UPSC CDS I 2024 Math Paper

Concept:The HCF (Highest Common Factor) is the greatest polynomial that divides all given expressions. Factorize each expression and find the common factor(s).Explanation:Factorize the third expression: $x^{3} - 125y^{3} = x^{3} - (5y)^{3} = (x - 5y)(x^{2} + 5xy + 25y^{2})$.Factorize the first expression: $x^{4} - 13x^{2}y^{2} - 300y^{4}$.Treat it as a quadratic in $x^{2}$: $(x^{2})^{2} - 13y^{2}(x^{2}) - 300y^{4} = (x^{2} - 25y^{2})(x^{2} + 12y^{2}) = (x - 5y)(x + 5y)(x^{2} + 12y^{2})$.Thus, $(x - 5y)$ is a common factor of the first and third expressions.Check if $(x - 5y)$ is also a factor of the second expression: substitute $x = 5y$ into $x^{3} - 4x^{2}y - 4xy^{2} - 5y^{3}$.$(5y)^{3} - 4(5y)^{2} y - 4(5y)y^{2} - 5y^{3} = 125y^{3} - 100y^{3} - 20y^{3} - 5y^{3} = 0$.Hence $(x - 5y)$ divides the second expression as well.No other common factor exists among all three; for example, $x+5y$ and $x^{2}+12y^{2}$ are not factors of the second or third expressions.Therefore, the HCF is $(x - 5y)$.Answer:Option A: $x - 5y$