Write down the number of terms of this expansion.

Find the coefficient of ${x^8}$.

11 terms     A1     N1

[1 mark]

valid approach     (M1)

eg$\,\,\,\,\,$$\left( {\begin{array}{*{20}{c}} {10} \\ r \end{array}} \right){({x^2})^{10 - r}}{\left( {\frac{2}{x}} \right)^r},{\text{ }}{a^{10}}{b^0} + \left( {\begin{array}{*{20}{c}} {10} \\ 1 \end{array}} \right){a^9}{b^1}\left( {\begin{array}{*{20}{c}} {10} \\ 2 \end{array}} \right){a^8}{b^2} + \ldots$

Pascal’s triangle to ${11^{th}}$ row

valid attempt to find value of $r$ which gives term in ${x^8}$     (M1)

eg$\,\,\,\,\,$${({x^2})^{10 - r}}\left( {\frac{1}{{{x^r}}}} \right) = {x^8},{\text{ }}{x^{2r}}{\left( {\frac{2}{x}} \right)^{10 - r}} = {x^8}$

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

eg$\,\,\,\,\,$$r = 6,{\text{ }}{{\text{5}}^{{\text{th}}}}{\text{ term, }}{{\text{7}}^{{\text{th}}}}{\text{ term}}$

correct working (may be seen in expansion)     (A1)

eg$\,\,\,\,\,$$\left( {\begin{array}{*{20}{c}} {10} \\ 6 \end{array}} \right){({x^2})^6}{\left( {\frac{2}{x}} \right)^4},{\text{ }}210 \times 16$

3360     A1     N3

[5 marks]

Although slightly challenging, this question aimed at assessing candidates’ fluency at using the binomial theorem to find the coefficient of a term.

In part a), most candidates realized that the expansion had 11 terms, although a few answered 10.

In part b), many candidates attempted to answer and knew what they needed to find. However, the execution of the plan was not always successful. A fair amount of students had difficulties with the powers of the factors of the required term and could only earn the first method mark for a valid approach. Some candidates gave the term instead of the coefficient as the answer. A few of them attempted to expand the binomial algebraically and very few added instead of multiplied, losing all marks.