If $2 \alpha = -1 - i \sqrt{3}$ and $2 \beta = -1 + i \sqrt{3}$,then $5 \alpha^4 + 5 \beta^4 + 7 \alpha^{-1} \beta^{-1}$ is equal to

If $\omega$ is a cube root of unity but not equal to $1$,then the minimum value of $|a + b\omega + c\omega^2|$ (where $a, b, c$ are integers but not all equal) is

If $1, \omega, \omega^2, \ldots, \omega^8$ are the roots of the equation $x^9-1=0$,then $\sum_{r=1}^8 \left(\omega^r\right)^{99} =$

Express $\frac{(\cos 2\theta - i\sin 2\theta)^4 (\cos 4\theta + i\sin 4\theta)^{-5}}{(\cos 3\theta + i\sin 3\theta)^{-2} (\cos 3\theta - i\sin 3\theta)^{-9}}$ in the form $x + iy$.

For $n \in Z^{+}$,$(1+\sin \theta+i \cos \theta)^n+(1+\sin \theta-i \cos \theta)^n=$

If $\alpha$ and $\beta$ are the roots of the equation $x^2-2x+4=0$,then $\alpha^9+\beta^9$ is equal to