| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2013 |
| Session | November |
| 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 to find area, algebraic rearrangement to derive the iterative form, and mechanical application of fixed-point iteration. All techniques are routine for P2 level with no novel insight required, making it slightly easier than average. |
| Spec | 1.08d Evaluate definite integrals: between limits1.09a Sign change methods: locate roots1.09b 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 to obtain terms \(4x^2\) and \(\frac{1}{2}e^t\) | B1 + B1 | |
| Substitute limits correctly | M1 | |
| Obtain correct equation in any form \(4a^2 + \frac{1}{2}e^a - \frac{1}{2} - \frac{1}{2}\) | A1 | |
| Rearrange to given answer correctly | A1 | [5] |
| (ii) Consider sign of \(\sqrt{\frac{2 - e^a}{8}} - a\), or equivalent | M1 | |
| Complete the argument correctly with appropriate calculations (\(f(0.2) = 0.112, f(0.3) = -0.015\)) | A1 | [2] |
| (iii) Use the iterative formula correctly at least once | M1 | |
| Obtain final answer 0.29 | A1 | |
| Show sufficient iterations to justify its accuracy to 2 d.p. | B1 | |
| or show there is a sign change in the interval \((0.285, 0.295)\) | [3] |
(i) Integrate to obtain terms $4x^2$ and $\frac{1}{2}e^t$ | B1 + B1 |
Substitute limits correctly | M1 |
Obtain correct equation in any form $4a^2 + \frac{1}{2}e^a - \frac{1}{2} - \frac{1}{2}$ | A1 |
Rearrange to given answer correctly | A1 | [5]
(ii) Consider sign of $\sqrt{\frac{2 - e^a}{8}} - a$, or equivalent | M1 |
Complete the argument correctly with appropriate calculations ($f(0.2) = 0.112, f(0.3) = -0.015$) | A1 | [2]
(iii) Use the iterative formula correctly at least once | M1 |
Obtain final answer 0.29 | A1 |
Show sufficient iterations to justify its accuracy to 2 d.p. | B1 |
or show there is a sign change in the interval $(0.285, 0.295)$ | [3]7\\
\includegraphics[max width=\textwidth, alt={}, center]{0900b607-6136-4bf7-a42e-6824d1a21e43-3_451_451_255_845}
The diagram shows part of the curve $y = 8 x + \frac { 1 } { 2 } \mathrm { e } ^ { x }$. The shaded region $R$ is bounded by the curve and by the lines $x = 0 , y = 0$ and $x = a$, where $a$ is positive. The area of $R$ is equal to $\frac { 1 } { 2 }$.\\
(i) Find an equation satisfied by $a$, and show that the equation can be written in the form
$$a = \sqrt { } \left( \frac { 2 - \mathrm { e } ^ { a } } { 8 } \right)$$
(ii) Verify by calculation that the equation $a = \sqrt { } \left( \frac { 2 - \mathrm { e } ^ { a } } { 8 } \right)$ has a root between 0.2 and 0.3.\\
(iii) Use the iterative formula $a _ { n + 1 } = \sqrt { } \left( \frac { 2 - \mathrm { e } ^ { a _ { n } } } { 8 } \right)$ to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P2 2013 Q7 [10]}}