Introduction to 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

Which one is true?

Find the complex number with the least argument among the points on the circular disc $\left|z-6 i\right|\le 3$.

Solve $z^2+\left|z\right|=0$, where $z$ is a complex number.

The points $A, B, C$ represent the complex numbers $z_1, z_2, z_3$ respectively in the Argand diagram. Show that $∆ABC$ is equilateral, if and only if

$\frac{1}{z_1-z_2}+\frac{1}{z_2-z_3}+\frac{1}{z_3-z_1}=0$

If $\alpha , \beta$ be any two complex numbers, prove that

$\left|\alpha +\sqrt{{\alpha }^2-{\beta }^2}\right|+\left|\alpha -\sqrt{{\alpha }^2-{\beta }^2}\right|=|\alpha +\beta |+|\alpha -\beta |$

Let $z$ be such a complex number that $\frac{z-1}{z-1}$ becomes purely imaginary. Show that $z$ lies on a circle with centre $\frac{1}{2}(1+i)$ and radius $\frac{1}{\sqrt{2}}$.

If $c$ is real, $b=p+iq z=x+iy$ show that the equation $b(z+\overline{z})+\overline{b}(\overline{z}-z)+c=0$ represents two straight lines in the $xy-$plane and find the angle between them.

If $i$ is a root of the equation$x+\frac{1}{x}=k$  where $k$ is a constant, find the value of $k\left[i^2=-1\right]$.

Do you think $4$ is a complex number ? If so, why ?

Find the value of $(i+\sqrt{3}{)}^{100}+(i-\sqrt{3}{)}^{100}+2^{100}$

If $n$ is any positive integer, then find the value of $\frac{1}{2}\left(i^{4n+1}-{\mathrm{i}}^{4n-1}\right)$.

If $x, y$ be real, $z=x+iy$ and $\frac{z-i}{z-1}$ be purely imaginary, then show that ${\left(x-\frac{1}{2}\right)}^2+{\left(y-\frac{1}{2}\right)}^2=\frac{1}{2}$.

If $z=x+iy$ and if $\frac{z-i}{z+1}$ be a purely imaginary number then show that the locus (in complex plane) of $z$ is a circle.

If $z=x+iy$ and if $\frac{z-i}{z+1}$ be a purely imaginary number then find the equation of locus of $z$ in the complex plane.

If $z=x+iy$ and if $\frac{z-1}{z+1}$ be a purely imaginary number then show that the locus of $z$ is a circle in the complex plane.

If $z$ be a complex number satisfying the condition $\left|z+5\right|\le 5$, then find the maximum and minimum value of $\left|z+2\right|$.

If $x+iy=\frac{3}{2+\cos \theta +i \sin \theta }$, prove that $x^2+y^2=4 x-3$
 

$1=\sqrt{1}=\sqrt{\left(-1\right)\left(-1\right)}=\sqrt{-1}\sqrt{-1}=i.i=i^2=-1$. Find out the wrong step(s) in this 'deduction'.