| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule applied to real-world data |
| Difficulty | Moderate -0.3 This is a straightforward C2 question combining trapezium rule application with basic interpretation. Part (i) is routine trapezium rule calculation, part (ii) tests understanding of over/underestimates with simple rectangle method, part (iii) is basic percentage calculation, and part (iv) is standard polynomial integration. All techniques are direct applications with no problem-solving insight required, making it slightly easier than average. |
| Spec | 1.08d Evaluate definite integrals: between limits1.09f Trapezium rule: numerical integration |
| Time \(( t\) seconds \()\) | 0 | 10 | 20 | 30 | 40 | 50 | 60 |
| Speed \(\left( v \mathrm {~m} \mathrm {~s} ^ { 1 } \right)\) | 28 | 19 | 14 | 11 | 12 | 16 | 22 |
**Question 1**
**i**
B1: 970 [m]
**ii**
B1: concave curve or line of traps is above curve
M3: for attempt at trap rule
- $\frac{1}{2} \times 10 \times (28 + 22 + 2[19 + 14 + 11 + 12 + 16])$
- M2 with 1 error, M1 with 2 errors
- Or M3 for 6 correct trapezia, M2 for 4 correct trapezia, M1 for 2 correct trapezia
A1: 830 to 880 incl. [m]
**iii**
B1: $t = 10$, $v = 19.5$
B1f.t.: Accept suitable sketch
M1: for 3 or more rectangles with values from curve
A1: model difference = 0.5 compared with 3% of 19 = 0.57
- Or $\frac{0.5}{19} \times 100 \approx 2.6$
**iv**
M1: $28t - \frac{1}{2}t^2 + 0.005t^3$ o.e.
- 2 terms correct, ignore $+ c$
M1: value at 60 [− value at 0]
- f.t. from integrated attempt with 3 terms
A1: 9601 Fig. 10 shows the speed of a car, in metres per second, during one minute, measured at 10 -second intervals.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{f56da008-e7f5-45b9-8db8-e2ba09ab0161-1_732_753_302_700}
\captionsetup{labelformat=empty}
\caption{Fig. 10}
\end{center}
\end{figure}
The measured speeds are shown below.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | }
\hline
Time $( t$ seconds $)$ & 0 & 10 & 20 & 30 & 40 & 50 & 60 \\
\hline
Speed $\left( v \mathrm {~m} \mathrm {~s} ^ { 1 } \right)$ & 28 & 19 & 14 & 11 & 12 & 16 & 22 \\
\hline
\end{tabular}
\end{center}
(i) Use the trapezium rule with 6 strips to find an estimate of the area of the region bounded by the curve, the line $t = 60$ and the axes. [This area represents the distance travelled by the car.]\\
(ii) Explain why your calculation in part (i) gives an overestimate for this area. Use appropriate rectangles to calculate an underestimate for this area.
The speed of the car may be modelled by $v = 28 - t + 0.015 t ^ { 2 }$.\\
(iii) Show that the difference between the value given by the model when $t = 10$ and the measured value is less than $3 \%$ of the measured value.\\
(iv) According to this model, the distance travelled by the car is
$$\int _ { 0 } ^ { 60 } \left( 28 \quad t + 0.015 t ^ { 2 } \right) \mathrm { d } t$$
Find this distance.
\hfill \mbox{\textit{OCR MEI C2 Q1 [12]}}