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

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 $|z|$?

Let $z_1$ and $z_2$ be two complex numbers such that $z_1 + z_2 = 5$ and $z_1^3 + z_2^3 = 20 + 15i$. Then $|z_1^4 + z_2^4|$ equals-

If $x = -2 + \sqrt{-3}$,then the value of $2x^4 + 5x^3 + 7x^2 - x + 38$ is equal to

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

If $z(1 + a) = b + ic$ and $a^2 + b^2 + c^2 = 1$,then $\frac{1 + iz}{1 - iz} = $