1. If tan^-1x + tan^-1y + tan^-1z = π, then x + y + z is equal to
a) xyz
b) 0
c) 1
d) 2xyz

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2. $tan^{-1}\left [\frac{\sqrt{1 + x^{2}} + \sqrt{1 - x^{2}}}{\sqrt{1 + x^{2}} - \sqrt{1 - x^{2}}} \right ]$ =
a) $\frac{\pi}{4} + \frac{1}{2}cos^{-1}x^{2}$
b) $\frac{\pi}{4} + cos^{-1}x^{2}$
c) $\frac{\pi}{4} + \frac{1}{2}cos^{-1}x$
d) $\frac{\pi}{4} - \frac{1}{2}cos^{-1}x^{2}$

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1. If the matrices A = $\begin{bmatrix} 2 &1 &3 \\ 4& 1& 0 \end{bmatrix}$ and B = $\begin{bmatrix} 1 &-1 \\ 0& 2\\ 5 &0 \end{bmatrix}$m then AB will be
a) $\begin{bmatrix} 17 &0 \\ 4 &-2 \end{bmatrix}$
b) $\begin{bmatrix} 4 &0 \\ 0 &4 \end{bmatrix}$
c) $\begin{bmatrix} 17 &4 \\ 0 &-2 \end{bmatrix}$
d) $\begin{bmatrix} 0 &0 \\ 0&0 \end{bmatrix}$

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2. The value of $\begin{vmatrix} ^{10}C_{4} &^{10}C_{5} & ^{11}C_{m}\\ ^{11}C_{6} &^{11}C_{7} &^{12}C_{m+2} \\ ^{12}C_{8}&^{12}C_{9} & ^{13}C_{m+4} \end{vmatrix}$ = 0 when m =
a) 6
b) 5
c) 4
d) 1

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1. $lim_{x \rightarrow 0}\frac{x tan2x - 2x tanx}{\left (1 - cos2x \right )^{2}}$ is
a) 2
b) -2
c) 1/2
d) -1/2

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2. If f(x) = $\frac{x^{2} - 10x + 25}{x^{2} - 7x + 10}$ for x ≠ 5 and f is continuous at x = 5, then f(5) =
a) 0
b) 5
c) 10
d) 25

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1. If y = $cot^{-1}\left [\frac{\sqrt{1 + sinx} + \sqrt{1 - sinx}}{\sqrt{1 + sinx} - \sqrt{1 - sinx}} \right ]$, then $\frac{dy}{dx}$ =
a) 1/2
b) 2/3
c) 3
d) 1

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2. If the straight line y - 2x + 1 = 0 is the tangent to the curve xy + ax + by = 0 x = 1, then the value of a and b are respectively
a) 1 and 2
b) 1 and -1
c) -1 and 2
d) -1 and -2
e) 1 and -2

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1. The integral $\int_{-1/2}^{1/2}\left \{ \left [x \right ] + log\left (\frac{1 + x}{1 - x} \right )\right \}$ dx equal (where [.] is the greatest integer function)
a) $-\frac{1}{2}$
b) 0
c) 1
d) $2log\frac{1}{2}$

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2. $\int_{0}^{1}\frac{x^{7}}{\sqrt{1 - x^{4}}}dx$ is equal to
a) 1
b) $\frac{1}{3}$
c) $\frac{2}{3}$
d) $\frac{\pi}{3}$

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1. Solution of the differential equation $\frac{dx}{y}$ + $\frac{dy}{y}$ = 0 is
a) xy = c
b) x + y = c
c) logx logy = c
d) x² + y² = c

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2. The degree of the differential equation $\frac{d^{2}y}{dx^{2}} + 3\left [\frac{dy}{dx} \right ]^{2}$ = $x^{2}log\left [\frac{d^{2}y}{dx^{2}} \right ]$ is
a) 1
b) 2
c) 3
d) None of these

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1. The area of a triangle whose vertices are A(1,-1,2), B(2,1,-1) and C(3,-1,2) is
a) 13
b) $\sqrt{13}$
c) 6
d) $\sqrt{6}$

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2. If M denotes the mid-point of the line joining $A\left (4\hat{i} + 5\hat{j} - 10\hat{k} \right )$ and $A\left (-\hat{i} + 2\hat{j} + \hat{k} \right )$, then equation of the plane through M and perpendicular to AB , is
a) $\vec{r}.\left (-5\hat{i} - 3\hat{j} + 11\hat{k} \right ) + \frac{135}{2}$ = 0
b) $\vec{r}.\left (\frac{3}{2}\hat{i} + \frac{7}{2}\hat{j} - \frac{9}{2}\hat{k} \right ) + \frac{135}{2}$ = 0
c) $\vec{r} . \left (4\hat{i} + 5\hat{j} - \hat{k} \right ) + 4$ = 0
d) $\vec{r} . \left (-\hat{i} + 2\hat{j} + \hat{k} \right ) + 4$ = 0

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1. The equation of the plane containing the lines
$\frac{x - 1}{2}$ = $\frac{y + 1}{-1}$= $\frac{z}{3}$ and $\frac{x}{2}$ = $\frac{y - 2}{-1}$ = $\frac{z + 1}{3}$ is
a) 8x - y + 5z - 8 = 0
b) 8x + y - 5z - 7 = 0
c) x - 8y + 3z + 6 = 0
d) 8x + y - 5z + 7 = 0
e) x + y + z - 6 = 0

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2. 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 k =
a) 2/9
b) 9/2
c) 0
d) None of these

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1. The minimum value of z = 2x₁ + 3x₂ subject to the constraints 2x₁ + 7x₂ ≥ 22, x₁ + x₂ ≥ 6, 5x₁ + x₂ ≥ 100
a) 14
b) 20
c) 10
d) 16

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2. A cold drink factory has two plants located at Bhopal and Gwalior. Each plant produces three diferent types of drinks A,B,C . The production capacity of the plants per day is as follows

Drinks	Plant at Bhopal	Plant at Gwalior
A	6,000 Bottles	2,000 Bottles
B	1,000 Bottles	2,500 Bottles
C	3,000 Bottles	3,000 Bottles

A demand of 80,000 bottles of A, 22,000 bottles of B and 40,000 bottles of C in the month of June is forecasted. The operating costs per day of plants at Bhopal and Gwalior are Rs. 6,000 and Rs. 4000 respectively. The number of days for which each plant must be run in June so as to minimize the operating costs in meeting the demand are
a) 12,4
b) 4,12
c) 40,0
d) None of these

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1. A coin is tossed 3 times by 2 persons. What is the probability that both get equal number of heads
a) $\frac{3}{8}$
b) $\frac{1}{9}$
c) $\frac{5}{16}$
d) None of these

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2. A binary number is made up of 16 bits. The probability of an incorrect bit appearing is p and the errors in different bits are independent of one another. The probability of forming an incorrect number is
a) $\frac{p}{16}$
b) p¹⁶
c) ¹⁶C₁p¹⁶
d) 1 - (1 - p)¹⁶

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1. $\int e^{xloga}.e^{x} dx$ is equal to
a) $\left (ae \right )^{c} + c$
b) $\frac{\left(ae \right )^{x}}{log\left (ae \right )} + c$
c) $\frac{e^{x}}{1 + log a} + c$
d) None of these

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2. $\int \sqrt{1 + cosx}dx$ equals
a) $2\sqrt{2}sin\frac{x}{2} + c$
b) $-2\sqrt{2}sin\frac{x}{2} + c$
c) $-2\sqrt{2}cos\frac{x}{2} + c$
d) $-2\sqrt{2}cos\frac{x}{2} + c$

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1. The magnetic field in a region is given by $\vec{B}= B_{0}\left ( 1+\frac{x}{a} \right )\hat{k}$. A square loop of edge length d is placed with its edge along the x- and y-axes. The loop is moved with a constant velocity $\vec{v}=v_{0}\hat{i}$. The emf induced in the loop is
a) $\frac{v_{0}B_{0}d^{2}}{a}$
b) $\frac{v_{0}B_{0}d^{3}}{a^{2}}$
c) v₀B₀d
d) zero

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2. The parabola y² = 4x and x² = 4y divide the square region bounded by the lines x = 4, y = 4 and the coordinate axes. If S₁, S₂,S₃ are respectively the areas of these parts numbered from top to bottom, then S₁ : S₂ : S₃ is
a) 2:1:2
b) 1:1:1
c) 1:2:1
d) 1:2:3

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