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Question

The equation 81sin2x+81cos2x=30 has two roots in the

A
I quadrant
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B
II quadrant
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C
III quadrant
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D
IV quadrant
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Solution

The correct options are
A I quadrant
B II quadrant
C III quadrant
D IV quadrant
Let 81sin2x=y.

Then 81cos2x=811sin2x=81.y1,

so that the given equation is equivalent to
y+81y1=30

y230y+81=0
solve the equation in y
y=3 or y=27

81sin2x=3 or 81sin2x=27
34sin2x=3 or 34sin2x=33
sin2x=14 or sin2x=34
sinx=±12 or sinx=±32
which gives two solutions of the given equation in each quadrant.

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