A) \[\left( 0,\frac{\pi }{6} \right]\]
B) \[\left( \frac{\pi }{9},\frac{\pi }{6} \right)\]
C) \[\left( \frac{\pi }{12},\frac{\pi }{9} \right]\]
D) \[\left( \frac{\pi }{3},\frac{\pi }{2} \right]\]
Correct Answer: A
Solution :
[a] \[\cot \left( \frac{\pi }{3\sqrt{3}}\sin 2x \right)=\sqrt{3}\] \[\Rightarrow \,\frac{\pi }{3\sqrt{3}}\sin 2x=n\pi +\frac{\pi }{6},n\in Z\] \[\Rightarrow \,\sin 2x=3\sqrt{3}n+\frac{\sqrt{3}}{2}\] For least positive solution, n=0 \[\Rightarrow \sin 2x=\frac{\sqrt{3}}{2}\] \[\Rightarrow 2x=\frac{\pi }{3}\Rightarrow x=\frac{\pi }{6}\]
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