If $\omega$ is a complex cube root of unity,then for any $n>1$,$\sum_{r=1}^{n-1} r(r+1-\omega)(r+1-\omega^2) =$

If $Z = r(\cos \theta + i \sin \theta), (\theta \neq -\pi / 2)$ is a solution of $x^3 = i$,then $r^9(\cos \theta + i \sin \theta)^9 = $

If $1, \omega, \omega^2$ are the cube roots of unity,$n \in \mathbb{N}$ and $n > 2$,then the least value of $n$ such that $1+\omega$ is a root of $x^n-x=0$ is

$\omega$ is a complex cube root of unity. Match the items of List-$I$ to the items of List-$II$.

List-$I$ (Expression)	List-$II$ (Value)
$A$. $\omega^{1010} + \omega^{2000}$	$I$. $0$
$B$. $(1 + \omega - \omega^2)(1 - \omega + \omega^2)$	$II$. $1$
$C$. $(2 + \omega^2 + \omega^4)^5$	$III$. $-1$
$D$. $(3 + 5\omega + 3\omega^2)^3$	$IV$. $4$
	$V$. $8$

The correct match is:

If $\omega$ is a non-real cube root of unity and $x = \omega^2 - \omega - 3$,then the value of $x^4 + 6x^3 + 10x^2 - 12x - 19$ is

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: