| Exam Board | Edexcel |
|---|---|
| Module | CP1 (Core Pure 1) |
| Session | Specimen |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Tank/reservoir mixing problems |
| Difficulty | Standard +0.8 This is a multi-step applied differential equations problem requiring students to (a) derive a DE from a word problem involving rates of change, (b) solve using integrating factor method with non-constant coefficients, and (c) critically evaluate the model. While the integrating factor technique is standard for Further Maths, the derivation and application context elevate it above routine exercises, though it remains accessible with systematic working. |
| Spec | 4.10g Damped oscillations: model and interpret |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Pond contains \(1000+5t\) litres after \(t\) days | M1 | Forms expression of form \(1000+kt\) for volume at time \(t\) |
| Rate of pollutant out \(=20\times\dfrac{x}{1000+5t}\) g per day | M1 | Expresses amount of pollutant out in terms of \(x\) and \(t\) |
| Rate of pollutant in \(=25\times2=50\) g per day | B1 | Correct interpretation for pollutant entering pond |
| \(\dfrac{dx}{dt}=50-\dfrac{4x}{200+t}\) | A1* | Puts all components together to form correct differential equation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(I=e^{\int\frac{4}{200+t}dt}=(200+t)^4\Rightarrow x(200+t)^4=\int50(200+t)^4\,dt\) | M1 | Uses model to find integrating factor and attempts solution |
| \(x(200+t)^4=10(200+t)^5+c\) | A1 | Correct solution |
| \(x=0,\;t=0\Rightarrow c=-3.2\times10^{12}\) | M1 | Interprets initial conditions to find constant of integration |
| \(t=8\Rightarrow x=10(200+8)-\dfrac{3.2\times10^{12}}{(200+8)^4}\) | M1 | Uses solution to find amount of pollutant after 8 days |
| \(=370\text{g}\) | A1 | Correct number of grams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| e.g. The model should account for pollutant not dissolving immediately upon entry; the rate of leaking could vary with volume of water in the pond | B1 | Suggests a suitable refinement to the model |
## Question 5:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Pond contains $1000+5t$ litres after $t$ days | M1 | Forms expression of form $1000+kt$ for volume at time $t$ |
| Rate of pollutant out $=20\times\dfrac{x}{1000+5t}$ g per day | M1 | Expresses amount of pollutant out in terms of $x$ and $t$ |
| Rate of pollutant in $=25\times2=50$ g per day | B1 | Correct interpretation for pollutant entering pond |
| $\dfrac{dx}{dt}=50-\dfrac{4x}{200+t}$ | A1* | Puts all components together to form correct differential equation |
**(4 marks)**
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $I=e^{\int\frac{4}{200+t}dt}=(200+t)^4\Rightarrow x(200+t)^4=\int50(200+t)^4\,dt$ | M1 | Uses model to find integrating factor and attempts solution |
| $x(200+t)^4=10(200+t)^5+c$ | A1 | Correct solution |
| $x=0,\;t=0\Rightarrow c=-3.2\times10^{12}$ | M1 | Interprets initial conditions to find constant of integration |
| $t=8\Rightarrow x=10(200+8)-\dfrac{3.2\times10^{12}}{(200+8)^4}$ | M1 | Uses solution to find amount of pollutant after 8 days |
| $=370\text{g}$ | A1 | Correct number of grams |
**(5 marks)**
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| e.g. The model should account for pollutant not dissolving immediately upon entry; the rate of leaking could vary with volume of water in the pond | B1 | Suggests a suitable refinement to the model |
**(1 mark)**
**(10 marks total)**\begin{enumerate}
\item A pond initially contains 1000 litres of unpolluted water.
\end{enumerate}
The pond is leaking at a constant rate of 20 litres per day.\\
It is suspected that contaminated water flows into the pond at a constant rate of 25 litres per day and that the contaminated water contains 2 grams of pollutant in every litre of water.
It is assumed that the pollutant instantly dissolves throughout the pond upon entry.\\
Given that there are $x$ grams of the pollutant in the pond after $t$ days,\\
(a) show that the situation can be modelled by the differential equation,
$$\frac { \mathrm { d } x } { \mathrm {~d} t } = 50 - \frac { 4 x } { 200 + t }$$
(b) Hence find the number of grams of pollutant in the pond after 8 days.\\
(c) Explain how the model could be refined.
\hfill \mbox{\textit{Edexcel CP1 Q5 [10]}}