1. If (tan^-1 x)² + (cot^-1x)² = $\frac{5\pi^{2}}{8}$, then x equals
a) -1
b) 1
c) 0
d) None of these

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2. If sin^-1x = $\frac{\pi}{5}$ for some x ε (-1,1), then the value of cos^-1x is
a) $\frac{3\pi}{10}$
b) $\frac{5\pi}{10}$
c) $\frac{7\pi}{10}$
d) $\frac{9s\pi}{10}$

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1. Let M and N be two even order non-singular skew-symmetric matrices such that MN = NM . If P^T denotes the transpose of P, then M²N²(M^TN)^-1(MN^-1)^T is equal to
a) M²
b) -N²
c) -M²
d) MN

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2. The sum of the products of the elements of any row of a determinant A with the co-factor of same row is always equal to
a) 1
b) 0
c) |A|
d) $\frac{1}{2}|A|$

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1. $lim_{n \rightarrow \infty} \left (\frac{n^{2} - 1 + 1}{n^{2} - n - 1} \right )^{n\left (n - 1 \right )}$ =
a) e
b) e²
c) 2^-1
d) 1

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2. The function f(x) = $log\left (\frac{1 + x}{1 - x} \right )$ satisfies the equation
a) f(x + 2) - 2f(x + 1) + f(x) = 0
b) f(x) + f(x + 1) = f(x(x + 1))
c) f(x) + f(y) = $f\left (\frac{x + y}{1 + xy} \right )$
d) f(x + y) = f(x)f(y)

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1. The radius of a cylinder is increasing at the rate of 3 m/sec and its altitude is decreasing at the rate of 4m/sec. The rate of change of volume when radius is 4 metres and altitude is 6 metres is
a) 80π cu. m/sec
b) 144π cu.m/sec
c) 80 cu. m/sec
d) 64 cu. m/sec
e) -80-π cu. m/sec

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2. If y = $e^{\sqrt{x}}$, then $\frac{dy}{dx}$ equals
a) $\frac{e^{\sqrt{x}}}{2\sqrt{x}}$
b) $\frac{\sqrt{x}}{e^{\sqrt{x}}}$
c) $\frac{x}{e^{\sqrt{x}}}$
d) $\frac{2\sqrt{x}}{e^{\sqrt{x}}}$

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1. $lim_{n \rightarrow \infty}\frac{1 + 2^{4} + 3^{4} + .... + n^{4}}{n^{5}}$ - $lim_{n \rightarrow \infty} \frac{1 + 2^{3} + 3^{3} + .... + n^{3}}{n^{5}}$ =
a) $\frac{1}{30}$
b) Zero
c) $\frac{1}{4}$
d) $\frac{1}{5}$

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2. Let I = $\int_{0}^{1}\frac{sinx}{\sqrt{x}}dx$ and J = $\int_{0}^{1}\frac{cosx}{\sqrt{x}}dx$. Then which one of the following is true
a) I < $\frac{2}{3}$ and J < 2
b) I < $\frac{2}{3}$ and J > 2
c) I > $\frac{2}{3}$ and J < 2
d) I > $\frac{2}{3}$ and J > 2

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1. To reduce the differential equation $\frac{dy}{dx}$ + P(x)y = Q(x).yⁿ to the linear form, the substitution is
a) v = $\frac{1}{y^{n}}$
b) v = $\frac{1}{y^{n-1}}$
c) v = yⁿ
d) v = y^n-1

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2. If the integrating factor of the differential equation $x\frac{dy}{dx}$ + my = x²e^x is x^-2, the value of m is
a) -1
b) 1
c) 2
d) -2

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1. If a ≠ 0, b ≠ 0 and |a + b| = |a - b|, then the vectors a and b are
a) Parallel to each other
b) Perpendicular to each other
c) Inclined at an angle of 60⁰
d) Neither perpendicular nor parallel

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2. If a and b are two non-zero vectors, then the component of b along a is
a) $\frac{\left (a.b \right )a}{b.b}$
b) $\frac{\left (a.b \right )b}{a.a}$
c) $\frac{\left (a.b \right )b}{a.b}$
d) $\frac{\left (a.b \right )a}{a.a}$

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1. Equation of the plane parallel to the planes x + 2y + 3z - 5 = 0, x + 2y + 3z - 7 = 0 and equidistant from them is
a) x + 2y + 3z - 6 = 0
b) x + 2y + 3z - 1 = 0
c) x + 2y + 3z - 8 = 0
d) x + 2y + 3z - 3 = 0

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2. The shortest distance from the plane 12x + 4y + 3z = 327 to the sphere x² + y² + 4x - 2y - 6z = 155 is
a) 26
b) $11\frac{4}{13}$
c) 13
d) 39

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1. In LPP, ΔJ for all basic variables is equal to
a) 1
b) -1
c) 0
d) None of these

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2. Variables of the objective function of the linear programming problem are
a) Zero
b) Zero or positive
c) Negative
d) Zero or negative

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1. A pair of a dice thrown, if appears on at least one of the dice, then the probability that the sum is 10 or greater is
a) $\frac{11}{36}$
b) $\frac{2}{9}$
c) $\frac{3}{11}$
d) $\frac{1}{12}$

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2. If n positive integers are taken at random and multiplied together, the probability that the last digit of the product is 2,4,6 or 8, is
a) $\frac{4^{n} + 2^{n}}{5^{n}}$
b) $\frac{4^{n} \times 2^{n}}{5^{n}}$
c) $\frac{4^{n} - 2^{n}}{5^{n}}$
d) None of these

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1. $\int \frac{secx}{\sqrt{sin\left (2x + \alpha \right ) + sin\alpha}}dx$ is equal to
a) $\sqrt{2sec\alpha \left (tanx + tan\alpha \right )}$
b) $\sqrt{2sec\alpha \left (tanx - tan\alpha \right )}$
c) $\sqrt{2sec\alpha \left (tan\alpha - tanx \right )}$
d) None of these

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2. $\int cos^{-3/7}xsin^{-11/7}xdx$ =
a) $log|sin^{4/7}x| + c$
b) $\frac{4}{7}tan^{4/7}x + c$
c) $\frac{-7}{4}tan^{4/7} x + c$
d) $log |cos^{3/7}x | + c$
e) $\frac{7}{4}tan^{4/7} x + c$

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1. On the interval $\left [\frac{5\pi}{3}, \frac{7\pi}{4} \right ]$ , the greatest value of the function f(x) = $\int_{6\pi/3}^{x}\left (6cost - 2sint \right )dt$ =
a) 3$\sqrt{2}$ + 2$\sqrt{2}$ + 1
b) 3$\sqrt{2}$ - 2$\sqrt{2}$ - 1
c) Does not exist
d) None of these

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2. The area bounded by the curves 4y = x² and 2y = 6 - x² is
a) 8
b) 6
c) 4
d) 10

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