HL AI Paper 3 Practice

Question 1 · Planning a coffee-shop network

Topics: Voronoi, perpendicular bisectors, graph theory, MST · 28 marks

A coffee-chain owner has three existing shops in a city at:

$A = (4, 2), B = (10, 4), C = (8, 10)$ (units = km).

She wants to open a fourth shop and to plan a delivery network between all four locations. Travel times (min) along direct roads are given later in the question.

(a) [3 marks]

Find the equation of the perpendicular bisector of $[A B]$ in the form $a x + b y + c = 0$ .

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Midpoint of $A B$ : $M = (7, 3)$ . (M1)

Gradient of $A B$ : $m_{A B} = \frac{4 - 2}{10 - 4} = \frac{1}{3}$ ; perpendicular gradient $= - 3$ . (M1)

y - 3 = - 3 (x - 7) \Rightarrow 3 x + y - 24 = 0.

(A1)

(b) [3 marks]

Find the equation of the perpendicular bisector of $[B C]$ .

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Hence find the point $P$ equidistant from $A$ , $B$ and $C$ , and calculate its distance from $A$ .

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(d) [3 marks]

Explain in Voronoi terms what point $P$ represents, and where the owner should place a fourth shop $D$ if she wants to maximise the size of shop $A$ 's Voronoi cell.

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(e) [5 marks]

The owner places the fourth shop at $D = (10, 10)$ . Travel times (min) between the four shops are: $A B = 12, A C = 20, B C = 15, A D = 22, B D = 10, C D = 8$ . Use Prim's algorithm starting at vertex $A$ to find the minimum spanning tree (MST). State the edges chosen in order and the total weight.

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(f) [5 marks]

The owner now wants a daily circuit: start at $A$ , visit every shop exactly once, return to $A$ . Use the nearest-neighbour algorithm starting at $A$ to find an upper bound for the shortest circuit.

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(g) [6 marks]

Now use the deleted-vertex algorithm (deleting $A$ ) to find a lower bound for the shortest Hamiltonian cycle. Compare with (f) and comment on whether the nearest-neighbour tour is optimal.

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Question 2 · Modelling a rumour spreading through a school

Topics: differential equations, logistic model, regression, R² · 27 marks

A rumour spreads through a school of $N = 800$ students. Let $x (t)$ be the number who know the rumour after $t$ hours. Assume the rate of spread is proportional both to the number who know and the number who don't:

\frac{d x}{d t} = k x (N - x), x (0) = 1.

Below is data recorded during one Wednesday morning.

$t$ (hours)	0	1	2	3	4	5
$x$	1	14	140	540	760	795

(a) [4 marks]

Solve the differential equation by separation of variables to show that $x (t) = \frac{N}{1 + A e^{- k N t}}$ for some constant $A$ you should identify in terms of $N$ and $x (0)$ .

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(b) [4 marks]

Use the data point $(t, x) = (3, 540)$ to estimate $k$ to 3 significant figures.

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Using the model with your $k$ , predict $x (4)$ and compare with the observed value $760$ .

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(d) [3 marks]

At what time is the rumour spreading fastest? Interpret this in the context.

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(e) [5 marks]

A researcher proposes an alternative model: exponential growth $y (t) = e^{c t}$ . Using GDC regression on all 6 data points, the researcher obtains $c \approx 1.90$ and $R^{2} \approx 0.72$ . Compare the two models. Which is better? Justify quantitatively and conceptually.

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(f) [4 marks]

Two students, initially resistant, refuse to believe the rumour and never adopt it. Modify the differential equation and describe qualitatively how the long-run behaviour changes.

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(g) [4 marks]

Suggest one further real-world factor not in the model that would change the shape (not just the ceiling) of the curve, and describe how the modified equation would look.

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Question 1 · Planning a coffee-shop network

Question 2 · Modelling a rumour spreading through a school