UPSC CDS I 2024 Math Paper

Concept:Find the total area of the shaded region by adding the area of a smaller semicircle and the area of a circular segment.Explanation:The large circle has centre $O$ and radius $R = 7$ cm.$CA$ and $BD$ are diameters, so $OC = OA = OD = OB = 7$ cm.The smaller shaded circular region (top semicircle) has radius $r = \frac{OC}{2} = \frac{7}{2}$ cm.Area of the smaller shaded region $= \pi r^2 = \pi \left(\frac{7}{2}\right)^2 = \frac{22}{7} \times \frac{7}{2} \times \frac{7}{2} = \frac{77}{2} = 38.5$ cm².The shaded segment of the larger region is the part of semicircle $DAB$ excluding triangle $DBA$.Area of semicircle $DAB = \frac{1}{2}\pi R^2 = \frac{1}{2} \times \frac{22}{7} \times 7^2 = 77$ cm².Area of triangle $DBA = \frac{1}{2} \times DB \times OA = \frac{1}{2} \times (2R) \times R = \frac{1}{2} \times 14 \times 7 = 49$ cm².Shaded segment area $= 77 - 49 = 28$ cm².Total shaded area $= 38.5 + 28 = 66.5$ cm².Answer:$66.5$ square cm (Option D).