Edexcel C3 — Question 6 15 marks

Exam BoardEdexcel
ModuleC3 (Core Mathematics 3)
Marks15
PaperDownload PDF ↗
TopicHarmonic Form
TypeApplied context modeling
DifficultyStandard +0.3 This is a standard C3 harmonic form question with routine application of R cos(θ - α) transformation. Part (a) uses standard formulas (R = √(a² + b²), tan α = b/a), part (b) applies basic properties of cosine range, and parts (c)-(d) are direct applications of the same technique to a context. All steps are algorithmic with no novel insight required, making it slightly easier than average.
Spec1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc
  1. 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]
  2. 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]
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.
  1. Calculate the maximum temperature predicted by this model and the value of \(t\) when this maximum occurs. [4]
  2. Calculate, to the nearest half hour, the times when the temperature is predicted to be \(12 °C\). [6]
\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  Q6 [15]}}
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