Consider the expansion of ${\left( {\frac{{{x^3}}}{2} + \frac{p}{x}} \right)^8}$. The constant term is $5103$. Find the possible values of $p$.

valid approach to find the required term     (M1)

eg$\;\;\;$$\left( {\begin{array}{*{20}{c}} 8 \\ r \end{array}} \right){\left( {\frac{{{x^3}}}{2}} \right)^{8 - r}}{\left( {\frac{p}{x}} \right)^r},{\text{ }}{\left( {\frac{{{x^3}}}{2}} \right)^8}{\left( {\frac{p}{x}} \right)^0} + \left( {\begin{array}{*{20}{c}} 8 \\ 1 \end{array}} \right){\left( {\frac{{{x^3}}}{2}} \right)^7}{\left( {\frac{p}{x}} \right)^1} +  \ldots$, Pascal’s triangle to required value

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

eg$\;\;\;$${7^{{\text{th}}}}{\text{ term, }}r = 6,{\text{ }}{\left( {\frac{1}{2}} \right)^2},{\text{ }}\left( {\begin{array}{*{20}{c}} 8 \\ 6 \end{array}} \right),{\text{ }}{\left( {\frac{{{x^3}}}{2}} \right)^2}{\left( {\frac{p}{x}} \right)^6}$

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

eg$\;\;\;$$\left( {\begin{array}{*{20}{c}} 8 \\ 6 \end{array}} \right){\left( {\frac{{{x^3}}}{2}} \right)^2}{\left( {\frac{p}{x}} \right)^6},{\text{ }}\frac{{8 \times 7}}{2} \times \frac{{{p^6}}}{{{2^2}}}$

setting up equation with their constant term equal to $5103$     M1

eg$\;\;\;$$\left( {\begin{array}{*{20}{c}} 8 \\ 6 \end{array}} \right){\left( {\frac{{{x^3}}}{2}} \right)^2}{\left( {\frac{p}{x}} \right)^6} = 5103,{\text{ }}{p^6} = \frac{{5103}}{7}$

$p =  \pm 3$     A1A1     N3

[6 marks]

Candidates tended to either do very well or very poorly in this question. Some had difficulty understanding what the constant term was, while others were unable to find the value of $r$ that led to the constant term. Many algebraic errors were seen in the calculation of the term, mostly having to do with forgetting to square $\frac{1}{2}$. Some missed the negative solution for p, despite the fact that the question asked for the “values” of $p$.