1. 1
   The angle $\theta$ satisfies $0<\theta<\frac{\pi}{2}$ and $\cos\theta=\frac{5}{13}$.

   (a)

   Find the value of $\cos(\pi-\theta)$.

   [1 mark]

   • A$-\frac{5}{13}$
   • B$\frac{5}{13}$
   • C$\frac{12}{13}$
   • D$-\frac{12}{13}$

   (b)

   Find the value of $\tan(\pi+\theta)$.

   [1 mark]

   • A$-\frac{12}{5}$
   • B$\frac{5}{12}$
   • C$\frac{12}{5}$
   • D$-\frac{5}{12}$

   (c)

   Find the exact value of $\sin(2\pi-\theta)+\cos\left(\frac{\pi}{2}+\theta\right)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   It is given that $\sin 20^\circ = p$, where $0<p<1$.

   (a)

   Which of the following is equal to $\sin 160^\circ$?

   [1 mark]

   • A$-p$
   • B$1-p$
   • C$\sqrt{1-p^{2}}$
   • D$p$

   (b)

   Which of the following is equal to $\cos 110^\circ$?

   [1 mark]

   • A$p$
   • B$-p$
   • C$\sqrt{1-p^{2}}$
   • D$-\sqrt{1-p^{2}}$

   (c)

   Find an expression for $\tan 200^\circ$ in terms of $p$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $f$ is defined by $f(x)=\sin x+\sin 3x$, for $x\in\mathbb{R}$.

   (a)

   Show that $f(\pi-x)=f(x)$ for all $x\in\mathbb{R}$.

   [3 marks]

   (b)

   (i) Show that $f$ is an odd function.
   (ii) Write down the equation of a vertical line of symmetry of the graph of $y=f(x)$.
   (iii) Hence find the exact values of $f\left(\frac{5\pi}{6}\right)$ and $f\left(-\frac{5\pi}{6}\right)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The height, $h$ metres, of a passenger above the ground on a Ferris wheel is modelled by $h(t)=20-18\cos\left(\frac{\pi t}{15}\right)$, where $t$ is the time in minutes after the passenger boards and $0\le t\le 30$. One revolution takes 30 minutes.

   (a)

   (i) Show that $h(30-t)=h(t)$, and interpret this result in context.
   (ii) Find the two times at which the passenger is 29 m above the ground.

   [6 marks]

   (b)

   (i) Show that $h(t+15)=40-h(t)$ for $0\le t\le15$.
   (ii) Hence, or otherwise, find the two times at which the passenger is 11 m above the ground.

   [6 marks]

   Total for question 4: 12 marks

End of questions