If $Z_1 = \sqrt{3} + i \sqrt{3}$ and $Z_2 = \sqrt{3} + i$,and $\left(\frac{Z_1}{Z_2}\right)^{50} = x + iy$,then the point $(x, y)$ lies in

For two non-zero complex numbers $z_1$ and $z_2$, if $\operatorname{Re}(z_1 z_2) = 0$ and $\operatorname{Re}(z_1 + z_2) = 0$, then which of the following are possible?
$(A) \operatorname{Im}(z_1) > 0$ and $\operatorname{Im}(z_2) > 0$
$(B) \operatorname{Im}(z_1) < 0$ and $\operatorname{Im}(z_2) > 0$
$(C) \operatorname{Im}(z_1) > 0$ and $\operatorname{Im}(z_2) < 0$
$(D) \operatorname{Im}(z_1) < 0$ and $\operatorname{Im}(z_2) < 0$
Choose the correct answer from the options given below:

If $\tan (u + iv) = i$,then the value of $v$ is

If $z_1, z_2, z_3$ are the roots of the equation $z^3 - z^2(4 + 3i) + z(3 + 8i) - 5i = 0$, then $Re(z_1) + Re(z_2) + Re(z_3)$ is -

Suppose $z$ is any root of $11 z^8 + 21 i z^7 + 10 i z - 22 = 0$ where $i = \sqrt{-1}$. Then,$S = |z|^2 + |z| + 1$ satisfies

$(r, \theta)$ denotes $r(\cos \theta + i \sin \theta)$. If $x = (1, \alpha)$,$y = (1, \beta)$,$z = (1, \gamma)$ and $x + y + z = 0$,then $\sum \cos (2\alpha - \beta - \gamma) = $