HSC Mathematics Standard 2 cheat sheet

Linear form y = mx + b: m is the gradient (rise/run) and b is the y-intercept where x = 0.

Gradient from two points: m = (y2 − y1)/(x2 − x1); a positive m rises left-to-right, a negative m falls.

Simultaneous linear equations: solve by substitution or elimination; the solution is the point of intersection of the two lines.

Quadratic form y = ax² + bx + c: axis of symmetry at x = −b/(2a); the turning point lies on this axis.

Zero-product rule: if (x − p)(x − q) = 0 then x = p or x = q; factorise before solving quadratics.

Quadratic formula: x = (−b ± √(b² − 4ac))/(2a); the discriminant b² − 4ac gives 0, 1 or 2 real roots.

Exponential growth/decay: y = A(1 + r)ⁿ grows when r > 0 and decays when r < 0; n counts compounding periods.

Rearranging formulas: perform the same inverse operation to both sides; to undo a square, take the square root of both sides.

Perimeter of a circle (circumference) C = 2πr = πd; arc length = (θ/360) × 2πr for θ in degrees.

Area of a circle A = πr²; area of a sector = (θ/360) × πr².

Area of a trapezium A = ½(a + b)h, where a and b are the parallel sides and h is the perpendicular height.

Volume of a prism = cross-sectional area × length; volume of a cylinder V = πr²h.

Volume of a pyramid or cone = ⅓ × base area × perpendicular height; a sphere has V = (4/3)πr³.

Surface area of a sphere S = 4πr²; curved surface of a cylinder = 2πrh, plus two circles if closed.

Pythagoras: c² = a² + b² for the hypotenuse; use it to find unknown sides, diagonals and slant heights.

Similar figures: if lengths scale by k, areas scale by k² and volumes by k³.

Unit conversion: 1 m = 100 cm, 1 m² = 10 000 cm², 1 m³ = 1 000 000 cm³, 1 L = 1000 cm³, 1 ha = 10 000 m².

Simple interest: I = PRN, where P is principal, R the annual rate as a decimal, and N the time in years.

Compound interest: A = P(1 + r)ⁿ, where r is the rate per period and n the number of compounding periods; interest = A − P.

Depreciation: straight-line value = P − Dn; declining-balance value = P(1 − r)ⁿ.

Changing compounding frequency: divide the annual rate by the number of periods per year and multiply the years by that number.

Inflation and appreciation use the same compound formula with a positive rate.

GST in Australia is 10%: GST-inclusive price = price × 1.1; GST component = inclusive price ÷ 11.

Shares and dividends: dividend per share × number of shares gives income; brokerage reduces the net proceeds of a sale.

Loans and annuities: the future value of an annuity uses repeated contributions, each earning interest for a different number of periods.

Mean = sum of scores ÷ number of scores; median is the middle score of ordered data; mode is the most frequent score.

Range = maximum − minimum; interquartile range IQR = Q3 − Q1 and measures the spread of the middle 50% of data.

Outliers by the 1.5 × IQR rule lie below Q1 − 1.5×IQR or above Q3 + 1.5×IQR.

Standard deviation measures spread about the mean; about 68% of roughly normal data lies within one SD, 95% within two.

z-score z = (x − μ)/σ states how many standard deviations a score lies from the mean.

Box plots show minimum, Q1, median, Q3 and maximum; parallel box plots compare two data sets.

Scatterplots: describe association as positive or negative and strong or weak; the line of best fit predicts values.

Correlation coefficient r lies between −1 and 1; r near 0 indicates little linear association, and correlation does not imply causation.

Relative frequency estimates probability: P(event) ≈ frequency of event ÷ total trials.

A network (graph) has vertices joined by edges; a weighted network labels each edge with a number such as distance or cost.

Degree of a vertex = number of edges meeting it; the sum of all degrees equals twice the number of edges.

A connected network has a path between every pair of vertices; a bridge is an edge whose removal disconnects the network.

Eulerian trail exists when exactly two vertices have odd degree; an Eulerian circuit exists when every vertex has even degree.

A tree is a connected network with no cycles; a spanning tree connects all vertices using the minimum number of edges.

Minimum spanning tree: use Kruskal's algorithm (choose cheapest edges that create no cycle) to minimise total edge weight.

Shortest path: Dijkstra's algorithm finds the least total weight from one vertex to another; the critical path is the longest path in a project network.

Adjacency matrix: entry aᵢⱼ counts the edges joining vertex i to vertex j; the matrix of a simple undirected network is symmetric.