UPSC CDS 2 2020 Math Paper

Concept:The four coins are placed at the vertices of a square such that each coin touches two adjacent coins.The distance between centers of adjacent coins equals twice the radius, so the side of the square is $2r$.Inside the square, each coin contributes a quarter‑circle area. The uncovered area is the square area minus the total area of the four quarter‑circles.Given uncovered area $= 42$ cm², we solve for the radius $r$.Explanation:Let the radius of each coin be $r$ cm and the side of the square be $a$ cm.Since each coin touches two others, the distance between adjacent centers is $2r$.Hence, the side of the square equals this distance: $a = 2r$.Area of the square $= a^2 = (2r)^2 = 4r^2$.Each coin lies at a vertex, so the part inside the square is a quarter‑circle of area $\frac{1}{4}\pi r^2$.Total area covered by the coins inside the square $= 4 \times \frac{1}{4}\pi r^2 = \pi r^2$.Uncovered area $= 4r^2 - \pi r^2 = r^2(4 - \pi)$.Substitute $\pi = \frac{22}{7}$: $4 - \frac{22}{7} = \frac{28-22}{7} = \frac{6}{7}$.Thus, $r^2 \times \frac{6}{7} = 42$.Multiply both sides by $\frac{7}{6}$: $r^2 = 42 \times \frac{7}{6} = 49$.Therefore, $r = 7$ cm (radius is positive).Answer:The radius of each coin is $7$ cm. (Option B)