| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Complete table then apply trapezium rule |
| Difficulty | Moderate -0.8 This is a straightforward numerical methods question requiring table completion by substituting values into a given function, applying the trapezium rule formula (a standard C2 technique), and stating whether the estimate is an over/under-estimate based on curve concavity. All steps are routine with no problem-solving or insight required beyond direct application of learned procedures. |
| Spec | 1.06d Natural logarithm: ln(x) function and properties1.09f Trapezium rule: numerical integration |
| \(x\) | 2 | 3 | 4 | 5 | 6 |
| \(y\) | 2.89 | 6.36 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Table with \(x\): 2, 3, 4, 5, 6 and \(y\): 2.89, 6.36, 11.55, 18.50, 27.27 | B2 | |
| (b) area \(= \frac{1}{2} \times 1 \times [2.89 + 27.27 + 2(6.36 + 11.55 + 18.50)]\) | B1 M1 A1 | |
| \(= 51.5\) (3sf) | A1 | |
| (c) over-estimate | B1 | |
| the curve passes below the top edge of each trapezium | B1 | (8 marks) |
**(a)** Table with $x$: 2, 3, 4, 5, 6 and $y$: 2.89, 6.36, 11.55, 18.50, 27.27 | B2 |
**(b)** area $= \frac{1}{2} \times 1 \times [2.89 + 27.27 + 2(6.36 + 11.55 + 18.50)]$ | B1 M1 A1 |
$= 51.5$ (3sf) | A1 |
**(c)** over-estimate | B1 |
the curve passes below the top edge of each trapezium | B1 | (8 marks)6.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{857bf144-b03e-4b46-b043-1119b30f9e78-3_572_954_246_497}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows the curve with equation $y = \left( x - \log _ { 10 } x \right) ^ { 2 } , x > 0$.
\begin{enumerate}[label=(\alph*)]
\item Copy and complete the table below for points on the curve, giving the $y$ values to 2 decimal places.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & 2 & 3 & 4 & 5 & 6 \\
\hline
$y$ & 2.89 & 6.36 & & & \\
\hline
\end{tabular}
\end{center}
The shaded area is bounded by the curve, the $x$-axis and the lines $x = 2$ and $x = 6$.
\item Use the trapezium rule with all the values in your table to estimate the area of the shaded region.
\item State, with a reason, whether your answer to part (b) is an under-estimate or an over-estimate of the true area.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q6 [8]}}