If ${x_r} = \cos \left( \frac{\pi }{2^r} \right) + i\sin \left( \frac{\pi }{2^r} \right)$,then the product ${x_1} \cdot {x_2} \cdot {x_3} \cdots \infty$ is:

If $\alpha$ and $\beta$ are the roots of the equation $x^2+x+1=0$, then the equation whose roots are $\alpha^{19}$ and $\beta^7$ is

If $\alpha, \beta, \gamma, \delta$ are the roots of the equation $x^4+x^2+1=0$ such that $\alpha+\beta=-1, \gamma+\delta=1, \alpha^2=\beta$ and $\gamma^2=-\delta$,then $\alpha^{2023}+\beta^{2023}+\gamma^{2022}+\delta^{2022}=$

The common roots of the equations $z^3+2z^2+2z+1=0$ and $z^{2018}+z^{2017}+1=0$ satisfy the equation

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

$\sum_{n=1}^{20} \left[ \sin \left( \frac{2n\pi}{21} \right) - i \cos \left( \frac{2n\pi}{21} \right) \right] = $