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Question-01 Considering only the principal values of the inverse trigonometric function, the value of \tan\left(\cos^{-1}\frac{1}{5\sqrt{2}}-\sin^{-1}\frac{4}{\sqrt{17}}\right) is
(A) \frac{3}{34}
(B) \frac{1}{34}
(C) \frac{3}{29}
(D) \frac{1}{29}
Answer : (C)
Question-02 The line L is passing through (1,2,3). The distance of any point on the line L from the line \vec r=(3\lambda-1)\hat i+(-2\lambda+3)\hat j+(4+\lambda)\hat k is constant. Then the line L does not pass through the point
(A) (4, 0, 4)
(B) (-2, 4, 2)
(C) (7, -2, 5)
(D) (-5, 6, 2)
Answer : (D)
Question-03 The distance of the plane \vec r=(\hat i-\hat j)+\lambda(\hat i+\hat j+\hat k)+\mu(\hat i-2\hat j+3\hat k) from the origin is
(A) \frac{7}{\sqrt{38}} units
(B) \frac{1}{\sqrt{38}} units
(C) \frac{5}{\sqrt{38}} units
(D) \frac{2}{\sqrt{38}} units
Answer : (A)
Question-04 If the angle between the line x=\frac{y-1}{2}=\frac{z-3}{\lambda} and the plane x+2y+3z=4 is \cos^{-1}\sqrt{\frac{5}{14}}, then the value of \lambda is
(A) \frac{1}{3}
(B) \frac{4}{5}
(C) \frac{2}{3}
(D) \frac{2}{5}
Answer : (C)
Question-05 If f(1)=3,\ f'(1)=2, then \frac{d}{dx}\left[\log\left(f(e^x+2x)\right)\right] at x=0 is
(A) \frac{2}{3}
(B) \frac{3}{2}
(C) 2
(D) 0
Answer : (C)
Question-06 If \frac{1}{6}\sin\theta,\ \cos\theta,\ \tan\theta are in G.P., then the general solution of \theta is
(A) 2n\pi\pm\frac{\pi}{3},\ n\in Z
(B) n\pi\pm\frac{\pi}{3},\ n\in Z
(C) n\pi\pm\frac{\pi}{4},\ n\in Z
(D) 2n\pi\pm\frac{\pi}{6},\ n\in Z
Answer : (A)
Question-07 Let f be a function which is continuous and differentiable for all x. If f(1)=1 and f'(x)\le5 for all x in [1,5], then the maximum value of f(5) is
(A) 5
(B) 20
(C) 6
(D) 21
Answer : (D)
Question-08 In a triangle ABC with usual notations, if \cot\frac{A}{2}=\frac{b+c}{a}, then the triangle ABC is
(A) an isosceles triangle.
(B) an equilateral triangle.
(C) a right angled triangle.
(D) an obtuse angled triangle.
Answer : (C)
Question-09 If matrix A=\frac{1}{11}\begin{bmatrix}-1&7&-24\\2&a&4\\2&-3&15\end{bmatrix} and A^{-1}=\begin{bmatrix}3&3&4\\2&-3&4\\b&-1&c\end{bmatrix}, then the values of a,b,c respectively are
(A) 3,1,0
(B) -\frac{6}{11},0,\frac{1}{11}
(C) -3,0,1
(D) -\frac{3}{11},0,\frac{1}{11}
Answer : (C)
Question-10 p: If 7 is an odd number then 7 is divisible by 2.
q: If 7 is prime number then 7 is an odd number.
If V_1 and V_2 are respective truth values of contrapositive of p and q then (V_1,V_2) =
(A) (T, T)
(B) (T, F)
(C) (F, T)
(D) (F, F)
Answer : (C)
Question-11 \lim_{x\to1}(\log_3 3x)^{\log_x 8}=
(A) e^{\log_3 8}
(B) \log_8 3
(C) e^{\log_8 3}
(D) \log_3 8
Answer : (A)
Question-12 The function f(x)=\sin^4x+\cos^4x increases if
(A) 0<x<\frac{\pi}{8}
(B) \frac{\pi}{4}<x<\frac{\pi}{2}
(C) \frac{3\pi}{8}<x<\frac{5\pi}{8}
(D) \frac{5\pi}{8}<x<\frac{3\pi}{4}
Answer : (B)
Question-13 The values of b and c for which the identity f(x+1)-f(x)=8x+3 is satisfied, where f(x)=bx^2+cx+d, are
(A) b=2,\ c=1
(B) b=4,\ c=-1
(C) b=1,\ c=2
(D) b=3,\ c=-1
Answer : (B)
Question-14 \int\frac{x^3}{x^4+5x^2+4}dx=
(A) \frac{1}{3}\log\left(\frac{(x^2+4)^2}{\sqrt{x^2+1}}\right)+c, where c is the constant of integration
(B) \log\left(\frac{(x^2+4)^2}{\sqrt{x^2+1}}\right)+c, where c is the constant of integration
(C) 3\log\left(\frac{(x^2+4)^2}{\sqrt{x^2+1}}\right)+c, where c is the constant of integration
(D) \frac{2}{3}\log\left(\frac{(x^2+4)^2}{\sqrt{x^2+1}}\right)+c, where c is the constant of integration
Answer : (A)
Question-15 z=\frac{3+2i\sin\theta}{1-2i\sin\theta},\ (i=\sqrt{-1}) will be purely imaginary if \theta=
(A) 2n\pi\pm\frac{\pi}{8},\ where\ n\in Z
(B) n\pi\pm\frac{\pi}{8},\ where\ n\in Z
(C) n\pi\pm\frac{\pi}{3},\ where\ n\in Z
(D) n\pi,\ where\ n\in Z
Answer : (C)
Question-16 The equations of the tangents to the circle x^2+y^2=36 which are perpendicular to the line 5x+y=2, are
(A) x+5y\pm6\sqrt{26}=0
(B) x-5y\pm6\sqrt{26}=0
(C) 5x-y\pm6\sqrt{26}=0
(D) 5x+y\pm6\sqrt{26}=0
Answer : (B)
Question-17 If \sin A=n\sin(A+2B), then \tan(A+B)=
(A) \frac{1+n}{2-n}\tan B
(B) \frac{1-n}{1+n}\tan B
(C) \frac{1-n}{2+n}\tan B
(D) \frac{1+n}{1-n}\tan B
Answer : (D)
Question-18 The number of integral values of p for which the vectors (p+1)\hat i-3\hat j+p\hat k, p\hat i+(p+1)\hat j-3\hat k and -3\hat i+p\hat j+(p+1)\hat k are linearly dependent vectors, are
(A) 0
(B) 1
(C) 2
(D) 3
Answer : (B)
Question-19 \int_{0}^{\frac{\pi}{4}}\left(\sqrt{\tan x}+\sqrt{\cot x}\right)dx=
(A) \sqrt{2\pi}
(B) \frac{\pi}{2}
(C) 2\pi
(D) \frac{\pi}{\sqrt{2}}
Answer : (D)
Question-20 If a curve y=a\sqrt{x}+bx passes through the point (1, 2) and the area bounded by this curve, line x=4 and the X-axis is 8 sq. units, then the value of a-b is
(A) -2
(B) 2
(C) -4
(D) 4
Answer : (D)
Question-21 The foci of a hyperbola coincide with the foci of the ellipse \frac{x^2}{25}+\frac{y^2}{9}=1. The equation of the hyperbola with eccentricity 2 is
(A) \frac{x^2}{12}-\frac{y^2}{4}=1
(B) \frac{x^2}{4}-\frac{y^2}{12}=1
(C) \frac{x^2}{12}-\frac{y^2}{16}=1
(D) \frac{x^2}{16}-\frac{y^2}{12}=1
Answer : (B)
Question-22 The differential equation satisfied by y=X\sin(6t+5)+Y\cos(6t+5) is (where X and Y are constants)
(A) \frac{d^2y}{dt^2}+6y=0
(B) \frac{d^2y}{dt^2}=0
(C) \frac{d^2y}{dt^2}+36y=0
(D) \frac{d^2y}{dt^2}+25y=0
Answer : (C)
Question-23 A wet substance in the open air loses its moisture at a rate proportional to the moisture content. If a sheet, hung in the open air, loses half its moisture during the first hour, then 90\% of the moisture will be lost in ____ hours.
(A) 2\log_2 10
(B) \frac{4\log 10}{\log 2}
(C) \log_2 10
(D) \frac{3\log 10}{\log 2}
Answer : (C)
Question-24 If a random variable X has p.d.f.
f(x)=\begin{cases}\frac{ax^2}{2}+bx, & 1\le x\le3 \\ 0, & otherwise\end{cases}and f(2)=2, then the values of a and b are, respectively
(A) 11,-10
(B) -9,10
(C) \frac{1}{6},\frac{5}{6}
(D) 9,-8
Answer : (B)
Question-25 If \vec p=2\hat i+\hat k,\ \vec q=\hat i+\hat j+\hat k,\ \vec r=4\hat i-3\hat j+7\hat k and a vector \vec m is such that \vec m\times\vec q=\vec r\times\vec q,\ \vec m\cdot\vec p=0, then \vec m=
(A) \hat i-8\hat j-2\hat k
(B) -10\hat i+3\hat j+7\hat k
(C) -\hat i-8\hat j+2\hat k
(D) 2\hat i+4\hat j+\hat k
Answer : (C)
Question-26 If the point (1, \alpha, \beta) lies on the line of the shortest distance between the lines \frac{x+2}{-3}=\frac{y-2}{4}=\frac{z-5}{2} and \frac{x+2}{-1}=\frac{y+6}{2},\ z=1, then \alpha+\beta=
(A) 3
(B) -3
(C) 1
(D) -7
Answer : (C)
Question-27 The angle between the lines x-3y-4=0,\ 4y-z+5=0 and x+3y-11=0,\ 2y-z+6=0 is
(A) \frac{\pi}{2}
(B) \frac{\pi}{4}
(C) \frac{\pi}{6}
(D) \frac{\pi}{3}
Answer : (A)
Question-28 If the area of parallelogram, whose diagonals are \hat i-\hat j+2\hat k and 2\hat i+3\hat j+\alpha\hat k is \frac{\sqrt{93}}{2} sq. units, then \alpha=
(A) -4,2
(B) -3,-2
(C) 2,1
(D) 4,2
Answer : (A)
Question-29 The correct constraints for the given feasible region are ….
(A) y-x\ge1,\ x+5y\le10,\ x+y\ge2,\ x,y\ge0
(B) y-x\le1,\ 2x+5y\le10,\ x+y\ge1,\ x,y\ge0
(C) y-x\ge1,\ 2x+5y\le10,\ x+y\ge1,\ x,y\ge0
(D) x-y\le1,\ 2x+5y\ge10,\ x+y\le1,\ x,y\ge0
Answer : (C)
Question-30 The circumradius of the triangle formed by the lines xy+2x+2y+4=0 and x+y+2=0 is
(A) 2 units
(B) 1 unit
(C) \sqrt{2} units
(D) \sqrt{3} units
Answer : (C)
Question-31 Derivative of x^{(x^x)} is
(A) x^{(x^x)}(x^x+1+\log x)
(B) x^{(x^x)}(x^x+\log x)
(C) x^{(x^x)}(x^x+x^{x-1}\log x(1+\log x))
(D) x^{(x^x)}(x^{x-1}+x^x\log(1+\log x))
Answer : (D)
Question-32 The number of solutions of \tan^{-1}\left(x+\frac{2}{x}\right)-\tan^{-1}\left(\frac{4}{x}\right)-\tan^{-1}\left(x-\frac{2}{x}\right)=0 are
(A) 1
(B) 2
(C) 3
(D) 0
Answer : (B)
Question-33 The derivative of \tan^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right) w.r.t. \tan^{-1}\left(\frac{2x\sqrt{1-x^2}}{1-2x^2}\right) at x=0 is
(A) \frac{1}{8}
(B) \frac{1}{4}
(C) \frac{1}{2}
(D) 1
Answer : (B)
Question-34 The normal to the curve x=9(1+\cos\theta),\ y=9\sin\theta at \theta always passes through the fixed point
(A) (9, 0)
(B) (8, 9)
(C) (0, 9)
(D) (9, 8)
Answer : (A)
Question-35 In a triangle ABC with usual notations, if 3a=b+c, then \cot\frac{B}{2}\cdot\cot\frac{C}{2}=
(A) 1
(B) \sqrt{2}
(C) 2
(D) 3
Answer : (C)
Question-36 If p: switch S_1 is closed, q: switch S_2 is closed, r: switch S_3 closed, then the symbolic form of the following switching circuit is equivalent to
(A) p\land(q\lor r)
(B) q\lor r
(C) p
(D) (\sim q\land\sim r)
Answer : (A)
Question-37 If
f(x)=\begin{cases}\frac{1-\cos4x}{x^2}, & x<0\\ a, & x=0\\ \frac{\sqrt{x}}{(16+\sqrt{x})^{\frac{1}{2}}-4}, & x>0\end{cases}is continuous at x=0, then a=
(A) 4
(B) 8
(C) -4
(D) -8
Answer : (B)
Question-38 \int \sec^{\frac{2}{3}}x\ \cosec^{\frac{4}{3}}x\ dx=
(A) 3\tan^{-\frac{1}{3}}x+c, where c is the constant of integration
(B) -3\tan^{-\frac{1}{3}}x+c, where c is the constant of integration
(C) -3\cot^{-\frac{1}{3}}x+c, where c is the constant of integration
(D) -\frac{3}{4}\tan^{-\frac{4}{3}}x+c, where c is the constant of integration
Answer : (B)
Question-39 If the lengths of three vectors \vec a,\vec b and \vec c are 5,12,13 units respectively, and each one is perpendicular to the sum of the other two, then |\vec a+\vec b+\vec c|=
(A) \sqrt{338}
(B) 169
(C) 338
(D) 676
Answer : (A)
Question-40 An open tank with a square bottom is to contain 4000 cubic cm. of liquid. The dimensions of the tank so that the surface area of the tank is minimum, is
(A) side =20 cm, height =10 cm
(B) side =10 cm, height =20 cm
(C) side =10 cm, height =40 cm
(D) side =20 cm, height =05 cm
Answer : (A)
Question-41 If four digit numbers are formed by using the digits 1,2,3,4,5,6,7 without repetition, then out of these numbers, the numbers exactly divisible by 25 are
(A) 20
(B) 40
(C) 50
(D) 51
Answer : (B)
Question-42 \int \frac{e^{2x}(\sin2x\cos2x-1)}{\sin^22x}dx=
(A) e^{2x}\cot(2x)+c, where c is the constant of integration
(B) -e^{2x}\cot(2x)+c, where c is the constant of integration
(C) 4e^{2x}\cot(2x)+c, where c is the constant of integration
(D) \frac{1}{2}e^{2x}\cot(2x)+c, where c is the constant of integration
Answer : (D)
Question-43 Three urns respectively contain 2 white and 3 black, 3 white and 2 black and 1 white and 4 black balls. If one ball is drawn from each urn, then the probability that the selection contains 1 black and 2 white balls is
(A) \frac{13}{125}
(B) \frac{37}{125}
(C) \frac{28}{125}
(D) \frac{33}{125}
Answer : (B)
Question-44 The lines x+2ay+a=0,\ x+3by+b=0,\ x+4cy+c=0 are concurrent then a,b,c are in
(A) Harmonic progression
(B) Geometric progression
(C) Arithmetic progression
(D) Arithmetico geometric progression
Answer : (A)
Question-45 In a box containing 100 apples, 10 are defective. The probability that in a sample of 6 apples, 3 are defective is
(A) 0.1548
(B) 0.1458
(C) 0.01854
(D) 0.01458
Answer : (D)
Question-46 The value of the integral \int_{1}^{2}\frac{x\ dx}{(x+2)(x+3)} is
(A) \log\left(\frac{125}{16}\right)
(B) \log\left(\frac{1024}{1125}\right)
(C) \log\left(\frac{16}{125}\right)
(D) \log\left(\frac{1125}{1024}\right)
Answer : (D)
Question-47 The general solution of the differential equation \frac{dy}{dx}+\sin\left(\frac{x+y}{2}\right)=\sin\left(\frac{x-y}{2}\right) is
(A) \log\left(\tan\left(\frac{y}{2}\right)\right)=c-2\sin\left(\frac{x}{2}\right), where c is the constant of integration
(B) \log\left(\tan\left(\frac{y}{4}\right)\right)=c-2\sin\left(\frac{x}{2}\right), where c is the constant of integration
(C) \log\left(\tan\left(\frac{y}{2}+\frac{\pi}{4}\right)\right)=c-2\sin x, where c is the constant of integration
(D) \log\left(\tan\left(\frac{y}{4}+\frac{\pi}{4}\right)\right)=c-2\sin\left(\frac{x}{2}\right), where c is the constant of integration
Answer : (B)
Question-48 Four defective oranges are accidentally mixed with sixteen good ones. Three oranges are drawn from the mixed lot. The probability distribution of defective oranges is
Answer : (B)
Question-49 The equation of the curve passing through (2, \frac{9}{2}) and having the slope \left(1-\frac{1}{x^2}\right) at (x, y) is
(A) xy=x^2+2x+1
(B) xy=x^2+x+2
(C) xy=x^2+x+5
(D) xy=x^2+2x+5
Answer : (A)
Question-50 The projection of the line segment joining P(2,-1,0) and Q(3,2,-1) on the line whose direction ratios are 1,2,2 is
(A) \frac{1}{3}
(B) \frac{2}{3}
(C) \frac{4}{3}
(D) \frac{5}{3}
Answer : (D)