JEE Math Practice Question

If $$f: \mathbb{R} \rightarrow \mathbb{R}$$ be a continuous function satisfying $$\int\limits_{0}^{\frac{\pi}{2}} f(\sin 2 x) \sin x d x+\alpha \int\limits_{0}^{\frac{\pi}{4}} f(\cos 2 x) \cos x d x=0$$, then the value of $$\alpha$$ is :

1. A.$$-\sqrt{3}$$
2. B.$$\sqrt{2}$$
3. C.$$-\sqrt{2}$$
4. D.$$\sqrt{3}$$