The $n^{th}$ roots of unity are in

If $n$ is a positive integer and $\omega \neq 1$ is a cube root of unity,the number of possible values of $\left|e^{\sum_{k=0}^n {^nC_k} \omega^k}\right|$ is

In the Argand plane,no value of $\sqrt[3]{1-i \sqrt{3}}$ lies in which quadrant?

If $\alpha, \beta$ are the roots of the equation $x^2-2x+4=0$ and for any $n \in N, \alpha^n+\beta^n=k \cos \frac{n \pi}{3}$, then $k=$

$f(x) = (\cos x + i \sin x) \cdot (\cos 3x + i \sin 3x) \cdots [\cos(2n-1)x + i \sin(2n-1)x]$,$n \in N$. Then $f''(x) = ?$ (Where $i = \sqrt{-1}$)

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