If $1, \omega, \omega^2$ are the cube roots of unity and $\alpha = \omega + 2\omega^2 - 3$,then $\alpha^3 + 12\alpha^2 + 48\alpha + 3$ equals

If $1, \omega, \omega^{2}$ are the cube roots of unity,then $(1+\omega)(1+\omega^{2})(1+\omega^{4})(1+\omega^{8})$ is equal to

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

If $\omega \neq 1$ is a cube root of unity,then the sum of the series $S = 1 + 2\omega + 3\omega^2 + \dots + 3n\omega^{3n-1}$ is

If $\alpha$ and $\beta$ are the roots of the quadratic equation $x^2+x+1=0$,then the equation whose roots are $\alpha^{2021}$ and $\beta^{2021}$ is given by $.......$

If $1, \omega, \omega^2$ are the cube roots of unity and $(x+y)(x \omega+y \omega^2)(x \omega^2+y \omega)=f(x, y)$,then $f(2, 3)=$