If $S = {x \in [0, 4 π] : \sin 2 x - \cos x = 0}$ and the sum of the elements of S is $k π$ , then k is

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Detailed Solution

$\sin 2 x - \cos x = 0 \Rightarrow 2 \sin x \cos x - \cos x = 0 \Rightarrow \cos x (2 \sin x - 1) = 0$
$\begin{array}{l} \Rightarrow \cos x = 0 or \sin x = \frac{1}{2} \Rightarrow x = 2 nπ \pm \frac{π}{2}, x = nπ + (- 1)^{n} \frac{π}{6} \\ x \in [0, 4 π] \Rightarrow x = \frac{π}{2}, \frac{3 π}{2}, \frac{5 π}{2}, \frac{7 π}{2} and x = \frac{π}{6}, \frac{5 π}{6}, \frac{13 π}{6}, \frac{17 π}{6} \\ ∴ S = \{\frac{π}{2}, \frac{3 π}{2}, \frac{5 π}{2}, \frac{7 π}{2}, \frac{π}{6}, \frac{5 π}{6}, \frac{13 π}{6}, \frac{17 π}{6}\} \end{array}$
Sum of elements of s is $k π \Rightarrow k π = \frac{π + 3 π + 5 π + 7 π}{2} + \frac{π + 5 π + 13 π + 17 π}{6}$
$\Rightarrow k π = 8 π + 6 π = 14 π \Rightarrow k = 14 .$