| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Tank/reservoir mixing problems |
| Difficulty | Moderate -0.3 This is a straightforward mixing problem with clear setup. Part (a) requires simple translation of the word problem into the given differential equation form. Part (b) is a standard integrating factor application with routine integration and boundary conditions. Part (c) asks for limit behavior which follows directly from the solution. While it requires multiple steps, each is procedural with no novel insight needed—slightly easier than average due to the scaffolded structure and explicit equation form provided. |
| Spec | 1.07t Construct differential equations: in context1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(a = 30\) and \(b = 0.01\) | B1 | |
| Total | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Separate variables and integrate one side | M1 | |
| Obtain terms \(-100\ln(30 - 0.01V)\) and \(t\), or equivalent | A1 FT + A1 FT | FT their \(a\) and \(b\) |
| Evaluate a constant, or use \(t=0\), \(V=0\) as limits, in a solution containing terms \(c\ln(30-0.01V)\) and \(dt\) where \(cd \neq 0\) | M1 | |
| Obtain solution \(100\ln 30 - 100\ln(30 - 0.01V) = t\), or equivalent | A1 | |
| Substitute \(V = 1000\) and obtain answer \(t = 40.5\) | A1 | |
| Total | 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Obtain \(V = 3000(1 - e^{-0.01t})\) | B1 | OE |
| State that \(V\) approaches 3000 | B1 | |
| Total | 2 |
## Question 10(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $a = 30$ and $b = 0.01$ | B1 | |
| **Total** | **1** | |
---
## Question 10(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Separate variables and integrate one side | M1 | |
| Obtain terms $-100\ln(30 - 0.01V)$ and $t$, or equivalent | A1 FT + A1 FT | FT their $a$ and $b$ |
| Evaluate a constant, or use $t=0$, $V=0$ as limits, in a solution containing terms $c\ln(30-0.01V)$ and $dt$ where $cd \neq 0$ | M1 | |
| Obtain solution $100\ln 30 - 100\ln(30 - 0.01V) = t$, or equivalent | A1 | |
| Substitute $V = 1000$ and obtain answer $t = 40.5$ | A1 | |
| **Total** | **6** | |
---
## Question 10(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Obtain $V = 3000(1 - e^{-0.01t})$ | B1 | OE |
| State that $V$ approaches 3000 | B1 | |
| **Total** | **2** | |
---10 A gardener is filling an ornamental pool with water, using a hose that delivers 30 litres of water per minute. Initially the pool is empty. At time $t$ minutes after filling begins the volume of water in the pool is $V$ litres. The pool has a small leak and loses water at a rate of $0.01 V$ litres per minute.
The differential equation satisfied by $V$ and $t$ is of the form $\frac { \mathrm { d } V } { \mathrm {~d} t } = a - b V$.
\begin{enumerate}[label=(\alph*)]
\item Write down the values of the constants $a$ and $b$.
\item Solve the differential equation and find the value of $t$ when $V = 1000$.
\item Obtain an expression for $V$ in terms of $t$ and hence state what happens to $V$ as $t$ becomes large.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2022 Q10 [9]}}