| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2024 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Find intersection point coordinates |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question requiring routine sketching, algebraic manipulation to derive an iterative formula, and mechanical application of fixed-point iteration. The algebra in part (b) is simple (equating the functions and rearranging), and part (c) requires only repeated substitution with no analysis of convergence or choice of rearrangement. Slightly easier than average due to its procedural nature. |
| Spec | 1.02m Graphs of functions: difference between plotting and sketching1.02n Sketch curves: simple equations including polynomials1.02s Modulus graphs: sketch graph of |ax+b|1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Show increasing curve above \(x\)-axis for \(y = 1 + e^{2x}\) | *B1 | And appearing in first and second quadrants. |
| Show V-shaped graph with vertex on positive \(x\)-axis and only one point of intersection with first curve | DB1 | With modulus graph crossing \(y\)-axis above first graph. |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or clearly imply \(1 + e^{2x} = 4 - x\) | B1 | |
| Arrange to confirm \(x = \frac{1}{2}\ln(3-x)\) | B1 | AG – necessary detail needed. Do not condone incorrect use of logs. |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use iterative process correctly at least once | M1 | |
| Obtain final answer \(0.465\) | A1 | Answer required to exactly 3 sf. |
| Show sufficient iterations to justify answer or show a sign change in the interval \([0.4645, 0.4655]\) | A1 | |
| Total | 3 |
## Question 4(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Show increasing curve above $x$-axis for $y = 1 + e^{2x}$ | *B1 | And appearing in first and second quadrants. |
| Show V-shaped graph with vertex on positive $x$-axis and only one point of intersection with first curve | DB1 | With modulus graph crossing $y$-axis above first graph. |
| **Total** | **2** | |
---
## Question 4(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or clearly imply $1 + e^{2x} = 4 - x$ | B1 | |
| Arrange to confirm $x = \frac{1}{2}\ln(3-x)$ | B1 | AG – necessary detail needed. Do not condone incorrect use of logs. |
| **Total** | **2** | |
---
## Question 4(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use iterative process correctly at least once | M1 | |
| Obtain final answer $0.465$ | A1 | Answer required to exactly 3 sf. |
| Show sufficient iterations to justify answer or show a sign change in the interval $[0.4645, 0.4655]$ | A1 | |
| **Total** | **3** | |
---4
\begin{enumerate}[label=(\alph*)]
\item Sketch the graphs of $y = 1 + \mathrm { e } ^ { 2 x }$ and $y = | x - 4 |$ on the same diagram.
\item The two graphs meet at the point $P$ .\\
Show that the $x$-coordinate of $P$ satisfies the equation $x = \frac { 1 } { 2 } \ln ( 3 - x )$ .\\
\includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-06_2716_38_109_2012}
\item Use an iterative formula, based on the equation in part (b), to find the $x$-coordinate of $P$ correct to 3 significant figures. Use an initial value of 0.45 and give the result of each iteration to 5 significant figures.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2024 Q4 [7]}}