UPSC CDS 2 2023 Math Paper

Concept:The area between two concentric circles equals $\pi$ times the difference of the squares of their radii, which can be found using the Pythagorean theorem from the given tangent chord.Explanation:Let $R$ be the radius of the outer circle and $r$ the radius of the inner circle.The chord of the outer circle of length $14\text{ cm}$ is tangent to the inner circle.Draw perpendicular $OM$ from centre $O$ to chord $AB$, where $M$ is the midpoint of $AB$.Since $AB = 14\text{ cm}$, we have $AM = \frac{14}{2} = 7\text{ cm}$.In right triangle $OAM$, $OA = R$ (radius of outer circle), $OM = r$ (radius of inner circle, because tangent distance from centre is the radius).By Pythagoras theorem: $R^{2} = r^{2} + 7^{2}$.Thus $R^{2} - r^{2} = 49$.Area between the circles = $\pi(R^{2} - r^{2}) = \frac{22}{7} \times 49 = 154\text{ square cm}$.Answer:154 square cm (Option D).