| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Modelling with logistic growth |
| Difficulty | Standard +0.8 This is a standard logistic growth differential equation requiring separation of variables with partial fractions, followed by applying two initial conditions to find constants. While the partial fractions decomposition and logarithmic manipulation require care, this is a well-practiced technique in Further Maths P3. The multi-step nature and algebraic manipulation elevate it slightly above average difficulty, but it follows a predictable solution path without requiring novel insight. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y)1.08l Interpret differential equation solutions: in context |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Separate variables correctly and integrate at least one side | M1 | |
| Integrate and obtain term \(kt\), or equivalent | A1 | |
| Carry out a relevant method to obtain \(A\) and \(B\) such that \(\dfrac{1}{x(4-x)}\equiv\dfrac{A}{x}+\dfrac{B}{4-x}\), or equivalent | M1* | |
| Obtain \(A=B=\dfrac{1}{4}\), or equivalent | A1 | |
| Integrate and obtain terms \(\dfrac{1}{4}\ln x - \dfrac{1}{4}\ln(4-x)\), or equivalent | A1\(\checkmark\) | |
| EITHER: Use a pair of limits in an expression containing \(p\ln x\), \(q\ln(4-x)\) and \(rt\) and evaluate a constant | DM1 | |
| Obtain correct answer in any form, e.g. \(\ln x - \ln(4-x) = 4kt - \ln 9\) | A1 | |
| or \(\ln\left(\dfrac{x}{4-x}\right) = 4kt - 8k\) | ||
| Use a second pair of limits and determine \(k\) | DM1 | |
| Obtain the given exact answer correctly | A1 | |
| OR: Use both pairs of limits in a definite integral | M1* | |
| Obtain the given exact answer correctly | A1 | |
| Substitute \(k\) and either pair of limits in an expression containing \(p\ln x\), \(q\ln(4-x)\) and \(rt\) and evaluate a constant | DM1 | |
| Obtain \(\ln\dfrac{x}{4-x} = t\ln 3 - \ln 9\) or equivalent | A1 | Total: [9] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Substitute \(x=3.6\) and solve for \(t\) | M1 | |
| Obtain answer \(t=4\) | A1 | Total: [2] |
# Question 10:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Separate variables correctly and integrate at least one side | M1 | |
| Integrate and obtain term $kt$, or equivalent | A1 | |
| Carry out a relevant method to obtain $A$ and $B$ such that $\dfrac{1}{x(4-x)}\equiv\dfrac{A}{x}+\dfrac{B}{4-x}$, or equivalent | M1* | |
| Obtain $A=B=\dfrac{1}{4}$, or equivalent | A1 | |
| Integrate and obtain terms $\dfrac{1}{4}\ln x - \dfrac{1}{4}\ln(4-x)$, or equivalent | A1$\checkmark$ | |
| **EITHER:** Use a pair of limits in an expression containing $p\ln x$, $q\ln(4-x)$ and $rt$ and evaluate a constant | DM1 | |
| Obtain correct answer in any form, e.g. $\ln x - \ln(4-x) = 4kt - \ln 9$ | A1 | |
| or $\ln\left(\dfrac{x}{4-x}\right) = 4kt - 8k$ | | |
| Use a second pair of limits and determine $k$ | DM1 | |
| Obtain the given exact answer correctly | A1 | |
| **OR:** Use both pairs of limits in a definite integral | M1* | |
| Obtain the given exact answer correctly | A1 | |
| Substitute $k$ and either pair of limits in an expression containing $p\ln x$, $q\ln(4-x)$ and $rt$ and evaluate a constant | DM1 | |
| Obtain $\ln\dfrac{x}{4-x} = t\ln 3 - \ln 9$ or equivalent | A1 | Total: [9] |
## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Substitute $x=3.6$ and solve for $t$ | M1 | |
| Obtain answer $t=4$ | A1 | Total: [2] |10 A large field of area $4 \mathrm {~km} ^ { 2 }$ is becoming infected with a soil disease. At time $t$ years the area infected is $x \mathrm {~km} ^ { 2 }$ and the rate of growth of the infected area is given by the differential equation $\frac { \mathrm { d } x } { \mathrm {~d} t } = k x ( 4 - x )$, where $k$ is a positive constant. It is given that when $t = 0 , x = 0.4$ and that when $t = 2 , x = 2$.\\
(i) Solve the differential equation and show that $k = \frac { 1 } { 4 } \ln 3$.\\
(ii) Find the value of $t$ when $90 \%$ of the area of the field is infected.
\hfill \mbox{\textit{CAIE P3 2016 Q10 [11]}}