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The number of ways of distributing $4$ blue balls $5$ yellow balls and $3$ red balls among $4$ children (considering ball of same colour as identical) is

If $\cot \theta =\frac{1}{2}$ and $\sec \varphi =-\frac{5}{3}$, where $\theta \in (\pi ,\frac{3\pi }{2})$ and $\varphi \in (\frac{\pi }{2},\pi )$, then the value of $\cot (\theta -\varphi )$ is

Prove the following,

$3si{n}^{-1}x=si{n}^{-1}(3x-4x^3),xϵ[-\frac{1}{2},\frac{1}{2}]$

1. 60.55

If $\frac{(2+1)(2^2+1)(2^4+1)(2^8+1)}{(2^8-1)}=4^n+1,$ then the value of $n$ is

If the lines $ax+y+1=0,x+by+1=0$ and $x+y+c=0(a,b,c$ being distinct and different from $1)$ are concurrent, then the value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is:

A question can be solved by $2$ methods. The probability of getting the correct answer using the method I and II are $\frac{1}{7}$ and $\frac{1}{8}$ respectively. If the problem was solved incorrectly then the probability that method I is

If $3x+5y+17=0$ is polar for the circle $x^2+y^2+4x+6y+9=0$, then the pole is

Given two sets $A=\{a,b,c,d\},B=\{b,c,d,e\}$, then $n\left[\right(A\times B)\cap (B\times A\left)\right]$ is

If the inequality $\sqrt{\frac{-3}{x^2+2x+10}}\ge 0$ holds good, then $x\in$

If $y=y\left(x\right)$ is the solution of the differential equation $\frac{dy}{dx}+\left(\tan x\right)y=\sin x,0\le x\le \frac{\pi }{3},$ with $y\left(0\right)=0,$ then $y\left(\frac{\pi }{4}\right)$ equal to :

If, $f\left(x\right)=cos\left[{\pi }^2\right]x+cos[-{\pi }^2]x,$ where [x] stands for greatest integer function, then

Multiplicative inverse of $\left(\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 4\end{array}\right)$is

A player tosses a coin and scores one point for every head and two point for every tail that truns up. He plays on until his scores reaches or psses $n$. $P_n$ denotes the probability of getting a scores of exactly $n$

$$\begin{array}{cc}\text{List I}& \text{List II}\\ \begin{array}{cc}\text{(a)}& \text{the value of}{P}_n\text{is}\end{array}& \text{(p)}1\\ \begin{array}{c}\text{(b) the value of}{P}_n+\frac{1}{2}{P}_{n-1}\end{array}& \text{(q)}\frac{5}{4}\\ \begin{array}{c}\text{(c)}2P_{101}+P_{100}\end{array}& \text{(r)}2\\ \begin{array}{c}\text{(d)}{P}_1+P_2\end{array}& \text{(s)}\frac{1}{2}[P_{n-1}+P_{n-2}]\end{array}$$

$\text{Which of the following is the only}\text{incorrect option?}$

On a multiple choice examination with three possible answer for each of the five questions, what is the probability that a candidate would get four or more correct answer just by guessing ?

The mean age of 50 persons was found to be 32 years. Later it was detected that the age 28 was wrongly noted as 35, the age 57 was wrongly noted as 30 and the age 60 was wrongly noted as 32. Then the correct mean age is

Let total revenue received from the sale of $x$ units of a product is given by $R\left(x\right)=12x+2x^2+6.$

Then the slope of marginal revenue is



[2 marks]

$\text{If}z+x+iy\text{then}\left(z\right)=\sqrt{x^2+y^2}{z}_1=2-i,z_2=1+i,\text{find}\left|\frac{z_1+z_2+1}{z_1-z_2+i}\right|.$

Calculate the mean deviation about median for the following data

$\begin{array}{cccccc}{X}_i& 3& 9& 10& 12& 13\\ f_i& 3& 4& 5& 2& 4\end{array}$

Two matrices $A=[a_{ij}{]}_{p\times q},B=[b_{ij}{]}_{m\times n}$ are equal if $p=m,q=n$ and for all $i,j$

Find the number of solutions for the equation $co{s}^{-1}\left(cosx\right)=-\frac{1}{2}$.

In $△$ ABC, if cot A, cot B, cot C be in A. P. then $a^2,b^2,c^2$ are in

The area of the triangle formed by the coordinate axes and a tangent to the curve $xy=a^2$ at the point $(x_1,y_1)$ on it is

The solution of $\frac{xdx+ydy}{ydx-xdy}=\frac{x\sin (x^2+y^2)}{y^3}$ is

(where $c$ is constant of integration)