| Exam Board | OCR MEI |
|---|---|
| Module | Further Numerical Methods (Further Numerical Methods) |
| Year | 2021 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule applied to real-world data |
| Difficulty | Standard +0.3 This question tests understanding of rounding vs truncation errors through a practical context. Parts (a)-(b) involve straightforward maximum error calculations (±0.50 per item). Parts (c)-(d) require recognizing that truncation creates systematic bias. Parts (e)-(f) extend the pattern but remain conceptually accessible. While multi-part, each step follows logically from understanding basic error propagation—no complex algebra or novel problem-solving required. Slightly easier than average due to the guided structure and real-world context that makes the concepts intuitive. |
| Spec | 1.09g Numerical methods in context |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (a) | 48×0.5 soi |
| £24 | M1 | |
| A1 | 3.3 |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (b) | consistent because 1.77 < 24 |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (c) | 52×0.495 |
| £25.74 | M1 | |
| A1 | 3.3 |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (d) | this could happen if a large number of items |
| Answer | Marks | Guidance |
|---|---|---|
| eg the mean error per item was 52.38p | B1 | 3.5a |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (e) | mpe = £0.99n |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (f) | expected error for Nina’s model is £0 since you |
| Answer | Marks |
|---|---|
| “estimated cost” + £0.495n | B1 |
| B1 | 2.4 |
| 3.5c | U6 |
Question 5:
5 | (a) | 48×0.5 soi
£24 | M1
A1 | 3.3
3.4
[2]
5 | (b) | consistent because 1.77 < 24 | B1 | 2.4 | allow consistent because error < mpe
[1]
5 | (c) | 52×0.495
£25.74 | M1
A1 | 3.3
3.4
[2]
5 | (d) | this could happen if a large number of items
eg cost less than £1
eg cost £1.99 or £2.99 etc
eg more than 50p over the pound
eg the mean error per item was 52.38p | B1 | 3.5a
[1]
5 | (e) | mpe = £0.99n | B1 | 3.4 | condone omission of units, allow
99n pence
[1]
5 | (f) | expected error for Nina’s model is £0 since you
would expect to round half the prices up and half
down oe
or
expected error in Kareem’s model is ‒£0.495n since
you would expect the average “chop” to be 49.5p oe
so new model should be
“estimated cost” + £0.495n | B1
B1 | 2.4
3.5c | U6
[2]5 When Nina does the weekly grocery shopping she models the total cost by adding up the cost of each item in her head as she goes along. To simplify matters she rounds the cost of each item to the nearest pound.
One week Nina buys 48 items.
\begin{enumerate}[label=(\alph*)]
\item Calculate the maximum possible error in Nina's model in this case.
Nina estimated the total cost of her shopping to be $\pounds 92$. The actual cost is $\pounds 90.23$.
\item Explain whether this is consistent with Nina's model.
The next week her husband, Kareem, does the weekly shopping. He models the total cost by chopping the cost of each item to the nearest pound as he goes along.
On this occasion Kareem buys 52 items.
\item Calculate the expected error in Kareem's model in this case.
Using his model Kareem estimates the total cost as $\pounds 76$.
The total cost of the shopping is $\pounds 103.24$.
\item Explain how such a large error could arise.
The next week Kareem buys $n$ items.
\item Write down a formula for the maximum possible error when Kareem uses his model to estimate the total cost of his shopping.
\item Explain how Kareem's model could be adapted so that his formula gives the same expected error as Nina's model when they are both used to estimate the total cost of the shopping.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Numerical Methods 2021 Q5 [9]}}