UPSC CDS 2 2022 Math Paper

Concept:In a cyclic quadrilateral, the exterior angle equals the interior opposite angle. For two triangles to be similar, corresponding angles must be equal.Explanation:Let $ABCD$ be cyclic with $AB$ and $DC$ extended to meet at $E$.Consider statement 1: $\triangle EBC \sim \triangle EAD$.Both triangles share $\angle E$ (common).From the cyclic property: $\angle EBC$ (exterior at $B$) equals $\angle CDA$.Also, $\angle EAD$ (exterior at $A$) equals $\angle BCD$.For similarity, we need $\angle EBC = \angle EAD$, i.e., $\angle CDA = \angle BCD$.But in a cyclic quadrilateral, adjacent interior angles $\angle CDA$ and $\angle BCD$ are not necessarily equal.Thus, the triangles are not similar generally. Statement 1 is false.Consider statement 2: $\angle CBE + \angle DAE = 180^{\circ}$.$\angle CBE$ (exterior at $B$) equals $\angle CDA$.$\angle DAE$ (exterior at $A$) equals $\angle BCD$.So the sum becomes $\angle CDA + \angle BCD$.These are two adjacent interior angles of the cyclic quadrilateral, not opposite angles.Only opposite angles sum to $180^{\circ}$; adjacent angles do not have a fixed sum.If $ABCD$ were a square, each would be $90^{\circ}$, sum = $180^{\circ}$, but then $AB \parallel DC$ and they never meet at $E$. Hence, under the given condition, the sum is not $180^{\circ}$.Thus, statement 2 is also false.Answer:Both statements are incorrect. The correct code is D (Neither 1 nor 2).