CTET Paper II (Mathematics and Science) 20 Aug 2023 Solved Paper

Concept:When a point on a side of a triangle is equidistant from all three vertices, it is the circumcenter. The side containing it becomes a diameter of the circumcircle, creating a right angle at the opposite vertex.Explanation:Point S lies on PQ and $RS = PS = QS$. So S is the midpoint of PQ and the circumcenter of $\triangle PQR$. Hence PQ is a diameter, making $\angle PRQ = 90°$ (angle in a semicircle).In $\triangle PQR$, $\angle QPR = 15°$. Using angle sum property: $\angle PQR = 180° - 90° - 15° = 75°$.Now compute $(2\angle PQR - \angle PRQ) = 2 \times 75° - 90° = 150° - 90° = 60°$.You can also use isosceles triangles: In $\triangle PRS$, $PS = RS$ gives $\angle SRP = \angle SPR = 15°$. In $\triangle QRS$, $QS = RS$ gives $\angle SQR = \angle SRQ = x$. Then in $\triangle PQR$: $15° + x + (15°+x) = 180°$ so $x = 75°$. Thus $\angle PQR = 75°$ and $\angle PRQ = 90°$, giving the same result.Answer:The required value is $60°$, so the correct option is B. 60°.