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$e_1$ and $e_2$ are the eccentricities of two conics S and $S^1$. If ${e_1}^2+{e_2}^2$ = 3 then both S and $S^1$ can be

The number of solutions of the equation $z^2$ + $\overline{z}$ = 0 is

An event A has the outcomes $w_1,w_2,w_3,and{w}_4$ with probabilities $\frac{1}{12},\frac{1}{12},\frac{1}{6}and\frac{1}{3}$.If the

probability of event A is P(A),find 24 P(A).

For ${\log }_{2x}{x}^2-1$ to be defined,

Which term of the A.P - 53, -47, -41, is the first positive term.

If $R=\left\{\right(x,y)|x,y\in Z,x^2+y^2\le 4\}$ is a relation in $Z$, then domain of $R$ is

Number of $4$ digit numbers of the form

$N=abcd,$ which satisfy following conditions :

$\left(i\right)4000\le N<6000$

$\left(ii\right)N$ is a multiple of $5$

$\left(iii\right)3\le b<c\le 6$ is equal to

For the given sequence $\log a,\log \left(ab\right),\log \left(a{b}^2\right)$, where $a,b,c>0$, the ${10}^{th}$ term is

If the $(r+1{)}^{th}$ term in the expansion of ${(\sqrt[3]{3\frac{a}{\sqrt{b}}}+\sqrt{\frac{b}{\sqrt[3]{3a}}})}^{21}$ has the same power of a and b, then value of r is

Check whether the following probabilities P(A) and P(B) are consistently defined
(i) $P\left(A\right)=0.5,P\left(B\right)=0.7,P(A\cap B)=0.6$
(ii) $P\left(A\right)=0.5,P\left(B\right)=0.4,P(A\cup B)=0.8$

The sum of the squares of perpendicular on any tangent of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}$ = 1 from two Points on the minor axis each one at a distance of units from the centre is

The value of $\underset{n\to \infty }{lim}\frac{3^n+2^n}{3^n-2^n}$ is

Let P(A)=P(B), prove that A=B

The equation of chord to the parabola $y^2=4x$ whose sum of ordinates and product of abscissas of the endpoints of the chord is $4$ and $9$ respectively, is :

The domain of the function $f\left(x\right)=\frac{1}{\sqrt{x+|x|}},$ is

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

In the xy plane, the segment with end points (3, 8) and (–5, 2) is the diameter of the circle. The point (k, 10) lies on the circle for

The straight lines joining origin and the point of intersections of the curve $y^2$ =4x with the line x+y+n = 0 are perpendicular. Find the value of $n^2$.

Let $F_1(x_1,0)$ and $F_2(x_2,0),$ where, $x_1<0$ and $x_2>0$ be the foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{8}=1$ suppose a parabola having vertex at the origin and focus at $F_2$ intersects the ellipse at point M in the first quadrant and at point N in the fourth quadrant.

If the tangents to the ellipse at M and N meet at R and the normal to the parabola at M meets the X-axis at Q then the ratio of area of $\Delta MQR$ to area of the quadrilateral $M{F}_{1}N{F}_2$ is

What is the equation of chord joining 2 points of a parabola given by parameters $t_{1}and{t}_2$.

Let $f=\left[\right(x,\frac{x^2}{1+x^2}):xϵR]$ be a function from R into R. Determine the range of f.

Out of the following which is a skew- symmetric matrix

For all real permissible values of $m,$ if the straight line $y=mx+\sqrt{9m^2-4}$ is tangent to a hyperbola, then equation of the hyperbola can be

How many triangles can be formed by joining 10 points on a circle?

The vertices of the ellipse $\frac{(x+1{)}^2}{25}+\frac{(y-3{)}^2}{16}$ = 1 is

The probability that a teacher will give an unannounced test during any class meeting is $\frac{1}{5}$.If a student is absent twice, the probability that he will miss atleast one test, is

If the third term in the binomial expansion of ${(1+x^{{\log }_{2}x})}^5$ equals $2560$, the a possible value of $x$ is :

From the sets given below, select equal sets :

A = {2, 4, 8, 12}, B = {1, 2, 3, 4},

C = {4, 8, 12, 14}, D = {3, 1, 4, 2},

E = {-1, 1}, F = {0, a}

G = {1, -1}, H = {0, 1}

The distance between the origin and the tangent to the curve $y=e^{2x}+x^2$ drawn at the point $x=0$ is

If $z_1$ and $z_2$ are complex numbers and $u=\sqrt{z_1{z}_2},$ then $|\frac{z_1+z_2}{2}+u|+|\frac{z_1+z_2}{2}-u|$ is equal to

The mean and variance of 7 observations are 8 and 16 respectively. If five of the observations are 2, 4, 10, 12, 14, find the remaining two observations.

If $A\left(z_1\right),B\left(z_2\right),C\left(z_3\right)$ be the vertices of triangle ABC in which $|\underset{–}{ABC}$ = $\frac{\pi }{4}$ and $\frac{AB}{BC}$ = $\sqrt{2}$ then $z_2$ is equal to

${}^n{C}_r+2{}^n{C}_{r-1}+{}^{n}Cr-2$ =

If the ratio of the $5^{\text{th}}$ term from the beginning to the $5^{\text{th}}$ term from the end in the expansion of ${(\sqrt[4]{42}+\frac{1}{\sqrt[4]{43}})}^n$ is $\sqrt{6}:1,$ then the value of $n$ is

Question 8

ABCD is a rectangle formed by points A (-1,-1), B(-1,4), C(5,4) and D(5,-1). P, Q, R, and S are mid-points of AB, BC, CD, and DA respectively. Is the quadrilateral PQRS a square, rectangle or rhombus? Justify your answer.

Let $z$ be a complex number satisfying $z+z^{-1}=1$. A possible value of $n$ when $z^n+z^{-n}$ is minimum, is

Draw the graph of logarithm function y= ${\log }_{a}x$ When x>0and a>1.