The real value of $\theta$ for which the expression $\frac{{1 + i\,\cos \theta }}{{1 - 2i\cos \theta }}$ is a real number is $\left( {n \in I} \right)$ 

For any complex number $z,\bar z = \left( {\frac{1}{z}} \right)$if and only if

If ${z_1} = 10 + 6i,{z_2} = 4 + 6i$ and $z$ is a complex number such that $amp\left( {\frac{{z - {z_1}}}{{z - {z_2}}}} \right) = \frac{\pi }{4},$ then the value of $|z - 7 - 9i|$ is equal to

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

If complex numbers $z_1$ and $z_2$ both satisfy $z + \overline z  = 2 | z -1 |$ and $arg(z_1 -z_2) = \frac{\pi}{3} ,$ then value of $Im (z_1 + z_2)$ is, where $Im (z)$ denotes imaginary part of $z$ -

Let $S=\left\{Z \in C: \bar{z}=i\left(z^2+\operatorname{Re}(\bar{z})\right)\right\}$. Then $\sum_{z \in S}|z|^2$ is equal to