If f(x) = ,using properties of determinants find the value of f(2x) – f(x).

Given,

f(x) =

Taking ‘a’ common from the first row

f(x) =

Applying C₂ → C₁ + C₂, we get-

f(x) =

Expanding about first R₁ we get –

f(x) = a{(a+x)a – (-1)(ax+x²)}

⇒ f(x) = a{a²+ax + ax + x²}

⇒ f(x) = a(a² + 2ax + x²) = a(a+x)²

∴ f(2x) = a(a+2x)²

Hence,

f(2x) – f(x) = a(a+2x)² – a(a+x)²

⇒ f(2x) - f(x) = a(a+2x-a-x)(a+2x+a+x) {∵ a²-b² = (a+b)(a-b)}

∴ f(2x)-f(x) = ax(2a+3x)