| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2004 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Tank/reservoir mixing problems |
| Difficulty | Standard +0.3 This is a structured multi-part differential equations question with significant scaffolding. Part (i) requires setting up a rate equation from a word problem (straightforward with given information). Part (ii) involves separable variables after substitution is provided, and part (iii) is direct substitution. While it requires multiple techniques, the heavy guidance and standard methods place it slightly above average difficulty. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or imply \(\frac{dV}{dt} = 1000\frac{dh}{dt}\) | B1 | |
| State or imply \(\frac{dV}{dt} = 30 - k\sqrt{h}\) or \(\frac{dh}{dt} = 0.03 - m\sqrt{h}\) | B1 | |
| Show that \(k = 10\) or \(m = 0.01\) and justify the given equation | B1 | [Allow the first B1 for the statement that \(0.03 = 30/1000\).] |
| Total: 3 marks | ||
| (ii) Separate variables and attempt integration of \(\frac{x-3}{x}\) with respect to \(x\) | M1* | |
| Obtain \(x - 3\ln x\), or equivalent | A1 | |
| Obtain \(0.005t\), or equivalent | A1 | |
| Use \(x = 3, t = 0\) in the evaluation of a constant or as limits in an answer involving \(\ln x\) and \(kt\) | M1(dep*) | |
| Obtain answer in any correct form e.g. \(t = 200(x-3-3\ln x+3\ln 3)\) | A1 | [To qualify for the first M mark, an attempt to solve the earlier differential equation in \(h\) and \(t\) must involve correct separation of variables, the use of a substitution such as \(\sqrt{h} = u\), and an attempt to integrate the resulting function of \(u\).] |
| Total: 5 marks | ||
| (iii) Substitute \(x = 1\) and calculate \(t\) | M1 | |
| Obtain answer \(t = 259\) correctly | A1 | Total: 2 marks |
**(i)** State or imply $\frac{dV}{dt} = 1000\frac{dh}{dt}$ | B1 | |
State or imply $\frac{dV}{dt} = 30 - k\sqrt{h}$ or $\frac{dh}{dt} = 0.03 - m\sqrt{h}$ | B1 | |
Show that $k = 10$ or $m = 0.01$ and justify the given equation | B1 | [Allow the first B1 for the statement that $0.03 = 30/1000$.] |
| | | **Total: 3 marks** |
**(ii)** Separate variables and attempt integration of $\frac{x-3}{x}$ with respect to $x$ | M1* | |
Obtain $x - 3\ln x$, or equivalent | A1 | |
Obtain $0.005t$, or equivalent | A1 | |
Use $x = 3, t = 0$ in the evaluation of a constant or as limits in an answer involving $\ln x$ and $kt$ | M1(dep*) | |
Obtain answer in any correct form e.g. $t = 200(x-3-3\ln x+3\ln 3)$ | A1 | [To qualify for the first M mark, an attempt to solve the earlier differential equation in $h$ and $t$ must involve correct separation of variables, the use of a substitution such as $\sqrt{h} = u$, and an attempt to integrate the resulting function of $u$.] |
| | | **Total: 5 marks** |
**(iii)** Substitute $x = 1$ and calculate $t$ | M1 | |
Obtain answer $t = 259$ correctly | A1 | **Total: 2 marks** |10 A rectangular reservoir has a horizontal base of area $1000 \mathrm {~m} ^ { 2 }$. At time $t = 0$, it is empty and water begins to flow into it at a constant rate of $30 \mathrm {~m} ^ { 3 } \mathrm {~s} ^ { - 1 }$. At the same time, water begins to flow out at a rate proportional to $\sqrt { } h$, where $h \mathrm {~m}$ is the depth of the water at time $t \mathrm {~s}$. When $h = 1 , \frac { \mathrm {~d} h } { \mathrm {~d} t } = 0.02$.\\
(i) Show that $h$ satisfies the differential equation
$$\frac { \mathrm { d } h } { \mathrm {~d} t } = 0.01 ( 3 - \sqrt { } h )$$
It is given that, after making the substitution $x = 3 - \sqrt { } h$, the equation in part (i) becomes
$$( x - 3 ) \frac { \mathrm { d } x } { \mathrm {~d} t } = 0.005 x$$
(ii) Using the fact that $x = 3$ when $t = 0$, solve this differential equation, obtaining an expression for $t$ in terms of $x$.\\
(iii) Find the time at which the depth of water reaches 4 m .
\hfill \mbox{\textit{CAIE P3 2004 Q10 [10]}}