| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2024 |
| Session | February |
| Marks | 11 |
| Topic | Partial Fractions |
| Type | Repeated linear factor with series expansion |
| Difficulty | Standard +0.8 This question combines partial fractions with a repeated linear factor (requiring differentiation or cover-up method) and binomial series expansion. Part (a) requires handling the repeated factor correctly, part (b) needs expanding three separate fractions and combining terms carefully, and part (c) requires understanding convergence conditions from multiple series. The repeated linear factor adds technical complexity beyond standard partial fractions, and the series manipulation requires careful algebraic work, placing this above average difficulty. |
| Spec | 1.02y Partial fractions: decompose rational functions4.04c Scalar product: calculate and use for angles4.08a Maclaurin series: find series for function4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n |
+ \mu \left( \begin{array} { c }
3 \\
1 \\
- 2
\end{array} \right)
$$
$P$ is the point of intersection of $l _ { 1 }$ and $l _ { 2 }$.\\
\begin{enumerate}[label=(\alph*)]
\item Find the position vector of $P$.
\item Find, correct to 1 decimal place, the acute angle between $/ _ { 1 }$ and $/ _ { 2 }$.\\
$Q$ is a point on $/ 1$ which is 12 metres away from $P \cdot R$ is the point on $/ 2$ such that $Q R$ is perpendicular to $/ 1$.
\item Determine the length $Q R$.\\[0pt]
\\
\end{enumerate}5.\\
\begin{enumerate}[label=(\alph*)]
\item Express $\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }$ in three partial fractions.
\item \\
Hence find the first three terms in the expansion of $\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }$ in ascending powers of $x$.
\item State the set of values for which the expansion in part (b) is valid.\\[0pt]
\\
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2024 Q5 [11]}}