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.