| Exam Board | OCR |
|---|---|
| Module | H240/01 (Pure Mathematics) |
| Year | 2018 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Topic | Differential equations |
| Type | Model comparison/critique |
| Difficulty | Standard +0.8 This question requires setting up and solving two different differential equations (one with separation of variables, one more complex), applying initial conditions, and critically comparing model behaviors. The inverse proportionality setup, the non-standard exponential term in part (ii), and the model comparison requiring asymptotic analysis make this significantly harder than routine DE questions, though it remains accessible to strong A-level students. |
| Spec | 1.07t Construct differential equations: in context1.08k Separable differential equations: dy/dx = f(x)g(y) |
13 A scientist is attempting to model the number of insects, $N$, present in a colony at time $t$ weeks. When $t = 0$ there are 400 insects and when $t = 1$ there are 440 insects.\\
(i) A scientist assumes that the rate of increase of the number of insects is inversely proportional to the number of insects present at time $t$.
\begin{enumerate}[label=(\alph*)]
\item Write down a differential equation to model this situation.
\item Solve this differential equation to find $N$ in terms of $t$.\\
(ii) In a revised model it is assumed that $\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { N ^ { 2 } } { 3988 \mathrm { e } ^ { 0.2 t } }$. Solve this differential equation to find $N$ in terms of $t$.\\
(iii) Compare the long-term behaviour of the two models.
\end{enumerate}
\hfill \mbox{\textit{OCR H240/01 2018 Q13 [13]}}