| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Rearrange to iterative form |
| Difficulty | Standard +0.3 This is a standard C3 question combining logarithms, normals, and iterative methods. Part (a) is routine sketching, part (b) requires finding the normal gradient and forming an equation (straightforward calculus), parts (c-d) involve algebraic rearrangement and calculator iteration. All techniques are standard textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 1.06d Natural logarithm: ln(x) function and properties1.07l Derivative of ln(x): and related functions1.07m Tangents and normals: gradient and equations1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
The curve with equation $y = \ln 3x$ crosses the $x$-axis at the point $P(p, 0)$.
\begin{enumerate}[label=(\alph*)]
\item Sketch the graph of $y = \ln 3x$, showing the exact value of $p$. [2]
\end{enumerate}
The normal to the curve at the point $Q$, with $x$-coordinate $q$, passes through the origin.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Show that $x = q$ is a solution of the equation $x^2 + \ln 3x = 0$. [4]
\item Show that the equation in part (b) can be rearranged in the form $x = \frac{1}{3}e^{-x^2}$. [2]
\item Use the iteration formula $x_{n + 1} = \frac{1}{3}e^{-x_n^2}$, with $x_0 = \frac{1}{3}$, to find $x_1, x_2, x_3$ and $x_4$. Hence write down, to 3 decimal places, an approximation for $q$. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q8 [11]}}