| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2008 |
| 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 +0.8 This is a multi-step applied differential equations problem requiring: (i) forming a DE from rates of change with V=4h³/3, (ii) algebraic manipulation with partial fractions, and (iii) separating variables and integrating. While the techniques are standard A-level (differentiation, separation of variables, partial fractions), the application context, multi-stage setup, and algebraic complexity place it moderately above average difficulty. |
| Spec | 1.02y Partial fractions: decompose rational functions1.07t Construct differential equations: in context1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or obtain \(\frac{dI}{dt} = 4h^2 \frac{dh}{dt}\) or \(\frac{dV}{dt} = 4h^2\), or equivalent | B1 | |
| State or imply \(\frac{dV}{dt} = 20 - kh^2\) | B1 | |
| Use the given values to evaluate \(k\) | M1 | |
| Show that \(k = 0.2\), or equivalent, and obtain the given equation | A1 | [4] |
| [The M1 is dependent on at least one B mark having been earned.] | ||
| (ii) Fully justify the given identity | B1 | [1] |
| (iii) Separate variables correctly and attempt integration of both sides | M1 | |
| Obtain terms \(-20t\) and \(t\), or equivalent | A1 | |
| Obtain terms \(\ln(10 + h)\) and \(\ln(10 - h)\), where \(ab \neq 0\), or \(k\ln\left | \frac{10+h}{10-h}\right | \) |
| Obtain correct terms, i.e. with \(a = 100\) and \(b = -100\), or \(k = 2000/20\), or equivalent | A1 | |
| Evaluate a constant and obtain a correct expression for \(t\) in terms of \(h\) | A1 | [5] |
**(i)** State or obtain $\frac{dI}{dt} = 4h^2 \frac{dh}{dt}$ or $\frac{dV}{dt} = 4h^2$, or equivalent | B1 |
State or imply $\frac{dV}{dt} = 20 - kh^2$ | B1 |
Use the given values to evaluate $k$ | M1 |
Show that $k = 0.2$, or equivalent, and obtain the given equation | A1 | [4] |
[The M1 is dependent on at least one B mark having been earned.] |
**(ii)** Fully justify the given identity | B1 | [1]
**(iii)** Separate variables correctly and attempt integration of both sides | M1 |
Obtain terms $-20t$ and $t$, or equivalent | A1 |
Obtain terms $\ln(10 + h)$ and $\ln(10 - h)$, where $ab \neq 0$, or $k\ln\left|\frac{10+h}{10-h}\right|$ | M1 |
Obtain correct terms, i.e. with $a = 100$ and $b = -100$, or $k = 2000/20$, or equivalent | A1 |
Evaluate a constant and obtain a correct expression for $t$ in terms of $h$ | A1 | [5]8\\
\includegraphics[max width=\textwidth, alt={}, center]{c687888e-bef0-4ea9-b5b3-e614028cc07c-3_654_805_274_671}
An underground storage tank is being filled with liquid as shown in the diagram. Initially the tank is empty. At time $t$ hours after filling begins, the volume of liquid is $V \mathrm {~m} ^ { 3 }$ and the depth of liquid is $h \mathrm {~m}$. It is given that $V = \frac { 4 } { 3 } h ^ { 3 }$.
The liquid is poured in at a rate of $20 \mathrm {~m} ^ { 3 }$ per hour, but owing to leakage, liquid is lost at a rate proportional to $h ^ { 2 }$. When $h = 1 , \frac { \mathrm {~d} h } { \mathrm {~d} t } = 4.95$.\\
(i) Show that $h$ satisfies the differential equation
$$\frac { \mathrm { d } h } { \mathrm {~d} t } = \frac { 5 } { h ^ { 2 } } - \frac { 1 } { 20 } .$$
(ii) Verify that $\frac { 20 h ^ { 2 } } { 100 - h ^ { 2 } } \equiv - 20 + \frac { 2000 } { ( 10 - h ) ( 10 + h ) }$.\\
(iii) Hence solve the differential equation in part (i), obtaining an expression for $t$ in terms of $h$.
\hfill \mbox{\textit{CAIE P3 2008 Q8 [10]}}