UPSC CDS 2 2022 Math Paper

Concept:The sign of a product depends on the sign of each factor. An even power of a real number is always non‑negative, an odd power retains the sign of the base.Explanation:Statement‑1: $x^{8}y^{9} < 0$. $x^{8} \ge 0$ for all $x$, and $x \neq 0$ because product is not zero. Since $x^{8} > 0$, the inequality forces $y^{9} < 0$, hence $y < 0$. But the sign of $x$ remains unknown. Thus Statement‑1 alone is insufficient to decide whether $xy > 0$. Statement‑2: $x^{9}y^{10} < 0$. $y^{10} \ge 0$ for all $y$, and $y \neq 0$ because product is not zero. Since $y^{10} > 0$, we get $x^{9} < 0$, hence $x < 0$. But the sign of $y$ remains unknown. Thus Statement‑2 alone is also insufficient. Combining both statements:From Statement‑1: $y < 0$. From Statement‑2: $x < 0$. Both $x$ and $y$ are negative, so their product $xy$ is positive: $xy > 0$. Therefore, the two statements together are sufficient to answer the question.Answer:Option C – Both Statement‑1 and Statement‑2 are sufficient to answer the question.