Q. If a binary operation is defined $a\star b=a^b$ then 2$\star 2$ is equal to:

1. $4$
2. $2$
3. $9$
4. $8$

Q. Show that :$\left[\begin{array}{ccc}1+x& 1& 1\\ 1& 1+y& 1\\ 1& 1& 1+z\end{array}\right]=xyz(1+\frac{1}{x}+\frac{1}{y}+\frac{1}{z})$

Q. If $y=(sinx{)}^x+(x{)}^{sinx}$ then find $\frac{dy}{dx}$

Q. A wire of length $25cm$ is to be cut off into two pieces. One piece is to be made onto a circle and other into a square. What should be the lengths of pieces so that combined area of circle and square is minimum?

Q. Using integration find the area of triangle whose sides are given by the equation $y=x+1,y=3x+1,x=5$

Q. Solve the following system of linear equations by matrix method:
$3x+x+z=10,2x-y-z=0,x-y+2z=1$

Q. Find the angel between plane $3x+4y-z=8$ and line $\frac{x-1}{2}=\frac{2-y}{7}=\frac{3z+6}{12}$

Q. Maximize $Z=12x+24y$ subject to the constraints $x+y\ge 5,5x+7y\le 35,x-y\ge 0,x,y\ge 0$ graphically.

Q. Using elementary transformations find the inverse of
$\left[\begin{array}{ccc}3& 2& 1\\ 2& 4& 3\\ 2& -1& 2\end{array}\right]$

Q. Adjacent sides of a parallelogram are given by the vector $2\hat{1}-\hat{j}+2\hat{k}and\hat{i}+5\hat{j}=\widehat{k.}$ . Find a unit vector in the direction of its diagonal. Also find the area of parallelogram.

Q. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius $20cm$ is $\frac{40}{\sqrt{3}}cm$.Also find the maximum volume.

Q. Find the images of the points$(5,-3,1)$ in the plane $2x-2y-3z=10$

Q. Find the shortest distance between the lines:
$\frac{x+1}{4}=\frac{y-3}{-6}=\frac{z+1}{1}$ and $\frac{x+3}{3}=\frac{y-5}{2}=\frac{z-7}{6}$

Q. Find particular solution of differential equation $x^{2}dy-(3x^2+xy+y^2)dx=0,y\left(1\right)=1$

Q. Bag $I$ contains $2$ blacks and $8$ red balls, bag $II$contains $7$ black and $3$ red balls and bag $III$ contains $5$ black and $5v$ red balls. One bage is chosen at random and a ball is drawn from it which is found to bered. Find the probability that the ball is drawn from bag $II$

Q. One kind of cake requires $300gm$ of flour and $15gm$ of fat and another kind of cake requires $150gm$ of flour and $30gm$ of fat. Find the maximum number of cake that can be made from $7.5kg$ of flour and $600gm$ of fat . Form a linear programming problem and solve it graphically.

Q. Show that function $f:R\to R,f\left(x\right)=\frac{2x+5}{8}$ is invertible. Also find inverse of f.

Q. If $A=\left[\begin{array}{c}2\\ -4\\ 1\end{array}\right],B=[53-1]$ then verify that $(AB{)}^{\prime }=B^{\prime }{A}^{\prime }$

Q. Vectors $\overrightarrow{a}=3\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=\hat{i}-\hat{j}+2\hat{k}and\overrightarrow{c}=2\hat{i}-\hat{j}-\hat{k}$. Find vector $\overrightarrow{d}$if $\overline{d}$ is perpendicular to $\overrightarrow{c}$ and $\overrightarrow{d}$. $\overrightarrow{a}=10,\overrightarrow{b}=1$

Q. Find particular solution of differential equation $cos\left(\frac{dy}{dx}\right)=\frac{1}{5},y\left(0\right)=2$

Q. If $y={\sin }^{-1}\left(\frac{2x}{1+x^2}\right)$ then find $\frac{dy}{dx}$

Q. Probabilities of A, B and C of solving a problem are $\frac{1}{3},\frac{1}{2}and\frac{1}{4}$ respectively. If they all try to solve the problem then find the probability that exactly one of them will solve the problem.

Q. Evaluate $\int {\sin }^{4}x{\cos }^{3}xdx.$

Q. Express$\left[\begin{array}{ccc}6& -4& 5\\ 1& 4& -2\\ 7& 5& 9\end{array}\right]$ as a sum of a symmetric matrix and a skew-symmetric matrix.

Q. Evaluate:${\int }_1^3(x^2+4)dx$.

Q. Using differentials, find approximate value of $\sqrt{360}$.

Q. Evaluate $\int \frac{dx}{x^2-4x+13}$

Q. Fine the integrating factor for the differential equation $cotx\frac{dy}{dx}+y=2x+x^2$

Q. Two cards are drawn (without replacement) from a well shuffled deck of $52$ cards. Find probability distribution and mean of number of cards numbered $4$

Q. Show that $ta{n}^{-1}\frac{1}{3}+ta{n}^{-1}\frac{1}{5}=\frac{1}{2}co{s}^{-1}\frac{33}{65}$