(i) The binomial expansion of \((1 + ax)^5\) is given by:

\(1 + 5(ax) + \frac{5 \times 4}{2}(ax)^2 + \cdots\)

\(= 1 + 5ax + 10a^2x^2\)

(ii) For the expansion of \((1 - 2x)(1 + ax)^5\), the term in \(x\) is:

\((1)(5ax) + (-2x)(1) = 5ax - 2x\)

Setting the coefficient of \(x\) to zero:

\(5a - 2 = 0\)

\(a = \frac{2}{5}\)

(iii) The coefficient of \(x^2\) in \((1 - 2x)(1 + ax)^5\) is:

\((-2x)(5ax) + (1)(10a^2x^2) = -10ax + 10a^2x^2\)

Substituting \(a = \frac{2}{5}\):

\(-10 \times \frac{2}{5} + 10 \left(\frac{2}{5}\right)^2 = -4 + 1.6 = -2.4\)