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Solve the following linear programming problem graphically :
Minimize
$Z = 13x – 15y$
Subject to constraints
$x + y \le 7$,
$2x – 3y + 6 \le 0$,
$x \ge 0, y \ge 0$

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

Find the general solution of the following differential equation: $x^{2}\frac{dy}{dx}=x^{2}+xy+y^{2}$.

If $I_{1}=\int_{-\pi/4}^{\pi/4}\frac{dx}{1+\cos 2x}$ and $I_{2}=\int_{-1/2}^{1/2}|x|\,dx $, then show that $I_{1}-4I_{2}=0$.

$\text{Find: }\int \frac{x^{2}}{(x^{2}+9)(x^{2}+16)}\,dx.$

$\text{Find }\int \sqrt{\frac{x+2}{x-2}}\,dx.$

$\text{Evaluate: }\int_{0}^{1} x\,\tan^{-1}x\,dx.$

$\text{Evaluate: }\tan\!\left(\sin^{-1}1-\cos^{-1}\!\left(-\frac{1}{2}\right)\right).$

$\text{Simplify: }\tan^{-1}\!\left(\frac{\cos 2x-\sin 2x}{\cos 2x+\sin 2x}\right),\quad 0<x<\frac{\pi}{4}.$

Vectors \(\vec{a}=3\hat{i}-2\hat{j}+2\hat{k}\) and \(\vec{b}=\hat{i}+2\hat{k}\) represent the two adjacent sides of a parallelogram. Find the vectors representing its diagonals and hence find their lengths.

Find the vector of magnitude \(14\) in the direction of \(\overrightarrow{QP}\), where \(P=(1,3,2)\) and \(Q=(-1,0,8)\) respectively.

A room freshner bottle in the shape of an inverted cone sprays the perfume at regular intervals such that volume of the perfume in the bottle decreases at the steady rate of 1 mm3/min. Find the rate at which level of perfume is dropping at an instant when level of perfume in the bottle is 10 mm, if the semi-vertical angle of conical bottle is $\frac{\pi}{6}$

If $ \sqrt{3}\,(x^2+y^2)=4xy$, then find \(\dfrac{dy}{dx}\) at $\left(\frac{1}{2},\,\frac{\sqrt{3}}{2}\right).$

Check whether function \(f(x)\) defined as

\[
f(x)=
\begin{cases}
\dfrac{|x-3|}{2(x-3)}, & x<3, \\[6pt]
\dfrac{x-6}{6}, & x\ge 3
\end{cases}
\]

is continuous at \(x=3\) or not?

The general solution for the differential equation $\frac{dy}{dx} = e^{3x-y}$ is

Evaluate the following integral $\int x^2 dx$

Which of the following cannot be the order of a row-matrix ?

If $2 \cos^{-1} x = y$, then

A matrix B = $[b_{ij}]_{m \times m}$ is said to be a diagonal matrix, if :

For any square matrix A with real entries, if $A + A'$ is a symmetric matrix then :