If $(1 + i\sqrt{3})^9 = a + ib$,then $b$ is equal to

Let $\omega$ be an imaginary cube root of unity. Then the value of $2(\omega + 1)(\omega^2 + 1) + 3(2\omega + 1)(2\omega^2 + 1) + \dots + (n + 1)(n\omega + 1)(n\omega^2 + 1)$ is

If the complex cube roots of $(-i)$ are $\alpha, \beta, \gamma$,then $\alpha^2+\beta^2+\gamma^2=$

If $1, \omega, \omega^2$ are the cube roots of unity,then $(1 - 2\omega + \omega^2)^6$ is equal to

Evaluate: $\frac{(\cos \theta + i\sin \theta)^4}{(\sin \theta + i\cos \theta)^5}$

If $\omega$ is a complex cube root of unity and $a, b, c$ are distinct real numbers,then $\frac{a+b \omega+c \omega^2}{c+a \omega+b \omega^2}+\frac{a+b \omega+c \omega^2}{b+c \omega+a \omega^2} = $