| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Simpson's rule application |
| Difficulty | Standard +0.3 This is a slightly easier than average C3 question. Part (i) requires standard differentiation of ln(composite function) and finding a tangent equation—routine techniques. Part (ii) is straightforward Simpson's Rule application with given strip count. Part (iii) is a simple geometric deduction (triangle area minus region A). No novel insight required, just methodical application of standard techniques. |
| Spec | 1.07l Derivative of ln(x): and related functions1.07m Tangents and normals: gradient and equations1.09f Trapezium rule: numerical integration |
\includegraphics{figure_8}
The diagram shows part of the curve $y = \ln(5 - x^2)$ which meets the $x$-axis at the point $P$ with coordinates $(2, 0)$. The tangent to the curve at $P$ meets the $y$-axis at the point $Q$. The region $A$ is bounded by the curve and the lines $x = 0$ and $y = 0$. The region $B$ is bounded by the curve and the lines $PQ$ and $x = 0$.
\begin{enumerate}[label=(\roman*)]
\item Find the equation of the tangent to the curve at $P$. [5]
\item Use Simpson's Rule with four strips to find an approximation to the area of the region $A$, giving your answer correct to 3 significant figures. [4]
\item Deduce an approximation to the area of the region $B$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR C3 Q8 [11]}}