1. 1
   Let $f(x) = 3\sin\left(2\left(x - \frac{\pi}{6}\right)\right) + 1$, for $x \in \mathbb{R}$.

   (a)

   Find the period of $f$.

   [1 mark]

   • A$2\pi$
   • B$4\pi$
   • C$\pi$
   • D$\frac{\pi}{2}$

   (b)

   Find the range of $f$.

   [1 mark]

   • A$-2 \le f(x) \le 4$
   • B$-3 \le f(x) \le 3$
   • C$-4 \le f(x) \le 2$
   • D$-1 \le f(x) \le 3$

   (c)

   Find the smallest positive value of $x$ at which $f(x)$ takes its maximum value.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The depth of water, $d$ metres, at the entrance to a harbour $t$ hours after midnight is modelled by $d(t) = 2.5\cos\left(\frac{\pi t}{6}\right) + 7$, for $0 \le t \le 24$.

   (a)

   Find the time between two consecutive high tides.

   [1 mark]

   • A6 hours
   • B12 hours
   • C$\frac{\pi}{6}$ hours
   • D$\frac{\pi^2}{3}$ hours

   (b)

   Find the depth of water at 04:00.

   [1 mark]

   • A8.25 m
   • B4.5 m
   • C$7 - \frac{5\sqrt{3}}{4}$ m
   • D5.75 m

   (c)

   Write down the minimum depth of water, and find the first time after midnight at which it occurs.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The graph of $y = \sin x$ is transformed into the graph of $y = g(x)$ by the following sequence of transformations: a vertical stretch with scale factor 3, then a horizontal stretch with scale factor $\frac{1}{2}$, then a translation of $\frac{\pi}{3}$ units in the positive $x$-direction, then a translation of 1 unit in the negative $y$-direction.

   (a)

   Write $g(x)$ in the form $a\sin\left(b(x + c)\right) + d$, stating the values of $a$, $b$, $c$ and $d$.

   [3 marks]

   (b)

   Find the exact $y$-intercept of the graph of $g$, and the exact coordinates of the maximum point of $g$ for $0 \le x \le \pi$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A Ferris wheel has a diameter of 40 m and its centre is 22 m above the ground. It rotates at a constant speed and completes one revolution every 10 minutes. At time $t = 0$ minutes, seat P is at the lowest point of the wheel. The height of P above the ground, $h$ metres, is modelled by $h(t) = a\cos(bt) + d$, where $a, b, d \in \mathbb{R}$ and $b > 0$.

   (a)

   Find the values of $a$, $b$ and $d$, and hence write down $h(t)$.

   [6 marks]

   (b)

   (i) Show that P is 32 m above the ground when $t = \frac{10}{3}$.
   (ii) Hence find the length of time, in minutes and seconds, during each revolution for which P is more than 32 m above the ground.

   [6 marks]

   Total for question 4: 12 marks

End of questions