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 –

The value of ${\left(\sqrt{3}+i\right)}^{14}+{\left(\sqrt{3}-i\right)}^{14}$ is -

The value of ${\left[i^{19}+{\left(\frac{1}{i}\right)}^{25}\right]}^2$ is

If $z$ is a complete number, then $z^2+{\overline{z}}^2=2$ represents

Represent the following complex number in the vector form in the complex plane.

$-3+i$

Represent the following complex number in the vector form in the complex plane.

$3+2i$

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

A complex number $z=1+i\sqrt{3}$.

The general argument of $z$ is

If $\mathrm{cos\alpha }+3\cos 3\mathrm{\beta }+5\cos 5\mathrm{\gamma }=0, \mathrm{sin\alpha }+3\sin 3\mathrm{\beta }+5\sin 5\mathrm{\gamma }=0$ and $\cos 3\alpha +$ $27\cos 9\mathrm{\beta }+125\cos 15\mathrm{\gamma }=\left({\mathrm{\lambda }}^2-4\right)\cos (\mathrm{\alpha }+3\mathrm{\beta }+5\mathrm{\gamma }),$ then $\lambda =$

$\left|\frac{1}{i^{2020}}+\frac{2}{i^{2021}}+\frac{3}{i^{2022}}+\frac{4}{i^{2023}}\right|=$

If $z_1,z_2$ are conjugate complex numbers. Match the items under the following columns?

$\frac{{\left(\sin \frac{\pi }{8}+i\cos \frac{\pi }{8}\right)}^8}{{\left(\sin \frac{\pi }{8}-i\cos \frac{\pi }{8}\right)}^8}=$

If ${\left(\frac{1+i}{1-i}\right)}^x=1$ then

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 $z=x+iy$ and $\left|z-2+2 i\right|=4$ then the locus of $z$ in the complex plane is

The modulus amplitude form of $i^{15}$ is

If $z$ is a purely imaginary number and $\mathrm{Im} \mathrm{z}<0$, then $\mathrm{amp} \mathrm{z}=$