Question

If $\alpha $ and $\beta $ are the solutions of $sinx=-\frac{1}{2}$ in $\left[0,2 \pi \right]$ and $\alpha $ and $\gamma $ are the solutions of $cos x=-\frac{\sqrt{3}}{2}$ in $\left[0,2 \pi \right]$ , then the value of $\frac{\alpha + \beta }{\left|\beta - \gamma \right|}$ is equal to

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

Answer: (C)

$sin x=-\frac{1}{2}\Rightarrow x=\frac{7 \pi }{6},\frac{11 \pi }{6}$
$cos x=-\frac{\sqrt{3}}{2}\Rightarrow x=\frac{5 \pi }{6},\frac{7 \pi }{6}$
$\Rightarrow \alpha =\frac{7 \pi }{6},\beta =\frac{11 \pi }{6},\gamma =\frac{5 \pi }{6}$
$\Rightarrow \alpha +\beta =\frac{7 \pi }{6}+\frac{11 \pi }{6}=3\pi ,\beta -\gamma =\frac{11 \pi }{6}-\frac{5 \pi }{6}=\pi $
$\Rightarrow \frac{\alpha + \beta }{\left|\beta - \gamma \right|}=\frac{3 \pi }{\pi }=3$