Free TCE General Mathematics Flashcard Review

Define the quantity to predict, choose variables and units, propose a relationship, state assumptions and restrict the domain. Fit or calculate the model, then test its output against data and the practical setting before drawing a conclusion.

With t minutes after draining begins, V=500−12t litres. The physical domain is 0≤t≤500/12≈41.67; beyond that the formula gives a negative volume, which is mathematically possible but meaningless for the tank.

Definitions make each number and parameter interpretable and expose unit errors. If distance d is kilometres and time t is hours, a gradient in d=vt must be kilometres per hour; an unexplained gradient cannot be checked in context.

It names a population, measurable variables and the comparison or relationship of interest. ‘Among Year 12 students at this school, is weekly paid-work time associated with sleep duration this term?’ is answerable from defined data.

Replace broad words with operational measures, a target population and a time frame. ‘Are phones bad for sleep?’ could become ‘For students aged 16–18, how is nightly screen time associated with self-reported sleep hours during one school week?’

The population is the full group the conclusion concerns; the sample is the observed subset. A result from 80 volunteers supports claims about those volunteers, but generalising to all Tasmanian students requires a sampling method that represents that wider population.

Use linear change for an approximately constant difference, exponential change for an approximately constant percentage, and a recurrence when each state depends on the previous state plus an update. Financial models add cash-flow timing and periodic interest conventions.

A first-order recurrence represents both mechanisms: P_(n+1)=1.04P_n+120. A pure exponential omits the fixed arrivals, while a linear model omits percentage growth on the changing population.

Check units, sign, order of magnitude, domain and relevant bounds, then compare with an estimate or known case. A calculated 720 km/h bicycle speed fails a scale check even if the calculator entry followed a written formula.

A positive value may still violate capacity, time, geometry or data-range constraints. For example, a regression may predict 140% participation; the arithmetic is positive, but a percentage above 100% signals that the model is unsuitable there.

An assumption states a condition treated as true so the model can operate. Saying a 3% growth rate remains constant makes an exponential forecast possible and also identifies a condition whose failure would weaken the forecast.

The periodic growth rate and compounding interval are assumed to remain stable, and external shocks or capacity limits are assumed not to change the process materially. These assumptions should be tested or qualified, especially for extrapolation.

Data limitations concern quality, sample size, bias or restricted range. Structural limitations concern the equation itself, such as a straight line ignoring saturation; collecting more data cannot repair a form that represents the wrong mechanism.

Few observations constrain the model near an edge, and extrapolation assumes the same pattern continues outside observed conditions. Thresholds, capacity limits or changing behaviour may appear after the measured range ends.

It answers the stated question, cites the relevant calculated or graphical evidence, uses context and acknowledges uncertainty or limitations. Its scope must match the population, period and variable definitions actually investigated.

Report the general pattern and quantify the fit, then identify the outlier and test how strongly it changes the fitted result. Do not silently remove it; investigate whether it is an error, an unusual valid case or evidence of another process.

State the question, data source and method, present labelled evidence, interpret results in context and give a conclusion with limitations. A coefficient or graph alone is incomplete because the reader must know what was measured and for whom.

Describe whether residuals are randomly scattered around zero and note any curvature, changing spread or unusual points. Then state what that pattern implies for the linear model rather than claiming a large correlation proves perfect prediction.

Keep full calculator precision or an exact expression through intermediate steps, then round once to the requested final accuracy. Early rounding accumulates error, especially when a payment or rate is multiplied across many periods.

The result is 18.75 because the third decimal digit is 6, so the second decimal rounds up. Writing 18.74 truncates rather than rounds, while excessive digits imply accuracy not supported by the instruction or data.