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The value of $\frac{1}{r_1^2}+\frac{1}{r_2^2}+\frac{1}{r_3^2}+\frac{1}{r^2}$ is

Find the modulus and argument of the following complex numbers and hence express each of then in polar form:
$\frac{(1-i)}{(cos\frac{\pi }{3}+isin\frac{\pi }{3})}$

The value of i + $i^2$ + $i^3$ + $i^4$ is __

Find the value of $\sum _{r=1}^n(-1{)}^{r+1}$${}^{n-1}{C}_{r-1}$

Find the mean deviation about the mean for the following data.

$\begin{array}{cccccc}{x}_i& 10& 30& 50& 70& 90\\ f_i& 4& 24& 28& 16& 8\end{array}$

Words are made with or without meaning from letters of the word AGAIN. If these words are written as in a dictionary , what is the ${55}^{th}$ word?

$\sqrt{3}cosec{20}^{\circ }-sec{20}^{\circ }$=

[IIT 1988]

Sets A and B have 3 and 6 elements respectively. What can be the minimum number of elements in A UB

If $\alpha$ and $\beta$ are imaginary cube roots of unity and x = a+b,y = a$\alpha$ + b$\beta$, z= a$\beta$ + b$\alpha$, then xyz =

If the aritmetic, geometric and harmonic menas between two positive real numbers be A, G and H, then

$li{m}_{x\to 1}$ $\frac{1}{|1-x|}$ =

If $A\cap B^{\prime }=\varphi$ then prove that $A=A\cap B$ and hence show that $A\subseteq B$.

Let $a_n$ be the $n^{th}$ term of a G. P. of positive terms. If $\sum _{n=1}^{100}{a}_{2n+1}=200$ and $\sum _{n=1}^{100}{a}_{2n}=100$, then $\sum _{n=1}^{200}{a}_n$ is equal to :

If $-4\le |x|\le 2$, then $x$ belongs to

The solution set of $x^2-4x+1<0$ is

For each of the following statements, determine whether an inclusive 'or' or exclusive 'or' is used. Give reasons for your answer.

(i) For identification you need a passport or an Ahar Card.

(ii) the school is closed if it is a holiday or a Sunday.

(iii) $\sqrt{3}$ is a rational number or an irrational number.

(iv) Two lines intersect at a point or are parallel.

(v) Students can take Sanskrit or French as their third language.

Find n, if the ratio of the fifth term from the beginning to fifth term from the end in the expansion of ${(\sqrt[4]{42}+\frac{1}{\sqrt[4]{43}})}^n$ is $\sqrt{6}:1.$

Or

Prove that the coeffficient of the middle term in the expansion of $(1+x{)}^{2n}$ is equal to the sum of the coefficient of middle terms in the expansion of $(1+x{)}^{2n-1}.$

If in the expansion of $(1+x{)}^n$, a, b, c are three

consecutive coefficients, then n =

The equation of the locus of foot of perpendiculars drawn from the origin to the line passing through a fixed point (a,b), is

The centroid of a triangle, whose vertices are (2,1), (5,2) and (3,4), is

1. 1.5

Find the general solution of each of the equations:

$\left(i\right)sin2x=-\frac{1}{2}$

(ii)tan 3x =-1

Solve : x ( $3^x$ - 1) ($4^x$ - 16 ) ( x- 5) < 0

Find the sum of 10 terms if the nth term of a sequence is $3^n$- 2.

A particle performing $SHM$ with frequency $10Hz$ and amplitude $5cm$ is initially in left extreme position. The equation of its displacement will be ($x$ is in metre)

Find the equation of the set of points which are equidistant from the points $A(1,2,3)$ and $B(3,2,-1).$

In the figure shown a liquid is flowing through a tube at the rate of 0.1 m³/sec. The tube is branched into two semi circular tubes of cross sectional area A/3 and 2A/3. The velocity of liquid at Q is (the cross-section of the main tube (A) = 10^-2 m² and V_P = 20 m/sec.):

In the polynomial

(x - 1)(x - 2)(x - 3)............... .........(x - 100), the coefficient of

$x^{99}$ is

Solve for x: $x^2$ - x - 6 > 0

Find the number of dissimilar terms in the expansion of $(a+b{)}^{100}$

If $z$ is a complex number of unit modulus and argument $\theta$, then arg$\left(\frac{1+z}{1+\overline{z}}\right)$ is equal to

Let A = {1, 2, 3}. The total number of distinct relations that can be defined over A is

If the sum of the n terms of G.P. is S product is P and sum of their inverse is R, than $P^2$ is equal to

Find the principal solution of sin x + sin 3x +sin 5x = 0

A birdlover at $p(x,y,z)$ from a house watches two birds sitting on the branches of another tree at $A(2,5,8)$ and $B(3,7,2)$ such that AP=BP , show that $2x+4y-2z+31=0$

Find the value of$\underset{x\to 2}{lim}\left[x\right]$, where [x] represents greatest integer less than or equal to x.

A random variable $X$ has probability distribution

$\begin{array}{ccccccccc}X& 1& 2& 3& 4& 5& 6& 7& 8\\ P\left(X\right)& 0.13& 0.22& 0.12& 0.21& 0.13& 0.08& 0.06& 0.05\end{array}$

If events are $E=\left\{x\text{is an odd number}\right\},$$F=\left\{x\text{is divisible by}3\right\}$ and $G=\left\{x\text{is less than}7\right\}$, then the value of $P(E\cup (F\cap G\left)\right)$ is

If $\omega$, ${\omega }^2$ are imaginary cube roots of unity and

$\frac{1}{a+\omega }$ + $\frac{1}{b+\omega }$ + $\frac{1}{c+\omega }$ = 2${\omega }^2$ and $\frac{1}{a+{\omega }^2}$ + $\frac{1}{b+{\omega }^2}$ + $\frac{1}{c+{\omega }^2}$ = 2$\omega$, then $\frac{1}{a+1}$ + $\frac{1}{b+1}$ + $\frac{1}{c+1}$ is equal to:

A letter is known to have come from either $\text{TATANAGAR}$ or $\text{CALCUTTA}$. On the envelope, just two consecutive letter $\text{TA}$ are visible. The probability that the letter has come from $\text{CALCUTTA}$ is

If S is the set of distinct values of b for which the folowing system of linear equations
$x+y+z=1$,
$x+ay+z=1$
and $ax+by+z=0$ has no solution, then S is

If $\alpha ,\beta and\gamma$ are the roots of the equation $x^3+3x+2=0and(\alpha -\beta )(\alpha -\gamma ),(\beta -\gamma )(\beta -\alpha ),(\gamma -\alpha )(\gamma -\beta )$ are the roots of equation $y^3-9y^2-216=0$ . Express y in terms of α.

If |x| < 1, then in the expansion of

$(1+2x+3x^2+4x^3+.......{)}^{\frac{1}{2}}$, the coefficient of $x^n$ is

If $cos3\theta$ = $\alpha cos\theta +\beta co{s}^3\theta$, then $(\alpha ,\beta )$ =

POQis a straight line through the origin O,P and Q represent the complex numbers a+ib andc+id respectively and OP=OQ, then

The strength in volumes of a solution containing 30.36 $gmlitr{e}^{-1}$ of $H_2{O}_2$ is