If $\omega$ is a complex cube root of unity,then $\sin \left[\left(\omega^{10}+\omega^{23}\right) \pi-\frac{\pi}{4}\right]=$

If $\alpha$ and $\beta$ are the roots of the equation $x^2-2x+4=0$,then $\alpha^{12}+\beta^{12}=$

The roots of the equation $x^3-3x^2+3x+7=0$ are $\alpha, \beta, \gamma$ and $\omega, \omega^2$ are complex cube roots of unity. If the terms containing $x^2$ and $x$ are missing in the transformed equation when each one of these roots is decreased by $h$,then $\frac{\alpha-h}{\beta-h}+\frac{\beta-h}{\gamma-h}+\frac{\gamma-h}{\alpha-h}=$

If $\alpha$ is a complex number such that $\alpha^{2}-\alpha+1=0$,then $\alpha^{2011}$ is equal to

Consider a regular $10$-gon with its vertices on the unit circle. With one vertex fixed,draw straight lines to the other $9$ vertices. Call them $L_1, L_2, \ldots, L_9$ and denote their lengths by $l_1, l_2, \ldots, l_9$ respectively. Then,the product $l_1 \times l_2 \times \ldots \times l_9$ is

Let ${\omega _n} = \cos \left( {\frac{{2\pi }}{n}} \right) + i\sin \left( {\frac{{2\pi }}{n}} \right)$ and ${i^2} = -1$. Then $(x + y{\omega _3} + z{\omega _3}^2)(x + y{\omega _3}^2 + z{\omega _3})$ is equal to: