UPSC CDS 1 2023 Math Paper

Concept:Data sufficiency problem: Determine if $m > n$ for non‑zero numbers $m$, $n$ using two conditional statements.Explanation:Statement I: $\frac{m}{n} > 1$.If both $m$ and $n$ are positive, then $m > n$.If both are negative, multiplying the inequality by the negative $n$ reverses it, giving $m < n$.Thus Statement I alone does not give a definite answer.Statement II: $m > 2n$.If $n > 0$, then $2n > n$, so $m > 2n > n$ implies $m > n$.If $n < 0$, then $2n < n$. For example, $n = -3$, $2n = -6$; $m = -5$ satisfies $m > 2n$ but $m = -5 < -3 = n$.Thus Statement II alone is also insufficient.Both statements together:When $m, n > 0$, both conditions lead to $m > n$.When $m, n < 0$, Statement I forces $m < n$, and Statement II only requires $m > 2n$ (which is automatically true if $m < n$ because $2n < n$).Since the signs of $m$ and $n$ are not fixed, we cannot decide whether $m > n$ or $m < n$.Answer:The question cannot be answered even by using both statements together.Hence option D is correct.