UPSC CDS 2 2022 Math Paper

Concept:Special properties of triangles, cyclic quadrilaterals, and parallelograms lead to the correctness of all three statements.Explanation:Statement 1: In an equilateral triangle, the medians, altitudes, and perpendicular bisectors all intersect at the same point.Hence the centroid (median intersection) and circumcenter (perpendicular bisector intersection) coincide.Therefore statement 1 is correct.Statement 2: Let $ABCD$ be a cyclic quadrilateral.The internal angle bisectors of $\angle A$, $\angle B$, $\angle C$, $\angle D$ meet to form quadrilateral $PQRS$.In $\triangle APD$ and $\triangle BQC$, using the angle sum property and summing the equations, we obtain:$\angle APD + \angle BQC = 180^{\circ}$.Thus opposite angles of $PQRS$ sum to $180^{\circ}$, so $PQRS$ is cyclic.Hence statement 2 is correct.Statement 3: In a cyclic parallelogram, opposite angles are equal (parallelogram property) and sum to $180^{\circ}$ (cyclic quadrilateral property).So $2\angle A = 180^{\circ} \Rightarrow \angle A = 90^{\circ}$.Thus all angles are $90^{\circ}$, making it a rectangle.Therefore statement 3 is correct.Answer:Statements 1, 2, and 3 are all correct. The correct option is D (1, 2 and 3).