| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core (Further Pure Core) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Taylor series |
| Type | Maclaurin series for composite exponential/root functions |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question requiring routine differentiation of a composite root function, standard Maclaurin series construction using f(0), f'(0), f''(0), and substitution of x=2 to approximate â5. While it involves multiple steps, each is a standard technique with no novel insight required, making it slightly easier than average even for Further Maths. |
| 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 |
|---|---|---|
| 4 | (a) | (i) |
| Answer | Marks |
|---|---|
| fâł(đĽ)= â(1+2đĽ) (cid:2879) (cid:2870) | B1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | (a) | (ii) |
| Answer | Marks |
|---|---|
| 2 | M1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| 1.1 | Their f(0), fâ(0) and fââ(0) evaluated and substituted into |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | (b) | (cid:2870) |
| Answer | Marks |
|---|---|
| (cid:2874)(cid:2872) | M1 |
| Answer | Marks |
|---|---|
| [2] | 3.1a |
| 2.2a | (cid:2869) |
Question 4:
4 | (a) | (i) | (cid:2869)
fâ˛(đĽ)= (1+2đĽ) (cid:2879) (cid:2870)
(cid:2871)
fâł(đĽ)= â(1+2đĽ) (cid:2879) (cid:2870) | B1
B1
[2] | 1.1
1.1
4 | (a) | (ii) | f(0)= 1, f(cid:4593)(0)= 1, f(cid:4593)(cid:4593)(0)= â1
1
1+đĽâ đĽ(cid:2870)
2 | M1
A1
[2] | 1.1
1.1 | Their f(0), fâ(0) and fââ(0) evaluated and substituted into
Maclaurin
Must come from correct expressions for f â˛(đĽ), f â˛â˛(đĽ); cannot
come from binomial expansion. Ignore subsequent terms.
4 | (b) | (cid:2870)
(cid:2869) (cid:2869) (cid:2869) (cid:2869)
(cid:3495)1+2Ă or 1+ â (cid:4672) (cid:4673)
(cid:2876) (cid:2876) (cid:2870) (cid:2876)
(cid:2869)(cid:2872)(cid:2871)
ď â5 â
(cid:2874)(cid:2872) | M1
A1
[2] | 3.1a
2.2a | (cid:2869)
using đĽ = in their expansion
(cid:2876)
AG4
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Given that $\mathrm { f } ( x ) = \sqrt { 1 + 2 x }$, find $\mathrm { f } ^ { \prime } ( x )$ and $\mathrm { f } ^ { \prime \prime } ( x )$.
\item Hence, find the first three terms of the Maclaurin series for $\sqrt { 1 + 2 x }$.
\end{enumerate}\item Hence, using a suitable value for $x$, show that $\sqrt { 5 } \approx \frac { 143 } { 64 }$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core 2023 Q4 [6]}}