If $Arg(z)$ denotes principal argument of a complex number $z$, then the value of expression $Arg\left( { - i{e^{i\frac{\pi }{9}}}.{z^2}} \right) + 2Arg\left( {2i{e^{-i\frac{\pi }{{18}}}}.\overline z } \right)$ is

If $z$ is a complex number, then the minimum value of $|z| + |z - 1|$ is

If $z$ is a complex number, then $z.\,\overline z = 0$ if and only if

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,

If $5 + ix^3y^2$ and $x^3 + y^2 + 6i$ are conjugate complex numbers and arg $(x + iy) = \theta $ , then ${\tan ^2}\,\theta $ is equal to

For the complex number $z$, one from $z + \bar z$ and $z\,\bar z$ is