| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2010 |
| Session | June |
| Marks | 8 |
| 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 requiring standard techniques: separate variables, integrate both sides (using a standard trigonometric integral for sec²x), apply initial conditions, and rearrange. Parts (ii) and (iii) involve basic limit analysis and sign reasoning. While it requires multiple steps and careful algebra, it follows a completely standard template with no novel insight required, making it slightly easier than average. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Separate variables correctly and attempt integration of both sides | B1 | |
| Obtain term \(\tan x\) | B1 | |
| Obtain term \(-\frac{1}{2}e^{-2t}\) | B1 | |
| Evaluate a constant or use limits \(x = 0, t = 0\) in a solution containing terms \(a \tan x\) and \(be^{-2t}\) | M1 | |
| Obtain correct solution in any form, e.g. \(\tan x = \frac{1}{2} - \frac{1}{4}e^{-2t}\) | A1 | |
| Rearrange as \(x = \tan^{-1}(\frac{1}{2} - \frac{1}{4}e^{-2t})\), or equivalent | A1 | [6] |
| (ii) State that \(x\) approaches \(\tan^{-1}(\frac{1}{2})\) | B1 | [1] |
| (iii) State that \(1 - e^{-2t}\) increases and so does the inverse tangent, or state that \(e^{-2t} \cos^2 x\) is positive | B1 | [1] |
**(i)** Separate variables correctly and attempt integration of both sides | B1 |
Obtain term $\tan x$ | B1 |
Obtain term $-\frac{1}{2}e^{-2t}$ | B1 |
Evaluate a constant or use limits $x = 0, t = 0$ in a solution containing terms $a \tan x$ and $be^{-2t}$ | M1 |
Obtain correct solution in any form, e.g. $\tan x = \frac{1}{2} - \frac{1}{4}e^{-2t}$ | A1 |
Rearrange as $x = \tan^{-1}(\frac{1}{2} - \frac{1}{4}e^{-2t})$, or equivalent | A1 | [6]
**(ii)** State that $x$ approaches $\tan^{-1}(\frac{1}{2})$ | B1 | [1]
**(iii)** State that $1 - e^{-2t}$ increases and so does the inverse tangent, or state that $e^{-2t} \cos^2 x$ is positive | B1 | [1]7 The variables $x$ and $t$ are related by the differential equation
$$\mathrm { e } ^ { 2 t } \frac { \mathrm {~d} x } { \mathrm {~d} t } = \cos ^ { 2 } x$$
where $t \geqslant 0$. When $t = 0 , x = 0$.\\
(i) Solve the differential equation, obtaining an expression for $x$ in terms of $t$.\\
(ii) State what happens to the value of $x$ when $t$ becomes very large.\\
(iii) Explain why $x$ increases as $t$ increases.
\hfill \mbox{\textit{CAIE P3 2010 Q7 [8]}}