If $\alpha$ satisfies the equation $x^2+x+1=0$ and $(1+\alpha)^7=A+B\alpha+C\alpha^2$,where $A, B, C \geq 0$,then $5(3A-2B-C)$ is equal to:

$(\sqrt{3}+i)^7+(\sqrt{3}-i)^7$ is equal to (in $\sqrt{3}$)

If $1, \omega, \omega^2, \ldots, \omega^{10}$ are the $11^{th}$ roots of unity,then the product of these roots is:

If $\alpha, \beta$ are the roots of the equation $x^2-4x+8=0$,then for any $n \in N$,$\alpha^{2n}+\beta^{2n}$ equals

If $z_1$ and $z_2$ are two of the $n^{\text{th}}$ roots of unity such that the line segment joining them subtends a right angle at the origin,then for a positive integer $k$,$n$ takes the form

If $n$ is an integer which leaves remainder $1$ when divided by $3$,then $(1+\sqrt{3}i)^n + (1-\sqrt{3}i)^n$ equals