| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2006 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Over/underestimate justification with graph |
| Difficulty | Moderate -0.8 This is a straightforward application of the trapezium rule formula with evenly-spaced data points provided in a table, requiring only substitution and arithmetic. The reasoning about over/underestimate requires recognizing the curve's concavity from the graph, which is a standard C2 skill but simpler than typical problem-solving questions. |
| Spec | 1.09f Trapezium rule: numerical integration |
| \(x\) | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 |
| \(y\) | 8.2 | 6.4 | 5.5 | 5.0 | 4.7 | 4.4 | 4.2 |
| Answer | Marks | Guidance |
|---|---|---|
| 16.1 | 4 marks | M3 for \(\frac{1}{4}[8.2 + 4.2 + 2(6.4 + 5.5 + 5 + 4.7 + 4.4)]\); M2 for one slip/error; M1 for two slips/errors |
| overestimate + expn eg sketch | 1 mark | 5 marks total |
16.1 | 4 marks | M3 for $\frac{1}{4}[8.2 + 4.2 + 2(6.4 + 5.5 + 5 + 4.7 + 4.4)]$; M2 for one slip/error; M1 for two slips/errors
overestimate + expn eg sketch | 1 mark | 5 marks total4
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{15b8f97b-c058-409f-907f-cb0a6102abc4-2_615_971_1457_539}
\captionsetup{labelformat=empty}
\caption{Fig. 4}
\end{center}
\end{figure}
Fig. 4 shows a curve which passes through the points shown in the following table.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | }
\hline
$x$ & 1 & 1.5 & 2 & 2.5 & 3 & 3.5 & 4 \\
\hline
$y$ & 8.2 & 6.4 & 5.5 & 5.0 & 4.7 & 4.4 & 4.2 \\
\hline
\end{tabular}
\end{center}
Use the trapezium rule with 6 strips to estimate the area of the region bounded by the curve, the lines $x = 1$ and $x = 4$, and the $x$-axis.
State, with a reason, whether the trapezium rule gives an overestimate or an underestimate of the area of this region.
\hfill \mbox{\textit{OCR MEI C2 2006 Q4 [5]}}