Complex Numbers

The value of the sum  $\sum _{n=1}^{13}\left(i^n\hspace{0.17em}+i^{n+1}\right),$ where  $i=\sqrt{-1}$ , equals

For positive integers  $n_1, n_2$  the value of expression ${\left(1+i\right)}^{n_1}+{\left(1+i^3\right)}^{n_1}+$${\left(1+i^5\right)}^{n_2}+{\left(1+i^7\right)}^{n_2}$ where  $i=\sqrt{-1}$ , is a real number if and only if

$i^{326}=?$

If $n$ is an integer and $z=cis\theta$, then $\frac{z^{2n}-1}{z^{2n}+1}=$

If $x+iy=\frac{1}{1+\cos \theta +i\sin \theta }$, then $4x^2-1=$

If $1, \omega , {\omega }^2$ are three cube roots of unity, then ${\left(1-\omega +{\omega }^2\right)}^3=$

Find the value of ${\left[\frac{1+\sin {\displaystyle \frac{\pi }{8}}+i \cos {\displaystyle \frac{\pi }{8}}}{1+ \sin \frac{\pi }{8}-i \cos \frac{\pi }{8}}\right]}^{\frac{8}{3}}$

$(1-i{)}^8=$

If $\theta \in \mathrm{ℝ}$ and $\frac{1-i\cos \theta }{1+2i\cos \theta }$ is real number, then $\theta$ will be (when $I:$ Set of integers)

If $x+iy=\frac{3}{2+\cos \left(\theta \right)+i\sin \left(\theta \right)},$ then $x^2+y^2=$________

A(z₁), B(z₂), C(z₃) are the vertices of a triangle ABC inscribed in the circle $\left|\text{z}\right|=2$. Internal angle bisector of the angle A meets the circum-circle again at D(z₄).

arg $\left[{\text{z}}_4/\left({\text{z}}_2-{\text{z}}_3\right)\right]$  is equal to

For two complex numbers $z_1=a-ib$ and $z_2=c-id$ then $z_1{z}_2$ is purely imaginary when 

If the complex number $\frac{2+3i}{4+5i}$ is equal to $\frac{a}{k}+i\frac{b}{k}$ then find the value of $a+b+k$.

If $(a+ib{)}^5=\alpha +i\beta ,$ then $(b+ia{)}^5=$

$2\sqrt{-4}\times \sqrt{-9}$ is equal to

The locus of $z$ represented by $Re\left(z^2-2z\right)=15$ is

The value of ${\left(\frac{-1-i}{\sqrt{2}}\right)}^{100}$ is equals to :

If $\frac{p+iq}{1+r}=\frac{1+iz}{1-iz}$ ,where $z=\frac{q+ir}{1+p},$ then 

Find the complex number satisfying the equation $z+\sqrt{2}\left|\right(z+1\left)\right|+i=0$.

Solve the equation $\left|z\right|=z+1+2i$ for $z$.