| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2016 |
| Session | November |
| Marks | 7 |
| 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 fixed-point iteration question requiring routine integration of exponentials, algebraic manipulation to isolate the variable, and mechanical application of an iterative formula. Part (i) involves standard integration and rearrangement (4 marks suggests multiple routine steps), while part (ii) is purely computational. No novel insight or complex problem-solving is required—slightly easier than average. |
| Spec | 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks |
|---|---|
| (ii) | Integrate to obtain 2e2x +5x |
| Answer | Marks |
|---|---|
| interval (1.8535,1.8545) | B1 |
| Answer | Marks |
|---|---|
| B1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Page 5 | Mark Scheme | Syllabus |
| Cambridge International AS Level – October/November 2016 | 9709 | 21 |
Question 4:
--- 4 (i)
(ii) ---
4 (i)
(ii) | Integrate to obtain 2e2x +5x
Apply limits correctly and equate to 100
Rearrange and apply logarithms correctly to reach a=...
Confirm given result a= 1ln(50+e−2a −5a)
2
Use the iterative formula correctly at least once
Obtain final answer 1.854
Show sufficient iterations to justify accuracy to 3 dp or show sign change in
interval (1.8535,1.8545) | B1
M1
M1
A1
M1
A1
B1 | [4]
[3]
Page 5 | Mark Scheme | Syllabus | Paper
Cambridge International AS Level – October/November 2016 | 9709 | 21It is given that the positive constant $a$ is such that
$$\int_{-a}^a (4e^{2x} + 5) dx = 100.$$
\begin{enumerate}[label=(\roman*)]
\item Show that $a = \frac{1}{4}\ln(50 + e^{-2a} - 5a)$. [4]
\item Use the iterative formula $a_{n+1} = \frac{1}{4}\ln(50 + e^{-2a_n} - 5a_n)$ to find $a$ correct to 3 decimal places. Give the result of each iteration to 5 decimal places. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2016 Q4 [7]}}