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VCE Specialist Mathematics formula sheet: all the sections, and what is missing
Maths Extension 2 & Specialist Maths
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Formula sheet
The card below opens today's paper with the sheet in a drawer. The contents follow in our own layout. The VCAA published the sheets for all VCE mathematics exams in August 2024 in a redesigned layout, and the sheet is on the VCAA Specialist Mathematics pages. Check any line against it before you rely on it.
Formula sheet
Open formula sheetMaths Extension 2 & Specialist Maths
Mensuration
A=\frac{r^2}{2}(\theta-\sin(\theta))area of a circle segment
V=\frac{4}{3}\pi r^3volume of a sphere
V=\pi r^2hvolume of a cylinder
A=\frac{1}{2}bc\sin(A)area of a triangle
V=\frac{1}{3}\pi r^2hvolume of a cone
\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}sine rule
V=\frac{1}{3}Ahvolume of a pyramid
c^2=a^2+b^2-2ab\cos(C)cosine rule
Algebra, number and structure (complex numbers)
z=x+iy=r(\cos(\theta)+i\sin(\theta))=r\text{cis}(\theta)forms of a complex number
|z|=\sqrt{x^2+y^2}=rmodulus
-\pi<\text{Arg}(z)\le\piprincipal 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
Data analysis, probability and statistics
E(aX_1+b)=aE(X_1)+bindependent random variables: expected value
E(a_1X_1+a_2X_2+\ldots+a_nX_n)=a_1E(X_1)+a_2E(X_2)+\ldots+a_nE(X_n)linear combination: expected value
Var(aX_1+b)=a^2Var(X_1)variance of a linear function
Var(a_1X_1+\ldots+a_nX_n)=a_1^2Var(X_1)+\ldots+a_n^2Var(X_n)variance of a linear combination
E(X_1+X_2+\ldots+X_n)=n\muidentically distributed variables: expected value
Var(X_1+X_2+\ldots+X_n)=n\sigma^2identically distributed variables: variance
\left(\overline{x}-z\frac{s}{\sqrt{n}},\ \overline{x}+z\frac{s}{\sqrt{n}}\right)approximate confidence interval for μ
E\left(\overline{X}\right)=\mudistribution of the sample mean: mean
Var\left(\overline{X}\right)=\frac{\sigma^2}{n}distribution of the sample mean: variance
Calculus: derivatives and integrals
\frac{d}{dx}\left(x^n\right)=nx^{n-1}\quad\int x^ndx=\frac{1}{n+1}x^{n+1}+c,\ n\ne-1power
\frac{d}{dx}\left(e^{ax}\right)=ae^{ax}\quad\int e^{ax}dx=\frac{1}{a}e^{ax}+cexponential
\frac{d}{dx}\left(\log_e(x)\right)=\frac{1}{x}\quad\int\frac{1}{x}dx=\log_e|x|+cnatural logarithm
\frac{d}{dx}(\sin(ax))=a\cos(ax)\quad\int\sin(ax)dx=-\frac{1}{a}\cos(ax)+csine
\frac{d}{dx}(\cos(ax))=-a\sin(ax)\quad\int\cos(ax)dx=\frac{1}{a}\sin(ax)+ccosine
\frac{d}{dx}(\tan(ax))=a\sec^2(ax)\quad\int\sec^2(ax)dx=\frac{1}{a}\tan(ax)+ctangent
\frac{d}{dx}(\cot(ax))=-a\text{cosec}^2(ax)\quad\int\text{cosec}^2(ax)dx=-\frac{1}{a}\cot(ax)+ccotangent
\frac{d}{dx}(\sec(ax))=a\sec(ax)\tan(ax)\quad\int\sec(ax)\tan(ax)dx=\frac{1}{a}\sec(ax)+csecant
\frac{d}{dx}(\text{cosec}(ax))=-a\text{cosec}(ax)\cot(ax)\quad\int\text{cosec}(ax)\cot(ax)dx=-\frac{1}{a}\text{cosec}(ax)+ccosecant
\frac{d}{dx}\left(\sin^{-1}(ax)\right)=\frac{a}{\sqrt{1-(ax)^2}}\quad\int\frac{1}{\sqrt{a^2-x^2}}dx=\sin^{-1}\left(\frac{x}{a}\right)+c,\ a>0inverse sine
\frac{d}{dx}\left(\cos^{-1}(ax)\right)=\frac{-a}{\sqrt{1-(ax)^2}}\quad\int\frac{-1}{\sqrt{a^2-x^2}}dx=\cos^{-1}\left(\frac{x}{a}\right)+c,\ a>0inverse cosine
\frac{d}{dx}\left(\tan^{-1}(ax)\right)=\frac{a}{1+(ax)^2}\quad\int\frac{a}{a^2+x^2}dx=\tan^{-1}\left(\frac{x}{a}\right)+cinverse tangent
\int(ax+b)^ndx=\frac{1}{a(n+1)}(ax+b)^{n+1}+c,\ n\ne-1power of a linear function
\int\frac{1}{ax+b}dx=\frac{1}{a}\log_e|ax+b|+creciprocal of a linear function
Calculus: rules and applications
\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
\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}chain rule
\int u\frac{dv}{dx}dx=uv-\int v\frac{du}{dx}dxintegration by parts
x_{n+1}=x_n+h,\quad y_{n+1}=y_n+h\times f(x_n,y_n)Euler's method, if dy/dx = f(x, y), x₀ = a and y₀ = b
\int_{t_1}^{t_2}\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\ dtarc length, parametric
\int_{x_1}^{x_2}2\pi y\sqrt{1+\left(\frac{dy}{dx}\right)^2}\ dxsurface area, Cartesian, about the x-axis
\int_{y_1}^{y_2}2\pi x\sqrt{1+\left(\frac{dx}{dy}\right)^2}\ dysurface area, Cartesian, about the y-axis
\int_{t_1}^{t_2}2\pi y\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\ dtsurface area, parametric, about the x-axis
\int_{t_1}^{t_2}2\pi x\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\ dtsurface area, parametric, about the y-axis
Kinematics
a=\frac{d^2x}{dt^2}=\frac{dv}{dt}=v\frac{dv}{dx}=\frac{d}{dx}\left(\frac{1}{2}v^2\right)acceleration
v=u+at,\quad s=ut+\frac{1}{2}at^2,\quad v^2=u^2+2as,\quad s=\frac{1}{2}(u+v)tconstant acceleration
Vectors in two and three dimensions
\mathbf{r}(t)=x(t)\mathbf{i}+y(t)\mathbf{j}+z(t)\mathbf{k}position vector
|\mathbf{r}(t)|=\sqrt{x(t)^2+y(t)^2+z(t)^2}magnitude
\frac{d\mathbf{r}}{dt}(t)=\frac{dx}{dt}\mathbf{i}+\frac{dy}{dt}\mathbf{j}+\frac{dz}{dt}\mathbf{k}derivative
\mathbf{r}_1\cdot\mathbf{r}_2=|\mathbf{r}_1||\mathbf{r}_2|\cos(\theta)=x_1x_2+y_1y_2+z_1z_2scalar product
\mathbf{r}_1\times\mathbf{r}_2=(y_1z_2-y_2z_1)\mathbf{i}+(x_2z_1-x_1z_2)\mathbf{j}+(x_1y_2-x_2y_1)\mathbf{k}vector cross product
\mathbf{r}(t)=\mathbf{r}_1+t\mathbf{r}_2=(x_1+x_2t)\mathbf{i}+(y_1+y_2t)\mathbf{j}+(z_1+z_2t)\mathbf{k}vector equation of a line
x(t)=x_1+x_2t,\quad y(t)=y_1+y_2t,\quad z(t)=z_1+z_2tparametric equation of a line
\mathbf{r}(s,t)=\mathbf{r}_0+s\mathbf{r}_1+t\mathbf{r}_2vector equation of a plane
x(s,t)=x_0+x_1s+x_2t,\quad y(s,t)=y_0+y_1s+y_2t,\quad z(s,t)=z_0+z_1s+z_2tparametric equation of a plane
ax+by+cz=dCartesian equation of a plane
Circular functions
\cos^2(x)+\sin^2(x)=1,\quad1+\tan^2(x)=\sec^2(x),\quad\cot^2(x)+1=\text{cosec}^2(x)Pythagorean identities
\sin(x\pm y)=\sin(x)\cos(y)\pm\cos(x)\sin(y)sine of a sum or difference
\cos(x\pm y)=\cos(x)\cos(y)\mp\sin(x)\sin(y)cosine of a sum or difference
\tan(x\pm y)=\frac{\tan(x)\pm\tan(y)}{1\mp\tan(x)\tan(y)}tangent of a sum or difference
\sin(2x)=2\sin(x)\cos(x),\quad\cos(2x)=\cos^2(x)-\sin^2(x)=2\cos^2(x)-1=1-2\sin^2(x),\quad\tan(2x)=\frac{2\tan(x)}{1-\tan^2(x)}double angles
\sin^2(ax)=\frac{1}{2}(1-\cos(2ax)),\quad\cos^2(ax)=\frac{1}{2}(1+\cos(2ax))squares of sine and cosine
How the two exams work with the sheet
The VCAA's 2025 Specialist Mathematics papers give these conditions:
| Item | Examination 1 | Examination 2 |
|---|---|---|
| Reading time | 15 minutes | 15 minutes |
| Writing time | 1 hour | 2 hours |
| Content | 9 questions, 40 marks | Section A: 20 multiple-choice questions, 20 marks. Section B: 6 questions, 60 marks |
| Technology | None: no calculator or software | One approved CAS calculator or CAS software, and one scientific calculator |
| Notes | None of any kind | One bound reference |
| Formula sheet | Supplied | Supplied |
Other states give different lengths of sheet. Compare the QCE, WACE, SACE and TCE guides, and read our Maths Extension 2 and Specialist Maths guide for the course.
What is not on the VCE Specialist formula sheet
- Matrices. No determinant, inverse or transformation matrices.
- Differential equation methods. Euler's method is given, but not separation of variables or the logistic solution.
- Proof templates. No structure for induction or contradiction.
- Roots of complex numbers and the conjugate rules. De Moivre's theorem is printed, but not the nth root formula.
- The quadratic formula and the discriminant. Not given.
- Series and Maclaurin expansions. None are printed.
- The distance from a point to a plane and vector projections. Dot and cross products are given but not projections.
How to use the sheet in the exam
- In Exam 1, use the reading time to find the pages for calculus and vectors, because answers must be exact.
- Write the formula from the sheet first, then substitute.
- For Euler's method, set up a small table of x, y and f(x, y) so each step is checked.
- In Exam 2, use your CAS for integrals, but show the integral you set up.
Example 1: Euler's method
Given dy/dx = x + y with x₀ = 0, y₀ = 1 and step h = 0.1, the sheet's Euler line gives the first two steps:
y_1=1+0.1\times(0+1)=1.1,\qquad y_2=1.1+0.1\times(0.1+1.1)=1.22
So y is approximately 1.22 when x = 0.2. The sheet gives the pattern, and you evaluate f(xₙ, yₙ) each time.
Example 2: surface area of revolution
The line y = x for 0 ≤ x ≤ 1 is rotated about the x-axis. With dy/dx = 1, the Cartesian surface area line gives:
\int_0^12\pi x\sqrt{1+1^2}\ dx=2\sqrt{2}\pi\left[\frac{x^2}{2}\right]_0^1=\sqrt{2}\pi
That is about 4.44 square units, which matches the curved surface area of a cone of radius 1 and slant height √2.
Example 3: a vector cross product
For r₁ = i + 2j + 3k and r₂ = 2i + k, the cross product line gives the components:
\mathbf{r}_1\times\mathbf{r}_2=(2\times1-0\times3)\mathbf{i}+(2\times3-1\times1)\mathbf{j}+(1\times0-2\times2)\mathbf{k}=2\mathbf{i}+5\mathbf{j}-4\mathbf{k}
Check it by taking the dot product with each original vector, which should give zero. Try your own in the Vectors Calculator.
Practise with the sheet open
Our Maths Extension 2 and Specialist Maths daily paper is a short timed paper of 8 questions worth 25 to 35 marks. It is not a VCAA paper. Sit it with no calculator for the Exam 1 style, and see the Specialist Maths mechanics paper for kinematics.
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 VCE Specialist Maths?
Yes. The VCAA supplies the same formula sheet with Examination 1 and Examination 2.
Can I bring notes into VCE Specialist Exam 1?
No. Students are not permitted to bring technology or notes of any kind into Exam 1.
What can I bring into Specialist Exam 2?
One bound reference, one approved CAS calculator or software, and one scientific calculator.
Is Euler's method on the sheet?
Yes, with the surface area and arc length formulas for parametric curves.
How long are the exams?
Exam 1 has 15 minutes of reading and 1 hour of writing. Exam 2 has 15 minutes of reading and 2 hours of writing.
Revision cards
Say the answer first, then flip the card to check it.
Where to go next
- Solve it
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Australia · VCEVCE Specialist Maths: Exam 1, Exam 2 and What to ReviseVCE Specialist Mathematics explained: Exam 1 (40 marks, no technology), Exam 2 (80 marks), the bound reference, weighting and worked questions.