| Exam Board | Edexcel |
|---|---|
| Module | CP2 (Core Pure 2) |
| Session | Specimen |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Systems of differential equations |
| Type | Predict population extinction or event time |
| Difficulty | Standard +0.8 This is a substantial coupled differential equations problem requiring multiple techniques: deriving a second-order DE from a system, solving it with auxiliary equation, finding the coupled solution, applying initial conditions, and critically interpreting the model. While the individual steps are standard Further Maths techniques, the multi-stage nature, coupled system context, and requirement for model critique make it moderately challenging—above average but not requiring novel mathematical insight. |
| Spec | 4.10d Second order homogeneous: auxiliary equation method4.10h Coupled systems: simultaneous first order DEs |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(r=10\frac{df}{dt}-2f \Rightarrow \frac{dr}{dt}=10\frac{d^2f}{dt^2}-2\frac{df}{dt}\) | M1 | Attempts to differentiate the first equation with respect to \(t\) |
| \(10\frac{d^2f}{dt^2}-2\frac{df}{dt}=-0.2f+0.4\!\left(10\frac{df}{dt}-2f\right)\) | M1 | Substitutes into the second equation |
| \(\frac{d^2f}{dt^2}-0.6\frac{df}{dt}+0.1f=0\) | A1* | Achieves printed answer with no errors |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(m^2-0.6m+0.1=0 \Rightarrow m=\frac{0.6\pm\sqrt{0.6^2-4\times0.1}}{2}\) | M1 | Uses the model to form and solve the auxiliary equation |
| \(m=0.3\pm0.1i\) | A1 | Correct values for \(m\) |
| \(f=e^{\alpha t}(A\cos\beta t+B\sin\beta t)\) | M1 | Uses the model to form the CF |
| \(f=e^{0.3t}(A\cos0.1t+B\sin0.1t)\) | A1 | Correct CF |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{df}{dt}=0.3e^{0.3t}(A\cos0.1t+B\sin0.1t)+0.1e^{0.3t}(B\cos0.1t-A\sin0.1t)\) | M1 | Differentiates the expression for number of foxes |
| \(r=10\frac{df}{dt}-2f = e^{0.3t}\!\left((3A+B)\cos0.1t+(3B-A)\sin0.1t\right)-2e^{0.3t}(A\cos0.1t+B\sin0.1t)\) | M1 | Uses result to find expression for number of rabbits |
| \(r=e^{0.3t}\!\left((A+B)\cos0.1t+(B-A)\sin0.1t\right)\) | A1 | Correct equation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(t=0,\, f=6 \Rightarrow A=6\) | M1 | Uses initial conditions in model for number of foxes |
| \(t=0,\, r=20 \Rightarrow B=14\) | M1 | Uses initial conditions in model for number of rabbits to find both unknown constants |
| \(r=e^{0.3t}(20\cos0.1t+8\sin0.1t)=0\) | M1 | Obtains expression for \(r\) in terms of \(t\) and sets \(=0\) |
| \(\tan0.1t=-2.5\) | A1 | Rearranges and obtains correct value for tan |
| \(2019\) | A1 | Identifies the correct year |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(3750\) foxes | B1 | Correct number of foxes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| e.g. the model predicts a large number of foxes on the island when the rabbits have died out and this may not be sensible | B1 | Makes a suitable comment on the outcome of the model |
## Question 7:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $r=10\frac{df}{dt}-2f \Rightarrow \frac{dr}{dt}=10\frac{d^2f}{dt^2}-2\frac{df}{dt}$ | M1 | Attempts to differentiate the first equation with respect to $t$ |
| $10\frac{d^2f}{dt^2}-2\frac{df}{dt}=-0.2f+0.4\!\left(10\frac{df}{dt}-2f\right)$ | M1 | Substitutes into the second equation |
| $\frac{d^2f}{dt^2}-0.6\frac{df}{dt}+0.1f=0$ | A1* | Achieves printed answer with no errors |
**(3 marks)**
---
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $m^2-0.6m+0.1=0 \Rightarrow m=\frac{0.6\pm\sqrt{0.6^2-4\times0.1}}{2}$ | M1 | Uses the model to form and solve the auxiliary equation |
| $m=0.3\pm0.1i$ | A1 | Correct values for $m$ |
| $f=e^{\alpha t}(A\cos\beta t+B\sin\beta t)$ | M1 | Uses the model to form the CF |
| $f=e^{0.3t}(A\cos0.1t+B\sin0.1t)$ | A1 | Correct CF |
**(4 marks)**
---
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{df}{dt}=0.3e^{0.3t}(A\cos0.1t+B\sin0.1t)+0.1e^{0.3t}(B\cos0.1t-A\sin0.1t)$ | M1 | Differentiates the expression for number of foxes |
| $r=10\frac{df}{dt}-2f = e^{0.3t}\!\left((3A+B)\cos0.1t+(3B-A)\sin0.1t\right)-2e^{0.3t}(A\cos0.1t+B\sin0.1t)$ | M1 | Uses result to find expression for number of rabbits |
| $r=e^{0.3t}\!\left((A+B)\cos0.1t+(B-A)\sin0.1t\right)$ | A1 | Correct equation |
**(3 marks)**
---
### Part (d)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $t=0,\, f=6 \Rightarrow A=6$ | M1 | Uses initial conditions in model for number of foxes |
| $t=0,\, r=20 \Rightarrow B=14$ | M1 | Uses initial conditions in model for number of rabbits to find both unknown constants |
| $r=e^{0.3t}(20\cos0.1t+8\sin0.1t)=0$ | M1 | Obtains expression for $r$ in terms of $t$ and sets $=0$ |
| $\tan0.1t=-2.5$ | A1 | Rearranges and obtains correct value for tan |
| $2019$ | A1 | Identifies the correct year |
---
### Part (d)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $3750$ foxes | B1 | Correct number of foxes |
---
### Part (d)(iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| e.g. the model predicts a large number of foxes on the island when the rabbits have died out and this may not be sensible | B1 | Makes a suitable comment on the outcome of the model |
**(7 marks for part (d); 17 marks total for Question 7)**\begin{enumerate}
\item At the start of the year 2000, a survey began of the number of foxes and rabbits on an island.
\end{enumerate}
At time $t$ years after the survey began, the number of foxes, $f$, and the number of rabbits, $r$, on the island are modelled by the differential equations
$$\begin{aligned}
& \frac { \mathrm { d } f } { \mathrm {~d} t } = 0.2 f + 0.1 r \\
& \frac { \mathrm {~d} r } { \mathrm {~d} t } = - 0.2 f + 0.4 r
\end{aligned}$$
(a) Show that $\frac { \mathrm { d } ^ { 2 } f } { \mathrm {~d} t ^ { 2 } } - 0.6 \frac { \mathrm {~d} f } { \mathrm {~d} t } + 0.1 f = 0$\\
(b) Find a general solution for the number of foxes on the island at time $t$ years.\\
(c) Hence find a general solution for the number of rabbits on the island at time $t$ years.
At the start of the year 2000 there were 6 foxes and 20 rabbits on the island.\\
(d) (i) According to this model, in which year are the rabbits predicted to die out?\\
(ii) According to this model, how many foxes will be on the island when the rabbits die out?\\
(iii) Use your answers to parts (i) and (ii) to comment on the model.
\hfill \mbox{\textit{Edexcel CP2 Q7 [17]}}