Let $\mathrm{z}$ be a complex number such that $|\mathrm{z}+2|=1$ and $\operatorname{Im}\left(\frac{z+1}{z+2}\right)=\frac{1}{5}$. Then the value of $|\operatorname{Re}(\overline{z+2})|$ is :

The amplitude of $\frac{{1 + \sqrt 3 \,i}}{{\sqrt 3 + i}}$ is

If the conjugate of $(x + iy)(1 - 2i)$ be $1 + i$, then

If $z$ is a complex number such that $| z | = 4$ and $arg \,(z) = \frac {5\pi }{6}$ , then $z$ is equal to

If ${z_1}$ and ${z_2}$ are two non-zero complex numbers such that $|{z_1} + {z_2}| = |{z_1}| + |{z_2}|,$then arg $({z_1}) - $arg $({z_2})$ is equal to

Let $\alpha=8-14 i , A=\left\{ z \in C : \frac{\alpha z -\bar{\alpha} \overline{ z }}{ z ^2-(\overline{ z })^2-112 i }=1\right\}$ and $B =\{ z \in C :| z +3 i |=4\}$ Then $\sum_{z \in A \cap B}(\operatorname{Re} z-\operatorname{Im} z)$ is equal to $...............$.