UPSC CDS 1 2023 Math Paper

Concept:The problem tests whether a unique value for $(x+y)$ can be found using either statement alone or both together.Explanation:Statement I:Given $(x+y)^4 = 256$.Since $256 = 4^4$, we have $(x+y)^4 = 4^4$.Taking the fourth root gives $x+y = \pm 4$.Thus, Statement I alone yields two possible values, so it is not sufficient to determine $(x+y)$ uniquely.Statement II:Given $(x+y)^3 < 16$.This inequality holds for infinitely many real numbers (e.g., $x+y = 0, 1, 2$, etc.), so it does not give a unique value.Combining both statements:From Statement I, $(x+y)$ is either $+4$ or $-4$.Check each in Statement II:• If $x+y = +4$, then $(4)^3 = 64$, and $64 < 16$ is false.• If $x+y = -4$, then $(-4)^3 = -64$, and $-64 < 16$ is true.Therefore, only $x+y = -4$ satisfies both statements.Hence, the value $(x+y) = -4$ is uniquely determined when both statements are used together.Thus, the question can be answered by using both statements together, but not by either alone.Answer:Option C.