UPSC CDS II 2025 Mathematics Paper

Concept:The expression is a quadratic in two variables. Substitute $a = 3x + y$ and $b = x + 5y$ to get a simple quadratic form. Factor it using the pattern $a^2 + ab - 20b^2 = (a + 5b)(a - 4b)$. Then replace back and check the given factors.Explanation:Let $a = 3x + y$ and $b = x + 5y$. The given expression becomes $a^2 + ab - 20b^2$.Factor this quadratic: $a^2 + ab - 20b^2 = (a + 5b)(a - 4b)$.Substitute back: $((3x + y) + 5(x + 5y)) \times ((3x + y) - 4(x + 5y))$.Simplify the first bracket: $3x + y + 5x + 25y = 8x + 26y = 2(4x + 13y)$.Simplify the second bracket: $3x + y - 4x - 20y = -x - 19y = -1(x + 19y)$.The overall expression is $2(4x + 13y) \times (-1)(x + 19y) = -2(4x + 13y)(x + 19y)$.Thus the algebraic factors are $(4x + 13y)$ and $(x + 19y)$ (apart from constant multipliers). Both I and II are factors.Answer:Both I and II (Option C).