If $z$ is a complex number such that $\frac{{z - 1}}{{z + 1}}$ is purely imaginary, then

If $a > 0$ and $z = \frac{{{{\left( {1 + i} \right)}^2}}}{{a - i}}$, has magnitude $\sqrt {\frac{2}{5}} $, then $\bar z$ 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

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

If $|z - 25i| \le 15$, then $|\max .amp(z) - \min .amp(z)| = $

$arg\left( {\frac{{3 + i}}{{2 - i}} + \frac{{3 - i}}{{2 + i}}} \right)$ is equal to