If $z$ is a complex number such that ${z^2} = {(\bar z)^2},$ then

$z_1$ and $z_2$ are two complex numbers such that $|z_1 + z_2|$ = $1$ and $\left| {z_1^2 + z_2^2} \right|$ = $25$ , then minimum value of $\left| {z_1^3 + z_2^3} \right|$ is

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 $...............$.

Let $S$ be the set of all complex numbers $z$ satisfying $\left|z^2+z+1\right|=1$. Then which of the following statements is/are $TRUE$?

$(A)$ $\left|z+\frac{1}{2}\right| \leq \frac{1}{2}$ for all $z \in S$  $(B)$ $|z| \leq 2$ for all $z \in S$

$(C)$ $\left|z+\frac{1}{2}\right| \geq \frac{1}{2}$ for all $z \in S$  $(D)$ The set $S$ has exactly four elements

Let $z_1$ and $z_2$ be any two non-zero complex numbers such that $3\left| {{z_1}} \right| = 4\left| {{z_2}} \right|$. If $z = \frac{{3{z_1}}}{{2{z_2}}} + \frac{{2{z_2}}}{{3{z_1}}}$ then

If $\bar z$ be the conjugate of the complex number $z$, then which of the following relations is false