| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 12 |
| 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 standard applied differential equations question requiring formation of a DE from a word problem, solving using separation of variables (straightforward after rearranging), then applying given conditions. Part (ii) involves routine iteration with a provided formula. The question is slightly easier than average because: the DE setup is guided, the solution form is given to verify rather than derive independently, and the iterative formula is provided rather than requiring the student to rearrange it themselves. The long-term behavior in (iii) is a standard limiting case. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y)1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State \(\frac{dV}{dt} = 80 - kV\) | B1 | |
| Correctly separate variables and attempt integration of one side | M1 | |
| Obtain \(a\ln(80 - kV) = t\) or equivalent | M1* | |
| Obtain \(-\frac{1}{k}\ln(80-kV) = t\) or equivalent | A1 | |
| Use \(t=0\) and \(V=0\) to find constant of integration or as limits | M1 (dep*) | |
| Obtain \(-\frac{1}{k}\ln(80-kV) = t - \frac{1}{k}\ln 80\) or equivalent | A1 | |
| Obtain given answer \(V = \frac{1}{k}(80 - 80e^{-kt})\) correctly | A1 | [7] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use iterative formula correctly at least once | M1 | |
| Obtain final answer \(0.14\) | A1 | |
| Show sufficient iterations to 4 s.f. to justify answer to 2 s.f. or show a sign change in the interval \((0.135, 0.145)\) | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State a value between \(530\) and \(540\text{ cm}^3\) inclusive | B1 | |
| State or imply that volume approaches \(569\text{ cm}^3\) (allowing any value between \(567\) and \(571\) inclusive) | B1 | [2] |
## Question 10(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State $\frac{dV}{dt} = 80 - kV$ | B1 | |
| Correctly separate variables and attempt integration of one side | M1 | |
| Obtain $a\ln(80 - kV) = t$ or equivalent | M1* | |
| Obtain $-\frac{1}{k}\ln(80-kV) = t$ or equivalent | A1 | |
| Use $t=0$ and $V=0$ to find constant of integration or as limits | M1 (dep*) | |
| Obtain $-\frac{1}{k}\ln(80-kV) = t - \frac{1}{k}\ln 80$ or equivalent | A1 | |
| Obtain given answer $V = \frac{1}{k}(80 - 80e^{-kt})$ correctly | A1 | [7] |
## Question 10(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use iterative formula correctly at least once | M1 | |
| Obtain final answer $0.14$ | A1 | |
| Show sufficient iterations to 4 s.f. to justify answer to 2 s.f. or show a sign change in the interval $(0.135, 0.145)$ | A1 | [3] |
## Question 10(iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State a value between $530$ and $540\text{ cm}^3$ inclusive | B1 | |
| State or imply that volume approaches $569\text{ cm}^3$ (allowing any value between $567$ and $571$ inclusive) | B1 | [2] |10 Liquid is flowing into a small tank which has a leak. Initially the tank is empty and, $t$ minutes later, the volume of liquid in the tank is $V \mathrm {~cm} ^ { 3 }$. The liquid is flowing into the tank at a constant rate of $80 \mathrm {~cm} ^ { 3 }$ per minute. Because of the leak, liquid is being lost from the tank at a rate which, at any instant, is equal to $k V \mathrm {~cm} ^ { 3 }$ per minute where $k$ is a positive constant.\\
(i) Write down a differential equation describing this situation and solve it to show that
$$V = \frac { 1 } { k } \left( 80 - 80 \mathrm { e } ^ { - k t } \right)$$
(ii) It is observed that $V = 500$ when $t = 15$, so that $k$ satisfies the equation
$$k = \frac { 4 - 4 e ^ { - 15 k } } { 25 }$$
Use an iterative formula, based on this equation, to find the value of $k$ correct to 2 significant figures. Use an initial value of $k = 0.1$ and show the result of each iteration to 4 significant figures.\\
(iii) Determine how much liquid there is in the tank 20 minutes after the liquid started flowing, and state what happens to the volume of liquid in the tank after a long time.
\hfill \mbox{\textit{CAIE P3 2013 Q10 [12]}}