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z₁ and z₂ are any two distinct complex numbers in an argand plane. If αβ |z₁|=γδ|z₂|,

then the complex number lies on the (α, β ϵ R)

The general solution(s) of the equation $\sec 4\theta -\sec 2\theta =2$ can be

Find the number of solutions of the equation $sin2\theta +cos2\theta +4sin\theta =1+4cos\theta$ lying in the interval $[-2\pi ,2\pi ].$___

1. 271

In $\Delta ABC,\frac{b+c}{11}=\frac{c+a}{12}=\frac{a+b}{13}$, then $\sin A:\sin B:\sin C=$

If $\overrightarrow{a}$ is unit vector and $(\overrightarrow{x}-\overrightarrow{a}).(\overrightarrow{x}+\overrightarrow{a})=8,then|\overrightarrow{x}|=$

A bag contains ${}^{\prime }{x}^{\prime }$ red balls ${}^{\prime }2x^{\prime }$ white balls and ${}^{\prime }3x^{\prime }$ black balls. $3$ balls are drawn at random. The probability that all the balls drawn are of different colors is $0.3$ .How many white balls are present in the bag?

Line x + 2y = 4 is translated by $\sqrt{5}$ units closer to the origin and then rotated by angle $ta{n}^{-1}\left(\frac{1}{2}\right)$ in the clockwise direction about the point where the shifted line cuts the x-axis. Find the distance of new line from point M(3, 3).___

The determinant
$\left|\begin{array}{ccc}{b}^2-ab& b-c& bc-ac\\ ab-a^2& a-b& b^2-ab\\ bc-ac& c-a& ab-a^2\end{array}\right|$ equals to:

(a) abc(b-c)(c-a)(a-b)
(b) (b-c)(c-a)(a-b)
(c) (a+b+c)(b-c)(c-a)(a-b)
(d) None of these

Let $f:N\to N$ be a function such that $f(m+n)=f\left(m\right)+f\left(n\right)$ for every $m,n\in N.$ If $f\left(6\right)=18,$ then $f\left(2\right)\cdot f\left(3\right)$ is equal to

Let $n$ be the number of ways in which $5$ boys and $5$ girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let $m$ be the number of ways in which $5$ boys and $5$ girls can stand in a queue in such a way that exactly four girls stand consecutively in the queue. Then the value of $\frac{m}{n}$ is

If the two curves $x=y^2$ and $xy=k$ cut each other orthogonally, then $k^2$ equals to

Let $E_1$ and $E_2$ be the two ellipses centred at origin. The major axis of $E_1$ and $E_2$ lie along the $x-$ axis and $y-$ axis respectively. Let $S$ be the circle $x^2+(y-1{)}^2=2,$ the straight line $x+y=3$ touches the curve $S,E_1$ and $E_2$ at $P,Q$ and $R$ respectively such that $PQ=PR=\frac{2\sqrt{2}}{3}.$ If $e_1$ and $e_2$ are the eccentricities of $E_1$ and $E_2,$ then which of the following is/are correct

Match the following by appropriately matching the lists based on the information given in $\text{Column I}$ and $\text{Column II}$.

$\begin{array}{cc}ColumnI& ColumnII\\ \text{a. Range of}f\left(x\right)={\sin }^{-1}x+{\cos }^{-1}x+{\cot }^{-1}x\text{is}& \text{p.}[0,\frac{\pi }{2})\cup (\frac{\pi }{2},\pi ]\\ \text{b. Range of}f\left(x\right)={\cot }^{-1}x+{\tan }^{-1}x+{\text{cosec}}^{-1}x\text{is}& \text{q.}[\frac{\pi }{2},\frac{3\pi }{2}]\\ \text{c. Range of}f\left(x\right)={\cot }^{-1}x+{\tan }^{-1}x+{\cos }^{-1}x\text{is}& \text{r.}\{0,\pi \}\\ \text{d. Range of}f\left(x\right)={\sec }^{-1}x+{\text{cosec}}^{-1}x+{\sin }^{-1}x\text{is}& \text{s.}[\frac{3\pi }{4},\frac{5\pi }{4}]\end{array}$

If $i=\sqrt{-1}$, then $4+5{(\frac{-1}{2}+i\frac{\sqrt{3}}{2})}^{334}-3{(\frac{1}{2}+i\frac{\sqrt{3}}{2})}^{365}$ is equal to

The vertices of a triangle are $(-1,\sqrt{3}),(2\cos \theta ,-2\sin \theta )$ and $(2\sin \theta ,-2\cos \theta )$ where $\theta \in R$. Then locus of orthocentre of the triangle is

Standard deviation about mean $\left(\overline{x}\right)$ for a given discrete frequency distribution $x_1,x_2,x_3,.....x_n$ with frequencies $f_1,f_2,f_3,...f_n$ is

Total number of ordered pairs (x,y) satisfying $|x|+|y|=2,sin\left(\frac{\pi x^2}{3}\right)=1$ is

If odd natural numbers are arranged in groups as $\left(1\right),(3,5),(7,9,11),...$ Then the sum of the numbers in the ${10}^{\text{th}}$ group is

Let two distinct numbers $a$ and $b$ are selected from the set $\{1,2,3,\dots ,9,10\}.$ Then the probability that the last digit of the number $a^b$ will be $6,$ is

'The sum of two and seven divided by three'



The given statement can be expressed mathematically as

Which of the following equations is not derived from the equation shown?

$2x+5=8$

Graph of $y=3|\frac{1}{2}x+2|-9$ is

If $(n+1)!=12\times (n-1)!$, then the value of $n$ can be

Fundamental period of $f\left(x\right)=|\sin 2x|$ is

Find the equation of a line that has y-intercept-4 and is parallel to the line joining (2, -5) and (1, 2).

The value of $3+\frac{1}{4+\frac{1}{3+\frac{1}{4+\frac{1}{3+\cdots \infty }}}}$ is equal to:

$\text{Let}(1+x+x^2{)}^{2014}=a_0+a_{1}x+a_2{x}^2+a_3{x}^3+....+a_{4028}{x}^{4028}\text{and let}A=a_0-a_3+a_6-......+a_{4026,}B=a_1-a_4+a_7-......-a_{4027,}C=a_2-a_5+a_8-......+a_{4028}\text{Then}$