| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2011 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Separable variables - standard (applied/contextual) |
| Difficulty | Standard +0.8 This is a substantial multi-part question requiring chain rule differentiation, separating variables, integration including partial fractions manipulation, and interpretation of limiting behavior. While individual techniques are standard C4 content, the algebraic manipulation in part (v) and the sustained reasoning across 6 parts with 18 marks elevates this above average difficulty. The question requires careful tracking of relationships between variables and non-trivial algebraic insight, but doesn't require novel problem-solving approaches beyond the syllabus. |
| Spec | 1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.08k Separable differential equations: dy/dx = f(x)g(y)1.08l Interpret differential equation solutions: in context |
Water is leaking from a container. After $t$ seconds, the depth of water in the container is $x$ cm, and the volume of water is $V$ cm$^3$, where $V = \frac{1}{3}x^3$. The rate at which water is lost is proportional to $x$, so that $\frac{dV}{dt} = -kx$, where $k$ is a constant.
\begin{enumerate}[label=(\roman*)]
\item Show that $x \frac{dx}{dt} = -k$. [3]
\end{enumerate}
Initially, the depth of water in the container is 10 cm.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Show by integration that $x = \sqrt{100 - 2kt}$. [4]
\item Given that the container empties after 50 seconds, find $k$. [2]
\end{enumerate}
Once the container is empty, water is poured into it at a constant rate of 1 cm$^3$ per second. The container continues to lose water as before.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{3}
\item Show that, $t$ seconds after starting to pour the water in, $\frac{dx}{dt} = \frac{1-x}{x^2}$. [2]
\item Show that $\frac{1}{1-x} - x - 1 = \frac{x^2}{1-x}$.
Hence solve the differential equation in part (iv) to show that
$$t = \ln\left(\frac{1}{1-x}\right) - \frac{1}{2}x^2 - x.$$ [6]
\item Show that the depth cannot reach 1 cm. [1]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C4 2011 Q8 [18]}}