Q. An open box with a square base is to be made out of a given quantity of cardboard of area $c^2$ square units. Show that the maximum volume of the box is $\frac{c^3}{6\sqrt{3}}$ cubic units.

Q. Find the coordinates of the point where line through the points $A=(3,4,1)$ and $B=(5,1,6)$ crosses the $xy$-plane.

Q. Find the particular solution of the differential equation $x(x^2-1)\frac{dy}{dx}=1,y=0$ when $x=2.$

Q. If $x=\sqrt{a^{{\sin }^{-1}t}},y=\sqrt{a^{{\cos }^{-1}t}}$, show that $\frac{dy}{dx}=-\frac{y}{x}.$

Q. Consider the binary operations $\ast :R\times R\to R$ and $o:R\times R\to R$ defined as $a\ast b=|a-b|$ and $aob=a$ for all $a,b\in R$. Show that ${}^{\prime }{\ast }^{\prime }$ is commutative but not associative, 'o' is associative but not commutative.

Q. Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.

Q. Find the principal value of ${\tan }^{-1}\sqrt{3}-{\sec }^{-1}(-2).$

Q. Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the cone.

Q. Evaluate: ${\int }_{-1}^2|x^3-x|dx.$

Q. Evaluate: ${\int }_0^1\frac{x{\sin }^{-1}x}{\sqrt{1-x^2}}dx.$

Q. Write the value of $(\hat{i}\times \hat{j})\cdot \hat{k}+\hat{i}\cdot \hat{j}.$

Q.

Q. Find the scalar components of the vector $\overrightarrow{AB}$ with initial point $A(2,1)$ and terminal point $B(-5,7).$

Q. Let $A$ be a square matrix of order $3\times 3$. Write the value of $|2A|$, where $|A|=4.$

Q. Using matrices, solve the following system of equations:
$2x+3y+3z=5,x-2y+z=-4,3x-y-2z=3.$

Q. Two cards are drawn simultaneously (without replacement) from a well-shuffled pack of $52$ cards. Find the mean and variance of the number of red cards.

1. Mean $=0.6$ and Variance $=0.3$
2. Mean $=0.1$ and Variance $=0.7$
3. Mean $=0.49$ and Variance $=0.37$
4. Mean $=0$ and Variance $=0.45$

Q. A $13m$ long ladder is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of $2cm/s$. How fast is it's height on the wall decreasing when the foot of the ladder is $5m$ away from the wall?

Q. Evaluate: ${\int }_0^2\sqrt{4-x^2}dx.$

Q. Prove the following:
$\cos ({\sin }^{-1}\frac{3}{5}+{\cot }^{-1}\frac{3}{2})=\frac{6}{5\sqrt{13}}.$

Q. Solve the following differential equation: $(1+x^2)dy+2xydx=\cot xdx;x\ne 0$

Q. The binary operation $\ast :R\times R\to R$ is defined as $a\ast b=2a+b$. Find $(2\ast 3)\ast 4.$

Q. Using properties of determinants, show that $\left|\begin{array}{ccc}b+c& a& a\\ b& c+a& b\\ c& c& a+b\end{array}\right|=4abc.$

Q. A dietician wishes to mix two types of foods in such away that the vitamin contents of the mixture contains at least $8$ units of vitamin A and $10$ units of vitamin C. Food I contains $2$ units/kg of vitamin A and $1$ unit/kg of vitamin C while Food II contains $1$ unit/kg of vitamin A and $2$ units/kg of vitamin $1$ unit/kg of vitamin C. It costs $Rs.5$ per kg to purchase food I and $Rs.7$ per kg to purchase Food II. Determine the maximum cost of such a mixture. Formulate the above as a LPP and solve it graphically.

Q. Find the value of $x+y$ from the following equation:
$2\left[\begin{array}{cc}x& 5\\ 7& y-3\end{array}\right]+\left[\begin{array}{cc}3& -4\\ 1& 2\end{array}\right]=\left[\begin{array}{cc}7& 6\\ 15& 14\end{array}\right]$

Q. Let $\overrightarrow{a}=\hat{i}+4\hat{j}+2\hat{k},\overrightarrow{b}=3\hat{i}-2\hat{j}+7\hat{k}$ and $\overrightarrow{c}=2\hat{i}-\hat{j}+4\hat{k}$. Find a vector $\overrightarrow{p}$ which is perpendicular to both $\overrightarrow{a}$ and $\overrightarrow{b}$ and $\overrightarrow{p}\cdot \overrightarrow{c}=18.$

Q. Find the distance of a point $3x-4y+122=3$ from the origin.

Q. Differentiate ${\tan }^{-1}\left[\frac{\sqrt{1+x^2}-1}{x}\right]$ with respect to $x.$

Q. Given $\int e^x(\tan x+1)\sec xdx=e^{x}f\left(x\right)+c$. Find $f\left(x\right).$

Q. If $A^T=\left[\begin{array}{cc}3& 4\\ -1& 2\\ 0& 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}-1& 2& 1\\ 1& 2& 3\end{array}\right]$, then find $A^T-B^T.$

Q. If $x=a(\cos t+t\sin t)$ and $y=a(\sin t-t\cos t),0<t<\frac{\pi }{2}$, find $\frac{d^{2}x}{d{t}^2},\frac{d^{2}y}{d{t}^2}$ and $\frac{d^{2}y}{d{x}^2}.$