UPSC CDS 2 2020 Math Paper

Concept:Both statements are properties of a trapezium derived from similarity of triangles formed by diagonals and parallel lines.Explanation:Statement 1: In trapezium $ABCD$ with $AB \parallel CD$, diagonals $AC$ and $BD$ intersect at $O$.Since $AB \parallel CD$, alternate interior angles are equal: $\angle OAB = \angle OCD$ and $\angle OBA = \angle ODC$.Thus $\triangle AOB \sim \triangle COD$ (AA similarity).Hence $\frac{AO}{OC} = \frac{BO}{OD}$ — the diagonals divide each other in the same ratio (proportionally).So statement 1 is correct.Statement 2: In trapezium $WXYZ$ with $WX \parallel ZY$, draw a line $PQ$ parallel to the bases, intersecting $WZ$ at $P$ and $XY$ at $Q$.In $\triangle WZX$, $PQ \parallel WX$, so by Basic Proportionality Theorem, $\frac{ZP}{PW} = \frac{ZQ}{QX}$.In $\triangle XYZ$, $PQ \parallel ZY$, so $\frac{XQ}{QY} = \frac{XR}{RZ}$ (where $R$ is intersection of diagonal $ZX$ with $PQ$).Combining the ratios yields $\frac{ZP}{PW} = \frac{YQ}{QX}$, proving that the line divides the non-parallel sides proportionally.Thus statement 2 is also correct.Answer:Both statements are correct. Hence option C is correct.