| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Chemical reaction kinetics |
| Difficulty | Standard +0.3 This is a straightforward first-order differential equation problem requiring substitution of the given x expression, separation of variables (dy/y = -3e^(-3t)dt), and direct integration. Part (ii) involves finding a limit as tââ, which is routine. The context is applied but the mathematics is standard A-level technique with no novel insight required. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Substitute for \(x\), separate variables correctly and attempt integration of both sides | M1 | |
| Obtain term in \(y\), or equivalent | A1 | |
| Obtain term \(e^{-3t}\), or equivalent | A1 | |
| Evaluate a constant, or use \(t = 0, y = 70\) as limits in a solution containing terms \(\ln y\) and \(be^{-3t}\) | M1 | |
| Obtain correct solution in any form, e.g. \(\ln y - \ln 70 = e^{-3t} - 1\) | A1 | |
| Rearrange and obtain \(y = 70\exp(e^{-3t} - 1)\), or equivalent | A1 | [6] |
| (ii) Using answer to part (i), either express \(p\) in terms of \(t\) or use \(e^{-3t} \to 0\) to find the limiting value of \(y\) | M1 | |
| Obtain answer \(\frac{100}{e}\) from correct exact work | A1 | [2] |
**(i)** Substitute for $x$, separate variables correctly and attempt integration of both sides | M1 |
Obtain term in $y$, or equivalent | A1 |
Obtain term $e^{-3t}$, or equivalent | A1 |
Evaluate a constant, or use $t = 0, y = 70$ as limits in a solution containing terms $\ln y$ and $be^{-3t}$ | M1 |
Obtain correct solution in any form, e.g. $\ln y - \ln 70 = e^{-3t} - 1$ | A1 |
Rearrange and obtain $y = 70\exp(e^{-3t} - 1)$, or equivalent | A1 | [6]
**(ii)** Using answer to part (i), either express $p$ in terms of $t$ or use $e^{-3t} \to 0$ to find the limiting value of $y$ | M1 |
Obtain answer $\frac{100}{e}$ from correct exact work | A1 | [2]
---5 In a certain chemical process a substance $A$ reacts with another substance $B$. The masses in grams of $A$ and $B$ present at time $t$ seconds after the start of the process are $x$ and $y$ respectively. It is given that $\frac { \mathrm { d } y } { \mathrm {~d} t } = - 0.6 x y$ and $x = 5 \mathrm { e } ^ { - 3 t }$. When $t = 0 , y = 70$.\\
(i) Form a differential equation in $y$ and $t$. Solve this differential equation and obtain an expression for $y$ in terms of $t$.\\
(ii) The percentage of the initial mass of $B$ remaining at time $t$ is denoted by $p$. Find the exact value approached by $p$ as $t$ becomes large.
\hfill \mbox{\textit{CAIE P3 2012 Q5 [8]}}