UPSC CDS II 2025 Mathematics Paper

Concept:The HCF of two polynomials is the highest-degree polynomial that divides both. Factor each polynomial and compare their factors.Explanation:First factor $f = x^{3} + y^{3} + 3xy - 1$. Use the identity $a^{3}+b^{3}+c^{3}-3abc = (a+b+c)(a^{2}+b^{2}+c^{2}-ab-bc-ca)$ with $a=x$, $b=y$, $c=-1$. We get $f = (x+y-1)(x^{2} - xy + x + y^{2} + y + 1)$.Now factor $g = (x-y)^{4} - 1$ as a difference of squares: $((x-y)^{2}-1)((x-y)^{2}+1)$.Further factor the first part: $(x-y-1)(x-y+1)$. Thus $g = (x-y-1)(x-y+1)(x^{2} - 2xy + y^{2} + 1)$.Compare the factors of $f$ and $g$. $f$ has factor $(x+y-1)$; $g$ has $(x-y-1)$ and $(x-y+1)$. These are different linear factors. The quadratic factors also do not match.Hence no common factor exists other than 1.Answer:D. 1