1. Let acute angle $\theta$ make the equation about $x$, $x^{2}+4 x \cos \theta+\cot \theta$ $=0$, have a repeated root. Then the radian measure of $\theta$ is $($ ).
(A) $\frac{\pi}{6}$
(B) $\frac{\pi}{12}$ or $\frac{5 \pi}{12}$
(C) $\frac{\pi}{6}$ or $\frac{5 \pi}{12}$
(D) $\frac{\pi}{12}$
$-1 . B$
Since the equation has a repeated root, we have $\Delta=16 \cos ^{2} \theta-4 \cot \theta=0$.
Given $0<\theta<\frac{\pi}{2}$, thus $4 \cos \theta(2 \sin 2 \theta-1)=0$, which gives $\sin 2 \theta=\frac{1}{2}$. Therefore, $\theta=\frac{\pi}{12}$ or $\theta=\frac{5 \pi}{12}$.