Q. Find $\int \frac{dx}{\sqrt{9+8x-x^2}}$.

Q. Prove that

Q. Find general solution of differential equation $\frac{dy}{dx}=\frac{1+y^2}{1+x^2}$

Q. Find the value of ${\sec }^{-1}(-2)-{\sin }^{-1}\left(\frac{1}{2}\right)$

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

Q. Find the direction cosines of x-axis.

Q. Show that the function $f\left(x\right)=\{\begin{array}{ccc}3-x,& if& x<1\\ 2,& if& x=1\\ 1+x,& if& x>1\end{array}$ is continuous at $x=1$

Q. Prove that ${\tan }^{-1}\left[\frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}-\sqrt{1-x}}\right]=\frac{\pi }{4}+\frac{1}{2}{\cos }^{-1}x,0<x<1$

Q. If functions $f,g:R\to R$ are defined as $f\left(x\right)=x^2+1,g\left(x\right)=2x-3$, then find $fog\left(x\right),gof\left(x\right)$ and $gog\left(3\right)$

Q. If $A=\left[123\right]$ and $B=\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$, then find $(AB{)}^{\prime }$

Q. Show the region of feasible solution under the following constraints:
$2x+y\le 6,x\ge 0,y\ge 0$

Q. Find $\int \sqrt{1+\cos 2x}dx$.

Q. A die is thrown twice and the sum of the numbers appearing is observed to be $7$. Find the conditional probability that the number $3$ has appeared at least once.

Q. From a pack of $52$ cards, two cards are drawn randomly one by one without replacement. Find the probability that both of them are red.

Q. If the radius of a sphere is measured as $9cm$ with an error of $0.02cm$, then find the approximate error in calculating its volume.

Q. If $A=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right]$ and $A^2-4A=k{I}_3$, then find the value of $k$.
[Where $I_3$ is the identity matrix of order $3$]

Q. Find the area of the region bounded by the two parabolas $x^2=4y$ and $y^2=4x$. (Draw the figure in answer-book)

Q. Find the area of the triangle whose vertices are $A(1,1,1),B(1,2,3)$ and $C(2,3,3)$

Q. Find: $\int x{\tan }^{-1}xdx$

Q. Find the distance of the plane $\overrightarrow{r}\cdot (2\hat{i}+\hat{j}+2\hat{k})=6$ from the origin.

Q. By graphical method solve the following linear programming problem for minimization:
Objective function constraints $Z=5x+y$
$3x+5y\ge 5$
$5x+2y\le 10$
$x\ge 0,y\ge 0$

Q. Find the area enclosed by the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$
(Draw the figure in answer-book)

Q. If $\overrightarrow{a}=2\hat{i}+2\hat{j}+3\hat{k},\overrightarrow{b}=-\hat{i}+2\hat{j}+\hat{k}$ and $\overrightarrow{c}=3\hat{i}+\hat{j}$ are such that $\overrightarrow{a}+\lambda \overrightarrow{b}$ is perpendicular to vector $\overrightarrow{c}$, then find the value of $\lambda$

Q. Find the absolute maximum and minimum values of function given by $f\left(x\right)=x^2-4x+8$ in the interval $[1,5]$

Q. If $\left|\begin{array}{ccc}a+b+2c& c& c\\ a& b+c+2a& a\\ b& b& c+a+2b\end{array}\right|=A(a+b+c{)}^3$, find the value of A.

Q. If a fair coin is tossed $10$ times, find the probability of appearing exactly four tails.

Q. Find the slope of the tangent to the curve $y=x^3-x+1$ at the point whose x-coordinate is $1$. Also find the equation of normal at the same point.

Q. If $2A+B=\left[\begin{array}{cc}3& -1\\ 2& 4\end{array}\right]$ and $B=\left[\begin{array}{cc}-1& -5\\ 0& 2\end{array}\right]$ , then find $A$.

Q. Prove that the relation $R$ defined on set $Z$ as $aRb\iff a-b$ is divisible by $3$, is an equivalence relation.

Q. If the magnitudes of vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are $1$ and $2$ respectively and $\overrightarrow{a}\cdot \overrightarrow{b}=1$, then find the angle between those vectors.