Question

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

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

Answer 1

Answer: (A)

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