Let $z =1+ i$ and $z _1=\frac{1+ i \overline{ z }}{\overline{ z }(1- z )+\frac{1}{ z }}$. Then $\frac{12}{\pi}$ $\arg \left(z_1\right)$ is equal to $..........$.

Let $S=\left\{z \in C : z^{2}+\bar{z}=0\right\}$. Then $\sum \limits_{z \in S}(\operatorname{Re}(z)+\operatorname{Im}(z))$ is equal to$......$

If $z$ and $w$ are two complex numbers such that $|zw| = 1$ and $arg(z) -arg(w) =\frac {\pi }{2},$ then

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

If ${z_1}$ and ${z_2}$ are two complex numbers, then $|{z_1} - {z_2}|$ is

If $z_1, z_2  $ are any two complex numbers, then $|{z_1} + \sqrt {z_1^2 - z_2^2} |$ $ + |{z_1} - \sqrt {z_1^2 - z_2^2} |$ is equal to