Exercise 1

If $z=x+i y$ show that $\left|x\right|+\left|y\right|\le \sqrt{2}\left|z\right|$.

Let $z$ be a complex number satisfying $\left|2z+10+10i\right|\le 5\sqrt{3}-5$. If $\arg$ denotes the principal argument lying in $(-\mathrm{\pi }, \mathrm{\pi }]$, then the least value of $\text{arg}\left(z\right)$ is

Express with rational denominator: $\frac{3\sqrt{-2}+2(-5)}{3\sqrt{-2}-2\sqrt{-2}}$

If $a=\cos \theta +i\sin \theta$, find the value of $\frac{1+a}{1-a}$.

Express the given complex number in the $r,\theta$ form :

$i\sqrt{3}$

If $\omega$, and ${\omega }^2$ are imaginary cube roots of unity then find the value of $\arg \omega +\arg {\omega }^2$.

The roots of $z^3-1=0$ are $1, \omega , {\omega }^2$. Determine the value of ${\omega }^3+{\omega }^4+{\omega }^5$. Otherwise, ${\omega }^3=1, {\omega }^4=\omega$ and ${\omega }^5={\omega }^2.$

$\frac{\left(5-i\right)}{\left(2-3i\right)}$, when expressed in polar form, is