| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2022 |
| Session | February |
| Marks | 13 |
| Topic | Polar coordinates |
| Type | Tangent parallel/perpendicular to initial line |
| Difficulty | Challenging +1.8 This is a challenging Further Maths polar coordinates question requiring: (a) implicit differentiation of polar curves to find tangent conditions, solving a transcendental inequality involving cosine; (b) sketching a limaçon; (c) computing volume using polar area integration then converting units. The conceptual demand (polar tangent perpendicularity condition, analyzing when exactly 4 solutions exist) and multi-step reasoning across parts elevate this significantly above standard A-level, though the individual techniques are within FM syllabus. |
| Spec | 1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^24.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
The curve $C$ has equation
$$r = a(p + 2\cos\theta) \quad 0 \leqslant \theta < 2\pi$$
where $a$ and $p$ are positive constants and $p > 2$
There are exactly four points on $C$ where the tangent is perpendicular to the initial line.
\begin{enumerate}[label=(\alph*)]
\item Show that the range of possible values for $p$ is
$$2 < p < 4$$ [5]
\item Sketch the curve with equation
$$r = a(3 + 2\cos\theta) \quad 0 \leqslant \theta < 2\pi \quad \text{where } a > 0$$ [1]
\end{enumerate}
John digs a hole in his garden in order to make a pond.
The pond has a uniform horizontal cross section that is modelled by the curve with equation
$$r = 20(3 + 2\cos\theta) \quad 0 \leqslant \theta < 2\pi$$
where $r$ is measured in centimetres.
The depth of the pond is 90 centimetres.
Water flows through a hosepipe into the pond at a rate of 50 litres per minute.
Given that the pond is initially empty,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item determine how long it will take to completely fill the pond with water using the hosepipe, according to the model. Give your answer to the nearest minute. [7]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2022 Q6 [13]}}