If the point $(x, y)$ satisfies the equation $\frac{x+i(x-2)}{3+i}-i=\frac{2y+i(1-3y)}{i-3}$,then $x+y=$

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

The values of $x$ for which $\sin x + i \cos 2x$ and $\cos x - i \sin 2x$ are conjugate to each other are

If $\cos A+\cos B+\cos C=0$ and $\sin A+\sin B+\sin C=0$,then $\cos (A-B)=$

The least value of $|z|$ where $z$ is a complex number satisfying the inequality $\exp \left(\frac{(|z|+3)(|z|-1)}{|z|+1} \log _{ e } 2\right) \geq \log _{\sqrt{2}}|5 \sqrt{7}+9 i |$,where $i=\sqrt{-1}$,is equal to:

If $z$ is a non-real complex number, then the minimum value of $\frac{\operatorname{Im}(z^5)}{(\operatorname{Im} z)^5}$ is