UPSC CDS 2 2023 Math Paper

Concept:The question reduces to comparing $a+d$ and $b+c$, where $a\le b\le c\le d$ are the four numbers.Explanation:Let the numbers be $a\le b\le c\le d$.Average of largest and smallest: $\frac{a+d}{2}$; average of all four: $\frac{a+b+c+d}{4}$.We need to check if $\frac{a+d}{2} > \frac{a+b+c+d}{4}$, i.e., $2(a+d) > a+b+c+d$, which simplifies to $a+d > b+c$.Statement I: $d-c < b-a$.Add $a+c$ to both sides: $(a+c)+(d-c) < (a+c)+(b-a)$ $\Rightarrow$ $a+d < b+c$.Thus the inequality $a+d > b+c$ is false. Hence the answer to the question is "No".Therefore Statement I alone is sufficient to answer the question.Statement II: $d-a > c-b$.This inequality does not imply $a+d > b+c$ or $a+d < b+c$.For example, numbers $(1,2,3,4)$ satisfy it but give $a+d = b+c$; $(1,2,3,5)$ also satisfy it and give $a+d > b+c$; $(1,3,4,5)$ satisfy it and give $a+d < b+c$.Thus Statement II alone is not sufficient.Since Statement I alone is sufficient but Statement II alone is not, the question can be answered using one statement alone.Answer:Option A: if the question can be answered by using one of the statements alone, but cannot be answered using the other statement alone.