If $\alpha$ and $\beta$ are the distinct roots of the equation $x^2 - x + 1 = 0$, then the value of $\alpha^{200} + \beta^{206} + 2$ is equal to

If $n$ is a positive integer, then $(1 + i\sqrt{3})^{2n} + (1 - i\sqrt{3})^{2n}$ is equal to .........

If $(1 + i) \cdot (1 + 2i) \ldots\ldots\ldots (1 + ni) = x + iy$ (Where $i = \sqrt{-1}$ ), then the value of $(2) \cdot (5) \cdot (10)\ldots\ldots\ldots(1 + n^2)$

If $\omega$ is a complex cube root of unity, then the value of $\sin\left[\pi(\omega^{10} + \omega^{23}) - \dfrac{\pi}{4}\right] =$