| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | First-order integration |
| Difficulty | Moderate -0.3 This is a straightforward C4 differential equations question requiring standard techniques: finding when dn/dt = 0 for part (a), integrating with an initial condition for part (b), and a simple interpretation for part (c). The integration is routine (exponential and constant), and no novel problem-solving is needed. Slightly easier than average due to the mechanical nature of all parts. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.08k Separable differential equations: dy/dx = f(x)g(y)1.08l Interpret differential equation solutions: in context |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\frac{dn}{dt} = 0 \Rightarrow e^{0.5t} = 5\) | M1 | |
| \(t = 2\ln 5 = 3.219\) mins \(= 3\) mins \(13\) secs | M1 A1 | |
| (b) \(\int dn = \int (e^{0.5t} - 5) \, dt\) | M1 A1 | |
| \(n = 2e^{0.5t} - 5t + c\) | M1 A1 | |
| \(t = 0, n = 20 \Rightarrow 20 = 2 + c, \quad c = 18\) | M1 | |
| \(n = 2e^{0.5t} - 5t + 18\) | A1 | |
| (c) as \(t\) increases, \(n\) rapidly becomes very large \(\therefore\) not realistic | B1 | (8) |
**(a)** $\frac{dn}{dt} = 0 \Rightarrow e^{0.5t} = 5$ | M1 |
$t = 2\ln 5 = 3.219$ mins $= 3$ mins $13$ secs | M1 A1 |
**(b)** $\int dn = \int (e^{0.5t} - 5) \, dt$ | M1 A1 |
$n = 2e^{0.5t} - 5t + c$ | M1 A1 |
$t = 0, n = 20 \Rightarrow 20 = 2 + c, \quad c = 18$ | M1 |
$n = 2e^{0.5t} - 5t + 18$ | A1 |
**(c)** as $t$ increases, $n$ rapidly becomes very large $\therefore$ not realistic | B1 | (8) |
---The number of people, $n$, in a queue at a Post Office $t$ minutes after it opens is modelled by the differential equation
$$\frac{dn}{dt} = e^{0.5t} - 5, \quad t \geq 0.$$
\begin{enumerate}[label=(\alph*)]
\item Find, to the nearest second, the time when the model predicts that there will be the least number of people in the queue. [3]
\item Given that there are 20 people in the queue when the Post Office opens, solve the differential equation. [4]
\item Explain why this model would not be appropriate for large values of $t$. [1]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q1 [8]}}