| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core (Further Pure Core) |
| Year | 2024 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Taylor series |
| Type | Direct substitution into standard series |
| Difficulty | Moderate -0.3 This is a straightforward application of a standard Maclaurin series with direct substitution. Part (a) requires recalling ln(1+x) and substituting x³; part (b) is arithmetic with x=0.5; part (c) tests understanding of convergence radius. All steps are routine with no problem-solving insight needed, making it slightly easier than average. |
| Spec | 4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n |
| Answer | Marks | Guidance |
|---|---|---|
| 10 | (a) | 𝑥6 𝑥9 |
| Answer | Marks | Guidance |
|---|---|---|
| 2 3 | B1 | |
| [1] | 1.1 | Must be simplified |
| 10 | (b) | let x = 12 |
| Answer | Marks |
|---|---|
| 1 5 3 6 | M1 |
| Answer | Marks |
|---|---|
| [3] | 3.1a |
| Answer | Marks |
|---|---|
| 1.1 | Or 𝑥3 = 0.125. Must be seen, can be implied by explicit |
| Answer | Marks | Guidance |
|---|---|---|
| 10 | (c) | Charlie takes x = 2 |
| The series only converges for −1 < x 1 | M1 | |
| A1 | 2.3 | |
| 2.4 | May be embedded |
| Answer | Marks | Guidance |
|---|---|---|
| Charlie takes 𝑥3 = 8 | M1 | May be embedded |
| The series only converges for −1 < 𝑥3 ≤ 1 | The series only converges for −1 < 𝑥3 ≤ 1 | A1 |
Question 10:
10 | (a) | 𝑥6 𝑥9
𝑥3− +
2 3 | B1
[1] | 1.1 | Must be simplified
10 | (b) | let x = 12
3 6 9
1 1 1 1 1
[ln1.125 ≈]( ) − ( ) + ( )
2 2 2 3 2
1 8 1
= [so n = 181]
1 5 3 6 | M1
A1
A1
[3] | 3.1a
1.1
1.1 | Or 𝑥3 = 0.125. Must be seen, can be implied by explicit
substitution.
oe correct substitution. If 𝑥3 = 0.125 used then
0.1252 0.1253
0.125− + . FT their three terms. Cannot be
2 3
implied.
cao
10 | (c) | Charlie takes x = 2
The series only converges for −1 < x 1 | M1
A1 | 2.3
2.4 | May be embedded
See appendix. Do not award if spoilt by an incorrect
statement.
Alternative method
Charlie takes 𝑥3 = 8 | M1 | May be embedded
The series only converges for −1 < 𝑥3 ≤ 1 | The series only converges for −1 < 𝑥3 ≤ 1 | A1 | A1 | See appendix. Do not award if spoilt by an incorrect
statement.
[2]10
\begin{enumerate}[label=(\alph*)]
\item Write down the first three terms of the Maclaurin series for $\ln \left( 1 + x ^ { 3 } \right)$.
\item Use these three terms to show that $\ln ( 1.125 ) \approx \frac { n } { 1536 }$, where $n$ is an integer to be determined.
\item Charlie uses the same first three terms of the series to approximate $\ln 9$ and gets an answer of 147, correct to 3 significant figures. However, $\ln 9 = 2.20$ correct to 3 significant figures. Explain Charlie's error.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core 2024 Q10 [6]}}