| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2025 |
| Session | January |
| Marks | 8 |
| Topic | Generalised Binomial Theorem and Partial Fractions |
| Type | Partial fractions showing coefficient is zero |
| Difficulty | Standard +0.3 This is a standard Further Maths partial fractions question followed by routine binomial expansion. Part (i) uses the cover-up method or substitution to find constants (straightforward algebra). Part (ii) applies standard binomial expansion formulas and combines series—all textbook techniques with no novel insight required. Slightly easier than average A-level due to its mechanical nature. |
| Spec | 1.02y Partial fractions: decompose rational functions1.04c Extend binomial expansion: rational n, |x|<1 |
\begin{enumerate}[label=(\roman*)]
\item Given that
$$\frac{3 + 2x^2}{(1 + x)^2(1 - 4x)} = \frac{A}{1 + x} + \frac{B}{(1 + x)^2} + \frac{C}{1 - 4x},$$
where $A$, $B$ and $C$ are constants, find $B$ and $C$, and show that $A = 0$. [4]
\item Given that $x$ is sufficiently small, find the first three terms of the binomial expansions of $(1 + x)^{-2}$ and $(1 - 4x)^{-1}$.
Hence find the first three terms of the expansion of $\frac{3 + 2x^2}{(1 + x)^2(1 - 4x)}$. [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2025 Q2 [8]}}