| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2007 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Rearrange to iterative form |
| Difficulty | Moderate -0.3 This is a standard fixed-point iteration question with routine steps: sketch graphs to show unique root, verify interval, algebraically rearrange equation (straightforward manipulation), then apply given iterative formula. All techniques are textbook exercises requiring no novel insight, though the multi-part structure and iteration to convergence adds slight complexity beyond pure recall. |
| Spec | 1.07t Construct differential equations: in context1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Make a recognisable sketch of an appropriate graph, e.g. \(y = \ln x\) | B1 | |
| Sketch an appropriate graph, e.g. \(y = 2 - x\), correctly and justify the given statement | B1 | [2] |
| (ii) Consider sign of \(2 - x - \ln x\) when \(x = 1.4\) and \(x = 1.7\), or equivalent | M1 | |
| Complete the argument with correct calculations | A1 | [2] |
| (iii) Rearrange the equation as \(x = \frac{1}{2}(4 + x - 2\ln x)\) as \(2 - x = \ln x\), or vice versa | B1 | [1] |
| (iv) Use the iterative formula correctly at least once | M1 | |
| Obtain final answer \(1.56\) | A1 | |
| Show sufficient iterations to 4 d.p. to justify its accuracy to 2 d.p., or show there is a sign change in the interval \([1.555, 1.565]\) | A1 | [3] |
**(i)** Make a recognisable sketch of an appropriate graph, e.g. $y = \ln x$ | B1 |
Sketch an appropriate graph, e.g. $y = 2 - x$, correctly and justify the given statement | B1 | [2]
**(ii)** Consider sign of $2 - x - \ln x$ when $x = 1.4$ and $x = 1.7$, or equivalent | M1 |
Complete the argument with correct calculations | A1 | [2]
**(iii)** Rearrange the equation as $x = \frac{1}{2}(4 + x - 2\ln x)$ as $2 - x = \ln x$, or vice versa | B1 | [1]
**(iv)** Use the iterative formula correctly at least once | M1 |
Obtain final answer $1.56$ | A1 |
Show sufficient iterations to 4 d.p. to justify its accuracy to 2 d.p., or show there is a sign change in the interval $[1.555, 1.565]$ | A1 | [3]6 (i) By sketching a suitable pair of graphs, show that the equation
$$2 - x = \ln x$$
has only one root.\\
(ii) Verify by calculation that this root lies between 1.4 and 1.7.\\
(iii) Show that this root also satisfies the equation
$$x = \frac { 1 } { 3 } ( 4 + x - 2 \ln x )$$
(iv) Use the iterative formula
$$x _ { n + 1 } = \frac { 1 } { 3 } \left( 4 + x _ { n } - 2 \ln x _ { n } \right)$$
with initial value $x _ { 1 } = 1.5$, to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P3 2007 Q6 [8]}}