| Exam Board | Edexcel |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2018 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Compare iteration convergence |
| Difficulty | Standard +0.3 This question requires showing α lies in an interval by substitution (routine), then using a cobweb diagram to determine convergence of an iteration. While it involves understanding fixed point iteration graphically, the reasoning is straightforward: checking if the iteration converges or diverges from the diagram. This is a standard Further Maths topic tested in a predictable way with minimal problem-solving required. |
| Spec | 1.06d Natural logarithm: ln(x) function and properties1.09a Sign change methods: locate roots1.09b Sign change methods: understand failure cases1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Attempts \(f(3) =\) and \(f(4) =\) where \(f(x) = \pm(2\ln(8-x)-x)\) | M1 | 2.1 |
| \(f(3) = (2\ln 5 - 3) = (+)0.22\) and \(f(4) = (2\ln 4 - 4) = -1.23\); change of sign and function continuous in \([3,4] \Rightarrow\) Root | A1* | 2.4 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Annotating graph with cobweb diagram starting at \(x_1 = 4\), at least two spirals | M1 | 2.4 |
| Deduces iteration formula can be used to find approximation for \(\alpha\) because the cobweb spirals inwards | A1 | 2.2a |
# Question 4(a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Attempts $f(3) =$ and $f(4) =$ where $f(x) = \pm(2\ln(8-x)-x)$ | M1 | 2.1 |
| $f(3) = (2\ln 5 - 3) = (+)0.22$ and $f(4) = (2\ln 4 - 4) = -1.23$; change of sign and function continuous in $[3,4] \Rightarrow$ Root | A1* | 2.4 |
**Notes:** M1 for attempting $f(3)$ and $f(4)$ — not routine, cannot be scored by substituting 3 and 4 in both functions separately. A1: Both values correct to at least 1 sf, correct explanation and conclusion. Allow reasons $2\ln 8 = 3.21 > 3$, $2\ln 4 = 2.77 < 4$ or similar.
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# Question 4(b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Annotating graph with cobweb diagram starting at $x_1 = 4$, at least two spirals | M1 | 2.4 |
| Deduces iteration formula **can be used** to find approximation for $\alpha$ because **the cobweb spirals inwards** | A1 | 2.2a |
**Notes:** M1 for 5 or more correct straight lines in cobweb diagram. If no graph then M0 A0. A1 requires correct attempt starting at 4, deducing it **can be used** as iterations converge to root. Suitable reasons: "spirals inwards", "gets closer to the root", "it converges".
---\begin{enumerate}
\item The curve with equation $y = 2 \ln ( 8 - x )$ meets the line $y = x$ at a single point, $x = \alpha$.\\
(a) Show that $3 < \alpha < 4$
\end{enumerate}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{b5f50f17-9f1b-4b4c-baf3-e50de5f2ea9c-08_666_1061_445_502}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Figure 2 shows the graph of $y = 2 \ln ( 8 - x )$ and the graph of $y = x$.\\
A student uses the iteration formula
$$x _ { n + 1 } = 2 \ln \left( 8 - x _ { n } \right) , \quad n \in \mathbb { N }$$
in an attempt to find an approximation for $\alpha$.\\
Using the graph and starting with $x _ { 1 } = 4$\\
(b) determine whether or not this iteration formula can be used to find an approximation for $\alpha$, justifying your answer.
\hfill \mbox{\textit{Edexcel Paper 1 2018 Q4 [4]}}