| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | Specimen |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Deriving the differential equation |
| Difficulty | Standard +0.5 This is a standard applied differential equations question requiring translation of a word problem into mathematical form, then solving using integrating factor method. Part (a) involves setting up the DE from the given conditions (routine modelling), part (b) is a standard integrating factor solution, and part (c) requires interpretation of the limiting behaviour. While it requires multiple steps and careful algebraic manipulation, it follows a well-established template for chemical reaction problems and doesn't require novel insight beyond applying standard techniques. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y)1.08l Interpret differential equation solutions: in context |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply \(\frac{dx}{dt} = k(10-x)(20-x)\) and show \(k = 0.01\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Separate variables and attempt integration of at least one side | M1 | |
| Carry out an attempt to find \(A\) and \(B\) such that \(\frac{1}{(10-x)(20-x)} \equiv \frac{A}{10-x} + \frac{B}{20-x}\) | M1 | |
| Obtain \(A = \frac{1}{10}\) and \(B = -\frac{1}{10}\), or equivalent | A1 | |
| Integrate and obtain \(-\frac{1}{10}\ln(10-x) + \frac{1}{10}\ln(20-x)\), or equivalent | A1FT | |
| Integrate and obtain term \(0.01t\), or equivalent | A1 | |
| Evaluate a constant, or use limits \(t=0\), \(x=0\) in a solution containing terms of the form \(a\ln(10-x)\) and \(ct\) | M1 | |
| Obtain answer in any form, e.g. \(-\frac{1}{10}\ln(10-x) + \frac{1}{10}\ln(20-x) = 0.01t + \frac{1}{10}\ln 2\) | A1FT | |
| Use laws of logarithms correctly to remove logarithms | M1 | |
| Rearrange and obtain \(x = \frac{20(e^{0.1t}-1)}{2e^{0.1t}-1}\), or equivalent | A1 | |
| Total | 9 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State that \(x\) approaches 10 | B1 |
## Question 10(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply $\frac{dx}{dt} = k(10-x)(20-x)$ and show $k = 0.01$ | B1 | |
## Question 10(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Separate variables and attempt integration of at least one side | M1 | |
| Carry out an attempt to find $A$ and $B$ such that $\frac{1}{(10-x)(20-x)} \equiv \frac{A}{10-x} + \frac{B}{20-x}$ | M1 | |
| Obtain $A = \frac{1}{10}$ and $B = -\frac{1}{10}$, or equivalent | A1 | |
| Integrate and obtain $-\frac{1}{10}\ln(10-x) + \frac{1}{10}\ln(20-x)$, or equivalent | A1FT | |
| Integrate and obtain term $0.01t$, or equivalent | A1 | |
| Evaluate a constant, or use limits $t=0$, $x=0$ in a solution containing terms of the form $a\ln(10-x)$ and $ct$ | M1 | |
| Obtain answer in any form, e.g. $-\frac{1}{10}\ln(10-x) + \frac{1}{10}\ln(20-x) = 0.01t + \frac{1}{10}\ln 2$ | A1FT | |
| Use laws of logarithms correctly to remove logarithms | M1 | |
| Rearrange and obtain $x = \frac{20(e^{0.1t}-1)}{2e^{0.1t}-1}$, or equivalent | A1 | |
| **Total** | **9** | |
## Question 10(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| State that $x$ approaches 10 | B1 | |10 In a chemical reaction, a compound $X$ is formed from two compounds $Y$ and $Z$.\\
The masses in grams of $X , Y$ and $Z$ present at time $t$ seconds after the start of the reaction are $x , 10 - x$ and $20 - x$ respectively. At any time the rate of formation of $X$ is proportional to the product of the masses of $Y$ and $Z$ present at the time. When $t = 0 , x = 0$ and $\frac { \mathrm { d } x } { \mathrm {~d} t } = 2$.\\
(a) Show that $x$ and $t$ satisfy the differential equation
$$\frac { \mathrm { d } x } { \mathrm {~d} t } = 0.01 ( 10 - x ) ( 20 - x ) .$$
(b) Solve this differential equation and obtain an expression for $x$ in terms of $t$.\\
(c) State what happens to the value of $x$ when $t$ becomes large.\\
\hfill \mbox{\textit{CAIE P3 2020 Q10 [11]}}