Q. Find the minimum value of $(ax+by)$, where $xy=c^2$.

Q.

Q. Given two independent events $A$ and $B$, such that $P\left(A\right)=0.3$ and $P\left(B\right)=0.6$. Find $P(A^{\prime }\cap B^{\prime })$.

Q. If $A$ is a skew symmetric matrix of order $3$, then the value of $|A|$ is

1. $3$
2. $0$
3. $9$
4. $27$

Q. If $\hat{i},\hat{j},\hat{k}$ are unit vectors along three mutually perpendicular directions, then

1. $\hat{i}.\hat{j}=1$
2. $\hat{i}\times \hat{j}=1$
3. $\hat{i}.\hat{j}=0$
4. $\hat{i}\times \hat{j}=0$

Q. A card is picked at random from a pack of $52$ playing cards. given that the picked card is a queen, the probability of this card to be a card of spade is

1. $\frac{1}{3}$
2. $\frac{4}{13}$
3. $\frac{1}{4}$
4. $\frac{1}{2}$

Q. If $A$ is a $3\times 3$ matrix such that $|A|=8$, then $|3A|$ equals.

1. $8$
2. $24$
3. $72$
4. $216$

Q. $\int x^2{e}^{x^3}dx$ equals

1. $\frac{1}{3}{e}^{x^3}+C$
2. $\frac{1}{3}{e}^{x^4}+C$
3. $\frac{1}{2}{e}^{x^3}+C$
4. $\frac{1}{2}{e}^{x^2}+C$

Q. If $y={\log }_e\left(\frac{x^2}{e^2}\right)$, then $\frac{d^{2}y}{d{x}^2}$ equals

1. $-\frac{1}{x}$
2. $-\frac{1}{x^2}$
3. $\frac{2}{x^2}$
4. $-\frac{2}{x^2}$

Q. $ABCD$ is a rhombus whose diagonals intersect at $E$. Then $\overrightarrow{EA}+\overrightarrow{EB}+\overrightarrow{EC}+\overrightarrow{ED}$ equals

1. $\overrightarrow{0}$
2. $\overrightarrow{AD}$
3. $2\overrightarrow{BC}$
4. $2\overrightarrow{AD}$

Q. The distance of the origin $(0,0,0)$ from the plane $-2x+6y-3z=-7$ is

1. $1unit$
2. $\sqrt{2}unit$
3. $2\sqrt{2}unit$
4. $3unit$

Q. The graph of the inequality $2x+3y>6$ is

1. half plane that contains the origin.
2. half plane that neither contains the origin nor the points of the line $2x+3y=6$
3. whole $XOY$ - plane excluding the points on the line $2x+3y=6$.
4. entire $XOY$ plane.

Q. If $A$ and $B$ are square matrices each of order $3$ and $|A|=5,|B|=3$, then the value of $|3AB|$ is ______

Q. The least vlue of the function $f\left(x\right)=ax+\frac{b}{x}(a>0,b>0,x>0)$ is _____

Q. The vector equation of a line which passes through the points $(3,4,-7)$ and $(1,-1,6)$ is ______.

Q. The line of shortest distance between two skew lines is _______ to both the lines.

Q. The integrating factor of the differential equation $x\frac{dy}{dx}+2y=x^2$ is _______.

Q. Find $\int \frac{x+1}{(x+2)(x+3)}dx$.

Q. The degree of the differential equation $1+{\left(\frac{dy}{dx}\right)}^2=x$ is _______.

Q. A relation in a set $A$ is called _______ relation, if each element of $A$ is related to itself.

Q. Find the cofactors of all the elements of $\left[\begin{array}{cc}1& -2\\ 4& 3\end{array}\right]$

Q. Let $f\left(x\right)=x|x|$, for all $x\in R$ check its differentiability at $x=0$.

Q. Find the value of ${\sin }^{-1}\left[\sin (-\frac{17\pi }{8})\right]$.

Q. Find the value of ${\int }_1^4|x-5|dx$.

Q. If $f\left(x\right)=x^4-10$, then find the approximate value of $f(2,1)$.

Q. Find the slope of the tangent to the curve $y=2{\sin }^2\left(3x\right)$ at $x=\frac{\pi }{6}$.

Q. If $f\left(x\right)=\frac{4x+3}{6x-4},x\ne \frac{2}{3}$, then show that $\left(fof\right)\left(x\right)=x$, for all $x\ne \frac{2}{3}$. Also, write inverse of $f$.

Q. Check if the relation $R$ in the set $R$ of real numbers defined as
$R=\left\{\right(a,b):a<b\}$ is
(i) symmetric;
(ii) transitive

Q. Solve the equation $x:{\sin }^{-1}\left(\frac{5}{x}\right)+{\sin }^{-1}\left(\frac{12}{x}\right)=\frac{\pi }{2}(x\ne 0)$

Q. Evaluate ${\int }_1^2[\frac{1}{x}-\frac{1}{2x^2}]e^{2x}dx$.