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.