| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | March |
| 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 separation, integration using standard forms (∫1/(1+4y²)dy = ½arctan(2y) and ∫e^(-x)dx), and applying initial conditions. Part (b) requires simple limit analysis. While it involves multiple steps and inverse trig integration, these are standard techniques for P3 level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks |
|---|---|
| Separate variables correctly and attempt integration of at least one side | B1 |
| Obtain term of the form \(a\tan^{-1}(2y)\) | M1 |
| Obtain term \(\frac{1}{2}\tan^{-1}(2y)\) | A1 |
| Obtain term \(-e^{-x}\) | B1 |
| Use \(x=1\), \(y=0\) to evaluate a constant or as limits in a solution containing terms of the form \(a\tan^{-1}(by)\) and \(ce^{\pm x}\) | M1 |
| Obtain correct answer in any form | A1 |
| Obtain final answer \(y=\frac{1}{2}\tan\!\left(2e^{-1}-2e^{-x}\right)\), or equivalent | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| State that \(y\) approaches \(\frac{1}{2}\tan(2e^{-1})\), or equivalent | B1FT | The FT is on correct work on a solution containing \(e^{-x}\) |
## Question 6(a):
| Separate variables correctly and attempt integration of at least one side | B1 | |
| Obtain term of the form $a\tan^{-1}(2y)$ | M1 | |
| Obtain term $\frac{1}{2}\tan^{-1}(2y)$ | A1 | |
| Obtain term $-e^{-x}$ | B1 | |
| Use $x=1$, $y=0$ to evaluate a constant or as limits in a solution containing terms of the form $a\tan^{-1}(by)$ and $ce^{\pm x}$ | M1 | |
| Obtain correct answer in any form | A1 | |
| Obtain final answer $y=\frac{1}{2}\tan\!\left(2e^{-1}-2e^{-x}\right)$, or equivalent | A1 | |
## Question 6(b):
| State that $y$ approaches $\frac{1}{2}\tan(2e^{-1})$, or equivalent | B1FT | The FT is on correct work on a solution containing $e^{-x}$ |
---6 The variables $x$ and $y$ satisfy the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 + 4 y ^ { 2 } } { \mathrm { e } ^ { x } }$$
It is given that $y = 0$ when $x = 1$.
\begin{enumerate}[label=(\alph*)]
\item Solve the differential equation, obtaining an expression for $y$ in terms of $x$.
\item State what happens to the value of $y$ as $x$ tends to infinity.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2020 Q6 [8]}}