1. 1
   A vertical tower $FT$ stands on horizontal ground, where $F$ is the foot and $T$ is the top. From a point $P$ on the ground, 40 m from $F$, the angle of elevation of $T$ is $30^\circ$.

   (a)

   Find the height of the tower.

   [1 mark]

   • A$\frac{40\sqrt3}{3}$ m
   • B$20$ m
   • C$40\sqrt3$ m
   • D$80$ m

   (b)

   A student walks from $P$ directly towards $F$ until the angle of elevation of $T$ is $60^\circ$. Find the distance the student walks.

   [1 mark]

   • A$\frac{40}{3}$ m
   • B$\frac{80}{3}$ m
   • C$20$ m
   • D$40-\frac{40\sqrt3}{3}$ m

   (c)

   A point $Q$ on the ground lies on the opposite side of the tower from $P$, with $P$, $F$ and $Q$ in a straight line. The angle of depression of $Q$ from $T$ is $45^\circ$. Find the exact distance $PQ$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A ship leaves harbour $H$ and sails 12 km on a bearing of $060^\circ$ to a buoy $A$. It then sails 5 km on a bearing of $150^\circ$ to a point $B$.

   (a)

   Find the size of angle $H\hat{A}B$.

   [1 mark]

   • A$30^\circ$
   • B$60^\circ$
   • C$90^\circ$
   • D$150^\circ$

   (b)

   Find the distance $HB$.

   [1 mark]

   • A$17$ km
   • B$\sqrt{119}$ km
   • C$7$ km
   • D$13$ km

   (c)

   Using a calculator, find the bearing of $B$ from $H$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Points $A$ and $B$ lie on horizontal ground in a straight line with the foot $F$ of a vertical mast $FT$, with $B$ between $A$ and $F$. $AB=60$ m. The angle of elevation of the top $T$ of the mast is $35^\circ$ from $A$ and $50^\circ$ from $B$. A calculator may be used.

   (a)

   Find $A\hat{T}B$, and show that $BT=\frac{60\sin35^\circ}{\sin15^\circ}$.

   [3 marks]

   (b)

   Hence find the height of the mast and the distance $BF$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A boat leaves port $P$ and sails 20 km on a bearing of $040^\circ$ to an island $Q$. It then sails 30 km on a bearing of $110^\circ$ to a lighthouse $R$. A calculator may be used.

   (a)

   (i) Show that $P\hat{Q}R=110^\circ$.
   (ii) Find the distance $PR$.
   (iii) Find how much shorter the direct route from $P$ to $R$ is than the route through $Q$.

   [6 marks]

   (b)

   Find the bearing of $R$ from $P$. Hence find the bearing on which the boat must sail to return directly from $R$ to $P$.

   [6 marks]

   Total for question 4: 12 marks

End of questions