| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2007 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Iterative/numerical methods |
| Difficulty | Standard +0.3 This is a straightforward application of two standard numerical methods (Euler and improved Euler) with explicit formulas provided. Students only need to substitute given values into the formulas and perform arithmetic calculations—no conceptual insight or problem-solving is required. While it's a Further Maths topic, the mechanical nature and provision of all formulas makes it easier than an average A-level question. |
| Spec | 1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y_1 = 2 + 0.1 \times \sqrt{1^2 + 2^2 + 3}\) | M1 | |
| \(y(1.1) = 2 + 0.1 \times \sqrt{8}\) | A1 | |
| \(y(1.1) = 2.28284\ldots = 2.2828\) to 4dp | A1 | 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(k_1 = 0.1 \times \sqrt{8} = 0.2828\) | M1, A1ft | PI |
| \(k_2 = 0.1 \times f(1.1,\ 2.2828\ldots)\) | M1 | |
| \(= 0.1 \times \sqrt{9.42137\ldots} = 0.3069(425\ldots)\) | A1 | PI |
| \(y(1.1) = y(1) + \frac{1}{2}[0.28284\ldots + 0.30694\ldots]\) | m1 | |
| \(2.29489\ldots = 2.2949\) to 4dp | A1 | 6 marks |
## Question 2:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y_1 = 2 + 0.1 \times \sqrt{1^2 + 2^2 + 3}$ | M1 | |
| $y(1.1) = 2 + 0.1 \times \sqrt{8}$ | A1 | |
| $y(1.1) = 2.28284\ldots = 2.2828$ to 4dp | A1 | **3 marks** |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $k_1 = 0.1 \times \sqrt{8} = 0.2828$ | M1, A1ft | PI |
| $k_2 = 0.1 \times f(1.1,\ 2.2828\ldots)$ | M1 | |
| $= 0.1 \times \sqrt{9.42137\ldots} = 0.3069(425\ldots)$ | A1 | PI |
| $y(1.1) = y(1) + \frac{1}{2}[0.28284\ldots + 0.30694\ldots]$ | m1 | |
| $2.29489\ldots = 2.2949$ to 4dp | A1 | **6 marks** |
---2 The function $y ( x )$ satisfies the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$
where
$$\mathrm { f } ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } + 3 }$$
and
$$y ( 1 ) = 2$$
\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 $y ( 1.1 )$, giving your answer to four decimal places.
\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 $h = 0.1$, to obtain an approximation to $y ( 1.1 )$, giving your answer to four decimal places.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2007 Q2 [9]}}