UPSC CDS 2 2021 Math Paper

Concept:The arithmetic mean is $\frac{P+Q}{2}$, but the values of $P$ and $Q$ depend on the chosen set of five consecutive integers whose product equals one of them.Explanation:Let the five consecutive integers be $n, n+1, n+2, n+3, n+4$.Their product $P_{rod} = n(n+1)(n+2)(n+3)(n+4)$.This product equals one of these integers.If any integer is $0$, the product is $0$, so the integer equal to the product must be $0$.Thus the set must contain $0$.Possible sets (including $0$):$(-4,-3,-2,-1,0)$, $(-3,-2,-1,0,1)$, $(-2,-1,0,1,2)$, $(-1,0,1,2,3)$, $(0,1,2,3,4)$.For each set, $P$ (smallest) and $Q$ (largest) give different means:$(-4+4)/2 = 0$, $(-3+1)/2 = -1$, $(-2+2)/2=0$, $(-1+3)/2 = 1$, $(0+4)/2 = 2$.Only sets with $P+Q = 0$ yield mean $0$, but other valid sets give $-1$, $1$, or $2$.Therefore the mean is not uniquely determined.Answer:D. Cannot be determined due to insufficient data