| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2008 |
| Session | November |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive equation from integral condition |
| Difficulty | Challenging +1.2 This question requires integration by parts (a standard P3 technique), then algebraic manipulation to derive the fixed point equation, followed by routine iterative methods. While it involves multiple steps and techniques, each component is straightforward application of syllabus content without requiring novel insight or particularly challenging problem-solving. |
| Spec | 1.08i Integration by parts1.09b Sign change methods: understand failure cases1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Integrate by parts and reach \(kxe^{\frac{1}{x}} - k\int e^{\frac{1}{x}} dx\) | M1 | |
| Obtain \(2xe^{\frac{1}{x}} - 2\int e^{\frac{1}{x}} dx\) | A1 | |
| Complete the integration, obtaining \(2xe^{\frac{1}{x}} - 4e^{\frac{1}{x}}\), or equivalent | A1 | |
| Substitute limits correctly and equate result to 6, having integrated twice | M1 | |
| Rearrange and obtain \(a = e^{-\frac{1}{a}} + 2\) | A1 | [5] |
| (ii) Make recognizable sketch of a relevant exponential graph, e.g. \(y = e^{-\frac{1}{x}} + 2\) | B1 | |
| Sketch a second relevant straight line graph, e.g. \(y = x\), or curve, and indicate the root | B1 | [2] |
| (iii) Consider sign of \(x - e^{-\frac{1}{x}} - 2\) at \(x = 2\) and \(x = 2.5\), or equivalent | M1 | |
| Justify the given statement with correct calculations and argument | A1 | [2] |
| (iv) Use the iterative formula \(x_{n+1} = 2 + e^{-\frac{1}{x_n}}\) correctly at least once, with \(2 \leq x_n \leq 2.5\) | M1 | |
| Obtain final answer 2.31 | A1 | |
| Show sufficient iterations to at least 4 d.p. to justify its accuracy to 2 d.p., or show there is a sign change in the interval \((2.305, 2.315)\) | A1 | [3] |
**(i)** Integrate by parts and reach $kxe^{\frac{1}{x}} - k\int e^{\frac{1}{x}} dx$ | M1 |
Obtain $2xe^{\frac{1}{x}} - 2\int e^{\frac{1}{x}} dx$ | A1 |
Complete the integration, obtaining $2xe^{\frac{1}{x}} - 4e^{\frac{1}{x}}$, or equivalent | A1 |
Substitute limits correctly and equate result to 6, having integrated twice | M1 |
Rearrange and obtain $a = e^{-\frac{1}{a}} + 2$ | A1 | [5]
**(ii)** Make recognizable sketch of a relevant exponential graph, e.g. $y = e^{-\frac{1}{x}} + 2$ | B1 |
Sketch a second relevant straight line graph, e.g. $y = x$, or curve, and indicate the root | B1 | [2]
**(iii)** Consider sign of $x - e^{-\frac{1}{x}} - 2$ at $x = 2$ and $x = 2.5$, or equivalent | M1 |
Justify the given statement with correct calculations and argument | A1 | [2]
**(iv)** Use the iterative formula $x_{n+1} = 2 + e^{-\frac{1}{x_n}}$ correctly at least once, with $2 \leq x_n \leq 2.5$ | M1 |
Obtain final answer 2.31 | A1 |
Show sufficient iterations to at least 4 d.p. to justify its accuracy to 2 d.p., or show there is a sign change in the interval $(2.305, 2.315)$ | A1 | [3]9 The constant $a$ is such that $\int _ { 0 } ^ { a } x \mathrm { e } ^ { \frac { 1 } { 2 } x } \mathrm {~d} x = 6$.\\
(i) Show that $a$ satisfies the equation
$$x = 2 + \mathrm { e } ^ { - \frac { 1 } { 2 } x } .$$
(ii) By sketching a suitable pair of graphs, show that this equation has only one root.\\
(iii) Verify by calculation that this root lies between 2 and 2.5.\\
(iv) Use an iterative formula based on the equation in part (i) to calculate the value of $a$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P3 2008 Q9 [12]}}