| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core (Further Pure Core) |
| Session | Specimen |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Taylor series |
| Type | Use series to find error or validity |
| Difficulty | Standard +0.3 This is a structured Further Maths question on Maclaurin series that guides students through standard applications. While it requires knowledge of series expansions and error analysis, each part is scaffolded with clear instructions. The most challenging aspect is part (iii) requiring algebraic manipulation of two series, but this is a routine technique at Further Maths level. The conceptual demand is moderate—understanding convergence and comparing approximations—but no novel insight is required. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (i) | 0.52 0.53 |
| Answer | Marks |
|---|---|
| 0.4167 | M1 |
| Answer | Marks |
|---|---|
| [2] | m |
| Answer | Marks |
|---|---|
| 1.1 | Substitute x(cid:32) 0.5 into |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (ii) | (A) |
| error 0.0112 | A1 | |
| [1] | 1.1 | Compare ln 1.5 = 0.4055 to |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (ii) | (B) |
| Answer | Marks | Guidance |
|---|---|---|
| convergence of the series | E1 | |
| [1] | 2.3 | |
| 7 | (iii) | 1(cid:14)x |
| Answer | Marks |
|---|---|
| (cid:169) (cid:185) | M1 |
| Answer | Marks |
|---|---|
| [2] | 3.1a |
| 1.1 | Attempt at series for |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (iv) | (A) |
| Answer | Marks |
|---|---|
| Using x(cid:32) 0.5, ln 3 (cid:32) 1.083. | M1 |
| Answer | Marks |
|---|---|
| [3] | m2.2a |
| Answer | Marks | Guidance |
|---|---|---|
| 1.1 | e | |
| 7 | (iv) | (B) |
| ln3: Inside range of convergence | c |
| Answer | Marks |
|---|---|
| [2] | i |
Question 7:
7 | (i) | 0.52 0.53
ln1.5(cid:32)0.5(cid:16) (cid:14) (cid:16)...
2 3
0.4167 | M1
c
A1
[2] | m
1.1
i
1.1 | Substitute x(cid:32) 0.5 into
series for ln(cid:11)1 (cid:14)x(cid:12) up to x3
Obtain 0.4167
7 | (ii) | (A) | e
error 0.0112 | A1
[1] | 1.1 | Compare ln 1.5 = 0.4055 to
4 d.p.
7 | (ii) | (B) | p
ln 3 would require x = 2, beyond the range of
convergence of the series | E1
[1] | 2.3
7 | (iii) | 1(cid:14)x
ln (cid:32)ln(1(cid:14)x)(cid:16)ln(1(cid:16)x)
1(cid:16)x
x2 x3
ln(1(cid:16)x)(cid:32)(cid:16)x(cid:16) (cid:16)
2 3
1(cid:14)x (cid:167) x3 (cid:183)
ln (cid:32)2(cid:168)x(cid:14) (cid:184)
1(cid:16)x 3
(cid:169) (cid:185) | M1
A1
[2] | 3.1a
1.1 | Attempt at series for
ln(cid:11)1 –x(cid:12)
n
7 | (iv) | (A) | 1(cid:14)x
(cid:32)1.5so x(cid:32) 0.2.
1(cid:16)x
ln 1.5 (cid:32) 0.4053
Using x(cid:32) 0.5, ln 3 (cid:32) 1.083. | M1
A1
B1
[3] | m2.2a
1.1
1.1 | e
7 | (iv) | (B) | (Much) better approximation to ln 1.5
ln3: Inside range of convergence | c
E1
E1
[2] | i
2.3
2.3\begin{enumerate}[label=(\roman*)]
\item Use the Maclaurin series for $\ln(1 + x)$ up to the term in $x^3$ to obtain an approximation to $\ln 1.5$. [2]
\item \begin{enumerate}[label=(\Alph*)]
\item Find the error in the approximation in part (i). [1]
\item Explain why the Maclaurin series in part (i), with $x = 2$, should not be used to find an approximation to $\ln 3$. [1]
\end{enumerate}
\item Find a cubic approximation to $\ln\left(\frac{1+x}{1-x}\right)$. [2]
\item \begin{enumerate}[label=(\Alph*)]
\item Use the approximation in part (iii) to find approximations to
• $\ln 1.5$ and
• $\ln 3$. [3]
\item Comment on your answers to part (iv) (A). [2]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core Q7 [11]}}