| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | June |
| Marks | 8 |
| Topic | Harmonic Form |
| Type | Applied context modeling |
| Difficulty | Standard +0.3 This is a standard A-level trigonometric modelling question requiring: (a) finding when cos = 1 (routine), (b) using amplitude to find k and stating minimum (straightforward), (c) solving a trigonometric inequality. All techniques are textbook exercises with no novel insight required. The 'no calculator' requirement adds minimal difficulty as the arithmetic is manageable. Slightly easier than average due to the structured, guided nature. |
| Spec | 1.02z Models in context: use functions in modelling1.05f Trigonometric function graphs: symmetries and periodicities1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals |
\includegraphics{figure_5}
A horizontal path connects an island to the mainland.
On a particular morning, the height of the sea relative to the path, $H$ m, is modelled by the equation
$$H = 0.8 + k \cos(30t - 70)°$$
where $k$ is a constant and $t$ is number of hours after midnight.
Figure 5 shows a sketch of the graph of $H$ against $t$.
Use the equation of the model to answer parts (a), (b) and (c).
(a) Find the time of day at which the height of the sea is at its maximum. [2]
Given that the maximum height of the sea relative to the path is 2 m,
(b) \begin{enumerate}[label=(\roman*)]
\item find a complete equation for the model,
\item state the minimum height of the sea relative to the path.
\end{enumerate} [2]
It is safe to use the path when the sea is 10 centimetres or more below the path.
(c) Find the times between which it is safe to use the path.
(Solutions relying entirely on calculator technology are not acceptable.) [4]
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q16 [8]}}