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The probability of simultaneous occurrence of at least one of the two events $X$ and$Y$ is $a$. If the probability that exactly one of the events $X, Y$ occurs is $b$, prove that $P(X') + P(Y') = 2 – 2a + b$.

Out of two bags, bag I contains 3 red and 4 white balls and bag II contains 8 red and 6 white balls. A die is thrown. If it shows a number less than 3 then a ball is drawn at random from bag I, otherwise a ball is drawn at random from bag II. Find the probability that the ball drawn from one of the bags is a red ball.

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