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$\sqrt[2]{2i}$=

In the expansion of $(\frac{1+x}{1-x}{)}^2$, the coefficient of $x^n$

10 students are participating in a debate on 'SAVE WILD LIFE'. Five students have to speak in favour and five students against it. These students are standing on two lines face-to-face whose equations are 3x - y - 4 = 0 and 6x - 2y -4 = 0 for a debate. Are the students standing on parallel lines ?

During the debate, which team you are in favour of? Why?

Show that the equation of the line passing through the origin and making an angle $\theta$ with the line y = mx + c is $\frac{y}{x}=\frac{m\pm tan\theta }{1\mp mtan\theta }$

If 'z' be any complex number such that |3z - 2| + |3z + 2| = 4, then locus of 'z' is:

For any two sets $A$ and $B,$ $A-(A\cap B)$ = ___.

The equation of the diameter of the circle $x^2+y^2+2x-4y-11=0$ which bisects the chords intercepted on the line $2x-y+3=0$ is

In a mathematics class, 20 children had forgotten their rulers and 17 forgotten their pencils, "Go and borrow them from someone at once”, said the teacher, 24 children left the room, then how many children had forgotten both is

If h denotes the number of honest people, p denotes the number of punctual people and a relation between honest people and punctual people is given as h = p + 16. If P denotes the number of people who progress in life and a relation between number of people who progress and honest people is given as

$P=\left(\frac{h}{8}\right)+5.$

Find the relation between number of people who progress in life and punctual people. How does the punctuality is important in the progress of life?

A survey shows that 63% of the people watch news whereas 76% watch sports. If $x$% of people watch both sports and news, then

If $a_1,a_2,a_3,a_4$ are the coefficients of any four

consecutive terms in the expansion of ($1+x{)}^n$, then

$\frac{a_1}{a_1+a_2}$ + $\frac{a_3}{a_3+a_4}$ =

Write the following intervals in set-builder form :

(i) (-3, 0) (ii) [6, 12]

(iii) [-23, 5]

An ellipse with centre at (0,0) cuts x axis at (3,0) and (-3,0). If its e=$\frac{1}{2}$ then the length of the Semiminor axis is

If $f\left(x\right)={\text{log}}_e(1-x)$ and $g\left(x\right)=\left[x\right]$ then find:

(i) $(f+g)\left(x\right)$

(ii) $\left(fg\right)\left(x\right)$

(iii) $\left(\frac{f}{g}\right)\left(x\right)$

(iv) $\left(\frac{g}{f}\right)\left(x\right)$

Also find $(f+g)(-1),\left(fg\right)\left(0\right),\left(\frac{f}{g}\right)(-1),\left(\frac{g}{f}\left(\frac{1}{2}\right)\right)$

If y = sin (sin x) and $\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}$ tan x+f(x) = 0, then f(x) equals.

If the distance between a focus and corresponding directrix of an ellipse be 8 and the eccentricity be $\frac{1}{2}$ ,

then length of the minor axis is

If tan A + tan B + tan C = tan A .tan B . tan C then

In a race between Achilles and tortoise, people assigned probability to Achilles winning and tortoise winning. These probability pairs are listed below. How many of these pairs satisfy the axiomatic approach, assuming only two results are tortoise wins and Achilles wins.

The derivative of log [log(logx)]w.r.t x is:

Let A = {x, y, z} and B = {1, 2}. Find the number of relations from to A to B.

If $|z+i|\le 2$, then the maximum value of $|z+12+6i|$ is

The coefficient of y in expansion of $(y^2+\frac{c}{y}{)}^5$ is :

In how many ways, a party of 5 men and 5 women be seated at a circular table, so that no two women are adjacent?

If x and y are the maximum and minimum values of sin $\theta$ + cos $\theta$, find the value of $x^2$ + $y^2$

$li{m}_{x\to 1}$ $\frac{logx}{x-1}$ =
[Rpet 1996; MP PET 1996; P.CET 2002]

$\frac{si{n}^{2}A-si{n}^{2}B}{sinAcosA-sinBcosB}$ = a when A = ${20}^{\circ }$ and B = ${25}^{\circ }$.Find the value of $\frac{1}{a^2}$.

The probabilities that a student passes in Mathematics, Physics and Chemistry are m, p and c, respectively. Of these subjects, the students has a 75% chance of passing in atleast one, a 50% chance of passing in atleast two and a 40% chance of passing in exactly two. Which of the following relations are true?

Simplify the expression (cosx - sinx) (1+4 sinxcosx)

E = $\frac{cos6x+6cos4x+15cos2x+10}{(cos5x+5cos3x+10cosx)}$ equals to

Find the equation of the line parallel to y-axis and drawn through the point of intersection of the lines x - 7y + 5 = 0 and 3x + y = 0.

Events E and F are such that P(not E or Not F) = 0.25 state whether E and F are mutually exclusive.

Positive numbers $x,y$ and $z$ satisfy
$xyz$ = ${10}^{81}$ and $\left({\log }_{10}x\right)\left({\log }_{10}yz\right)+\left({\log }_{10}y\right)\left({\log }_{10}z\right)=468$, then the value of $({\log }_{10}x{)}^2+({\log }_{10}y{)}^2+({\log }_{10}z{)}^2$ is -

P(a, b) is the mid point of a line segmen between axis. Show that equation of the line is $\frac{x}{a}+\frac{y}{b}=2$

If $(10{)}^9+2(11{)}^1(10{)}^8+3(11{)}^2(10{)}^7+\cdots +10(11{)}^9=k(10{)}^9,$ then $k$ is equal to

A person writes a letter of them to copy the letter and mail to four different persons with instruction that they move the chain similarly. Assuming that the chain is not broken and that it costs 50 paise to mail one letter. Find the amount spent on the postage when 8th set of letter is mailed.

If $A$ and $B$ are two independent events such that $P\left(A\right)>0.5,P\left(B\right)>0.5$, $P(A\cap \overline{B})=\frac{3}{25},P(\overline{A}\cap B)=\frac{8}{25},$ then the value of $P(A\cap B)$ is

If α, β are the roots of a$x^2$ +2bx+c=0 and a+δ, β+δ are the roots of A$x^2$ + 2Bx + C=0 then $\frac{b^2–ac}{B^2-Ac}$ is

Find the number of integral terms in the expansion of $(\sqrt{3}+\sqrt[8]{85}{)}^{256}$

A is a set with 6 elements. So, the number of subsets is:

The sum of 1 + $\frac{2}{5}$ + $\frac{3}{5^2}$ + $\frac{4}{5^3}$ + .............upto n terms is

The value of ${(}^{21}{C}_1{-}^{10}{C}_1)+{(}^{21}{C}_2{-}^{10}{C}_2)+{(}^{21}{C}_3{-}^{10}{C}_3)+{(}^{21}{C}_4{-}^{10}{C}_4)+...+{(}^{21}{C}_{10}{-}^{10}{C}_{10})$ is

The solution set of the inequality $\frac{(x-1{)}^2(x-2)}{(x^2+1)x}$ > 0 is

A body radiates energy 5 W at a temperature of ${127}^{\circ }C$. If the temperature is increased to ${927}^{\circ }C$, then it radiates energy at the rate of

If $A(a{t}^2,2at),B(\frac{a}{t^2},\frac{-2a}{t})$ and C(a, 0), then 2a is equal to

$\frac{1}{tan3A-tanA}-\frac{1}{cot3A-cotA}=$

If sets A and B are defined as
$A=\left\{\right(x,y)|y=\frac{1}{x},x\ne 0,xϵR\},B=\left\{\right(x,y)|y=-x,xϵR\}$, then