| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2016 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Taylor series |
| Type | Match series to function form |
| Difficulty | Standard +0.8 This requires knowing the Taylor series for sin(2x), expanding a product involving (1-x²)^(-1) as a geometric series, then algebraically manipulating to match coefficients up to x^5. While systematic, it demands careful bookkeeping across multiple series expansions and is more involved than standard single-series questions, placing it moderately above average difficulty. |
| Spec | 4.08a Maclaurin series: find series for function4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\sin 2x = 2x - \frac{(2x)^3}{3!} + \frac{(2x)^5}{5!} \cdots = 2x - \frac{4}{3}x^3 + \frac{4}{15}x^5 \cdots\) | B1 | Correct expansion. ISW in higher powers |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((1 - x^2)^{-1} = 1 + x^2 + x^4 \cdots\) | B1 | Correct simplified expansion seen or used. ISW in higher powers |
| Series expansions for \(\sin 2x\) and \((1-x^2)^{-1}\) attempted and used in given function | M1 | Series expansions for \(\sin 2x\) and \((1-x^2)^{-1}\) attempted and used in the given function |
| \(\left(-\frac{4}{3} - 2 + 2p\right) = 0; \quad \left(\frac{4}{15} - 2 + 2p\right) = q\) | m1 | c's coefficient of \(x^3\) equated to 0 and c's coeff of \(x^5\) equated to \(q\) (PI) and an attempt to solve as far as reaching a value for \(p\) and a value for \(q\) |
| \(p = \frac{5}{3}; \quad q = \frac{8}{5}\) | A1 | ACF of both exact values |
# Question 2:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\sin 2x = 2x - \frac{(2x)^3}{3!} + \frac{(2x)^5}{5!} \cdots = 2x - \frac{4}{3}x^3 + \frac{4}{15}x^5 \cdots$ | B1 | Correct expansion. ISW in higher powers |
**Total: 1 mark**
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(1 - x^2)^{-1} = 1 + x^2 + x^4 \cdots$ | B1 | Correct simplified expansion seen or **used**. ISW in higher powers |
| Series expansions for $\sin 2x$ and $(1-x^2)^{-1}$ attempted and used in given function | M1 | Series expansions for $\sin 2x$ and $(1-x^2)^{-1}$ attempted and used in the given function |
| $\left(-\frac{4}{3} - 2 + 2p\right) = 0; \quad \left(\frac{4}{15} - 2 + 2p\right) = q$ | m1 | c's coefficient of $x^3$ equated to 0 **and** c's coeff of $x^5$ equated to $q$ (PI) and an attempt to solve as far as reaching a value for $p$ and a value for $q$ |
| $p = \frac{5}{3}; \quad q = \frac{8}{5}$ | A1 | ACF of both exact values |
**Total: 4 marks**
---2
\begin{enumerate}[label=(\alph*)]
\item Write down the expansion of $\sin 2 x$ in ascending powers of $x$ up to and including the term in $x ^ { 5 }$.
\item It is given that the first non-zero term in the expansion of
$$\sin 2 x - 2 x \left( 1 - p x ^ { 2 } \right) \left( 1 - x ^ { 2 } \right) ^ { - 1 }$$
in ascending powers of $x$ is $q x ^ { 5 }$.\\
Find the values of the rational numbers $p$ and $q$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2016 Q2 [5]}}