Q. Express $A=\left[\begin{array}{ccc}2& 5& -1\\ 3& 1& 5\\ 7& 6& 9\end{array}\right]$as sum of symmetric and skew- symmetric matrices.

Q. If is a binary operation such that $a\ast b=a^2+b^2$ then $3\ast 5$ is

1. 9
2. 34
3. 8
4. 25

Q. A window is in the form of rectangle surmounted by a semi-circular opening. The perimeter of window is 30 M . Find the dimensions of window so that it can admit maximum light through the whole opening
or
Prove that volume of largest cone, which can be inscribed in a sphere, is 8/27 part of sphere

Q. Prove that : ${\sin }^{-1}\left(\frac{5}{13}\right)+{\cos }^{-1}\left(\frac{4}{5}\right)=\frac{1}{2}{\sin }^{-1}\left(\frac{3696}{4225}\right)$

Q. If matrix A = $[aij{]}_{3\times 2},$ and $aij=(3i-2j{)}^2$, then find the matrix A.

Q. If y = $(x{)}^{tanx}+(tanx{)}^{x}thenfind\frac{dy}{dx}$

Q. Two cards are drawn (without replacement) from a well-shuffled deck of $52$ cards. Find the probability distribution table and mean.

Q. Integrating factor of different equation $\frac{dy}{dx}+y=3$ is

1. e
2. log x
3. $e^x$
4. x

Q. From differential equation representing the family of lines making equal intercepts on the co-ordinate axes.

Q. Find the Integrating factor of differential equation given that $\tan x\frac{dy}{dx}+y=2x\tan x+x^2$

Q. Using differentials, find approximate value of $(0.37{)}^{1/2}$.

Q. The two adjacent sides of a parallelogram are $2\hat{i}-4\hat{j}+5\hat{k}$ and $\hat{i}-2\hat{j}-3\hat{k}$.Find the unit vector parallel to its diagonal.Also, find its area.

Q. Distance between plane $3x+4y-20=0$ and point $(0,0,-7)$ is

1. $4$ units
2. $2$ units
3. $3$ units
4. $1$ units

Q. Find the distance between the point $(2,3,-1)$ and foot or perpendicular drawn from $(3,1,-1)$ to the plane $x-y+3z=10$

Q. Check whether Lagrange's mean value theorem is applicable on f(x) = sin x + cos x interval $[0,\frac{\pi }{2}]$

Q. Prove that function $f:R\to R,F\left(x\right)=\frac{3-2x}{7}$ in - one - one and onto. Also find $f^{-1}$

Q. Find the angle between the plane 2x+3 y-5z = 10 and the line passing from the point (2, 3, -1) or (1, 2, 1)

Q. If $P\left(A\right)=\frac{7}{13}$, $P\left(B\right)=\frac{9}{13}$ , $P(A\cup B)=\frac{12}{13}$, then find (A|B).

Q. If p(E) denotes probability of occurrence of event E then

1. $P\left(E\right)\in [-1,1]$
2. $P\left(E\right)\in (1,2)$
3. $P\left(E\right)\in (0,1)$
4. $P\left(E\right)\in [0,1]$

Q. $\int e^x(logx+\frac{1}{x})$ dx is equal to

1. log x + c
2. $\frac{e^x}{x}+c$
3. $e^x+c$
4. $e^{x}logx+c$

Q. This inequality $|a\cdot b|\le \left|a\right|\cdot \left|b\right|$ is called

1. Cauchy - schwartz inequality
2. Triangle inequality
3. Rolle's Theorem
4. Lagrange's mean value theorem

Q. If y = sin $(si{n}^{-1}x+co{s}^{-1}x),x\in [-1,1]then\frac{dy}{dx}$ is

1. $\frac{-\pi }{2}$
2. 1
3. $\frac{\pi }{2}$
4. 0

Q. Evaluate $\int \frac{7dx}{x(x^7-1)}$

Q. Differentiate ${\left(x\right)}^{tanx}+(tanx{)}^x$ w.r.t $x$

Q. If $f\left(x\right)=\{\begin{array}{cc}\frac{\sin x}{x}& x\le 0\\ k-1& x>0\end{array}$ is continuous at $x=0$ then the value of $k$ is-

1. 2
2. 0
3. -1
4. 1

Q. Evaluate

Q. If $x,y,z$ are different and $\left|\begin{array}{ccc}x& x^2& 1+x^3\\ y& y^2& 1+y^3\\ z& z^2& 1+z^3\end{array}\right|=0$ then prove that $xyz=-1$

Q. Find the area of region bounded by the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$

Q. Find particular solution of different equation $\frac{dy}{dx}=\frac{1+y^2}{1+x^2}$ given that x = 0 y = 1

Q. If $co{s}^{-1}x=y$ then

1. $-\pi \le y\le \pi$
2. $\frac{-\pi }{2}\le y\le \frac{\pi }{2}$
3. $0\le y\le \frac{\pi }{2}$
4. $0\le y\le \pi$