Question 2 - A-Level Maths

AQA FP1 2006 June — Question 2 6 marks

Exam Board	AQA
Module	FP1 (Further Pure Mathematics 1)
Year	2006
Session	June
Marks	6
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Fixed Point Iteration
Type	Apply basic Euler method for differential equations
Difficulty	Moderate -0.3 This is a straightforward application of Euler's method with a given step length and starting point. It requires only routine calculation over two steps (0.2 each from x=2 to x=2.4), evaluating log₁₀ at simple values, and basic arithmetic. While it's a Further Maths topic, the execution is mechanical with no conceptual challenges or problem-solving required, making it slightly easier than average.
Spec	1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams

2 A curve satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \log _ { 10 } x$$ Starting at the point $( 2,3 )$ on the curve, use a step-by-step method with a step length of 0.2 to estimate the value of $y$ at $x = 2.4$. Give your answer to three decimal places.

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Question 2:

Answer	Marks	Guidance
Working	Marks	Guidance
1st increment is $0.2\lg 2\ldots$	M1	or $0.2\lg 2.1$ or $0.2\lg 2.2$
$\approx 0.06021$	A1	PI
$x=2.2 \Rightarrow y \approx 3.06021$	A1$\checkmark$	PI; ft numerical error
2nd increment is $0.2\lg 2.2$	m1	consistent with first one
$\approx 0.06848$	A1	PI
$x=2.4 \Rightarrow y \approx 3.12869 \approx 3.129$	A1$\checkmark$ (6)	ft numerical error

## Question 2:
| Working | Marks | Guidance |
|---------|-------|----------|
| 1st increment is $0.2\lg 2\ldots$ | M1 | or $0.2\lg 2.1$ or $0.2\lg 2.2$ |
| $\approx 0.06021$ | A1 | PI |
| $x=2.2 \Rightarrow y \approx 3.06021$ | A1$\checkmark$ | PI; ft numerical error |
| 2nd increment is $0.2\lg 2.2$ | m1 | consistent with first one |
| $\approx 0.06848$ | A1 | PI |
| $x=2.4 \Rightarrow y \approx 3.12869 \approx 3.129$ | A1$\checkmark$ (6) | ft numerical error |

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2 A curve satisfies the differential equation

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \log _ { 10 } x$$

Starting at the point $( 2,3 )$ on the curve, use a step-by-step method with a step length of 0.2 to estimate the value of $y$ at $x = 2.4$. Give your answer to three decimal places.

\hfill \mbox{\textit{AQA FP1 2006 Q2 [6]}}

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