From my diagram,(Fig 1)
As PQRS is a square,
PQ=QR=RS=SP=a
Also,PQ=BC=a ; where BC is the diameter of the circle
ED , which is the diagonal of the inner square is also the diameter of the circle.
So,BC=ED=a
Thus,Radius of the circle,r=a/2
Now,from Fig 2 ,
In square HDGE,
∆DGE,is a right-angled triangle.
Applying Pythagoras theorem for right-angled triangle,
DG^2+EG^2=ED^2
We can get the side of the inner square.
DG^2+DG^2=ED^2
or,2DG^2=a^2
or,DG^2=(a^2)/2
or,DG=a/√2
Now,Area of the shaded region,
=Area of the circle-Area of the inner square
=π*r*r-(side)*(side)
=π(a/2)^2 - (a/√2)*(a/√2)
=π(a^2)/4 - (a^2)/2
=(a^2)*((π/4)-(1/2))
=(a^2)*((π-2)/4)
[ANS: (A)]