If $x^{2}+x+1=0$, then the value of $(x+\frac{1}{x})^{4}+(x^{2}+\frac{1}{x^{2}})^{4}+(x^{3}+\frac{1}{x^{3}})^{4}+\dots+(x^{25}+\frac{1}{x^{25}})^{4}$ is:

The number of complex numbers $p$ such that $|p|=1$ and the imaginary part of $p^{4}$ is $0$ 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 $\omega$ is a complex cube root of unity,then $\omega^{\left(\frac{1}{3}+\frac{2}{9}+\frac{4}{27}+\ldots \infty\right)}+\omega^{\left(\frac{1}{2}+\frac{3}{8}+\frac{9}{32}+\ldots \infty\right)}$ is equal to

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

If $\alpha$ and $\beta$ are the roots of the equation $x^2-x+1=0$,then $\alpha^{2009}+\beta^{2009}$ is equal to