| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2003 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Separable variables |
| Difficulty | Standard +0.3 This is a straightforward separable variables question with a simple substitution (u = P - A). All parts follow standard procedures: separate and integrate, apply initial conditions, solve for constants, and rearrange. The square root adds minimal complexity compared to typical A-level differential equations questions, making it slightly easier than average. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Separate variables and attempt to integrate \(\frac{1}{\sqrt{(P-A)}}\) | M1 | |
| Obtain term \(2\sqrt{(P-A)}\) | A1 | |
| Obtain term \(-kt\) | A1 | For the M1, \(\sqrt{(P-A)}\) must be treated correctly |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use limits \(P = 5A\), \(t = 0\) and attempt to find constant \(c\) | M1 | |
| Obtain \(c = 4\sqrt{A}\), or equivalent | A1 | |
| Use limits \(P = 2A\), \(t = 2\) and attempt to find \(k\) | M1 | |
| Obtain given answer \(k = \sqrt{A}\) correctly | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Substitute \(P = A\) and attempt to calculate \(t\) | M1 | |
| Obtain answer \(t = 4\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Using answers to part (ii), attempt to rearrange solution to give \(P\) in terms of \(A\) and \(t\) | M1 | |
| Obtain \(P = \frac{1}{4}A(4 + (4-t)^2)\), or equivalent, having squared \(\sqrt{A}\) | A1 |
## Question 9(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Separate variables and attempt to integrate $\frac{1}{\sqrt{(P-A)}}$ | M1 | |
| Obtain term $2\sqrt{(P-A)}$ | A1 | |
| Obtain term $-kt$ | A1 | For the M1, $\sqrt{(P-A)}$ must be treated correctly |
**Total: [3]**
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## Question 9(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use limits $P = 5A$, $t = 0$ and attempt to find constant $c$ | M1 | |
| Obtain $c = 4\sqrt{A}$, or equivalent | A1 | |
| Use limits $P = 2A$, $t = 2$ and attempt to find $k$ | M1 | |
| Obtain given answer $k = \sqrt{A}$ correctly | A1 | |
**Total: [4]**
---
## Question 9(iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Substitute $P = A$ and attempt to calculate $t$ | M1 | |
| Obtain answer $t = 4$ | A1 | |
**Total: [2]**
---
## Question 9(iv):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Using answers to part (ii), attempt to rearrange solution to give $P$ in terms of $A$ and $t$ | M1 | |
| Obtain $P = \frac{1}{4}A(4 + (4-t)^2)$, or equivalent, having squared $\sqrt{A}$ | A1 | |
**Total: [2]**
---9 Compressed air is escaping from a container. The pressure of the air in the container at time $t$ is $P$, and the constant atmospheric pressure of the air outside the container is $A$. The rate of decrease of $P$ is proportional to the square root of the pressure difference ( $P - A$ ). Thus the differential equation connecting $P$ and $t$ is
$$\frac { \mathrm { d } P } { \mathrm {~d} t } = - k \sqrt { } ( P - A )$$
where $k$ is a positive constant.\\
(i) Find, in any form, the general solution of this differential equation.\\
(ii) Given that $P = 5 A$ when $t = 0$, and that $P = 2 A$ when $t = 2$, show that $k = \sqrt { } A$.\\
(iii) Find the value of $t$ when $P = A$.\\
(iv) Obtain an expression for $P$ in terms of $A$ and $t$.
\hfill \mbox{\textit{CAIE P3 2003 Q9 [11]}}