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If $z_1,z_2,z_3$ are vertices of an equilateral triangle $ABC$ such that $|z_1-i|=|z_2-i|=|z_3-i|$, then $|z_1+z_2+z_3|$ equals to

Let the circles $C_1:x^2+y^2=9$ and $C_2:(x-3{)}^2+(y-4{)}^2=16,$ intersect at the points $X$ and $Y.$ Suppose that another circle $C_3:(x-h{)}^2+(y-k{)}^2=r^2$ satisfies the following conditions:



$\left(i\right)$ centre of $C_3$ is collinear with the centres of $C_1$ and $C_2.$



$\left(ii\right)$$C_1$ and $C_2$ both lie inside $C_3$, and



$\left(iii\right)$ $C_3$ touches $C_1$ at $M$ and $C_2$ at $N$



Let the line through $X$ and $Y$ intersect $C_3$ at $Z$ and $W$, and let a common tangent of $C_1$ and $C_3$ be the tangent to the parabola $x^2=8\alpha y.$

There are some expressions given in $List-I$ whose values are given in $List-II$ below:



$\begin{array}{cccc}& \text{List}I& & \text{List}II\\ \left(I\right)& 2h+k& \left(P\right)& 6\\ \left(II\right)& \frac{\text{length of}ZW}{\text{length of}XY}& \left(Q\right)& \sqrt{6}\\ \left(III\right)& \frac{\text{Area of triangle}MZN}{\text{Area of triangle}ZMW}& \left(R\right)& \frac{5}{4}\\ \left(IV\right)& \alpha & \left(S\right)& \frac{21}{5}\\ & & \left(T\right)& 2\sqrt{6}\\ & & \left(U\right)& \frac{10}{3}\end{array}$



Which of the following is the only INCORRECT combination?

Two dice are thrown. Describe the sample space of this experiment.

A solution of differential equation $\left(se{c}^{2}y\right)\frac{dy}{dx}+2xtany=x^3$, is

A ladder $12$ units long slides in a vertical plane with its ends in contact with a vertical wall and a horizontal floor along $x-$axis. Then the locus of a point on the ladder $4$ units from its foot, is

Solve the given inequality for real x:

If $AB$ is a double ordinate of the hyperbola

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ such that ∆ OAB (O is the origin) is an equilateral triangle, then the eccentricity $e$ of the hyperbola satisfies

Let there exist a unique point $P$ inside a $△ABC$ such that $\angle PAB=\angle PBC=\angle PCA=\alpha .$ If $PA=x,PB=y,PC=z,\Delta =$ area of $△ABC$ and $a,b,c,$ are the sides opposite to the angle $A,B,C$ respectively, then $\tan \alpha$ is equal to

If $z+\frac{1}{z}=2\cos \theta$, where $z$ is complex number and $i=\sqrt{-1}$, then $\frac{z^n-1}{z^n+1}$ is

If roots of the given quadratic $2x^2-9x+7=0$ are $p,q.$ Then the equation whose roots are

If the line $y=mx+c$ touches $x^2-y^2=1$ and $y^2=4x,$ then $m^2$ is equal to

Let $I=\underset{a}{\overset{b}{\int }}(x^4-2x^2)dx$. If $I$ is minimum then the ordered pair $(a,b)$ is :

In a $△ABC,(a+b+c)(tan\frac{A}{2}+tan\frac{B}{2})$ is

Suppose X follows a binomial distribution with parameters n and p, where 0<p<1. If $\frac{P(X=r)}{P(X=n-r)}$ is independent of n for every r, then p=

If the major axis of a vertical ellipse is three times the minor axis, then its eccentricity is equal to

If $z_1$ and $z_2$ are two non-zero complex numbers such that $|z_1-z_2|=||z_1|-|z_2||$ then $\arg \left(z_1\right)-\arg \left(z_2\right)=$

In a triangle $(1-\frac{r_1}{r_2})(1-\frac{r_1}{r_3})=2$ then the triangle is

Evaluate the following limit:
$\underset{x\to 1}{\text{lim}}\frac{a{x}^2+bx+c}{c{x}^2+bx+a},a+b+c\ne 0$

The line joining the origin to the points of intersection of the curves $a{x}^2+2hxy+b{y}^2+2gx=0$ and $a^{{}^{\prime }}{x}^2+2h^{{}^{\prime }}xy+b^{{}^{\prime }}{y}^2+2g^{{}^{\prime }}x=0$ will be mutually perpendicular, if

Write the maximum and minimum values of sin (sin x).

Suppose that $f\left(x\right)$ is continuous in $[1,2]$ such that $f\left(x\right)=0$ has atleast one real solution in $(1,2)$, then

Find the value of (${\frac{\sqrt{3}}{2}+\frac{i}{2})}^5$ + (${\frac{\sqrt{3}}{2}-\frac{i}{2})}^5$

Find the area bouded by the curves $y=2x-x^2,4y=(x-2{)}^2$ and $y=0$

The root of the function $f\left(x\right)=x^3+x-1$ obtaind after first iteration on application of Newton Raphson scheme using an initial guess of $x_0=1$