Q. Find the value of $\int \frac{{\sec }^{2}x}{cose{c}^{2}dx}dx$

Q. Find $A$, if $A+B=\left[\begin{array}{cc}5& 2\\ 0& 9\end{array}\right]$ and $A-B=\left[\begin{array}{cc}-3& -6\\ 4& -1\end{array}\right]$

Q. If $\overrightarrow{a}=\hat{i}+4\hat{j}+2\hat{k},\overrightarrow{b}=3\hat{i}-2\hat{j}+7\hat{k}$ and $\overrightarrow{c}=2\hat{i}-\hat{j}+4\hat{k}$, then find a vector $\overrightarrow{d}$ which is perpendicular to both $\overrightarrow{a}$ and $\overrightarrow{b}$ and $\overrightarrow{c}\cdot \overrightarrow{d}=15$

Q. An experiment succeeds twice as often as it fails. Find the probability that in the next six trials, there will at least $4$ successes.

Q. Find the angle between planes $2x+y-2z=5$ and $3x-6y-2z=7$

Q. If $\left|\begin{array}{ccc}1+a& 1& 1\\ 1& 1+b& 1\\ 1& 1& 1+c\end{array}\right|=Xabc(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c})$. Find the value of $X$.

Q. If a line makes angles ${90}^{\circ },{60}^{\circ }$ and ${30}^{\circ }$ with the positive direction of $x,y$ and z-axes respectively, find its direction cosines

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

Q. One kind of cake requires $200gm$ of flour and $25gm$ of fat, and another kind of cake requires $100gm$ of flour and $50gm$ of fat. Find the maximum number of cakes which can be made from $5kg$ of flour and $1kg$ of fat assuming that there is no shortage of other ingredients used to making the cakes.

Q. Find the points on the curve $\frac{x^2}{4}+\frac{y^2}{25}=1$ at which the tangents are parallel to x-axis

Q. If $A$ and $B$ are independent events with $P\left(A\right)=0.3$ and $P\left(B\right)=0.4$, then find the value of $P(A\cup B)$.

Q. If $\hat{i}+\hat{j}+\hat{k},2\hat{i}+5\hat{j},3\hat{i}+2\hat{j}-3\hat{k}$ and $\hat{i}-6\hat{j}-\hat{k}$ are the position vectors of points $A,B,C$ and $D$ respectively, then find the angle between $\overrightarrow{AB}$ and $\overrightarrow{CD}$

Q. Show that the relation $R$ in the set $R$ of real numbers, defined as $R=\left\{\right(a,b):a\le b^2\}$ is neither reflexive nor symmetric nor transitive

Q. If $A=\left[\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right]$, then find the value of $A^2-5A+6I$

Q. If $A=\left[\begin{array}{c}-2\\ 4\\ 6\end{array}\right]$ and $B=[14-6]$ then find $AB$

Q. Find the area of the region bounded by the curve $y=x^2$ and the line $y=4$.

Q. Solve the following linear programming problem by graphical method:
Maximize $Z=3x+2y$
subject to the in constraints
$x+2y\le 10,3x+y\le 15,x,y\ge 0$

Q. Find $\frac{dy}{dx}$, if $x^y+y^x=b^a+a^b$

Q. Let $f:N\to Y$ be a function defined as $f\left(x\right)=4x+3$, where, $Y=\{yϵN:y=4x+3\text{for some}xϵN\}$. Show that $f$ is invertible. Find the inverse function

Q. Show the feasible solution region under the following constraints:
$2x+3y\le 18,x\ge 0,y\ge 0$

Q. Using integration find the area of triangular region whose sides have the equation $y=2x+1,y=3x+1$ and $x=4$

Q. Write the function ${\tan }^{-1}\left(\frac{\cos x-\sin x}{\cos x+\sin x}\right),x<\pi$ in the simplest form

Q. Find general solution of differential equation $\frac{dy}{dx}+y=1(y\ne 1)$.

Q. Using elementary transformations, find the inverse of the matrix $\left[\begin{array}{cc}3& 10\\ 2& 7\end{array}\right]$

Q. Find the variance of numbers obtained on thrown an unbiased die.

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

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

Q. Find the values of $a$ and $b$ such that the function defined by $f\left(x\right)=\{\begin{array}{ccc}5,& if& x\le 2\\ ax+b,& if& 2<x<10\\ 21,& if& x\ge 10\end{array}$ is a continuous function.

Q. Find unit vector of a given vector $\overrightarrow{a}=2\hat{i}-3\hat{j}+4\hat{k}$

Q. Prove that ${\tan }^{-1}\frac{63}{16}={\sin }^{-1}\frac{5}{13}+{\cos }^{-1}\frac{3}{5}$