UPSC CDS 2 2021 Math Paper

Concept:A solid formed by combining a cone and a hemisphere has volume equal to that of a sphere of the same radius.Explanation:Radius of both cone and hemisphere is $x$.Height of cone is given as $2x$.Volume of cone = $\frac{1}{3}\pi x^2 (2x) = \frac{2}{3}\pi x^3$.Volume of hemisphere = $\frac{2}{3}\pi x^3$ (since hemisphere volume is half of sphere volume).Total volume of solid = $\frac{2}{3}\pi x^3 + \frac{2}{3}\pi x^3 = \frac{4}{3}\pi x^3$.Volume of a sphere of radius $x$ is exactly $\frac{4}{3}\pi x^3$.Thus the solid's volume equals that of a sphere of radius $x$.Answer:Option A. sphere of radius $x$