UPSC CDS II 2024 Math Paper

Concept:Determine the average of five distinct integers with given extremes, using the conditions in the statements.Explanation:The five distinct integers include 2 (least) and 8 (largest).The other three must be chosen from {3,4,5,6,7}.List all 10 possible sets and their sums:{2,3,4,5,8}: sum = 22{2,3,4,6,8}: sum = 23{2,3,4,7,8}: sum = 24{2,3,5,6,8}: sum = 24{2,3,5,7,8}: sum = 25{2,3,6,7,8}: sum = 26{2,4,5,6,8}: sum = 25{2,4,5,7,8}: sum = 26{2,4,6,7,8}: sum = 27{2,5,6,7,8}: sum = 28Statement‑I: The sum is a multiple of 5.Only the sets {2,3,5,7,8} and {2,4,5,6,8} satisfy this (sum = 25).Both have the same sum, so the average is $25/5 = 5$.Thus Statement‑I alone gives a unique answer.Statement‑II: The number of odd integers is odd.Check each set for odd count (odd numbers: 3,5,7):• {2,3,4,6,8}: 1 odd → sum=23• {2,3,5,7,8}: 3 odds → sum=25• {2,4,5,6,8}: 1 odd → sum=25• {2,4,6,7,8}: 1 odd → sum=27These give different sums (23,25,27), so averages differ.Thus Statement‑II alone is not sufficient.Therefore the question can be answered by using Statement‑I alone, but not by Statement‑II alone.Answer:Option A.