Complex Numbers

If $\sinh ^{-1}(2)+\sinh ^{-1}(3)=\alpha$, then $\sinh \alpha=$

If $w_1$ and $w_2$ are two non-zero complex numbers and ${ }a, b$ are non-zero real numbers such that $\left|a w_1+b w_2\right|=\left|a w_1-b w_2\right|$, then $\frac{w_1}{w_2}$ is

Let $z$ satisfy $|z|=1, z=1-\bar{z}$ and $\operatorname{Im}(z)>0$
Statement $\mathbf{I} z$ is a real number
Statement II Principal argument of $z$ is $\frac{\pi}{3}$.
Then,

If $z$ and $w$ are two non-zero complex numbers such that $|z w|=1$ and $\arg z-\arg w=\frac{\pi}{2}$, then $\bar{z} w=$

By taking $\sqrt{a \pm i b}=x \pm i y, x>0$, if we get $\frac{\sqrt{21+12 \sqrt{2 i}}}{\sqrt{21-12 \sqrt{2 i}}}=a+i b$, then $\frac{b}{a}=$

Two values of $(-8-8 \sqrt{3} i)^{1 / 4}$ are

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

$z_1, z_2, z_3$ represent the vertices $A, B, C$ of a $\triangle A B C$ respectively in the argand plane. If $\left|z_1-z_2\right|=\sqrt{25-12 \sqrt{3}},\left|\frac{z_1-z_3}{z_2-z_3}\right|=\frac{3}{4}$ and $\angle A C B=30^{\circ}$, then t...

The product of the four values of the complex number $(1+i)^{3 / 4}$ is

If $x=3-2 \sqrt{3} \mathrm{i}$, then $x^4-12 x^3+54 x^2-108 x-54=$

\((1+\sqrt{5}+i \sqrt{10-2 \sqrt{5}})^5=\)

If $(\sqrt{3}-i)^n=2^n, n \in N$, then the least possible value of $n$ is

If the point $P$ denotes the complex number $z=x+i y$ in the argand plane and $\frac{z-(2-i)}{z+(1+2 i)}$ is purely imaginary number, then the locus of $P$ is

For any two non-zero complex numbers $z_1$ and $z_2$, if $\left|z_1+z_2\right|^2=\left|z_1\right|^2+\left|z_2\right|^2$, then

\((1+\sqrt{3} i)^6-(\sqrt{3}+i)^6=\)

If $1, \omega, \omega^2$ are the cube roots of unity, then
$$ 1\left(2+\frac{1}{\omega}\right)\left(2+\frac{1}{\omega^2}\right)+2\left(3+\frac{1}{\omega}\right)\left(3+\frac{1}{\omega^2}\right) +3\left(4+\frac{1}{\omega}\right)\left(4+\frac...

If the least positive integer $n$ satisfying the equation $\left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)^n=-1$ is $p$ and the least positive integer $m$ satisfying the equation $\left(\frac{1-\sqrt{3 i}}{1+\sqrt{3} i}\right)^m=\operatorname{ci...

If $z$ is a complex number such that $\frac{z-1}{z-i}$ is purely imaginary and locus of $z$ represents a circle with centre $(\alpha, \beta)$ and radius $r$, then $\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=$

Sum of the squares of the imaginary roots of the equation $z^8-20 z^4+64=0$ is

If $\cosh 2 x=199$, then $\cot h x=$

The minimum value of $|z-1|+|z-5|$ is

If $z$ is a non-real root of $x^7=1$, then $1+3 z+5 z^2+7 z^3+9 z^4+11 z^5+13 z^6=$

If $z=x+i y$ and if the point $P$ in the argand diagram represents $z$, then the locus of the point $P$ satisfying the equation $2|z-2-3 i|=3|z+i-2|$ is a circle with centre

If $x^6=(\sqrt{3}-i)^5$, then the product of all of its roots is

If $z=x+i y$ and $x^2+y^2=1$, then $\frac{1+x+i y}{1+x-i y}=$

If $(3+4 i)^{2025}=5^{2023}(x+i y)$, then $\sqrt{x^2+y^2}=$

If $a=\operatorname{Im}\left(\frac{1+z^2}{2 i z}\right)$ and $z$ is any non-zero complex number such that $|z|=1$, then $a=$

If $a \pm i b$ and $b \pm a i$ are the roots of $x^4-10 x^3+50 x^2-130 x+169=0$, then $\frac{a}{b}+\frac{b}{a}=$

If $\left(\frac{\cos \theta+i \sin \theta}{\sin \theta+i \cos \theta}\right)^{2024}+\left(\frac{1+\cos \theta+i \sin \theta}{1-\cos \theta+i \sin \theta}\right)^{2025}=x+i y$ then the value of $x+y$ at $\theta=\frac{\pi}{2}$ is

If $i=\sqrt{-1}$, then $\sum\limits_{n=2}^{30} i^n+\sum\limits_{n=30}^{65} i^{n+3}=$

\((\sqrt{\sqrt{2}+1}+i \sqrt{\sqrt{2}-1})^8=\)

If $z_1$ and $z_2$ are two of the $n$th roots of unity such that the line segment joining them subtends at a right angle at the origin, then for a positive integer $k, n$ takes the form

$(r, \theta)$ denotes $r(\cos \theta+i \sin \theta)$. If $x=(1, \alpha), y=(1, \beta), z=(1, \gamma)$ and $x+y+z=0$, then $\Sigma \cos (2 \alpha-\beta-\gamma)$ is equal to

$\omega$ is a complex cube root of unity and if $z$ is a complex number satisfying $|z-1| \leq 2$ and $\left|\omega^2 z-1-\omega\right|=a$, then the set of possible values of $a$ is

If the roots of the equation $z^3+i z^2+2 i=0$ are the vertices of a $\triangle A B C$, then that $\triangle A B C$ is

If the number of real roots of $x^9-x^5+x^4-1=0$ is $n$, the number of complex roots having argument on imaginary axis is $m$ and the number of complex roots having argument in 2nd quadrant is $K, m \cdot n \cdot k=$

All the values of $(8 i)^{\frac{1}{3}}$ are

The locus of the complex number $Z$ such that $\arg \left(\frac{Z-1}{Z+1}\right)=\frac{\pi}{4}$ is

If $Z$ is a complex number such that $|Z| \leq 3$ and $\frac{-\pi}{2} \leq \operatorname{amp} Z \leq \frac{\pi}{2}$, then the area of the region formed by locus of $Z$ is

$\arg \left[\frac{(1+i \sqrt{3})(-\sqrt{3}-i)}{(1-i)(-i)}\right]$ is equal to

If $P(x, y)$ represents the complex number $z=x+iy$ in the argand plane and $\arg \left(\frac{z-3 i}{z+4}\right)=\frac{\pi}{2}$, then the equation of the locus of $P$ is

If $\alpha_1, \alpha_2, \alpha_3, \alpha_4$ and $\alpha_5$ are the roots of $x^5-5 x^4+9 x^3-9 x^2+5 x-1=0$, then $\frac{1}{\alpha_1^2}+\frac{1}{\alpha_2^2}+\frac{1}{\alpha_3^2}+\frac{1}{\alpha_4^2}+\frac{1}{\alpha_5^2}$ is equal to

The complex conjugate of $(4-3 i)(2+3 i)(1+4 i)$ is.

If $\omega$ is the cube root of unity,

\(\frac{a+b \omega+c \omega^2}{c+a \omega+b \omega^2}+\frac{a+b \omega+c \omega^2}{b+c \omega+b \omega^2}=\)

If $(3+i)$ is a root of $x^2+a x+b=0$, then $a=$

If the amplitude of $(z-2)$ is $\frac{\pi}{2}$, then the locus of $z$ is

The square root of $7+24 i$

Imaginary part of $\frac{(1-i)^3}{(2-i)(3-2 i)}$ is

If $n$ is an integer and $Z=\cos \theta+i \sin \theta, \theta \neq(2 n+1) \frac{\pi}{2}$, then $\frac{1+Z^{2 n}}{1-Z^{2 n}}=$

$S=\{z \in C /|z+1-i|=1\}$ represents