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Question

Complete the set of values of x in (0,π) satisfying the inequation 1+log2sinx+log2sin3x0 is

A
[0,π6]
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B
[π2,3π4]
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C
[π6,π4][3π4,5π6]
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D
[5π6,π]
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Solution

The correct option is C [π6,π4][3π4,5π6]
x(0,π)3x(0,3π)
For the inequality to be defined,
sinx>0 and sin3x>0
x(0,π3)(2π3,π) (1)

1+log2sinx+log2sin3x0
log2(2sinxsin3x)0
2sinxsin3x1
2sinx(3sinx4sin3x)1
8sin4x6sin2x+10

Let sin2x=t
Then 8t26t+10
(t12)(t14)0
14t12
14sin2x12
12sinx12
x[π6,π4][3π4,5π6] (2)

From (1) and (2),
x[π6,π4][3π4,5π6]

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