The third term in the expansion of ${(2x + p)^6}$ is $60{x^4}$ . Find the possible values of p .

attempt to expand binomial     (M1)

e.g. ${(2x)^6}{p^0} + \left( {\begin{array}{*{20}{c}} 6\\ 1 \end{array}} \right){(2x)^5}{(p)^1} +  \ldots$ , $\left( {\begin{array}{*{20}{c}} n\\ r \end{array}} \right){(2x)^r}{(p)^{n - r}}$

one correct calculation for term in ${x^4}$ in the expansion for power 6     (A1)

e.g. 15 , $16{x^4}$

correct expression for term in ${x^4}$    (A1)

e.g. $\left( {\begin{array}{*{20}{c}} 6\\ 2 \end{array}} \right){(2x)^4}{(p)^2}$ , ${15.2^4}{p^2}$

Notes: Accept sloppy notation e.g. omission of brackets around $2x$ .

Accept absence of $x$ in middle factor.

correct term     (A1)

e.g. $240{p^2}{x^4}$ (accept absence of ${x^4}$ )

setting up equation with their coefficient equal to 60     M1

e.g. $\left( {\begin{array}{*{20}{c}} 6\\ 2 \end{array}} \right){(2)^4}{(p)^2} = 60$ , $240{p^2}{x^4} = 60{x^4}$ , ${p^2} = \frac{{60}}{{240}}$

$p = \pm \frac{1}{2}(p = \pm 0.5)$     A1A1     N3

[7 marks]

This question proved challenging for many students. Most candidates recognized the need to expand a binomial but many executed this task incorrectly by selecting the wrong term, omitting brackets, or ignoring the binomial coefficient. Other candidates did not recognize that there were two values for p when solving their quadratic equation.