If $z$ and $\omega$ are two complex numbers such that $|z \omega|=1$ and $\arg (z)-\arg (\omega)=\frac{3 \pi}{2}$, then $\arg \left(\frac{1-2 \bar{z} \omega}{1+3 \bar{z} \omega}\right)$ is:

(Here arg(z) denotes the principal argument of complex number $z$ )

Let $z$ be a complex number, then the equation ${z^4} + z + 2 = 0$ cannot have a root, such that

For any two complex numbers ${z_1},{z_2}$we have $|{z_1} + {z_2}{|^2} = $ $|{z_1}{|^2} + |{z_2}{|^2}$ then

The conjugate of a complex number is $\frac{1}{{i - 1}}$ then that complex number is

The set of all $\alpha  \in R$, for which $w = \frac{{1 + \left( {1 - 8\alpha } \right)z}}{{1 - z}}$ is a purely imaginary number, for all $z \in C$ satisfying $\left| z \right| = 1$ and ${\mathop{\rm Re}\nolimits} \,z \ne 1$,  is

Number of complex numbers $z$ such that $\left| z \right| + z - 3\bar z = 0$ is equal to