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Trigonometrical Equations
hard
Let $\theta, 0 < \theta < \pi / 2$, be an angle such that the equation $x ^2+4 x \cos \theta+\cot \theta=0$ has equal roots for $x$. Then $\theta$ in radians is
A
$\frac{\pi}{6}$ only
B
$\frac{\pi}{12}$ or $\frac{5 \pi}{12}$
C
$\frac{\pi}{6}$ or $\frac{5 \pi}{12}$
D
$\frac{\pi}{12}$ only
(KVPY-2021)
Solution
(b)
$x^2+4 x \cos \theta+\cot \theta=0$
$D =0 \Rightarrow 16 \cos ^2 \theta=4 \cot \theta$
$\Rightarrow 4 \cos ^2 \theta=\frac{\cos \theta}{\sin \theta}$
$\Rightarrow \sin 2 \theta=\frac{1}{2}$
$\Rightarrow 2 \theta=\frac{\pi}{6}, \frac{5 \pi}{6}$
$\Rightarrow \theta=\frac{\pi}{12}, \frac{5 \pi}{12}$
Standard 11
Mathematics
