If $1, \omega, \omega^2$ are the cube roots of unity and $1, \alpha, \alpha^2, \alpha^3$ are the fourth roots of unity in usual notation, then $\alpha+\alpha \omega-\alpha^3 \omega^2=$

If $z=\alpha+i \beta$ satisfies the equation $|z|-z=1+2 i$ and $|z|=\sqrt{\alpha^2+\beta^2}$, then $z \bar{z}=$

If $-i$ and $\alpha$ are the roots of the equation $i z^2-2(i+1) z+(2-i)=0, \tan \theta=\frac{-1}{2}$ and $\theta \in 4$ th quadrant, then $5^3 \cos 6 \theta=$

If $1, \alpha_1, \alpha_2, \alpha_3, \ldots \alpha_{n-1}$ are $n$th roots of unity then $\sum\limits_{1 \le i < f \le n - 1}^{} {} {a_i}{a_j} = $