§-General Mathematics syllabus
VIC · VCAA← General Mathematics
General Mathematics syllabus, dot point by dot point
Every dot point in the VIC General Mathematics syllabus, with a focused answer for each. Click any dot point for a worked explainer, past exam questions and links to related points.
Unit 3 Recursion and financial modelling
Module overview →How do annuity investments and savings plans grow when regular payments are added to a compounding balance, and how is the finance solver set up for them?
Model an annuity investment or savings plan where a regular payment is added to a compounding balance using the recurrence relation, analyse it on a finance solver, and find the final balance, total interest and required payment
How do you investigate the association between two categorical variables, or between a numerical and a categorical variable, using tables and grouped displays?
Construct and interpret a two-way frequency table, convert it to percentages to investigate association between two categorical variables, use a segmented or side-by-side bar chart, and compare a numerical variable across categories with parallel boxplots
How do you build a boxplot from a five-number summary, test for outliers, and compare two groups using parallel boxplots?
The five-number summary, construction and interpretation of boxplots, the use of the lower and upper fences to identify outliers, and comparison of distributions using parallel boxplots
How do you measure the association between two numerical variables and fit a least-squares line to model and predict?
Investigate the association between two numerical variables using a scatterplot, the correlation coefficient and the coefficient of determination, fit a least-squares regression line, and interpret its slope, intercept and residuals
When a scatterplot is curved, how do the squared, log and reciprocal transformations straighten the data so a least-squares line can be fitted?
Recognise non-linear association from a scatterplot and residual plot, apply the squared, logarithmic or reciprocal transformation to the explanatory or response variable to linearise the data, fit a least-squares line to the transformed data, and use it to predict
How do you classify data, choose the right display, and describe the shape, centre and spread of a distribution?
Types of data (categorical and numerical), appropriate graphical displays, and describing a numerical distribution in terms of shape, centre and spread using the mean, median, range, interquartile range and standard deviation
How do you summarise, display and describe the distribution of a single numerical or categorical variable in VCE General Mathematics?
Display and describe the distribution of a numerical variable using a histogram, dot plot, stem plot or boxplot, summarise it with measures of centre and spread, and identify outliers using the lower and upper fences
How does the normal distribution describe data, and how do the 68-95-99.7 rule and standardised z-scores let you compare and find percentages?
Use the normal distribution and the 68-95-99.7 rule to estimate the percentage of values within a number of standard deviations of the mean, and standardise a value to a z-score to compare values from different distributions
What is a perpetuity, how does the balance stay constant when the payment exactly equals the interest, and how do you find the payment or the principal?
Model a perpetuity as a special annuity in which the regular payment equals the interest earned each period so the balance never changes, and find the perpetual payment, the required principal or the interest rate
How are compound interest investments, reducing-balance loans and annuities modelled with recurrence relations and analysed on a financial solver?
Model and analyse compound interest investments, reducing-balance loans, annuities and perpetuities using a first-order recurrence relation and a finance solver, and interpret balance, repayment, interest and the effect of changing parameters
How do seasonal indices quantify a repeating pattern, and how do you deseasonalise and reseasonalise time series data?
Calculate seasonal indices from time series data, interpret an index as a percentage above or below the seasonal average, deseasonalise data by dividing by the index, and reseasonalise a forecast by multiplying by the index
How do first-order recurrence relations generate arithmetic and geometric sequences, and how are their terms and sums found?
Use a first-order linear recurrence relation to generate a sequence, recognise arithmetic and geometric sequences, find the nth term with an explicit rule, and compute the sum of an arithmetic or geometric series
How do simple and compound interest differ, how does each follow a recurrence relation, and how do you compare them with effective rates?
Model simple interest with a linear recurrence and compound interest with a geometric recurrence, find balances and interest earned, convert between nominal and effective annual rates, and compare simple and compound growth
How do you display and describe a time series, and how do moving-mean and moving-median smoothing reveal the underlying trend?
Construct and interpret a time series plot, describe its features (trend, seasonality, cycles, irregular fluctuations), smooth it using moving-mean and moving-median smoothing, and fit a least-squares trend line for forecasting
Unit 4 Matrices
Module overview →How do communication and dominance matrices use a matrix of zeros and ones to count direct and indirect links, and how do you rank competitors?
Build a binary communication or dominance matrix, square it to count two-step links, add the matrix and its square to combine one-step and two-step connections, and rank competitors by their total
How is a project scheduled with critical path analysis, and how are maximum flow and bipartite allocation problems solved?
Construct an activity network, use forward and backward scanning to find earliest and latest start times, floats and the critical path, determine the maximum flow through a network using the minimum cut, and solve an allocation problem with the Hungarian algorithm
What is the difference between Eulerian and Hamiltonian travel, and how do vertex degrees decide whether an Eulerian trail or circuit exists?
Distinguish walks, trails, paths, cycles, Eulerian trails and circuits, and Hamiltonian paths and cycles, and use the number of odd-degree vertices to decide whether an Eulerian trail or circuit exists in a connected graph
What are the key terms of graph theory, and how are spanning trees and shortest paths found in a weighted network?
Use graph and network terminology (vertices, edges, degree, connected, planar), apply Euler's formula and the handshake result, find a minimum spanning tree, and determine the shortest path through a weighted network
How does a Leslie matrix model an age-structured population, and how do birth and survival rates project the population forward?
Set up a Leslie matrix from age-specific birth (fecundity) and survival rates, multiply it by an age-structure state matrix to project a population forward, and interpret the long-term growth and age distribution
How are matrices added, multiplied and inverted, and how is a matrix inverse used to solve a system of linear equations?
Perform matrix addition, scalar multiplication and matrix multiplication, find the determinant and inverse of a 2x2 matrix, and use the inverse to solve a system of simultaneous linear equations
How do you find the greatest flow a network can carry from source to sink, and how does the minimum cut prove the maximum flow?
Model a directed capacitated network with a source and sink, find the maximum flow from source to sink, identify cuts and their capacities, and use the minimum cut to confirm the maximum flow
What is a spanning tree, and how does Prim's algorithm find the minimum spanning tree that connects every vertex at least cost?
Identify a tree and a spanning tree in a connected network, apply Prim's algorithm to build the minimum spanning tree of a weighted graph, and find its total weight
How does a permutation matrix rearrange the rows of a state matrix, and how are binary matrices used to encode and reorder information?
Recognise and use a permutation matrix as a binary matrix with exactly one 1 in each row and column, apply it to reorder the entries of a state matrix, and identify the matrix that reverses or repeats the reordering
How does the Hungarian algorithm assign tasks to workers so the total cost is the lowest possible, using row and column reduction?
Model an allocation problem with a cost matrix and a bipartite graph, apply the Hungarian algorithm using row reduction, column reduction and line covering to find the minimum-cost one-to-one allocation
How do transition matrices model step-by-step change in a system, and how are future states and steady states found?
Set up a transition matrix to model a Markov system, use it with an initial state matrix to find the state after n steps, identify the steady-state or long-run distribution, and apply a recurrence including the case with additions or removals