If the cube root of $1$ is $\omega$,then the value of $(3 + \omega + 3\omega^2)^4$ is

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 $.......$

For any real number $n \in \mathbb{R}$,$(\cosh x + \sinh x)^n =$

$\sum_{k=1}^6\left(\sin \frac{2 \pi k}{7}-i \cos \frac{2 \pi k}{7}\right)=$

Express the following expression in the form $A + iB$: $(\cos 2\theta + i\sin 2\theta )^{ - 5} (\cos 3\theta - i\sin 3\theta )^6 (\sin \theta - i\cos \theta )^3$.

If $1, \omega, \omega^2$ are the cube roots of unity,then the value of $(x+y)^2+(x \omega+y \omega^2)^2+(x \omega^2+y \omega)^2$ is