The expression $\frac{(1+i)^{n}}{(1-i)^{n-2}}$ equals

If $1+2i$ is a root of the equation $x^4-3x^3+8x^2-7x+5=0$,then the sum of the squares of the other roots is

If $i=\sqrt{-1}$,then $\sum_{n=0}^{\infty}\left(\frac{i}{3}\right)^n=$

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:

If $\frac{3x + 2iy}{5i - 2} = \frac{15}{8x + 3iy}$,then

Solve: $i x^2 - 3 x - 2 i = 0$