of equations:

$$x + y + z = 0$$

$$\alpha x + \beta y + \gamma z = 0$$

$$\alpha^2 x + \beta^2 y + \gamma^2 z = 0$$

Then the system of equations has

(A) A unique solution for all values of $$\alpha ,\beta ,\gamma$$

(B) Infinite numbers of solutions if any two of $$\alpha ,\beta ,\gamma$$ are equal

(C) A unique solution if $$\alpha ,\beta , \gamma$$ are distinct

(D) More than one, but finite number of solutions depending on values of $$\alpha ,\beta ,\gamma$$

Ans: (B)

78.  The equations of the circles which touch both the axis and the line $$4x + 3y = 12$$ 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 $$\sin \alpha$$, $$\cos \alpha$$ be the roots of the equation $$x^2 - bx + c = 0$$. Then which of the following statements is/are correct ?

(A) $$c \le \frac{1}{2}$$         (B) $$b \le \sqrt{2}$$         (C) $$c > \frac{1}{2}$$         (D) $$b > \sqrt{2}$$

Ans: (A)

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