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

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

Answer: (C)

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