| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2005 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Polar curve intersection points |
| Difficulty | Standard +0.8 This FP2 polar coordinates question requires converting between polar and Cartesian forms, sketching curves, and finding intersection points. While the conversions are standard (circle and line), part (c) requires solving a non-trivial trigonometric equation involving sec and cos, demanding algebraic manipulation and careful angle consideration within given domains. This is moderately challenging for Further Maths students. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Using polar/Cartesian relationships to form Cartesian equation | M1 | |
| \(x^2 + y^2 = 6x\) | A1 | Equation in any form e.g. \((x-3)^2 + y^2 = 9\) from sketch |
| For D: \(r\cos\left(\frac{\pi}{3} - \theta\right) = 3\) and attempt to expand | M1 | |
| \(\frac{x}{2} + \frac{\sqrt{3}y}{2} = 3\) | M1A1 | Any form; 5 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| "Circle", symmetric in initial line passing through pole | B1 | |
| Straight line | B1 | |
| Both passing through \((6, 0)\) | B1 | 3 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Polars: Meet where \(6\cos\theta\cos\left(\frac{\pi}{3} - \theta\right) = 3\) | M1 | |
| \(\sqrt{3}\sin\theta\cos\theta = \sin^2\theta\) | M1 | |
| \(\sin\theta = 0\) or \(\tan\theta = \sqrt{3}\) \(\left[\theta = 0 \text{ or } \frac{\pi}{3}\right]\) | M1 | |
| Points are \((6, 0)\) and \(\left(3, \frac{\pi}{3}\right)\) | B1, A1 | 5 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Eliminate \(x\) or \(y\) to form quadratic in one variable | M1 | |
| \([2x^2 - 15x + 18 = 0,\ 4y^2 - 6\sqrt{3}\ y = 0]\) Solve to find values of \(x\) or \(y\) | M1 | |
| \(\left[x = \frac{3}{2}\) or \(6;\ y = 0\) or \(\frac{3\sqrt{3}}{2}\right]\) | B1A1 | |
| Points must be \((6, 0)\) and \(\left(3, \frac{\pi}{3}\right)\) | B1A1 |
# Question 4:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Using polar/Cartesian relationships to form Cartesian equation | M1 | |
| $x^2 + y^2 = 6x$ | A1 | Equation in any form e.g. $(x-3)^2 + y^2 = 9$ from sketch |
| For D: $r\cos\left(\frac{\pi}{3} - \theta\right) = 3$ and attempt to expand | M1 | |
| $\frac{x}{2} + \frac{\sqrt{3}y}{2} = 3$ | M1A1 | Any form; 5 marks total |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| "Circle", symmetric in initial line passing through pole | B1 | |
| Straight line | B1 | |
| Both passing through $(6, 0)$ | B1 | 3 marks total |
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Polars: Meet where $6\cos\theta\cos\left(\frac{\pi}{3} - \theta\right) = 3$ | M1 | |
| $\sqrt{3}\sin\theta\cos\theta = \sin^2\theta$ | M1 | |
| $\sin\theta = 0$ or $\tan\theta = \sqrt{3}$ $\left[\theta = 0 \text{ or } \frac{\pi}{3}\right]$ | M1 | |
| Points are $(6, 0)$ and $\left(3, \frac{\pi}{3}\right)$ | B1, A1 | 5 marks total |
### Alternatives for Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Eliminate $x$ or $y$ to form quadratic in one variable | M1 | |
| $[2x^2 - 15x + 18 = 0,\ 4y^2 - 6\sqrt{3}\ y = 0]$ Solve to find values of $x$ or $y$ | M1 | |
| $\left[x = \frac{3}{2}$ or $6;\ y = 0$ or $\frac{3\sqrt{3}}{2}\right]$ | B1A1 | |
| Points must be $(6, 0)$ and $\left(3, \frac{\pi}{3}\right)$ | B1A1 | |
---4. The curve $C$ has polar equation $\quad r = 6 \cos \theta , \quad - \frac { \pi } { 2 } \leq \theta < \frac { \pi } { 2 }$, and the line $D$ has polar equation $\quad r = 3 \sec \left( \frac { \pi } { 3 } - \theta \right) , \quad - \frac { \pi } { 6 } < \theta < \frac { 5 \pi } { 6 }$.
\begin{enumerate}[label=(\alph*)]
\item Find a cartesian equation of $C$ and a cartesian equation of $D$.
\item Sketch on the same diagram the graphs of $C$ and $D$, indicating where each cuts the initial line.
The graphs of $C$ and $D$ intersect at the points $P$ and $Q$.
\item Find the polar coordinates of $P$ and $Q$.\\
(5)(Total 13 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 2005 Q4 [13]}}