UPSC CDS 2 2023 Math Paper

Concept:According to Archimedes' principle, the volume of water displaced equals the volume of the submerged object.The rise in water level is the height of the cylinder formed by this displaced water.Explanation:The sphere has radius $5$ cm, so its volume is $\frac{4}{3}\pi (5)^3 = \frac{4}{3}\pi \times 125 = \frac{500\pi}{3}$ cm$^3$.The cylindrical vessel has a base radius of $10$ cm, so its base area is $\pi (10)^2 = 100\pi$ cm$^2$.When the sphere is completely submerged, the water rises by a height $h$.Volume of water displaced equals base area times $h$, i.e., $100\pi h$.Setting this equal to the sphere's volume: $100\pi h = \frac{500\pi}{3}$.Cancel $\pi$ and solve for $h$: $100h = \frac{500}{3}$ → $h = \frac{500}{3 \times 100} = \frac{5}{3}$ cm.Answer:The water level rises by $\frac{5}{3}$ cm, which corresponds to option A.