UPSC CDS II 2024 Math Paper

Concept:The question requires determining three distinct integer weights whose sum is a prime number, using the given relationships between them.Explanation:Using Statement-I alone: Let the weights be $x$, $2x$, and $y$, where $x$ and $y$ are integers and $x \neq y$. The sum is $3x + y$, a prime. There are infinitely many possibilities (e.g., $x=1$, $y=2$ gives sum 5, prime, but weights 1,2,2 are not distinct). Hence Statement-I alone is insufficient.Using Statement-II alone: Let the weights be $x$, $3x$, and $y$. Sum is $4x + y$, a prime. Again, infinite possibilities, so Statement-II alone is insufficient.Using both statements together: There are two possible cases.Case 1: The three weights are in the ratio $1:2:3$. Let them be $x$, $2x$, $3x$. Sum = $6x$. For sum to be prime, $6x$ must be prime. But $6x$ is always divisible by 2 and 3 (for $x \ge 1$), so it is composite (e.g., $x=1$ gives 6, not prime). No integer $x$ satisfies.Case 2: The three weights are $x$, $2x$, $6x$ (since one is twice another and another is thrice a different weight). Sum = $9x$. For sum to be prime, $9x$ must be prime. But $9x$ is divisible by 3 (for $x \ge 1$), so it is composite (e.g., $x=1$ gives 9, not prime). No integer $x$ satisfies.Both cases lead to no valid set of distinct integer weights with prime sum. Thus the information is contradictory and the question cannot be answered.Answer:D