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

Q. Find the coordinates of the foot of perpendicular drawn from the point $A(-1,8,4)$ to the line joining the points $B(0,-1,3)$ and $C(2,-3,-1).$ Hence, find the image of the point $A$ in the line $BC.$

Q. A bag $X$ contains $4$ white balls and $2$ black balls, while another bag $Y$ contains $3$ white balls and $3$ black balls. Two balls are drawn (without replacement) at random from one of the bags and were found to be one white and one black. Find the probability that the balls were drawn from bag $Y.$

Q. i) Solve for $x:{\tan }^{-1}(x-1)+{\tan }^{-1}x+{\tan }^{-1}(x+1)={\tan }^{-1}3x$.

Q. Let $A=R\times R$ and $\ast$ be a binary operation on A defined by
$(a,b)\ast (c,d)=(a+c,b+d)$
Show that $\ast$ is commutative and associative. Find the identity element for $\ast$ on A. Also find the inverse of every element $(a,b)\in A$.

Q. Find : $\int \frac{x^2+x+1}{(x^2+1)(x+2)}dx$

Q. Find the equation of tangents to the curve $y=x^3+2x-4$, which are perpendicular to the line $x+14y+3=0$.

Q. Using the method of integration, the area of the triangular region whose vertices are $A(2,-2),B(4,3),C(1,2)$ comes out to be $\frac{k}{2}$, find $k$

Q. Prove that $y=\frac{4\sin \theta }{2+\cos \theta }$ is an increasing function of $\theta$ on $[0,\frac{\pi }{2}]$.

Q. Find : $\int \frac{(2x-5)e^{2x}}{{(2x-3)}^3}dx.$

Q. A retired person wants to invest an amount of Rs. $50,000.$ His broker recommends investing in two type of bonds A and B yielding $10\%$ and $9\%$ return respectively on the invested amount. He decides to invest at least $Rs.20,000$ in bond A and at least $Rs.10,000$ in bond B. He also wants to invest at least as much in bond A as in bond B. Solve this linear programming problem graphically to maximize his returns.

1. $4900$
2. $2900$
3. $5400$
4. $4000$

Q. Find $\frac{dy}{dx}$ if $y={sin}^{-1}\left[\frac{6x-4\sqrt{1-4x^2}}{5}\right].$

Q. Three numbers are selected at random (without replacement) from first six positive integers. Let $X$ denote the largest of the three numbers obtained. Find the probability distribution of $X$. Also, find the mean and variance of the distribution.

Q. If $A$ is a $3\times 3$ matrix $\left|3A\right|=k\left|A\right|$, then write the value of $k.$

Q. If $f\left(x\right)=\frac{\sin (a+1)x+2\sin x}{x},x<0=2,x=0=\frac{\sqrt{1+bx}-1}{x},x>0$

Q. If $\overrightarrow{a}=4\hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{b}=2\hat{i}-2\hat{j}+\hat{k}$, then find a unit vector parallel to the vector $\overrightarrow{a}+\overrightarrow{b}$.

Q. Show that the four points $A(4,5,1),B(0,-1,-1),C(3,9,4)$ and $D(-4,4,4)$ are coplanar.

Q. Find : $\int (x+3)\sqrt{3-4x-x^2}dx.$

Q. Show that semi-vertical angle of a cone of maximum volume and given slant height is ${\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)$.

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

Q. If $A=\left(\begin{array}{c}\cos \alpha \\ -\sin \alpha \end{array}\begin{array}{c}\sin \alpha \\ \cos \alpha \end{array}\right)$, find $\alpha$ satisfying $0<\alpha <\frac{\pi }{2}$ when $A+A^T=\sqrt{2}{I}_2$; where $A^T$ is transpose of $A.$

Q. Find the particular solution of differential equation :
$\frac{dy}{dx}=-\frac{x+y\cos x}{1+\sin x}$ given that $y=1$ when $x=0.$

Q. Using properties of determinants, prove that
$\left|\begin{array}{c}{(x+y)}^2\\ zx\\ zy\end{array}\begin{array}{c}zx\\ {(z+y)}^2\\ xy\end{array}\begin{array}{c}zy\\ xy\\ {(z+x)}^2\end{array}\right|=2xyz{(x+y+z)}^3.$

Q. For what values of $k,$ the system of linear equations
$x+y+z=2$
$2x+y-z=3$
$3x+2y+kz=4$
has a unique solution?

Q. Find the equation of the plane which contains the line of intersection of the planes $\overrightarrow{r}.(\hat{i}-2\hat{j}+3\hat{k})-4=0$ and $\overrightarrow{r}.(-2\hat{i}+\hat{j}+\hat{k})+5=0$ and whose intercept on x-axis is equal to that of on y-axis.

Q. Evaluate : ${\int }_{-2}^2\frac{x^2}{1+5^x}dx.$

Q. A typist charges Rs. $145$ for typing $10$ English and $3$ Hindi pages, while charges for typing $3$ English and $10$ Hindi pages are Rs. $180.$ Using matrices, find the charges of typing one English and one Hindi page separately. However typist charged only Rs. $2$ per page from a poor student Shyam for $5$ Hindi pages. How much less was charged from this poor boy? Which values are reflected in this problem?

Q. Find $\lambda$ and $\mu$ if $(\hat{i}+3\hat{j}+9k)\times (3\hat{i}-\lambda \hat{j}+\mu k)=0.$

Q. If $A=\left(\begin{array}{c}1\\ 0\\ 2\end{array}\begin{array}{c}0\\ 2\\ 0\end{array}\begin{array}{c}2\\ 1\\ 3\end{array}\right)$ and $A^3-6A^2+7A+k{I}_3=O$ find $k.$

Q. Write the sum of intercepts cut off by the plane $\overrightarrow{r}.(2\hat{i}+\hat{j}-\hat{k})-5=0$ on the three axes.