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Question

If the equation (sin1x)3+(cos1x)3=aπ3, has a solution, then 'a' lies in the interval

A
[13,78]
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B
[132,2732]
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C
[132,78]
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D
[111,98]
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Solution

The correct option is C [132,78]
Let f(x)=(sin1x)3+(cos1x)3
Put sin1x=t
Then f(t)=t3+(π2t)3
f(t)=3[t2(π2t)2]
f(t)=3π2(2tπ2)

Critical point: f(t)=0t=π4
Also, π2tπ2

f(t=π4)=(π4)3+(π4)3=π332 minimum
f(t=π2)=7π38 maximum
f(t=π2)=π38
Range of f is [132π3,78π3]




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