June 2018 Paper 1 Q4
4. The curve with equation \(y = 2\ln(8 - x)\) meets the line \(y = x\) at a single point, \(x = \alpha\).
Figure 2 shows the graph of \(y = 2\ln(8 - x)\) and the graph of \(y = x\).
A student uses the iteration formula
\[x_{n+1} = 2\ln(8 - x_n), \quad n \in \mathbb{N}\]in an attempt to find an approximation for \(\alpha\).
Using the graph and starting with \(x_1 = 4\)
| Scheme | Marks | AO |
|---|---|---|
| Attempts \(\mathrm{f}(3) =\) and \(\mathrm{f}(4) =\) where \(\mathrm{f}(x) = \pm\left(2\ln(8 - x) - x\right)\) | M1 | 2.1 |
| \(\mathrm{f}(3) = \left(2\ln(5) - x\right) = (+)0.22\) and \(\mathrm{f}(4) = \left(2\ln(4) - 4\right) = -1.23\) Change of sign and function continuous in interval \([3, 4] \Rightarrow\) Root * | A1* | 2.4 |
| (2) |
Notes
M1: Attempts \(\mathrm{f}(3) =\) and \(\mathrm{f}(4) =\) where \(\mathrm{f}(x) = \pm\left(2\ln(8 - x) - x\right)\) or alternatively compares \(2\ln 5\) to 3 and \(2\ln 4\) to 4. This is not routine and cannot be scored by substituting 3 and 4 in both functions
A1: Both values (calculations) correct to at least 1 sf with correct explanation and conclusion. (See underlined statements)
When comparing terms, allow reasons to be \(2\ln 5 = 3.21 \gt 3\), \(2\ln 4 = 2.77 \lt 4\) or similar
(corrected from the printed mark scheme: it prints \(2\ln 8 = 3.21 \gt 3\); the value 3.21 is \(2\ln 5\))
| Scheme | Marks | AO |
|---|---|---|
| For annotating the graph by drawing a cobweb diagram starting at \(x_1 = 4\) It should have at least two spirals | M1 | 2.4 |
| Deduces that the iteration formula can be used to find an approximation for \(\alpha\) because the cobweb spirals inwards for the cobweb diagram | A1 | 2.2a |
| (2) | ||
| (4 marks) |
Notes
M1: For an attempt at using a cobweb diagram. Look for 5 or more correct straight lines. It may not start at 4 but it must show an understanding of the method. If there is no graph then it is M0 A0
A1: For a correct attempt starting at 4 and deducing that the iteration can be used as the iterations converge to the root. You must statement that it can be used with a suitable reason. Suitable reasons could be " it spirals inwards", it gets closer to the root", it converges "
