| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2006 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Complete table then apply trapezium rule |
| Difficulty | Moderate -0.8 This is a straightforward application of the trapezium rule with a given formula and table structure. Part (a) requires simple calculator substitution, and part (b) is a routine trapezium rule calculation with 7 ordinates at equal intervals—a standard C2 exercise with no conceptual challenges beyond following the formula. |
| Spec | 1.09f Trapezium rule: numerical integration |
| \(t\) | 0 | 5 | 10 | 15 | 20 | 25 | 30 |
| \(v\) | 0 | 1.22 | 2.28 | 6.11 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(t = 15, 25, 30\) | B1 B1 B1 | (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| (b) \(S \approx \frac{1}{2} \times 5[0 + 15.37 + 2(1.22 + 2.28 + 3.80 + 6.11 + 9.72)]\) | B1 [M1] | |
| \(= \frac{5}{2}[61.63] = 154.075 = \) AWRT 154 | A1 | (3 marks) |
(a) $t = 15, 25, 30$ | B1 B1 B1 | (3 marks)
$v = 3.80, 9.72, 15.37$
(b) $S \approx \frac{1}{2} \times 5[0 + 15.37 + 2(1.22 + 2.28 + 3.80 + 6.11 + 9.72)]$ | B1 [M1]
$= \frac{5}{2}[61.63] = 154.075 = $ AWRT 154 | A1 | (3 marks)
**Guidance:**
- (a) S.C. Penalise AWRT these values once at first offence, thus the following marks could be AWRT 2 dp (Max 2/3)
**Total: 6 marks**
---\begin{enumerate}
\item The speed, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, of a train at time $t$ seconds is given by
\end{enumerate}
$$v = \sqrt { } \left( 1.2 ^ { t } - 1 \right) , \quad 0 \leqslant t \leqslant 30$$
The following table shows the speed of the train at 5 second intervals.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | }
\hline
$t$ & 0 & 5 & 10 & 15 & 20 & 25 & 30 \\
\hline
$v$ & 0 & 1.22 & 2.28 & & 6.11 & & \\
\hline
\end{tabular}
\end{center}
(a) Complete the table, giving the values of $v$ to 2 decimal places.
The distance, $s$ metres, travelled by the train in 30 seconds is given by
$$s = \int _ { 0 } ^ { 30 } \sqrt { } \left( 1.2 ^ { t } - 1 \right) \mathrm { d } t$$
(b) Use the trapezium rule, with all the values from your table, to estimate the value of $s$.\\
(3)\\
\hfill \mbox{\textit{Edexcel C2 2006 Q6 [6]}}