UPSC CDS I 2019 Math Paper

Concept:An isosceles trapezium is always cyclic. A cyclic parallelogram must be a rectangle because opposite angles are equal and sum to $180^{\circ}$.Explanation:Statement 1: In an isosceles trapezium, the legs are equal. Draw perpendiculars from the vertices of the shorter base to the longer base. The two right triangles formed are congruent by hypotenuse-leg test, so base angles are equal. Let $\text{angle }B = \text{angle }C$. Using interior angles on the same side of a transversal, $\text{angle }A + \text{angle }B = 180^{\circ}$. Thus $\text{angle }A + \text{angle }C = 180^{\circ}$. Hence, by the converse of the cyclic quadrilateral theorem, the trapezium is cyclic. So statement 1 is correct.Statement 2: Let $\text{ABCD}$ be a cyclic parallelogram. In a parallelogram, opposite angles are equal: $\text{angle }A = \text{angle }C$ and $\text{angle }B = \text{angle }D$. In a cyclic quadrilateral, opposite angles sum to $180^{\circ}$: $\text{angle }A + \text{angle }C = 180^{\circ}$. Substituting $\text{angle }A = \text{angle }C$ gives $2\text{angle }A = 180^{\circ}$, so $\text{angle }A = 90^{\circ}$. Thus all angles are $90^{\circ}$, making it a rectangle. So statement 2 is also correct.Answer:Both statements 1 and 2 are correct, i.e., option C.