I. Multiple Choice (This question is worth 36 points, 6 points per sub-question)
1. Let the acute angle $\theta$ make the equation $x^{2}+4 x \cos \theta+\cot \theta=0$ 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) $\frac{\pi}{12}$
(B)
I. Multiple Choice Question
1.【Analysis and Solution】Since the equation $x^{2}+4 x \cos \theta+\cot \theta=0$ has a repeated root, we have $\Delta=16 \cos ^{2} \theta-4 \cot \theta=0$. $\because 0<\theta<\frac{\pi}{2}, \therefore 4 \cot \theta(2 \sin 2 \theta-1)=0$ which gives $\sin 2 \theta=\frac{1}{2}$.
$\therefore 2 \theta=\frac{\pi}{6}$ or $2 \theta=\frac{5 \pi}{6}$, thus $\theta=\frac{\pi}{12}$ or $\frac{5 \pi}{12}$. Therefore, the answer is B.