Question 6 - A-Level Maths

CAIE P2 2021 November — Question 6 8 marks

Exam Board	CAIE
Module	P2 (Pure Mathematics 2)
Year	2021
Session	November
Marks	8
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Fixed Point Iteration
Type	Show convergence to specific root
Difficulty	Standard +0.3 This is a standard A-level fixed point iteration question with routine components: sketching graphs to show existence of a root, verifying bounds by substitution, showing convergence algebraically (straightforward rearrangement), and applying iteration. All steps follow predictable patterns with no novel insight required, making it slightly easier than average.
Spec	1.06a Exponential function: a^x and e^x graphs and properties 1.06d Natural logarithm: ln(x) function and properties 1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams

By sketching a suitable pair of graphs on the same diagram, show that the equation $$\ln x = 2 \mathrm { e } ^ { - x }$$ has exactly one root.
Verify by calculation that the root lies between 1.5 and 1.6.
Show that if a sequence of values given by the iterative formula $$x _ { n + 1 } = \mathrm { e } ^ { 2 \mathrm { e } ^ { - x _ { n } } }$$ converges, then it converges to the root of the equation in part (a).
Use the iterative formula in part (c) to determine the root correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

Show mark scheme Show mark scheme source

Question 6(a):

Answer	Marks	Guidance
Answer	Mark	Guidance
Draw correct sketch of $y=\ln x$ or $y=2e^{-x}$	B1	$y=\ln x$ must extend into 1st and 4th quadrants. $y=2e^{-x}$ must extend into 1st and 2nd quadrants
Draw correct sketch of second curve and indicate one root	B1	Point of intersection must be circled/identified or a statement such that 'there is only one point of intersection so one root only' or similar
Total	2

Question 6(b):

Answer	Marks	Guidance
Answer	Mark	Guidance
Consider sign of $\ln x-2e^{-x}$, or equivalent, for $1.5$ and $1.6$	M1
Obtain $-0.04...$ and $0.06...$ or equivalents and justify conclusion	A1
Total	2

Question 6(c):

Answer	Marks	Guidance
Answer	Mark	Guidance
Replace $x_{n+1}$ and $x_n$ by $x$ and apply logarithms to confirm result	B1	AG. Allow if done 'in reverse' but $x_n$ and $x_{n+1}$ need to be seen in the final statement
Total	1

Question 6(d):

Answer	Marks	Guidance
Answer	Mark	Guidance
Use iteration process correctly at least once	M1	Need to see 3 correct values
Obtain final answer $1.54$	A1	Answer required to exactly 3sf
Show sufficient iterations to 5sf to justify answer or show sign change in interval $[1.535, 1.545]$	A1
Total	3

## Question 6(a):

| Answer | Mark | Guidance |
|--------|------|----------|
| Draw correct sketch of $y=\ln x$ or $y=2e^{-x}$ | B1 | $y=\ln x$ must extend into 1st and 4th quadrants. $y=2e^{-x}$ must extend into 1st and 2nd quadrants |
| Draw correct sketch of second curve and indicate one root | B1 | Point of intersection must be circled/identified or a statement such that 'there is only one point of intersection so one root only' or similar |
| **Total** | **2** | |

---

## Question 6(b):

| Answer | Mark | Guidance |
|--------|------|----------|
| Consider sign of $\ln x-2e^{-x}$, or equivalent, for $1.5$ and $1.6$ | M1 | |
| Obtain $-0.04...$ and $0.06...$ or equivalents and justify conclusion | A1 | |
| **Total** | **2** | |

---

## Question 6(c):

| Answer | Mark | Guidance |
|--------|------|----------|
| Replace $x_{n+1}$ and $x_n$ by $x$ and apply logarithms to confirm result | B1 | AG. Allow if done 'in reverse' but $x_n$ and $x_{n+1}$ need to be seen in the final statement |
| **Total** | **1** | |

---

## Question 6(d):

| Answer | Mark | Guidance |
|--------|------|----------|
| Use iteration process correctly at least once | M1 | Need to see 3 correct values |
| Obtain final answer $1.54$ | A1 | Answer required to exactly 3sf |
| Show sufficient iterations to 5sf to justify answer or show sign change in interval $[1.535, 1.545]$ | A1 | |
| **Total** | **3** | |

---

Show LaTeX source

6
\begin{enumerate}[label=(\alph*)]
\item By sketching a suitable pair of graphs on the same diagram, show that the equation

$$\ln x = 2 \mathrm { e } ^ { - x }$$

has exactly one root.
\item Verify by calculation that the root lies between 1.5 and 1.6.
\item Show that if a sequence of values given by the iterative formula

$$x _ { n + 1 } = \mathrm { e } ^ { 2 \mathrm { e } ^ { - x _ { n } } }$$

converges, then it converges to the root of the equation in part (a).
\item Use the iterative formula in part (c) to determine the root correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
\end{enumerate}

\hfill \mbox{\textit{CAIE P2 2021 Q6 [8]}}

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