Let $\frac{π}{2} < x < π$ be such that $\cot x = \frac{- 5}{\sqrt{11}}$ . Then $(\sin \frac{11 x}{2}) (\sin 6 x - \cos 6 x) + (\cos \frac{11 x}{2}) (\sin 6 x + \cos 6 x)$ is equal to

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$\frac{\sqrt{11} + 1}{2 \sqrt{3}}$

$\frac{\sqrt{11} - 1}{3 \sqrt{2}}$

$\frac{\sqrt{11} + 1}{3 \sqrt{2}}$

$\frac{\sqrt{11} - 1}{2 \sqrt{3}}$

answer is B.

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Detailed Solution

Given that $\cot x = \frac{- 5}{\sqrt{11}}$ $∴ \sin \frac{11 x}{2} (\sin 6 x - \cos 6 x) + \cos \frac{11 x}{2} (\sin 6 x + \cos 6 x)$
$= \cos (6 x - \frac{11 x}{2}) + \sin (6 x - \frac{11 x}{2}) = \cos \frac{x}{2} + \sin \frac{x}{2} = \sqrt{1 + \sin x} = \sqrt{1 + \frac{\sqrt{11}}{6}} = \frac{\sqrt{11} + 1}{2 \sqrt{3}}$