UPSC CDS 2 2023 Math Paper

Concept:Use algebraic identities to check if each statement forces $x=y=z$.Explanation:Statement-I: $x^{2}+y^{2}+z^{2}-xy-yz-zx = \frac{1}{2}\left[(x-y)^{2}+(y-z)^{2}+(z-x)^{2}\right]$.If this sum equals $0$, each squared term must be $0$, so $x=y=z$.Thus Statement-I alone answers the question with a definite "yes".Statement-II: $x^{3}+y^{3}+z^{3}-3xyz = (x+y+z)(x^{2}+y^{2}+z^{2}-xy-yz-zx)$.The product is $0$ if either factor is $0$.Case 1: $x^{2}+y^{2}+z^{2}-xy-yz-zx = 0$ implies $x=y=z$ as above.Case 2: $x+y+z = 0$ does not require equality.For example, $x=1$, $y=1$, $z=-2$ satisfy $x+y+z=0$ but $x\neq y\neq z$.Hence Statement-II alone is insufficient to conclude $x=y=z$.Therefore, the question can be answered using Statement-I alone, but not using Statement-II 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.