| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Complete table then apply trapezium rule |
| Difficulty | Easy -1.2 This is a straightforward C2 question requiring basic substitution into a simple function, standard application of the trapezium rule with given ordinates, and recognition that the trapezium rule overestimates for a concave function. All parts are routine textbook exercises with no problem-solving or insight required. |
| Spec | 1.02b Surds: manipulation and rationalising denominators1.09f Trapezium rule: numerical integration |
| \(x\) | 2 | 4 | 6 | 8 | 10 |
| \(y\) | \(\sqrt { } 3\) | \(\sqrt { } 11\) | \(\sqrt { } 19\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sqrt{7}\) and \(\sqrt{15}\) | B1 | Both \(\sqrt{7}\) and \(\sqrt{15}\). Allow awrt 2.65 and 3.87 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\text{Area}(R)\approx\frac{1}{2}\times2\times\left\{\sqrt{3}+2\left(\sqrt{7}+\sqrt{11}+\sqrt{15}\right)+\sqrt{19}\right\}\) | B1; M1 | Outside brackets \(\frac{1}{2}\times2\) or 1 (may be implied); For structure of \(\{\ldots\ldots\ldots\}\) |
| \(=1\times25.76166865\ldots=25.76\quad(2\text{dp})\) | A1 cao | 25.76 |
| Answer | Marks | Guidance |
|---|---|---|
| underestimate | B1 | Accept 'under', 'less than' etc. |
# Question 3(a):
| $\sqrt{7}$ and $\sqrt{15}$ | B1 | Both $\sqrt{7}$ and $\sqrt{15}$. Allow awrt 2.65 and 3.87 |
**[1]**
---
# Question 3(b):
| $\text{Area}(R)\approx\frac{1}{2}\times2\times\left\{\sqrt{3}+2\left(\sqrt{7}+\sqrt{11}+\sqrt{15}\right)+\sqrt{19}\right\}$ | B1; M1 | Outside brackets $\frac{1}{2}\times2$ or 1 (may be implied); For structure of $\{\ldots\ldots\ldots\}$ |
| $=1\times25.76166865\ldots=25.76\quad(2\text{dp})$ | A1 cao | 25.76 |
**[3]**
---
# Question 3(c):
| underestimate | B1 | Accept 'under', 'less than' etc. |
**[1] [Total 5]**
---3.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{f07cc9ed-a820-46c8-a3a3-3c780cf20fa7-05_821_1273_118_338}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows a sketch of part of the curve with equation $y = \sqrt { } ( 2 x - 1 ) , x \geqslant 0.5$
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 $x = 10$.
The table below shows corresponding values of $x$ and $y$ for $y = \sqrt { } ( 2 x - 1 )$.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & 2 & 4 & 6 & 8 & 10 \\
\hline
$y$ & $\sqrt { } 3$ & & $\sqrt { } 11$ & & $\sqrt { } 19$ \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Complete the table with the values of $y$ corresponding to $x = 4$ and $x = 8$.
\item Use the trapezium rule, with all the values of $y$ in the completed table, to find an approximate value for the area of $R$, giving your answer to 2 decimal places.
\item State whether your approximate value in part (b) is an overestimate or an underestimate for the area of $R$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 2014 Q3 [5]}}