| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Differential equations |
| Type | Exponential growth/decay - direct proportionality (dN/dt = kN) |
| Difficulty | Moderate -0.8 This is a standard textbook radioactive decay question requiring straightforward application of separable differential equations. Part (a) is direct translation to dN/dt = -kN, part (b) is routine separation and integration, parts (c-d) involve simple substitution and calculation with given values. No novel insight or complex problem-solving required—purely procedural application of a well-practiced technique. |
| Spec | 1.08l Interpret differential equation solutions: in context |
A radioactive isotope decays in such a way that the rate of change of the number $N$ of radioactive atoms present after $t$ days, is proportional to $N$.
\begin{enumerate}[label=(\alph*)]
\item Write down a differential equation relating $N$ and $t$. [2]
\item Show that the general solution may be written as $N = Ae^{-kt}$, where $A$ and $k$ are positive constants. [5]
\end{enumerate}
Initially the number of radioactive atoms present is $7 \times 10^{18}$ and 8 days later the number present is $3 \times 10^{17}$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the value of $k$. [3]
\item Find the number of radioactive atoms present after a further 8 days. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q12 [12]}}