The model assumes that each town is positioned at a single point. Describe possible circumstances in which this modelling assumption is reasonable.

Sketch a Voronoi diagram showing the regions within the metropolitan area that are closest to each town.

Find the equation of the perpendicular bisector of $\left[\text{IF}\right]$.

Given that the coordinates of one vertex of the new Voronoi diagram are $(20, 37.5)$, find the coordinates of the other two vertices within the metropolitan area.

Sketch this new Voronoi diagram showing the regions within the metropolitan area which are closest to each town.

A car departs from a point due north of Hamilton. It travels due east at constant speed to a destination point due North of Gaussville. It passes through the Edison, Isaacopolis and Fermitown districts. The car spends $30\%$ of the travel time in the Isaacopolis district.

Find the distance between Gaussville and the car’s destination point.

Find the location of the toxic waste dump, given that this location is not on the edge of the metropolitan area.

Make one possible criticism of the council’s choice of location.

Find which town is last to be polluted. Justify your answer.

Write down the number of days it takes for the pollution to reach the last town.

A sewer inspector needs to plan the shortest possible route through each of the connections between different locations. Determine an appropriate start point and an appropriate end point of the inspection route.

Note that the fact that each location is connected to itself does not correspond to a sewer that needs to be inspected.

the size of each town is small (in comparison with the distance between the towns)
OR
if towns have an identifiable centre
OR
the centre of the town is at that point       R1

Note: Accept a geographical landmark in place of “centre”, e.g. “town hall” or “capitol”.

[1 mark]

       A1

Note: There is no need for a scale / coordinates here. Condone boundaries extending beyond the metropolitan area.

[1 mark]

the gradient of $\text{IF}$ is $\frac{40-20}{40-30}=2$          (A1)

negative reciprocal of any gradient          (M1)

gradient of perpendicular bisector $=\frac{1}{2}$

Note: Seeing $-\frac{2}{3}$ (for example) used clearly as a gradient anywhere is evidence of the “negative reciprocal” method despite being applied to an inappropriate gradient.

midpoint is $\left(\frac{40+30}{2}, \frac{40+20}{2}\right)=\left(35, 30\right)$          (A1)

equation of perpendicular bisector is $y-30=-\frac{1}{2}\left(x-35\right)$         A1

Note: Accept equivalent forms e.g. $y=-\frac{1}{2}x+\frac{95}{2}$  or  $2y+x-95=0$.
Allow FT for the final A1 from their midpoint and gradient of perpendicular bisector, as long as the M1 has been awarded

[4 marks]

the perpendicular bisector of $\text{EH}$ is $y=20$          (A1)

Note: Award this A1 if seen in the $y$-coordinate of any final answer or if $20$ is used as the $y$-value in the equation of any other perpendicular bisector.

attempt to use symmetry OR intersecting two perpendicular bisectors         (M1)

$\left(\frac{25}{3}, 20\right)$         A1

$\left(20, 2.5\right)$         A1

[4 marks]

         M1A1

Note: Award M1 for exactly four perpendicular bisectors around $\text{I}$ ($\text{IE}$, $\text{IF}$, $\text{IG}$ and $\text{IH}$) seen, even if not in exactly the right place.

Award A1 for a completely correct diagram. Scale / coordinates are NOT necessary. Vertices should be in approximately the correct positions but only penalized if clearly wrong (condone northern and southern vertices appearing to be very close to the boundary).

Condone the Voronoi diagram extending outside of the square.

Do not award follow-though marks in this part.

[2 marks]

$30\%$ of $40$ is $12$         (A1)

recognizing line intersects bisectors at $y=c$ (or equivalent) but different $x$-values          (M1)

$c=\frac{3}{2}{x}_1+\frac{15}{2}$  and $c=-\frac{1}{2}{x}_2+\frac{95}{2}$

finding an expression for the distance in Isaacopolis in terms of one variable         (M1)

$x_2-x_1=\left(95-2c\right)-\frac{2c-15}{3}=100-\frac{8c}{3}$

equating their expression to $12$

$100-\frac{8c}{3}=0.3\times 40=12$

$c=33$

distance $=33 \left(\text{km}\right)$        A1

[4 marks]

must be a vertex (award if vertex given as a final answer)         (R1)

attempt to calculate the distance of at least one town from a vertex         (M1)

Note: This must be seen as a calculation or a value.

correct calculation of distances        A1

$\frac{65}{3}$  OR  $21.7$  AND  $\sqrt{406.25}$  OR  $20.2$

$\left(\frac{25}{3}, 20\right)$        A1

Note: Award R1M0A0A0 for a vertex written with no other supporting calculations.
Award R1M0A0A1 for correct vertex with no other supporting calculations.
The final A1 is not dependent on the previous A1. There is no follow-through for the final A1.

Do not accept an answer based on “uniqueness” in the question.

[4 marks]

For example, any one of the following:

decision does not take into account the different population densities

closer to a city will reduce travel time/help employees

it is closer to some cites than others        R1

Note: Accept any correct reason that engages with the scenario.
Do not accept any answer to do with ethical issues about whether toxic waste should ever be dumped, or dumped in a metropolitan area.

[1 mark]

METHOD 1

attempting ${\mathit{M}}^3$         M1

attempting ${\mathit{M}}^4$         M1

e.g.

last row/column of ${\mathit{M}}^3=\left(3 5 1 6 0 7\right)$

last row/column of ${\mathit{M}}^4=\left(10 12 4 16 1 18\right)$

hence Isaacopolis is the last city to be polluted          A1

Note: Do not award the A1 unless both ${\mathit{M}}^3$ and ${\mathit{M}}^4$ are considered.
Award M1M0A0 for a claim that the shortest distance is from $T$ to $I$ and that it is $4$, without any support.

METHOD 2

attempting to translate $\mathit{M}$ to a graph or a list of cities polluted on each day         (M1)

correct graph or list         A1

hence Isaacopolis is the last city to be polluted          A1

Note: Award M1A1A1 for a clear description of the graph in words leading to the correct answer.

[3 marks]

it takes $4$ days        A1

[1 mark]

EITHER

the orders of the different vertices are:

$\text{E 2}$
$\text{F 1}$
$\text{G 2}$
$\text{H 2}$
$\text{I \hspace{0.05em}\hspace{0.05em}\hspace{0.05em}1}$
$\text{T 2}$           (A1)

Note: Accept a list where each order is $2$ greater than listed above.

OR

a correct diagram/graph showing the connections between the locations           (A1)

Note: Accept a diagram with loops at each vertex.
This mark should be awarded if candidate is clearly using their correct diagram from the previous part.

THEN

“Start at F and end at I”  OR  “Start at I and end at F”            A1

Note: Award A1A0 for “it could start at either F or I”.
Award A1A1 for “IGEHTF” OR “FTHEGI”.
Award A1A1 for “F and I” OR “I and F”.

[2 marks]

Question 2 was based on Voronoi diagrams, but a substantial number of candidates appeared to have not met this topic.

2(a) was generally well answered, although a strangely common answer was to claim that the towns must be 1km by 1km! Perhaps this came from thinking of coordinates as representing pixels rather than points.

2(b) was done well by most candidates who knew what a Voronoi diagram was.

2(c)(i) showed some poor planning skills by many candidates who found a line perpendicular to the given Voronoi edge rather than finding the gradient of IF.

In 2(c)(ii) candidates would have benefited from taking a step back and thinking about the symmetry of the situation before unthinkingly intersecting a lot of lines.

2(c)(iii) was done well by some, but many did not seem to have any intuition for the effect of adding a point to a Voronoi diagram.

2(d) was probably the worst answered question. Candidates did not seem to have the problem-solving tools to deal with this slightly unfamiliar situation.

In 2(e) many candidates clearly did not know that the solution to the toxic waste problem (as described in the syllabus) occurs at a vertex of the Voronoi diagram.

A pleasing number of candidates were able to approach 2(f) even if they were unable to do earlier parts of the question, and in these very long questions candidates should be advised that sometimes later parts are not necessarily harder or impossible to access. Very few who attempted the matrix power approach could interpret what zeroes meant in the matrices produced, but a good number successfully turned the matrix into a graph and proceeded well from there.