Question

If $\alpha $ and $\beta $ are the solutions of $cot x=-\sqrt{3}$ in $\left[0,2 \pi \right]$ and $\alpha $ and $\gamma $ are the solutions of $cosec x=-2$ in $\left[0,2 \pi \right],$ then the value of $\frac{\left|\alpha - \beta \right|}{\beta + \gamma }$ is equal to

• A. $\frac{1}{2}$
• B. $2$
• C. $\frac{1}{3}$
• D. $3$

Answer 1

Answer: (A)

Solutions of $cot x=-\sqrt{3}$ on the interval $\left[0 , 2 \pi \right]$ are
$x=\frac{5 \pi }{6},\frac{11 \pi }{6}$
The solutions for $cosec x=-2$ on the same interval $\left[0 , 2 \pi \right]$ are
$x=\frac{7 \pi }{6},\frac{11 \pi }{6}$
So, $\alpha =\frac{11 \pi }{6},\beta =\frac{5 \pi }{6},\gamma =\frac{7 \pi }{6}$
$\Rightarrow \alpha -\beta =\pi $ and $\beta +\gamma =2\pi $
$\Rightarrow \frac{\left|\alpha - \beta \right|}{\beta + \gamma }=\frac{\pi }{2 \pi }=\frac{1}{2}$