The sum of the series $i - 2 - 3i + 4 + 5i - 6 - 7i + 8 + \dots$ up to $100$ terms,where $i = \sqrt{-1}$,is:

For a complex number $z$, let $\operatorname{Re}(z)$ denote the real part of $z$. Let $S$ be the set of all complex numbers $z$ satisfying $z^4 - |z|^4 = 4iz^2$, where $i = \sqrt{-1}$. Then the minimum possible value of $|z_1 - z_2|^2$, where $z_1, z_2 \in S$ with $\operatorname{Re}(z_1) > 0$ and $\operatorname{Re}(z_2) < 0$, is:

The minimum degree of a polynomial equation with rational coefficients having $\sqrt{3}+\sqrt{27}$ and $\sqrt{2}+5i$ as two of its roots is

If $(3 + i)z = (3 - i)\bar{z}$,then the complex number $z$ is

If $z_1$ and $z_2$ are any two complex numbers,then $|z_1 + \sqrt{z_1^2 - z_2^2}| + |z_1 - \sqrt{z_1^2 - z_2^2}|$ is equal to

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