| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2011 |
| Session | November |
| Marks | 7 |
| 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 differential equation with standard integration techniques (exponential and power functions). The separation is immediate, integration is routine, and applying initial conditions to find the constant is mechanical. Part (ii) requires only basic limit analysis. Slightly above average difficulty due to the fractional power and exponential, but still a standard textbook exercise. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| (i) Separate variables and attempt integration on both sides | M1* | |
| Obtain \(2N^{0.5}\) on left-hand side or equivalent | A1 | |
| Obtain \(-60e^{-0.02t}\) on right-hand side or equivalent | A1 | |
| Use 0 and 100 to evaluate a constant or as limits in a solution containing terms \(aN^{0.5}\) and \(be^{-0.02t}\) | DM1* | |
| Obtain \(2N^{0.5} = -60e^{-0.02t} + 80\) or equivalent | A1 | |
| Conclude with \(N = (40 - 30e^{-0.02t})^2\) or equivalent | A1 | [6] |
| (ii) State number approaches 1600 or equivalent, following expression of form \((c + de^{-0.02t})^n\) | B1\(\sqrt{}\) | [1] |
## Question 4:
| Answer/Working | Mark | Guidance |
|---|---|---|
| **(i)** Separate variables and attempt integration on both sides | M1* | |
| Obtain $2N^{0.5}$ on left-hand side or equivalent | A1 | |
| Obtain $-60e^{-0.02t}$ on right-hand side or equivalent | A1 | |
| Use 0 and 100 to evaluate a constant or as limits in a solution containing terms $aN^{0.5}$ and $be^{-0.02t}$ | DM1* | |
| Obtain $2N^{0.5} = -60e^{-0.02t} + 80$ or equivalent | A1 | |
| Conclude with $N = (40 - 30e^{-0.02t})^2$ or equivalent | A1 | [6] |
| **(ii)** State number approaches 1600 or equivalent, following expression of form $(c + de^{-0.02t})^n$ | B1$\sqrt{}$ | [1] |
---4 During an experiment, the number of organisms present at time $t$ days is denoted by $N$, where $N$ is treated as a continuous variable. It is given that
$$\frac { \mathrm { d } N } { \mathrm {~d} t } = 1.2 \mathrm { e } ^ { - 0.02 t } N ^ { 0.5 }$$
When $t = 0$, the number of organisms present is 100 .\\
(i) Find an expression for $N$ in terms of $t$.\\
(ii) State what happens to the number of organisms present after a long time.
\hfill \mbox{\textit{CAIE P3 2011 Q4 [7]}}