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.

Solve the equation $21 x^{2}-28 x+10=0$

If $4 x+i(3 x-y)=3+i(-6),$ where $x$ and $y$ are real numbers, then find the values of $x$ and $y.$

The least positive integer $\mathrm{n}$ such that $\frac{(2 \mathrm{i})^{\mathrm{n}}}{(1-\mathrm{i})^{\mathrm{n}-2}}, \mathrm{i}=\sqrt{-1}$ is a positive integer, is ….. .

The number of real values of $a$ satisfying the equation ${a^2} – 2a\sin x + 1 = 0$ is