In the expansion of \(a{x^3}{(2 + ax)^{11}}\), the coefficient of the term in \({x^5}\) is 11880. Find the value of \(a\).

valid approach for expansion (must have correct substitution for parameters, but accept an incorrect value for \(r\))     (M1)

eg\(\,\,\,\,\,\)\(\left( {\begin{array}{*{20}{c}} {11} \\ r \end{array}} \right){(2)^{11 - r}}a{x^r},{\text{ }}\left( {\begin{array}{*{20}{c}} {11} \\ 3 \end{array}} \right){(2)^8}{(ax)^3},{\text{ }}{2^{11}} + \left( {\begin{array}{*{20}{c}} {11} \\ 1 \end{array}} \right){(2)^{10}}{(ax)^1} + \left( {\begin{array}{*{20}{c}} {11} \\ 2 \end{array}} \right){(2)^9}(ax) + \ldots \)

recognizing need to find term in \({x^2}\) in binomial expansion     (A1)

eg\(\,\,\,\,\,\)\(r = 2,{\text{ }}{(ax)^2}\)

correct term or coefficient in binomial expansion (may be seen in equation)     (A1)

eg\(\,\,\,\,\,\)\(\left( {\begin{array}{*{20}{c}} {11} \\ 2 \end{array}} \right){(ax)^2}{(2)^9},{\text{ }}55({a^2}{x^2})(512),{\text{ }}28160{a^2}\)

setting up equation in \({x^5}\) with their coefficient/term (do not accept other powers of \(x\))     (M1)

eg\(\,\,\,\,\,\)\(a{x^3}\left( {\begin{array}{*{20}{c}} {11} \\ 2 \end{array}} \right){(ax)^2}{(2)^9} = 11880{x^5}\)

correct equation     (A1)

eg\(\,\,\,\,\,\)\(28160{a^3} = 11880\)

\(a = \frac{3}{4}\)     A1     N3

[6 marks]