Let $\frac{1 - ix}{1 + ix} = a - ib$ and $a^2 + b^2 = 1$,where $a$ and $b$ are real,then $x = $

If $z_n = (1 + i \sqrt{2})^n$, $n \in Z$, then $\frac{1}{9} \operatorname{Re}(z_4 \bar{z}_5) = $

If $\frac{2+3i \sin \theta}{1-2i \sin \theta}$ is purely imaginary,then $\cos^2 \theta=$

If $z = 3 - 4i$,then ${z^4} - 3{z^3} + 3{z^2} + 99z - 95$ is equal to

If $\alpha$ and $\beta$ are the roots of the equation $2z^2 - 3z - 2i = 0$,where $i = \sqrt{-1}$,then $16 \cdot \operatorname{Re}\left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}}\right) \cdot \operatorname{Im}\left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}}\right)$ is equal to

Suppose $n$ is a natural number such that $|i + 2i^2 + 3i^3 + \ldots + ni^n| = 18\sqrt{2}$,where $i = \sqrt{-1}$. Then,$n$ is