| 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 | Moderate -0.3 This is a structured multi-part differential equations question with clear scaffolding through parts (i)-(iv). Part (i) is a standard substitution integral, part (ii) is routine partial fractions, part (iii) applies these results to solve a separable DE (though the algebra requires care), and part (iv) involves simple substitution and limit evaluation. While it requires multiple techniques, the question guides students through each step methodically, making it slightly easier than average for C4 level. |
| 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
$$\frac{dM}{dt} = \frac{M}{t(1+t^2)}.$$
\begin{enumerate}[label=(\roman*)]
\item Find $\int \frac{t}{1+t^2} dt$. [3]
\item Find constants $A$, $B$ and $C$ such that
$$\frac{1}{t(1+t^2)} = \frac{A}{t} + \frac{Bt+C}{1+t^2}.$$ [5]
\item Use integration, together with your results in parts (i) and (ii), to show that
$$M = \frac{Kt}{\sqrt{1+t^2}},$$
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 Q1 [18]}}