| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Complete table then apply trapezium rule |
| Difficulty | Moderate -0.3 This is a straightforward application of the trapezium rule requiring table completion with calculator work and a standard concavity argument. While it involves logarithms and composite functions, the question is entirely procedural with no problem-solving or insight required—slightly easier than a typical C2 question due to its routine nature. |
| 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 |
\includegraphics{figure_4}
The diagram shows the curve with equation $y = (x - \log_{10} x)^2$, $x > 0$.
\begin{enumerate}[label=(\roman*)]
\item Copy and complete the table below for points on the curve, giving the $y$ values to 2 decimal places.
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$x$ & 2 & 3 & 4 & 5 & 6 \\
\hline
$y$ & 2.89 & 6.36 & & & \\
\hline
\end{tabular}
[2]
\end{enumerate}
The shaded region is bounded by the curve, the $x$-axis and the lines $x = 2$ and $x = 6$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Use the trapezium rule with all the values in your table to estimate the area of the shaded region. [3]
\item State, with a reason, whether your answer to part (b) is an under-estimate or an over-estimate of the true area. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR C2 Q4 [7]}}