| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2010 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive equation from integral condition |
| Difficulty | Standard +0.3 This question involves integration by parts (a standard P3 technique) followed by straightforward fixed-point iteration with a given formula. Part (i) requires routine application of integration by parts to derive the equation, while part (ii) is purely mechanical iteration. The integration is standard bookwork and the iteration requires no insight beyond repeatedly substituting values into a calculator. |
| Spec | 1.08i Integration by parts1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Attempt integration by parts | M1 | |
| Obtain \(-x^{-1}\ln x + \int \frac{1}{x^2}dx\), \(x\ln x - x - 2\int\frac{\ln x}{x^2}dx - 2\int\frac{1}{x^2}dx\) or equivalent | A1 | |
| Obtain \(-x^{-1}\ln x - x^{-1}\) or equivalent | A1 | |
| Use limits correctly, equate to \(\frac{2}{e}\) and attempt rearrangement to obtain \(a\) in terms of \(\ln a\) | M1 | |
| Obtain given answer \(a = \frac{2}{3}(1 + \ln a)\) correctly | A1 | [5] |
| (ii) Use valid iterative formula correctly at least once | M1 | |
| Obtain final answer \(3.96\) | A1 | |
| Show sufficient iterations to \(> 4\) dp to justify accuracy to \(2\) dp or show sign change in interval \((3.955, 3.965)\) [4 → 3.9772 → 3.9676 → 3.9636 → 3.9619] | A1 | [3] |
**(i)** Attempt integration by parts | M1 |
Obtain $-x^{-1}\ln x + \int \frac{1}{x^2}dx$, $x\ln x - x - 2\int\frac{\ln x}{x^2}dx - 2\int\frac{1}{x^2}dx$ or equivalent | A1 |
Obtain $-x^{-1}\ln x - x^{-1}$ or equivalent | A1 |
Use limits correctly, equate to $\frac{2}{e}$ and attempt rearrangement to obtain $a$ in terms of $\ln a$ | M1 |
Obtain given answer $a = \frac{2}{3}(1 + \ln a)$ correctly | A1 | [5]
**(ii)** Use valid iterative formula correctly at least once | M1 |
Obtain final answer $3.96$ | A1 |
Show sufficient iterations to $> 4$ dp to justify accuracy to $2$ dp or show sign change in interval $(3.955, 3.965)$ [4 → 3.9772 → 3.9676 → 3.9636 → 3.9619] | A1 | [3]
SR: Use of $a_{n+1} = e^{(a_n-1)}$ to obtain 0.50 also earns 3/3.7 (i) Given that $\int _ { 1 } ^ { a } \frac { \ln x } { x ^ { 2 } } \mathrm {~d} x = \frac { 2 } { 5 }$, show that $a = \frac { 5 } { 3 } ( 1 + \ln a )$.\\
(ii) Use an iteration formula based on the equation $a = \frac { 5 } { 3 } ( 1 + \ln a )$ to find the value of $a$ correct to 2 decimal places. Use an initial value of 4 and give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P3 2010 Q7 [8]}}