"HSC maths formula sheet" is searched by students in the week before the trial and the week before the HSC, usually to find out whether a particular formula will be given. The honest answer is that NESA's reference sheet is printed at the back of every mathematics paper, it is the same sheet for Advanced, Extension 1 and Extension 2 (four pages) and a different, shorter one for Standard 1 and 2 (one page), and the skill is not finding it but knowing what is on it and what is not. This page lists both sheets section by section, with the formulas named in words, then the list of everything that is not there and must be memorised, and a printable annotated version to practise with from Term 1.
Download the annotated HSC maths reference sheet (PDF) (both sheets section by section, the memorise list for each course, and the exam-day habits; two pages, free)
The two sheets
| Mathematics Standard 1 and 2 | Mathematics Advanced, Extension 1, Extension 2 | |
|---|---|---|
| Length | 1 page | 4 pages |
| Where | printed at the back of the paper | printed at the back of the paper |
| Since | the 2019 HSC (the first exam of the current syllabus) | the 2020 HSC |
| Shared | Standard 1 and Standard 2 see the same sheet | all three courses see the same sheet, so an Advanced student sees integration by parts and complex numbers and can ignore them |
| Source | NESA, "Mathematics Standard 1 and 2 reference sheet" | NESA, "Mathematics Advanced, Extension 1 and 2 reference sheet" |
NESA's own note on the Standard sheet says it so students "don't have to memorise a range of formulae" and that the terms in the formulae are not defined: you must know what each letter means.
The Standard sheet, section by section
| Section | What is given |
|---|---|
| Measurement: limits of accuracy | absolute error is half the precision; upper and lower bounds |
| Length and area | arc length as a fraction of the circumference (degrees); sector area; the trapezium; the trapezoidal-rule approximation A is about h over 2 times the sum of the two end widths |
| Surface area and volume | cylinder (closed) and sphere surface area; pyramid or cone volume (a third of base times height); sphere volume |
| Trigonometry | sine, cosine and tangent as ratios; area of a triangle (half ab sin C); the sine rule; the cosine rule in both forms |
| Financial mathematics | future value with compound interest; straight-line depreciation; declining-balance depreciation |
| Statistical analysis | the outlier rule (1.5 times the IQR beyond the quartiles); the z-score; the normal distribution with 68, 95 and 99.7 percent within one, two and three standard deviations |
Not on the Standard sheet: Pythagoras; simple interest; the rates, ratio and speed-distance-time work of Financial and Measurement questions; fuel consumption, heart-rate and dosage formulas (given in the question when needed); the mean, median, mode and range; the five-number summary; network definitions (the critical path method); unit conversions; the probability rules. The sheet is a memory aid for formulas, not a method guide.
The Advanced, Extension 1 and Extension 2 sheet, page by page
Page 1: measurement, financial maths, series, functions, logs
| Section | What is given |
|---|---|
| Measurement | arc length and sector area (degree form); the trapezium; cylinder and sphere surface area; cone or pyramid and sphere volume |
| Financial mathematics | compound interest, A equals P times one plus r to the n |
| Sequences and series | the nth term and the sum of an arithmetic series (both forms); the nth term and sum of a geometric series (both forms); the limiting sum for a ratio between minus one and one |
| Functions | the quadratic formula; for a cubic, the sum of the roots, the sum of the products in pairs, and the product of the roots; the circle with centre (h, k) |
| Logarithmic and exponential functions | the inverse relation between logs and powers; the change-of-base rule; a to the x equals e to the x ln a |
Page 2: trigonometry, statistics, probability
| Section | What is given |
|---|---|
| Trigonometric functions | the three ratios; half ab sin C; the sine rule; the cosine rule (both forms); arc length and sector area in radians; the two exact-value triangles (45 degrees with root 2; 30 and 60 degrees with 2, 1 and root 3) |
| Identities | secant, cosecant and cotangent; cos squared plus sin squared equals one |
| Compound angles | sin, cos and tan of A plus B; the t-formulae (t equals tan of half A); the four products-to-sums results; sin squared and cos squared of nx in terms of cos 2nx |
| Statistical analysis | the z-score; the outlier rule; the normal distribution and its percentages; expectation and variance of a random variable |
| Probability | the product rule for independent events; the addition rule; conditional probability |
| Continuous random variables | probability as the integral of the density function |
| Binomial distribution | the probability of r successes; the notation X is Bin(n, p); expectation np and variance np(1 minus p) |
Page 3: every derivative and integral
The left column lists the derivative of fourteen forms: a power of a function, a product, a composite (the chain rule), a quotient, sin, cos and tan of a function, e to a function, ln of a function, a to a function, log base a of a function, and the three inverse trig functions. The right column lists the matching integrals: the power of a function, sin, cos, sec squared, e, the reciprocal (ln), a to a function, the two inverse-trig forms (sin inverse and tan inverse over a), integration by parts, and the trapezoidal rule for a definite integral with n strips.
Advanced students use the top ten lines; the inverse trig lines are Extension 1; integration by parts is Extension 2 (and a useful shortcut for strong Extension 1 students).
Page 4: combinatorics, vectors, complex numbers, mechanics
| Section | What is given | Course |
|---|---|---|
| Combinatorics | permutations, combinations, the binomial expansion of (x plus a) to the n | Extension 1 |
| Vectors | the magnitude of a 2D vector; the dot product as a product of magnitudes and cos theta, and as x1x2 plus y1y2; the vector equation of a line r equals a plus lambda b | Extension 1 (3D forms are not written out: Extension 2 students extend them) |
| Complex numbers | polar and exponential form; de Moivre's theorem in both forms | Extension 2 |
| Mechanics | the four equivalent forms of acceleration; the two displacement forms of simple harmonic motion; the SHM equation x double dot equals minus n squared (x minus c) | Extension 2 (and the SHM forms for Extension 1 as background) |
What is not on the sheet: the memorise list
For Advanced:
- Pythagoras; the distance, midpoint and gradient formulas; parallel and perpendicular gradients; the point-gradient and general forms of a line.
- The index laws and the log laws (log of a product, a quotient, a power); the log of 1 and of the base.
- Simple interest; the recurrence approach to loans and annuities (there is no annuity formula on the sheet).
- The mean, median, mode, range, quartiles, IQR, variance and standard deviation from a data set; the shape of a box plot.
- Differentiation from first principles; the second-derivative test; the meaning of stationary, inflexion and concavity; the derivative of a constant and of x alone.
- The area between a curve and the axis and between two curves (set up from the sheet's integrals); the definite integral as signed area.
- The graphs and transformations (translation, dilation, reflection) of the standard functions; even and odd functions; the absolute value function.
- The exact values themselves (the triangles are given; the signs by quadrant and the values at 0, 90, 180 are not).
- Radian-degree conversion (180 degrees equals pi radians).
Extension 1 adds: the projectile equations (derive from acceleration minus g with the given s, v forms), related rates set-ups, the inverse functions and their domains, the binomial theorem's general term (the expansion is given; the general term form is not written separately), the methods of proof by induction, and the integration-by-substitution procedure.
Extension 2 adds: the 3D dot and vector forms, the integration techniques (partial fractions, t-substitution, reduction formulas), the proof structures, the complex roots of unity and the triangle inequality, and the Euler and conjugate relations.
How to use the sheet in the exam
- Know its layout before the trial. Print the real NESA sheet in Term 1 and do every homework set with it beside you, so that in the exam the cosine rule is found in two seconds, not two minutes.
- Read the whole formula, including the condition: the limiting sum needs the ratio between minus one and one; the integral of a power excludes n equals minus one; the t-formulae need the half-angle.
- Use the radian form in calculus questions: arc length l equals r theta and sector area half r squared theta are on page 2; the degree forms on page 1 are for measurement questions.
- In multiple choice, check against the sheet rather than from memory: a sign in a compound-angle identity decides the answer.
- Write the formula down before substituting, in the sheet's letters, then define yours: markers award method marks for the right formula with a wrong substitution.
Where to get help
For a Year 11 or 12 student in Sydney, Melbourne or anywhere on the Australian east coast, a 1-on-1 online class in HSC mathematics, Standard, Advanced or Extension 1, taught in the school's order with past HSC questions marked to NESA's marking guidelines from the first term; Australian after-school hours (4 to 7 pm Sydney time is early afternoon for the tutor), every class recorded, A$14 to A$19 an hour. The free 30-minute first class is a real lesson on the topic the student is on.
Download the annotated reference sheet (PDF)
Questions parents ask
1Is there a formula sheet in the HSC maths exam?
2What is on the HSC Advanced maths reference sheet?
3What is not on the HSC maths formula sheet?
4What is on the Mathematics Standard reference sheet?
5Can I use the reference sheet during the whole exam?
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