If the complex numbers $z_1, z_2$ and the origin form an equilateral triangle,then $z_1^2 + z_2^2 = $

Let $f(x) = x^4 + ax^3 + bx^2 + c$ be a polynomial with real coefficients such that $f(1) = -9$. Suppose that $i\sqrt{3}$ is a root of the equation $4x^3 + 3ax^2 + 2bx = 0$,where $i = \sqrt{-1}$. If $\alpha_1, \alpha_2, \alpha_3$,and $\alpha_4$ are all the roots of the equation $f(x) = 0$,then $|\alpha_1|^2 + |\alpha_2|^2 + |\alpha_3|^2 + |\alpha_4|^2$ is equal to:

If $z$ is a complex number satisfying $|z|^2 - |z| - 2 < 0$,then the value of $|z^2 + z \sin \theta|$,for all values of $\theta$,is

If $z = 2 + 3i$,then $z^{5} + (\bar{z})^{5}$ is equal to:

If $(x-iy)^{\frac{1}{3}} = a+ib$,then $\frac{ax-by}{a-b} = $

Let $z$ be a complex number with $\operatorname{Im}(z)=10$ and satisfying $\frac{2z-n}{2z+n}=2i-1$, where $i=\sqrt{-1}$, for some natural number $n$. Then: