UPSC CDS 2 2021 Math Paper

Concept:The total surface area of the solid is the sum of the cube's surface area and the hemisphere's curved surface area, minus the area of the hemisphere's base (which is attached to the cube).Explanation:Step 1: Surface area of cube = $6a^{2} = 6 \times 14^{2} = 6 \times 196 = 1176\ \mathrm{cm}^{2}$.Step 2: Curved surface area of hemisphere = $2\pi r^{2} = 2 \times \frac{22}{7} \times 7^{2} = 2 \times \frac{22}{7} \times 49 = 2 \times 22 \times 7 = 308\ \mathrm{cm}^{2}$.Step 3: Base area of hemisphere = $\pi r^{2} = \frac{22}{7} \times 7^{2} = \frac{22}{7} \times 49 = 22 \times 7 = 154\ \mathrm{cm}^{2}$.Step 4: The hemisphere rests on the cube's top face, so its base is not exposed. The cube's top face originally had area $196\ \mathrm{cm}^{2}$, but the portion under the hemisphere ($154\ \mathrm{cm}^{2}$) is hidden, leaving $42\ \mathrm{cm}^{2}$ exposed.Step 5: Total surface area = Cube surface area ($1176$) + Curved surface area of hemisphere ($308$) - Base area of hemisphere ($154$) = $1176 + 308 - 154 = 1330\ \mathrm{cm}^{2}$.Answer:$1330\ \mathrm{cm}^{2}$ (option A).