Mathematics Question from JEE Main 2024 · 31 January, Shift 1

Let $f : \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = \dfrac{4^{x}}{4^{x} + 2}$ and

$$M = \int_{f(a)}^{f(1-a)} x \sin^4\big(x(1 - x)\big)\, dx, \qquad N = \int_{f(a)}^{f(1-a)} \sin^4\big(x(1 - x)\big)\, dx; \qquad a \ne \tfrac{1}{2}.$$

If $\alpha M = \beta N$, $\alpha, \beta \in \mathbb{N}$, then the least value of $\alpha^2 + \beta^2$ is equal to: