UPSC CDS II 2025 Mathematics Paper

Concept:LCM of algebraic expressions is the product of each distinct factor raised to its highest power found in any of the expressions.Explanation:Factor each expression completely.$A = x^4 + x^2 y^2 + y^4 = (x^2 + xy + y^2)(x^2 - xy + y^2).$$B = x^3 y + y^4 = y(x^3 + y^3) = y(x + y)(x^2 - xy + y^2).$$C = x^4 y^2 - x^3 y^3 = x^3 y^2 (x - y).$List all distinct factors: $x$, $y$, $(x - y)$, $(x + y)$, $(x^2 + xy + y^2)$, $(x^2 - xy + y^2)$.Take the highest power of each: $x^3$ (from $C$), $y^2$ (from $C$), and each factor to the first power.Multiply them: LCM = $x^3 y^2 (x - y)(x + y)(x^2 + xy + y^2)(x^2 - xy + y^2)$.Note that $(x - y)(x + y)(x^2 + xy + y^2)(x^2 - xy + y^2) = x^6 - y^6$.Thus LCM = $x^3 y^2 (x^6 - y^6)$.Answer:Option B: $x^3 y^2 (x^6 - y^6)$