The imaginary part of $(\sqrt{3}-i)^{2016}+(-\sqrt{3}-i)^{2019}$ is

If $\alpha, \beta$ are the roots of $x^2-x+1=0$,then the quadratic equation whose roots are $\alpha^{2015}$ and $\beta^{2015}$ is

The product of the distinct $(2n)^{\text{th}}$ roots of $1+i\sqrt{3}$ is equal to:

Let $\alpha$ and $\beta$ be the roots of $x^{2}+x+1=0$. If $n$ is a positive integer,then $\alpha^{n}+\beta^{n}$ is

If $\omega$ is a complex cube root of unity and $a, b, c$ are distinct real numbers,then $\frac{a+b \omega+c \omega^2}{c+a \omega+b \omega^2}+\frac{a+b \omega+c \omega^2}{b+c \omega+a \omega^2} = $

Let ${z_1}$ and ${z_2}$ be $n^{th}$ roots of unity which are ends of a line segment that subtend a right angle at the origin. Then $n$ must be of the form