Question

Let $S$ be the sum of all $x$ in the interval $[0,2\pi ]$ such that $3{\cot }^{2}x+8\cot x+3=0$, then the value of $\frac{S}{\pi }$ is

• A. $3$
• B. $4$
• C. $5$
• D. $6$

Answer 1

The correct option is C $5$

$3{\cot }^{2}x+8\cot x+3=0$
$\Rightarrow \cot x=\frac{-4\pm \sqrt{7}}{3}$
Both roots are negative.
$\therefore \frac{\pi }{2}<x_1,x_2<\pi$ or $\frac{3\pi }{2}<x_1,x_2<2\pi$
$\therefore \pi <x_1+x_2<2\pi$ or $3\pi <x_1+x_2<4\pi$

Product of roots $=\cot x_1\cot x_2=1$
$\Rightarrow \cot x_1\cot (\frac{3\pi }{2}-x_1)=1$
$\Rightarrow x_1+x_2=\frac{3\pi }{2}$
Similarly, $x_1+x_2=\frac{7\pi }{2}$

S = $\frac{3\pi }{2}+\frac{7\pi }{2}=5\pi$