1. If 2sin^-1x - cos^-1x = $\frac{\pi}{2}$, then x is equal to
a) 1/$\sqrt{2}$
b) -1/$\sqrt{2}$
c) $-\sqrt{3/2}$
d) $\sqrt{3}2$
e) 1/2

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2. The solution of sin^-1x - sin^-12x = $\frac{\pm \pi}{3}$ is
a) $\pm\frac{1}{3}$
b) $\pm\frac{1}{4}$
c) $\pm \frac{\sqrt{3}}{2}$
d) $\pm\frac{1}{2}$

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1. If a ≠ p, b ≠ 1, c ≠ r and $\begin{vmatrix} p& q& r\\ p+a& q+b &2c \\ a & b &r \end{vmatrix}$ = 0, then $\frac{p}{p - a} + \frac{q}{q - b} + \frac{r}{r - c}$ =
a) 3
b) 2
c) 1
d) 0

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2. l,m,n are the p^th, q^th and r^th term of a G.P. , all positive then $\begin{vmatrix} log l & p &1 \\ logm& q & 1\\ log n& r& 1 \end{vmatrix}$ equals
a) -1
b) 2
c) 1
d) 0

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1. The value of $lim_{x \rightarrow 0}\frac{\sqrt{x^{2} + 1} - 1}{\sqrt{x^{2} + 9} - 3}$ is
a) 3
b) 4
c) 1
d) 2

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2. The function f(x) = $\frac{1 - sinx + cosx}{1 + sin x cosx}$ is not defined at x = π. The value of f(π) , so that f(x) is continuous at x = π, is
a) -1/2
b) 1/2
c) -1
d) 1

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1. The maximum value of xe^-x is
a) $-\frac{1}{e}$
b) e
c) $\frac{1}{e}$
d) -e

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2. The derivative of f(x) = |x² - x| at x = 2 is
a) -3
b) 0
c) 3
d) Not defined

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1. The part of circle x² + y² = 9 in between y = 0 and y = 2 is revolved about y -axis. The volume of generating solid will
a) $\frac{46}{3}\pi$
b) 12π
c) 16π
d) 28π

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2. The value of the definite integral $\int_{0}^{1}\frac{xdx}{x^{3} + 16}$ lies in the interval [a,b]. The smallest such interval is
a) $\left [0,\frac{1}{17} \right ]$
b) [0,1]
c) $\left [0,\frac{1}{27} \right ]$
d) None of these

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1. Solution of $\frac{dy}{dx}$ = $\frac{xlogx^{2} + x}{siny + ycosy}$ is
a) ysiny = x²logx + c
b) ysiny = x² + c
c) ysiny = x² + logx + c
d) ysiny = xlogx + c

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2. The order and degree of the differential equation $x\left (\frac{d^{3}y}{dx^{3}} \right )^{2} + y\left (\frac{dy}{dx} \right )^{4} + y^{2}$ = 0 is
a) Order2, degree 3
b) Order 3, degree 2
c) Order 2, degree 2
d) None of these

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1. If the position vectors of A, B, C, D are 2i + j, i - 3j, 3i + 2j and i + λj respectively and $\overrightarrow{AB}$ || $\overrightarrow{CD}$, then λ will be
a) -8
b) -6
c) 8
d) 6

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2. If ABCDEF is regular hexagon, then $\overrightarrow{AD}$ + $\overrightarrow{EB}$ + $\overrightarrow{FC}$ =
a) 0
b) 2$\overrightarrow{AB}$
c) $3\overrightarrow{AB}$
d) $4\overrightarrow{AB}$

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1. In the space the equation by + cz + d = 0 represents a plane perpendicular to the plane
a) YOZ
b) Z = k
c) ZOX
d) XOY

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2. From a point P(λ, λ, λ) perpendiculars PQ and PR are drawn respectively on the lines y = x, z = 1 and y = -x , z = -1. If P is such that ∠QPR is a right angle, then the possible value (s) of λ is (are)
a) $\sqrt{2}$
b) 1
c) -1
d) $-\sqrt{2}$

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1. The point which provides the solution of the linear programming problem Max. (45x + 55y) subject to constraint x,y ≥ 0.6x + 4y ≥ 120 . 3x + 10y ≤ 180 is
a) (15,10)
b) (10,15)
c) (0,18)
d) (20,0)

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2. The maximum value of z = 4x + 2y subject to the constraints 2x + 3y ≤ 18, x + y ≥ 10 ; x , y ≥ 0 is
a) 36
b) 40
c) 20
d) None of these

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1. One bag contains 5 white and 4 black balls. Another bag contains 7 white and 9 black balls . A balls is transferred from the first bag to the second and then a ball is drawn from second. The probability that the ball is white, is
a) 8/17
b) 40/153
c) 5/9
d) 4/9

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2. Let A and B be events for which P(A) = x, P(B) = y, P(A ∩ B) = z, then P($\bar{A}$ ∩ B) equals
a) (1 - x)y
b) 1 - x + y
c) y - z
d) 1 - x + y - z

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1. $\int \frac{\sqrt{x}}{1 + x}dx$ =
a) $\sqrt{x} - tan^{-1}\sqrt{x} + c$
b) $2\left (\sqrt{x} - tan^{-1}\sqrt{x} \right ) + c$
c) $2\left (\sqrt{x} + tan^{-1}\sqrt{x} \right ) + c$
d) $\sqrt{1 + x}+ c$

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2. $\int \frac{cos2x + x + 1}{x^{2} + sin2x + 2x}dx$ =
a) $log\left (x^{2} + sin2x + 2x \right ) + c$
b) $-log\left (x^{2} + sin2x + 2x \right ) + c$
c) $\frac{1}{2}log\left (x^{2} + sin2x + 2x \right ) + c$
d) None of these

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1. The value of the integral $\int_{-\pi/2}^{\pi/2}\left (x^{2} + ln\frac{\pi + x}{\pi - x} \right )cosx dx$ is
a) 0
b) $\frac{\pi^{2}}{2} - 4$
c) $\frac{\pi^{2}}{2} + 4$
d) $\frac{\pi^{2}}{2}$

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2. The area of the region between the curves y = $\sqrt{\frac{1 + sinx}{cosx}}$ and y = $\sqrt{\frac{1 - sinx}{cosx}}$ bounded by the lines x = 0 and x = $\frac{\pi}{4}$ is
a) $\int_{0}^{\sqrt{2}-1}\frac{t}{\left (1 + t^{2} \right )\sqrt{1 - t^{2}}}dt$
b) $\int_{0}^{\sqrt{2}-1}\frac{4t}{\left (1 + t^{2} \right )\sqrt{1 - t^{2}}}dt$
c) $\int_{0}^{\sqrt{2}+1}\frac{4t}{\left (1 + t^{2} \right )\sqrt{1 - t^{2}}}dt$
d) $\int_{0}^{\sqrt{2}+1}\frac{t}{\left (1 + t^{2} \right )\sqrt{1 - t^{2}}}dt$

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