Three semicircles of radius 1 are constructed on diameter AB of a semi

My query is how to solve this question in under 2minutes, and that too in exam ?

rsama

Important lesson here: Always look at the answer choices.

The area of the shaded region will be: Area of large semi-circle - Area of the region created by the 3 smaller semi-circles

We know that the big semi-circle has a radius of 2, which means it's area is:
\(\frac{πr^2}{2}\)

\(= \frac{π(2)^2}{2}\)

\(= 2π\)

In answer choices A & B the larger semi-circle has an area of π, which means it has a radius of \(\sqrt{2}\). However, we know that the semi-circle has a radius of 2, and that the area is 2π which means that these answer choices have to be wrong.Eliminate both answer choices.

C & D cannot be the area of the shaded area for one simple reason: we subtract the area of the smaller semi-circles from the area of the big semi-circle. We do not add. Eliminate both choices.

Now we may not know how to get to E, or don't want to waste time with solving to get to E, but we know with 100% certainty that the answer cannot be A, B, C or D. So E has to be the answer.

Answer E.