Complex Numbers

Let $z=x+i y$ represent a point of $P(x, y)$ in the argand plane. If $z$ satisfies the condition that amplitude of $\frac{z-3}{z-2 i}=-\frac{\pi}{2}$ then the locus of $P$ is

\((1-i \sqrt{3})^{2025}=\)

\(\left(\frac{1+i}{1-i}\right)^{228}=\)

One of the roots of the equation $(x+1)^4+81=0$ is

If $1+2 i$ is a root of the equation $x^4-3 x^3+8 x^2-7 x+5=0$, then sum of the squares of the other roots is

If $\omega \neq 1$ is a cube root of unity, then one root among the 7th roots of $(1+\omega)$ is

If $\left|Z_1-3-4 i\right|=5$ and $\left|Z_2\right|=15$, then the sum of the maximum and minimum values of $\left|Z_1-Z_2\right|$ is

If the eight vertices of a regular octagon are given by the complex number $\frac{1}{x_j-2 i}(j=1,2,3,4,5,6,7,8)$, then the radius of the circumcircle of the octagon is

If $Z=r(\cos \theta+i \sin \theta),\left(\theta \neq-\frac{\pi}{2}\right)$ is solution of $x^3=i$, then $r^9(\cos \theta+i \sin \theta)^9=x^{3-}=i$

If $\alpha$ is a root of the equation $x^2-x+1=0$, then
$\left(\alpha+\frac{1}{\alpha}\right)^3+\left(\alpha^2+\frac{1}{\alpha^2}\right)^3+\left(\alpha^3+\frac{1}{\alpha^3}\right)^3+\left(\alpha^4+\frac{1}{\alpha^4}\right)^3+\ldots$ to 12 ...

The set of all values of $\theta$ such that $\frac{1-i \cos \theta}{1+2 i \sin \theta}$ is purely imaginary is

One of the values of $\sqrt{24-70 i}+\sqrt{-24+70 i}$ is

Number of real values of $(-1-\sqrt{3 i})^{3 / 4}$ is

If a complex number $z=x+i y$ represents a point $p(x, y)$ in the argand plane and $z$ satisfies the condition that the imaginary part of $\frac{z-3}{z+3 i}$ is zero, then the locus of the point $P$ is

\((\sqrt{3}+i)^{10}+(\sqrt{3}-i)^{10}=\)

The amplitude of the complex number $\frac{(\sqrt{3}+i)(1-\sqrt{3} i)}{(-1+i)(-1-i)}$ is

The product of all the values of $(\sqrt{3}-i)^{\frac{3}{7}}$ is

If $\omega$ is a complex cube root of unity and $x=\omega^2-\omega+2$, then

Let $z=x+i y$ and $P(x, y)$ be a point on the argand plane. If $z$ satisfies the condition $\arg \left(\frac{z-3 i}{z+2 i}\right)=\frac{\pi}{4}$, then the locus of $P$ is

If $\frac{2+3 i}{i-2}-\frac{4 i-3}{3+4 i}=x+i y$, then $3 x+y=$

If $n, K \in N$ such that $n \neq 3 K$, then $(\sqrt{3}+i)^{2 n}+(\sqrt{3}-i)^{2 n}=$

If $|Z|=2, Z_1=\frac{Z}{2} e^{i \alpha}$ and $\theta$ is the $\operatorname{amp}(Z)$, then $\frac{Z_1^n-Z_1^{-n}}{Z_1^n+Z_1^{-n}}=$

$\omega$ is a complex cube root of unity and $Z$ is a complex number satisfying $|Z-1| \leq 2$. The possible values of $r$ such that $|Z-1| \leq 2$ and $\left|\omega Z-1-\omega^2\right|=r$ have no common solution are

In argand plane, no value of $\sqrt[3]{1-i \sqrt{3}}$ lie in

The point $P$ denotes the complex number $z=x+i y$ in the argand plane. If $\frac{2 z-i}{z-2}$ is a purely real number, then the equation of the locus of $P$ is

One of the roots of the equation $x^{14}+x^9-x^5-1=0$ is

If $\sqrt{5}-i \sqrt{15} \doteqdot r(\cos \theta+i \sin \theta),-\pi<\theta<\pi$, then $r^2\left(\sec \theta+3 \operatorname{cosec}^2 \theta\right)=$

$x$ and $y$ are two complex numbers such that $|x|=|y|=1$.
If $\arg (x)=2 \alpha, \arg (y)=3 \beta$ and $\alpha+\beta=\frac{\pi}{36}$, then $x^6 y^4+\frac{1}{x^6 y^4}=$

If $\frac{(2-i) x+(1+i)}{2+i}+\frac{(1-2 i) y+(1-i)}{1+2 i}=1-2 i$, then $2 x+4 y=$

The number of common roots among the 12 th and 30th roots of unity is

The product of all the values of $(\sqrt{3}-i)^{\frac{2}{5}}$ is

If $z=1-\sqrt{3} i$, then $z^3-3 z^2+3 z=$

If $z=\frac{(2-i)(1+i)^{3}}{(1-i)^{2}}$, then $\arg (z)=$

$\alpha, \beta$ are the roots of the equation $x^{2}+2 x+4=0$. If the point representing $\alpha$ in the argand diagram lies in the 2nd quadrant and $\alpha^{2024}-\beta^{2024}=i k,(i=\sqrt{-1})$, then $k=$

$z=x+i y$ and the point $P$ represents $z$ in the argand plane. If the amplitude of $\left(\frac{2 z-i}{z+2 i}\right)$ is $\frac{\pi}{4}$, then the equation of the locus of $P$ is

One of the values of $(-64 i)^{5 / 6}$ is

If $z=x+i y$ and if the point $P$ represents $z$ in the argand plane, then the locus of $z$ satisfying the equation $|z-1|+|z+i|=2$ is

If $x$ and $y$ are two positive real numbers such that $x+i y=\frac{13 \sqrt{-5+12 i}}{(2-3 i)(3+2 i)}$, then $13 y-26 x=$

The roots of the equation $x^{3}-3 x^{2}+3 x+7=0$ are $\alpha, \beta, \lambda$ and $\omega, \omega^{2}$ are complex cube roots of unity, If the terms containing $x^{2}$ and $x$ are missing in the transformed equation when each one of these ...

If $z_{1}=\sqrt{3}+i \sqrt{3}$ and $z_{2}=\sqrt{3}+i$, and $\left(\frac{z_{1}}{z_{2}}\right)^{50}=x+i y$, then the point $(x, y)$ lies in

If $z=x+i y$ satisfies the equation $z^{2}+a z+a^{2}=0, a \in R$, then

If $\omega$ is the complex cube root of unity and$\left(\frac{a+b \omega+c \omega^{2}}{c+a \omega+b \omega^{2}}\right)^{k}+\left(\frac{a+b \omega+c \omega^{2}}{b+a \omega^{2}+c \omega}\right)^{l}=2$, then $2 k+l$ is always

If $z_{1}, z_{2}, z_{3}$ are three complex numbers with unit modulus such that $\left|z_{1}-z_{2}\right|^{2}+\left|z_{1}-z_{3}\right|^{2}=4$, then $z_{1} \bar{z}_{2}+\bar{z}_{1} z_{2}+z_{1} \bar{z}_{3}+\bar{z}_{1} z_{3}=$

The locus of $z$ such that $\left|\frac{z-i}{z+i}\right|=2$, where $z=x+i y$, is

If the roots of the equation $z^2-i=0$ are $\alpha$ and $\beta$, then $|\arg \beta-\arg \alpha|=$

If $x=\log \left(y+\sqrt{y^2+1}\right)$, then $y=$

If $x_n=\cos \frac{\pi}{2^n}+i \sin \frac{\pi}{2^n}$, then $\prod_{n=1}^{\infty} x_n=$

If $i=\sqrt{-1}$, then $\operatorname{Arg}\left[\frac{(1+i)^{2025}}{(1-i)^{2022}}\right]=$

$\alpha, \beta, \gamma$ are the roots of the equation $x^3+2 x^2-x-2=0$, then $\alpha^6+\beta^6+\gamma^6=$

If $\frac{3 x+2}{(x+1)\left(2 x^2+3\right)}=\frac{A}{x+1}+\frac{B x+C}{2 x^2+3}$, then $A-B+C=$