IB Maths AI HL AI 3.4 Graph theory questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

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Question 1

Paper 1Medium8 marks

A courier moves between three depots, $A$ $B$ and $C$ The directed routes leaving each depot are shown below.

Depot	Depots reachable in one leg
$A$	$B, C$
$B$	$A, C$
$C$	$A$

At each depot, the courier chooses uniformly from the routes leaving that depot. Use the convention that the entry in row $i$ column $j$ of a transition matrix is the probability of moving from depot $i$ to depot $j$

(a)

Write down the adjacency matrix $A$ for this network, using the order $A,B,C$ for both rows and columns.

[2]

(b)

Find the number of two-leg walks from $A$ back to $A$ using the appropriate entry of $A^2$

[2]

(c)

Construct the transition matrix $T$ for the courier’s movements, using the order $A,B,C$ for both rows and columns.

[2]

(d)

Using your transition matrix, find the probability that the courier is at $A$ after exactly two legs, starting from $A$

[2]

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