| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2024 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Simpson's rule application |
| Difficulty | Moderate -0.8 This is a straightforward numerical methods question requiring routine application of Simpson's rule with provided values and a simple logarithm manipulation. Part (a) is trivial substitution, part (b) is standard Simpson's rule application (though requiring careful arithmetic), and part (c) uses the property ln(√u) = ½ln(u). No problem-solving insight or novel techniques required—purely procedural execution of a standard FP1 topic. |
| Spec | 1.06d Natural logarithm: ln(x) function and properties1.09f Trapezium rule: numerical integration |
| \(\boldsymbol { x }\) | 2 | 2.5 | 3 | 3.5 | 4 | 4.5 | 5 |
| \(\boldsymbol { y }\) | 1.946 | 2.225 | 2.725 | 2.944 | 3.146 | 3.332 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| awrt \(2.485\) | B1 | Correct value, seen in table or in the work. Accept awrt. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(h = 0.5\) | B1 | Correct step length used, stated or clearly implied in working. |
| \(\frac{1}{3} \times 0.5 \left[1.946 + 3.332 + 2(2.485 + 2.944) + 4(2.225 + 2.725 + 3.146)\right]\) \(= \frac{1}{6} \times 48.52\) | M1 | Correct structure for Simpson's rule \(\frac{1}{3}h[\text{ends} + 2\text{evens} + 4\text{odds}]\) using their value of \(h\). Insides must be correct. Condone missing closing bracket. Condone minor miscopies of ordinates as long as they are in correct positions. |
| \(8.09\) cao | A1 | For \(8.09\), correct answer only, must be to 3 s.f. Must have scored the M. Note: calculator gives \(8.086\ldots\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(0.5 \times \text{"} 8.09\text{"} = 4.045\) or \(4.04\) or \(4.05\) (awrt either) | B1ft | Deduces the value by finding \(0.5 \times\) their answer to (b). Allow awrt 3 s.f. Accept as a fraction e.g. \(\frac{1213}{300}\) as long as it is half their answer to (b), or rounds correctly to it. |
## Question 1:
### Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| awrt $2.485$ | **B1** | Correct value, seen in table or in the work. Accept awrt. |
**(1 mark)**
---
### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $h = 0.5$ | **B1** | Correct step length used, stated or clearly implied in working. |
| $\frac{1}{3} \times 0.5 \left[1.946 + 3.332 + 2(2.485 + 2.944) + 4(2.225 + 2.725 + 3.146)\right]$ $= \frac{1}{6} \times 48.52$ | **M1** | Correct structure for Simpson's rule $\frac{1}{3}h[\text{ends} + 2\text{evens} + 4\text{odds}]$ using their value of $h$. Insides must be correct. Condone missing closing bracket. Condone minor miscopies of ordinates as long as they are in correct positions. |
| $8.09$ cao | **A1** | For $8.09$, correct answer only, must be to 3 s.f. Must have scored the M. Note: calculator gives $8.086\ldots$ |
**(3 marks)**
---
### Part (c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $0.5 \times \text{"} 8.09\text{"} = 4.045$ or $4.04$ or $4.05$ (awrt either) | **B1ft** | Deduces the value by finding $0.5 \times$ their answer to (b). Allow awrt 3 s.f. Accept as a fraction e.g. $\frac{1213}{300}$ as long as it is half their answer to (b), or rounds correctly to it. |
**(1 mark)**
---
**(5 marks total)**\begin{enumerate}
\item (a) Given that
\end{enumerate}
$$y = \ln \left( 3 + x ^ { 2 } \right)$$
complete the table with the value of $y$ corresponding to $x = 3$, giving your answer to 4 significant figures.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | }
\hline
$\boldsymbol { x }$ & 2 & 2.5 & 3 & 3.5 & 4 & 4.5 & 5 \\
\hline
$\boldsymbol { y }$ & 1.946 & 2.225 & & 2.725 & 2.944 & 3.146 & 3.332 \\
\hline
\end{tabular}
\end{center}
In part (b) you must show all stages of your working.
\section*{Solutions relying entirely on calculator technology are not acceptable.}
(b) Use Simpson's rule with all the values of $y$ in the completed table to estimate, to 3 significant figures, the value of
$$\int _ { 2 } ^ { 5 } \ln \left( 3 + x ^ { 2 } \right) \mathrm { d } x$$
(c) Using your answer to part (b) and making your method clear, estimate the value of
$$\int _ { 2 } ^ { 5 } \ln \sqrt { \left( 3 + x ^ { 2 } \right) } \mathrm { d } x$$
\hfill \mbox{\textit{Edexcel FP1 2024 Q1 [5]}}