Australian examsQCE
QCE Specialist Mathematics formula book: everything in it and what is not
Maths Extension 2 & Specialist Maths
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Formula sheet
The card below opens today's paper with the book in a drawer. The contents follow in our own layout, in the order the QCAA prints them. The book is a Creative Commons PDF dated June 2026 for the 2025 syllabus, so check it against the QCAA Specialist Mathematics page before you depend on a line.
Formula sheet
Open formula sheetMaths Extension 2 & Specialist Maths
Mensuration
C=2\pi rcircumference of a circle
A=\pi r^2area of a circle
A=bharea of a parallelogram
A=\frac{1}{2}(a+b)harea of a trapezium
A=\frac{1}{2}bharea of a triangle
S=\pi rs+\pi r^2total surface area of a cone
S=2\pi rh+2\pi r^2total surface area of a cylinder
S=4\pi r^2surface area of a sphere
V=\frac{1}{3}\pi r^2hvolume of a cone
V=\pi r^2hvolume of a cylinder
V=Ahvolume of a prism
V=\frac{1}{3}Ahvolume of a pyramid
V=\frac{4}{3}\pi r^3volume of a sphere
Graph equations
y=a(x-h)^2+kquadratic, turning point form
y=a(x-x_1)(x-x_2)quadratic, factorised form
y=a(x-h)^3+kcubic, turning point form
y=a(x-x_1)(x-x_2)(x-x_3)cubic, factorised form
(x-h)^2+(y-k)^2=r^2circle
y=a\sqrt{x-h}+ksquare root
y=\frac{a}{x-h}+kreciprocal
y=r^{(x-h)}+kexponential, where r > 0
y=\log_a(x-h)+klogarithmic, where a > 1
y=a\sin(b(x-h))+ksine
y=a\cos(b(x-h))+kcosine
Logarithms
a^x=b\Leftrightarrow x=\log_a(b)exponents and logarithms
\log_a(x)+\log_a(y)=\log_a(xy)sum of logarithms
\log_a(x)-\log_a(y)=\log_a\left(\frac{x}{y}\right)difference of logarithms
\log_a\left(x^n\right)=n\log_a(x)power of a logarithm
\log_a(x)=\frac{\log_b(x)}{\log_b(a)}change of base
\log_a(1)=0logarithm of 1
\log_a(a)=1logarithm of the base
Calculus
f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}definition of the derivative
\int_a^bf(x)dx\approx\text{limit of sums }\sum_if(x_i)\delta x_iintegral as a limit of sums
\int_a^bf(x)dx=F(b)-F(a)fundamental theorem
\frac{d}{dx}\left(f(x)+g(x)\right)=\frac{d}{dx}f(x)+\frac{d}{dx}g(x)derivative of a sum
\int\left(f(x)+g(x)\right)dx=\int f(x)dx+\int g(x)dxintegral of a sum
\int kf(x)dx=k\int f(x)dxconstant multiple of an integral
\frac{d}{dx}x^n=nx^{n-1}power rule
\int x^ndx=\frac{x^{n+1}}{n+1}+cintegral of a power, n not equal to −1
\frac{d}{dx}e^x=e^x,\quad\frac{d}{dx}e^{f(x)}=f'(x)e^{f(x)},\quad\int e^xdx=e^x+cexponential
\frac{d}{dx}\ln(x)=\frac{1}{x},\quad\frac{d}{dx}\ln(f(x))=\frac{f'(x)}{f(x)},\quad\int\frac{1}{x}dx=\ln(x)+c\ \text{for}\ x>0natural logarithm
\frac{d}{dx}\sin(x)=\cos(x),\quad\frac{d}{dx}\sin(f(x))=f'(x)\cos(f(x)),\quad\int\sin(x)dx=-\cos(x)+csine
\frac{d}{dx}\cos(x)=-\sin(x),\quad\frac{d}{dx}\cos(f(x))=-f'(x)\sin(f(x)),\quad\int\cos(x)dx=\sin(x)+ccosine
\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}chain rule, if y = f(u) and u = g(x)
\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}product rule
\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}quotient rule
\int_a^bf(x)dx\approx\frac{w}{2}\left[f(x_0)+2\left(f(x_1)+f(x_2)+f(x_3)+\ldots+f(x_{n-1})\right)+f(x_n)\right]trapezoidal rule, where w = (b − a)/n
Trigonometry
c^2=a^2+b^2-2ab\cos(C)cosine rule
\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}sine rule
\text{area}=\frac{1}{2}bc\sin(A)area of a triangle
\sin^2(A)+\cos^2(A)=1Pythagorean identity
Statistics
(x+y)^n=x^n+\binom{n}{1}x^{n-1}y+\ldots+\binom{n}{r}x^{n-r}y^r+\ldots+y^nbinomial theorem
P(X=r)=\binom{n}{r}p^r(1-p)^{n-r}binomial probability
E(X)=\mu=\sum p_ix_idiscrete random variable, mean
Var(X)=\sum p_i(x_i-\mu)^2discrete random variable, variance
\sqrt{Var(X)}discrete random variable, standard deviation
E(X)=\mu=\int_{-\infty}^{\infty}xp(x)dxcontinuous random variable, mean
Var(X)=\sigma^2=\int_{-\infty}^{\infty}(x-\mu)^2p(x)dxcontinuous random variable, variance
pBernoulli distribution, mean
p(1-p)Bernoulli distribution, variance
npbinomial distribution, mean
np(1-p)binomial distribution, variance
psample proportion, mean
\sqrt{\frac{p(1-p)}{n}}sample proportion, standard deviation
\left(\hat{p}-z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}},\ \hat{p}+z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\right)approximate confidence interval for p
z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}approximate margin of error
P\left(\overline{A}\right)=1-P(A)complementary probability
P(A\cup B)=P(A)+P(B)-P(A\cap B)general addition rule
P(A\cap B)=P(A)P(B)independent events
P(A\cap B)=P(A|B)P(B)conditional probability
Additional calculus for Specialist Mathematics
\frac{d}{dx}\ln(f(x))=\frac{f'(x)}{f(x)},\quad\int\frac{1}{x}dx=\ln|x|+c\ \text{for}\ x\ne0derivative and integral of the natural logarithm
\int\frac{f'(x)}{f(x)}dx=\ln|f(x)|+c\ \text{for}\ f(x)\ne0integral of a logarithmic form
\frac{d}{dx}\tan(x)=\sec^2(x),\quad\int\sec^2(x)dx=\tan(x)+ctangent
\frac{d}{dx}\sin^{-1}\left(\frac{x}{a}\right)=\frac{1}{\sqrt{a^2-x^2}},\quad\int\frac{1}{\sqrt{a^2-x^2}}dx=\sin^{-1}\left(\frac{x}{a}\right)+cinverse sine
\frac{d}{dx}\cos^{-1}\left(\frac{x}{a}\right)=\frac{-1}{\sqrt{a^2-x^2}},\quad\int\frac{-1}{\sqrt{a^2-x^2}}dx=\cos^{-1}\left(\frac{x}{a}\right)+cinverse cosine
\frac{d}{dx}\tan^{-1}\left(\frac{x}{a}\right)=\frac{a}{a^2+x^2},\quad\int\frac{a}{a^2+x^2}dx=\tan^{-1}\left(\frac{x}{a}\right)+cinverse tangent
\int u\frac{dv}{dx}dx=uv-\int v\frac{du}{dx}dxintegration by parts
V=\pi\int_a^b\left[f(x)\right]^2dxvolume of a solid of revolution about the x-axis
V=\pi\int_a^b\left[f(y)\right]^2dyvolume of a solid of revolution about the y-axis
\int_a^bf(x)dx\approx\frac{w}{3}\left[f(x_0)+4\left[f(x_1)+f(x_3)+\ldots\right]+2\left[f(x_2)+f(x_4)+\ldots\right]+f(x_n)\right]Simpson's rule, where w = (b − a)/n
\frac{d^2x}{dt^2}=-\omega^2x\ \text{gives}\ x=A\sin(\omega t+\alpha)\ \text{or}\ x=A\cos(\omega t+\beta)simple harmonic motion
v^2=\omega^2\left(A^2-x^2\right)simple harmonic motion, speed
T=\frac{2\pi}{\omega},\quad f=\frac{1}{T}period and frequency
a=\frac{dv}{dt}=\frac{d^2x}{dt^2}=v\frac{dv}{dx}=\frac{d}{dx}\left(\frac{1}{2}v^2\right)acceleration in four forms
Additional trigonometry for Specialist Mathematics
\tan^2(A)+1=\sec^2(A)Pythagorean identity, tangent and secant
\cot^2(A)+1=\text{cosec}^2(A)Pythagorean identity, cotangent and cosecant
\sin(A+B)=\sin(A)\cos(B)+\cos(A)\sin(B)sine of a sum
\sin(A-B)=\sin(A)\cos(B)-\cos(A)\sin(B)sine of a difference
\cos(A+B)=\cos(A)\cos(B)-\sin(A)\sin(B)cosine of a sum
\cos(A-B)=\cos(A)\cos(B)+\sin(A)\sin(B)cosine of a difference
\sin(2A)=2\sin(A)\cos(A)double angle, sine
\cos(2A)=\cos^2(A)-\sin^2(A)=1-2\sin^2(A)=2\cos^2(A)-1double angle, cosine
\sin(A)\sin(B)=\frac{1}{2}\left(\cos(A-B)-\cos(A+B)\right)product of sines
\cos(A)\cos(B)=\frac{1}{2}\left(\cos(A-B)+\cos(A+B)\right)product of cosines
\sin(A)\cos(B)=\frac{1}{2}\left(\sin(A+B)+\sin(A-B)\right)sine times cosine
\cos(A)\sin(B)=\frac{1}{2}\left(\sin(A+B)-\sin(A-B)\right)cosine times sine
Graph equations for Specialist Mathematics
(x-h)^2+(y-k)^2+(z-l)^2=r^2sphere
(x-h)^2+(y-k)^2=r^2circle
\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1ellipse
\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1hyperbola, opening left and right
\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1hyperbola, opening up and down
Additional statistics for Specialist Mathematics
|A\cup B|=|A|+|B|-|A\cap B|inclusion-exclusion, two sets
|A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|inclusion-exclusion, three sets
^nP_r=\frac{n!}{(n-r)!}=n\times(n-1)\times(n-2)\times\ldots\times(n-r+1)permutation
^nC_r=\binom{n}{r}=\frac{n!}{r!(n-r)!}combination
f(t)=\lambda e^{-\lambda t}\ \text{for}\ t\ge0,\ \lambda>0exponential distribution, density
\frac{1}{\lambda}exponential distribution, mean
\frac{1}{\lambda}exponential distribution, standard deviation
\musample means, mean
\frac{\sigma}{\sqrt{n}}sample means, standard deviation
E=z\frac{s}{\sqrt{n}}approximate margin of error
\left(\overline{x}-z\frac{s}{\sqrt{n}},\ \overline{x}+z\frac{s}{\sqrt{n}}\right)approximate confidence interval for the mean
Real and complex numbers
z=a+bi=r\left(\cos(\theta)+i\sin(\theta)\right)=r\text{cis}(\theta)forms of a complex number
|z|=r=\sqrt{a^2+b^2}modulus
\text{Arg}(z)=\theta,\quad\tan(\theta)=\frac{b}{a},\ -\pi<\theta\le\pi,\ a\ne0principal argument
\arg(z)=\text{Arg}(z)+2\pi n,\ n\text{ an integer}argument
z_1z_2=r_1r_2\ \text{cis}(\theta_1+\theta_2)product
\frac{z_1}{z_2}=\frac{r_1}{r_2}\ \text{cis}(\theta_1-\theta_2)quotient
z^n=r^n\text{cis}(n\theta)de Moivre's theorem
Matrices
A+B=B+Acommutative law for addition
A+0=Aadditive identity
A+(-A)=0additive inverse
AI=A=IAmultiplicative identity
AA^{-1}=I=A^{-1}Amultiplicative inverse
A(B+C)=AB+ACleft distributive law
(B+C)A=BA+CAright distributive law
\det(A)=ad-bcdeterminant of a 2 by 2 matrix with rows (a, b) and (c, d)
A^{-1}=\frac{1}{\det(A)}\times\text{the matrix with rows }(d,\ -b)\text{ and }(-c,\ a),\ \det(A)\ne0inverse of that matrix
Linear transformations
| Transformation | Matrix, written row by row |
|---|---|
| dilation by factor a parallel to the x-axis and factor b parallel to the y-axis | (a, 0) then (0, b) |
| rotation by angle θ anticlockwise about the origin | (cos θ, −sin θ) then (sin θ, cos θ) |
| reflection in the line y = x tan θ | (cos 2θ, sin 2θ) then (sin 2θ, −cos 2θ) |
Vectors
|\mathbf{a}|=\sqrt{a_1^2+a_2^2},\quad|\mathbf{a}|=\sqrt{a_1^2+a_2^2+a_3^2}magnitude in two and three dimensions
\tan(\theta)=\frac{y}{x},\ x\ne0direction
\hat{\mathbf{n}}=\frac{\mathbf{n}}{|\mathbf{n}|}unit vector
\mathbf{a}\cdot\mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos(\theta)scalar (dot) product
\mathbf{a}\cdot\mathbf{b}=a_1b_1+a_2b_2+a_3b_3dot product in components
\mathbf{r}=\mathbf{a}+t\mathbf{d}vector equation of a line
x=a_1+td_1,\quad y=a_2+td_2,\quad z=a_3+td_3parametric equations of a line
\frac{x-a_1}{d_1}=\frac{y-a_2}{d_2}=\frac{z-a_3}{d_3}Cartesian equation of a line
\mathbf{a}\times\mathbf{b}=|\mathbf{a}||\mathbf{b}|\sin(\theta)\hat{\mathbf{n}}vector (cross) product
\mathbf{a}\times\mathbf{b}=\left(a_2b_3-a_3b_2,\ a_3b_1-a_1b_3,\ a_1b_2-a_2b_1\right)cross product in components
\mathbf{a}\text{ on }\mathbf{b}=|\mathbf{a}|\cos(\theta)=\mathbf{a}\cdot\hat{\mathbf{b}}scalar projection
\mathbf{a}\text{ on }\mathbf{b}=\left(\mathbf{a}\cdot\hat{\mathbf{b}}\right)\hat{\mathbf{b}}=\left(\frac{\mathbf{a}\cdot\mathbf{b}}{\mathbf{b}\cdot\mathbf{b}}\right)\mathbf{b}vector projection
\mathbf{r}\cdot\mathbf{n}=\mathbf{a}\cdot\mathbf{n}vector equation of a plane
ax+by+cz+d=0Cartesian equation of a plane
Physical constant
| Quantity | Symbol | Value | Unit |
|---|---|---|---|
| magnitude of mean acceleration due to gravity on Earth | g | 9.8 | m s⁻² |
When you get the book
Specialist Mathematics has one external examination of two papers, worth 50% of your final result. The QCAA provides the book in both:
| Paper | Weight | Working time | Perusal | Technology |
|---|---|---|---|---|
| Paper 1 | 25% | 90 minutes | 5 minutes | None: no calculator, technology or notes |
| Paper 2 | 25% | 90 minutes | 5 minutes | QCAA-approved graphics calculator without CAS, or a scientific calculator; no notes |
Both papers have short response and multiple-choice questions on Units 3 and 4, and may draw on Units 1 and 2 and on Mathematical Methods. Our Specialist Maths mechanics guide and QCE Specialist Maths guide cover the topics.
What is not in the Specialist book
- Exact trigonometric values. The identities are there, but the values at 30°, 45° and 60° are not.
- Proof by induction and contradiction. The book gives no templates.
- Integration by substitution and partial fractions. Only integration by parts is listed.
- Differential equations. The book has no separation of variables, no integrating factors and no Euler's method step.
- Roots of complex numbers. De Moivre's theorem is given, but you must work out the nth root formula.
- Mechanics beyond acceleration forms and simple harmonic motion. There is no Newton's second law, no friction model and no momentum line.
- The quadratic formula. Still absent from the book.
How to use the book in the exam
- Spend the perusal time marking the page for vectors and complex numbers, the two sections students usually need to find fastest.
- In Paper 1 write every formula before you substitute. You have no calculator, so exact values matter.
- In Paper 2 write the setup, then let the graphics calculator do the arithmetic.
- For integration by parts, write u and dv/dx first, since the book gives only the pattern.
Example 1: Simpson's rule
Estimate\int_0^2x^2dxwith 4 strips, so w = 0.5 and the ordinates are 0, 0.25, 1, 2.25 and 4. The book's Simpson line gives:
\frac{0.5}{3}\left[0+4(0.25+2.25)+2(1)+4\right]=\frac{0.5}{3}\times16=2.667
This equals 8/3 exactly, because Simpson's rule is exact for quadratics. The trapezoidal rule in the Methods part would give 2.75 for the same strips.
Example 2: de Moivre's theorem
Find (1 + i)⁸. In polar form 1 + i has r = √2 and θ = π/4, so the book's de Moivre line gives:
(1+i)^8=\left(\sqrt{2}\right)^8\text{cis}\left(8\times\frac{\pi}{4}\right)=16\ \text{cis}(2\pi)=16
Check the answer in the Complex Numbers Calculator.
Example 3: a scalar projection
Let a = (2, 1, 2) and b = (3, 0, 4). The dot product is 2 × 3 + 1 × 0 + 2 × 4 = 14 and |b| = 5, so the scalar projection of a on b is:
\mathbf{a}\cdot\hat{\mathbf{b}}=\frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|}=\frac{14}{5}=2.8
Try other vectors in the Vectors Calculator.
Practise with the book open
Our Maths Extension 2 and Specialist Maths daily paper is a short timed paper of 8 questions worth 25 to 35 marks, not a QCAA paper. The Specialist Maths mechanics paper covers the mechanics section. See the QCE Maths Methods formula sheet guide for the first half of this book.
Exam answers
The answers and a worked mark scheme come with today's paper once it is marked. They show which line of the sheet each question wanted. We do not publish official answers, past paper answers or leaked papers.
Common questions
Do you get a formula sheet in QCE Specialist Maths?
Yes. The QCAA provides the Specialist Mathematics formula book in both external papers, according to the syllabus.
Is a calculator allowed in Specialist Paper 1?
No. Paper 1 is technology-free. Paper 2 allows an approved graphics calculator without CAS.
Does the book include Mathematical Methods formulas?
Yes. The first part repeats the Methods book, then Specialist sections follow.
Are complex number formulas in the book?
Yes: polar form, modulus, argument, product, quotient and de Moivre's theorem.
How long is each exam?
5 minutes of perusal and 90 minutes of working time per paper.
Revision cards
Say the answer first, then flip the card to check it.
Where to go next
- Solve it
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