| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2008 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Convert Cartesian to polar equation |
| Difficulty | Standard +0.8 This Further Maths question requires converting between polar and Cartesian forms using the identity sin 2θ = 2sin θ cos θ, then manipulating to get y = 2/x (a rectangular hyperbola). Part (c) requires solving simultaneous polar equations. While systematic, it demands fluency with multiple polar identities and algebraic manipulation beyond standard A-level, placing it moderately above average difficulty. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta) |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(r^2 2\sin\theta\cos\theta = 8\) | M1 | |
| \(x = r\cos\theta\); \(y = r\sin\theta\) | M1 | |
| \(xy = 4\); \(y = \frac{4}{x}\) | A1 | Total: 3 |
| (b) [Graph showing rectangular hyperbola with branches in first and third quadrants] | B1 | Total: 1 |
| (c) \(r = 2\sec\theta\) is \(x = 2\) | B1 | |
| Sub \(x = 2\) in \(xy = 4 \Rightarrow 2y = 4\) | M1 | |
| In cartesian, A(2, 2) | ||
| \(\Rightarrow \tan\theta = \frac{y}{x} = 1 \Rightarrow \theta = \frac{\pi}{4}\) | M1 | |
| \(\Rightarrow r = \sqrt{x^2+y^2} = \sqrt{8}\) | ||
| \(\theta = \frac{\pi}{4}; r = \sqrt{8}\) | A1 | Total: 4 |
| Altm2: Eliminating r to reach eqn. in \(\cos\theta\) and \(\sin\theta\) only (M1) | ||
| Substitution \(r=2\sec(\frac{\pi}{4})\) (m1) | ||
| \(r = \sqrt{8}\) (A1) OE surd |
**(a)** $r^2 2\sin\theta\cos\theta = 8$ | M1 | | sin 2θ = 2 sin θ cos θ used
$x = r\cos\theta$; $y = r\sin\theta$ | M1 | | Either one stated or used
$xy = 4$; $y = \frac{4}{x}$ | A1 | Total: 3 | Either OE e.g. $y = \frac{8}{2x}$
**(b)** [Graph showing rectangular hyperbola with branches in first and third quadrants] | B1 | Total: 1 |
**(c)** $r = 2\sec\theta$ is $x = 2$ | B1 | |
Sub $x = 2$ in $xy = 4 \Rightarrow 2y = 4$ | M1 | |
In cartesian, A(2, 2) | | |
$\Rightarrow \tan\theta = \frac{y}{x} = 1 \Rightarrow \theta = \frac{\pi}{4}$ | M1 | | Used either $\tan\theta = \frac{y}{x}$ or $r = \sqrt{x^2+y^2}$
$\Rightarrow r = \sqrt{x^2+y^2} = \sqrt{8}$ | | |
$\theta = \frac{\pi}{4}; r = \sqrt{8}$ | A1 | Total: 4 | r must be given in surd form
Altm2: Eliminating r to reach eqn. in $\cos\theta$ and $\sin\theta$ only (M1) | | | Altm3: $\sin\theta = 2$ (B1); Solving $r\cos\theta = 2$ and $r\sin\theta = 2$ simultaneously (M1); $\tan\theta = 1$ or $r^2=2+2^2$ (M1); $\theta = \frac{\pi}{4}; r = \sqrt{8}$ (A1) need both
Substitution $r=2\sec(\frac{\pi}{4})$ (m1) | | |
$r = \sqrt{8}$ (A1) OE surd | | |6 A curve $C$ has polar equation
$$r ^ { 2 } \sin 2 \theta = 8$$
\begin{enumerate}[label=(\alph*)]
\item Find the cartesian equation of $C$ in the form $y = \mathrm { f } ( x )$.
\item Sketch the curve $C$.
\item The line with polar equation $r = 2 \sec \theta$ intersects $C$ at the point $A$. Find the polar coordinates of $A$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2008 Q6 [8]}}