| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2006 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Separable variables - standard (polynomial/exponential x-side) |
| Difficulty | Standard +0.3 This is a standard separable differential equations question requiring students to form the DE from a worded statement, then solve by separation of variables and apply two boundary conditions. While it involves multiple steps (8 marks total), the techniques are routine C4 material with no novel insight required—slightly easier than average due to the straightforward setup and algebraic manipulation. |
| Spec | 1.07t Construct differential equations: in context1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{dA}{dt}\) or \(kA^2\) seen | M1 | |
| \(\frac{dA}{dt} = kA^2\) | A1 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Separate variables + attempt to integrate | *M1 | Accept if based on \(\frac{dA}{dr} = kA^2\) or \(A^2\) |
| \(-\frac{1}{A} = kt + c\) or \(-\frac{1}{kA} = t + c\) or \(-\frac{1}{A} = t + c\) | A1 | |
| Subst one of (0,0), (1,1000) or (2,2000) into eqn. | dep*M1 | Equation must contain \(k\) and/or \(c\) |
| Subst another of (0,0), (1,1000) or (2,2000) into eqn | dep*M1 | This equation must contain \(k\) and \(c\) |
| Substitute \(A = 3000\) into eqn with \(k\) and \(c\) subst | dep*M1 | |
| \(t = \frac{7}{3}\) | A1 | 6 Accept 2.33, 2h 20 m |
| ISW |
### (i)
$\frac{dA}{dt}$ or $kA^2$ seen | M1 |
$\frac{dA}{dt} = kA^2$ | A1 | 2
---
### (ii)
Separate variables + attempt to integrate | *M1 | Accept if based on $\frac{dA}{dr} = kA^2$ or $A^2$
$-\frac{1}{A} = kt + c$ or $-\frac{1}{kA} = t + c$ or $-\frac{1}{A} = t + c$ | A1 |
Subst one of (0,0), (1,1000) or (2,2000) into eqn. | dep*M1 | Equation must contain $k$ and/or $c$
Subst another of (0,0), (1,1000) or (2,2000) into eqn | dep*M1 | This equation must contain $k$ and $c$
Substitute $A = 3000$ into eqn with $k$ and $c$ subst | dep*M1 |
$t = \frac{7}{3}$ | A1 | 6 Accept 2.33, 2h 20 m
ISW | |
---A forest is burning so that, $t$ hours after the start of the fire, the area burnt is $A$ hectares. It is given that, at any instant, the rate at which this area is increasing is proportional to $A^2$.
\begin{enumerate}[label=(\roman*)]
\item Write down a differential equation which models this situation. [2]
\item After 1 hour, 1000 hectares have been burnt; after 2 hours, 2000 hectares have been burnt. Find after how many hours 3000 hectares have been burnt. [6]
\end{enumerate}
\hfill \mbox{\textit{OCR C4 2006 Q5 [8]}}