| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2010 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Separable variables - standard (applied/contextual) |
| Difficulty | Standard +0.3 This is a straightforward separable variables question with standard steps: form the differential equation from a verbal statement, separate variables, integrate (using a simple substitution u = 2A - 5), apply initial conditions to find the constant, then use a second condition to find the proportionality constant. All techniques are routine for P3 level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 4.10a General/particular solutions: of differential equations |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State \(\frac{dA}{dt} = k\sqrt{2A - 5}\) | B1 | [1] |
| (ii) Separate variables correctly and attempt integration of each side | M1 | |
| Obtain \((2A - 5)^{\frac{1}{2}} = \ldots\) or equivalent | A1 | |
| Obtain \(= kt\) or equivalent | A1 | |
| Use \(t = 0\) and \(A = 7\) to find value of arbitrary constant | M1 | |
| Obtain \(C = 3\) or equivalent | A1 | |
| Use \(t = 10\) and \(A = 27\) to find \(k\) | M1 | |
| Obtain \(k = 0.4\) or equivalent | A1 | |
| Substitute \(t = 20\) and values for \(C\) and \(k\) to find value of \(A\) | M1 | |
| Obtain \(63\) | A1 | cwo |
**(i)** State $\frac{dA}{dt} = k\sqrt{2A - 5}$ | B1 | [1]
**(ii)** Separate variables correctly and attempt integration of each side | M1 |
Obtain $(2A - 5)^{\frac{1}{2}} = \ldots$ or equivalent | A1 |
Obtain $= kt$ or equivalent | A1 |
Use $t = 0$ and $A = 7$ to find value of arbitrary constant | M1 |
Obtain $C = 3$ or equivalent | A1 |
Use $t = 10$ and $A = 27$ to find $k$ | M1 |
Obtain $k = 0.4$ or equivalent | A1 |
Substitute $t = 20$ and values for $C$ and $k$ to find value of $A$ | M1 |
Obtain $63$ | A1 | cwo | [9]9 A biologist is investigating the spread of a weed in a particular region. At time $t$ weeks after the start of the investigation, the area covered by the weed is $A \mathrm {~m} ^ { 2 }$. The biologist claims that the rate of increase of $A$ is proportional to $\sqrt { } ( 2 A - 5 )$.\\
(i) Write down a differential equation representing the biologist's claim.\\
(ii) At the start of the investigation, the area covered by the weed was $7 \mathrm {~m} ^ { 2 }$ and, 10 weeks later, the area covered was $27 \mathrm {~m} ^ { 2 }$. Assuming that the biologist's claim is correct, find the area covered 20 weeks after the start of the investigation.
\hfill \mbox{\textit{CAIE P3 2010 Q9 [10]}}