| Exam Board | OCR MEI |
|---|---|
| Module | Further Numerical Methods (Further Numerical Methods) |
| Year | 2019 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Numerical integration comparison |
| Difficulty | Standard +0.3 This is a straightforward numerical methods question requiring standard application of Simpson's rule using given midpoint and trapezium rule values. The formula M_n + (M_n - T_n)/3 is a standard result, and the question guides students through the process. While it requires careful calculation and understanding of error estimation for precision, it involves no novel problem-solving or conceptual difficulty beyond routine application of taught techniques. |
| Spec | 1.09f Trapezium rule: numerical integration |
| A | B | C | ||
| 1 | \(n\) | \(M _ { n }\) | \(T _ { n }\) | |
| 2 | 1 | 1.632527 | 1 | |
| 3 | 2 | 1.641461 | 1.316263 | |
| 4 | 4 | 1.623053 | 1.478862 | |
| 5 | 8 | 1.610295 | 1.550957 | |
| 6 | 16 | 1.604132 | 1.580626 | |
| 7 | 32 | 1.601505 | 1.592379 |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (a) | evidence of (M + T )/2 |
| Answer | Marks |
|---|---|
| =(B2+C2)/2 or =0.5*(B2+C2) | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| [2] | 1.1 | |
| 1.1 | need to see = for A1 | |
| 6 | (b) | extrapolation attempted with their S , their last |
| Answer | Marks |
|---|---|
| 1.60 is certain or 1.600 is probable | M1 |
| Answer | Marks |
|---|---|
| [8] | 3.1a |
| Answer | Marks |
|---|---|
| 2.2b | Simpson’s estimates |
| Answer | Marks |
|---|---|
| ratios | For method marks, at |
Question 6:
6 | (a) | evidence of (M + T )/2
1 1
=(B2+C2)/2 or =0.5*(B2+C2) | M1
A1
[2] | 1.1
1.1 | need to see = for A1
6 | (b) | extrapolation attempted with their S , their last
64
difference and their r
r = 0.37 to 0.4, difference = 0.0021665 to
0.00217 and S = 1.59846(3)
64
1.599735…to 1.59991
1.60 is certain or 1.600 is probable | M1
A1
M1
M1
M1
A1
M1
A1
[8] | 3.1a
1.1
2.1
1.1
3.2a
1.1
1.1
2.2b | Simpson’s estimates
All correct to 5 dp or better
Differences
ratios | For method marks, at
least two correct in each
case: estimates to 5 or 6
dp, differences and
ratios awrt to 3 dp or
better
if M0, allow SC2 for
1.598607 to 1.58961
obtained from
16×1.598463‒1.596297
oe
15
PPMMTT
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© OCR 20196 The spreadsheet output in Fig. 6 shows approximations to $\int _ { 0 } ^ { 1 } x ^ { - \sqrt { x } } \mathrm {~d} x$ found using the midpoint rule, denoted by $M _ { n }$, and the trapezium rule, denoted by $T _ { n }$.
\begin{table}[h]
\begin{center}
\begin{tabular}{ | c | c | c | c | l | }
\hline
& A & B & C & \\
\hline
1 & $n$ & $M _ { n }$ & $T _ { n }$ & \\
\hline
2 & 1 & 1.632527 & 1 & \\
\hline
3 & 2 & 1.641461 & 1.316263 & \\
\hline
4 & 4 & 1.623053 & 1.478862 & \\
\hline
5 & 8 & 1.610295 & 1.550957 & \\
\hline
6 & 16 & 1.604132 & 1.580626 & \\
\hline
7 & 32 & 1.601505 & 1.592379 & \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Fig. 6}
\end{center}
\end{table}
\begin{enumerate}[label=(\alph*)]
\item Write down an efficient spreadsheet formula for cell C3.
\item By first completing the table in the Printed Answer Booklet using the Simpson's rule, calculate the most accurate estimate of $\int _ { 0 } ^ { 1 } x ^ { - \sqrt { x } } \mathrm {~d} x$ that you can, justifying the precision quoted.
\section*{END OF QUESTION PAPER}
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Numerical Methods 2019 Q6 [10]}}