IMO: D

Consider the following fig:

Attachment:

Capture.JPG [ 15.19 KiB | Viewed 1240 times ]
Question Stem: Area of the shaded region = \(\frac{[Area of the square - Area of the cirlce]}{4}\)

Area of square = 2r*2r = \(4r^2\)

Area of Circle = \(πr^2\)

Area of the shaded region = \(\frac{[4r^2 - πr^2]}{4}\)

IS \(\frac{r^2[4 - π]}{4}\) > 4 ?

IS \(r^2[4 - π]\) > 16 ? -- (i)

St 1: The area of the large rectangle equals 64.I think the question is flawed. PSTU is a rectangle and PQRS is also a rectangle(Every square is a rectangle). The use of word large is ambiguous. It can either represent PSTU or PQRS. If largest is used instead then PQRS can be considered which would be sufficient alone as well. Area of PSTU = 64

Where PS = 2r & SU = r

Thus, 2r*r = 64

\(r^2\)= 32 ---(ii)

Sub in Eq - (i)

\(32[4 - π]\) > 16 ?

YESHence SuffSt 2: The perimeter of the shaded region equals 8 + 2πPerimeter of the shaded region =

Length of the arc TV + Side TQ + Side QVLength of the arc = \(\frac{θ}{360} * 2πr\)

= \(\frac{90}{360} * 2πr\)

= \(\frac{πr}{2}\)

Thus,

\(8 + 2π = \frac{πr}{2} + r + r\)

r = 4

Sub in eq--(i)

\(r^2[4 - π]\) > 16 ?

\(16[4 - π]\) > 16 ?

YESHence Suff
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