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Two dice
are thrown. The events A, B and C are as follows:

A: getting an even number on the first die.

B: getting an odd number on the first die.

C: getting the sum of the numbers on the dice ≤
5

Describe the events

(i) (ii) not
B (iii) A or B

(iv) A and B (v) A but not C (vi) B or C

(vii) B and C (viii)

The sum of the series $2C_0+\frac{C_1}{2}\cdot 2^2+\frac{C_2}{3}\cdot 2^3+\frac{C_3}{4}\cdot 2^4+\cdots +\frac{C_n}{n+1}\cdot 2^{n+1}$ is equal to , where$\left(C_r{=}^n{C}_r\right)$

The coefficient of the term independent of $x$ in the expansion ${(\sqrt{\frac{x}{3}}+\frac{\sqrt{3}}{x^2})}^{10}$ is

Evaluate the Given limit:

There are 30 pens in a box. Nine are yellow, 4 are black and 7 are green. What's the probability that a pen which is taken from the box without looking will be green?

cot
4x
(sin 5x
+ sin 3x)
= cot x
(sin 5x
– sin 3x)

Two dice are thrown simultaneously. Find the probability of getting a sum of 12.

The option(s) with the value of $a$ and $L$ that satisfy the following equation is(are)

$\frac{\underset{0}{\overset{4\pi }{\int }}{e}^t({\sin }^{6}at+{\cos }^{4}at)dt}{\underset{0}{\overset{\pi }{\int }}{e}^t({\sin }^{6}at+{\cos }^{4}at)dt}=L?$

Match the column

$\begin{array}{cc}\text{Column A}& \text{Column B}\\ \text{1.cos}\frac{2\pi }{7}\text{+ cos}\frac{4\pi }{7}\text{+ cos}\frac{6\pi }{7}& \text{A.0}\\ \text{2.cos}\frac{\pi }{7}\text{+ cos}\frac{2\pi }{7}\text{+cos}\frac{3\pi }{7}\text{+ cos}\frac{4\pi }{7}\text{+ cos}\frac{5\pi }{7}\text{+ cos}\frac{6\pi }{7}& \text{B.}\frac{1}{2}\\ \text{3.sin}\frac{\pi }{11}\text{+ sin}\frac{3\pi }{11}\text{+ sin}\frac{5\pi }{11}\text{+ sin}\frac{7\pi }{11}\text{+ sin}\frac{9\pi }{11}& \text{C.}\frac{-1}{2}\end{array}$

The equation of the line parallel to $Y-$axis and $3$units to the right of it, is

The sum of the roots of equation ${\log }_2(3^{2+x}-6^x)=3+x{\log }_2\left(\frac{3}{2}\right)$ is

If $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{c},\overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{a}$ and a,b,c be the moduli of the vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ respectively, then

If $\int \frac{1}{x(x^5-1)(x^5+1)}dx=A\ln |x|+B\ln |x^5-1|+C\ln |x^5+1|+D,$ then which of the following is/ are correct

(where $A,B,C$ are fixed constants and $D$ is constant of integration)

Given 7 flags of different colours, how many different signals can be generated if a signal requires the use of two flags, one below the other ?

If $\sqrt{3}\cos \theta -\sin \theta =1,$ then $\theta =$

Consider the system of equations ax + by = 0, cx + dy = 0 where a, b, c, d $\epsilon$ {0, 1}. The probability that the system of equations has unique solution is

If $si{n}^{-1}x+si{n}^{-1}y+si{n}^{-1}z=\pi$ then $x^4+y^4+z^4+4x^2{y}^2{z}^2=k(x^2{y}^2+y^2{z}^2+z^2{x}^2)$ where k is equal to

If the coefficient of variation and standard deviation of a distribution are 2 and 0.4 respectively, then mean of the distribution is

If $\cos (\alpha -\beta )=1$ and $\cos (\alpha +\beta )=\frac{1}{2},$ where $\alpha ,\beta \in (-\pi ,\pi ),$ then the number of ordered pairs $(\alpha ,\beta )$ satisfying both the equations is

Number of solution(s) of $2^{\sin |x|}=4^{|\cos x|}$ in $[-\pi ,\pi ]$ is equal to :

The distance between the origin and the point nearest it on the surface $z^2=1+xy$ is

1. 0.125

If the angles of a triangle are in the ratio 3 : 4 : 5, then the sides are in the ratio

Values of x for which the sixth term of the expansion of

$E={(3^{lo{g}_3\sqrt{9^{|x-2|}}}+7^{\left(\frac{1}{5}\right)lo{g}_7\left[\right(4){.3}^{|x-2|}-9]})}^7$ is 567, are

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two vectors such that $|2\overrightarrow{a}+3\overrightarrow{b}|=|3\overrightarrow{a}+\overrightarrow{b}|$ and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ${60}^{\circ }.$ If $\frac{1}{8}\overrightarrow{a}$ is a unit vector, then $\left|\overrightarrow{b}\right|$ is equal to: