If $z = x + iy$ $(x, y \in R, x \neq -1/2)$,the number of values of $z$ satisfying $|z|^n = z^2|z|^{n-2} + z|z|^{n-2} + 1$ $(n \in N, n > 1)$ is

Let $z$ and $w$ be two complex numbers such that $|z| \le 1$,$|w| \le 1$ and $|z + iw| = |z - i\overline{w}| = 2$. Then $z$ is equal to

If $x_r = \cos(\pi/3^r) - i\sin(\pi/3^r)$ (where $i = \sqrt{-1}$),then the value of $x_1 \cdot x_2 \cdot x_3 \cdots \infty$ is:

If $x+\frac{1}{x}=2 \sin \alpha$ and $y+\frac{1}{y}=2 \cos \beta$,then $x^3 y^3+\frac{1}{x^3 y^3}=$

The value of $\left(\frac{1+\sin \frac{2 \pi}{9}+i \cos \frac{2 \pi}{9}}{1+\sin \frac{2 \pi}{9}-i \cos \frac{2 \pi}{9}}\right)^3$ is

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: