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^x. 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^st, 3^rd and 4^th terms are in G.P., then

(A) the 5^th term is always 0

(B) the 1^st 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)

****