1.1. Express $2cos \theta -sin \theta$ in the form $Rcos( \theta + \alpha )$, 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.

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

$H=3+4cos(0.5t)-2sin(0.5t)$

where $t$ is the time in seconds after the wheel starts rotating.

Using the model, find

(i) the maximum height of $P$ above the water level,

(ii) the value of $t$ when this maximum height first occurs, giving your answer to one decimal place.

1.3. 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.)