Basics of 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

Let $z$and $w$be two non-zero complex numbers such that  $\left|z\right|=\left|w\right|$  and  $\arg z+\arg w=\pi$  then $z$equals –

In the complex plane, let $z_1=\sqrt{3}+i$ and $z_2=\sqrt{3}-i$ be two adjacent vertices of an $n$-sided regular polygon centered at the origin. Then, $n$ equals

The polar form of the complex number $\frac{1+i}{1-i}$ is equal to

If $z$ is a complex number of unit modulus and argument $\theta ,$ then the real part of $\frac{z(1-\overline{z})}{\overline{z}(1+z)}$ is

If $\left(1+\mathrm{i}\right)\mathrm{z}=\left(1-\mathrm{i}\right)\stackrel{-}{\mathrm{z}}$ , then $\mathrm{z}$ is -

The conjugate of complex number $\frac{2-3i}{4-i}$  is

For two complex numbers $z_1,z_2$ the relation $|z_1+z_2|=\left|z_1\right|+\left|z_2\right|$ hold, if

A function $f$ is defined by $f\left(z\right)=i\overline{z}$ , where $i=\sqrt{-1}$ and $\overline{z}$ is the complex conjugate of $z$. The number of values of $z$ which satisfies both $\left|z\right|=5$ and $f\left(z\right)=z$ , is:

In the Argand plane if O, P, Q represent respectively the origin, the complex number z and z + iz, then angle OPQ is

A real value of $x$ satisfies the equation $\left(\frac{3-4ix}{3+4ix}\right)=\alpha -i\beta (\alpha , \beta \in R)$ if ${\alpha }^2+{\beta }^2$ is equal to

Which of the following is correct for any two complex numbers $z_1$ and $z_2$?

Let $z$ be a complex number such that $\left|\frac{z-i}{z+2i}\right|=1$ and $\left|z\right|=\frac{5}{2}$ . Then the value of $\left|z+3i\right|$ is

If $a=\cos \alpha +i\sin \alpha , b=\cos \beta +i\sin \beta$, $c=\cos \gamma +i\sin \gamma$ and $\frac{b}{c}+\frac{c}{a}+\frac{a}{b}=1$, then $\cos \left(\beta -\gamma \right)+\cos \left(\gamma -\alpha \right)+\cos (\alpha -\beta )$ is equal to $(\mathrm{where}\mathit{ }i=\sqrt{-1})$

The value of $\sum _{n=0}^{100}(i{)}^{n!}$, where $i=\sqrt{-1}$, is equal to_____

If $\left(1+\mathrm{i}\right)\mathrm{z}=\left(1-\mathrm{i}\right)\stackrel{-}{\mathrm{z}}$ , then $\mathrm{z}$ is -

If $P$ represents $z=x+iy$ in the argand plane ${\left|z-1\right|}^2+{\left|z+1\right|}^2=4$, then locus of $P$ is (where $i=\sqrt{-1}$)

If $z_1,z_2$ and $z_3,z_4$ are two pairs of complex conjugate numbers, then $\arg \left(\frac{z_1}{z_4}\right)+\arg \left(\frac{z_2}{z_3}\right)$ can be equal to