| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2015 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule applied to real-world data |
| Difficulty | Moderate -0.8 This is a straightforward application of the trapezium rule with clearly provided data points. Students simply need to recognize they should find the area between two curves by calculating (upper surface area) - (lower surface area), or equivalently sum the differences in y-values. The arithmetic is routine with no conceptual challenges beyond basic trapezium rule recall. |
| Spec | 1.05b Sine and cosine rules: including ambiguous case1.05c Area of triangle: using 1/2 ab sin(C)1.09f Trapezium rule: numerical integration |
| Upper surface | Lower surface | ||
| \(x\) | \(y\) | \(x\) | \(y\) |
| 0 | 0 | 0 | 0 |
| 4 | 1.45 | 4 | -0.85 |
| 8 | 1.56 | 8 | -0.76 |
| 12 | 1.27 | 12 | -0.55 |
| 16 | 1.04 | 16 | -0.30 |
| 20 | 0 | 20 | 0 |
9
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{5c7ac296-a911-451b-ad18-5ade3ac23e74-3_253_1486_328_292}
\captionsetup{labelformat=empty}
\caption{Fig. 9.1}
\end{center}
\end{figure}
(i) Jean is designing a model aeroplane. Fig. 9.1 shows her first sketch of the wing's cross-section. Calculate angle A and the area of the cross-section.\\
(ii) Jean then modifies her design for the wing. Fig. 9.2 shows the new cross-section, with 1 unit for each of $x$ and $y$ representing one centimetre.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{5c7ac296-a911-451b-ad18-5ade3ac23e74-3_431_1682_970_194}
\captionsetup{labelformat=empty}
\caption{Fig. 9.2}
\end{center}
\end{figure}
Here are some of the coordinates that Jean used to draw the new cross-section.
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
\multicolumn{2}{|l|}{Upper surface} & \multicolumn{2}{|l|}{Lower surface} \\
\hline
$x$ & $y$ & $x$ & $y$ \\
\hline
0 & 0 & 0 & 0 \\
\hline
4 & 1.45 & 4 & -0.85 \\
\hline
8 & 1.56 & 8 & -0.76 \\
\hline
12 & 1.27 & 12 & -0.55 \\
\hline
16 & 1.04 & 16 & -0.30 \\
\hline
20 & 0 & 20 & 0 \\
\hline
\end{tabular}
\end{center}
Use the trapezium rule with trapezia of width 4 cm to calculate an estimate of the area of this cross-section.
\hfill \mbox{\textit{OCR MEI C2 2015 Q9 [11]}}