$\frac{{\sqrt {5 + 12i} + \sqrt {5 – 12i} }}{{\sqrt {5 + 12i} – \sqrt {5 – 12i} }} = $

${(1 + i)^{10}}$, where ${i^2} = – 1,$ is equal to

The smallest positive integer $n$for which ${(1 + i)^{2n}} = {(1 – i)^{2n}}$is

$\sum\limits_{n – 1}^{50} {{i^{(2n – 1)!}}} $ is equal to (where $i = \sqrt {-1}$ )

Let $S$ be the set of all $(\alpha, \beta), \pi<\alpha, \beta<2 \pi$, for which the complex number $\frac{1-i \sin \alpha}{1+2 i \sin \alpha}$ is purely imaginary and $\frac{1+i \cos \beta}{1-2 i \cos \beta}$ is purely real. Let $Z_{\alpha \beta}=\sin 2 \alpha+i \cos 2 \beta,(\alpha, \beta) \in S$ . Then $\sum_{(\alpha, \beta) \in s }\left(i Z_{\alpha \beta}+\frac{1}{i \bar{Z}_{\alpha \beta}}\right)$ is equal to.