HSC Mathematics Standard 1 cheat sheet

Linear equation: solve by inverse operations, keeping both sides balanced; e.g. 3x − 7 = 11 gives x = 6.

Gradient of a line: m = (y₂ − y₁)/(x₂ − x₁); horizontal lines have m = 0, vertical lines have undefined gradient.

Gradient–intercept form y = mx + b: m is the gradient, b is the y-intercept where x = 0.

Point–gradient form y − y₁ = m(x − x₁) builds a line from one point and its gradient.

Parallel lines have equal gradients; perpendicular lines have gradients with product −1 (m₁m₂ = −1).

Simultaneous equations: substitute or eliminate to reduce two equations to one variable, then back-substitute.

Quadratic equations: factorise, or use x = (−b ± √(b² − 4ac))/(2a) for ax² + bx + c = 0.

Direct variation y = kx and inverse variation y = k/x; find k from a given pair, then substitute to predict.

Perimeter is the distance around a shape; area is measured in square units (cm², m²).

Rectangle area = length × width; triangle area = ½ × base × height (height perpendicular to base).

Circle: circumference C = 2πr = πd; area A = πr².

Trapezium area = ½(a + b)h, where a and b are the parallel sides and h their perpendicular separation.

Prism volume = cross-sectional area × length; cylinder volume = πr²h.

Pythagoras' theorem: c² = a² + b² for the hypotenuse c of a right-angled triangle.

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

Similar figures: matching lengths scale by factor k, areas by k², volumes by k³.

Speed, distance, time: speed = distance ÷ time; convert km/h to m/s by dividing by 3.6.

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

Compound interest: A = P(1 + r)ⁿ, with r the rate per period as a decimal and n the number of periods.

Compound interest earned = A − P; simple interest earned = I, so compare the two to see the compounding effect.

Depreciation: value after n periods = P(1 − r)ⁿ for declining-balance depreciation.

Straight-line depreciation subtracts the same dollar amount each year: value = P − nD.

Weekly, fortnightly or monthly earnings: annual salary ÷ 52, 26 or 12 respectively.

Overtime: time-and-a-half = 1.5 × hourly rate; double time = 2 × hourly rate.

GST of 10%: GST amount = price × 0.10; to remove GST from a GST-inclusive price, divide by 1.1.

Reducing-balance loans: interest each period is charged on the outstanding balance, so interest falls as the balance falls.

Mean = sum of values ÷ number of values; it is affected by outliers.

Median is the middle value of ordered data (average the two middle values if n is even); resistant to outliers.

Mode is the most frequent value; data sets can be bimodal or have no mode.

Range = highest − lowest; interquartile range IQR = Q₃ − Q₁ and contains the middle 50% of data.

Standard deviation measures spread about the mean: larger values mean more variable data.

Outlier rule: a value is an outlier if it lies below Q₁ − 1.5×IQR or above Q₃ + 1.5×IQR.

z-score = (x − mean) ÷ standard deviation; it states how many standard deviations a value sits from the mean.

For a normal distribution about 68% of values lie within 1 standard deviation of the mean, about 95% within 2.

Relative frequency = frequency ÷ total; it estimates probability and should sum to 1 across all categories.