| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2014 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Iterative formula from integral equation |
| Difficulty | Standard +0.8 This question requires integration by parts of ln(2x), algebraic manipulation to derive the given form, then iterative numerical methods. While each component is standard P3 material, the combination of analytical derivation followed by numerical iteration, plus the need to handle exponential/logarithmic expressions carefully, makes this moderately challenging—above average but not exceptionally difficult for Further Maths Pure 3 level. |
| Spec | 1.08i Integration by parts1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Integrate and reach \(b\ln 2x - c\left[x - \frac{1}{x}dx\right]\), or equivalent | M1* | |
| Obtain \(\ln 2x - \int x - \frac{1}{x}dx\), or equivalent | A1 | |
| Obtain integral \(\ln 2x - x\), or equivalent | A1 | |
| Substitute limits correctly and equate to 1, having integrated twice. Obtain a correct equation in any form, e.g. \(a\ln 2a - a + 1 - \ln 2 = 1\) | M1(dep*) | |
| Obtain the given answer | A1 | [6] |
| (ii) Use the iterative formula correctly at least once | M1 | |
| Obtain final answer 1.94 | A1 | |
| Show sufficient iterations to 4 d.p. to justify 1.94 to 2d.p. or show that there is a sign change in the interval (1.935, 1.945). | A1 | [3] |
**(i)** Integrate and reach $b\ln 2x - c\left[x - \frac{1}{x}dx\right]$, or equivalent | M1* |
Obtain $\ln 2x - \int x - \frac{1}{x}dx$, or equivalent | A1 |
Obtain integral $\ln 2x - x$, or equivalent | A1 |
Substitute limits correctly and equate to 1, having integrated twice. Obtain a correct equation in any form, e.g. $a\ln 2a - a + 1 - \ln 2 = 1$ | M1(dep*) |
Obtain the given answer | A1 | [6]
**(ii)** Use the iterative formula correctly at least once | M1 |
Obtain final answer 1.94 | A1 |
Show sufficient iterations to 4 d.p. to justify 1.94 to 2d.p. or show that there is a sign change in the interval (1.935, 1.945). | A1 | [3]6 It is given that $\int _ { 1 } ^ { a } \ln ( 2 x ) \mathrm { d } x = 1$, where $a > 1$.\\
(i) Show that $a = \frac { 1 } { 2 } \exp \left( 1 + \frac { \ln 2 } { a } \right)$, where $\exp ( x )$ denotes $\mathrm { e } ^ { x }$.\\
(ii) Use the iterative formula
$$a _ { n + 1 } = \frac { 1 } { 2 } \exp \left( 1 + \frac { \ln 2 } { a _ { n } } \right)$$
to determine the value of $a$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P3 2014 Q6 [9]}}