| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2008 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive equation from integral condition |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring standard integration (polynomial and logarithmic), basic algebraic manipulation to rearrange an equation, interval verification by substitution, and application of a given iterative formula. All techniques are routine for P2 level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.09b Sign change methods: understand failure cases1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks |
|---|---|
| (i) Obtain terms \(\frac{1}{2}x^2\) and \(\ln x\) | B1 + B1 |
| Substitute limits correctly | M1 |
| Obtain correct equation in any form, e.g. \(\frac{1}{2}a^2 + \ln a - \frac{1}{2} = 6\) | A1 |
| Obtain given answer correctly | A1 |
| [5] | |
| (ii) Consider sign of \(a - \sqrt{(13 - 2\ln a)}\) at \(a = 3\) and \(a = 3.5\), or equivalent | M1 |
| Complete the argument correctly with correct calculations | A1 |
| [2] | |
| (iii) Use the iterative formula correctly at least once | M1 |
| Obtain final answer 3.26 | A1 |
| Show sufficient iterations to justify its accuracy to 2 d.p. or show there is a sign change in the interval (3.255, 3.265) | B1 |
| [3] |
**(i)** Obtain terms $\frac{1}{2}x^2$ and $\ln x$ | B1 + B1 |
Substitute limits correctly | M1 |
Obtain correct equation in any form, e.g. $\frac{1}{2}a^2 + \ln a - \frac{1}{2} = 6$ | A1 |
Obtain given answer correctly | A1 |
| [5] |
**(ii)** Consider sign of $a - \sqrt{(13 - 2\ln a)}$ at $a = 3$ and $a = 3.5$, or equivalent | M1 |
Complete the argument correctly with correct calculations | A1 |
| [2] |
**(iii)** Use the iterative formula correctly at least once | M1 |
Obtain final answer 3.26 | A1 |
Show sufficient iterations to justify its accuracy to 2 d.p. or show there is a sign change in the interval (3.255, 3.265) | B1 |
| [3] |8 The constant $a$, where $a > 1$, is such that $\int _ { 1 } ^ { a } \left( x + \frac { 1 } { x } \right) \mathrm { d } x = 6$.\\
(i) Find an equation satisfied by $a$, and show that it can be written in the form
$$a = \sqrt { } ( 13 - 2 \ln a )$$
(ii) Verify, by calculation, that the equation $a = \sqrt { } ( 13 - 2 \ln a )$ has a root between 3 and 3.5.\\
(iii) Use the iterative formula
$$a _ { n + 1 } = \sqrt { } \left( 13 - 2 \ln a _ { n } \right)$$
with $a _ { 1 } = 3.2$, 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 P2 2008 Q8 [10]}}