| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Topic | Differential equations |
| Type | Exponential growth/decay - approach to limit (dN/dt = k(N - Nâ‚€)) |
| Difficulty | Standard +0.3 This is a standard C4 differential equations question with clear scaffolding. Part (a) tests understanding of rate equations (2 marks, routine explanation). Part (b) is a standard separable DE requiring integration and applying initial conditions to find constants—textbook procedure. Part (c) uses given information to find k then evaluate V(10). While multi-step (13 marks total), each step follows established techniques with no novel insight required. Slightly easier than average due to heavy scaffolding and standard method application. |
| Spec | 4.10a General/particular solutions: of differential equations4.10c Integrating factor: first order equations |
Liquid is pouring into a container at a constant rate of $20\text{ cm}^3\text{s}^{-1}$ and is leaking out at a rate proportional to the volume of the liquid already in the container.
\begin{enumerate}[label=(\alph*)]
\item Explain why, at time $t$ seconds, the volume, $V\text{ cm}^3$, of liquid in the container satisfies the differential equation
$$\frac{dV}{dt} = 20 - kV,$$
where $k$ is a positive constant. [2]
\end{enumerate}
The container is initially empty.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item By solving the differential equation, show that
$$V = A + Be^{-kt},$$
giving the values of $A$ and $B$ in terms of $k$. [6]
\end{enumerate}
Given also that $\frac{dV}{dt} = 10$ when $t = 5$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item find the volume of liquid in the container at 10 s after the start. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q8 [13]}}