1. 1
   The point $P$ lies on the unit circle $x^2+y^2=1$ such that the line $OP$, where $O$ is the origin, makes an angle of $\frac{2\pi}{3}$ measured anticlockwise from the positive $x$-axis.

   (a)

   Write down the exact coordinates of $P$.

   [1 mark]

   • A$\left(\frac{\sqrt3}{2},\,-\frac12\right)$
   • B$\left(-\frac{\sqrt3}{2},\,\frac12\right)$
   • C$\left(\frac12,\,\frac{\sqrt3}{2}\right)$
   • D$\left(-\frac12,\,\frac{\sqrt3}{2}\right)$

   (b)

   Find the equation of the line $OP$.

   [1 mark]

   • A$y=\sqrt3x$
   • B$y=-\frac{1}{\sqrt3}x$
   • C$y=-\sqrt3x$
   • D$y=-\frac{\sqrt3}{2}x$

   (c)

   The point $Q$ lies on the unit circle such that $OQ$ makes an angle of $-\frac{2\pi}{3}$ with the positive $x$-axis. Find the exact distance $PQ$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The angle $\alpha$ satisfies $0<\alpha<\frac{\pi}{2}$, with $\sin\alpha=p$ and $\cos\alpha=q$.

   (a)

   Find $\sin(\pi+\alpha)$ in terms of $p$ or $q$.

   [1 mark]

   • A$p$
   • B$-p$
   • C$q$
   • D$-q$

   (b)

   Find $\sin(\pi-\alpha)$ in terms of $p$ or $q$.

   [1 mark]

   • A$p$
   • B$-p$
   • C$q$
   • D$-q$

   (c)

   Find $\tan(2\pi-\alpha)$ in terms of $p$ and $q$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   In triangle $ABC$, $AB=10$ cm, $AC=6$ cm and $A\hat{B}C=30^\circ$. There are two possible triangles that fit this information.

   (a)

   Using a calculator, find the two possible sizes of $A\hat{C}B$, and verify that both give a valid triangle.

   [3 marks]

   (b)

   Let $BC=x$ cm. By using the cosine rule in triangle $ABC$, show that $x^2-10\sqrt3\,x+64=0$. Hence find the exact difference between the two possible lengths of $BC$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A Ferris wheel has radius 20 m and its centre is 22 m above the ground. A seat starts level with the centre, on the right-hand side, and the wheel turns anticlockwise. After the wheel has turned through an angle $\theta$ radians, the seat is $20\cos\theta$ m to the right of the centre and its height above the ground is $h=22+20\sin\theta$ m. (A negative horizontal value means the seat is to the left of the centre.)

   (a)

   (i) Find the exact height of the seat when $\theta=\frac{7\pi}{6}$.
   (ii) Find the exact horizontal distance of the seat from the centre when $\theta=\frac{5\pi}{4}$, and state whether it is to the left or right of the centre.
   (iii) Find the values of $\theta$, for $0\le\theta<2\pi$, at which the seat is 32 m above the ground.

   [6 marks]

   (b)

   (i) Two seats are fixed at diametrically opposite points of the wheel. Show that the sum of their heights is always 44 m.
   (ii) A seat is $22+10\sqrt3$ m above the ground and to the left of the centre. Find the value of $\theta$, where $0\le\theta<2\pi$, for this seat, justifying your answer.

   [6 marks]

   Total for question 4: 12 marks

End of questions