General MathematicsQ&A by dot point

Construct a scatterplot, describe the association between two numerical variables in terms of direction, form and strength, calculate and interpret Pearson's correlation coefficient $r$ and the coefficient of determination $r^2$, and recognise that correlation does not imply causation

Apply a square, logarithmic or reciprocal transformation to one variable to linearise a non-linear association, fit a least-squares line to the transformed data, use the transformed equation to predict, and choose the transformation that best straightens the scatter

Model depreciation of an asset using flat-rate (straight-line), reducing-balance and unit-cost methods with recurrence relations and rules, compute book value and scrap value, and compare the methods over the life of the asset

Locate a position on the Earth using latitude and longitude, define great circles and small circles, calculate the distance along a meridian or the equator using the angular separation, and convert between nautical miles, minutes of arc and kilometres

Fit a least-squares regression line to bivariate data, interpret the slope and intercept in context, use the line to make predictions through interpolation and extrapolation, and assess the fit using a residual plot and the coefficient of determination

Calculate residuals for a least-squares line, construct and interpret a residual plot, use the pattern in the residual plot to decide whether a linear model is appropriate, and identify when a transformation is needed because the residuals show curvature

Use arithmetic and geometric sequences and first-order recurrence relations to model linear growth or decay and geometric growth or decay, find the nth term and partial sums, and apply these models to simple interest, reducing-balance depreciation and compound contexts

Construct and interpret time series plots, identify trend, seasonality, cyclical and irregular variation, smooth a series using moving averages, calculate and apply seasonal indices to deseasonalise data, and fit a trend line to forecast future values

Relate longitude to local time using the rule that the Earth turns 15 degrees of longitude per hour, calculate the time difference between two locations from their longitudes, apply the conventions of east being ahead and west being behind, and combine this with flight times

Model an annuity investment (regular deposits earning compound interest) and an annuity that pays a regular income (drawing down a lump sum) using first-order recurrence relations, compute the future value of contributions and the duration a payout annuity lasts, and apply this to superannuation

Represent an allocation as a bipartite graph or a cost matrix, solve a small assignment problem to minimise total cost or time using the Hungarian algorithm, handle maximisation by converting it to a minimisation, and interpret the optimal allocation

Distinguish nominal and effective annual interest rates, calculate the effective annual rate for a given nominal rate and compounding frequency, and use the effective rate to compare investments or loans that compound at different frequencies

Represent a project as an activity network with durations and dependencies, perform forward and backward scanning to find earliest and latest start times, identify the critical path and minimum completion time, and calculate float for non-critical activities

Model a flow problem on a directed network with edge capacities, find the maximum flow from source to sink by inspection, identify cuts and their capacities, and use the maximum-flow minimum-cut theorem to confirm the maximum flow

Represent a situation as a graph or network using vertices and edges, determine the degree of vertices and verify the handshaking result, classify graphs as connected, simple, complete, bipartite or planar, apply Euler's formula, and identify Eulerian and Hamiltonian trails and circuits

Model compound interest investments, reducing-balance loans and annuities using first-order recurrence relations, compute future value, repayment and balance using technology, build and interpret an amortisation table, and analyse the effect of changing the rate, repayment or compounding period

Define a tree and a spanning tree, identify the minimum spanning tree of a weighted connected graph using Prim's algorithm, calculate its total weight, and apply minimum spanning trees to minimum connector problems such as cabling or pipelines

Find minimum spanning trees using Prim's algorithm, determine shortest paths through a weighted network, calculate maximum flow using the maximum-flow minimum-cut idea, and schedule a project using critical path analysis to find the earliest completion time and float

Find the shortest path between two vertices in a weighted network by inspection and by systematic labelling, distinguish the shortest path from the minimum spanning tree, and apply shortest-path methods to travel-time and cost problems