UPSC CDS 2 2023 Math Paper

Concept:The LCM of algebraic expressions is the product of the highest power of every distinct irreducible factor present in the expressions.Explanation:Given expressions: $x^{4} + x^{2}y^{2} + y^{4}$, $x^{3} + y^{3}$, $x^{3} - y^{3}$.Factor each expression using standard identities:$x^{4} + x^{2}y^{2} + y^{4} = (x^{2} + xy + y^{2})(x^{2} - xy + y^{2})$$x^{3} + y^{3} = (x + y)(x^{2} - xy + y^{2})$$x^{3} - y^{3} = (x - y)(x^{2} + xy + y^{2})$The distinct factors are $(x + y)$, $(x - y)$, $(x^{2} - xy + y^{2})$, and $(x^{2} + xy + y^{2})$.Each factor appears only once. Hence the LCM is their product.LCM $= (x + y)(x - y)(x^{2} - xy + y^{2})(x^{2} + xy + y^{2})$Multiply $(x + y)(x - y) = x^{2} - y^{2}$.Multiply $(x^{2} - xy + y^{2})(x^{2} + xy + y^{2}) = x^{4} + x^{2}y^{2} + y^{4}$.Thus LCM $= (x^{2} - y^{2})(x^{4} + x^{2}y^{2} + y^{4})$Using the identity $(a - b)(a^{2} + ab + b^{2}) = a^{3} - b^{3}$ with $a = x^{2}$, $b = y^{2}$, we get:$(x^{2} - y^{2})(x^{4} + x^{2}y^{2} + y^{4}) = (x^{2})^{3} - (y^{2})^{3} = x^{6} - y^{6}$.Therefore the LCM simplifies to $x^{6} - y^{6}$.Answer:$(x^{6} - y^{6})$ (Option C).