| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Iterative/numerical methods |
| Difficulty | Standard +0.8 This is a substantial multi-part question requiring: (a) setting up a differential equation from a word problem with proportionality, (b) solving using separation of variables with partial fractions and logarithms, then rearranging to a specific form, (c) applying iterative methods, and (d) finding a limiting value. The separation and algebraic manipulation in part (b) is non-trivial, and the question tests multiple techniques across 4 parts. However, each individual step follows standard A-level methods without requiring novel insight. |
| Spec | 1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams4.10a General/particular solutions: of differential equations4.10b Model with differential equations: kinematics and other contexts4.10c Integrating factor: first order equations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or imply equation of the form \(\frac{\mathrm{d}x}{\mathrm{d}t} = k\,\frac{x}{20-x}\) | M1 | |
| Obtain \(k = 19\) | A1 | AG |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Separate variables and integrate at least one side | M1 | |
| Obtain terms \(20\ln x - x\) and \(19t\), or equivalent | A1 A1 | |
| Evaluate a constant or use \(t = 0\) and \(x = 1\) as limits in a solution containing terms \(a\ln x\) and \(bt\) | M1 | |
| Substitute \(t = 1\) and rearrange the equation in the given form | A1 | AG |
| Total | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use \(x_{n+1} = e^{0.9+0.05x_n}\) correctly at least once | M1 | |
| Obtain final answer \(x = 2.83\) | A1 | |
| Show sufficient iterations to 4 decimal places to justify 2.83 to 2 d.p. or show there is a sign change in the interval \((2.825,\, 2.835)\) | A1 | |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Set \(x = 20\) and obtain answer \(t = 2.15\) | B1 | |
| Total | 1 |
## Question 10(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply equation of the form $\frac{\mathrm{d}x}{\mathrm{d}t} = k\,\frac{x}{20-x}$ | M1 | |
| Obtain $k = 19$ | A1 | AG |
| **Total** | **2** | |
## Question 10(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Separate variables and integrate at least one side | M1 | |
| Obtain terms $20\ln x - x$ and $19t$, or equivalent | A1 A1 | |
| Evaluate a constant or use $t = 0$ and $x = 1$ as limits in a solution containing terms $a\ln x$ and $bt$ | M1 | |
| Substitute $t = 1$ and rearrange the equation in the given form | A1 | AG |
| **Total** | **5** | |
## Question 10(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use $x_{n+1} = e^{0.9+0.05x_n}$ correctly at least once | M1 | |
| Obtain final answer $x = 2.83$ | A1 | |
| Show sufficient iterations to 4 decimal places to justify 2.83 to 2 d.p. or show there is a sign change in the interval $(2.825,\, 2.835)$ | A1 | |
| **Total** | **3** | |
## Question 10(d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Set $x = 20$ and obtain answer $t = 2.15$ | B1 | |
| **Total** | **1** | |10 A large plantation of area $20 \mathrm {~km} ^ { 2 }$ is becoming infected with a plant disease. At time $t$ years the area infected is $x \mathrm {~km} ^ { 2 }$ and the rate of increase of $x$ is proportional to the ratio of the area infected to the area not yet infected.
When $t = 0 , x = 1$ and $\frac { \mathrm { d } x } { \mathrm {~d} t } = 1$.
\begin{enumerate}[label=(\alph*)]
\item Show that $x$ and $t$ satisfy the differential equation
$$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 19 x } { 20 - x }$$
\item Solve the differential equation and show that when $t = 1$ the value of $x$ satisfies the equation $x = \mathrm { e } ^ { 0.9 + 0.05 x }$.
\item Use an iterative formula based on the equation in part (b), with an initial value of 2 , to determine $x$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\item Calculate the value of $t$ at which the entire plantation becomes infected.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q10 [11]}}