Q. Two tailors, $A$ and $B$, earn Rs. $300$ and Rs. $400$ per day, respectively. $A$ can stitch $6$ shirts and $4$ pairs of trousers while $B$ can stitch $10$ shirts and $4$ pairs of trousers per day. To find how many days should each of them work and if it is desired to produce at least $60$ shirts and $32$ pairs of trousers at a minimum labour cost, formulate this as an LPP and find the number of days $A$ and $B$ worked.

Q. Determine the value of 'k' for which the following function is continuous at $x=3$:
$f\left(x\right)=\{\begin{array}{cc}\frac{(x+3{)}^2-36}{x-3}& ,x\ne 3\\ k& ,x=3\end{array}$

Q. If $x^y+y^x=a^b$, then find $\frac{dy}{dx}$.

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

Q. Find the distance between the planes $2x-y+2z=5$ and $5x-2.5y+5z=20$.

Q. Find: $\int \frac{e^{x}dx}{(e^x-1{)}^2(e^x+2)}$

Q. Of the students in a school, it is known that 30% have 100% attendance and 70% students are irregular. Previous year results report that 70% of all students who have 100% attendance attain A grade and 10% irregular students attain A grade in their annual examination. At the end of the year, one student is chosen at random from the school and he was found to have an A grade. What is the probability that the student has 100% attendance? Is regularity required only in school? Justify your answer.

Q. If $e^y(x+1)=1$, then show that $\frac{d^{2}y}{d{x}^2}={\left(\frac{dy}{dx}\right)}^2$.

Q. Find matrix $A$ such that $\left[\begin{array}{cc}2& -1\\ 1& 0\\ -3& 4\end{array}\right]A=\left[\begin{array}{cc}-1& -8\\ 1& -2\\ 9& 22\end{array}\right]$

Q. Evaluate: ${\int }_1^4\{|x-1|+|x-2|+|x-4|\}dx$

Q. Find : $\int \frac{dx}{5-8x-x^2}$

Q. Find : $\int \frac{si{n}^{2}x-co{s}^{2}x}{sinxcosx}dx$

Q. Find the area of the $△ABC$, coordinates of whose vertices are $A(4,1),B(6,6)$ and $C(8,4)$.

Q. A die, whose faces are marked $1,2,3$ in red and $4,5,6$ in green, is tossed. Let $A$ be the event "number obtained is even" and $B$ be the event "number obtained is red". Find if $A$ and $B$ are independent events.

Q. There are $4$ cards numbered $1,3,5$ and $7$, one number on one card. Two cards are drawn at random without replacement. Let $X$ denote the sum of the numbers on the two drawn cards. Find the mean and variance of $X$.

Q. If $A$ is a skew-symmetric matrix of order $3$, then prove that det $A=0$.

Q. Solve the following linear programming problem graphically:
Maximize $Z=7x+10y$
subject to the constraints
$4x+6y\le 240$
$6x+3y\le 240$
$x\ge 10$
$x\ge 0,y\ge 0$

Q. If $ta{n}^{-1}\frac{x-3}{x-4}+ta{n}^{-1}\frac{x+3}{x+4}=\frac{3}{4}$, then find the value of $x$.

Q. A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is $10m$. Find the dimensions of the window to admit maximum light trough the whole opening.

Q. The x-coordinate of a point on the line joining the points P(2, 2, 1) and Q(5, 1, -2) is 4. Find its z-coordinate.

Q. Find the area enclosed between the parabola $4y=3x^2$ and the straight line $3x-2y+12=0$

Q. If for any $2\times 2$ square matrix A, $A\left(adjA\right)=\left[\begin{array}{cc}8& 0\\ 0& 8\end{array}\right]$, them write the value of det[A].

Q. Show that the points A, B, C with position vectors $2\hat{i}-\hat{j}+\hat{k},\hat{i}-3\hat{j}-5\hat{k}$ and $3\hat{i}-4\hat{j}-4\hat{k}$ respectively, are the vertices of a right angled triangle. Hence find the area of the triangle.

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

Q. Show that the function $f\left(x\right)=x^3-3x^2+6x-100$ is increasing on R.

Q. Find the value of c in Rolle's theorem for the function $f\left(x\right)=x^3-3x$ in $[-\sqrt{3},0]$.

Q. Using properties of determinants, prove that

Q. The volume of a sphere is increasing at the rate of $8c{m}^3/s$. Find the rate at which its surface area is increasing when the radius of the sphere is $12cm$.

Q. Evaluate: ${\int }_0^x\frac{x\tan x}{\sec x+\tan x}dx$

Q. If $\overrightarrow{a}=2\hat{i}-\hat{j}-2\hat{k}$ and $\overrightarrow{b}=7\hat{i}+2\hat{j}-3\hat{k}$, then express $\overrightarrow{b}$ in the form of $\overrightarrow{b}=\overrightarrow{b_1}+\overrightarrow{b_2}$, where $\overrightarrow{b_1}$ is parallel to $\overrightarrow{a}$ and $\overrightarrow{b_2}$ is perpendicular to $\overrightarrow{a}$.