The multiplicative inverse of $z$ is

If $x+iy = \frac{a+ib}{a-ib}$,prove that $x^{2}+y^{2}=1$.

The complex number $\frac{1+2i}{1-i}$ lies in

If $\frac{c + i}{c - i} = a + ib$,where $a, b, c$ are real,then $a^2 + b^2 = $

Find the set $\{x \in [0, 2\pi] \mid \sin x + i \cos 2x \text{ and } \cos x - i \sin 2x \text{ are conjugate to each other}\}$.

If $m_1, m_2, m_3$ and $m_4$ respectively denote the moduli of the complex numbers $1+4i, 3+i, 1-i$ and $2-3i$,then the correct relation among the following is: