If $y = \cos \theta + i\sin \theta$,then the value of $y + \frac{1}{y}$ is

If $\alpha$ and $\beta$ are imaginary cube roots of unity,then the value of $\alpha^4 + \beta^{28} + \frac{1}{\alpha \beta}$ is

All the values of $(8i)^{\frac{1}{3}}$ are

Let $z = \left( \frac{\sqrt{3}}{2} + \frac{i}{2} \right)^5 + \left( \frac{\sqrt{3}}{2} - \frac{i}{2} \right)^5$. If $R(z)$ and $I(z)$ respectively denote the real and imaginary parts of $z$,then:

If $w = \frac{-1 + i \sqrt{3}}{2}$,where $i = \sqrt{-1}$,then the value of $(3 + w + 3 w^2)^4$ 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