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. $\frac{\pi}{12}$
Solve:
$\Delta=16 \cos ^{2} \theta-4 \cot \theta=0 \Rightarrow 4 \sin \theta \cos \theta=2 \sin 2 \theta=1 \Rightarrow 2 \theta=\frac{\pi}{6}$ or $\frac{5 \pi}{6}$
$\Rightarrow \theta=\frac{\pi}{12}$ or $\frac{5 \pi}{12}$. Therefore, choose $B$.