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Question

For x(0,π), the equation sinx+2sin2xsin3x=3 has

A
no solution
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B
one solution
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C
two solutions
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D
three solutions
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Solution

The correct option is A no solution
sinx+2sin2xsin3x=3
sinx+4sinxcosx3sinx+4sin3x=3sinx(1+4cosx3+4sin2x)=3sinx(4sin2x+4cosx2)=3sinx(4cos2x4cosx2)=3sinx[(2cosx1)23]=3sinx[3(2cosx1)2]=3

Maximum value of 3(2cosx1)2 is 3 when cosx=12 but at this value, sinx1
Hence, sinx[3(2cosx1)2]=3 is not possible.



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