 Solution

If every point of $E[n]$ is rational, the Weil pairingand its nondegeneracy imply that every $n$th root of unity belongs to $\mathbb{Q}$. The only roots of unity in $\mathbb{Q}$ are $\pm 1$, so $n\le 2$. Since the question assumes $n\ge 2$, only $n=2$ is possible. It does occur: any nonsingular equationwith distinct $a,b,c\in \mathbb{Q}$ has all four points of $E[2]$ rational.