(i) The binomial expansion of \((1 + ax)^{-2}\) is given by:

\(1 - 2(ax) + \frac{(-2)(-3)}{2!}(ax)^2 + \frac{(-2)(-3)(-4)}{3!}(ax)^3 + \ldots\)

The coefficient of \(x\) is \(-2a\) and the coefficient of \(x^3\) is \(-4a^3\).

Equating these coefficients: \(-2a = -4a^3\)

\(2a = 4a^3\)

\(a^2 = \frac{1}{2}\)

\(a = \frac{1}{\sqrt{2}}\)

(ii) Substitute \(a = \frac{1}{\sqrt{2}}\) into the expansion:

\((1 + ax)^{-2} = 1 - 2\left(\frac{1}{\sqrt{2}}x\right) + \frac{3}{2}\left(\frac{1}{\sqrt{2}}x\right)^2\)

\(= 1 - \sqrt{2}x + \frac{3}{2}x^2\)