| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Chemical reaction kinetics |
| Difficulty | Standard +0.3 This is a standard C4 differential equations question with routine integration techniques. Part (i) is a simple substitution, part (ii) is textbook partial fractions, part (iii) applies separation of variables with given results, and part (iv) is straightforward evaluation. All techniques are core syllabus with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08k Separable differential equations: dy/dx = f(x)g(y) |
In a chemical process, the mass $M$ grams of a chemical at time $t$ minutes is modelled by the differential equation
$$\frac{dM}{dt} = -\frac{M}{2(1 + \frac{t}{2})}$$
\begin{enumerate}[label=(\roman*)]
\item Find $\int \frac{1}{1 + \frac{t}{2}} dt$ [3]
\item Find constants $A$, $B$ and $C$ such that
$$\frac{1}{t(1 + \frac{t}{2})} = \frac{A}{t} + \frac{Bt + C}{1 + \frac{t}{2}}$$ [5]
\item Use integration, together with your results in parts (i) and (ii), to show that
$$M \sim \frac{K}{.1 + \frac{t}{2}}$$
where $K$ is a constant. [6]
\item When $t = 1$, $M = 25$. Calculate $K$
What is the mass of the chemical in the long term? [4]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C4 Q3 [18]}}