| Exam Board | Edexcel |
|---|---|
| Module | FP1 AS (Further Pure 1 AS) |
| Year | 2018 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Newton's law of cooling |
| Difficulty | Moderate -0.5 This is a straightforward application of Euler's method with only two iterations and simple arithmetic (k=0.1 makes calculations easy). Part (b) requires minimal conceptual understanding of the cooling constant. Easier than average A-level as it's purely procedural with no problem-solving or insight required. |
| Spec | 1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams4.10a General/particular solutions: of differential equations |
| V349 SIHI NI IMIMM ION OC | VJYV SIHIL NI LIIIM ION OO | VJYV SIHIL NI JIIYM ION OC |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Uses \(h = 1.5\), \(\theta_0 = 80\), \(k = 0.1\) in complete strategy to find \(\theta_1\) | M1 | Two iterations from \(t=0\) to \(t=3\), so \(h=1.5\) |
| \(\left(\frac{d\theta}{dt}\right)_0 = -0.1(80-20) = -6\) | M1 | Uses model to evaluate initial value of \(\frac{d\theta}{dt}\) using \(k=0.1\) and \(\theta_0 = 80\) |
| \(\frac{\theta_1 - 80}{1.5} = -6 \Rightarrow \theta_1 = 80 + (1.5)(-6)\) | M1 | Applies approximation formula with \(\theta_0=80\), \(k=0.1\) and their \(h\) |
| \(\theta_1 = 71\) | A1 | Finds approximation for \(\theta\) at 1.5 minutes as 71 |
| \(\left(\frac{d\theta}{dt}\right)_1 = -0.1(71-20) = -5.1\) | M1 | Uses their 71 and \(k=0.1\) to find \(\frac{d\theta}{dt}\) |
| \(\theta_2 = 71 + (1.5)(-5.1) = 63.35\ (°C)\) | A1 | Applies approximation formula again to give \(63.35°C\) or awrt \(63°C\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Decrease \(k\) to become a smaller positive value | B1 | Allow B1 for "the value of \(k\) should satisfy \(0 < k < 0.1\)"; condone "the value of \(k\) would need to be decreased". Give B0 for "change \(k\) to become negative" |
# Question 2:
## Part (a)
| Working/Answer | Marks | Guidance |
|---|---|---|
| Uses $h = 1.5$, $\theta_0 = 80$, $k = 0.1$ in complete strategy to find $\theta_1$ | M1 | Two iterations from $t=0$ to $t=3$, so $h=1.5$ |
| $\left(\frac{d\theta}{dt}\right)_0 = -0.1(80-20) = -6$ | M1 | Uses model to evaluate initial value of $\frac{d\theta}{dt}$ using $k=0.1$ and $\theta_0 = 80$ |
| $\frac{\theta_1 - 80}{1.5} = -6 \Rightarrow \theta_1 = 80 + (1.5)(-6)$ | M1 | Applies approximation formula with $\theta_0=80$, $k=0.1$ and their $h$ |
| $\theta_1 = 71$ | A1 | Finds approximation for $\theta$ at 1.5 minutes as 71 |
| $\left(\frac{d\theta}{dt}\right)_1 = -0.1(71-20) = -5.1$ | M1 | Uses their 71 and $k=0.1$ to find $\frac{d\theta}{dt}$ |
| $\theta_2 = 71 + (1.5)(-5.1) = 63.35\ (°C)$ | A1 | Applies approximation formula again to give $63.35°C$ or awrt $63°C$ |
**(6 marks)**
## Part (b)
| Working/Answer | Marks | Guidance |
|---|---|---|
| Decrease $k$ to become a smaller positive value | B1 | Allow B1 for "the value of $k$ should satisfy $0 < k < 0.1$"; condone "the value of $k$ would need to be decreased". Give B0 for "change $k$ to become negative" |
**(1 mark)**\begin{enumerate}
\item The temperature, $\theta ^ { \circ } \mathrm { C }$, of coffee in a cup, $t$ minutes after the cup of coffee is put in a room, is modelled by the differential equation
\end{enumerate}
$$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = - k ( \theta - 20 )$$
where $k$ is a constant.\\
The coffee has an initial temperature of $80 ^ { \circ } \mathrm { C }$\\
Using $k = 0.1$\\
(a) use two iterations of the approximation formula $\left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) _ { 0 } = \frac { y _ { 1 } - y _ { 0 } } { h }$ to estimate the temperature of the coffee 3 minutes after it was put in the room.
The coffee in a different cup, which also had an initial temperature of $80 ^ { \circ } \mathrm { C }$ when it was put in the room, cools more slowly.\\
(b) Use this information to suggest how the value of $k$ would need to be changed in the model.
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\hfill \mbox{\textit{Edexcel FP1 AS 2018 Q2 [7]}}