UPSC CDS 1 2023 Math Paper

Concept:The area of a circle can be found using its radius. The radius can be determined from either an arc length or a chord length when the subtended angle is known.Explanation:From Statement I: The arc length formula is $\text{Arc length} = 2\pi r \times \frac{\theta}{360^\circ}$.Given arc length $= 7$ cm and $\theta = 30^\circ$, we have $7 = 2\pi r \times \frac{30}{360} = 2\pi r \times \frac{1}{12}$.Solving gives $r = \frac{42}{\pi}$ cm.Then area $= \pi r^2 = \pi \left(\frac{42}{\pi}\right)^2 = \frac{1764}{\pi}$ sq cm.Thus Statement I alone is sufficient.From Statement II: A chord of length 10 cm subtends $90^\circ$ at the centre. Let O be centre, A and B ends of chord. Then $OA = OB = r$, and chord $AB = 10$ cm.By Pythagoras theorem in right triangle OAB (since $\text{angle}\ AOB = 90^\circ$), $r^2 + r^2 = 10^2$ ⟹ $2r^2 = 100$ ⟹ $r = 5\sqrt{2}$ cm.Area $= \pi r^2 = \pi (5\sqrt{2})^2 = 50\pi$ sq cm.Thus Statement II alone is also sufficient.Since either statement alone can answer the question, the correct choice is option B.Answer:Option B.