| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Chemical reaction kinetics |
| Difficulty | Standard +0.8 This is a substantial C4 differential equations problem requiring partial fractions decomposition, separation of variables, integration, and manipulation of logarithms to reach a specific form, followed by applying initial conditions and finding limits. While the techniques are all standard C4 content, the multi-step nature (13 marks total), algebraic manipulation required, and need to work toward a given answer make this moderately harder than average. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y) |
In a chemical reaction two substances combine to form a third substance. At time $t$, $t \geq 0$, the concentration of this third substance is $x$ and the reaction is modelled by the differential equation
$$\frac{dx}{dt} = k(1 - 2x)(1 - 4x), \text{ where } k \text{ is a positive constant.}$$
\begin{enumerate}[label=(\alph*)]
\item Solve this differential equation and hence show that
$$\ln \left| \frac{1 - 2x}{1 - 4x} \right| = 2kt + c, \text{ where } c \text{ is an arbitrary constant.}$$ [7]
\item Given that $x = 0$ when $t = 0$, find an expression for $x$ in terms of $k$ and $t$. [4]
\item Find the limiting value of the concentration $x$ as $t$ becomes very large. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q8 [13]}}