If for $z=\alpha+i \beta$,$|z+2|=z+4(1+i)$,then $\alpha+\beta$ and $\alpha \beta$ are the roots of the equation

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

Let $z$ be a complex number such that $|z+2|=1$ and $\operatorname{Im}\left(\frac{z+1}{z+2}\right)=\frac{1}{5}$. Then the value of $|\operatorname{Re}(\overline{z+2})|$ is:

Let $z_1, z_2 \in \mathbb{C}$ be the distinct solutions of the equation $z^2 + 4z - (1 + 12i) = 0$. Then $|z_1|^2 + |z_2|^2$ is equal to:

If $x+iy = (1+i)^6 - (1-i)^6$,then which one of the following is true?

The number of complex numbers $z$ satisfying $\overline{z} = i z^2$ is