Unit 3

9 dot points, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

When can a network be drawn with no crossings, and how do its faces, edges and vertices relate?

Recognise planar graphs and apply Euler's formula relating vertices, edges and faces.

Planar graphs, faces, and Euler's formula v - e + f = 2, with worked checks and applications in TCE Mathematics Applications.

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When can we trace every edge, or visit every vertex, of a network exactly once?

Identify and find Eulerian trails and circuits and Hamiltonian paths and cycles in networks.

Walks, trails, paths, Eulerian trails and circuits, Hamiltonian paths and cycles, and the vertex-degree conditions that decide them in TCE Mathematics Applications.

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How do we detect and describe association between two categorical variables?

Construct and interpret two-way frequency tables to investigate association between categorical variables.

Constructing two-way frequency tables, calculating row and column percentages, and judging association between categorical variables in TCE Mathematics Applications.

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How do we describe and sum sequences that grow by a constant amount or a constant factor?

Use arithmetic and geometric sequences, including their nth-term and sum rules, to model practical situations.

The nth-term and sum formulas for arithmetic and geometric sequences, with applications to growth, decay, and accumulation in TCE Mathematics Applications.

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How do we model a process that both multiplies and adds a fixed amount each step?

Use first-order linear recurrence relations of the form V(n+1) = R V(n) + d to model and analyse practical situations.

The general first-order linear recurrence with a multiplier and a constant, its long-term behaviour, equilibrium value, and financial and population applications in TCE Mathematics Applications.

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How do we check whether a straight line is the right model for bivariate data?

Calculate residuals and use a residual plot to assess the appropriateness of a linear model.

Calculating residuals, constructing and reading residual plots, and judging whether a linear model fits bivariate data in TCE Mathematics Applications.

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How can we find the best decision when choices are limited by constraints?

Formulate and solve linear programming problems graphically to optimise an objective function.

Defining variables, writing constraints, graphing feasible regions, and using the corner-point method to maximise or minimise an objective in TCE Mathematics Applications.

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How do matrices and networks model connected systems?

Perform matrix operations and analyse graphs, paths and networks

Matrix arithmetic, multiplication and inverses plus graph theory, adjacency matrices and shortest paths for TCE Mathematics Applications, with worked examples.

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How does money grow or shrink over time, and how are loans and investments modelled?

Apply simple and compound interest, depreciation, and recurrence relations to financial situations.

Simple and compound interest, depreciation, effective rates, and recurrence relations for loans and investments in TCE Mathematics Applications.

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