IB Mathematics Applications & Interpretation Topic AHL 3.15—adjacency Matrices and Tables Questionbank (HL)

IB Mathematics Applications & Interpretation Topic AHL 3.15—adjacency Matrices and Tables Questionbank (HL) | RevisionDojo

Practice IB Mathematics Applications & Interpretation (AI) Topic AHL 3.15—adjacency Matrices and Tables with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 3.15—adjacency Matrices and Tables and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

HLPaper 2

A communication system has nodes $U, V, W, X, Y$ , and $Z$ , with directed edges representing signal paths, some with self-loops. The adjacency matrix is:

$M = 100001110000010000001100000110000011$

Draw the directed graph, including self-loops.

[2]

Find the number of walks of length 4 from $U$ to $Z$ .

[3]

Determine how many distinct nodes are reachable from $U$ by a walk of length exactly 5.

[4]

A new node $K$ is added, with directed edges $K \to V$ , $K \to X$ , and $Z \to K$ . Update the adjacency matrix and graph.

[3]

In the updated system, find the number of cycles of length 3 including $K$ .

[4]

Question 2

HLPaper 2

A transport network has hubs $A, B, C, D, E$ , and $F$ , with directed edges representing one-way routes. The adjacency matrix includes self-loops (hubs retaining cargo):

$M = 100001110000011000001100000110000011$

Draw the directed graph, including self-loops.

[2]

Find the number of walks of length 5 from $A$ to $F$ .

[3]

A cargo starts at $A$ . List the hubs that can be reached after exactly 4 steps.

[4]

A new hub $G$ is added with directed edges $G \to B$ , $G \to F$ , and $E \to G$ . Update the adjacency matrix and graph.

[3]

In the updated network, find the number of walks of length 3 from $G$ to $A$ , and interpret in the context of cargo transport.

[4]

Question 3

HLPaper 2

A directed graph models a data network with nodes $P, Q, R, S$ , and $T$ , representing servers, and directed edges indicating one-way data transmission. ![](A directed graph with nodes $P, Q, R, S, T$ and directed edges $P \to Q$ , $Q \to R$ , $Q \to S$ , $R \to S$ , $S \to T$ , $T \to P$ .)

Write down the adjacency matrix $M$ for this graph, with rows and columns ordered $P, Q, R, S, T$ .

[2]

Find the number of distinct walks of length 5 from $P$ to $R$ .

[3]

Determine the number of servers reachable from $P$ in exactly 3 steps.

[3]

A new server $U$ is added. Add directed edges $Q \to U$ and $U \to R$ . Update the adjacency matrix and redraw the graph, with rows and columns ordered $P, Q, R, S, T, U$ .

[3]

In the updated network, calculate the number of walks of length 4 from $U$ to $P$ .

[4]

Question 4

HLPaper 2

A graph represents a pipeline network with vertices $A, B, C, D, E$ , and edges with weights as maintenance costs (in thousands of USD). The adjacency matrix is:

$M = 0508050304030608060704070$

Draw the weighted graph corresponding to $M$ .

[2]

Use Prim's algorithm, starting at $A$ , to find the minimum spanning tree, indicating the order of edge selection.

[5]

A maintenance team must visit all vertices, starting and ending at $A$ . Use the nearest neighbor algorithm to find an upper bound for the total cost.

[4]

Find a lower bound for the total cost using the MST from Part 1.

[3]

If the edge $A - B$ cost increases to $x$ thousand USD, find the range of $x$ for which $A - B$ remains in the MST.

[4]

Question 5

HLPaper 2

A logistics company operates a network of warehouses labeled

A, B, C, D

, and

E

, connected by direct delivery routes with weights representing distances (in km). The weighted graph is shown below.

Write down the adjacency matrix $M$ for this graph, with rows and columns ordered $A, B, C, D, E$ .

[2]

Use Kruskal's algorithm to find the minimum spanning tree (MST), listing the edges in the order they are selected.

[5]

Calculate the total distance of the MST.

[2]

A delivery van must visit all warehouses, starting and ending at $A$ . Use the nearest neighbor algorithm to find an upper bound for the total distance.

[4]

Determine a lower bound for the van's total distance using the MST from Part 1.

[3]

IB Mathematics Applications & Interpretation AHL 3.15—adjacency Matrices and Tables Questionbank

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IB Mathematics Applications & Interpretation AHL 3.15—adjacency Matrices and Tables Questionbank

IB Mathematics Applications & Interpretation (AI) Topic AHL 3.15—adjacency Matrices and Tables Questionbank

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