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Question-01 Argument of the complex number z=\frac{13-5i}{4-9i}, i=\sqrt{-1} is
(A) \frac{\pi}{4}
(B) \frac{\pi}{2}
(C) \frac{\pi}{2}
(D) \frac{\pi}{3}
Answer : (A)
Question-02 If \sin\theta=\frac{1}{2}\left(x+\frac{1}{x}\right), then \sin3\theta+\frac{1}{2}\left(x^3+\frac{1}{x^3}\right)=
(A) 0
(B) 1
(C) \frac{1}{4}
(D) 2
Answer : (A)
Question-03 Let A=(0,0), B=(3,0), C=(0,-4) are vertices of \Delta ABC, then the co-ordinates of incentre of \Delta ABC is
(A) (1,-1)
(B) \left(\frac{45}{14},\frac{-3}{14}\right)
(C) \left(\frac{3}{14},\frac{45}{14}\right)
(D) \left(\frac{-3}{14},\frac{45}{14}\right)
Answer : (A)
Question-04 The equations of the tangents to the circle x^2+y^2=36 which are perpendicular to the line 5x+y-2=0 are
(A) x-5y\pm\sqrt{26}=0
(B) x+5y\pm\sqrt{26}=0
(C) x-5y\pm\sqrt{26}=0
(D) x+5y\pm\sqrt{26}=0
Answer : (A)
Question-05 f(x)=\begin{cases}3-x,\ -1\le x<0\\1+\frac{5x}{3},\ -3\le x\le2\end{cases} and
g(x)=\begin{cases}-x,\ -2\le x\le3\\x,\ 0\le x\le1\end{cases}then range of (fog)(x) is
(A) \left[1,\frac{8}{3}\right]
(B) \left[-4,\frac{8}{3}\right]
(C) \left[-4,\frac{13}{3}\right]
(D) \left[\frac{8}{3},\frac{10}{3}\right]
Answer : (C)
Question-06 The foci of the conic 25x^2+16y^2-150x=175 are
(A) (0,\pm3)
(B) (3,\pm3)
(C) (0,\pm5)
(D) (5,\pm5)
Answer : (B)
Question-07 Let A=\lim_{x\to0^+}(1+\tan^2\sqrt{x})^{\frac{1}{2x}}, then \log_e A=
(A) 2
(B) 1
(C) \frac{1}{2}
(D) \frac{1}{4}
Answer : (C)
Question-08 The function f(x)=2x-|x-x^2| is
(A) continuous at x=1
(B) discontinuous at x=1
(C) not defined at x=1
(D) discontinuous at x=0
Answer : (A)
Question-09 Consider the statements given by following
(A) If 4+3=8, then 5+3=9
(B) If 6+4=10, then moon is flat
(C) If both (A) and (B) are true, then 6+5=17
Then which of the following statement is correct?
(A) (A) is true while (B) and (C) are false
(B) (A) and (B) are false, while (C) is true
(C) (A) and (C) are true, while (B) is false
(D) (A) is false, but (B) and (C) are true
Answer : (C)
Question-10 If A=\begin{bmatrix}1&-1&1\\0&2&-3\\2&1&0\end{bmatrix}, B=adjA and C=5A, then \frac{|adjB|}{|C|}=
(A) 2
(B) 4
(C) 1
(D) 5
Answer : (C)
Question-11 The number of values of x in the interval [0,3\pi] satisfying the equation 2\sin^2x+5\sin x-3=0 is
(A) 4
(B) 6
(C) 2
(D) 1
Answer : (A)
Question-12 The equation x^2-3xy+\lambda y^2+3x-5y+2=0, where \lambda is real number represents pair of lines. If \theta is acute angle between the lines, then \frac{cosec^2\theta}{\sqrt{10}} =
(A) 10
(B) \frac{1}{\sqrt{10}}
(C) 2
(D) \sqrt{10}
Answer : (D)
Question-13 If \theta is the angle between the lines whose direction cosines are given by 6mn-2nl+5lm=0 and 3l+m+5n=0, then \sin\theta=
(A) \frac{\sqrt{35}}{6}
(B) \frac{1}{6}
(C) \frac{\sqrt{37}}{6}
(D) \frac{5}{6}
Answer : (A)
Question-14 If X \sim B\left(6,\frac{1}{2}\right), then P(|X-2|\le1) =
(A) \frac{31}{32}
(B) \frac{41}{64}
(C) \frac{51}{64}
(D) \frac{63}{64}
Answer : (B)
Question-15 Let \vec a=\hat i+\hat j,\ \vec b=2\hat i-\hat k,\ \vec c=3\hat i-\hat j+\hat k, then vector \vec p satisfying \vec p\cdot\vec a=0 and \vec p\times\vec b=\vec c\times\vec b is
(A) \hat i-\hat j+\hat k
(B) \hat i-2\hat j+\hat k
(C) -\hat i+\hat j+\hat k
(D) \hat i-\hat j+2\hat k
Answer : (D)
Question-16 A random variable X has following p.d.f. f(x)=kx(1-x),\ 0\le x\le1 and P(x>a)=\frac{20}{27}, then a=
(A) \frac{1}{3}
(B) \frac{2}{3}
(C) \frac{1}{2}
(D) \frac{1}{4}
Answer : (A)
Question-17 The probability distribution of a random variable X is given by
| X = x_i | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| P(X = x_i) | 0.4 | 0.3 | 0.1 | 0.1 | 0.1 |
Then the variance of X is
(A) 1.76
(B) 2.45
(C) 3.2
(D) 4.8
Answer : (A)
Question-18 If (\cos^{-1}x)^2-(\sin^{-1}x)^2>0, then
(A) x<\frac{1}{2}
(B) -1<x<\sqrt{2}
(C) 0\le x<\frac{1}{\sqrt{2}}
(D) -1\le x<\frac{1}{\sqrt{2}}
Answer : (D)
Question-19 If \vec a=2\hat i+3\hat j+4\hat k,\ \vec b=\hat i-2\hat j-2\hat k,\ \vec c=-\hat i+4\hat j+3\hat k and if \vec d is vector perpendicular to both \vec b and \vec c, \vec a\cdot\vec d=18, then |\vec a\times\vec d|^2=
(A) 640
(B) 680
(C) 720
(D) 740
Answer : (C)
Question-20 The rate of change of volume of spherical balloon at any instant is directly proportional to its surface area. If initially its radius is 3 cm, after 2 minutes its radius becomes 9 cm, then radius of balloon after 4 minutes is
(A) 12 cm
(B) 14 cm
(C) 15 cm
(D) 18 cm
Answer : (C)
Question-21 The line passing through the points (a, 1, 6) and (3, 4, b) crosses the yz-plane at \left(0,\frac{17}{2},-\frac{13}{2}\right), then the value of (3a + 4b) is
(A) 19
(B) 16
(C) 21
(D) 23
Answer : (A)
Question-22 Solution of (2y-x)frac{dy}{dx}=1 is
(A) x=2(y-1)+ce^{-y}
(B) x=2(y-1)+ce^{-x}
(C) y=2(x-1)+ce^{-x}
(D) y=2(x-1)+ce^{-y}
Answer : (A)
Question-23 The integrating factor of y+\frac{d}{dx}(xy)=x(\sin x+\log x) is
(A) x
(B) \log x^2
(C) x^2
(D) x^3
Answer : (C)
Question-24 The differential equation whose solution is Ax^2+By^2=1, where A and B are arbitrary constants is of
(A) degree 1 and order 2
(B) degree 2 and order 1
(C) degree 3 and order 2
(D) degree 1 and order 3
Answer : (A)
Question-25 The angle between the lines x = y, z = 0 and y = 0, z = 0 is
(A) 30°
(B) 45°
(C) 60°
(D) 90°
Answer : (B)
Question-26 Let \vec a,\vec b,\vec c be three vectors such that \vec a+\vec b+\vec c=0, |\vec a|=3,\ |\vec b|=4,\ |\vec c|=5, then \vec a\cdot\vec b+\vec b\cdot\vec c+\vec c\cdot\vec a=
(A) 25
(B) -25
(C) 50
(D) -50
Answer : (C)
Question-27 The area bounded by the curve y=x^2+3, y=x, x=3 and y-axis is
(A) \frac{9}{2}\ sq.\ units
(B) 18 sq. units
(C) \frac{27}{2}\ sq.\ units
(D) \frac{27}{3}\ sq.\ units
Answer : (C)
Question-28 \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(x^2+\log\left(\frac{\pi-x}{\pi+x}\right)\cos x\right)dx
(A) 0
(B) \frac{\pi^3}{12}
(C) \frac{\pi^2}{2}-4
(D) \frac{\pi^2}{2}+4
Answer : (B)
Question-29 \int_{0}^{1}\log\left(\frac{1}{x}-1\right)dx=
(A) \frac{1}{2}
(B) 1
(C) 2
(D) 0
Answer : (D)
Question-30 If the foot of the perpendicular drawn from the origin to a plane is P(2,-1,4), then the equation of the plane is
(A) 2x+y+4z-19=0
(B) x+y+z-5=0
(C) 2x-2y-3z+6=0
(D) 2x-y+4z-21=0
Answer : (D)
Question-31 \int\frac{(5\sin\theta-2)\cos\theta}{(5-\cos^2\theta-4\sin\theta)}d\theta=
(A) \log(5\sin\theta-2)+c
(B) 5\log(5\sin\theta-2)-\frac{8}{(\sin\theta-2)}+c
(C) \log(5\sin\theta-2)+\frac{8}{(\sin\theta-2)}+c
(D) \log(5\sin\theta-2)+\frac{1}{(\sin\theta-2)}+c
Answer : (B)
Question-32 Number of switches in alternative equivalent simple circuit for the circuit is (are)
(A) 0
(B) 1
(C) 2
(D) 3
Answer : (B)
Question-33 If 0\le\cos^{-1}x\le\pi and -\frac{\pi}{2}\le\sin^{-1}x\le\frac{\pi}{2}, then at x=\frac{1}{5} the value of \cos(2\cos^{-1}x+\sin^{-1}x) is
(A) -\sqrt{\frac{24}{25}}
(B) \sqrt{\frac{24}{25}}
(C) \frac{\sqrt{24}}{25}
(D) -\frac{\sqrt{24}}{25}
Answer : (A)
Question-34 In a triangle ABC with usual notations if b\sin C(b\cos C+c\cos B)=42, then area of triangle ABC =
(A) 42 sq. units
(B) 21 sq. units
(C) 24 sq. units
(D) 12 sq. units
Answer : (B)
Question-35 \int\frac{x}{1+x^4}dx=
(A) \frac{1}{2}\tan^{-1}x^2+c
(B) 2\tan^{-1}x+c
(C) \frac{1}{2}\tan^{-1}x+c
(D) \tan^{-1}x^2+c
Answer : (A)
Question-36 Let the plane passing through point (2, 1, -1)containing line joining the points (1, 3, 2) and (1, 2, 1) makes intercepts p,q,r on co-ordinate axes, then p+q+r=
(A) 0
(B) 3
(C) 2
(D) -2
Answer : (A)
Question-37 \int\sqrt{x^2+3x}dx=
(A) \sqrt{x^2+3x}+\log\sqrt{x^2+3x}+c
(B) \frac{2x+3}{4}\sqrt{x^2+3x}-\frac{9}{8}\log(x+\sqrt{x^2+3x})+c
(C) x\sqrt{x^2+3x}+\log(x+\sqrt{x^2+3x})+c
(D) \frac{2x+3}{4}\sqrt{x^2+3x}-\frac{9}{8}\log(x+\frac{3}{2}+\sqrt{x^2+3x})+c
Answer : (D)
Question-38 If the line ax+by+c=0 is normal to the curve xy=1, then
(A) a>0,\ b>0
(B) a>0,\ b<0
(C) a<0,\ b\ge0
(D) a<0,\ b<0
Answer : (B)
Question-39 The sum of two nonzero numbers is 4. The minimum value of the sum of their reciprocals is
(A) \frac{3}{4}
(B) \frac{6}{5}
(C) 1
(D) \frac{4}{5}
Answer : (C)
Question-40 The combined equation of the tangent and normal to the curve xy=15 at the point (5, 3) is
(A) 15x^2-15y^2+16xy=480
(B) 15x^2+16xy-198x+10y+480-15y^2=0
(C) 15x^2-16xy+9x-10y-480+15y^2=0
(D) 15x^2+16xy+198x-10y-480+15y^2=0
Answer : (B)
Question-41 The angle between the line x=\frac{y-1}{2}=\frac{z-3}{\lambda} and the plane x+2y+3z=6 is \cos^{-1}\sqrt{\frac{5}{14}}, then the value of \lambda is
(A) \frac{2}{3}
(B) \frac{4}{3}
(C) \frac{1}{3}
(D) \frac{5}{3}
Answer : (A)
Question-42 The length and breadth of a rectangle are x cm and y cm respectively. If the length decreases at the rate of 5 cm/minute and the breadth increases at the rate of 3 cm/minute, then the rates of change of the perimeter and area respectively when the length is 5 cm and breadth is 2 cm, are
(A) -4 and 5
(B) -5 and 3
(C) 3 and 5
(D) 3 and -5
Answer : (A)
Question-43 If f(x)=3x^3+2x^2f(1)+xf''(2)+f'''(3) then f(x)=
(A) \frac{1}{7}(3x^3-90x^2+72x+18)
(B) \frac{1}{7}(21x^3-90x^2+72x+126)
(C) 3x^3-90x^2+72x+18
(D) 3x^3-45x^2+36x+9
Answer : (B)
Question-44 If x=\sin\theta,\ y=\sin^3\theta, then \frac{d^2y}{dx^2} at \theta=\frac{\pi}{6} is
(A) \frac{1}{2}
(B) \frac{\sqrt{3}}{2}
(C) 3
(D) 6
Answer : (C)
Question-45 If x = e^{\tan^{-1}\left(\frac{y - x^{2}}{x^{2}}\right)}, then \frac{dy}{dx} at x=1 is
(A) 1
(B) 0
(C) 2
(D) 3
Answer : (D)
Question-46 The difference between the maximum value and minimum value of objective function z=3x+5y subject to constraints x+3y\le60,\ x+y\ge10,\ x-y\ge0,\ x,y\ge0 is
(A) 60
(B) 20
(C) 40
(D) 80
Answer : (D)
Question-47 If \vec a=\frac{1}{\sqrt{10}}(3\hat i+\hat k) and \vec b=\frac{1}{7}(2\hat i+3\hat j-6\hat k), then the value of (2\vec a-\vec b)\cdot((\vec a\times\vec b)\times(\vec a+2\vec b)) =
(A) 3
(B) -3
(C) 5
(D) -5
Answer : (D)
Question-48 In a triangle with one of the angles 120^\circ, the lengths of the sides form an A.P. If length of the greatest side is 7 m, then the area of the triangle is
(A) \frac{15\sqrt{3}}{4}\ m^2
(B) \frac{15\sqrt{3}}{2}\ m^2
(C) \frac{15}{2}\ m^2
(D) \frac{15}{4}\ m^2
Answer : (A)
Question-49 If ^{15}C_4+^{15}C_5+^{16}C_6+^{17}C_7+^{18}C_8=^{19}C_r, then the value of r is equal to
(A) 9 or 10
(B) 7 or 12
(C) 8 or 10
(D) 8 or 11
Answer : (D)
Question-50 If A and B are independent events such that P(A\cap B')=\frac{3}{25} and P(A'\cap B)=\frac{8}{25}, then P(A)=
(A) \frac{3}{8}
(B) 4
(C) \frac{1}{5}
(D) \frac{2}{5}
Answer : (C)