Question

Let $\theta$ be an acute angle such that the equation $x^3+4x^2\cos \theta +x\cot \theta =0$ has multiple roots. Then the value of $\theta$ (in radians) is

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{8}$
• C. $\frac{\pi }{12}$ or $\frac{5\pi }{12}$
• D. $\frac{\pi }{6}$ or $\frac{5\pi }{12}$

Answer 1

Answer: (C)

Given, $x^3+4x^2\cos \theta +x\cot \theta =0$
$\Rightarrow x\left(x^2+4x\cos \theta +\cot \theta \right)=0$
$\Rightarrow x=0$ is root of equation.
$\Rightarrow x^2+4x\cos \theta +\cot \theta =0$ has multiple roots
So, $\Delta =0$
$\Rightarrow (4\cos \theta {)}^2-4(1)(\cot \theta )=0$
$\Rightarrow 16{\cos }^2\theta =4\cot \theta$
$\Rightarrow 4{\cos }^2\theta =\cot \theta$
$\Rightarrow 4{\cos }^2\theta =\frac{\cos \theta }{\sin \theta }$
$\Rightarrow \cos \theta =0$ as $4\cos \theta =\frac{1}{\sin \theta }$
$\Rightarrow 2\sin \theta \cos \theta =\frac{1}{2}$
$\Rightarrow \sin 2\theta =\frac{1}{2}$
$\Rightarrow 2\theta =\frac{\pi }{6}$ or $\frac{5\pi }{6}$
$\Rightarrow \theta =\frac{\pi }{12}$ or $\frac{5\pi }{12}$