VCE General Mathematics Units 1 and 2 – Discrete Mathematics

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Description

3.1 Matrices
3.2 Understanding Networks
3.3 Number Sequences




Additional information

Outcome

Matrices
Key knowledge
• the concept of a matrix and its use to store, display and manipulate information
• types of matrices (row, column, square, zero, identity) and the order of a matrix
• matrix arithmetic: the definition of addition, subtraction, multiplication by a scalar, multiplication, the power of a square matrix, and the conditions for their use
• determinant and inverse of a matrix.
Key skills
• use matrices to store and display information that can be presented in rows and columns
• identify row, column, square, zero, and identity matrices and determine their order
• add and subtract matrices, multiply a matrix by a scalar or another matrix, raise a matrix to a power and determine its inverse, using technology as applicable
• use matrix sums, difference, products, powers and inverses to model and solve practical problems.

Graphs and networks
Key knowledge
• the language, properties and types of graphs, including edge, face, loop, vertex, the degree of a vertex, isomorphic and connected graphs, and the adjacency matrix, Euler’s formula for planar graphs, and walks, trails, paths, circuits, bridges and cycles in the context of traversing a graph
• weighted graphs and networks, and the shortest path problem
• trees, minimum spanning trees and Prim’s algorithm.
Key skills
• describe a planar graph in terms of the number of faces (regions), vertices and edges and apply Euler’s formula to solve associated problems
• apply the concepts of connected graphs: trails, paths, circuits, bridges and cycles to model and solve practical problems related to traversing a graph
• find the shortest path in a weighted graph (solution by inspection only)
• apply the concepts of trees and minimum spanning trees to solve practical problems using Prim’s algorithm when appropriate.

Number patterns and recursion
Key knowledge
• the concept of sequence as a function and its recursive specification
• the use of a first-order linear recurrence relation to generate the terms of a number sequence including the special cases of arithmetic and geometric sequences; and the rule for the nth term, tn, of an arithmetic sequence and a geometric sequence and their evaluation
• the use of a first-order linear recurrence relation to model linear growth and decay, including the rule for evaluating the term after n periods of linear growth or decay
• the use of a first-order linear recurrence relation to model geometric growth and decay, including the use of the rule for evaluating the term after n periods of geometric growth or decay
• Fibonacci and related sequences and their recursive specification.
Key skills
• use a given recurrence relation to generate an arithmetic or a geometric sequence, deduce the rule for the nth term from the recursion relation and evaluate
• use a recurrence relation to model and analyse practical situations involving discrete linear and geometric growth or decay
• formulate the recurrence relation to generate the Fibonacci sequence and use this sequence to model and analyse practical situations.

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