| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2006 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Trapezium rule estimation |
| Difficulty | Moderate -0.8 This is a straightforward application of the trapezium rule with clear parameters given (five ordinates, four strips). Part (a) requires only substitution into the standard formula with simple function evaluations, and part (b) tests basic understanding that more strips improve accuracy. Below average difficulty as it's purely procedural with no problem-solving or conceptual challenges. |
| Spec | 1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(h = 1\) | B1 | PI |
| Integral \(= \frac{h}{2}\{\ldots\}\) | M1 | OE summing of areas of the four trapezia. \([0.75+0.35+0.15+0.079\ldots]\) |
| \(\{\ldots\} = f(0) + f(4) + 2[f(1) + f(2) + f(3)]\) | ||
| \(= \left[1 + \frac{1}{17} + 2\left(\frac{1}{2} + \frac{1}{5} + \frac{1}{10}\right)\right]\) | A1 | Exact or to 3dp values. Condone one numerical slip |
| Integral \(= 1.329\) | A1 | CSO. Must be 1.329 |
| Total (a) | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Increase the number of ordinates | E1 | OE |
| Total (b) | 1 |
## Question 2(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $h = 1$ | B1 | PI |
| Integral $= \frac{h}{2}\{\ldots\}$ | M1 | OE summing of areas of the four trapezia. $[0.75+0.35+0.15+0.079\ldots]$ |
| $\{\ldots\} = f(0) + f(4) + 2[f(1) + f(2) + f(3)]$ | | |
| $= \left[1 + \frac{1}{17} + 2\left(\frac{1}{2} + \frac{1}{5} + \frac{1}{10}\right)\right]$ | A1 | Exact or to 3dp values. Condone one numerical slip |
| Integral $= 1.329$ | A1 | **CSO**. Must be 1.329 |
| **Total (a)** | **4** | |
## Question 2(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Increase the number of ordinates | E1 | OE |
| **Total (b)** | **1** | |
---2
\begin{enumerate}[label=(\alph*)]
\item Use the trapezium rule with five ordinates (four strips) to find an approximate value for
$$\int _ { 0 } ^ { 4 } \frac { 1 } { x ^ { 2 } + 1 } \mathrm {~d} x$$
giving your answer to four significant figures.
\item State how you could obtain a better approximation to the value of the integral using the trapezium rule.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2006 Q2 [5]}}