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In a traingle $ABC,2a^2+4b^2+c^2=4ab+2ac,$ then numerical value of $\cos B$ is equal to

Let $H:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,\text{where}a>b>0,$be a hyperbola in the $xy-$plane whose conjugate axis $LM$ subtends an angle of ${60}^{\circ }$ at one of its vertices $N.$ Let the area of the triangle $LMN$ be $4\sqrt{3}.$ sq. unit



$\begin{array}{cccc}& \text{LIST-I}& & \text{LIST-II}\\ P.& \text{The length of the conjugate axis of}H\text{is}& 1.& 8\\ Q.& \text{The eccentricity of}H\text{is}& 2.& \frac{4}{\sqrt{3}}\\ R.& \text{The distance between the foci of}H\text{is}& 3.& \frac{2}{\sqrt{3}}\\ S.& \text{The length of the latus rectum of}H\text{is}& 4.& 4\end{array}$



The correct option is

The number of complex numbers $z_1$ which can simultaneously satisfy both the equations $|$z - 2$|$ = 2 and z(1 - i) + $\overline{z}(1+i)$ = 4 is equal to

A particle is projected from point $\text{G}$ such that it touches the points $\text{B, C, D}$ and $\text{E}$ of a regular hexagon of side $a$. Its horizontal range $\text{GH}$ is

1. 36

A bag contains 10 balls. 2 red, 3 blue and 5 black. Three are drawn at random. The probability that the three balls are of different colours is

If the pair of straight lines $a{x}^2+2hxy+b{y}^2=0$ is rotated about the origin through ${90}^{\circ },$ then the equations in the new position is

Every _______ number of the Fibonacci sequence is a multiple of 8

The set of values of $a$ for which the equation $\sqrt{a}\cos x-2\sin x=\sqrt{2}+\sqrt{2-a}$ possesses a solution, is

If the graph of $2x+3y=8$ is translated $5$ units down, then which of the following denotes the equation of the new graph?

${lim}_{x\to 1}\frac{1+cos\pi x}{(1-x{)}^2}$

Given angle $A={60}^{\circ },c=\sqrt{3}-1,b=\sqrt{3}+1$. Solve the triangle

If $\left[\begin{array}{ccc}1& -1& x\\ 1& x& 1\\ x& -1& 1\end{array}\right]$ has no inverse, then the possible real value of $x$ is

The real number k for which the equation, $2x^3+3x+k=0$ has two distinct real roots in [0, 1]

If $a$ is equal to number of solution(s) of the equation $|\log |x||=|\tan x|$, where $x\in (-\frac{\pi }{2},\frac{\pi }{2})$, then domain of the function $f\left(x\right)=\log \left({\tan }^{-1}\left(\frac{a-x}{a+x}\right)\right)$ is equal to

Let x, y, z be positive reals. Which of the following implies x = y = z?

(I) x${}^3$ + y${}^3$ + z${}^3$ = 3xyz

(II) x${}^3$ + y${}^2$ z + yz${}^2$ = 3xyz

(III) x${}^3$ + y${}^2$ z + z${}^2$ x = 3xyz

(IV) (x + y + z)${}^3$ = 27xyz

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are non-zero and non-collinear vectors, then $\left[\overrightarrow{a}\overrightarrow{b}\hat{i}\right]\hat{i}+\left[\overrightarrow{a}\overrightarrow{b}\hat{j}\right]\hat{j}+\left[\overrightarrow{a}\overrightarrow{b}\hat{k}\right]\hat{k}$ is equal to

Probability fo solving specific problem independently by A and B are $\frac{1}{2}and\frac{1}{3}$ respectively. If both try to solve the problem independently, find the probability that
Exactly one of them solves the problem

If $f\left(x\right)=(a-x^n{)}^{\frac{1}{n}},a>0$ and $nϵN$, then prove that $f\left(f\right(x\left)\right)=x$ for all $x$.

In a single throw of two dice, find the probability that neither a doublet nor a total of 9 will appear.

Two fair dice are rolled simultaneously. It is found that one of the dice show odd prime number. The probability that the remaining dice also show an odd prime number, is equal to

Rearrange the following sentences (A), (B), (C), (D), (E) and (F) to make a meaningful paragraph and answer the questions which follow:

Find the sum to n
terms in the geometric progression

Consider a function $f:C\to C$ defined as $f\left(z\right)=z^{12}+2z^{11}+3z^{10}+...+12z+13$. If $\alpha =\cos \frac{2\pi }{13}+i\sin \frac{2\pi }{13},$ where $i=\sqrt{-1},$ then the value of $f\left(\alpha \right)f\left({\alpha }^2\right)f\left({\alpha }^3\right)\dots f\left({\alpha }^{12}\right)$ is

If $\left(\begin{array}{cc}1& -\tan \theta \\ \tan \theta & 1\end{array}\right)\left(\begin{array}{cc}1& \tan \theta \\ -\tan \theta & 1\end{array}\right)=\left(\begin{array}{cc}a& -b\\ -b& a\end{array}\right)$, then the values of $a$ and $b$ are:

If $f\left(x\right)=9^{-2/x^2},x\ne 0$ is continuous at $x=0,$ then the value of $f\left(0\right)$ is

Two vectors $\overrightarrow{A}\&\overrightarrow{B}$ are such that $|\overrightarrow{A}|=|\overrightarrow{B}|$. Magnitude of ($\overrightarrow{A}+\overrightarrow{B}$) is $\sqrt{3}$ times of magnitude of ( $\overrightarrow{A}-\overrightarrow{B}$). The angle between $\overrightarrow{A}\&\overrightarrow{B}$ is