If $arg\, z < 0$ then $arg\, (-z)\, -arg(z)$ is equal to

Let ${z_1}$ and ${z_2}$ be two complex numbers with $\alpha $ and $\beta $ as their principal arguments such that $\alpha + \beta > \pi ,$ then principal $arg\,({z_1}\,{z_2})$ is given by

Argument of $ - 1 - i\sqrt 3 $ is

 Find the conjugate of $\frac{(3-2 i)(2+3 i)}{(1+2 i)(2-i)}$.

The inequality $|z - 4|\, < \,|\,z - 2|$represents the region given by

Lets $S=\{z \in C:|z-1|=1$ and $(\sqrt{2}-1)(z+\bar{z})-i(z-\bar{z})=2 \sqrt{2}\}$. Let $\mathrm{z}_1, \mathrm{z}_2$ $\in S$ be such that $\left|z_1\right|=\max _{z \in S}|z|$ and $\left|z_2\right|=\min _{z \in S}|z|$. Then $\left|\sqrt{2} z_1-z_2\right|^2$ equals :