| Exam Board | AQA |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Session | Specimen |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Systems of differential equations |
| Type | Interpret model parameters from equations |
| Difficulty | Challenging +1.8 This is a Further Maths predator-prey modelling question requiring construction of coupled differential equations from verbal descriptions, then numerical solution. While the setup is lengthy and requires careful translation of rates into mathematical form, the actual techniques (forming DEs, Euler's method or numerical integration) are standard Further Maths content. The 9-mark allocation reflects extended working rather than exceptional conceptual difficulty, placing it well above average but below the most challenging proof-based questions. |
| Spec | 4.10b Model with differential equations: kinematics and other contexts4.10h Coupled systems: simultaneous first order DEs |
| Answer | Marks | Guidance |
|---|---|---|
| 15(a)(i) | Forms a differential equation | |
| for the foxes. | AO3.3 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| for the rabbits. | AO3.3 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| terms correct. | AO1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| linear differential equation. | AO3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| order differential equation. | AO1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| been awarded | AO1.1b | A1F |
| Answer | Marks | Guidance |
|---|---|---|
| their general solution. | AO3.4 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| from ‘their’ general solution. | AO3.4 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| rabbits. | AO1.1a | M1 |
| (a)(ii) | Substitutes 0.7 and obtains | |
| approximately 160. CAO | AO3.4 | A1 |
| (b) | States a suitable refinement |
| Answer | Marks | Guidance |
|---|---|---|
| or other valid refinement | AO3.5c | B1 |
| Answer | Marks |
|---|---|
| Total | 11 |
| Total | 100 |
Question 15:
--- 15(a)(i) ---
15(a)(i) | Forms a differential equation
for the foxes. | AO3.3 | B1 | dy dy
∝ x⇒ =kx
dt dt
dy
=20,x =200⇒k =0.1
dt
dy
=0.1x
dt
dx
=1.2x−1.1y
dt
1.2x 1 dx
y = −
1.1 1.1dt
dy 1.2 dx 1 d2x
= −
dt 1.1 dt 1.1dt2
1.2 dx 1 d2x
0.1x= −
1.1 dt 1.1dt2
d2x dx
−1.2 +0.11x=0
dt2 dt
λ2 −1.2λ+0.11=0
λ=0.1 or 1.1
x= Ae0.1t +Be1 . 1t
t =0,x=80
A+B=80
dx
t =0, =74
dt
74=0.1A+1.1B
74=0.1(80−B)+1.1B
B=66, A=14
x=14e0.1t +66e1.1t
Forms a differential equation
for the rabbits. | AO3.3 | B1
Differentiates ‘their’ equation
that contains y and obtains
expression with at least two
terms correct. | AO1.1a | M1
Formulates a second order
linear differential equation. | AO3.1a | M1
Obtains roots of auxiliary
equation for ‘their’ second
order differential equation. | AO1.1a | M1
States correct general
solution.
FT provided all M marks have
been awarded | AO1.1b | A1F
Uses initial population to find
equation linking constants for
their general solution. | AO3.4 | M1
Obtains initial rate of change
for rabbits from ‘their’ DE and
differentiates and obtains a
second equations for A and B
from ‘their’ general solution. | AO3.4 | M1
Finds A and B and states the
model for the number of
rabbits. | AO1.1a | M1
(a)(ii) | Substitutes 0.7 and obtains
approximately 160. CAO | AO3.4 | A1 | 14e0.07 +66e0.77 =157.6
(b) | States a suitable refinement
about the fact that an
increased rabbit population
will require more food supply
or other valid refinement | AO3.5c | B1 | Take account of the food available
for the rabbits as this may limit
population growth.
Total | 11
Total | 100An isolated island is populated by rabbits and foxes. At time $t$ the number of rabbits is $x$ and the number of foxes is $y$.
It is assumed that:
• The number of foxes increases at a rate proportional to the number of rabbits. When there are 200 rabbits the number of foxes is increasing at a rate of 20 foxes per unit period of time.
• If there were no foxes present, the number of rabbits would increase by 120% in a unit period of time.
• When both foxes and rabbits are present the foxes kill rabbits at a rate that is equal to 110% of the current number of foxes.
• At time $t = 0$, the number of foxes is 20 and the number of rabbits is 80.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Construct a mathematical model for the number of rabbits.
[9 marks]
\item Use this model to show that the number of rabbits has doubled after approximately 0.7 units of time.
[1 mark]
\end{enumerate}
\item Suggest one way in which the model that you have used for the number of rabbits could be refined.
[1 mark]
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 1 Q15 [11]}}