Q. Using properties of definite integrals, evaluate:
${\int }_0^{\frac{\pi }{2}}\frac{\sin x-\cos x}{1+\sin x\cos x}dx$

Q. Find the area of the region bound by the curves $y=6x-x^2$ and $y=x^2-2x$.

Q. The two lines of regressions are $x+2y-5=0$ and $2x+3y-8=0$ and the variance of $x$ is $12$. Find the variance of $y$ and the coefficient of correlation.

Q. An urn contains $10$ white and $3$ black balls while another urn contains $3$ white and $5$ black balls. Two balls are drawn from the first urn and put into the second urn then a ball is drawn from the latter. Find the probability that it is a white ball.

Q. Evaluate: $\int {\tan }^{3}xdx$

Q. Find the Cartesian equation of the plane, passing through the line of intersection of the planes: $\overrightarrow{r}.(2\hat{i}+3\hat{j}-4\hat{k})+5=0$ and $\overrightarrow{r}.(\hat{i}-5\hat{j}+7\hat{k})+2=0$ and intersecting $y$-axis at $(0,3)$

Q. Calculate Karl Pearson's coefficient of correlation between $x$ and $y$ for the following data and interpret the result:
$(1,6),(2,5),(3,7),(4,9),(5,8),(6,10),(7,11),(8,13),(9,12)$

Q. A company manufactures two types of products $A$ and $B$. Each unit of $A$ requires $3$ grams of nickel and $1$ gram of chromium, while each unit of $B$ requires $1$ gram of nickel and $2$ grams of chromium. The firm can produce $9$ grams of nickel and $8$ grams of chromium. The profit is Rs. $40$ on each unit of product of type $A$ and Rs. $50$ on each unit of type $B$. How many units of each type should the company manufacture so as to earn maximum profit? Use linear programming to find the solution

Q. A pair of dice is thrown. What is the probability of getting an even number on the first die or a total of $8$?

Q. Find the equation of the standard ellipse, taking its axes as the coordinate axes, whose minor axis is equal to the distance between the foci and whose length of latus rectum is $10$. Also, find its eccentricity.

Q. Solve for $x$, if
$\tan \left({\cos }^{-1}x\right)=\frac{2}{\sqrt{5}}$

Q. A rectangle is inscribed in a semicircle of radius $r$ with one of its sides on the diameter of the semicircle. Find the dimensions of the rectangle to get maximum area. Also, find the maximum area.

Q. Find the locus of a complex number, $z=x+iy$, satisfying the relation $\left|\frac{z-3i}{z+3i}\right|\le \sqrt{2}$
Illustrate the locus of $z$ in the Argand plane.

Q. Express $\frac{2+i}{(1+i)(1-2i)}$ in the form of $a+ib$. Find its modulus and argument.

Q. A bill of Rs. $1800$ drawn on 10th September, $2010$ at $6$ months was discounted for Rs. $1782$ at a bank. If the rate of interest was $5\%$ per annum, on what date was the bill discounted.

Q. Solve the following differential equation:
$x^{2}dy+(xy+y^2)dx=0$, when $x=1$ and $y=1$

Q. A committee of $4$ persons has to be chosen from $8$ boys and $6$ girls, consisting of at least one girl. Find the probability that the committee consists of more girls than boys.

Q. Write the Boolean function corresponding to the switching circuit given below:

Q. Solve the differential equation:
$x\frac{dy}{dx}+y=3x^2-2$

Q. Evaluate: $\int \frac{\sin x+\cos x}{\sqrt{9+16\sin 2x}}dx$

Q. The demand function is $x=\frac{24-2p}{3}$ where $x$ is the number of units demanded and $p$ is the price per unit. Find:
$\left(i\right)$ The revenue function $R$ in terms of $p$.
$\left(ii\right)$ The price and the number of units demanded for which the revenue is maximum.

Q. The marks obtained by $10$ candidates in English and Mathematics are given below:

Marks in English	$20$	$13$	$18$	$21$	$11$	$12$	$17$	$14$	$19$	$15$
Marks in Mathematics	$17$	$12$	$23$	$25$	$14$	$8$	$19$	$21$	$22$	$19$

Estimate the probable score for Mathematics if the marks obtained in English are $24$.

Q. The difference between mean and variance of a binomial distribution is $1$ and the difference of their squares is $11$. Find the distribution.

Q. Find the image of the point $(2,-1,5)$ in the line $\frac{x-11}{10}=\frac{y+2}{-4}=\frac{z+8}{-11}$
Also, find the length of the perpendicular from the point $(2,-1,5)$ to the line.

Q. In an automobile factory, certain parts are to be fixed into the chassis in a section before it moves into another section. On a given day, one of the three persons $A,B$ and $C$ carries out this task. $A$ has $45$% chance, $B$ has $35$% chance and $C$ has $20$% chance of doing the task. The probability that $A,B$ and $C$ will take more than the allotted time is $\frac{1}{6},\frac{1}{10}$ and $\frac{1}{20}$ respectively. If it is found that the time taken is more than the allotted time, what is the probability that $A$ has done the task?

Q. Find a unit vector perpendicular to each of the vectors $\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{a}-\overrightarrow{b}$ where $\overrightarrow{a}=3\hat{i}+2\hat{j}+2\hat{k}$ and $\overrightarrow{b}=\hat{i}+2\hat{j}-2\hat{k}$.

Q. Using L' Hospital rule, evaluate: $\underset{x\to 0}{lim}(\frac{1}{x^2}-\frac{\cot x}{x})$

Q. Find the matrix for which:
$\left[\begin{array}{cc}5& 4\\ 1& 1\end{array}\right]X=\left[\begin{array}{cc}1& -2\\ 1& 3\end{array}\right]$

Q. A man borrows Rs. $20,000$ at $12\%$ per annum, compounded semi-annually and agrees to pay it in $10$ equal semi-annual installments. Find the value of each installment, if the first payment is due at the end of two years.

Q. Solve the following system of linear equations using matrix method:
$3x+y+z=1,2x+2z=0,5x+y+2z=2$