| Exam Board | OCR MEI |
|---|---|
| Module | Further Numerical Methods (Further Numerical Methods) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule applied to real-world data |
| Difficulty | Standard +0.8 This question tests understanding of numerical error propagation and floating-point arithmetic limitations. Part (a)(ii) requires conceptual insight about how relative errors behave under different operations without calculation. Part (b)(ii) demands understanding of machine precision and catastrophic cancellation in the computation of 10^x - 1 for very small x, which is a subtle numerical analysis concept beyond standard A-level content. |
| A | B | C | ||
| 1 | \(x\) | \(10 ^ { x }\) | \(\log _ { 10 } 10 ^ { x }\) | |
| 2 | \(1 \mathrm { E } - 12\) | 1 | \(1.00001 \mathrm { E } - 12\) | |
| 3 | \(1 \mathrm { E } - 11\) | 1 | \(9.99998 \mathrm { E } - 12\) |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (a) | (i) |
| Answer | Marks |
|---|---|
| awrt ‒0.000 843 cao | M1 |
| A1 | 1.1 |
| 1.1 | allow omission of 2 in numerator; allow bracket error; |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (a) | (ii) |
| operations is different | B1 | 2.4 |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (b) | (i) |
| 1.00001×10 | [1] | do not allow just eg the spreadsheet stores values to greater |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (b) | (ii) |
| Answer | Marks |
|---|---|
| to compute the values in C2 and C3 | B1 |
| B1 | 2.4 |
Question 6:
6 | (a) | (i) | or oe
2 2 2
�3.14 +2�−�𝜋𝜋 +2� �3.14 +2�
2 2
(𝜋𝜋 +2) (𝜋𝜋 +2) −1
awrt ‒0.000 843 cao | M1
A1 | 1.1
1.1 | allow omission of 2 in numerator; allow bracket error;
may be embedded in modulus
NB
allow B2 for cor−re4ct answer unsupported
6 | (a) | (ii) | it would be different as the order of
operations is different | B1 | 2.4
[1]
6 | (b) | (i) | −12 | B1 | 1.1 | or 0.000 000 000 001 000 01
1.00001×10 | [1] | do not allow just eg the spreadsheet stores values to greater
accuracy than it displays
do not allow just eg the spreadsheet stores values to greater
accuracy than it displays
6 | (b) | (ii) | not zero because the value stored in B2 and
B3 is not stored as 1, even though they are
displayed as 1
not equal to the [values in A2 and A3]
because the values in B2 and B3 are
different to each other (even though they are
displayed as 1), so different values are used
to compute the values in C2 and C3 | B1
B1 | 2.4
2.4
[2]6
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Calculate the relative error when $\pi$ is chopped to $\mathbf { 2 }$ decimal places in approximating
$$\pi ^ { 2 } + 2 .$$
\item Without doing any calculation, explain whether the relative error would be the same when $\pi$ is chopped to 2 decimal places when approximating $( \pi + 2 ) ^ { 2 }$.
The table shows some spreadsheet output. The values of $x$ in column A are exact.
\begin{center}
\begin{tabular}{ | c | c | c | c | l | }
\hline
& A & B & C & \\
\hline
1 & $x$ & $10 ^ { x }$ & $\log _ { 10 } 10 ^ { x }$ & \\
\hline
2 & $1 \mathrm { E } - 12$ & 1 & $1.00001 \mathrm { E } - 12$ & \\
\hline
3 & $1 \mathrm { E } - 11$ & 1 & $9.99998 \mathrm { E } - 12$ & \\
\hline
\end{tabular}
\end{center}
The formula in cell B2 is $= 10 ^ { \wedge } \mathrm { A } 2$.\\
This has been copied down to cell B3.\\
The formula in cell C2 is $\quad =$ LOG(B2) .\\
This formula has been copied down to cell C3.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Write the value displayed in cell C 2 in standard mathematical notation.
\item Explain why the values in cells C 2 and C 3 are neither zero nor the same as the values in cells A2 and A3 respectively.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Numerical Methods 2023 Q6 [6]}}