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Problem 679

AMC 10/12, early questions

Algebra Difficulty 3.9 Multiple choice China Mathematical Competition (Hainan) · China

Let $\theta$ be an acute angle such that the equation $x^2 + 4x\cos\theta + c\cot\theta = 0$ involving variable $x$ has multiple roots. Then the measure of $\theta$ in radians is:

Pick one

Official solution

Since the equation $x^2 + 4x\cos\theta + c\cot\theta = 0$ has multiple roots, we have
$\Delta = 16\cos^2\theta - 4\cot\theta = 0,$
or
$4\cot\theta(2\sin 2\theta - 1) = 0.$

It follows that
$2\theta = \frac{\pi}{6} \text{ or } 2\theta = \frac{5\pi}{6}.$
Thus $\theta = \frac{\pi}{12}$ or $\theta = \frac{5\pi}{12}$ . Answer: B.

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