| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 1 (Further Pure Core 1) |
| Year | 2021 |
| Session | June |
| Marks | 16 |
| Topic | Systems of differential equations |
| Type | Determine long-term or comparative behaviour |
| Difficulty | Challenging +1.2 This is a structured Further Maths question on coupled differential equations with clear steps: differentiate to eliminate variables (routine), solve a second-order linear DE with constant coefficients (standard FM technique), and apply initial conditions. Part (d) requires interpretation but follows directly from the solutions. While FM content is inherently harder than standard A-level, this is a textbook application with scaffolding rather than requiring novel insight. |
| Spec | 4.10h Coupled systems: simultaneous first order DEs |
5 In a predator-prey environment the population, at time $t$ years, of predators is $x$ and prey is $y$. The populations of predators and prey are measured in hundreds.
The populations are modelled by the following simultaneous differential equations.\\
$\frac { \mathrm { d } x } { \mathrm {~d} t } = y \quad \frac { \mathrm {~d} y } { \mathrm {~d} t } = 2 y - 5 x$
\begin{enumerate}[label=(\alph*)]
\item Show that $\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } - 5 x$.
\item \begin{enumerate}[label=(\roman*)]
\item Find the general solution for $x$.
\item Find the equivalent general solution for $y$.
Initially there are 100 predators and 300 prey.
\end{enumerate}\item Find the particular solutions for $x$ and $y$.
\item Determine whether the model predicts that the predators will die out before the prey.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 1 2021 Q5 [16]}}