If $z+\frac{1}{z}=1$,then $\frac{\left(z^{20}+1\right)\left(z^{40}+1\right)\left(z^{60}+1\right)}{z^{60}}=$

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 $2 \alpha = -1 - i \sqrt{3}$ and $2 \beta = -1 + i \sqrt{3}$,then $5 \alpha^4 + 5 \beta^4 + 7 \alpha^{-1} \beta^{-1}$ is equal to

The cube roots of unity are the vertices of a/an ......... which is inscribed in a circle of unit radius,with its centre at the origin.

If $1, \omega, \omega^2$ denote the cube roots of unity,then the value of $(1-\omega+\omega^2)^5+(1+\omega-\omega^2)^5$ is

If $\omega$ is a complex cube root of unity, then the value of the expression $2(1 + \frac{1}{\omega})(1 + \frac{1}{\omega^2}) + 3(2 + \frac{1}{\omega})(2 + \frac{1}{\omega^2}) + ... + (n + 1)(n + \frac{1}{\omega})(n + \frac{1}{\omega^2})$ is...