The number of all possible solutions of the equation $z^3+\overline{z}=0$ is

If $3+i$ and $2-\sqrt{3}$ are the roots of the equation $f(x)=a_0+a_1 x+a_2 x^2+\ldots+a_{n} x^{n}$ where $a_0, a_1, \ldots, a_{n} \in \mathbb{Z}$,then the least value of $n$ and the value of $a_0$ are respectively:

The sum of the squares of the modulus of all complex numbers $z$ satisfying $\bar{z} = i z^{2} + z^{2} - z$ is equal to

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

If $\left(\frac{\cos \theta+i \sin \theta}{\sin \theta+i \cos \theta}\right)^{2024}+\left(\frac{1+\cos \theta+i \sin \theta}{1-\cos \theta+i \sin \theta}\right)^{2025}=x+i y$,then the value of $x+y$ at $\theta=\frac{\pi}{2}$ 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 $z$)