General MathematicsQ&A by dot point

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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