Q. Form the differential equation representing the family of curves $$y^{2} = m (a^{2} - x^{2})$$, where $$a$$ and $$m$$ are parameters.

Q. If A is a square matrix of order $3$ with $|A|=4$, then write the value of $|-2A|$.

Q. If $y=si{n}^{-1}x+co{s}^{-1}x$ , find $\frac{dy}{dx}$.

Q. Write the order and the degree of the differential equation ${\left(\frac{d^{4}y}{d{x}^4}\right)}^2={[x+{\left(\frac{dy}{dx}\right)}^2]}^3$

Q. If a line has the direction ratios $-18,12,-4$, then what are its direction cosines ?

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. If $\ast$ is defined on the set $R$ of all real numbers by $\ast :a\ast b=\sqrt{a^2+b^2}$, find the identify element, if it exists in $R$ with respect to $\ast$.

Q. If $A=\left[\begin{array}{cc}0& 2\\ 3& -4\end{array}\right]$ and $kA=\left[\begin{array}{cc}0& 3a\\ 2b& 24\end{array}\right]$, then find the values of $k$, $a$ and $b$.

Q. Find : $\int \frac{sinx-cosx}{\sqrt{1+sin2x}}dx,0<x<\frac{\pi }{2}$

Q. Find : $\int (logx{)}^{2}dx$

Q. Find a unit vector perpendicular to both the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, where $\overrightarrow{a}=\hat{i}-7\hat{j}+7\hat{k}$ and $\overrightarrow{b}=3\hat{i}-2\hat{j}+2\hat{k}$.

Q. Show that the vectors $\hat{i}-2\hat{j}+3\hat{k},-2\hat{i}+3\hat{j}-4\hat{k}$ and $\hat{i}-3\hat{j}+5\hat{k}$ are coplanar.

Q. Let $X$ be a random variable which assumes values $x_1,x_2,x_3,x_4$ such that $2P(X=x_1)=3P(X=x_2)=P(X=x_3)=5P(X=x_4)$.
Find the probability distribution of $X$.

Q. A coin is tossed $5$ times. Find the probability of getting (i) at least $4$ heads, and (ii) at most $4$ heads.

Q. Show that the relation $R$ on the set $Z$ of all integer, given by $R=\left\{\right(a,b):2$ divides $(a-b)\}$ is an equivalence relation.

Q. If $ta{n}^{-1}x-co{t}^{-1}x=ta{n}^{-1}\left(\frac{1}{\sqrt{3}}\right),x>0$, find the value of $x$ and hence find the value of $se{c}^{-1}\left(\frac{2}{x}\right)$

Q. Using properties of determinants, find
$\left|\begin{array}{ccc}1& a& a^2\\ 1& b& b^2\\ 1& c& c^2\end{array}\right|$

Q. If $(sinx{)}^y=x+y$, find $\frac{dy}{dx}$.

Q. Find : $\int \frac{sin2x}{(si{n}^{2}x+1)(si{n}^{2}x+3)}dx$

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

Q. Solve the differential equation : $(1+x^2)dy+2xydx=cotxdx$

Q. Let $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ be three vectors such that $|\overrightarrow{a}|=1,|\overrightarrow{b}|=2$ and $|\overrightarrow{c}|=3$. If the projection of $\overrightarrow{b}$ along $\overrightarrow{a}$ is equal to the projection of $\overrightarrow{c}$ along $\overrightarrow{a}$ ; and $\overrightarrow{b},\overrightarrow{c}$ are perpendicular to each other, then find $\cos (a,b):\cos (a,c)$

Q. Find the value of $\lambda$ for which the following lines are perpendicular to each other :
$\frac{x-5}{5\lambda +2}=\frac{2-y}{5}=\frac{1-z}{-1};\frac{x}{1}=\frac{y+\frac{1}{2}}{2\lambda }=\frac{z-1}{3}$
Hence, find whether the lines intersect or not.

Q. If $A=\left[\begin{array}{ccc}1& 1& 1\\ 0& 1& 3\\ 1& -2& 1\end{array}\right]$, find $A^{-1}$.
Hence, solve the following system of equations :
$x+y+z=6$
$y+3z=11$
and $x-2y+z=0$

Q. Find the inverse of the following matrix, using elementary transformations :
$A=\left[\begin{array}{ccc}2& 3& 1\\ 2& 4& 1\\ 3& 7& 2\end{array}\right]$

Q. Find the area of the triangle whose vertices are $(-1,1),(0,5)$ and $(3,2)$

Q. Find the vector and cartesian equations of the plane passing through the points $(2,5,-3),(-2,-3,5)$ and $(5,3,-3)$. Also find the point of intersection of this plane with the line passing through points $(3,1,5)$ and $(-1,-3,-1)$

Q. Find the equation of the plane passing through the intersection of the planes $\overrightarrow{r}$. $(\hat{i}+\hat{j}+\hat{k})=1$ and $\overrightarrow{r}.(2\hat{i}+3\hat{j}-\hat{k})+4=0$ and parallel to x-axis. Hence, find the distance of the plane from x-axis.

Q. There are two boxes $I$ and $II$. Box $I$ contains $3$ red and $6$ black balls. Box $II$ contains $5$ red and 'n' black balls. One of the two boxes, box $I$ and box $II$ is selected at random and a ball is drawn at random. The ball drawn is found to be red. If the probability that this red ball comes out from box $II$ is $\frac{3}{5}$, find the value of 'n'.

Q. A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A require $5$ minutes each for cutting and $10$ minutes each for assembling. Souvenirs of type $B$ require $8$ minutes each for cutting and $8$ minutes each for assembling. There are $3$ hours and $20$ minutes available for cutting and $4$ hours available for assembling. The profit is $Rs.5$ each for type $A$ and $Rs.6$ each for type $B$ souvenirs. How many souvenirs of each type should the company manufacture in order to maximize profit ? Formulate the above $LPP$ and solve it graphically and also find the maximum profit.