| Exam Board | Edexcel |
|---|---|
| Module | FD1 AS (Further Decision 1 AS) |
| Year | 2019 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule then exact integration comparison |
| Difficulty | Moderate -0.3 This question requires tracing through a numerical algorithm (trapezium rule) with given values and completing a table, then calculating percentage error. While it involves multiple steps, it's primarily mechanical execution requiring careful arithmetic rather than problem-solving or conceptual insight. The algorithm structure is explicitly given, making this easier than average A-level questions that require independent method selection or mathematical reasoning. |
| Spec | 1.08d Evaluate definite integrals: between limits1.09f Trapezium rule: numerical integration |
| Step 1 | Start |
| Step 2 | Input the values of A, B and N |
| Step 3 | Let H = (B - A) / N |
| Step 4 | Let C = H / 2 |
| Step 5 | Let D = 0 |
| Step 6 | Let D = D + A\(^4\) + B\(^4\) |
| Step 7 | Let E = A |
| Step 8 | Let E = E + H |
| Step 9 | If E = B go to Step 12 |
| Step 10 | Let D = D + 2 × E\(^4\) |
| Step 11 | Go to Step 8 |
| Step 12 | Let F = C × D |
| Step 13 | Output F |
| Step 14 | Stop |
| Answer | Marks |
|---|---|
| 2(a) | (i) |
| Answer | Marks |
|---|---|
| (ii) Final output = 50.5625 | M1 |
| Answer | Marks |
|---|---|
| A1 | 1.1b |
| Answer | Marks |
|---|---|
| (b) | 3 |
| Answer | Marks |
|---|---|
| 4.47% | B1 |
| Answer | Marks |
|---|---|
| A1ft | 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| A | B | N |
Question 2:
--- 2(a) ---
2(a) | (i)
A B N H C D E F
1 3 4 0.5 0.25 0
82 1
1.5
92.125 2
124.125 2.5
202.25 3 50.5625
(ii) Final output = 50.5625 | M1
A1
A1
A1 | 1.1b
1.1b
1.1b
1.1b
(4)
(b) | 3
∫ x4dx=48.4
1
50.5625−48.4
×100
48.4
4.47% | B1
M1
A1ft | 1.1b
1.1a
3.2b
(3)
(7 marks)
Notes
(a)(i)
M1: At least three rows of cells completed (so at least two values of D and E given) with either a
correct first row or 82 found for D – condone repeated values in all columns or a single value in each
row
A1: CAO – the values in the second, third and fourth rows correct (so up to the 92.125 in column D
and the 2 in column E) – accept exact equivalent fractions
A1: CAO – all values correct in columns A to E – accept exact equivalent fractions
(ii)
9
A1: CAO (output = 50.5625) (or equivalent e.g. 50 ) – allow if stated only in column F
16
(b)
B1: CAO (48.4)
M1: Correct method (including multiplying by 100) using candidate’s final output from (a)(ii) and
their value for I
A1ft: Follow through their final output from (a)(ii) (for reference: 4.4679752…) must be using 48.4
- dependent on M mark in (a) and percentage error being < 10% (answer must be given to 3
significant figures)
A | B | N | H | C | D | E | FThe following algorithm produces a numerical approximation for the integral
$$I = \int_A^B x^4 \, dx$$
\begin{tabular}{ll}
Step 1 & Start \\
Step 2 & Input the values of A, B and N \\
Step 3 & Let H = (B - A) / N \\
Step 4 & Let C = H / 2 \\
Step 5 & Let D = 0 \\
Step 6 & Let D = D + A$^4$ + B$^4$ \\
Step 7 & Let E = A \\
Step 8 & Let E = E + H \\
Step 9 & If E = B go to Step 12 \\
Step 10 & Let D = D + 2 × E$^4$ \\
Step 11 & Go to Step 8 \\
Step 12 & Let F = C × D \\
Step 13 & Output F \\
Step 14 & Stop
\end{tabular}
For the case when A = 1, B = 3 and N = 4,
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item complete the table in the answer book to show the results obtained at each step of the algorithm.
\item State the final output. [4]
\end{enumerate}
\item Calculate, to 3 significant figures, the percentage error between the exact value of $I$ and the value obtained from using the approximation to $I$ in this case. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FD1 AS 2019 Q2 [7]}}