Question 5 - A-Level Maths

CAIE P3 2024 November — Question 5 7 marks

Exam Board	CAIE
Module	P3 (Pure Mathematics 3)
Year	2024
Session	November
Marks	7
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Fixed Point Iteration
Type	Solve exponential equation via iteration
Difficulty	Moderate -0.3 This is a routine multi-part question on numerical methods requiring standard techniques: sketching graphs to show uniqueness of root, interval testing with substitution, and applying a given iterative formula. All steps are procedural with no novel insight required, making it slightly easier than average A-level material.
Spec	1.02n Sketch curves: simple equations including polynomials 1.09a Sign change methods: locate roots 1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams

By sketching a suitable pair of graphs, show that the equation $2 + \mathrm { e } ^ { - 0.2 x } = \ln ( 1 + x )$ has only one root.
Show by calculation that this root lies between 7 and 9 . \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-08_2716_40_109_2009}
Use the iterative formula $$x _ { n + 1 } = \exp \left( 2 + \mathrm { e } ^ { - 0.2 x _ { n } } \right) - 1$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places. $\left[ \exp ( x ) \right.$ is an alternative notation for $\left. \mathrm { e } ^ { x } .\right]$ \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-10_481_789_262_639} The diagram shows the curve $y = \sin 2 x ( 1 + \sin 2 x )$, for $0 \leqslant x \leqslant \frac { 3 } { 4 } \pi$, and its minimum point $M$. The shaded region bounded by the curve that lies above the $x$-axis and the $x$-axis itself is denoted by $R$.

Show mark scheme Show mark scheme source

Question 5(a):

Answer	Marks	Guidance
Answer	Mark	Guidance
Sketch a relevant graph, e.g. $y = 2+e^{-0.2x}$; correct curvature, intersections with $y$-axis approximately correct, horizontal asymptote approximately correct	B1	Need not draw in; allow scale not marked and implied by sketch
Sketch a second relevant graph, e.g. $y=\ln(1+x)$ and justify the given statement	B1

Question 5(b):

Answer	Marks	Guidance
Answer	Mark	Guidance
Calculate the value of a relevant expression or values of a relevant pair of expressions at $x=7$ and $x=9$	M1
Complete the argument correctly with correct calculated values	A1	E.g. $2.079 < 2.246$ and $2.302 > 2.165$, or $0.167>0$ and $-0.137<0$

Question 5(c):

Answer	Marks	Guidance
Answer	Mark	Guidance
Use the iterative process correctly at least once	M1	Obtain one value and substitute that value back into the formula
Obtain final answer $8.03$	A1
Show sufficient iterations to at least 4 decimal places to justify $8.03$ to 2 decimal places, or show sign change in interval $(8.025, 8.035)$	A1	E.g. 7, 8.4555, 7.8846, 8.0849, 8.0115, 8.0380, 8.0283, 8.0318, 8, 8.0421, 8.0268, 8.0324, 9, 7.7172, 8.1490, 7.9887, 8.0463, 8.0253, 8.0329

## Question 5(a):

| Answer | Mark | Guidance |
|--------|------|----------|
| Sketch a relevant graph, e.g. $y = 2+e^{-0.2x}$; correct curvature, intersections with $y$-axis approximately correct, horizontal asymptote approximately correct | B1 | Need not draw in; allow scale not marked and implied by sketch |
| Sketch a second relevant graph, e.g. $y=\ln(1+x)$ and justify the given statement | B1 | |

---

## Question 5(b):

| Answer | Mark | Guidance |
|--------|------|----------|
| Calculate the value of a relevant expression or values of a relevant pair of expressions at $x=7$ and $x=9$ | M1 | |
| Complete the argument correctly with correct calculated values | A1 | E.g. $2.079 < 2.246$ and $2.302 > 2.165$, or $0.167>0$ and $-0.137<0$ |

---

## Question 5(c):

| Answer | Mark | Guidance |
|--------|------|----------|
| Use the iterative process correctly at least once | M1 | Obtain one value and substitute that value back into the formula |
| Obtain final answer $8.03$ | A1 | |
| Show sufficient iterations to at least 4 decimal places to justify $8.03$ to 2 decimal places, or show sign change in interval $(8.025, 8.035)$ | A1 | E.g. 7, 8.4555, 7.8846, 8.0849, 8.0115, 8.0380, 8.0283, 8.0318, 8, 8.0421, 8.0268, 8.0324, 9, 7.7172, 8.1490, 7.9887, 8.0463, 8.0253, 8.0329 |

---

Show LaTeX source

5
\begin{enumerate}[label=(\alph*)]
\item By sketching a suitable pair of graphs, show that the equation $2 + \mathrm { e } ^ { - 0.2 x } = \ln ( 1 + x )$ has only one root.
\item Show by calculation that this root lies between 7 and 9 .\\

\includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-08_2716_40_109_2009}
\item Use the iterative formula

$$x _ { n + 1 } = \exp \left( 2 + \mathrm { e } ^ { - 0.2 x _ { n } } \right) - 1$$

to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.\\
$\left[ \exp ( x ) \right.$ is an alternative notation for $\left. \mathrm { e } ^ { x } .\right]$\\

\includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-10_481_789_262_639}

The diagram shows the curve $y = \sin 2 x ( 1 + \sin 2 x )$, for $0 \leqslant x \leqslant \frac { 3 } { 4 } \pi$, and its minimum point $M$. The shaded region bounded by the curve that lies above the $x$-axis and the $x$-axis itself is denoted by $R$.
\end{enumerate}

\hfill \mbox{\textit{CAIE P3 2024 Q5 [7]}}

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