If $z = \frac{\sqrt{3} + i}{2}$,then the value of $z^{69}$ is

Let $\alpha, \beta$ be the roots of the equation $x^2-\sqrt{2} x+2=0$. Then $\alpha^{14}+\beta^{14}$ is equal to

If $\cos \theta + i \sin \theta, \theta \in R$, is a root of the equation $a_0 x^n + a_1 x^{n-1} + \ldots + a_{n-1} x + a_n = 0$, where $a_0, a_1, \ldots, a_n \in R$ and $a_0 \neq 0$, then the value of $a_1 \sin \theta + a_2 \sin 2 \theta + \ldots + a_n \sin n \theta$ is:

The least positive integral value of $n$ such that $\left[\frac{1+\sin \frac{2 \pi}{9}+i \cos \frac{2 \pi}{9}}{1+\sin \frac{2 \pi}{9}-i \cos \frac{2 \pi}{9}}\right]^n=1$ is

The common roots of the equations $z^3+2z^2+2z+1=0$ and $z^{2014}+z^{2015}+1=0$ are

If $i$ is the root of the equation $x^2+1=0$,then $(1+\sqrt{3}i)^{2023}+(1-\sqrt{3}i)^{2023}=$