(i) (a) The expansion of \((1-x)^6\) using the binomial theorem is:

\((1-x)^6 = 1 - 6x + 15x^2 + \ldots\)

(b) The expansion of \((1+2x)^6\) using the binomial theorem is:

\((1+2x)^6 = 1 + 12x + 60x^2 + \ldots\)

(ii) To find the coefficient of \(x^2\) in \([(1-x)(1+2x)]^6\), we consider the product of the expansions:

\((1-x)^6 = 1 - 6x + 15x^2\)

\((1+2x)^6 = 1 + 12x + 60x^2\)

The coefficient of \(x^2\) is obtained by considering the terms:

\(1 \cdot 60x^2 = 60x^2\)

\(-6x \cdot 12x = -72x^2\)

\(15x^2 \cdot 1 = 15x^2\)

Adding these gives the coefficient of \(x^2\):

\(60 - 72 + 15 = 3\)