Let $Z_1, Z_2, Z_3$ be three non-zero complex numbers such that $a = |Z_1|, b = |Z_2|, c = |Z_3|$. If the determinant $\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} = 0$,then:

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:

The biquadratic equation,two of whose roots are $1+i$ and $1-\sqrt{2}$,is

Convert the complex number $z = \frac{i-1}{\cos \frac{\pi}{3} + i \sin \frac{\pi}{3}}$ into polar form.

If $(2+i)$ is a root of the equation $x^3-5x^2+9x-5=0$,then the other roots are

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