For $k>0$,if $k \sqrt{-1}$ is a root of the equation $x^4+6 x^3-16 x^2+24 x-80=0$,then $k^2=$

Let $S = \{z \in \mathbb{C} : \bar{z} = i(z^2 + \operatorname{Re}(\bar{z}))\}$. Then $\sum_{z \in S} |z|^2$ is equal to

If ${z_1} = 1 + 2i$ and ${z_2} = 3 + 5i$, then $\operatorname{Re} \left( \frac{{\bar{z}_2}{z_1}}{{z_2}} \right)$ is equal to

If the complex numbers $z_1, z_2$ and the origin form an equilateral triangle,then $z_1^2 + z_2^2 = $

Let $S$ be the set of all complex numbers $z$ satisfying $|z^2+z+1|=1$. Then which of the following statements is/are $TRUE$?
$(A) |z+\frac{1}{2}| \leq \frac{1}{2}$ for all $z \in S$
$(B) |z| \leq 2$ for all $z \in S$
$(C) |z+\frac{1}{2}| \geq \frac{1}{2}$ for all $z \in S$
$(D)$ The set $S$ has exactly four elements

Let a complex number $z$,$|z| \neq 1$,satisfy $\log_{\frac{1}{\sqrt{2}}} \left( \frac{|z|+11}{(|z|-1)^2} \right) \leq 2$. Then,the largest value of $|z|$ is equal to ............