Let $z, w \in \mathbb{C}$ satisfy $z^2 + \bar{w} = z$ and $w^2 + \bar{z} = w$. Then,the number of ordered pairs of complex numbers $(z, w)$ is equal to

If $z = x + iy$,${z^{1/3}} = a - ib$ and $\frac{x}{a} - \frac{y}{b} = k({a^2} - {b^2})$,then the value of $k$ is equal to:

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

Let $S$ be the set of all complex numbers $z$ satisfying $|z^2+z+1|=1$. Then which of the following statements is/are $TRUE$?
$(A) |z+\frac{1}{2}| \leq \frac{1}{2}$ for all $z \in S$
$(B) |z| \leq 2$ for all $z \in S$
$(C) |z+\frac{1}{2}| \geq \frac{1}{2}$ for all $z \in S$
$(D)$ The set $S$ has exactly four elements

If $z_1$ and $z_2$ are two unimodular complex numbers that satisfy $z_1^2 + z_2^2 = 5,$ then $(z_1 - \bar{z}_1)^2 + (z_2 - \bar{z}_2)^2$ is equal to -

Let $\bar{z}$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of $(\bar{z})^2+\frac{1}{z^2}$ are integers,then which of the following is/are possible value$(s)$ of $|z|$?