If a is a positive integer, then the number of values of a satisfying ∫0π2{a2cos3x4+34cosx+asinx-20cosx}dx≤-a23

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one

two

three

four

answer is D.

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Detailed Solution

The L.H.S. of the above inequality is equal to a2sin⁡3x12+34sin⁡x−acos⁡x−20sin⁡x0π/2=a2−112+34−a(0−1)−20=2a23+a−20.Thus the given inequality is 2a2/3+a−20≤−a2/3i.e., a2+a−20≤0⇔−5≤a≤4Since a is a positive integer so a=1,2,3,4.

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