If $\omega$ is a complex cube root of unity,then the value of $\frac{a+b \omega+c \omega^2}{c+a \omega+b \omega^2}+\frac{a+b \omega+c \omega^2}{b+c \omega+a \omega^2}$ 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 $z = \cos \theta + i \sin \theta$,then $z^r + (\bar{z})^r = $

If $\omega ( \neq 1)$ is a cube root of unity and $(1 + \omega )^7 = A + B\omega$,then $A$ and $B$ are respectively,the numbers

If $x = a + b$,$y = a\alpha + b\beta$,and $z = a\beta + b\alpha$,where $\alpha$ and $\beta$ are complex cube roots of unity,then $xyz$ =

The value of $\left(\frac{1+\sqrt{3} i}{1-\sqrt{3} i}\right)^{64}+\left(\frac{1-\sqrt{3} i}{1+\sqrt{3} i}\right)^{64}$ is