The third term in the expansion of \({(x + k)^8}\) is \(63{x^6}\). Find the possible values of \(k\).

valid approach to find the required term     (M1)

eg\(\;\;\;\left( {\begin{array}{*{20}{c}} 8 \\ r \end{array}} \right){x^{8 - r}}{k^r}\), Pascal’s triangle to \({{\text{8}}^{{\text{th}}}}\) row, \({x^8} + 8{x^7}k + 28{x^6}{k^2} +  \ldots \)

identifying correct term (may be indicated in expansion)     (A1)

eg\(\;\;\;\left( {\begin{array}{*{20}{c}} 8 \\ 2 \end{array}} \right){x^6}{k^2},{\text{ }}\left( {\begin{array}{*{20}{c}} 8 \\ 6 \end{array}} \right){x^6}{k^2},{\text{ }}r = 2\)

setting up equation in \(k\) with their coefficient/term     (M1)

eg\(\;\;\;28{k^2}{x^6} = 63{x^6},{\text{ }}\left( {\begin{array}{*{20}{c}} 8 \\ 6 \end{array}} \right){k^2} = 63\)

\(k =  \pm 1.5{\text{ (exact)}}\)     A1A1     N3

[5 marks]

Candidates who recognized that the third term is required usually completed the question successfully, although some candidates only gave a single value for \(k\). A few candidates attempted to fully expand algebraically, which proved to be a fruitless enterprise.