设方程组 $\begin{cases}\sqrt{x(1-y)}+\sqrt{y(1-x)}=\dfrac 1 2,\\\sqrt{x(1-x)}+\sqrt{y(1-y)}=\dfrac{\sqrt 3}4\end{cases}$ 共有 $n$ 组解，则 $n=$（ ）
A．$0$
B．$2$
C．$4$
D．$6$
答案 C．
解析 设 $x=\cos^2\alpha$，$y=\cos^2\beta$，其中 $\alpha,\beta\in \left[0,\dfrac{\pi}2\right]$，则\[\begin{cases} \cos\alpha\sin\beta+\cos\beta\sin\alpha=\dfrac 12,\\ \cos\alpha\sin\alpha+\cos\beta\sin\beta=\dfrac{\sqrt 3}4,\end{cases}\iff \begin{cases} \sin(\alpha+\beta)=\dfrac 12,\\ \sin(\alpha+\beta)\cos(\alpha-\beta)=\dfrac{\sqrt 3}4,\end{cases}\]而 $\alpha+\beta \in [0,\pi]$，$\alpha-\beta \in \left[-\dfrac{\pi}2,\dfrac{\pi}2\right]$，因此\[\begin{cases} \alpha+\beta=\dfrac{\pi}6,\dfrac{5\pi}6,\\ \alpha-\beta=\pm \dfrac{\pi}6,\end{cases}\]从而 $(\alpha,\beta)=\left(\dfrac{\pi}6,0\right),\left(0,\dfrac{\pi}6\right),\left(\dfrac{\pi}2,\dfrac{\pi}3\right),\left(\dfrac{\pi}3,\dfrac{\pi}2\right)$，共计 $4$ 组解．