Which of the following are correct for any two complex numbers $z_1$ and $z_2$?

Let $z \in \mathbb{C}$ be such that $\frac{z^2+3i}{z-2+i}=2+3i$. Then the sum of all possible values of $z^2$ is

If $P, Q$ and $R$ are angles of an isosceles triangle and $\angle P = \frac{\pi}{2}$, then the value of $\left(\cos \frac{P}{3} - i \sin \frac{P}{3}\right)^3 + (\cos Q + i \sin Q) (\cos R - i \sin R) + (\cos P - i \sin P) (\cos Q - i \sin Q) (\cos R - i \sin R)$ is:

If $z$ and $\omega$ are two complex numbers such that $|z \omega|=1$ and $\arg(z) - \arg(\omega) = \frac{3 \pi}{2}$,then $\arg \left(\frac{1-2 \bar{z} \omega}{1+3 \bar{z} \omega}\right)$ is:
(Here $\arg(z)$ denotes the principal argument of complex number $z$)

If $a_k = \cos \alpha_k + i \sin \alpha_k$ for $k = 1, 2, 3$ and $a_1, a_2, a_3$ are the roots of the equation $x^3 + bx + c = 0$,then the real part of $b$ is:

If $z$ and $w$ are two complex numbers such that $|zw| = 1$ and $\arg(z) - \arg(w) = \frac{\pi}{2}$,then