| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core (Further Pure Core) |
| Year | 2021 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Taylor series |
| Type | Use series to find error or validity |
| Difficulty | Standard +0.8 This question tests Maclaurin series derivation, error calculation, and crucially, understanding of convergence/validity conditions. Part (c) requires recognizing that x=1 lies outside the radius of convergence for ln(1+2x), which demands conceptual understanding beyond routine application. The multi-step nature and the conceptual trap in part (c) place this above average difficulty. |
| Spec | 4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (a) | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | B1 | |
| [1] | 1.1 | |
| 5 | (b) | ln (1 .2 ) 0 .1 8 |
| Answer | Marks |
|---|---|
| = (−)1.27% | B1ft |
| Answer | Marks |
|---|---|
| [3] | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (c) | −12x1−1 x 1 |
| Answer | Marks |
|---|---|
| approximation when x = 1 | M1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| 2.3 | substitute x = 1 in quadratic |
Question 5:
5 | (a) | 1
ln(1+2x)2x− (2x)2 =2x−2x2
2 | B1
[1] | 1.1
5 | (b) | ln (1 .2 ) 0 .1 8
0.18−ln(1.2)
% error = 100
ln(1.2)
= (−)1.27% | B1ft
M1
A1
[3] | 1.1
1.1
1.1
5 | (c) | −12x1−1 x 1
2 2
the Maclaurin series is not convergent for this
approximation when x = 1 | M1
A1
[2] | 1.1
2.3 | substitute x = 1 in quadratic5
\begin{enumerate}[label=(\alph*)]
\item Use a Maclaurin series to find a quadratic approximation for $\ln ( 1 + 2 x )$.
\item Find the percentage error in using the approximation in part (a) to calculate $\ln ( 1.2 )$.
\item Jane uses the Maclaurin series in part (a) to try to calculate an approximation for $\ln 3$.
Explain whether her method is valid.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core 2021 Q5 [6]}}