Q. Evaluate: $\int \frac{\sin 2x}{(1+\sin x)(2+\sin x)}dx$

Q. If the matrix $\left(\begin{array}{cc}6& -x^2\\ 2x-15& 10\end{array}\right)$ is symmetric, find the value of $x$.

Q. Given that: $A=\left(\begin{array}{ccc}1& -1& 0\\ 2& 3& 4\\ 0& 1& 2\end{array}\right)$ and $B=\left(\begin{array}{ccc}2& 2& -4\\ -4& 2& -4\\ 2& -1& 5\end{array}\right)$, find AB.
Using this result, solve the following system of equation:
$x-y=3,2x+3y+4z=17$ and $y+2x=7$

Q. Find the equation of the plane passing through the point $(1,-2,1)$ and perpendicular to the line joining the points $A(3,2,1)$ and $B(1,4,2)$.

Q. Find the coordinates of the centre, foci and equation of directrix of the hyperbola $x^2-3y^2-4x=8$.

Q. Show that the surface area of a closed cuboid with square base and given volume is minimum when it is a cube.

Q.

Calculate the Spearman's ranks correlation coefficient for the following data and interpret the result:

$\begin{array}{ccccccccccc}X& 35& 54& 80& 95& 73& 73& 35& 91& 83& 81\\ Y& 40& 60& 75& 90& 70& 75& 38& 95& 75& 71\end{array}$

Q. If $a+ib=\frac{x+iy}{x-iy}$ prove that $a^2+b^2=1$ and $\frac{b}{a}=\frac{2xy}{x^2-y^2}$

Q. Verify Langrange's mean value theorem for the function:
$f\left(x\right)=x(1-\log x)$ and find the value of '$c$' in the interval [1, 2].

Q. If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are three mutually perpendicular vectors of equal magnitude, prove that $(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})$ is equally inclined with vectors $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$.

Q. Evaluate: ${\int }_0^{\pi /4}\log (1+\tan \theta )d\theta$

Q. Using L'Hospital's Rule, evaluate:
$\underset{x\to \frac{\pi }{2}}{lim}x\tan x-\frac{\pi }{4}\sec x$

Q. Prove that $\frac{1}{2}{\cos }^{-1}\left(\frac{1-x}{1+x}\right)={\tan }^{-1}\sqrt{x}$

Q. In a class of $60$ students, $30$ opted for Mathematics, $32$ opted for Biology and $24$ opted for both Mathematics and Biology. If one of these students is selected at random, find the probability that:
(i) The student opted for Mathematics or Biology.
(ii) The student has opted neither Mathematics nor Biology.
(iii) The student has opted Mathematics but not Biology.

Q. A problem is given to three students whose chances of solving it are $\frac{1}{4},\frac{1}{5}$ and $\frac{1}{3}$ respectively. Find the probability that the problem is solved.

Q. Solve the equation for x:
${\sin }^{-1}x+{\sin }^{-1}(1-x)={\cos }^{-1}x,x\ne 0$

Q. Bag $A$ contains $1$ white, $2$ blue and $3$ red balls. Bag $B$ contains $3$ white, $3$ blue and $2$ red balls. Bag $C$ contains $2$ white, $3$ blue and $4$ red balls. One bag is selected at random and then two balls are drawn from the selected bag. Find the probability that the balls drawn are white and red.

Q. Solve: $(x^2-y{x}^2)dy+(y^2+x{y}^2)dx=0$

Q. Using properties of determinates, prove that:
$\left|\begin{array}{ccc}a& b& b+c\\ c& a& c+a\\ b& c& a+b\end{array}\right|=(a+b+c)(a-c{)}^2$

Q. Solve: $\frac{dy}{dx}=1-xy+y-x$

Q. By using the data $\overline{x}=25,\overline{y}=30,b_{yx}=1.6$ and $b_{xy}=0.4$, find:
(a) The regression equation $y$ on $x$.
(b) What is the most likely value of $y$ when $x=60$?
(c) What is the coefficient of correlation between $x$ and $y$?

Q. Evaluate: $\int \frac{1}{x^2}{\sin }^2\left(\frac{1}{x}\right)dx$

Q. Find the value of $\lambda$ for which the four points with position vectors $6\hat{i}-7\hat{j},16\hat{i}-19\hat{j}-4\hat{k},\lambda \hat{j}-6\hat{k}$ and $2\hat{i}-5\hat{j}+10\hat{k}$ are coplanar.

Q. Prove that locus of $z$ is circle and find its centre and radius if $\frac{z-i}{z-1}$ is purely imaginary.

Q. If $y=\cos \left(\sin x\right)$, show that:
$\frac{d^{2}y}{d{x}^2}+\tan x\frac{dy}{dx}+y{\cos }^{2}x=0$

Q. Draw a rough sketch of the curve $y^2=4x$ and find the area of the region enclosed by the curve and the line $y=x$.

Q. If $A,B$ and $C$ are the elements of Boolean algebra, simplify the expression $(A^{\prime }+B^{\prime })(A+C^{\prime })+B^{\prime }(B+C)$.
Draw the simplified circuit.

Q. Show that the lines $\frac{x-4}{1}=\frac{y+3}{-4}=\frac{z+1}{7}$ and $\frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+10}{8}$ intersect. Find the coordinates of their point of intersection.

Q. Find the line of best fit for the following data, treating x as dependent variable (Regression equation $x$ on $y$):

$X$	14	12	13	14	16	10	13	12
$Y$	14	23	17	24	18	25	23	24

Hence, estimate the value of $x$ when $y=16$.

Q. If $y-2x-k=0$ touches the conic $3x^2-5y^2=15$, find the value of $k$.