1. 1
   In triangle $ABC$, $AB = 9$ cm, $AC = 7$ cm and $B\hat{A}C = 55°$.

   (a)

   Find the length of $BC$, correct to 3 significant figures.

   [1 mark]

   • A$7.60$ cm
   • B$14.2$ cm
   • C$9.69$ cm
   • D$11.4$ cm

   (b)

   Find the size of $A\hat{B}C$, correct to 1 decimal place.

   [1 mark]

   • A$49.0°$
   • B$76.0°$
   • C$131.0°$
   • D$55.0°$

   (c)

   Find the area of triangle $ABC$, correct to 3 significant figures.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A sector $OAB$ of a circle with centre $O$ and radius 9 cm has $A\hat{O}B = \dfrac{2\pi}{3}$ radians.

   (a)

   Find the length of the arc $AB$, giving your answer as an exact multiple of $\pi$.

   [1 mark]

   • A$3\pi$ cm
   • B$12\pi$ cm
   • C$6\pi$ cm
   • D$81\pi$ cm

   (b)

   Find the area of the sector $OAB$, giving your answer as an exact multiple of $\pi$.

   [1 mark]

   • A$13.5\pi$ cm$^2$
   • B$54\pi$ cm$^2$
   • C$81\pi$ cm$^2$
   • D$27\pi$ cm$^2$

   (c)

   Find the perimeter of the sector $OAB$, correct to 3 significant figures.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The angle $\theta$ is obtuse, and $\sin\theta = \dfrac35$.

   (a)

   Find the exact value of $\cos\theta$, and hence the exact value of $\tan\theta$.

   [3 marks]

   (b)

   Find the exact value of $\sin2\theta$ and the exact value of $\cos2\theta$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A vertical flagpole $FT$ stands on horizontal ground at point $F$, where $T$ is the top of the pole. Points $P$ and $Q$ lie on the ground with $FP = 25$ m, $FQ = 18$ m and $P\hat{F}Q = 70°$. From $P$, the angle of elevation of $T$ is $32°$. Separately, a storage container in the shape of a cuboid has a rectangular base measuring 6 m by 4 m and a height of 3 m; a support cable runs from one bottom corner of the container to the diagonally opposite top corner.

   (a)

   Find the height $FT$ of the flagpole, and find the distance $PQ$, each correct to 3 significant figures.

   [6 marks]

   (b)

   Find the length of the support cable on the storage container, and the angle it makes with the base of the container, each correct to 3 significant figures.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Consider the angle $\theta = \dfrac{5\pi}{6}$.

   (a)

   Find the exact value of $\cos\theta$.

   [1 mark]

   • A$-\dfrac{\sqrt3}{2}$
   • B$\dfrac{\sqrt3}{2}$
   • C$-\dfrac12$
   • D$\dfrac12$

   (b)

   Find the exact value of $\sin\theta$.

   [1 mark]

   • A$-\dfrac{\sqrt3}{2}$
   • B$\dfrac12$
   • C$-\dfrac12$
   • D$\dfrac{\sqrt3}{2}$

   (c)

   Find the exact value of $\tan\theta$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Let $f(x) = 4\cos\left(3\left(x+\dfrac{\pi}{4}\right)\right) - 2$, for $x \in \mathbb{R}$.

   (a)

   Find the period of $f$.

   [1 mark]

   • A$6\pi$
   • B$\dfrac{\pi}{3}$
   • C$\dfrac{2\pi}{3}$
   • D$2\pi$

   (b)

   Find the range of $f$.

   [1 mark]

   • A$-2\le f(x)\le4$
   • B$-4\le f(x)\le2$
   • C$-3\le f(x)\le3$
   • D$-6\le f(x)\le2$

   (c)

   Find the minimum value of $f(x)$, and the smallest positive value of $x$ at which it occurs.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Consider the equation $2\cos^2x - \sin x - 1 = 0$, for $0 \le x \le 2\pi$.

   (a)

   Show that the equation can be written as $2\sin^2x + \sin x - 1 = 0$, and hence find the values of $\sin x$ that satisfy the original equation.

   [3 marks]

   (b)

   Hence find all solutions of the equation for $0 \le x \le 2\pi$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The vertical displacement, $d$ cm, of a buoy from its mean position, $t$ seconds after release, is modelled by $d(t) = 6\sin\left(\dfrac{\pi}{4}t\right)$, for $t \ge 0$.

   (a)

   State the amplitude of the motion, and find its period. Find the first time $t>0$ at which the buoy reaches its maximum displacement, and state this maximum displacement.

   [6 marks]

   (b)

   Find the first two values of $t>0$ for which the buoy's displacement is 3 cm above its mean position.

   [6 marks]

   Total for question 8: 12 marks

End of questions