UPSC CDS 2 2023 Math Paper

Concept:Use algebraic identities to simplify expressions, then analyze whether the given statements determine the relation between $m$ and $n$.Explanation:From Statement I: $m = (1-p)(p^2+p+1) = 1-p^3$ (using $a^3-b^3 = (a-b)(a^2+ab+b^2)$).$n = (p+1)(p^2-p+1) = p^3+1$ (using $a^3+b^3 = (a+b)(a^2-ab+b^2)$).Thus $m = 1-p^3$ and $n = p^3+1$.The value of $p$ can be any real number; for example, if $p=0$, then $m=1$ and $n=1$ (so $m=n$); if $p>0$, then $n>m$; if $p<0$, then $m>n$.So Statement I alone is insufficient.From Statement II: $m = pn$. This gives $m/n = p$, but $p$ can be any real number. Without knowing $p$, we cannot decide if $m>n$ (e.g., $p=2$ gives $m>n$, $p=0$ gives $m=n$, $p=-1$ gives $m < n$).Thus Statement II alone is also insufficient.Combining both: we have $m = 1-p^3$, $n = p^3+1$, and $m = pn$.Substitute: $1-p^3 = p(p^3+1) \implies 1-p^3 = p^4+p \implies p^4+p+p^3-1=0$.This equation has multiple real roots (e.g., $p=1$ gives $1+1+1-1=2\neq0$, $p=-1$ gives $1-1-1-1=-2\neq0$, $p=0$ gives $-1=0$ false).Even if we find a $p$, the relation $m>n$ would depend on that specific $p$, and not be uniquely determined for all real $p$.Therefore, even using both statements together, we cannot answer whether $m>n$ for all real numbers $m,n$.Answer:Option D – The question cannot be answered even by using both statements together.