SACE Stage 2 General Mathematics flashcards

33 cards. Read the term and the definition here. The deck at the top of this page is optional.

Simple interest formula

Interest computed only on the original principal: I = P·r·t, where r is the annual decimal rate and t is time in years. The total amount is A = P(1 + rt).

Compound interest formula

Interest earned on principal plus previously earned interest: A = P(1 + r)^n, where r is the rate per compounding period and n is the total number of periods.

Effective annual rate

The true yearly growth rate once compounding is included: EAR = (1 + r/m)^m − 1, where r is the nominal annual rate and m the number of compounds per year.

Reducing-balance depreciation

A recurrence where each period's value is the previous value reduced by a fixed percentage: V(n+1) = V(n)(1 − r). Unlike flat-rate depreciation, the dollar loss shrinks each period.

Annuity future value

The accumulated value of regular equal payments: FV = M[((1 + r)^n − 1)/r], where M is each payment and r the rate per period.

Loan repayment from present value

The regular repayment on a reducing-balance loan is M = PV·r / (1 − (1 + r)^(−n)), obtained by rearranging the annuity present value formula.

Real value after inflation

To express a future nominal amount in today's purchasing power, divide by the inflation factor: real value = nominal value ÷ (1 + i)^n, where i is the per-period inflation rate.

Statistical investigation stages

A cycle of posing a question, collecting data, displaying and summarising it, interpreting results in context, and communicating conclusions while acknowledging uncertainty and limitations.

Interquartile range rule

IQR = Q3 − Q1. Values more than 1.5 IQR below Q1 or above Q3 are flagged as outliers, which are then reported rather than silently deleted.

Pearson correlation coefficient

A number between −1 and 1 describing the strength and direction of a linear association. It is symmetric, so swapping the two variables does not change r.

Least-squares regression line

The line minimising the sum of squared vertical residuals, with gradient b = r(s_y/s_x) and intercept a = ȳ − b·x̄, so it always passes through (x̄, ȳ).

Coefficient of determination

The value r², the proportion of variation in the response variable accounted for by the least-squares linear model. Values near 1 indicate a strong linear fit.

Correlation versus causation

An association between two variables does not prove one causes the other; a lurking or confounding variable, or a biased sample, may explain the observed relationship.

Five-number summary

The minimum, first quartile, median, third quartile and maximum of a data set. It supports box plots and is resistant to the influence of extreme values.

Break-even point

The output level where total cost equals total revenue, found by solving C(x) = R(x). Below it the business makes a loss; above it, a profit.

Equal first differences rule

If successive y-values in a table differ by a constant amount as x increases by equal steps, the relationship is linear with gradient equal to that constant difference.

Quadratic vertex formula

For y = ax² + bx + c, the turning point occurs at x = −b/(2a); substituting this x gives the maximum or minimum value of the model.

Exponential growth recognition

In a table, if successive y-values share a constant ratio as x increases by equal steps, the model is y = A·b^x, with b equal to that ratio.

Linearising a power model

Taking logarithms of y = A·x^n gives log y = log A + n·log x, so plotting log y against log x produces a straight line of gradient n.

Linearising an exponential model

Taking logarithms of y = A·b^x gives log y = log A + x·log b, so a semi-log plot is straight with gradient log b and intercept log A.

Inverse variation model

A relationship of the form y = k/x, whose graph is a decreasing curve with the axes as asymptotes. The constant k comes from any known pair of values.

Vertex degree

The number of edges meeting a vertex, with loops counted twice. In a directed network, indegree and outdegree count incoming and outgoing edges separately.

Handshake lemma

In any network, the sum of all vertex degrees equals twice the number of edges. A consequence is that the count of odd-degree vertices must be even.

Eulerian circuit condition

A closed trail using every edge exactly once exists only if the network is connected and every vertex has even degree. If exactly two vertices are odd, only an open trail exists.

Spanning tree edge count

A tree is a connected network containing no cycles. Any spanning tree on n vertices therefore contains exactly n − 1 edges, whatever the original network size.

Kruskal's algorithm

To find a minimum spanning tree, sort edges from cheapest to most expensive and repeatedly add an edge provided it does not create a cycle, stopping after n − 1 edges.

Dijkstra's algorithm

A shortest-path method that repeatedly selects the unfixed vertex with the smallest tentative distance, then relaxes its neighbours' distances. It requires non-negative edge weights.

Adjacency matrix powers

For an adjacency matrix A, the entry in row i, column j of A^n counts the number of distinct walks of length n from vertex i to vertex j.

Matrix multiplication condition

The product AB exists only when A's column count equals B's row count. The result has A's row count and B's column count, with entries formed from row-by-column products.

First-order linear recurrence

A rule tₙ₊₁ = k·tₙ + c with a starting value. Its fixed point is c/(1 − k) when k ≠ 1, and the sequence converges to that point only if |k| < 1.

Arithmetic sequence sum

For a sequence with first term t₁ and common difference d, the nth term is t₁ + (n − 1)d and the sum of n terms is (n/2)(2t₁ + (n − 1)d).

Geometric sequence sum

For a sequence with first term t₁ and common ratio r, the nth term is t₁·r^(n−1) and the sum of n terms is t₁(r^n − 1)/(r − 1) when r ≠ 1.

Transition matrix steady state

When columns of a transition matrix T give proportions moving between states, the steady-state vector s satisfies T·s = s and has entries summing to 1.