| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2012 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule with stated number of strips |
| Difficulty | Moderate -0.8 This is a straightforward application of the trapezium rule with clearly specified ordinates and strip width. Part (a) requires only substitution into the standard formula with basic calculator work, while part (b) tests recall of how to improve trapezium rule accuracy (use more strips). No problem-solving or conceptual insight needed—purely procedural execution of a standard numerical method. |
| Spec | 1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(h = 1\) | B1 | \(h=1\) stated or used (PI by x-values 0,1,2,3,4 provided no contradiction) |
| \(f(x) = \frac{2^x}{x+1}\); \(I \approx \frac{h}{2}\{f(0)+f(4)+2[f(1)+f(2)+f(3)]\}\) | M1 | OE summing of areas of the 'trapezia' |
| \(= 1 + \frac{16}{5} + 2\left(\frac{2}{2}+\frac{4}{3}+\frac{8}{4}\right) = 1+3.2+2(1+1.33\ldots+2)\) | A1 | OE Accept 1dp evidence. Can be implied by later correct work provided >1 term or a single term which rounds to 6.43 |
| \((I \approx) 0.5[4.2+2\times4.333..] = 6.43\) (to 3sf) | A1 (Total: 4) | CAO Must be 6.43 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Increase the number of ordinates | E1 (Total: 1) | OE e.g. increase the number of strips |
## Question 2:
**Part (a):**
| Working | Mark | Guidance |
|---------|------|----------|
| $h = 1$ | B1 | $h=1$ stated or used (PI by x-values 0,1,2,3,4 provided no contradiction) |
| $f(x) = \frac{2^x}{x+1}$; $I \approx \frac{h}{2}\{f(0)+f(4)+2[f(1)+f(2)+f(3)]\}$ | M1 | OE summing of areas of the 'trapezia' |
| $= 1 + \frac{16}{5} + 2\left(\frac{2}{2}+\frac{4}{3}+\frac{8}{4}\right) = 1+3.2+2(1+1.33\ldots+2)$ | A1 | OE Accept 1dp evidence. Can be implied by later correct work provided >1 term or a single term which rounds to 6.43 |
| $(I \approx) 0.5[4.2+2\times4.333..] = 6.43$ (to 3sf) | A1 (Total: 4) | CAO Must be 6.43 |
**Part (b):**
| Working | Mark | Guidance |
|---------|------|----------|
| Increase the number of ordinates | E1 (Total: 1) | OE e.g. increase the number of strips |
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\begin{enumerate}[label=(\alph*)]
\item Use the trapezium rule with five ordinates (four strips) to find an approximate value for
$$\int _ { 0 } ^ { 4 } \frac { 2 ^ { x } } { x + 1 } \mathrm {~d} x$$
giving your answer to three 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 2012 Q2 [5]}}