Q.

Consider all rectangles lying in the region (x,y)∈ℝ×ℝ:0≤x≤π2 and 0≤y≤2sin⁡(2x)and having one side on the x-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is

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a

3π2

b

π

c

π23

d

π32

answer is C.

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Detailed Solution

Perimeter =2(2α+2cos⁡2α)     since if x=π4+α⇒y=2 sin2π4+α=2 cos 2αP=4(α+cos⁡2α)dPdα=4(1−2sin⁡2α)=0sin⁡2α=122α=π6,5π6d2Pdα2=−16cos⁡2α for maximum α=π12  if 2α=π6⇒d2Pdα2=−16cos⁡π6=-83<0 Area =(2α)(2cos⁡2α)=π6×2×32=π23
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