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

If $a \pm ib$ and $b \pm ai$ are the roots of $x^4-10x^3+50x^2-130x+169=0$,then $\frac{a}{b}+\frac{b}{a}=$

If the set $R = \{(a, b) : a + 5b = 42, a, b \in N\}$ has $m$ elements and $\sum_{n=1}^m (1 - i^{n!}) = x + iy$,where $i = \sqrt{-1}$,then the value of $m + x + y$ is:

Let $z=x+iy$ and $w=u+iv$ be complex numbers on the unit circle such that $z^2+w^2=1$. Then the number of ordered pairs $(z, w)$ is

If $x = -5 + 2\sqrt{-4}$,then the value of the expression $x^4 + 9x^3 + 35x^2 - x + 4$ is

Let $\omega = e^{i \pi / 3}$,and $a, b, c, x, y, z$ be non-zero complex numbers such that $a+b+c = x$,$a+b \omega + c \omega^2 = y$,and $a+b \omega^2 + c \omega = z$. Then the value of $\frac{|x|^2+|y|^2+|z|^2}{|a|^2+|b|^2+|c|^2}$ is