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Question-01 A fair coin is tossed 100 times. The chance of getting a head even number of times is
(A) \frac{1}{8}
(B) \frac{3}{8}
(C) \frac{1}{2}
(D) \frac{3}{4}
Answer : (C)
Question-02 Let \vec a,\vec b,\vec c,\vec d are vectors such that \vec a\times\vec b=2\hat i+3\hat j-\hat k and \vec c\times\vec d=3\hat i+2\hat j+\lambda\hat k and if \begin{vmatrix} \vec{a}\cdot\vec{c} & \vec{b}\cdot\vec{c} \\ \vec{a}\cdot\vec{d} & \vec{b}\cdot\vec{d} \end{vmatrix} = 0
then \lambda=
(A) 6
(B) -6
(C) 12
(D) -12
Answer : (C)
Question-03 In \triangle ABC, with usual notations, if \cos\frac{B}{2}=\frac{c+a}{2a}, then a^2=
(A) b^2-c^2
(B) b+c
(C) b^2+c^2
(D) b-c
Answer : (C)
Question-04 The joint equation of two lines passing through (-2,3) and parallel to the bisectors of the angle between the co-ordinate axes is
(A) x^2-y^2+4x+6y-4=0
(B) x^2+y^2+4x+6y-5=0
(C) x^2-y^2+4x+6y-5=0
(D) x^2+y^2+4x+6y+4=0
Answer : (C)
Question-05 Let the circle with centre at origin pass through the vertices of an equilateral triangle ABC. If A=(2,4), then the length of the median through A is
(A) 2\sqrt{5} units
(B) 3\sqrt{5} units
(C) 4\sqrt{5} units
(D) 6\sqrt{5} units
Answer : (B)
Question-06 Let \vec a=\hat i+\hat j+\hat k,\ \vec b and \vec c=\hat j-\hat k be three vectors such that \vec a\times\vec b=\vec c and \vec a\cdot\vec c=1. If the length of projection vector of the vector \vec b on the vector \vec a\times\vec c is l, then the value of 3l^2 is
(A) 1
(B) 2
(C) 4
(D) 6
Answer : (B)
Question-07 The distance of the point (1,2) from the line x+y=0 measured parallel to the line 3x-y=2 is
(A) \frac{3\sqrt{2}}{8} units
(B) \frac{3\sqrt{10}}{4} units
(C) 10 units
(D) 5\sqrt{5} units
Answer : (B)
Question-08 \lim_{x\to2}\frac{x+3x^2+5x^3+7x^4-166}{x-2}=
(A) 167
(B) 267
(C) 287
(D) 297
Answer : (D)
Question-09 If f(x)=\frac{10^x+7^x-14^x-5^x}{1-\cos x},\ x\ne0 is continuous at x=0, then the value of f(0) is
(A) \log2\left[\log\left(\frac{5}{7}\right)\right]
(B) \log4\left[\log\left(\frac{5}{7}\right)\right]
(C) \log2\left[\log\left(\frac{7}{5}\right)\right]
(D) \log4\left[\log\left(\frac{7}{5}\right)\right]
Answer : (B)
Question-10 If A and B are non-singular matrices of order 2 such that (AB)^{-1}=\frac{1}{6}\begin{bmatrix}-7&-3\\2&3\end{bmatrix} and A^{-1}=\frac{1}{3}\begin{bmatrix}4&3\\-1&0\end{bmatrix} then B^{-1}=
(A) \frac{1}{2}\begin{bmatrix}2&3\\1&-1\end{bmatrix}
(B) \frac{1}{2}\begin{bmatrix}3&1\\2&4\end{bmatrix}
(C) \frac{1}{2}\begin{bmatrix}-1&3\\1&2\end{bmatrix}
(D) \frac{1}{6}\begin{bmatrix}1&1\\2&3\end{bmatrix}
Answer : (C)
Question-11 If \sin A+\sin B=x and \cos A+\cos B=y, then \sin(A+B)=
(A) \frac{2xy}{x^2+y^2}
(B) \frac{xy}{x^2+y^2}
(C) \frac{2xy}{y^2-x^2}
(D) \frac{xy}{y^2-x^2}
Answer : (A)
Question-12 Let mean and standard deviation of probability distribution
| X = x | -3 | 0 | 1 | \alpha |
|---|---|---|---|---|
| P(X = x) | \frac{1}{4} | K | \frac{1}{4} | \frac{1}{3} |
be \mu and \sigma respectively and if \sigma-\mu=2 then \sigma=
(A) \frac{3}{2}
(B) \frac{5}{2}
(C) \frac{7}{2}
(D) \frac{9}{2}
Answer : (C)
Question-13 In a game a man wins ₹ 40 if he gets 5 or 6 on a throw of a fair die and loses ₹ 20 for getting any other number on the die. If he decides to throw the die either till he gets a five or six or to a maximum of three throws, then his expected maximum gain/loss (in rupees) is
(A) -10
(B) 10
(C) 0
(D) 1
Answer : (C)
Question-14 The rate at which a substance cools in moving air, is proportional to the difference between the temperature of the substance and that of air. The temperature of air is 290\ K and the substance cools from 370\ K to 330\ K in 10 minutes. Then the time to cool the substance upto 295\ K is
(A) 40 min
(B) 95 min
(C) 50 min
(D) 60 min
Answer : (A)
Question-15 If x\frac{dy}{dx}=y(\log y-\log x+1), then the solution of the equation is
(A) \log\frac{x}{y}=cy, where c is the constant of integration
(B) \log\frac{y}{x}=cy, where c is the constant of integration
(C) \log\frac{x}{y}=cx, where c is the constant of integration
(D) \log\frac{y}{x}=cx, where c is the constant of integration
Answer : (D)
Question-16 The order and degree of the differential equation \sqrt{\frac{dy}{dx}}-4\frac{dy}{dx}-7x=0 is respectively
(A) 1,2
(B) 2,1
(C) 2,2
(D) 3,1
Answer : (A)
Question-17 The differential equation of all circles touching the Y-axis at the origin and centre on the X-axis is
(A) x^2+y^2+2xy\frac{dy}{dx}=0
(B) x^2-y^2+2xy\frac{dy}{dx}=0
(C) 2x^2+y^2+xy\frac{dy}{dx}=0
(D) x^2-2y^2+2xy\frac{dy}{dx}=0
Answer : (B)
Question-18 The area bounded by the curve x^2=8y and the straight line x-8y+2=0 is
(A) \frac{9}{8} sq. units
(B) \frac{15}{16} sq. units
(C) \frac{9}{16} sq. units
(D) \frac{15}{8} sq. units
Answer : (C)
Question-19 In a triangle ABC with usual notations,
\cot\frac{A}{2}+\cot\frac{B}{2}+\cot\frac{C}{2}=(A) \frac{s^2}{\Delta}, where \Delta is the area of the triangle ABC
(B) \frac{s}{\Delta}, where \Delta is the area of the triangle ABC
(C) \frac{\Delta}{s}, where \Delta is the area of the triangle ABC
(D) \Delta, where \Delta is the area of the triangle ABC
Answer : (A)
Question-20 The area of the rectangle having vertices P,Q,R,S with position vectors \hat i+\hat j+\hat k,\ \hat i+\hat j+\hat k,\ \hat i-\hat j+\hat k,\ -\hat i-\hat j+\hat k respectively is
(A) 1 square unit
(B) 2 square units
(C) 3 square units
(D) 4 square units
Answer : (D)
Question-21 The value of \int_{0}^{1}\tan^{-1}(1-x+x^2)\,dx is
(A) \frac{\pi}{2}-\log2
(B) \frac{\pi}{2}+\log2
(C) \log2
(D) 0
Answer : (C)
Question-22 \int_{3}^{5}\frac{\sqrt{x}\,dx}{\sqrt{8-x}+\sqrt{x}}=
(A) 0
(B) 1
(C) 2
(D) 3
Answer : (B)
Question-23 \int\frac{dx}{\sqrt{x}+x}=
(A) \log\sqrt{x}+c, where c is the constant of integration
(B) \log(\sqrt{x}+x)+c, where c is the constant of integration
(C) \log(1+\sqrt{x})+c, where c is the constant of integration
(D) 2\log(1+\sqrt{x})+c, where c is the constant of integration
Answer : (D)
Question-24 With usual notations, in \triangle ABC, the lengths of two sides are 10 cm and 9 cm respectively. If angles A,B,C are in A.P. then perimeter of \triangle ABC is
(A) 24+2\sqrt{6} cm
(B) 24+\sqrt{6} cm
(C) 24-2\sqrt{6} cm
(D) 22-\sqrt{6} cm
Answer : (B)
Question-25 \int \frac{dx}{e^x-1}=
(A) \log(e^x-1)+x+c, where c is the constant of integration.
(B) \log(e^x-1)-x+c, where c is the constant of integration.
(C) x-\log(e^x-1)+c, where c is the constant of integration.
(D) \log(e^x-1)-xe^x+c, where c is the constant of integration.
Answer : (B)
Question-26 The maximum value of x^{\frac{2}{3}}+(x-2)^{\frac{2}{3}} is
(A) 0
(B) 2
(C) 2^{\frac{2}{3}}
(D) 1
Answer : (B)
Question-27 \int\left(\frac{x-3}{x^2+9}\right)^2 dx=
(A) \frac{1}{3}\tan^{-1}\left(\frac{x}{3}\right)-\frac{3}{x^2+9}+c, where c is the constant of integration.
(B) \frac{1}{3}\tan^{-1}\left(\frac{x}{3}\right)-\frac{1}{x^2+9}+c, where c is the constant of integration.
(C) \frac{1}{3}\tan^{-1}\left(\frac{x}{3}\right)+\frac{3}{x^2+9}+c, where c is the constant of integration.
(D) \frac{1}{3}\tan^{-1}\left(\frac{x}{3}\right)-\frac{1}{x^2+9}+c, where c is the constant of integration.
Answer : (C)
Question-28 The point on the curve 4y^2-4y+2x-1=0 at which the tangent becomes parallel to Y-axis is
(A) (1, \frac{1}{2})
(B) (\frac{1}{2}, 1)
(C) (-1, -\frac{1}{2})
(D) (\frac{1}{2}, 0)
Answer : (A)
Question-29 The value of \tan^2(\sec^{-1}4)+\cot^2(\cosec^{-1}3) is
(A) 15
(B) 25
(C) 23
(D) 7
Answer : (C)
Question-30 In a triangle ABC, with usual notations, if a=5,\ b=4,\ \cos(A-B)=\frac{31}{32}, then c=
(A) 6
(B) 7
(C) 5
(D) 2
Answer : (A)
Question-31 Which of the following is the negation of the statement “For all M>0, there exist x\in S such that x\ge M“?
(A) \exists M>0\ \text{such that}\ x\ge M\ \text{for all}\ x\in S
(B) \exists M>0,\ \exists x\in S\ \text{such that}\ x\ge M
(C) \exists M>0\ \text{such that}\ x<M\ \text{for all}\ x\in S
(D) \exists M>0,\ \exists x\in S\ \text{such that}\ x<M
Answer : (C)
Question-32 The equation of the line passing through the point of intersection of \frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4} and \frac{x-4}{5}=\frac{y-1}{2}=z and also through the point (2,1,-2) is
(A) \vec r=(-\hat i-\hat j-\hat k)+\lambda(\hat i+2\hat j+\hat k)
(B) \vec r=(-\hat i-\hat j+\hat k)+\lambda(2\hat i+2\hat j+\hat k)
(C) \frac{x+1}{3}=\frac{y+1}{2}=\frac{z+1}{-1}
(D) \frac{x-1}{3}=\frac{y-1}{2}=\frac{z+1}{1}
Answer : (C)
Question-33 A particle moves along a curve y=\frac{2x^3-1}{3}. The points on the curve at which the y-coordinate is changing 18 times the x-coordinate are
(A) (-3, -\frac{55}{3}) , \ (3,-\frac{53}{3})
(B) (-3, \frac{53}{3}) , \ (3,\frac{55}{3})
(C) (-3, -\frac{53}{3}) , \ (3,\frac{55}{3})
(D) (-3, -\frac{55}{3}) , \ (3,\frac{53}{3})
Answer : (D)
Question-34 The equation of motion of the particle is s=at^2+bt+c. If the displacement after 1 second is 20 m, velocity after 2 seconds is 30 m/s and the acceleration is 10 m/s², then
(A) a+c=2b
(B) a+c=b
(C) a-c=b
(D) a+c=3b
Answer : (B)
Question-35 If y=\tan^{-1}\left(\frac{4\sin2x}{\cos2x-6\sin^2x}\right), then \frac{dy}{dx} at x=0 is
(A) \frac{1}{8}
(B) -8
(C) 8
(D) -\frac{1}{8}
Answer : (C)
Question-36 If x=a\sin2t(1+\cos2t),\ y=b\cos2t(1-\cos2t), then \frac{dy}{dx} is equal to
(A) \frac{b}{a}\tan t
(B) \frac{a}{b}\tan t
(C) \frac{b}{a\tan t}
(D) \frac{a}{b\tan t}
Answer : (A)
Question-37 The contrapositive of the statement \sim p\lor(q\land\sim r) is
(A) p\rightarrow(q\land r)
(B) (q\land r)\rightarrow p
(C) \sim q\lor\sim r\rightarrow p
(D) (r\lor\sim q)\rightarrow\sim p
Answer : (D)
Question-38 If the lines \frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4} and \frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1} intersect, then the value of k is
(A) \frac{3}{2}
(B) -\frac{3}{2}
(C) \frac{9}{2}
(D) -\frac{2}{9}
Answer : (C)
Question-39 If y=\sqrt{x+\sqrt{y+\sqrt{x+\sqrt{y+\dots\infty}}}}, then \frac{dy}{dx}=
(A) 2y-1
(B) \frac{1}{2y-1}
(C) \frac{y^2-x}{2y^3-2xy-1}
(D) ^{14}C_6
Answer : (C)
Question-40 The graph with correct feasible region of L.P.P. for the constraints 2x+y\le10,\ y\le x,\ y\le2x,\ y\ge0 is
Answer : (A)
Question-41 The distance of the point P(3,4,4) from the point of intersection of the line joining the points Q(3,-4,-5),\ R(2,-3,1) and the plane 2x+y+z=7 is
(A) 7 units
(B) 9 units
(C) 11 units
(D) 6 units
Answer : (A)
Question-42 If f(x)=3x+10,\ g(x)=x^2-1, then (fog)^{-1}(x)=
(A) \frac{x-7}{3}
(B) \left(\frac{x-7}{3}\right)^{\frac{1}{2}}
(C) \left(\frac{x-7}{3}\right)^{\frac{1}{3}}
(D) \left(\frac{3}{x-7}\right)^{\frac{3}{2}}
Answer : (B)
Question-43 A family consisting of a mother, father and their 8 children (4 boys and 4 girls) are to be seated at a round table in a party. How many ways can this be done if the mother and father sit together and the males and females alternate?
(A) 567
(B) 765
(C) 657
(D) 576
Answer : (D)
Question-44 If \vec a,\vec b,\vec c are three vectors such that |\vec a|=\sqrt{31},\ 4|\vec b|=|\vec c|=2 and 2(\vec a\times\vec b)=3(\vec c\times\vec a) and if the angle between \vec b and \vec c is \frac{2\pi}{3} then
\left|\frac{\vec a\times\vec c}{\vec a\cdot\vec b}\right|^2=(A) 1
(B) 2
(C) 3
(D) 11
Answer : (C)
Question-45 Let \vec a=2\hat i+\hat j+\hat k,\ \vec b=\hat i+2\hat j-\hat k and vector \vec c be coplanar. If \vec c is perpendicular to \vec a, then \vec c is
(A) -\hat i+2\hat k
(B) -\hat i+\hat j+\hat k
(C) \hat i-2\hat j
(D) -\hat j+\hat k
Answer : (D)
Question-46 If x=-2+\sqrt{3}, then the value of 2x^4+5x^3+7x^2-x+38 is
(A) 1
(B) -2
(C) 3
(D) 5
Answer : (C)
Question-47 Bag I contains 3 red and 2 green balls and Bag II contains 5 red and 3 green balls. A ball is drawn from one of the bag at random and it is found to be green. Then the probability that it is drawn from Bag I is
(A) \frac{8}{31}
(B) \frac{12}{31}
(C) \frac{14}{31}
(D) \frac{16}{31}
Answer : (D)
Question-48 If the tangent at the point (2\sec\theta,3\tan\theta) to the hyperbola \frac{x^2}{4}-\frac{y^2}{9}=1 is parallel to 3x-y+4=0, then the value of \theta is
(A) 45^\circ
(B) 60^\circ
(C) 30^\circ
(D) 90^\circ
Answer : (C)
Question-49 The equation of the plane containing the line \frac{x}{1}=\frac{y}{2}=\frac{z}{3} and perpendicular to the plane containing the lines \frac{x}{2}=\frac{y}{3}=\frac{z}{1} and \frac{x}{3}=\frac{y}{2}=\frac{z}{1} is
(A) x-13y+z=0
(B) 13x-8y+5z=0
(C) 13x-8y+z=0
(D) 13x-y+z=0
Answer : (C)
Question-50 The equation of the plane containing the line \frac{x+1}{2}=\frac{y+2}{1}=\frac{z-2}{3} and the point (1,-1,3) is
(A) x-2y-3=0
(B) 2x+y-1=0
(C) 3x-2z+3=0
(D) 2x-y-z=0
Answer : (A)