| Exam Board | WJEC |
|---|---|
| Module | Further Unit 4 (Further Unit 4) |
| Year | 2024 |
| Session | June |
| Marks | 21 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Tangent parallel/perpendicular to initial line |
| Difficulty | Challenging +1.8 This is a challenging Further Maths question requiring multiple sophisticated techniques: De Moivre's theorem manipulation to express powers of cosine, polar area integration with trigonometric simplification, and finding tangent conditions in polar coordinates using implicit differentiation. While each technique is standard for Further Maths, the combination and algebraic complexity (especially parts b and c requiring careful calculus and exact values) place this well above average difficulty. |
| Spec | 1.05l Double angle formulae: and compound angle formulae4.02q De Moivre's theorem: multiple angle formulae4.09c Area enclosed: by polar curve |
\begin{enumerate}[label=(\alph*)]
\item Given that $z^n + \frac{1}{z^n} = 2\cos n\theta$, where $z = \cos\theta + \mathrm{i}\sin\theta$, express $16\cos^4\theta$ in the form
$$a\cos 4\theta + b\cos 2\theta + c,$$
where $a$, $b$, $c$ are integers whose values are to be determined. [5]
\end{enumerate}
The diagram below shows a sketch of the curve C with polar equation
$$r = 3 - 4\cos^2\theta, \quad \text{where } \frac{\pi}{6} \leq \theta \leq \frac{5\pi}{6}.$$
\includegraphics{figure_4}
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Calculate the area of the region enclosed by the curve C. [8]
\item Find the exact polar coordinates of the points on C at which the tangent is perpendicular to the initial line. [8]
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 4 2024 Q4 [21]}}