| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Applied context modeling |
| Difficulty | Standard +0.3 This is a standard C3 harmonic form question with routine application to a real-world context. Part (a) uses the textbook R cos(θ - α) method, parts (b-c) apply it directly, and part (d) requires solving a trigonometric equation. All techniques are well-practiced in C3 with no novel insight required, making it slightly 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]
\item Find the maximum and minimum values of $2\cos\theta + 5\sin\theta$ and the smallest possible value of $\theta$ for which the maximum occurs. [2]
\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}{2}
\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{Edexcel C3 Q8 [15]}}