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

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:

The number of solutions for $z^3+\bar{z}=0$ is

If ${z_1} = a + ib$ and ${z_2} = c + id$ are complex numbers such that $|{z_1}| = |{z_2}| = 1$ and $R({z_1}\overline {{z_2}} ) = 0,$ then the pair of complex numbers ${w_1} = a + ic$ and ${w_2} = b + id$ satisfies

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

Let $z=x+iy$ and $w=u+iv$ be complex numbers on the unit circle such that $z^2+w^2=1$. Then the number of ordered pairs $(z, w)$ is