Q. Form the differential equation of the family of circles having centre on $y$-axis and radius 3 units

Q. Given that the two numbers appearing on throwing two dice are different. Find the probability of the event ' the sum of numbers on the dice is $4$'.

Q. Define collinear vectors.

Q. Evaluate ${\int }_0^{2/3}\frac{dx}{(4+9x^2)}$

Q. Which rules did he break as a school boy?

Q. Find the position vector of a point $R$ which divides the line joining two points $P$ and $Q$ whose position vectors are $\hat{i}+2\hat{j}-\hat{k}$ and $-\hat{i}+\hat{j}+\hat{k}$ respectively in the ratio $2:1$

Q. If $P\left(A\right)=\frac{3}{5}$ and $P\left(B\right)=\frac{1}{5}$, then find $P(A\cap B)$, if $A$ and $B$ are independent events.

Q. Let $\ast$ be the binary operation on $N$ given by $a\ast b=$ LCM of $a$ and $b$. Find
$5\ast 7$ $20\ast 16$

Q. Drive the equation of a plane perpendicular to a given vector and passing through a given point both in vector and Cartesian form.

Q. Find the principle value of ${\text{cosec}}^{-1}(-\sqrt{2})$

Q. Define feasible region in LPP.

Q. Find the probability distribution of number of heads in two tosses of a coin.

Q. Find the approximate change in the volume of a cube of a side $x$ meters caused by increasing the side by $3$%.

Q. What was the consequence of buying jalebis with the fees money?

Q. If $A$ is a square matrix with $\left|A\right|=8$. Find the value of $\left|A{A}^{-1}\right|$.

Q. The probability that a bulb produced by a factory will fuse after 150 days of use is 0.05. Find the probability that out of 5 such bulbs.
(i) none
(ii) not more than one
(iii) more than one
(iv) at least one
will fuse after 150 days of use.

Q. Let $f:N\to R$ be defined by $f\left(x\right)=4x^2+12x+15$. show that $f:N\to S$ where S is the range of fuction $f$, is invertible. Also find the inverse of $f$.

Q. Find the values of $K$ if Area of the triangle is $4$ sq. units and vertices are $\left(k0\right)$ $\left(40\right)$ $\left(02\right)$ using determinants.

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

Q. If $:R\to R$ defined by $f\left(x\right)=1+x^2$, then show that $f$ is neither $1-1$ nor onto

Q. If $ax+b{y}^2=\cos y$, find $\frac{dy}{dx}$.

Q. Integrate $\int \frac{{\tan }^4\sqrt{x}{\sec }^2\sqrt{x}}{\sqrt{x}}dx$ with respect to $x$.

Q. Solve the equation ${\tan }^{-1}\left(\frac{1-x}{1+x}\right)=\frac{1}{2}{\tan }^{-1}x,x>0$

Q. Find the order and degree of the differential equation ${\left(\frac{dy}{dx}\right)}^2+\frac{dy}{dx}-{\sin }^{2}y=0$

Q. Find the area of the parallelogram whose adjacent sides are determined by the vector.
$\overrightarrow{a}=\hat{i}-\hat{j}-3\hat{k}$ and $\overrightarrow{b}=2\hat{i}-7\hat{j}+\hat{k}$

Q. Find the direction collinear of a line which makes equal angles with the positive co-ordinate axes.

Q. If $y=\cos \sqrt{x}$ find $\frac{dy}{dx}$

Q. Show that: ${\sin }^{-1}\left(2x\sqrt{1-x^2}\right)=2{\cos }^{-1}x,\frac{1}{\sqrt{2}}\le x\le 1$

Q. Construct $2\times 2$ matrix $A=\left|a_{ij}\right|$ where elements are given by $a_{ij}=\frac{i}{j}$

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