| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Show root in interval |
| Difficulty | Standard +0.3 This is a structured multi-part question covering standard C3 techniques: verifying a root lies in an interval, rearranging equations, iterative methods, and integration. Each part guides students through the process with clear instructions. While it requires competence across several topics (exponentials, logarithms, numerical methods, integration), no individual step requires novel insight or particularly challenging problem-solving. It's slightly easier than a typical C3 question due to its scaffolded nature. |
| Spec | 1.06g Equations with exponentials: solve a^x = b1.08d Evaluate definite integrals: between limits1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
\includegraphics{figure_8}
The diagram shows part of each of the curves $y = e^{\frac{1}{3}x}$ and $y = \sqrt[3]{(3x + 8)}$. The curves meet, as shown in the diagram, at the point $P$. The region $R$, shaded in the diagram, is bounded by the two curves and by the $y$-axis.
\begin{enumerate}[label=(\roman*)]
\item Show by calculation that the $x$-coordinate of $P$ lies between 5.2 and 5.3. [3]
\item Show that the $x$-coordinate of $P$ satisfies the equation $x = \frac{2}{3} \ln(3x + 8)$. [2]
\item Use an iterative formula, based on the equation in part (ii), to find the $x$-coordinate of $P$ correct to 2 decimal places. [3]
\item Use integration, and your answer to part (iii), to find an approximate value of the area of the region $R$. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR C3 Q8 [16]}}