UPSC CDS 1 2022 Math Paper

Concept:Triangle inequality and properties of medians: The sum of two sides of a triangle is always greater than twice the median to the third side.Explanation:Consider statement 1: "Sum of any two sides is less than twice the median to the third side."In any triangle, by vector or geometric proof, $AB + AC > 2AD$ (where $AD$ is median from $A$).Thus the sum is actually greater than twice the median, not less. Hence statement 1 is false.Now statement 2: "Perimeter of a triangle is greater than the sum of its three medians."Using the above property for each median:$AB + AC > 2AD$ (median from $A$ to $BC$)$BC + BA > 2BE$ (median from $B$ to $CA$)$CA + CB > 2CF$ (median from $C$ to $AB$)Adding all three inequalities:$2(AB + BC + CA) > 2(AD + BE + CF)$Dividing by 2: $AB + BC + CA > AD + BE + CF$Thus perimeter > sum of medians. Hence statement 2 is correct.Answer:Only statement 2 is correct. (Option B)