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

Q. If ${\tan }^{-1}\left(\frac{x-2}{x-4}\right)+{\tan }^{-1}\left(\frac{x+2}{x+4}\right)=\frac{\pi }{4}$, find the value of $x.$

Q. A manufacturing company makes two types of teaching aids $A$ and $B$ of Mathematics for Class XII. Each type of $A$ requires $9$ labour hours of fabricating and $1$ labour hour for finishing. Each type of $B$ requires $12$ labour hours for fabricating and finishing, the maximum labour hours available per week are $180$ and $30$ respectively. The company makes a profit of $Rs.80$ on each piece of type $A$ and $Rs.120$ on each piece of type $B$. How many pieces of type $B$ should be manufactured per week to get a maximum profit? Make it as an LPP and solve graphically. What is the maximum profit per week?

Q. If $R=\left\{\right(x,y):x+2y=8\}$ is a relation on $N,$ write the range of $R.$

Q. Show that the altitude of the right circular cone of maximum volume that can be described in a sphere of radius $r$ is $\frac{4r}{3}$. Also show that the maximum volume of the cone is $\frac{8}{27}$ of the volume of the sphere.

Q. Evaluate: $\int \frac{x+2}{\sqrt{x^2+5x+6}}dx.$

Q. If $f\left(x\right)={\int }_0^{x}t\sin tdt$, then write the value of $f^{\prime }\left(x\right).$

Q. Evaluate: ${\int }_0^{\pi }\frac{4x\sin x}{1+{\cos }^{2}x}dx.$

Q. A dealer in rual area wishes to purchase a number of sewing machines. He has only $Rs5,760$ to invest and has space for at most $20$ items for storage. An electronic sewing machine costs him $Rs360$ and a manually operated sewing machine $Rs240$. He can sell an electronic sewing machine at a profit of $Rs22$ and a manually operated sewing machine at a profit of $Rs18$. Assuming that he can sell all the items that he can buy how should he invest his money in order to maximize his profit? Formulate this problem mathematically and then solve it.

Q. If $\left[\begin{array}{cc}x-y& z\\ 2x-y& w\end{array}\right]=\left[\begin{array}{cc}-1& 4\\ 0& 5\end{array}\right]$, find the value of $x+y.$

Q. Find the value of $\frac{dy}{dx}$ at $\theta =\frac{\pi }{4}$ if $x=a{e}^{\theta }(\sin \theta -\cos \theta )$ and $y=a{e}^{\theta }(\sin \theta +\cos \theta ).$

Q. Evaluate: ${\int }_2^4\frac{x}{x^2+1}dx$

Q. If ${\tan }^{-1}x+{\tan }^{-1}y=\frac{\pi }{4},xy<1$, then write the value of $x+y+xy.$

Q. Two school $A$ and $B$ want to award their selected students on the values of sincerity, truthfulness and helpfulness. The school $A$ wants to award $Rs.x$ each, $Rs.y$ each and $Rs.z$ each for the three respective values to $3,2$ and $1$ students respectively with a total award money of $Rs.1,600$. School $B$ wants to spend $Rs.2,300$ to award its $4,1$ and $3$ students on the respective values (by giving the same award money to the three values as before). If the total amount of award for one prize on each value is $Rs.900$, using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for award.

Q. Find the value of ${}^{\prime }{p}^{\prime }$ for which the vectors $3\hat{i}+2\hat{j}+9\hat{k}$ and $\hat{i}+p\hat{j}+3\hat{k}$ are parallel.

Q. Find the particular solution of the differential equation $\frac{dy}{dx}=\frac{x(2\log x+1)}{\sin y+y\cos y}$ given that $y=\frac{\pi }{2}$ when $x=1$.

Q. A line passes through $(2,-1,3)$ and it is perpendicular to the lines $\overrightarrow{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda (2\hat{i}-2\hat{j}+\hat{k})$ and $\overrightarrow{r}=(2\hat{i}-\hat{j}-3\hat{k})+\mu (\hat{i}+2\hat{j}+2\hat{k})$. Obtain its equation in vector and Cartesian form.

Q. Find the equation of the tangent and normal to the curve $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ at point $(\sqrt{2}a,b).$

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

Q. Find the value(s) of $x$ for which $y=\left[x\right(x-2){]}^2$ is a increasing function.

Q. Solve the differential equation $(1+x^2)\frac{dy}{dx}+y=e^{{\tan }^{-1}x}.$

Q. Prove that: ${\tan }^{-1}\left[\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}\right]=\frac{\pi }{4}-\frac{1}{2}{\cos }^{-1}x,\frac{-1}{\sqrt{2}}\le x\le 1.$

Q. Using properties of determinants, prove that $\left[\begin{array}{ccc}x+y& x& x\\ 5x+4y& 4x& 2x\\ 10x+8y& 8x& 3x\end{array}\right]=x^3$

Q. If $\left|\begin{array}{cc}3x& 7\\ -2& 4\end{array}\right|=\left|\begin{array}{cc}8& 7\\ 6& 4\end{array}\right|$, find the value of $x.$

Q. Find $\overrightarrow{a}\cdot (\overrightarrow{b}\times \overrightarrow{c})$, if $\overrightarrow{a}=2\hat{i}+\hat{j}+3\hat{k},\overrightarrow{b}=-\hat{i}+2\hat{j}+\hat{k}$ and $\overrightarrow{c}=3\hat{i}+\hat{j}+2\hat{k}.$

Q. If $A$ is a square matrix such that $A^2=A$, then write the value of $7A-(I+A{)}^3$, where $I$ is an identity matrix.

Q. Show that the four points $A,B,C$ and $D$ with position vectors $4\hat{i}+5\hat{j}+\hat{k},-\hat{j}-\hat{k},3\hat{i}+9\hat{j}+4\hat{k}$ and $4(-\hat{i}+\hat{j}+\hat{k})$ respectively are coplanar.

Q. If $y=P{e}^{ax}+Q{e}^{bx}$, show that $\frac{d^{2}y}{d{x}^2}-(a+b)\frac{dy}{dx}+aby=0.$

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

Q. The scalar product of the vector $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}$ with a unit vector along the sum of vectors $\overrightarrow{b}=2\hat{i}+4\hat{j}-5\hat{k}$ and $\overrightarrow{c}=\lambda \hat{i}+2\hat{j}+3\hat{k}$ is equal to one. Find the value of $\lambda$ and hence find the unit vector along $\overrightarrow{b}+\overrightarrow{c}.$