A) \[9,\,\,x\in \,\,\left[ 0,\,\frac{\pi }{2} \right]\]
B) \[10,\,x\,\,\in \,\,[0,\,\,\pi ]\]
C) \[8,\,\,x\,\in \,\,\left[ -\frac{\pi }{2},\,\frac{\pi }{2} \right]\]
D) \[12,\,\,x\,\,\in \,\,[-\pi ,\,\,\pi ]\]
Correct Answer: D
Solution :
\[3\,\sqrt{{{[32\,{{\cos }^{6}}x-\,48\,{{\cos }^{4}}\,x\,+18\,{{\cos }^{2}}\,x-1]}^{2}}}\] \[=\sin \,x\] \[=3\,\sqrt{{{[2\,(16\,{{\cos }^{6}}x-24\,{{\cos }^{4}}x+9\,{{\cos }^{2}}\,x)-1]}^{2}}}\] \[=3\,\sqrt{2{{(4\,{{\cos }^{3}}x-3\,\cos \,x)}^{2}}-1}\] \[=3\,\sqrt{2\,{{\cos }^{2}}3x-1}\,\] \[3\,\sqrt{{{\cos }^{2}}\,6x}=\,\sin \,x\] \[3\,|\cos \,6x|\,=\,\sin \,x\] \[6,\,x\in \,\left[ 0,\,\frac{\pi }{2} \right]\] \[12,\,\,x\,\in \,[0,\pi ]\] Clearly, number of solutions, \[=\,6,\,x\in \,\left[ -\frac{\pi }{2},\,\frac{\pi }{2} \right]\] \[=12,\,x\,\in \,[-\pi ,\,\pi ]\]You need to login to perform this action.
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