Q. Find the magnitude of each of the two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, having the same magnitude such that the angle between them is ${60}^o$ and their scalar product is $\frac{9}{2}$.

Q. Using integration, find the area of the region in the first quadrant enclosed by the $x-axis$, the line $y=x$ and the circle $x^2+y^2=32$.

Q. Given $A=\left[\begin{array}{cc}2& -3\\ -4& 7\end{array}\right]$, compute $A^{-1}$ ans show that $2A^{-1}=9I-A$.

Q. Find the differential equation representing the family of curves $y=a{e}^{bx+5}$, where a and b are arbitrary constants.

Q. An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when depth of the tank is half of its width. If the cost is to be born by nearby settled lower income families, for whom water will be provided, what kind of value is hidden in this question?

Q. If $\theta$ is the angle between two vectors $\hat{i}-2\hat{j}+3\hat{k}$ and $3\hat{i}-2\hat{j}+\hat{k}$, find $\theta$.

Q. Suppose a girl throws a die. If she gets $1$ or $2$, she tosses a coin three times and notes the number of tails. If she gets $3,4,5$ or $6$, she tosses a coin once and notes whether a 'head' or 'tail' is obtained. If she obtained exactly one 'tail', what is the probability that she threw $3,4,5$ or $6$ with the die?

Q. If $x=a(2\theta -\sin 2\theta )$ and $y=a(1-\cos 2\theta )$, find $\frac{dy}{dx}$ when $\theta =\frac{\pi }{3}$.

Q. A black and a red die are rolled together. Find the conditional probability of obtaining the sum $8$, given that the red die resulted in a number less than $4$.

Q. Find the particular solution of the differential equation $\frac{dy}{dx}+2y\tan x=\sin x$, given that $y=0$ when $x=\frac{\pi }{3}$.

Q. Find the intervals in which the function $f\left(x\right)=\frac{x^4}{4}-x^3-5x^2+24x+12$ is
(a) strictly increasing, (b) strictly decreasing.

Q. If $a\ast b$ denotes the larger of ${}^{\prime }{a}^{\prime }$ and ${}^{\prime }{b}^{\prime }$ and if $aob=(a\ast b)+3,$ then write the value of $(5$ $o$ $10),$ where $\ast$ and $o$ are binary operations.

Q. Differentiate ${\tan }^{-1}\left(\frac{1+\cos x}{\sin x}\right)$ with respect to $x$.

Q. Prove that $3{\sin }^{-1}x={\sin }^{-1}(3x-4x^3),$ x$ϵ[-\frac{1}{2},\frac{1}{2}]$

Q. The total cost C(x) associated with the production of x units of an item is given by $C\left(x\right)=0.05x^3-0.02x^2+30x+5000$. Find the marginal cost when $3$ units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output.

Q. Let $\overrightarrow{a}=4\hat{i}+5\hat{j}-\hat{k},\overrightarrow{b}=\hat{i}-4\hat{j}+5\hat{k}$ and $\overrightarrow{c}=3\hat{i}+\hat{j}-\hat{k}$. Find a vector $\overrightarrow{d}$ which is perpendicular to both $\overrightarrow{c}$ and $\overrightarrow{b}$ and $\overrightarrow{d}\cdot \overrightarrow{a}=21$.

Q. A factory manufactures two types of screws $A$ and $B$, each type requiring the use of two machines, an automatic and a hand-operated. It takes $4$ minutes on the automatic and $6$ minutes on the hand-operated machines to manufacture a packet of screws ${}^{\prime }{A}^{\prime }$ while it takes $6$ minutes on the automatic and $3$ minutes on the hand-operated machine to manufacture a packet of screws 'B'. Each machine is available for at most $4$ hours on any day. The manufacturer can sell a packet of screws ${}^{\prime }{A}^{\prime }$ at a profit of $70$ paise and screws ${}^{\prime }{B}^{\prime }$ at a profit of $Rs.1$. Assuming that he can sell all the screws he manufactures, how many packets of each type should the factory owner produce in a day in order to maximize his profit? Formulate the given LLP and solve it graphically and find the maximum profit.

Q. Find the equations of the tangent and the normal, to the curve $16x^2+y^2=145$ at the point $(x_1,y_1)$, where $x_1=2$ and $y_1>0$.

Q. If $y=\sin \left(\sin x\right)$, prove that $\frac{d^{2}y}{d{x}^2}+\tan x\frac{dy}{dx}+y{\cos }^{2}x=0$.

Q. Find the value of: $\int \frac{2\cos x}{(1-\sin x)(1+{\sin }^{2}x)}dx$.

Q. Find the shortest distance between the lines $\overrightarrow{r}=(4\hat{i}-\hat{j})+\lambda (\hat{i}+2\hat{j}-3\hat{k})$ and $\overrightarrow{r}=(\hat{i}-\hat{j}+2\hat{k})+\mu (2\hat{i}+4\hat{j}-5\hat{k})$.

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

Q. Using properties of determinants, prove that $\left|\begin{array}{ccc}1& 1& 1+3x\\ 1+3y& 1& 1\\ 1& 1+3z& 1\end{array}\right|=9(3xyz+xy+yz+zx)$.

Q. If $(x^2+y^2{)}^2=xy$, find $\frac{dy}{dx}$.

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

Q. Two numbers are selected at random(without replacement) from the first five positive integers. Let X denote the larger of the two numbers obtained. Find the mean and variance of X.

Q. Evaluate: $\int \frac{\cos 2x+2{\sin }^{2}x}{{\cos }^{2}x}dx$.

Q. If the matrix A$=\left[\begin{array}{ccc}0& a& -3\\ 2& 0& -1\\ b& 1& 0\end{array}\right]$ is skew symmetric, find the values of 'a' and 'b'.

Q. Evaluate: ${\int }_0^{\pi /4}\frac{\sin x+\cos x}{16+9\sin 2x}dx$.

Q. Find the particular solution of the differential equation $e^x\tan ydx+(2-e^x)se{c}^{2}ydy=0$, given that $y=\frac{\pi }{4}$ when $x=0$.