| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Tangent parallel/perpendicular to initial line |
| Difficulty | Challenging +1.2 This is a structured multi-part question on polar coordinates requiring standard techniques: area integration using the polar formula, finding tangent points by converting to Cartesian coordinates and using calculus, and geometric calculations. While it involves several steps and Further Maths content (polar curves), each part follows well-established methods with clear guidance, making it moderately above average difficulty but not requiring novel insight. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
\includegraphics{figure_1}
Figure 1 shows a sketch of the cardioid $C$ with equation $r = a(1 + \cos \theta)$, $-\pi < \theta \leq \pi$. Also shown are the tangents to $C$ that are parallel and perpendicular to the initial line. These tangents form a rectangle $WXYZ$.
\begin{enumerate}[label=(\alph*)]
\item Find the area of the finite region, shaded in Fig. 1, bounded by the curve $C$. [6]
\item Find the polar coordinates of the points $A$ and $B$ where $WZ$ touches the curve $C$. [5]
\item Hence find the length of $WX$. [2]
Given that the length of $WZ$ is $\frac{3\sqrt{3}a}{2}$,
\item find the area of the rectangle $WXYZ$. [1]
A heart-shape is modelled by the cardioid $C$, where $a = 10$ cm. The heart shape is cut from the rectangular card $WXYZ$, shown in Fig. 1.
\item Find a numerical value for the area of card wasted in making this heart shape. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q16 [16]}}