| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2023 |
| Session | January |
| Marks | 7 |
| Topic | Generalised Binomial Theorem |
| Type | Product with quadratic or higher term |
| Difficulty | Moderate -0.3 Part (i) is a standard binomial expansion with negative index requiring routine application of the formula and validity condition (|x/2| < 1). Part (ii) requires multiplying the expansion by (1+x²) and collecting terms, which is straightforward but adds a small problem-solving element. Overall, this is slightly easier than average due to being mostly procedural with clear steps. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
\begin{enumerate}[label=(\roman*)]
\item Expand $(2+x)^{-2}$ in ascending powers of $x$ up to and including the term in $x^3$, and state the set of values of $x$ for which the expansion is valid. [5]
\item Hence find the coefficient of $x^3$ in the expansion of $\frac{1+x^2}{(2+x)^2}$. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2023 Q5 [7]}}