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In how many ways 4 Indian and 4 Pakistani army Generals can be seated, half on each side of a long table, so that no two Indian Generals may be together?

(i) Find the number of sitting arrangements.

(ii) Do you feel that periodic meeting between senior officers and politicians are in the interest of both the countries for keeping peace and goodwill in the region? Express your views briefly.

If $\alpha {=}^m{C}_2$, Then ${}^{\alpha }{C}_2$ is equal to ..........

The number of ways in which any four letters can be selected from the word "EXAMINATION” is ______.

The point A(sin $\theta$, cos$\theta$) is 3 units away from the point B (2 cos ${75}^{\circ }$, 2 sin ${75}^{\circ }$). If $0^{\circ }$ $\le \theta <$ ${360}^{\circ }$, then $\theta$ is __ degree

If z = $i\log (2-\sqrt{3})$, then value of cos z is ______ [use $e^{\left({\log }_{e}A\right)}=A$]

How many of the following pairs of function are identcal

(a) $f\left(x\right)=\frac{x^2}{x},g\left(x\right)=x$

(b) $f\left(x\right)=\frac{x^2-1}{x-1},g\left(x\right)=x+1$

(c) $f\left(x\right)={\sec }^{2}x-{\tan }^{2}x,g\left(x\right)=1$

(d) $f\left(x\right)=\frac{x}{x^2},g\left(x\right)=\frac{1}{x}$

If z is a complex number such that $z^2$ = ($\overline{z}{)}^2$,then

$li{m}_{x\to \infty }$$\frac{\left(\sqrt{n}\right)}{(\sqrt{n}+(\sqrt{n+1}\left)\right)}$ =

Find the general solution of the equation $sinmx+sinnx=0$

Or

Find the general solution of the equations $sin2x+sin4x+sin6x=0$

Solve the inequalities: $-3\le 4-\frac{7x}{2}\ge 18$

Given standard equation of ellipse,$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b$, with eccentricity e.

Match the following
$\begin{array}{cc}\text{a) Major axis}& \text{i) 2a(1}-{\text{e}}^2)\\ \text{b) Minor axis}& \text{ii) y=0}\\ \text{c) Double ordinate}& \text{iii) x=0}\\ \text{d) Latus Rectum length}& \text{iv) x=}\frac{\text{-a}}{\text{e}}\\ & \text{v)}\sqrt{1-\frac{{\text{b}}^{\text{2}}}{{\text{a}}^{\text{2}}}}\end{array}$

The term independent of x in the expansion of $(x+\frac{1}{x}+2{)}^m$ is:

An equilateral triangle is inscribed in the parabola $y^2=4ax$ so that one angular point of the triangle is at the vertex of the parabola. Find the length of each side of the triangle.

A, B, C in order to cut a pack of cards replace them after each cut, on the condition that the first who cuts a spade shall win a prize. Find their respective chances.

In a hyperbola e=2 and the length of semitransverse axis is 3 and the length of conjugate axis is

Out of 900 modules tested, 400 modules are from mathematics and 200 are from physics, how many modules are tested from both the subjects:

A function $f\left(x\right)$ is given by $f\left(x\right)=\frac{5^x}{5^x+5},$ then the sum of the series $f\left(\frac{1}{20}\right)+f\left(\frac{2}{20}\right)+f\left(\frac{3}{20}\right)+.....+f\left(\frac{39}{20}\right)$ is equal to:

One urn contains two black balls (labelled $B_1$ and $B_2$). Suppose, the following experiment is performed. One of the two urns is chosen at random. Next a ball is randomly chosen from the urn. Then, a second ball is chosen at random from the same urn without replacing the first ball.

(i) Write the sample space showing all possible outcomes.

(ii) What is the probability that two black balls are chosen ?

(iii) What is the probability that two balls of opposite colour are chosen ?

Find the mean, standard deviation and variance of first 10 multiples of 3.

How many numbers can be formed using the digits 1, 2, 3, 4, 3, 2, 1 so that the odd digits always occupy the odd places?

Form the biconditional statement p $\leftrightarrow$ q, where

p : A natural number n is odd.

q : Natural number n is not divisible by 2.

Find the domain and range of the function given by $f\left(x\right)=\frac{1}{\sqrt{x-\left[x\right]}}$, where [x] is greatest integer function.

The set of real values of x satisfying the equation
${|x-1|}^{lo{g}_3\left(x^2\right)-2lo{g}_x\left(9\right)}=(x-1{)}^7$
is

$f\left(x\right)=\left[x\right]$ is discontinuous at $x=1$ because

$\underset{x\to 0}{lim}[\frac{1}{x}-\frac{lo{g}_e(1+x)}{x^2}]=$

Solve $tanx+tan2x+tan3x=tanxtan2xtan3x.$

OR

prove that $cot\frac{\pi }{24}=\sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{6}.$

The derivative of an even function if exists is

The range of the function $f\left(x\right)=x^2-6x+7$ is

The sum of $n$ terms of the series $1+3+7+15+31+...n$ terms is .

If the sum of the deviations of $50$ observations from $30$ is $50$, then the mean of these observations is:

Solution set of the inequality ${\log }_3(x+2)(x+4)+{\log }_{\frac{1}{3}}(x+2)<\frac{1}{2}{\log }_{\sqrt{3}}7$ is -

$Letf\left(x\right)=\left[x\right]cos\left(\frac{\pi }{[x+2]}\right)where,\left[\right]denotesthegreatestintegerfunction.Then,thedomainoffis$

If $sinx+cosecx=2$, then $si{n}^{n}x+cose{c}^{n}x$ is equal to

[UPSEAT 2002]

If $r(1-m^2)+m(p-q)=0$, then a bisector of the angle between the lines represented by the equation $p{x}^2-2rxy+q{y}^2=0$, is

The value of $co{s}^{-1}x+co{s}^{-1}(\frac{x}{2}+\frac{1}{2}\sqrt{3-3x^2})$ where $(\frac{1}{2}\le x\le 1)$ is equal to

The sum to (n+1) terms of the following series

$\frac{C_0}{2}-\frac{C_1}{3}+\frac{C_2}{4}-\frac{C_3}{5}$ + ......... is

What is the dimensional formula for permittivity of free space?

Find the coefficient of $x^6{y}^3$ in the expansion of $(x+2y{)}^9$

Find the function which closely describes the given graph.

You are given $cosx=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}......;$

$sinx=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}......$

$tanx=x+\frac{x^3}{3}+\frac{2.x^5}{15}......$

Find the value of $\underset{x\to 0}{lim}\frac{xcosx+sinx}{x^2+tanx}$

$P\left(n\right):1+3+5+...+2n-1=n^2$

The statement P(n) is

$\underset{n\to \infty }{lim}\frac{1-2+3-4+5-6+.......2n}{\sqrt{n^2+1}+\sqrt{4n^2-1}}\text{is equal to:}$

Find the coordinates of the points which trisect the line segment joining the points $P(4,2,-6)$ and $Q(10,-16,6).$

If $\alpha ,\beta$ are the roots of the equation $4x^2+3x+7=0$. Then find the value of ${\alpha }^2\beta +{\beta }^2\alpha$

If A+B =${225}^{\circ }$, (1+cotA)(1+cotB) equal to

If a, b, c are in AP, show that

$\frac{1}{(\sqrt{b}+\sqrt{c})},\frac{1}{(\sqrt{c}+\sqrt{a})},\frac{1}{(\sqrt{a}+\sqrt{b})}$ are in AP.

Express $\left[\right(x,y):x^2+y^2=25,\text{where}x,yϵW]$ as a set of ordered pairs.

6 women and 5 men are to be seated in a row so that no 2 men can sit together. Number of ways they can be seated is