1. 1
   A quadrilateral $PQRS$ has $PQ$ parallel to $SR$, $PQ\neq SR$, and $PS=QR$ but $PS$ is not parallel to $QR$.

   (a)

   What type of quadrilateral is $PQRS$?

   [1 mark]

   • AParallelogram
   • BKite
   • CRhombus
   • DIsosceles trapezium

   (b)

   A separate solid has 5 faces: 2 triangular and 3 rectangular. What is this solid?

   [1 mark]

   • ACone
   • BTriangular prism
   • CSquare pyramid
   • DCuboid

   (c)

   A third solid tapers from a circular base to a single point (apex). What is this solid?

   [1 mark]

   • ACylinder
   • BFrustum
   • CCone
   • DSphere

   Total for question 1: 3 marks

2. 2
   In triangle $XYZ$, angle $X=55°$ and angle $Y=72°$. Side $YZ$ is extended beyond $Z$ to point $W$.

   (a)

   Find angle $XZY$, the interior angle of the triangle at $Z$.

   [2 marks]

   (b)

   Find the exterior angle $XZW$, formed by extending side $YZ$ beyond $Z$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A hiker walks from base camp $A$ to checkpoint $B$ on a bearing of $048°$, then from $B$ to camp $C$ on a bearing of $138°$.

   (a)

   Find the bearing of $A$ from $B$ (the back bearing).

   [3 marks]

   (b)

   Given that angle $ABC$ (the angle at $B$ between paths $BA$ and $BC$) can be found using the bearings, calculate angle $ABC$.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   Points $E$, $F$, $G$ and $H$ lie on a circle with centre $O$, in that order, forming a cyclic quadrilateral $EFGH$. Angle $EOG$ (the reflex-free angle at the centre standing on the same arc as angle $EFG$) is $136°$. Chord $EG$ is drawn, and angle $FEG=35°$.

   (a)

   Find angle $EFG$, using the circle theorem relating angles at the centre and circumference.

   [4 marks]

   (b)

   Find angle $EHG$, the opposite angle to $EFG$ in the cyclic quadrilateral $EFGH$.

   [4 marks]

   (c)

   In triangle $EFG$, given angle $FEG=35°$ and angle $EFG=68°$ (from part (a)), find angle $EGF$ and hence state, with a reason, whether triangle $EFG$ is scalene, isosceles or right-angled.

   [5 marks]

   Total for question 4: 13 marks

5. 5
   On a map with scale $1:50000$, the distance between two villages, $M$ and $N$, is measured as 6 cm.

   (a)

   What is the real distance between $M$ and $N$, in kilometres?

   [1 mark]

   • A$0.3$ km
   • B$3$ km
   • C$30$ km
   • D$300$ km

   (b)

   A third village, $P$, is on a bearing of $205°$ from $M$. Which compass direction is $P$ approximately from $M$?

   [1 mark]

   • ANorth-east
   • BSouth-east
   • CSouth-west
   • DNorth-west

   (c)

   What is the bearing of $M$ from $P$ (the back bearing of $205°$)?

   [1 mark]

   • A$025°$
   • B$155°$
   • C$335°$
   • D$385°$

   Total for question 5: 3 marks

6. 6
   A circle has centre $O$. A tangent touches the circle at point $T$, and $OT=5$ cm. A point $S$ outside the circle is such that $ST$ is part of the tangent line and $OS=13$ cm.

   (a)

   State the size of angle $OTS$, giving a reason.

   [2 marks]

   (b)

   Find the length $TS$, using Pythagoras' theorem.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A regular polygon has an interior angle of $156°$.

   (a)

   Find the exterior angle of the polygon, and hence find the number of sides.

   [3 marks]

   (b)

   Two parallel lines are crossed by a transversal. One angle formed is co-interior with an angle equal to the exterior angle found in part (a) (i.e. $24°$). Find this co-interior angle.

   [3 marks]

   Total for question 7: 6 marks

8. 8
   A landscape designer plans a triangular garden plot $XYZ$ on a scale drawing using a scale of $1:400$. On the drawing, $XY=6$ cm, $YZ=8$ cm and $XZ=10$ cm. Corner $X$ is on a bearing of $062°$ from the site office $B$. A circular flower bed at corner $Z$ has centre $O$, with $ZW$ a diameter and $V$ a third point on the circumference; a tangent to the circle touches at $V$.

   (a)

   Find the real-life length of each side of the garden plot $XYZ$, in metres, and state the type of triangle $XYZ$ is, giving a reason based on its side lengths.

   [4 marks]

   (b)

   Given that angle $ZVW=90°$ (angle in a semicircle) and a tangent to the circle touches at $V$, name the circle theorem that gives angle $ZVW$, and find the angle between this tangent and the radius $OV$.

   [4 marks]

   (c)

   The garden design also includes a regular polygon flower border with an exterior angle of $36°$. Find the number of sides of this polygon, its interior angle, and name the shape.

   [5 marks]

   Total for question 8: 13 marks

End of questions