The $n^{th}$ roots of unity are in

If $\omega$ is a complex cube root of unity,then $\cos \left(\sum_{k=1}^7(k-\omega)(k-\omega^2) \frac{\pi}{175}\right) =$

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...

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

If $z = {\left( {\frac{{\sqrt 3 }}{2} + \frac{i}{2}} \right)^5} + {\left( {\frac{{\sqrt 3 }}{2} - \frac{i}{2}} \right)^5}$,then

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