If $\left|z-\frac{2}{z}\right|=2$,then the greatest value of $|z|$ is

If $(\cos \theta + i\sin \theta )(\cos 2\theta + i\sin 2\theta ) \dots (\cos n\theta + i\sin n\theta ) = 1$,then the value of $\theta$ is

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 $z$)

If $Z_1, Z_2, Z_3$ are three complex numbers with unit modulus such that $|Z_1-Z_2|^2+|Z_1-Z_3|^2=4$,then $Z_1 \overline{Z_2}+\overline{Z_1} Z_2+Z_1 \overline{Z_3}+\overline{Z_1} Z_3=$

If $x_n = \cos \left(\frac{\pi}{4^n}\right) + i \sin \left(\frac{\pi}{4^n}\right)$,then the product $x_1 x_2 x_3 \ldots \infty$ is equal to

If $z=\cos 6^{\circ}+i \sin 6^{\circ}$,then $\sum_{n=1}^{20} \operatorname{Im}\left(z^{2 n-1}\right)=$