If $\alpha$ is a non-real root of $x^7=1$,then $\alpha(1+\alpha)(1+\alpha^2+\alpha^4) = $

Let $A_r = \left(x+\frac{1}{x}\right)^3 \cdot \left(x^2+\frac{1}{x^2}\right)^3 \cdot \left(x^3+\frac{1}{x^3}\right)^3 \cdots \left(x^r+\frac{1}{x^r}\right)^3$. If $x^2+x+1=0$,then $\frac{1}{A_3}+\frac{1}{A_6}+\frac{1}{A_9}+\frac{1}{A_{12}}+\cdots \infty =$

If $w$ is a complex cube root of unity, then the value of $w^{10} - w^7 + w^5 - w^2 + 1$ is:

The number of complex roots of the equation $x^{11}-x^7+x^4-1=0$ whose arguments lie in the first quadrant is

What is the value of $(1-i \sqrt{3})^9$?

If $2+2 \sqrt{3} i=k(\cos \theta+i \sin \theta)$ where $k > 0$,then find the value of $\frac{1}{\sqrt{3}}[\cos 6 \theta+i \sin 6 \theta]$.