| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2012 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Parametric curves and Cartesian conversion |
| Type | Convert to Cartesian (sin/cos identities) |
| Difficulty | Standard +0.3 This is a standard C4 parametric equations question requiring routine techniques: chain rule for dy/dx, double angle formula (sin 2θ = 2sin θ cos θ), and algebraic manipulation using cos²θ + sin²θ = 1. All steps are predictable textbook methods with no novel insight required, making it slightly easier than average. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation |
5 A curve is defined by the parametric equations
$$x = 2 \cos \theta , \quad y = 3 \sin 2 \theta$$
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Show that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = a \sin \theta + b \operatorname { cosec } \theta$$
where $a$ and $b$ are integers.
\item Find the gradient of the normal to the curve at the point where $\theta = \frac { \pi } { 6 }$.
\end{enumerate}\item Show that the cartesian equation of the curve can be expressed as
$$y ^ { 2 } = p x ^ { 2 } \left( 4 - x ^ { 2 } \right)$$
where $p$ is a rational number.
\end{enumerate}
\hfill \mbox{\textit{AQA C4 2012 Q5 [9]}}