| Exam Board | OCR MEI |
|---|---|
| Module | Further Numerical Methods (Further Numerical Methods) |
| Year | 2024 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule applied to real-world data |
| Difficulty | Standard +0.8 This Further Maths numerical methods question requires knowledge of central difference formulas and Taylor series error estimation. Part (a) is straightforward application of the central difference formula, but part (b) requires understanding of truncation error in linear approximation using derivatives, which goes beyond standard A-level and involves multi-step reasoning with error analysis concepts specific to Further Maths numerical methods. |
| Spec | 1.09f Trapezium rule: numerical integration |
| \(x\) | 1.9 | 2 | 2.1 |
| \(\mathrm { f } ( x )\) | 0.5842 | 0.6309 | 0.6753 |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | (a) | 0.6753−0.5842 |
| Answer | Marks |
|---|---|
| 0.4555 or 0.456 or 0.46 | M1 |
| Answer | Marks |
|---|---|
| [2] | 1.1a |
| 1.1 | central difference method |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | (b) | 0.05×their 0.4555 |
| ‒ 0.023 to ‒ 0.022775 | M1 | |
| A1FT | 1.1 | |
| 1.1 | may see [f(2.05) ≈] 0.6309 + 0.05 × their 0.4555 ; |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | 0.612547 | 1 |
Question 1:
1 | (a) | 0.6753−0.5842
oe
2.1−1.9
0.4555 or 0.456 or 0.46 | M1
A1
[2] | 1.1a
1.1 | central difference method
if M0 allow SC1 for 0.444 from forward difference method
allow B2 for correct answer unsupported
1 | (b) | 0.05×their 0.4555
‒ 0.023 to ‒ 0.022775 | M1
A1FT | 1.1
1.1 | may see [f(2.05) ≈] 0.6309 + 0.05 × their 0.4555 ;
may be implied by
f(2.05) ≈ 0.653675, 0.6537 or 0.6539
accept 0.022775 to 0.023; B2 for correct answer unsupported
[2]
1 | 0.612547 | 1 | 0.741701 The table shows some values of $x$, together with the associated values of a function, $\mathrm { f } ( x )$.
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
$x$ & 1.9 & 2 & 2.1 \\
\hline
$\mathrm { f } ( x )$ & 0.5842 & 0.6309 & 0.6753 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Use the information in the table to calculate the most accurate estimate of $f ^ { \prime } ( 2 )$ possible.
\item Calculate an estimate of the error when $f ( 2 )$ is used as an estimate of $f ( 2.05 )$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Numerical Methods 2024 Q1 [4]}}