If $\omega$ is a complex cube root of unity,then $\sum_{k=1}^6\left(\omega^k+\frac{1}{\omega^k}\right)^2=$

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

For $n > 1$ and $n \in N$, if $z_1, z_2, \ldots, z_n$ are the roots of the equation $(z+1)^n = z^n$, then $\sum_{i=1}^{n-1} \frac{\cot^{-1}(2|\operatorname{Im} z_i|) - 1}{2 \operatorname{Re} z_i} = $

If $z = \sin \theta - i \cos \theta,$ then for any integer $n$

If $\alpha$ and $\beta$ are imaginary cube roots of unity,then $\alpha^4 + \beta^4 + \frac{1}{\alpha\beta} = $

The real part of $z$ that satisfies $i z^4+1=0$ is