A) \[\frac{\pi }{4},\frac{\pi }{10}\]
B) \[\frac{\pi }{6},\frac{\pi }{3}\]
C) \[\frac{\pi }{4},\frac{\pi }{2}\]
D) \[\frac{\pi }{8},\frac{\pi }{16}\]
Correct Answer: B
Solution :
\[\sin \,\,x+\,\sin \,5x=\,\sin \,3x\] \[\Rightarrow \] \[2\,\,\sin \,3x\,cos\,2x\,=\sin \,3x\] \[\Rightarrow \] \[\sin 3x(2\cos 2x-1)=0\] \[\Rightarrow \] \[\sin \,\,3x=0\] or \[2\,\cos \,2x-1=0\] if \[\sin \,\,3x=0\] \[\Rightarrow \] \[3x=0,\] or \[\pi \] \[\Rightarrow \] \[x=0\] or \[x=\frac{\pi }{3}\] And if \[2\,\,\cos \,2x-1=0\] \[\Rightarrow \] \[\cos \,\,2x=\frac{1}{2}\,=\,\cos \,\frac{\pi }{3}\] \[\Rightarrow \] \[2x=\frac{\pi }{3}\Rightarrow x=\frac{\pi }{6}\] \[\therefore \] Solutions in \[\left( 0,\frac{\pi }{2} \right)\] are \[\frac{\pi }{3},\frac{\pi }{6}.\]
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