Standard +0.3 This is a standard harmonic form question requiring routine application of R cos(θ + α) transformation, finding maximum values, and solving trigonometric equations. Part (a) is textbook procedure, parts (b)-(c) follow directly from the harmonic form with straightforward calculations, and part (d) requires minimal insight. The multi-part structure adds length but not conceptual difficulty—all techniques are standard A-level pure maths with no novel problem-solving required.
Express \(2 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\) and the value of \(\alpha\) in radians to 3 decimal places.
\begin{figure}[h]
\end{figure}
Figure 6 shows the cross-section of a water wheel.
The wheel is free to rotate about a fixed axis through the point \(C\).
The point \(P\) is at the end of one of the paddles of the wheel, as shown in Figure 6.
The water level is assumed to be horizontal and of constant height.
The vertical height, \(H\) metres, of \(P\) above the water level is modelled by the equation
$$H = 3 + 4 \cos ( 0.5 t ) - 2 \sin ( 0.5 t )$$
where \(t\) is the time in seconds after the wheel starts rotating.
Using the model, find
the maximum height of \(P\) above the water level,
the value of \(t\) when this maximum height first occurs, giving your answer to one decimal place.
In a single revolution of the wheel, \(P\) is below the water level for a total of \(T\) seconds. According to the model,
find the value of \(T\) giving your answer to 3 significant figures.
(Solutions based entirely on calculator technology are not acceptable.)
In reality, the water level may not be of constant height.
Explain how the equation of the model should be refined to take this into account.
9.
\begin{enumerate}[label=(\alph*)]
\item Express $2 \cos \theta - \sin \theta$ in the form $R \cos ( \theta + \alpha )$, where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$ Give the exact value of $R$ and the value of $\alpha$ in radians to 3 decimal places.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{ede204ac-09c3-486b-8877-df935e6ed015-18_456_1150_488_584}
\captionsetup{labelformat=empty}
\caption{Figure 6}
\end{center}
\end{figure}
Figure 6 shows the cross-section of a water wheel.\\
The wheel is free to rotate about a fixed axis through the point $C$.\\
The point $P$ is at the end of one of the paddles of the wheel, as shown in Figure 6.\\
The water level is assumed to be horizontal and of constant height.\\
The vertical height, $H$ metres, of $P$ above the water level is modelled by the equation
$$H = 3 + 4 \cos ( 0.5 t ) - 2 \sin ( 0.5 t )$$
where $t$ is the time in seconds after the wheel starts rotating.\\
Using the model, find
\item \begin{enumerate}[label=(\roman*)]
\item the maximum height of $P$ above the water level,
\item the value of $t$ when this maximum height first occurs, giving your answer to one decimal place.
In a single revolution of the wheel, $P$ is below the water level for a total of $T$ seconds. According to the model,
\end{enumerate}\item find the value of $T$ giving your answer to 3 significant figures.\\
(Solutions based entirely on calculator technology are not acceptable.)
In reality, the water level may not be of constant height.
\item Explain how the equation of the model should be refined to take this into account.
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2024 Q9 [11]}}