1. (2004 National High School Mathematics Competition) 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. $\frac{\pi}{12}$
1. B $\Delta=16 \cos ^{2} \theta-4 \cot \theta=0$, we get $2 \sin 2 \theta \cos \theta=\cos \theta$ so $\sin 2 \theta=\frac{1}{2}$, thus $2 \theta$ is $\frac{\pi}{6}$ or $\frac{5 \pi}{6}$, $\theta$ is $\frac{\pi}{12}$ or $\frac{5 \pi}{12}$.