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Question-01 If f(x)=\log\left(\frac{1+x}{1-x}\right) and g(x)=\frac{3x+x^3}{1+3x^2}, then (fog)(x)=
(A) 2f(x)
(B) 3f(x)
(C) 4f(x)
(D) -f(x)
Answer : (B)
Question-02 If \sec x+\tan x=2,\ 0<x<\frac{\pi}{2} then \sin\frac{x}{4}=
(A) \frac{1}{\sqrt{10+3\sqrt{10}}}
(B) \frac{1}{\sqrt{2(10+3\sqrt{10})}}
(C) \frac{1}{\sqrt{10-3\sqrt{10}}}
(D) \frac{1}{2\sqrt{10-3\sqrt{10}}}
Answer : (B)
Question-03 The eccentricity of the curve represented by x=3(\cos t+\sin t),\ y=4(\cos t-\sin t) is
(A) \frac{\sqrt{7}}{4}
(B) \frac{7}{16}
(C) \frac{\sqrt{7}}{3}
(D) \frac{\sqrt{8}}{4}
Answer : (A)
Question-04 The rate at which the population of a city increases varies as the population. In a period of 20 years, the population increased from 4 lakhs to 6 lakhs. In another 20 years the population will be
(A) 8 lakhs
(B) 12 lakhs
(C) 9 lakhs
(D) 10 lakhs
Answer : (C)
Question-05 If \vec a=\hat i+\hat j+\hat k,\ \vec b=\hat j-\hat k then a vector \vec c such that \vec a\times\vec c=\vec b and \vec a\cdot\vec c=3 is
(A) \frac{5}{3}\hat i+\frac{2}{3}\hat j+\frac{2}{3}\hat k
(B) \hat i-2\hat j+4\hat k
(C) \hat i+2\hat k
(D) 2\hat i-3\hat j+4\hat k
Answer : (A)
Question-06 The solution set of the constraints |x-y|\le1,\ x,y\ge0 is
(A) a finite set
(B) an unbounded set
(C) a convex polygon
(D) such that feasible region does not exist
Answer : (B)
Question-07 The lines \frac{x-0}{1}=\frac{y-2}{2}=\frac{z+3}{\lambda} and \frac{x-2}{2}=\frac{y-6}{3}=\frac{z-3}{\lambda} are coplanar and p is the plane containing these lines, then which of the following point does not lie on the plane.
(A) (1, 6, 4)
(B) (2, 8, 7)
(C) (1, 2, 3)
(D) (4, 10, 9)
Answer : (A)
Question-08 \int_{1}^{3}\frac{\log x^2}{\log(16x^2-8x^3+x^4)}dx=
(A) 1
(B) 3
(C) \log2
(D) \frac{1}{2}
Answer: A
Question-09 If \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, then \frac{d^2y}{dx^2} is
(A) -\frac{b^4}{a}
(B) \frac{b^4}{a^2}
(C) -\frac{b^4}{y^3}
(D) -\frac{b^4}{a^2y^3}
Answer : (D)
Question-10 If Rolle’s theorem holds for the function x^3+ax^2+bx,\ 1\le x\le2 at the point \frac{4}{3}, then the values of a and b are respectively
(A) 5, 8
(B) -8, 5
(C) 8, -5
(D) -5, 8
Answer : (D)
Question-11 \int\frac{1}{e^x+1}dx=
(A) x+\log(e^x+1)+c
(B) x-\log(e^x+1)+c
(C) \log(e^x-1)+x+c
(D) \log(e^x-1)-x+c
Answer : (B)
Question-12 If y=\tan^{-1}\left(\frac{4x}{1+5x^2}\right)+\cot^{-1}\left(\frac{3-2x}{2+3x}\right), then \frac{dy}{dx} is equal to
(A) \frac{5}{1+25x^2}
(B) \frac{1}{1+25x^2}
(C) \frac{1}{1+5x^2}
(D) \frac{5}{1+5x^2}
Answer : (A)
Question-13 The differential equation x\frac{dy}{dx}=2y represents
(A) a family of circles with radius c.
(B) a family of parabolas with vertex at the origin and axis along the positive Y-axis
(C) a family of parabolas with vertex at origin and axis along the positive X-axis
(D) a family of ellipses
Answer : (B)
Question-14 \int e^x\left(\frac{x+5}{(x+6)^2}\right)dx is
(A) \frac{e^x}{(x+6)^2}+c
(B) \frac{e^x}{x+5}+c
(C) \frac{e^x}{(x+5)^2}+c
(D) \frac{e^x}{x+6}+c
Answer : (D)
Question-15 The principal solution of (5+3\sin\theta)(2\cos\theta+1)=0 are
(A) -\frac{\pi}{3},\frac{2\pi}{3}
(B) \frac{2\pi}{3},\frac{5\pi}{3}
(C) \frac{2\pi}{3},\frac{4\pi}{3}
(D) \frac{2\pi}{3},\frac{7\pi}{3}
Answer : (C)
Question-16 Let X denote the number of hours you study on a Sunday. It is known that
P(X=x)=\begin{cases}0.1,&x=0\\kx,&x=1\ or\ 2\\k(5-x),&x=3\ or\ 4\\0,&otherwise\end{cases}where k is constant. Then the probability that you study at least two hours on a Sunday is
(A) 0.55
(B) 0.15
(C) 0.75
(D) 0.3
Answer : (C)
Question-17 The principal value of \cos^{-1}\left[\frac{1}{\sqrt{2}}\left(\cos\frac{9\pi}{10}-\sin\frac{9\pi}{10}\right)\right] is
(A) \frac{3\pi}{20}
(B) \frac{17\pi}{20}
(C) \frac{7\pi}{10}
(D) \frac{\pi}{10}
Answer : (B)
Question-18 The length of the foot of the perpendicular from the point \left(1,\frac{3}{2},2\right) to the plane 2x-2y+4z+17=0 is
(A) \sqrt{6} units
(B) 3\sqrt{3} units
(C) 4\sqrt{3} units
(D) 2\sqrt{6} units
Answer : (D)
Question-19 A tetrahedron has vertices O(0,0,0),A(1,2,1),B(2,1,3),C(-1,1,2). Then the angle between the faces OAB and ABC will be
(A) \cos^{-1}\left(\frac{19}{35}\right)
(B) \cos^{-1}\left(\frac{1}{35}\right)
(C) \cos^{-1}\left(\frac{9}{35}\right)
(D) \cos^{-1}\left(\frac{4}{35}\right)
Answer : (A)
Question-20 If A=\begin{bmatrix}3&-3&4\\2&-3&4\\0&-1&1\end{bmatrix}, then A^{-1}=
(A) A
(B) A^2
(C) A^3
(D) A^4
Answer : (C)
Question-21 The area bounded by the parabolas y=9x^2, y=\frac{x^2}{16} and the line y=1 is
(A) \frac{22}{9} sq. units
(B) \frac{44}{9} sq. units
(C) \frac{8}{9} sq. units
(D) \frac{26}{9} sq. units
Answer : (B)
Question-22 \int\frac{dx}{3\cos2x+5} equals
(A) \frac{1}{2}\tan^{-1}(\tan x)+c
(B) \frac{1}{2}\tan^{-1}\left(\frac{\tan x}{2}\right)+c
(C) \frac{1}{4}\tan^{-1}\left(\frac{1}{2}\tan x\right)+c
(D) \frac{1}{4}\tan^{-1}(\tan x)+c
Answer : (C)
Question-23 The solution of the equation x^2y-x^3\frac{dy}{dx}=y^4\cos x, where y(0)=1, is
(A) y^3=3x^2\sin x
(B) x^3=3y^3\sin x
(C) x^3=y^3\sin x
(D) y^3=4x^3\sin x
Answer : (B)
Question-24 If the statements p, q and r are true, false and true statements respectively, then the truth value of the statement pattern [\sim q\land(p\lor\sim q)\land\sim r]\lor p and the truth value of its dual statement respectively are
(A) T, T
(B) F, T
(C) T, F
(D) F, F
Answer : (A)
Question-25 If the lines \frac{1-x}{2}=\frac{7y+4}{2\lambda}=\frac{2z-5}{2} and \frac{7-7x}{3\lambda}=\frac{y-1}{7}=\frac{6-z}{5} are at right angle, then the value of \lambda is
(A) \frac{4}{7}
(B) \frac{7}{4}
(C) \frac{20}{7}
(D) \frac{5}{4}
Answer : (B)
Question-26 The negation of the statement “The triangle is an equilateral or isosceles triangle and the triangle is not isosceles and it is right angled” is
(A) The triangle is not an equilateral or not an isosceles triangle or it is not an isosceles triangle or it is not right angled
(B) The triangle is not an equilateral triangle or not isosceles triangle and it is isosceles or it is not right angled
(C) If the triangle is an equilateral triangle or an isosceles triangle then it is an isosceles triangle or not right angled
(D) If the triangle is an equilateral triangle or an isosceles triangle then it is not isosceles triangle and it is not right angled
Answer : (C)
Question-27 f(x)=\frac{x}{2}+\frac{2}{x},\ x\ne0 is strictly decreasing in
(A) (2, 3)
(B) (1, 3)
(C) (-2, 2)
(D) (1, 2)
Answer : (D)
Question-28 If f(x)=\frac{\sin(\pi\cos^2 x)}{3x^2},\ x\ne0 is continuous at x=0 then f(0)=
(A) 0
(B) \frac{\pi}{3}
(C) -\frac{\pi}{3}
(D) \frac{3}{\pi}
Answer : (B)
Question-29 Let z be the complex number with \text{Im}(z)=10 and satisfying \frac{2z-n}{2z+n}=2i-1, where i=\sqrt{-1}, for some natural number n then
(A) n=20 and \text{Re}(z)=10
(B) n=20 and \text{Re}(z)=-10
(C) n=40 and \text{Re}(z)=10
(D) n=40 and \text{Re}(z)=-10
Answer : (D)
Question-30 The rate of change of the volume of a sphere with respect to its surface area, when the radius is 5 m is
(A) \frac{5}{2}\ m
(B) \frac{2}{5}\ m
(C) \frac{1}{2}\ m
(D) \frac{1}{3}\ m
Answer : (A)
Question-31 The maximum value of the function a\sin x+b\cos x is
(A) \sqrt{a^2+b^2}
(B) \sqrt{a^2-b^2}
(C) a^2+b^2
(D) a^2-b^2
Answer : (A)
Question-32 The number of values of x in the interval [0,3\pi] satisfying the equation 2\sin^2 x+5\sin x-3=0 is
(A) 6
(B) 1
(C) 2
(D) 4
Answer : (D)
Question-33 \lim_{x\to5}\frac{\sqrt{2-2\cos(x^2-12x+35)}}{x-5}=
(A) \frac{2}{-5}
(B) -2
(C) -\frac{1}{2}
(D) -5
Answer : (B)
Question-34 If ^{n}C_{0}+\frac{1}{2}\ ^{n}C_{1}+\frac{1}{3}\ ^{n}C_{2}+…+\frac{1}{n}\ ^{n}C_{n-1}+\frac{1}{n+1}\ ^{n}C_{n}=\frac{1023}{10} then n=
(A) 7
(B) 8
(C) 9
(D) 10
Answer : (C)
Question-35 A pair of fair dice is thrown 4 times. If getting the same number on both dice is considered as a success, then the probability of two successes are
(A) \frac{25}{216}
(B) \frac{25}{36}
(C) \frac{25}{108}
(D) \frac{25}{104}
Answer : (A)
Question-36 The position vectors of the points A, B, C are \hat i+2\hat j-\hat k,\ \hat i+\hat j+\hat k,\ 2\hat i+3\hat j+2\hat k respectively. If A is chosen as the origin, then the cross product of position vectors of B and C are
(A) -5\hat i+2\hat j+\hat k
(B) -\hat i+0\hat j-\hat k
(C) \hat i-\hat k
(D) 5\hat i-2\hat j-\hat k
Answer : (A)
Question-37 If the area of a parallelogram whose diagonals are represented by vectors 3\hat i+\lambda\hat j+2\hat k and \hat i-2\hat j+3\hat k is \frac{\sqrt{117}}{2} sq. units, then \lambda=
(A) -1
(B) -2
(C) -3
(D) -4
Answer : (D)
Question-38 y=e^x(A\cos x+B\sin x) is the solution of the differential equation
(A) x^2\frac{d^2y}{dx^2}+(1+y^2)=0
(B) \frac{d^2y}{dx^2}-\frac{dy}{dx}+y=0
(C) \frac{d^2y}{dx^2}-2\frac{dy}{dx}+2y=0
(D) x\frac{d^2y}{dx^2}-2\frac{dy}{dx}+2y=0
Answer : (C)
Question-39 A family has 3 children. The probability that all the three children are girls, given that at least one of them is a girl is
(A) \frac{7}{8}
(B) \frac{1}{8}
(C) \frac{1}{7}
(D) \frac{2}{7}
Answer : (C)
Question-40 In a triangle ABC, with usual notations.
\frac{2\cos A}{a}+\frac{\cos B}{b}+\frac{2\cos C}{c}=\frac{a}{bc}+\frac{b}{ca}Then \angle A=
(A) \frac{\pi}{4}
(B) \frac{\pi}{6}
(C) \frac{\pi}{2}
(D) \frac{\pi}{3}
Answer : (C)
Question-41 A line passes through P(-4,1) and meets the co-ordinate axes at points A and B. If P divides the segment AB internally in the ratio 1:2, then the equation of the line is
(A) x-2y+6=0
(B) x+10y-6=0
(C) 2x+y+4=0
(D) x-y+5=0
Answer : (A)
Question-42 In 3-dimensional space, the equation x^2-8x+12=0 represents
(A) two straight lines
(B) a pair of straight lines passing through the origin
(C) 2 planes parallel to YZ-plane
(D) 2 planes parallel to XZ-plane
Answer : (C)
Question-43 A pair of tangents are drawn to the circle x^2+y^2+6x-4y-12=0 from a point P(-4,-5), then the area enclosed between these tangents and the area of the circle is
(A) 25\left(\frac{4+\pi}{4}\right) sq. units
(B) 25\left(\frac{4+\pi}{2}\right) sq. units
(C) 25\left(\frac{4-\pi}{2}\right) sq. units
(D) 25\left(\frac{4-\pi}{4}\right) sq. units
Answer : (D)
Question-44 If \vec a,\vec b,\vec c are non coplanar unit vectors such that \vec a\times(\vec b\times\vec c)=\frac{\vec b+\vec c}{\sqrt{2}} then the angle between \vec a and \vec b is
(A) \frac{\pi}{2}
(B) \frac{\pi}{4}
(C) \frac{\pi}{3}
(D) \frac{3\pi}{4}
Answer : (D)
Question-45 The joint equation of the bisector of the angle between the lines 2x^2+11xy+3y^2=0 is
(A) 11x^2+2xy-11y^2=0
(B) x^2+2xy-y^2=0
(C) 3x^2-11xy+2y^2=0
(D) 11x^2-2xy-11y^2=0
Answer : (A)
Question-46 If a random variable X has the following probability distribution of X
| X = x | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|---|
| P(X = x) | 0 | k | 2k | 2k | 3k | k^2 | 2k^2 | 7k^2 + k |
Then P(X\ge6)=
(A) \frac{19}{100}
(B) \frac{81}{100}
(C) \frac{9}{100}
(D) \frac{91}{100}
Answer : (A)
Question-47 \int_{0}^{1}\frac{1}{2+\sqrt{x}}dx=
(A) 2\log\left(\frac{2e}{3}\right)
(B) 2\log\left(\frac{4e}{9}\right)
(C) \log\left(\frac{2e}{3}\right)
(D) \log\left(\frac{4e}{9}\right)
Answer : (B)
Question-48 Let M and N be foots of the perpendiculars drawn from the point P(a,a,a) on the lines x-y=0,\ z=1 and x+y=0,\ z=-1 respectively and if \angle MPN=90^\circ then a^2=
(A) 1
(B) 4
(C) 6
(D) 9
Answer : (A)
Question-49 If y=\log_3(\log_3 x) then \frac{dy}{dx} at x=3 is
(A) \frac{1}{3}(\log3)^{-3}
(B) \frac{1}{3}(\log3)
(C) \frac{1}{3}(\log3)^{-3}
(D) \frac{1}{3}(\log3)^{-2}
Answer : (D)
Question-50 If in triangle ABC, with usual notations
\sin\frac{A}{2}\cdot\sin\frac{C}{2}=\sin\frac{B}{2}and 2s is the perimeter of the triangle, then the value of s is
(A) 2b
(B) b
(C) 4b
(D) \frac{b}{2}
Answer : (A)