Let $p, q \in \mathbb{R}$ and $(1-\sqrt{3}i)^{200} = 2^{199}(p + iq)$,where $i = \sqrt{-1}$. Then $p + q + q^2$ and $p - q + q^2$ are roots of the equation:

If $z$ is a complex number such that $z^2+z+1=0$,then $\left(z+\frac{1}{z}\right)^3+\left(z^2+\frac{1}{z^2}\right)^3+\left(z^3+\frac{1}{z^3}\right)^3+\ldots+\left(z^{2020}+\frac{1}{z^{2020}}\right)^3=$

For the equation $x^2 + x + 1 = 0$,if $\alpha$ and $\beta$ are the roots,then which of the following equations has roots $\alpha^{19}$ and $\beta^{7}$?

If $(\sqrt{3}-i)^{n}=2^{n}, n \in N$,then the least possible value of $n$ is

If $f(x)$ and $g(x)$ are two polynomials such that the polynomial $P(x) = f(x^3) + xg(x^3)$ is divisible by $x^2 + x + 1$,then $P(1)$ is equal to ....... .

If the number of real roots of $x^9-x^5+x^4-1=0$ is $n$,the number of complex roots having argument on the imaginary axis is $m$,and the number of complex roots having argument in the $2^{nd}$ quadrant is $k$,then $m \cdot n \cdot k = $