| Exam Board | Edexcel |
|---|---|
| Module | C34 (Core Mathematics 3 & 4) |
| Year | 2016 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule then exact integration comparison |
| Difficulty | Standard +0.3 This is a straightforward three-part question testing standard integration techniques. Part (a) requires routine application of the trapezium rule formula with given strip width. Part (b) involves a standard integration using substitution or recognition of a derivative form (integrating (2x+5)^{-1/2}). Part (c) is simple arithmetic comparing the two answers. While it requires multiple techniques, each step is procedural with no novel insight needed, making it slightly easier than average. |
| Spec | 1.08d Evaluate definite integrals: between limits1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Strip width \(= 1\) | B1 | May be implied by trapezium rule |
| \(\text{Area} \approx \frac{1}{2}\left(\frac{1}{\sqrt{9}} + \frac{1}{\sqrt{15}} + 2\left(\frac{1}{\sqrt{11}} + \frac{1}{\sqrt{13}}\right)\right)\) | M1 A1 | M1: Correct structure for \(y\) values: \((y\) at \(x=2) + (y\) at \(x=5) + 2(\text{sum of other } y \text{ values})\). A1: Correct numerical expression. If decimals used, look for awrt 1dp |
| Awrt \(0.875\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\int \frac{1}{\sqrt{2x+5}}\,dx = (2x+5)^{\frac{1}{2}}\) | M1 A1 | M1: \(\int \frac{1}{\sqrt{2x+5}}\,dx = k(2x+5)^{\frac{1}{2}}\). A1: \(k=1\) |
| \(\int_2^5 \frac{1}{\sqrt{2x+5}}\,dx = (2(5)+5)^{\frac{1}{2}} - (2(2)+5)^{\frac{1}{2}}\) | dM1 | Substitutes 5 and 2 and subtracts correct way round. May be implied by correct exact answer but not by decimal answer only e.g. \(0.8729\ldots\) |
| \(= \sqrt{15} - \sqrt{9}\ (= \sqrt{15} - 3)\) | A1 | \(\sqrt{15} - \sqrt{9}\) or \(\sqrt{15} - 3\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\pm(\text{correct}(a) - \text{correct}(b)) = \pm 0.002\) or \(\pm\frac{\text{correct}(a) - \text{correct}(b)}{\text{correct}(b)} \times 100 = \pm 0.2\%\) | B1 | Finds magnitude of error and writes as \(\pm 0.002\) or \(\pm 2\times 10^{-3}\) or \(\pm 0.2\%\) |
| Total: 1 mark | Total: 9 marks |
## Question 7:
### Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Strip width $= 1$ | B1 | May be implied by trapezium rule |
| $\text{Area} \approx \frac{1}{2}\left(\frac{1}{\sqrt{9}} + \frac{1}{\sqrt{15}} + 2\left(\frac{1}{\sqrt{11}} + \frac{1}{\sqrt{13}}\right)\right)$ | M1 A1 | M1: Correct structure for $y$ values: $(y$ at $x=2) + (y$ at $x=5) + 2(\text{sum of other } y \text{ values})$. A1: Correct numerical expression. If decimals used, look for awrt 1dp |
| Awrt $0.875$ | A1 | |
**Total: 4 marks**
### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\int \frac{1}{\sqrt{2x+5}}\,dx = (2x+5)^{\frac{1}{2}}$ | M1 A1 | M1: $\int \frac{1}{\sqrt{2x+5}}\,dx = k(2x+5)^{\frac{1}{2}}$. A1: $k=1$ |
| $\int_2^5 \frac{1}{\sqrt{2x+5}}\,dx = (2(5)+5)^{\frac{1}{2}} - (2(2)+5)^{\frac{1}{2}}$ | dM1 | Substitutes 5 and 2 and subtracts correct way round. May be implied by correct exact answer but **not** by decimal answer only e.g. $0.8729\ldots$ |
| $= \sqrt{15} - \sqrt{9}\ (= \sqrt{15} - 3)$ | A1 | $\sqrt{15} - \sqrt{9}$ or $\sqrt{15} - 3$ |
**Total: 4 marks**
### Part (c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\pm(\text{correct}(a) - \text{correct}(b)) = \pm 0.002$ or $\pm\frac{\text{correct}(a) - \text{correct}(b)}{\text{correct}(b)} \times 100 = \pm 0.2\%$ | B1 | Finds magnitude of error and writes as $\pm 0.002$ or $\pm 2\times 10^{-3}$ or $\pm 0.2\%$ |
**Total: 1 mark | Total: 9 marks**
---7.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{8d67f716-c8af-4460-8a6b-62073ba9b825-13_695_986_121_497}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Diagram not drawn to scale
Figure 1 shows a sketch of part of the curve with equation $y = \frac { 1 } { \sqrt { 2 x + 5 } } , x > - 2.5$\\
The finite region $R$, shown shaded in Figure 1, is bounded by the curve, the $x$-axis and the lines with equations $x = 2$ and $x = 5$
\begin{enumerate}[label=(\alph*)]
\item Use the trapezium rule with three strips of equal width to find an estimate for the area of $R$, giving your answer to 3 decimal places.
\item Use calculus to find the exact area of $R$.
\item Hence calculate the magnitude of the error of the estimate found in part (a), giving your answer to one significant figure.
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel C34 2016 Q7 [9]}}