Q. Using integration find the area of a triangular region whose sides have the equation $y=x+1,y=2x+1$ and $x=2$.
(Draw the figure in answer book)

Q. Prove that ${\tan }^{-1}\left(\frac{2}{9}\right)+{\tan }^{-1}\left(\frac{1}{4}\right)=\frac{1}{2}{\sin }^{-1}\left(\frac{4}{5}\right)$.

Q. The radius of a circle is increasing uniformly at the rate of 5cm/sec. Find the rate at which the area of the circle is increasing when the radius is 6 cm.

Q. Find x, if ${\tan }^{-1}3+{\cot }^{-1}x=\frac{\pi }{2}$.

Q. Show the region of feasible solution under the following constraints $2x+y\le 8,x\ge 0,y\ge 0$ in answer book.

Q. Find $\int \frac{\tan x}{\cot x}dx$.

Q. Construct a 2 $\times$ 2 matrix $A=\left[a_{ij}\right]$, whose elements are given by $a_{ij}=|-5i+2j|$.

Q. Find $\int \log (x^2+1)dx$.

Q. Find $\int \frac{1}{3x^2+6x+2}dx$

Q. Find a unit vector perpendicular to each of the vectors $2\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{a}-2\overrightarrow{b}$, where $\overrightarrow{a}=\hat{i}+2\hat{j}-\hat{k},\overrightarrow{b}=\hat{i}+\hat{j}+\hat{k}$.

Q. Prove that the relation R in set of real number R defined as $R=\left\{\right(a,b):a\ge b\}$ is reflexive and transitive but not symmetric.

Q. By graphical method solve the following linear programming problem for maximization.
Objective function Z = 1000 x + 600 y
Constraints $x+y\le 200$
$4x-y\le 0$
$x\ge 20,x\ge 0,y\ge 0$.

Q. $f\left(x\right)=\{\begin{array}{c}\frac{K\cos x}{\pi -2x};x\ne \frac{\pi }{2}\\ 5;x=\frac{\pi }{2}\end{array}$

Q. Find the area bounded by the parabola $y^2=4x$ and the straight line y = x. (Draw the figure in answer book)

Q. Find the intervals in which the function f given by $f\left(x\right)=x^2-6x+5$ is
(a) Strictly increasing
(b) Strictly decreasing

Q. Find $y=(si{n}^{-1}x{)}^2$, then show that $(1-x^2)\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}=2$.

Q. If $\overrightarrow{a}=2\hat{i}-\hat{j}+5\hat{k}$ and $\overrightarrow{b}=4\hat{i}-2\hat{j}+\lambda \hat{k}$ such that $\overrightarrow{a}||\overrightarrow{b}$, find the value of $\lambda$.

Q. Find the value of x:

Q. From a lot of 30 bulbs which include 6 defectives, a sample of 2 bulbs are drawn at random with replacement. Find the probability distribution of the number of defective bulbs.

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

Q. Find the direction cosine of the line $\frac{x}{4}=\frac{y}{7}=\frac{z}{4}$

Q. Find the angle between planes $\overrightarrow{r}.(\hat{i}-\hat{j}+\hat{k})=5$ and $\overrightarrow{r}.(2\hat{i}+\hat{j}-\hat{k})=7$

Q. Find $\int \frac{(x-1)(x-logx{)}^3}{x}dx$.

Q. If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are unit vectors such that $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0$, find the value of $\overrightarrow{a}.\overrightarrow{b}+\overrightarrow{b}.\overrightarrow{c}+\overrightarrow{c}.\overrightarrow{a}$.

Q. If $A$ and $B$ are independent events with $P\left(A\right)=0.2$ and $P\left(B\right)=0.5$, then find the value of $P(A\cup B)$.

Q. Consider $f:R\to R$ given by $f\left(x\right)=2x+3$. Show that f is invertible. Find also the inverse of function f.

Q. Prove that $\left|\begin{array}{ccc}a& a^2& b+c\\ b& b^2& c+a\\ c& c^2& a+b\end{array}\right|=(a+b+c)(a-b)(b-c)(c-a)$.

Q. Find the general solution of the differential equation $\frac{dy}{dx}-\frac{y}{x}=0$.

Q. Bag A contains 2 red and 3 black balls while another bag B contains 3 red and 4 black balls. One ball is drawn at random from one of the bag and it is found to be red. Find the probability that it was drawn from bag B.

Q. If $[x-3]\left[\begin{array}{c}2x\\ 6\end{array}\right]=0$, then find the value of x.