Explanation

(d)
Since, $y^{2} - y + a = \left( y - \frac{1}{2} \right)^{2} + a - \frac{1}{4}$
and $- \sqrt{2} \leq \sin{x + \cos{x\ \leq \sqrt{2}},}$ given equation will have no real values of $x$ for any $y$, if $a - \frac{1}{4} > \sqrt{2}$
$ie,\ a \in \left( \ \sqrt{2} + \frac{1}{4},\infty \right)$
$\Rightarrow a \in (\sqrt{3},\infty)$ (as $\sqrt{2} + \frac{1}{4} < \sqrt{3})$