| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2005 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Rearrange to iterative form |
| Difficulty | Standard +0.3 This is a straightforward fixed-point iteration question requiring standard techniques: sketching graphs to show uniqueness, verifying a root interval by substitution, algebraic rearrangement (trivial exponential manipulation), and applying a given iterative formula. All steps are routine with no novel insight required, making it slightly easier than average. |
| Spec | 1.02m Graphs of functions: difference between plotting and sketching1.06d Natural logarithm: ln(x) function and properties1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Make recognizable sketch of a relevant graph, e.g. \(y = 1/x\) | B1 | |
| (ii) Sketch an appropriate second graph, e.g. \(y = \ln x\), correctly and justify the given statement | B1 | 2 |
| (iii) Consider the sign of \(x\) in \(x = 1\) and \(x = 2\), or equivalent | M1 | |
| Complete the argument correctly with appropriate calculations | A1 | 2 |
| (iii) Show that the given equation is equivalent to \(1/x = \ln x\), or vice versa | B1 | 1 |
| (iv) Use the iterative formula correctly at least once | M1 | |
| Obtain final answer 1.76 | A1 | |
| Show sufficient iterations to justify its accuracy to 2 d.p., or show there is a sign change in (1.755, 1.765) | B1 | 3 |
(i) Make recognizable sketch of a relevant graph, e.g. $y = 1/x$ | B1 |
(ii) Sketch an appropriate second graph, e.g. $y = \ln x$, correctly and justify the given statement | B1 | 2
(iii) Consider the sign of $x$ in $x = 1$ and $x = 2$, or equivalent | M1 |
Complete the argument correctly with appropriate calculations | A1 | 2
(iii) Show that the given equation is equivalent to $1/x = \ln x$, or vice versa | B1 | 1
(iv) Use the iterative formula correctly at least once | M1 |
Obtain final answer 1.76 | A1 |
Show sufficient iterations to justify its accuracy to 2 d.p., or show there is a sign change in (1.755, 1.765) | B1 | 35 (i) By sketching a suitable pair of graphs, show that there is only one value of $x$ that is a root of the equation
$$\frac { 1 } { x } = \ln x$$
(ii) Verify by calculation that this root lies between 1 and 2 .\\
(iii) Show that this root also satisfies the equation
$$x = \mathrm { e } ^ { \frac { 1 } { x } }$$
(iv) Use the iterative formula
$$x _ { n + 1 } = \mathrm { e } ^ { \frac { 1 } { x _ { n } } }$$
with initial value $x _ { 1 } = 1.8$, to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P2 2005 Q5 [8]}}