| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2021 |
| Session | March |
| Marks | 13 |
| Topic | Harmonic Form |
| Type | Applied context modeling |
| Difficulty | Standard +0.3 This is a standard R-cos(θ-α) transformation question with straightforward applications. Part (a) is routine bookwork requiring only formula application. Parts (b) and (c) involve direct substitution and solving trigonometric equations within a given domain—all standard A-level techniques with no novel insight required. The multi-part structure and context add slight complexity but this remains easier than average. |
| 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*), resume]
\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]}}