| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2021 |
| Session | April |
| Marks | 13 |
| Topic | Harmonic Form |
| Type | Applied context modeling |
| Difficulty | Standard +0.3 This is a standard Further Maths harmonic form question with routine application. Part (a) uses the textbook R cos(θ-α) transformation with straightforward calculation. Parts (b) and (c) apply this to a modeling context requiring max/min finding and equation solving, but all techniques are standard. The multi-part structure and modeling context add slight complexity beyond basic C3/C4 trigonometry, but this remains a well-practiced question type with no novel insights required. |
| Spec | 1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals |
\begin{enumerate}[label=(\alph*)]
\item Express $2 \cos \theta + 5 \sin \theta$ in the form $R \cos (\theta - \alpha)$, where $R > 0$ and $0 < \alpha < \frac{\pi}{2}$
Give the values of $R$ and $\alpha$ to 3 significant figures. [3]
\end{enumerate}
The temperature $T$ °C, of an unheated building is modelled using the equation
$$T = 15 + 2 \cos \left( \frac{\pi t}{12} \right) + 5\sin \left( \frac{\pi t}{12} \right), \quad 0 \leq t < 24,$$
where $t$ hours is the number of hours after 1200.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Calculate the maximum temperature predicted by this model and the value of $t$ when this maximum occurs. [4]
\item Calculate, to the nearest half hour, the times when the temperature is predicted to be 12 °C. [6]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2021 Q5 [13]}}