| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2016 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Separable variables - standard (polynomial/exponential x-side) |
| Difficulty | Standard +0.8 Part (a) requires implicit differentiation with a trigonometric function, which is moderately challenging but follows standard C4 techniques. Part (b) is a separable differential equation requiring integration of cosec(3y) and connection to the inverse tan result from part (a), demanding careful algebraic manipulation and integration skills across 7 marks. The multi-step nature and need to recognize the connection between parts elevates this above average difficulty. |
| Spec | 1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.07s Parametric and implicit differentiation1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\theta = \tan^{-1}\left(\frac{3x}{2}\right) \Rightarrow 2\tan\theta = 3x\) | B1 | |
| \(2\sec^2\theta \cdot \frac{d\theta}{dx} = 3\) | M1 | Use of correct identity to get \(\sec^2\theta\) in terms of \(x\) ; condone missing bracket. |
## Part (a)
$\theta = \tan^{-1}\left(\frac{3x}{2}\right) \Rightarrow 2\tan\theta = 3x$ | B1 |
$2\sec^2\theta \cdot \frac{d\theta}{dx} = 3$ | M1 | Use of correct identity to get $\sec^2\theta$ in terms of $x$ ; condone missing bracket.
$\sec^2\theta = 1 + \tan^2\theta = 1 + \leftIt is given that $\theta = \tan^{-1}\left(\frac{3x}{2}\right)$.
\begin{enumerate}[label=(\alph*)]
\item By writing $\theta = \tan^{-1}\left(\frac{3x}{2}\right)$ as $2\tan\theta = 3x$, use implicit differentiation to show that
$$\frac{d\theta}{dx} = \frac{k}{4 + 9x^2}$$, where $k$ is an integer.
[3 marks]
\item Hence solve the differential equation
$$9y(4 + 9x^2)\frac{dy}{dx} = \cosec 3y$$
given that $x = 0$ when $y = \frac{\pi}{3}$. Give your answer in the form $\mathbf{g}(y) = \mathbf{h}(x)$.
[7 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA C4 2016 Q8 [10]}}