| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| 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 separation, integration of basic trigonometric functions (∫sin 2θ/(3+cos 2θ) dθ via substitution u=3+cos 2θ), and applying an initial condition. The second part requires simple analysis of the resulting expression. Slightly above average due to the trigonometric manipulation, but still a routine textbook exercise with no novel insight required. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Separate variables correctly and attempt integration of at least one side | B1 | |
| Obtain term \(\ln x\) | B1 | |
| Obtain term of the form \(k \ln(3 + \cos 2\theta)\), or equivalent | M1 | |
| Obtain term \(-\frac{1}{2}\ln(3 + \cos 2\theta)\), or equivalent | A1 | |
| Use \(x = 3, y = \frac{\pi}{3}\) to evaluate a constant or as limits in a solution | M1 | |
| with terms \(a \ln x\) and \(b \ln(3 + \cos 2\theta)\) where \(ab \neq 0\) | ||
| State correct solution in any form, e.g. \(\ln x = -\frac{1}{2}\ln(3 + \cos 2\theta) + \frac{1}{2}\ln 3\) | A1 | |
| Rearrange in a correct form, e.g. \(x = \sqrt{\frac{27}{3 + \cos 2\theta}}\) | A1 | [7] |
| (ii) State answer \(x = 3\sqrt{3}/2\), or exact equivalent (accept decimal answer in [2.59, 2.60]) | B1 | [1] |
**(i)** Separate variables correctly and attempt integration of at least one side | B1 |
Obtain term $\ln x$ | B1 |
Obtain term of the form $k \ln(3 + \cos 2\theta)$, or equivalent | M1 |
Obtain term $-\frac{1}{2}\ln(3 + \cos 2\theta)$, or equivalent | A1 |
Use $x = 3, y = \frac{\pi}{3}$ to evaluate a constant or as limits in a solution | M1 |
with terms $a \ln x$ and $b \ln(3 + \cos 2\theta)$ where $ab \neq 0$ | | |
State correct solution in any form, e.g. $\ln x = -\frac{1}{2}\ln(3 + \cos 2\theta) + \frac{1}{2}\ln 3$ | A1 |
Rearrange in a correct form, e.g. $x = \sqrt{\frac{27}{3 + \cos 2\theta}}$ | A1 | [7]
**(ii)** State answer $x = 3\sqrt{3}/2$, or exact equivalent (accept decimal answer in [2.59, 2.60]) | B1 | [1]6 The variables $x$ and $\theta$ satisfy the differential equation
$$( 3 + \cos 2 \theta ) \frac { \mathrm { d } x } { \mathrm {~d} \theta } = x \sin 2 \theta$$
and it is given that $x = 3$ when $\theta = \frac { 1 } { 4 } \pi$.\\
(i) Solve the differential equation and obtain an expression for $x$ in terms of $\theta$.\\
(ii) State the least value taken by $x$.
\hfill \mbox{\textit{CAIE P3 2016 Q6 [8]}}