UPSC CDS 2 2021 Math Paper

Concept:The angle subtended by a chord at the center is twice the angle subtended at any point on the remaining part of the circle.Explanation:Assume the given diagram shows $AB$ as the diameter and $P$ is a point such that the central angle $\angle BOP = 90^{\circ}$ and $\angle AOP = 90^{\circ}$.Chord $BP$ subtends $\angle BOP$ at the centre $O$ and $\angle BAP$ at point $A$ on the circle.Hence $\angle BAP = \frac{1}{2} \angle BOP = \frac{1}{2} \times 90^{\circ} = 45^{\circ}$.Similarly, chord $AP$ subtends $\angle AOP$ at the centre and $\angle AQP$ at point $Q$ on the circle.Therefore $\angle AQP = \frac{1}{2} \angle AOP = \frac{1}{2} \times 90^{\circ} = 45^{\circ}$.Answer:$45^{\circ}$ (Option C).