| Exam Board | Edexcel |
|---|---|
| Module | PMT Mocks (PMT Mocks) |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Staircase/cobweb diagram |
| Difficulty | Standard +0.3 This is a standard fixed-point iteration question requiring students to (a) verify a root lies in an interval using sign changes or direct substitution, and (b) use a cobweb/staircase diagram to determine convergence. Both parts are routine A-level techniques with clear graphical support provided, making it slightly easier than average. |
| Spec | 1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks |
|---|---|
| Both values correct to at least 1 significant figure with correct explanation and conclusion | A1 |
| M1 | Attempts \(f(2) = \ldots\) and \(f(3) = \ldots\) where \(f(x) = \pm(2 + \ln(4-x) - x)\) |
| Answer | Marks | Guidance |
|---|---|---|
| The cobweb spirals inwards / converges to the root | A1 | For a correct attempt starting at 3 and deducing that the iteration formula can be used to find an approximation for \(\beta\) because 'the cobweb spirals inwards' or 'the cobweb gets closer to the root' or 'the cobweb converges' |
| M1 | For an attempt at using a cobweb diagram starting at \(x_1 = 3\). It should have at least two spirals. |
### Part a:
Both values correct to at least 1 significant figure with correct explanation and conclusion | A1 |
| M1 | Attempts $f(2) = \ldots$ and $f(3) = \ldots$ where $f(x) = \pm(2 + \ln(4-x) - x)$
### Part b:
The cobweb spirals inwards / converges to the root | A1 | For a correct attempt starting at 3 and deducing that the iteration formula can be used to find an approximation for $\beta$ because 'the cobweb spirals inwards' or 'the cobweb gets closer to the root' or 'the cobweb converges'
| M1 | For an attempt at using a cobweb diagram starting at $x_1 = 3$. It should have at least two spirals.4. The curve with equation $y = 2 + \ln ( 4 - x )$ meets the line $y = x$ at a single point, $x = \beta$.\\
a. Show that $2 < \beta < 3$
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{63d85737-99d4-4916-a479-fe44f77b1505-07_961_1002_296_513}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Figure 2 shows the graph of $y = 2 + \ln ( 4 - x )$ and the graph of $y = x$.\\
A student uses the iteration formula
$$x _ { n + 1 } = 2 + \ln \left( 4 - x _ { n } \right) , \quad n \in N ,$$
in an attempt to find an approximation for $\beta$.\\
Using the graph and starting with $x _ { 1 } = 3$,\\
b. determine whether the or not this iteration formula can be used to find an approximation for $\beta$, justifying your answer.\\
\hfill \mbox{\textit{Edexcel PMT Mocks Q4 [4]}}