UPSC CDS 2 2021 Math Paper

We have,
Cone : r1=x and h=2x
Hemisphere :r2=xTotal surface area of solid = curved surface area of cone + curved surface area of hemisphere
=πr1l+2πr22
=π⋅x⋅AB+2πx2
In right angled triangle ABC,
AB2‌‌=OA2+OB2
⇒AB2‌‌=(2x)2+x2
‌‌=4x2+x2=5x2
⇒AB‌‌=√5x
Put AB=√5x into Eq (i), we get
Total surface area of solid
‌‌=π⋅x⋅√5x+2πx2
‌‌=3.14×2.23x2+6.28x2
‌‌=7.0022x2+6.28x2
‌‌=13.2822x2≈13.3x2
Hence, the approximate total surface area of the solid is 13.3x2.