UPSC CDS 1 2023 Math Paper

Concept:The sum of two numbers can be found directly using the algebraic identity $(a+b)^2 = a^2 + b^2 + 2ab$ without needing the individual numbers or the given HCF.Explanation:Given: product of the two numbers $= 2160$, sum of their squares $= 4896$, and their HCF is $12$.Use the identity: $(\text{sum})^2 = \text{sum of squares} + 2 \times \text{product}$.Substitute the given values: $(\text{sum})^2 = 4896 + 2 \times 2160 = 4896 + 4320 = 9216$.Therefore, the sum of the two numbers is $\sqrt{9216} = 96$.The arithmetic mean of two numbers is half of their sum: mean $= 96 \div 2 = 48$.Notice that the HCF $= 12$ is not required for this calculation; it is redundant and does not affect the result.This method is faster and avoids finding the individual numbers.Answer:$48$ (Option C)