Explanation

(a)
Given, $\sin{2x + \cos{4x = 2}}$
$\Rightarrow \ \sin{2x + 1 - 2\sin^{2}{2x = 2}}$
$\Rightarrow \ \ 2\sin^{2}{2x - \sin{2x + 1 = 0}}$
Now, Discriminant, $D = ( - 1)^{2} - 4.2.1 = - 7 < 0$
Hence, it is an imaginary equation, so the real root does not exist.