Q. If matrix $A=\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$ and $A^2=KA$, then write the value of $K.$

Q. Write the principle value of ${\tan }^{-1}\left(\sqrt{3}\right)-{\cot }^{-1}(-\sqrt{3})$.

Q. Evaluate: $\int \frac{\cos 2x-\cos 2a}{\cos x-\cos a}dx$.

Q. Consider $f:R+\to [4,\infty ]$ given by $f\left(x\right)=x^2+4$. Show
that $f$ is invertible with the inverse $f^{-1}$ of $f$ given by $f^{-1}\left(y\right)=\sqrt{y-4}$ where $R_{+}$ is the set of all non-negative real numbers.

Q. Find the area of the greatest rectangle that can be inscribed in a ellipes $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.

Q. Find the area of the region bounded by the parabola $y=x^2$ and $y=|x|$.

Q. If $y^x=e^{y-x}$, prove that $\frac{dy}{dx}=\frac{(1+\log y{)}^2}{\log y}$.

Q. A manufacturer considers that men and women workers are equally efficient and so he pays them at the same rate. He has $30$ workers (male and female) and $17$ units capital; which he uses to produce two types of goods $A$ and $B$. To produce one unit of $A$, $2$ workers and $3$ units of capital are required while $3$ workers and $1$ unit of capital is required to produce one unit of $B$. If $A$ and $B$ are priced at $Rs.100$ and $Rs.120$ per unit respectively, how should he use his resources to maximise the total revenue? Form the LPP and solve graphically.
Do you agree with this view of the manufacturer that men and women workers are equally efficient and so should be paid at the same rate?

1. $1290$
2. $1260$
3. $1130$
4. $3421$

Q. The probabilities of two students A and B coming to the school in time are $\frac{3}{7}$ and $\frac{5}{7}$ respectively. Assuming that the events' A coming in time' and 'B coming in time' are independent. Find the probability of only one of them coming to the school in time. Write atleast one advantages of coming to school in time.

Q. If matrix $A=\left[\begin{array}{cc}2& -2\\ -2& 2\end{array}\right]$ and $A^2=pA$, then write the value of $p$.

Q. Find the Cartesian equation of the line which passes through the point (-2, 4, -5) and is parallel to the line $\frac{x+3}{3}=\frac{4-y}{5}=\frac{z+8}{6}$.

Q. The money to be spent for the walfare of the employees of a firm is proportional to the rate of change of its total revenue (marginal revenue). If the total revenue (in rupees) received from the sales of $x$ units of a product is given by $R\left(x\right)=3x^2+36x+5$, find the marginal revenue, when $x=5,$ and write which value does the question indicates.

Q. Find the length of the perpendicular drawn from the origin to the plane $2x-3y+6z+21=0$.

Q. Find the particular solution of the differential equation $(ta{n}^{-1}y-x)dy=(1+y^2)dx$, given that when $x=0,y=0$.

Q. Write the differential equation representing the family of curves $y=mx$, where $m$ is an arbitrary constant.

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

Q. Evaluate $\int \frac{dx}{x(x^5+3)}.$

Q. For what value of $x$, is the matrix $A=\left[\begin{array}{ccc}0& 1& -1\\ -1& 0& 3\\ x& -3& 0\end{array}\right]$ a skew symmetric matrix?

Q. Using properties of determinants prove the following:
$\left|\begin{array}{ccc}x& x+y& x+2y\\ x+2y& x& x+y\\ x+y& x+2y& x\end{array}\right|=9y^2(x+y).$

Q. Show that : $tan\left(\begin{array}{c}\frac{1}{2}si{n}^{-1}\frac{3}{4}\end{array}\right)=\frac{4-\sqrt{7}}{3}$.

Q. In a hockey match, both teams $A$ and $B$ scored same number of goals up to the end of the game, so to decide the winner, the referee asked both the captains to throw a die alternately and decided that the team, whose captain gets a six first, will be declared the winner. If the captain of team $A$ was asked to start, find their respective probabilities of winning the match and state whether the decision of the referee was fair or not.

Q. Write the value of $ta{n}^{-1}\left[2\sin \right(2{\cos }^{-1}\frac{\sqrt{3}}{2}\left)\right]$.

Q. Show that the lines
$\overrightarrow{r}=3\hat{i}+2\hat{j}-4\hat{k}+\lambda (\hat{i}+2\hat{j}+2\hat{k})$
$\overrightarrow{r}=5\hat{i}-2\hat{j}+\mu (3\hat{i}+2\hat{j}+6\hat{k})$
are intersecting. Hence find their point of intersection.

Q. Differentiate the following with respect to $x$: ${\sin }^{-1}\left(\begin{array}{c}\frac{2^{x+1}{3}^x}{1+(36{)}^x}\end{array}\right).$

Q. If $A_{ij}$ is the cofactor of the element $a_{ij}$ of the determinant $\left[\begin{array}{ccc}2& -3& 5\\ 6& 0& 4\\ 1& 5& -7\end{array}\right]$ then write the value of $a_{32}.A_{32}.$

Q. Find the equation of the plane passing through the line of intersection of the planes $\overrightarrow{r}.(\hat{i}+3\hat{j})-6=0$ and $\overrightarrow{r}.(3\hat{i}-\hat{j}-4\hat{k})=0$ whose perpendicular distance from origin is unity.

Q. $P$ and $Q$ are two points with position vectors $3\overrightarrow{a}-2\overrightarrow{b}$ and $\overrightarrow{a}+\overrightarrow{b}$ respectively. Write the position vector of a point R which divides the line segment $PQ$ in the ratio $2:1$ externally.

Q. If $\overrightarrow{a}=\hat{i}-\hat{j}+7\hat{k}$ and $\overrightarrow{b}=5\hat{i}-\hat{j}+\lambda \hat{k}$, then find the value of $\lambda$, so that $\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{a}-\overrightarrow{b}$ are perpendicular vectors.

Q. Find the value of $k$, for which $f\left(x\right)=\left\{\begin{array}{cc}\frac{\sqrt{1+kx}-\sqrt{1-kx}}{x}& if-1\le x<0\\ \frac{2x+1}{x-1}& if0\le x<1\end{array}\right\}$ is continuous at $x=0$.

Q. Find $|\overrightarrow{x}|$ if for a unit vector $\overrightarrow{a},(\overrightarrow{x}-\overrightarrow{a}).(\overrightarrow{x}+\overrightarrow{a})=15.$