| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2006 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Separable variables - standard (applied/contextual) |
| Difficulty | Moderate -0.3 This is a straightforward separable variables question with standard integration of a power function. Part (a) requires routine separation and integration with substitution u = x - 6, while part (b) involves basic interpretation and substitution into the solution. The techniques are standard C4 material with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y)1.08l Interpret differential equation solutions: in context |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\int \frac{dx}{\sqrt{x-6}} = \int -2\ dt\) | M1 | Attempt to separate and integrate |
| \(2\sqrt{x-6} = -2t + c\) | A1A1 | \(c\) on either side |
| \(t=0\), \(x=70 \Rightarrow c=16\) | m1A1F | Follow on \(c\) from sensible attempt at integrals (\(\sqrt{}\) not \(\ln\)) |
| \(t = 8 - \sqrt{x-6}\) | A1 | CAO (or AEF) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| The liquid level stops falling/flowing at minimum depth | B1 | |
| \(x=22\): \(t = 8 - \sqrt{22-6}\) | M1 | Use \(x=22\) in their equation provided there is a \(c\); Or start again using limits; M1 \(2\sqrt{64} - 2\sqrt{16} = \pm 2t\), A1 \(t=4\) |
| \(t = 4\) | A1 | CAO |
## Question 8:
### Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\int \frac{dx}{\sqrt{x-6}} = \int -2\ dt$ | M1 | Attempt to separate and integrate |
| $2\sqrt{x-6} = -2t + c$ | A1A1 | $c$ on either side |
| $t=0$, $x=70 \Rightarrow c=16$ | m1A1F | Follow on $c$ from sensible attempt at integrals ($\sqrt{}$ not $\ln$) |
| $t = 8 - \sqrt{x-6}$ | A1 | CAO (or AEF) |
### Part (b)(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| The liquid level stops falling/flowing at minimum depth | B1 | |
| $x=22$: $t = 8 - \sqrt{22-6}$ | M1 | Use $x=22$ in their equation provided there is a $c$; Or start again using limits; M1 $2\sqrt{64} - 2\sqrt{16} = \pm 2t$, A1 $t=4$ |
| $t = 4$ | A1 | CAO |8
\begin{enumerate}[label=(\alph*)]
\item Solve the differential equation
$$\frac { \mathrm { d } x } { \mathrm {~d} t } = - 2 ( x - 6 ) ^ { \frac { 1 } { 2 } }$$
to find $t$ in terms of $x$, given that $x = 70$ when $t = 0$.
\item Liquid fuel is stored in a tank. At time $t$ minutes, the depth of fuel in the tank is $x \mathrm {~cm}$. Initially there is a depth of 70 cm of fuel in the tank. There is a tap 6 cm above the bottom of the tank. The flow of fuel out of the tank is modelled by the differential equation
$$\frac { \mathrm { d } x } { \mathrm {~d} t } = - 2 ( x - 6 ) ^ { \frac { 1 } { 2 } }$$
\begin{enumerate}[label=(\roman*)]
\item Explain what happens when $x = 6$.
\item Find how long it will take for the depth of fuel to fall from 70 cm to 22 cm .
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C4 2006 Q8 [9]}}