Free reference guide · HSC
Worked examples: HSC Maths Advanced Formula Sheet
Use the Advanced formulas from the shared NESA sheet, with worked examples in calculus, statistics and financial mathematics.
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An available formula is a starting point, not a worked solution. Show the function, substitution or differentiation step and any domain restriction. These examples are original practice, not official questions or a claim that every displayed relationship is printed on the sheet.
For the other part of the shared NESA sheet, see the HSC Maths Extension 1 Formula Sheet.
Download the worked examples (PDF) →
Formula index with worked examples
Measurement · arcs and sectors
Formula or reference: For degrees, ℓ = (θ/360°) × 2πr and A = (θ/360°) × πr². For radians, ℓ = rθ and A = r²θ/2.
Units and conditions: Match the angle units to the formula. Use square units for area.
Worked example: A 60° sector with r = 3 cm has arc length π cm and area 3π/2 cm². Using θ = π/3 radians gives the same results.
Measurement · trapezium area
Formula or reference: A = h(a + b)/2
Units and conditions: a and b are the parallel sides, and h is perpendicular to them.
Worked example: Parallel sides of 5 m and 9 m and a height of 4 m give 28 m².
Measurement · surface areas
Formula or reference: Closed cylinder: A = 2πr² + 2πrh. Sphere: A = 4πr².
Units and conditions: Use square units. The cylinder formula includes both ends.
Worked example: At radius 2 cm and cylinder height 3 cm, the cylinder has area 20π cm² and the sphere has area 16π cm².
Measurement · volumes
Formula or reference: Pyramid or cone: V = Ah/3. Sphere: V = 4πr³/3.
Units and conditions: Use base area A, perpendicular height h and cubic units.
Worked example: A base of 15 cm² and height 4 cm give volume 20 cm³. A sphere of radius 3 cm has volume 36π cm³.
Functions · quadratic roots
Formula or reference: For ax² + bx + c = 0, x = [−b ± √(b² − 4ac)]/(2a).
Units and conditions: a ≠ 0. Real roots require a non-negative discriminant.
Worked example: For x² − 5x + 6 = 0, x = (5 ± 1)/2, giving 2 and 3.
Relations · equation of a circle
Formula or reference: (x − h)² + (y − k)² = r²
Units and conditions: Centre (h,k), radius r > 0. Use the same units on both axes.
Worked example: The equation (x − 1)² + (y + 2)² = 9 has centre (1,−2) and radius 3.
Finance · compound growth
Formula or reference: A = P(1 + r)ⁿ
Units and conditions: P and A are monetary amounts. Use a decimal rate r per period.
Worked example: $1000 at 4% per year for two years becomes $1081.60.
Sequences · arithmetic term and sum
Formula or reference: T(n) = a + (n − 1)d. S(n) = n[2a + (n − 1)d]/2 = n(a + ℓ)/2, where ℓ is the last term.
Units and conditions: n is a positive integer. a, d, ℓ and the sum use the same units.
Worked example: For 4, 7, 10, 13, 16 the fifth term is 16 and the sum is 5(4 + 16)/2 = 50.
Sequences · geometric term and sums
Formula or reference: T(n) = ar^(n−1). S(n) = a(1 − rⁿ)/(1 − r) = a(rⁿ − 1)/(r − 1) for r ≠ 1. S∞ = a/(1 − r) for |r| < 1.
Units and conditions: The common ratio r is unitless. An infinite sum requires convergence.
Worked example: For a = 8 and r = 1/2, T(3) = 2, S(3) = 14 and S∞ = 16.
Logarithms · inverse and change of base
Formula or reference: logₐ(aˣ) = x and a^(logₐ x) = x. logₐ x = logᵦ x/logᵦ a. aˣ = e^(x ln a).
Units and conditions: Bases are positive and different from 1. Logarithm arguments are positive.
Worked example: log₂ 8 = 3, which can also be evaluated as ln(8)/ln(2). Then 2³ = e^(3 ln 2) = 8.
Trigonometry · right triangles and reciprocals
Formula or reference: sin A = opp/hyp, cos A = adj/hyp, tan A = opp/adj. sec A = 1/cos A, cosec A = 1/sin A, cot A = cos A/sin A.
Units and conditions: Ratios are unitless. Denominators must be non-zero.
Worked example: For an angle opposite the side of length 3 in a 3–4–5 triangle, sin A = 3/5, cos A = 4/5, tan A = 3/4, sec A = 5/4, cosec A = 5/3 and cot A = 4/3.
Trigonometry · standard triangles and identity
Formula or reference: 45°–45°–90° sides are in ratio 1:1:√2. 30°–60°–90° sides are 1:√3:2. cos² x + sin² x = 1.
Units and conditions: Match each side to the opposite angle.
Worked example: sin 30° = 1/2 and cos 30° = √3/2. Their squares add to 1. At 45°, both ratios equal 1/√2.
Trigonometry · non-right triangles
Formula or reference: A = ab sin C/2. a/sin A = b/sin B = c/sin C. c² = a² + b² − 2ab cos C and cos C = (a² + b² − c²)/(2ab).
Units and conditions: Match each side to its opposite angle. The area uses the angle between a and b.
Worked example: For sides a = 3 cm, b = 4 cm with C = 90°, the area is 6 cm² and c = 5 cm. The sine rule gives sin A = 3/5.
Statistics · standard score and outlier fences
Formula or reference: z = (x − μ)/σ. Lower and upper fences are Q₁ − 1.5 IQR and Q₃ + 1.5 IQR.
Units and conditions: z is unitless. The fences have the data’s units.
Worked example: At x = 74, μ = 62 and σ = 8, z = 1.5. If Q₁ = 58 and Q₃ = 66, the fences are 46 and 78.
Statistics · normal distribution bands
Formula or reference: About 68%, 95% and 99.7% of a normal population lie within 1, 2 and 3 standard deviations of the mean, respectively.
Units and conditions: Percentages are approximations for a normal distribution.
Worked example: For a normal distribution with mean 50 and standard deviation 4, the two-standard-deviation interval is 42 to 58 and contains about 95%.
Statistics · mean and variance
Formula or reference: E(X) = μ. Var(X) = E[(X − μ)²] = E(X²) − μ².
Units and conditions: The mean uses the original units and variance uses squared units.
Worked example: If X is 0 or 4 with equal probability, E(X) = 2 and E(X²) = 8, so Var(X) = 4.
Probability · independent, combined and conditional events
Formula or reference: For independent A and B, P(A ∩ B) = P(A)P(B). P(A ∪ B) = P(A) + P(B) − P(A ∩ B). P(A | B) = P(A ∩ B)/P(B) for P(B) > 0.
Units and conditions: Probabilities are unitless. Only the product expression requires independence.
Worked example: If independent events have probabilities 0.5 and 0.4, their intersection is 0.2, union is 0.7, and P(A | B) is 0.5.
Probability · continuous random variables
Formula or reference: If a is the lower endpoint of the support, P(X ≤ r) = ∫ from a to r of f(x) dx. P(a < X < b) = ∫ from a to b of f(x) dx.
Units and conditions: Use a valid density f and integrate over the event interval.
Worked example: A uniform density of 1/4 from 0 to 4 gives P(X ≤ 1) = 1/4 and P(1 < X < 3) = 1/2.
Calculus · derivative rules
Formula or reference: d[f(x)ⁿ]/dx = nf′(x)f(x)^(n−1). (uv)′ = u′v + uv′. (u/v)′ = (vu′ − uv′)/v². dy/dx = (dy/du)(du/dx).
Units and conditions: Use each rule where its component functions are differentiable and v ≠ 0 in the quotient rule.
Worked example: Differentiating (3x + 1)² gives 6(3x + 1). For x²(x + 1), the product rule gives 3x² + 2x. For x/(x + 1), the quotient rule gives 1/(x + 1)².
Calculus · trigonometric derivatives
Formula or reference: d[sin f]/dx = f′ cos f. d[cos f]/dx = −f′ sin f. d[tan f]/dx = f′ sec² f.
Units and conditions: Use radians and avoid tangent poles.
Worked example: At x = 0, the derivatives of sin(2x), cos(2x) and tan(2x) are 2, 0 and 2.
Calculus · exponential and logarithmic derivatives
Formula or reference: d(e^f)/dx = f′e^f. d(a^f)/dx = (ln a)f′a^f. d(ln f)/dx = f′/f. d(logₐ f)/dx = f′/(f ln a).
Units and conditions: Logarithm arguments must be positive. Use a valid positive base different from 1.
Worked example: At x = 1, the derivatives of e^(2x), 2^(2x), ln(2x) and log₂(2x) are 2e², 8 ln 2, 1 and 1/ln 2.
Calculus · power antiderivatives
Formula or reference: ∫ f′(x)f(x)ⁿ dx = f(x)^(n+1)/(n + 1) + C for n ≠ −1.
Units and conditions: Use an interval where the functions are defined. Include the integration constant.
Worked example: ∫ 3x² dx = x³ + C. With f(x) = 2x + 1, ∫ 2(2x + 1)³ dx = (2x + 1)⁴/4 + C.
Calculus · trigonometric antiderivatives
Formula or reference: ∫ f′ sin f dx = −cos f + C. ∫ f′ cos f dx = sin f + C. ∫ f′ sec² f dx = tan f + C.
Units and conditions: Use radians and work between poles for the last expression.
Worked example: With f(x) = 3x, antiderivatives of 3 sin(3x), 3 cos(3x) and 3 sec²(3x) are −cos(3x), sin(3x) and tan(3x), respectively, each plus C.
Calculus · exponential and logarithmic antiderivatives
Formula or reference: ∫ f′e^f dx = e^f + C. ∫ f′a^f dx = a^f/ln a + C. ∫ f′/f dx = ln|f| + C.
Units and conditions: Use a > 0 with a ≠ 1. Work where f is non-zero in the logarithmic form.
Worked example: With f(x) = 2x, antiderivatives of 2e^(2x), 2 × 3^(2x) and 1/x are e^(2x), 3^(2x)/ln 3 and ln|x|, each plus C.
Calculus · trapezoidal approximation
Formula or reference: For n equal strips, multiply (b − a)/(2n) by the sum of the endpoint heights and twice each interior height.
Units and conditions: Use equal subintervals. The integral has units of input times output.
Worked example: On [0,2] for f(x) = x², two strips give (0 + 2 × 1 + 4)/2 = 3.
More worked examples
NESA publishes a combined Mathematics Advanced, Extension 1 and Extension 2 reference sheet. Use the parts relevant to your course. The examples below focus on choosing a calculus method and interpreting the answer.
Chain rule
Relationship: For y = [g(x)]^n, dy/dx = n[g(x)]^(n−1)g′(x).
Differentiate y = (3x + 1)^4.
Show worked response and error check
The outside derivative is 4(3x + 1)^3 and the inside derivative is 3. Multiplying gives y′ = 12(3x + 1)^3.
Check: Omitting the inside derivative gives a result too small by a factor of three.
Signed area versus total area
Relationship: A definite integral adds signed contributions; split at zeros to find total area.
Find the total area between y = x and the x-axis for −1 ≤ x ≤ 1.
Show worked response and error check
The integral from −1 to 1 of x is zero, but the area is not. Each triangle has area ½ × 1 × 1, so the total area is 1 square unit. Equivalently integrate −x on [−1,0] and x on [0,1].
Check: A zero definite integral can reflect cancellation of equal positive and negative contributions.
Exponential rate of change
Relationship: If P(t) = Ae^(kt), then P′(t) = kAe^(kt).
A model has P(t) = 100e^(0.2t), with t in hours. Find its initial rate of change.
Show worked response and error check
P′(t) = 20e^(0.2t), so P′(0) = 20 units per hour. P(0) = 100 is the starting amount, not the rate.
Check: Differentiate before substituting if the question asks for a rate; include the time unit.
What to work out beyond the sheet
This guide follows the reference sheet for the HSC through 2026. Check NESA’s materials for later cohorts. For geometric area, split the interval at sign changes and add the positive areas. The definite integral gives signed area.
Subject resources · Official past papers · ATAR calculator guide