| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2015 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Deduce related integral from numerical approximation |
| Difficulty | Moderate -0.3 Part (a) is a standard trapezium rule application with values provided in a table—pure procedural calculation. Part (b) requires recognizing that the integrand can be split and that the second term is half the original function, making this a straightforward manipulation once part (a) is complete. This is slightly easier than average due to the mechanical nature and clear structure, though it does require basic integral properties. |
| Spec | 1.09f Trapezium rule: numerical integration |
| \(x\) | 2 | 5 | 8 | 11 |
| \(y\) | 8.485 | 2.502 | 1.524 | 1.100 |
| Answer | Marks | Guidance |
|---|---|---|
| \(x\) | 2 | 5 |
| \(y\) | 8.485 | 2.502 |
| State \(h=3\), or use of \(\frac{1}{2}\times 3\) | B1 aef | |
| \(\{8.485 + 1.100 + 2(2.502 + 1.524)\}\) | M1A1 | For structure of \(\{\ldots\}\) |
| \(\frac{1}{2}\times 3 \times\{17.637\} = 26.4555 \approx 26.46\) | A1 [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Adds \(9+\ldots\) | M1 | |
| Half of their answer from (a) | M1 | Allow use of half of 26.4555 |
| \(9 + 13.23 = 22.23\) | A1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| \(x\) | 2 | 5 |
| \(y\) | 5.2425 | 2.251 |
| Uses \(\frac{1}{2}\times 3\times\{5.2425+1.550+2(2.251+1.762)\}\) | M1, M1 | |
| \(= 22.23\) | A1 [3] |
# Question 2:
## Part (a)
| $x$ | 2 | 5 | 8 | 11 |
| $y$ | 8.485 | 2.502 | 1.524 | 1.100 |
State $h=3$, or use of $\frac{1}{2}\times 3$ | B1 aef |
$\{8.485 + 1.100 + 2(2.502 + 1.524)\}$ | M1A1 | For structure of $\{\ldots\}$
$\frac{1}{2}\times 3 \times\{17.637\} = 26.4555 \approx 26.46$ | A1 [4] |
## Part (b) Way 1:
Adds $9+\ldots$ | M1 |
Half of their answer from (a) | M1 | Allow use of half of 26.4555
$9 + 13.23 = 22.23$ | A1 [3] |
## Part (b) Way 2:
| $x$ | 2 | 5 | 8 | 11 |
| $y$ | 5.2425 | 2.251 | 1.762 | 1.550 |
Uses $\frac{1}{2}\times 3\times\{5.2425+1.550+2(2.251+1.762)\}$ | M1, M1 |
$= 22.23$ | A1 [3] |
---\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{3b99072a-cd16-4c1d-9e44-085926a3ba24-03_473_654_233_603}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows a sketch of part of the graph of $y = \frac { 12 } { \sqrt { \left( x ^ { 2 } - 2 \right) } } , x \geqslant 2$\\
The table below gives values of $y$ rounded to 3 decimal places.
\begin{center}
\begin{tabular}{ | l | l | l | l | l | }
\hline
$x$ & 2 & 5 & 8 & 11 \\
\hline
$y$ & 8.485 & 2.502 & 1.524 & 1.100 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Use the trapezium rule with all the values of $y$ from the table to find an approximate value, to 2 decimal places, for
$$\int _ { 2 } ^ { 11 } \frac { 12 } { \sqrt { \left( x ^ { 2 } - 2 \right) } } \mathrm { d } x$$
\item Use your answer to part (a) to estimate a value for
$$\int _ { 2 } ^ { 11 } \left( 1 + \frac { 6 } { \sqrt { \left( x ^ { 2 } - 2 \right) } } \right) d x$$
\end{enumerate}
\hfill \mbox{\textit{Edexcel C12 2015 Q2 [7]}}