| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2024 |
| Session | January |
| Marks | 13 |
| Topic | Generalised Binomial Theorem and Partial Fractions |
| Type | Partial fractions with repeated linear factor |
| Difficulty | Standard +0.8 This is a Further Maths question requiring partial fractions with a repeated linear factor, binomial expansion of multiple terms, and careful algebraic manipulation. The repeated factor (3x+2)² adds complexity beyond standard A-level, and combining multiple expansions to find coefficients requires systematic work. However, it follows a predictable structure with clear steps, making it challenging but not requiring novel insight. |
| Spec | 4.05c Partial fractions: extended to quadratic denominators4.08a Maclaurin series: find series for function4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n |
Let
$$f(x) = \frac{27x^2 + 32x + 16}{(3x + 2)^2(1 - x)}$$
\begin{enumerate}[label=(\alph*)]
\item Express $f(x)$ in terms of partial fractions [5]
\item Hence, or otherwise, find the series expansion of $f(x)$, in ascending powers of $x$, up to and including the term in $x^2$. Simplify each term. [6]
\item State, with a reason, whether your series expansion in part (b) is valid for $x = \frac{1}{2}$. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2024 Q5 [13]}}