| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2023 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Solve exponential equation via iteration |
| Difficulty | Standard +0.3 This is a multi-part question involving standard A-level techniques: sketching an absolute value function, verifying root location by sign change (straightforward substitution), and applying a given iterative formula. While it requires careful execution across multiple parts, each component uses routine methods with no novel problem-solving or proof required. Slightly easier than average due to the structured guidance. |
| Spec | 1.02s Modulus graphs: sketch graph of |ax+b|1.06a Exponential function: a^x and e^x graphs and properties1.09b Sign change methods: understand failure cases1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Draw (more or less) correct sketch with vertex on positive \(x\)-axis | *B1 | Crossing \(y\)-axis above given graph, may be implied by extrapolation |
| Indicate in some way the two roots | DB1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Consider sign of \(3 - e^{-\frac{1}{2}x} + 5x - 4\) or of \(3 - e^{-\frac{1}{2}x} - \ | 5x-4\ | \) for 0.36 and 0.37 |
| Obtain \(-0.035\ldots\) and \(0.018\ldots\), or equivalents, and justify conclusion | A1 | AG necessary detail needed |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use iteration process correctly at least once | M1 | |
| Obtain final answer 1.295 | A1 | Answer required to exactly 4 sf |
| Show sufficient iterations to 6 sf to justify answer or show sign change in interval \([1.2945, 1.2955]\) | A1 |
## Question 4(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Draw (more or less) correct sketch with vertex on positive $x$-axis | *B1 | Crossing $y$-axis above given graph, may be implied by extrapolation |
| Indicate in some way the two roots | DB1 | |
## Question 4(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Consider sign of $3 - e^{-\frac{1}{2}x} + 5x - 4$ or of $3 - e^{-\frac{1}{2}x} - \|5x-4\|$ for 0.36 and 0.37 | M1 | But not for sign of $3 - e^{-\frac{1}{2}x} - 5x + 4$. May be implied by $-0.035\ldots$ and $0.018\ldots$, or equivalents |
| Obtain $-0.035\ldots$ and $0.018\ldots$, or equivalents, and justify conclusion | A1 | AG necessary detail needed |
## Question 4(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use iteration process correctly at least once | M1 | |
| Obtain final answer 1.295 | A1 | Answer required to exactly 4 sf |
| Show sufficient iterations to 6 sf to justify answer or show sign change in interval $[1.2945, 1.2955]$ | A1 | |
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\begin{enumerate}[label=(\alph*)]
\item \\
\includegraphics[max width=\textwidth, alt={}, center]{4ce3208e-8ceb-4848-a9c7-fcda166319f4-05_753_944_278_630}
The diagram shows the graph of $y = 3 - \mathrm { e } ^ { - \frac { 1 } { 2 } x }$.\\
On the diagram, sketch the graph of $y = | 5 x - 4 |$, and show that the equation $3 - e ^ { - \frac { 1 } { 2 } x } = | 5 x - 4 |$ has exactly two real roots.
It is given that the two roots of $3 - \mathrm { e } ^ { - \frac { 1 } { 2 } x } = | 5 x - 4 |$ are denoted by $\alpha$ and $\beta$, where $\alpha < \beta$.
\item Show by calculation that $\alpha$ lies between 0.36 and 0.37 .
\item Use the iterative formula $x _ { n + 1 } = \frac { 1 } { 5 } \left( 7 - \mathrm { e } ^ { - \frac { 1 } { 2 } x _ { n } } \right)$ to find $\beta$ correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2023 Q4 [7]}}