1. 1
   A regular polygon has interior angles that are each $150°$.

   (a)

   (a) How many sides does this polygon have?

   [1 mark]

   • A12
   • B10
   • C8
   • D15

   (b)

   (b) What is the sum of the interior angles of this polygon?

   [1 mark]

   • A$2160°$
   • B$1800°$
   • C$1620°$
   • D$1980°$

   (c)

   (c) A different quadrilateral has three interior angles of $80°$, $95°$ and $110°$. What is its fourth angle?

   [1 mark]

   • A$85°$
   • B$70°$
   • C$75°$
   • D$65°$

   Total for question 1: 3 marks

2. 2
   A trapezium has parallel sides of length $9$ cm and $15$ cm, and a perpendicular height of $8$ cm. Separately, a composite shape is formed by attaching a semicircle of radius $6$ cm to one side of a rectangle measuring $12$ cm by $6$ cm, where the semicircle's diameter equals the rectangle's $12$ cm side.

   (a)

   (a) Calculate the area of the trapezium.

   [2 marks]

   (b)

   (b) Calculate the area of the composite shape (rectangle plus semicircle), correct to 1 decimal place.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Triangle $PQR$ has a right angle at $Q$. $PQ = 7$ cm and $PR = 25$ cm.

   (a)

   (a) Show that $QR = 24$ cm.

   [3 marks]

   (b)

   (b) Hence show that $\tan(\angle QPR) = \frac{24}{7}$, and find the size of angle $QPR$, correct to 1 decimal place.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   Square $S$ has an area formed by sides of length $4$ cm, and is enlarged by scale factor $\frac{3}{2}$, centre a fixed point, to form square $T$. Separately, $\mathbf{u} = \binom{3}{-1}$ and $\mathbf{v} = \binom{-1}{4}$, and two canoeists paddle away from the same jetty: the first on a bearing of $062°$, the second on a bearing of $148°$.

   (a)

   (a) Calculate the area of square $T$, and state the ratio of the area of $S$ to the area of $T$ in its simplest form.

   [4 marks]

   (b)

   (b) Point $X$ has position vector $2\mathbf{u}+\mathbf{v}$ relative to the origin, and point $Y$ has position vector $5\mathbf{u}-\mathbf{v}$. Find $\vec{XY}$ in column vector form, and calculate its magnitude, correct to 1 decimal place.

   [4 marks]

   (c)

   (c) Calculate the angle between the two canoeists' paths, measured at the jetty. Then find the bearing the first canoeist would need to paddle on to return directly to the jetty from her current position.

   [5 marks]

   Total for question 4: 13 marks

5. 5
   Triangle $DEF$ has a right angle at $E$. Angle $D = 42°$ and $DE = 15$ cm.

   (a)

   (a) Calculate the length $EF$, correct to 1 decimal place.

   [1 mark]

   • A11.1 cm
   • B20.2 cm
   • C10.0 cm
   • D13.5 cm

   (b)

   (b) Calculate the length of the hypotenuse $DF$, correct to 1 decimal place.

   [1 mark]

   • A20.2 cm
   • B13.5 cm
   • C11.1 cm
   • D16.7 cm

   (c)

   (c) What is the size of angle $F$?

   [1 mark]

   • A$42°$
   • B$48°$
   • C$58°$
   • D$38°$

   Total for question 5: 3 marks

6. 6
   Quadrilateral $ABCD$ has $\vec{AB} = \binom{5}{2}$ and $\vec{DC} = \binom{5}{2}$.

   (a)

   (a) State, with a reason, what type of quadrilateral $ABCD$ must be, given that $\vec{AB} = \vec{DC}$.

   [2 marks]

   (b)

   (b) Given also that $\vec{AD} = \binom{-1}{4}$, find $\vec{BC}$ in column vector form.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A lighthouse $L$ is on a bearing of $048°$ from a harbour $H$. A buoy $B$ is on a bearing of $138°$ from the same harbour $H$.

   (a)

   (a) Show that angle $LHB$ (the angle between the lighthouse and the buoy, measured at the harbour) is $90°$.

   [3 marks]

   (b)

   (b) Find the bearing of the harbour $H$ from the lighthouse $L$ (the back bearing).

   [3 marks]

   Total for question 7: 6 marks

8. 8
   A regular hexagon and a regular pentagon are placed so that one side of the hexagon coincides exactly with one side of the pentagon. Both shapes have side length $10$ cm. (The area of a regular hexagon with side $a$ is $\frac{3\sqrt3}{2}a^2$.)

   (a)

   (a) Calculate the interior angle of a regular hexagon and the interior angle of a regular pentagon. Hence find the size of the combined angle formed at a point where one vertex of each shape meets (the two interior angles added together).

   [4 marks]

   (b)

   (b) Calculate the perimeter of the pentagon, and calculate the area of the regular hexagon, correct to 1 decimal place.

   [4 marks]

   (c)

   (c) The hexagon is enlarged by scale factor $0.4$, centre a fixed point, to form a smaller hexagon. Calculate the area of the smaller hexagon, correct to 1 decimal place. Then calculate the scale factor that would be needed to enlarge the ORIGINAL hexagon so that its area becomes exactly double, correct to 3 significant figures.

   [5 marks]

   Total for question 8: 13 marks

End of questions