If $\alpha$ and $\beta$ are the roots of the equation $x^2-2x+4=0$,then $\alpha^{12}+\beta^{12}=$

Let $\alpha, \beta$ be the roots of the equation $x^2-\sqrt{6}x+3=0$ such that $\operatorname{Im}(\alpha)>\operatorname{Im}(\beta)$. Let $a, b$ be integers not divisible by $3$ and $n$ be a natural number such that $\frac{\alpha^{99}}{\beta}+\alpha^{98}=3^n(a+ib)$,where $i=\sqrt{-1}$. Then $n+a+b$ is equal to:

For any real number $n \in \mathbb{R}$,$(\cosh x + \sinh x)^n =$

If $1, \omega$ and $\omega^2$ are the cube roots of unity,then $(a+b+c)(a+b \omega+c \omega^2)(a+b \omega^2+c \omega) = $

If $1, z_1, z_2, \ldots, z_{n-1}$ are the $n$th roots of unity,then the value of $(1-z_1)(1-z_2) \ldots (1-z_{n-1})$ is equal to

If $z = \frac{\sqrt{3} + i}{-2}$,then $z^{69}$ is equal to