The conjugate of complex number $\frac{{2 - 3i}}{{4 - i}},$ is

Let $z_1, z_2 \in C$ such that $| z_1 + z_2 |= \sqrt 3$ and $|z_1| = |z_2| = 1,$ then the value of $|z_1 - z_2|$ is

The conjugate of $\frac{{{{(2 + i)}^2}}}{{3 + i}},$ in the form of $a + ib$, is

Let $z$ be a complex number such that $\left| z \right| + z = 3 + i$ (where $i = \sqrt { - 1} $). Then $\left| z \right|$ is equal to

For $a \in C$, let $A =\{z \in C: \operatorname{Re}( a +\overline{ z }) > \operatorname{Im}(\bar{a}+z)\}$ and $B=\{z \in C: \operatorname{Re}(a+\bar{z}) < \operatorname{Im}(\bar{a}+z)\}$. Then among the two statements :

$(S 1)$ : If $\operatorname{Re}(A), \operatorname{Im}(A) > 0$, then the set $A$ contains all the real numbers

$(S2)$: If $\operatorname{Re}(A), \operatorname{Im}(A) < 0$, then the set $B$ contains all the real numbers,

The number of solutions of the equation ${z^2} + \bar z = 0$ is