| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Sketch graphs to show root existence |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question on standard A-level techniques: sketching y=ln(x) and y=3x-x² to show intersection (routine), verifying the root location by substitution (trivial calculation), and applying a given iterative formula (mechanical process requiring no derivation or convergence analysis). All parts are textbook exercises with no problem-solving or novel insight required, making it slightly easier than average. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.06d Natural logarithm: ln(x) function 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 |
|---|---|---|
| Sketch a relevant graph, e.g. \(y = \ln x\) | B1 | \(\ln(x)\): sketch should imply \(y\)-axis is an asymptote. Through \((1, 0)\) if marked. Correct shape |
| Sketch a second relevant graph, e.g. \(y = 3x - x^2\), and justify the given statement by marking the root on the sketch or by use of a suitable comment | B1 | \(3x - x^2\): Symmetrical. Through \((0,0)\) and \((3,0)\) if marked. If \(\ln(x)\) correct accept parabola for \(+ve\) \(y\) only. If \(\ln(x)\) incorrect then need parabola in 3 quadrants |
| Answer | Marks | Guidance |
|---|---|---|
| Calculate the values of a relevant expression or pair of expressions at \(x = 2\) and \(x = 2.8\) | M1 | Allow for a smaller interval. At least one value correct if comparing with 0. If using pairs then the pairing must be clear |
| Complete the argument correctly with correct calculated values | A1 | e.g. \(0.693 < 2\) and \(1.03 > 0.56\) or \(1.307 > 0, -0.47 < 0\); using \(\sqrt{3x} - \ln x\): \(0.304 > 0, -0.085 < 0\). Need to have calculated values to at least 2 sf |
| Answer | Marks | Guidance |
|---|---|---|
| Use the iterative process correctly at least once | M1 | |
| Obtain final answer \(2.63\) | A1 | |
| Show sufficient iterations to at least 4 dp to justify \(2.63\) to 2 dp, or show there is a sign change in the interval \((2.625, 2.635)\) | A1 | SC Allow M1 A1 A0 to a candidate who starts at a point in the interval and reaches a premature conclusion |
## Question 5(a):
| Sketch a relevant graph, e.g. $y = \ln x$ | B1 | $\ln(x)$: sketch should imply $y$-axis is an asymptote. Through $(1, 0)$ if marked. Correct shape |
|---|---|---|
| Sketch a second relevant graph, e.g. $y = 3x - x^2$, and justify the given statement by marking the root on the sketch or by use of a suitable comment | B1 | $3x - x^2$: Symmetrical. Through $(0,0)$ and $(3,0)$ if marked. If $\ln(x)$ correct accept parabola for $+ve$ $y$ only. If $\ln(x)$ incorrect then need parabola in 3 quadrants |
**Total: 2 marks**
---
## Question 5(b):
| Calculate the values of a relevant expression or pair of expressions at $x = 2$ and $x = 2.8$ | M1 | Allow for a smaller interval. At least one value correct if comparing with 0. If using pairs then the pairing must be clear |
|---|---|---|
| Complete the argument correctly with correct calculated values | A1 | e.g. $0.693 < 2$ and $1.03 > 0.56$ or $1.307 > 0, -0.47 < 0$; using $\sqrt{3x} - \ln x$: $0.304 > 0, -0.085 < 0$. Need to have calculated values to at least 2 sf |
**Total: 2 marks**
---
## Question 5(c):
| Use the iterative process correctly at least once | M1 | |
|---|---|---|
| Obtain final answer $2.63$ | A1 | |
| Show sufficient iterations to at least 4 dp to justify $2.63$ to 2 dp, or show there is a sign change in the interval $(2.625, 2.635)$ | A1 | SC Allow M1 A1 A0 to a candidate who starts at a point in the interval and reaches a premature conclusion |
**Total: 3 marks**5
\begin{enumerate}[label=(\alph*)]
\item By sketching a suitable pair of graphs, show that the equation $\ln x = 3 x - x ^ { 2 }$ has one real root.
\item Verify by calculation that the root lies between 2 and 2.8.
\item Use the iterative formula $x _ { n + 1 } = \sqrt { 3 x _ { n } - \ln x _ { n } }$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2022 Q5 [7]}}