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Question

Let x,y[0,2π] and satisfying the equation sin3x+cos3y+6sinxcosy=8. If a=x+y and b=xy, then

A
minimum value of a is π2
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B
maximum value of a is 5π2
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C
minimum value of b is 3π2
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D
maximum value of b is π
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Solution

The correct options are
A minimum value of a is π2
B maximum value of a is 5π2
C minimum value of b is 3π2
We know that,
a3+b3+c33abc=(a+b+c)(a2+b2+c2abbcca)
If a+b+c=0, then
a3+b3+c3=3abc

Now, sin3x+cos3y+(2)3=3sinxcosy(2)
sinx+cosy2=0
sinx+cosy=2
sinx=1,cosy=1
x=π2, y=0,2π
a=x+y=π2,5π2
b=xy=π2,3π2

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