| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2009 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Apply improved Euler method for differential equations |
| Difficulty | Standard +0.3 This is a straightforward application of standard numerical methods (Euler and improved Euler) with clear formulas provided. Students must substitute given values and perform arithmetic calculations with no conceptual challenges or novel problem-solving required. Slightly above average difficulty only due to the algebraic manipulation of the rational function and the two-part structure. |
| Spec | 1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(y_1 = 3 + 0.2 \times \left[\frac{1^2 + 3^2}{1+3}\right]\) | M1A1 | |
| \(= 3.5\) | A1 | Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(k_1 = 0.2 \times 2.5 = 0.5\) | B1ft | PI; ft from (a) |
| \(k_2 = 0.2 \times f(1.2,\ 3.5)\) | M1 | ft on (a) |
| \(\ldots = 0.2 \times \frac{1.2^2 + 3.5^2}{1.2 + 3.5} = 0.5825(53\ldots)\) | A1ft | PI; condone 3dp |
| \(y(1.2) = y(1) + \frac{1}{2}[0.5 + 0.5825(53\ldots)]\) | m1 | |
| \(= 3.54127\ldots = 3.5413\) to 4dp | A1ft | Total: 5; ft one slip; if answer not to 4dp withhold this mark |
## Question 1:
### Part (a):
| Working | Marks | Guidance |
|---------|-------|---------|
| $y_1 = 3 + 0.2 \times \left[\frac{1^2 + 3^2}{1+3}\right]$ | M1A1 | |
| $= 3.5$ | A1 | **Total: 3** |
### Part (b):
| Working | Marks | Guidance |
|---------|-------|---------|
| $k_1 = 0.2 \times 2.5 = 0.5$ | B1ft | PI; ft from (a) |
| $k_2 = 0.2 \times f(1.2,\ 3.5)$ | M1 | ft on (a) |
| $\ldots = 0.2 \times \frac{1.2^2 + 3.5^2}{1.2 + 3.5} = 0.5825(53\ldots)$ | A1ft | PI; condone 3dp |
| $y(1.2) = y(1) + \frac{1}{2}[0.5 + 0.5825(53\ldots)]$ | m1 | |
| $= 3.54127\ldots = 3.5413$ to 4dp | A1ft | **Total: 5**; ft one slip; if answer not to 4dp withhold this mark |
---1 The function $y ( x )$ satisfies the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$
where
$$\mathrm { f } ( x , y ) = \frac { x ^ { 2 } + y ^ { 2 } } { x + y }$$
and
$$y ( 1 ) = 3$$
\begin{enumerate}[label=(\alph*)]
\item Use the Euler formula
$$y _ { r + 1 } = y _ { r } + h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$
with $h = 0.2$, to obtain an approximation to $y ( 1.2 )$.
\item Use the improved Euler formula
$$y _ { r + 1 } = y _ { r } + \frac { 1 } { 2 } \left( k _ { 1 } + k _ { 2 } \right)$$
where $k _ { 1 } = h \mathrm { f } \left( x _ { r } , y _ { r } \right)$ and $k _ { 2 } = h \mathrm { f } \left( x _ { r } + h , y _ { r } + k _ { 1 } \right)$ and $h = 0.2$, to obtain an approximation to $y ( 1.2 )$, giving your answer to four decimal places.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2009 Q1 [8]}}