| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2009 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Apply basic Euler method for differential equations |
| Difficulty | Standard +0.3 This is a straightforward application of two standard numerical methods (Euler's method and midpoint method) with given formulas. Students simply substitute values into provided formulas requiring only arithmetic and one square root calculation per step. No derivation, proof, or problem-solving insight needed—purely procedural execution, making it slightly easier than average. |
| Spec | 1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y(3.1) = y(3) + 0.1\sqrt{3^2 + 2 + 1}\) | M1A1 | |
| \(= 2 + 0.1 \times \sqrt{12} = 2.3464\) | A1 | Total: 3 marks. Condone > 4dp if correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y(3.2) = y(3) + 2(0.1)[f(3.1, y(3.1))]\) | M1 | |
| \(= 2 + 2(0.1)[\sqrt{3.1^2 + 2.3464 + 1}]\) | A1F | ft on candidate's answer to (a) |
| \(= 2 + 0.2 \times 3.599499 = 2.720\) | A1 | Total: 3 marks. CAO Must be 2.720 |
# Question 1:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y(3.1) = y(3) + 0.1\sqrt{3^2 + 2 + 1}$ | M1A1 | |
| $= 2 + 0.1 \times \sqrt{12} = 2.3464$ | A1 | Total: 3 marks. Condone > 4dp if correct |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y(3.2) = y(3) + 2(0.1)[f(3.1, y(3.1))]$ | M1 | |
| $= 2 + 2(0.1)[\sqrt{3.1^2 + 2.3464 + 1}]$ | A1F | ft on candidate's answer to (a) |
| $= 2 + 0.2 \times 3.599499 = 2.720$ | A1 | Total: 3 marks. CAO Must be 2.720 |
---1 The function $y ( x )$ satisfies the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$
where
$$\mathrm { f } ( x , y ) = \sqrt { x ^ { 2 } + y + 1 }$$
and
$$y ( 3 ) = 2$$
\begin{enumerate}[label=(\alph*)]
\item Use the Euler formula
$$y _ { r + 1 } = y _ { r } + h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$
with $h = 0.1$, to obtain an approximation to $y ( 3.1 )$, giving your answer to four decimal places.
\item Use the formula
$$y _ { r + 1 } = y _ { r - 1 } + 2 h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$
with your answer to part (a), to obtain an approximation to $y ( 3.2 )$, giving your answer to three decimal places.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2009 Q1 [6]}}