Curriculum

VCE Specialist Mathematics

11, 12

Community-maintained — may not exactly match your study design

Unit 1 — Algebra, complex numbers, vectors and proof

Algebra and structure

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Quadratic Formula $x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$ Practise
Discriminant $Δ = b^{2} - 4 a c$ Practise
Sum of Roots $α + β = - \frac{b}{a}$ Practise
Product of Roots $α β = \frac{c}{a}$ Practise
Product of Powers $a^{m} \cdot a^{n} = a^{m + n}$ Practise
Quotient of Powers $\frac{a ^{m}}{a ^{n}} = a^{m - n}$ Practise
Power of a Power $(a^{m})^{n} = a^{mn}$ Practise
Negative Exponent $a^{- n} = \frac{1}{a ^{n}}$ Practise
Zero Exponent $a^{0} = 1 (a \neq = 0)$ Practise
Pascal's Rule $(r n + 1) = (r n) + (r - 1 n)$ Practise
Binomial Theorem $(a + b)^{n} = \sum_{r = 0}^{n} (r n) a^{n - r} b^{r}$ Practise
Middle Term of Binomial $T_{\frac{n}{2} + 1} = (n /2 n) a^{n /2} b^{n /2} (n even)$ Practise
nth term of arithmetic sequence $T_{n} = a + (n - 1) d$ Practise
nth term of geometric sequence $T_{n} = a \cdot r^{n - 1}$ Practise
Sum of arithmetic series $S_{n} = \frac{n}{2} (a + l)$ Practise
Sum of geometric series $S_{n} = a (\frac{1 - r ^{n}}{1 - r}) (r \neq = 1)$ Practise
Sum to infinity $S_{\infty} = \frac{a}{1 - r} (∣ r ∣ < 1)$ Practise

Not yet covered

Partial fraction decomposition (no formula in corpus)
Proof by mathematical induction (conceptual, not formula-driven)
Rational root theorem and factorisation of polynomials over real/complex numbers (no specific formula)
Simultaneous equations in three variables (procedural, no dedicated formula)

Complex numbers — introduction

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Real Part from Polar Form $a = r cos θ$ Practise
Imaginary Part from Polar Form $b = r sin θ$ Practise
Modulus of a Complex Number $∣ z ∣ = a^{2} + b^{2}$ Practise
Argument of a Complex Number $ar g (z) = atan2 (b, a)$ Practise
Polar Form of a Complex Number $z = r (cos θ + i sin θ) = r cis θ$ Practise
Complex Conjugate $\overline{z} = a - bi$ Practise
Product with Conjugate $z \overline{z} = ∣ z ∣^{2} = a^{2} + b^{2}$ Practise
Reciprocal of a Complex Number $\frac{1}{z} = \frac{z}{∣ z ∣ ^{2}}$ Practise

Not yet covered

Arithmetic of complex numbers — addition, multiplication, division in Cartesian form (procedural, no dedicated formula)
Complex conjugate properties (|z|^2 = z * conjugate(z)) (no specific formula in corpus)
Locus and regions in the complex plane (no formula — geometry concept)

Vectors in two dimensions

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Dot Product — Component Form $a \cdot b = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}$ Practise
Dot Product — Geometric Form $a \cdot b = ∣ a ∣ ∣ b ∣ cos θ$ Practise
Vector Magnitude $∣ a ∣ = a_{1}^{2} + a_{2}^{2} + a_{3}^{2}$ Practise
Unit Vector $\hat{a} = \frac{a}{∣ a ∣}$ Practise
Scalar Projection $comp_{b} a = \frac{a \cdot b}{∣ b ∣}$ Practise
Vector Projection $proj_{b} a = \frac{a \cdot b}{∣ b ∣ ^{2}} b$ Practise
Perpendicular Component $perp_{b} a = a - proj_{b} a$ Practise

Not yet covered

Vector addition, subtraction and scalar multiplication (procedural, no dedicated formula)
Position vectors and vector magnitude in 2D (covered by dot-product.magnitude)
Parallel and perpendicular vector conditions (no specific formula — application of dot product)

Unit 2 — Vectors, complex numbers, circular functions and calculus

Vectors in three dimensions — cross product and lines

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Cross Product $a \times b = (a_{2} b_{3} - a_{3} b_{2}) i - (a_{1} b_{3} - a_{3} b_{1}) j + (a_{1} b_{2} - a_{2} b_{1}) k$ Practise
Cross Product Magnitude $∣ a \times b ∣ = ∣ a ∣ ∣ b ∣ sin θ$ Practise
Area of Triangle $Area = \frac{1}{2} ∣ a \times b ∣$ Practise
Vector Equation of a Line $r = a + λ d$ Practise
Parametric Equations of a Line $x = a_{1} + λ d_{1}, y = a_{2} + λ d_{2}, z = a_{3} + λ d_{3}$ Practise
Scalar Projection $comp_{b} a = \frac{a \cdot b}{∣ b ∣}$ Practise
Vector Projection $proj_{b} a = \frac{a \cdot b}{∣ b ∣ ^{2}} b$ Practise
Perpendicular Component $perp_{b} a = a - proj_{b} a$ Practise

Not yet covered

3D coordinate geometry fundamentals (no specific formula)
Distance between two points in 3D (no dedicated formula — extension of 2D distance)
Shortest distance from a point to a line in 3D (no specific formula in corpus)
Angle between two lines in 3D (application of dot product)

Complex numbers — polar form and de Moivre

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Polar Form of a Complex Number $z = r (cos θ + i sin θ) = r cis θ$ Practise
Euler's Formula $e^{i θ} = cos θ + i sin θ$ Practise
Exponential Form $z = r e^{i θ}$ Practise
Argument of a Complex Number $ar g (z) = atan2 (b, a)$ Practise
Modulus of a Complex Number $∣ z ∣ = a^{2} + b^{2}$ Practise
De Moivre's Theorem $(r cis θ)^{n} = r^{n} cis (n θ)$ Practise
De Moivre's Theorem — Trig Form $(cos θ + i sin θ)^{n} = cos (n θ) + i sin (n θ)$ Practise

Not yet covered

Multiplication and division of complex numbers in polar form (procedural from de Moivre — no separate formula)
Powers of complex numbers using de Moivre (application of de Moivre, no separate formula)

Circular functions — compound and double angle

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sin(A + B) = sin A cos B + cos A sin B $sin (A + B) = sin A cos B + cos A sin B$ Practise
cos(A + B) = cos A cos B - sin A sin B $cos (A + B) = cos A cos B - sin A sin B$ Practise
tan x = sin x / cos x $tan x = \frac{s i n x}{c o s x}$ Practise
sin(2x) = 2 sin x cos x $sin (2 x) = 2 sin x cos x$ Practise
cos(2x) = cos^2(x) - sin^2(x) $cos (2 x) = cos^{2} x - sin^{2} x$ Practise
Unit Circle: sin, cos, tan $P (θ) = (cos θ, sin θ)$ Practise
sin^2 x + cos^2 x = 1 $sin^{2} x + cos^{2} x = 1$ Practise
1 + tan^2(x) = sec^2(x) $1 + tan^{2} x = sec^{2} x$ Practise
sin(-x) = -sin(x) $sin (- x) = - sin x$ Practise
cos(-x) = cos(x) $cos (- x) = cos x$ Practise

Not yet covered

Half-angle formulas for sine, cosine and tan (no formula in corpus)
Sum-to-product and product-to-sum identities (no formula in corpus)
Reciprocal circular functions sec, cosec, cot (no formulas in corpus)
Inverse circular functions arcsine, arccosine, arctan (no formulas in corpus)

Unit 3 — Complex numbers, differential equations and vectors

Complex numbers — roots, Euler and advanced topics

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nth Roots of a Complex Number $z_{k} = r^{1/ n} cis (\frac{θ + 2 π k}{n}), k = 0, 1, \dots, n - 1$ Practise
nth Roots of Unity $z_{k} = cis (\frac{2 π k}{n}), k = 0, 1, \dots, n - 1$ Practise
Complex Conjugate $\overline{z} = a - bi$ Practise
Euler's Formula $e^{i θ} = cos θ + i sin θ$ Practise
Exponential Form $z = r e^{i θ}$ Practise
De Moivre's Theorem $(r cis θ)^{n} = r^{n} cis (n θ)$ Practise
De Moivre's Theorem — Trig Form $(cos θ + i sin θ)^{n} = cos (n θ) + i sin (n θ)$ Practise

Not yet covered

nth roots of unity — primitive roots and geometry on Argand diagram (conceptual, formula covers roots-of-unity)
Factorisation of polynomials over complex numbers (no specific formula — extension of conjugate root theorem)

Differential equations

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General Solution of dy/dx = f(x) $y = \int f (x) d x + C$ Practise
Proportional Rate of Change $\frac{d N}{d t} = k N$ Practise
Exponential Growth and Decay $y = A e^{k x}$ Practise
Newton's Law of Cooling $\frac{d T}{d t} = - k (T - T_{a})$ Practise

Not yet covered

Separation of variables technique (procedural, not a specific formula)
First-order linear differential equations — integrating factor method (no formula in corpus)
Second-order homogeneous differential equations with constant coefficients (no formula in corpus)
Logistic growth model (no formula in corpus)
Slope fields and Euler's method (conceptual/procedural, no formula)

Vectors — planes and geometry in 3D

Community-maintained — may not exactly match your study design

Plane — Normal Form $n \cdot (r - a) = 0$ Practise
Plane — Cartesian Form $a x + b y + cz = d$ Practise
Distance from Point to Plane $d = \frac{∣ a x _{0} + b y _{0} + c z _{0} + d ∣}{a ^{2} + b ^{2} + c ^{2}}$ Practise
Parametric Equations of a Line $x = a_{1} + λ d_{1}, y = a_{2} + λ d_{2}, z = a_{3} + λ d_{3}$ Practise
Vector Equation of a Line $r = a + λ d$ Practise
Cross Product $a \times b = (a_{2} b_{3} - a_{3} b_{2}) i - (a_{1} b_{3} - a_{3} b_{1}) j + (a_{1} b_{2} - a_{2} b_{1}) k$ Practise
Cross Product Magnitude $∣ a \times b ∣ = ∣ a ∣ ∣ b ∣ sin θ$ Practise
Area of Triangle $Area = \frac{1}{2} ∣ a \times b ∣$ Practise
Dot Product — Component Form $a \cdot b = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}$ Practise
Dot Product — Geometric Form $a \cdot b = ∣ a ∣ ∣ b ∣ cos θ$ Practise

Not yet covered

Intersection of two lines in 3D (no specific formula)
Intersection of a line and a plane (no specific formula)
Intersection of two planes — line of intersection (no specific formula)
Shortest distance from a point to a line in 3D (no formula in corpus)
Angle between a line and a plane (application of dot product, no separate formula)
Angle between two planes (application of dot product of normals, no separate formula)

Unit 4 — Mechanics, advanced calculus and statistical inference

Mechanics — kinematics, forces and moments

Community-maintained — may not exactly match your study design

Squared Distance from Origin $∣ r ∣^{2} = x^{2} + y^{2} + z^{2}$ Practise
Speed Squared from Velocity Components $∣ v ∣^{2} = v_{x}^{2} + v_{y}^{2}$ Practise
s = (u + v)/2 * t $s = \frac{u + v}{2} t$ Practise
v = u + a t $v = u + a t$ Practise
v^2 = u^2 + 2 a s $v^{2} = u^{2} + 2 a \cdot s$ Practise
r = r0 + u t + 1/2 a t^2 $r = r_{0} + u t + \frac{1}{2} a t^{2}$ Practise
F = m a $F_{net} = m a$ Practise
Sum of Forces in One Direction = Sum in Opposite Direction $\sum F_{rightward} = \sum F_{leftward}$ Practise
f_s_max = mu_s * N $f_{s, m a x} = μ_{s} N$ Practise
f_k = mu_k * N $f_{k} = μ_{k} N$ Practise
W = F d cos theta $W = F \cdot s = F s cos θ$ Practise
P = W / t $P_{avg} = \frac{W}{t} = \frac{Δ E}{Δ t}$ Practise
M = F d $M = F d_{⊥}$ Practise
M = r F sin theta $∣ M ∣ = ∣ r \times F ∣ = r F sin θ$ Practise
Clockwise Moments = Anticlockwise Moments $\sum M_{anticlockwise} = \sum M_{clockwise}$ Practise

Not yet covered

Variable acceleration — displacement/velocity from integration (covered by calculus, no separate formula)
Constant force vector applications (application of F=ma, no separate formula)
Inclined plane mechanics — resolving forces parallel and perpendicular (no formula in corpus)
Connected bodies — tension in strings and pulleys (no formula in corpus)
Circular motion in mechanics (corpus has circular motion in physics only, not in mechanics-maths)

Advanced calculus — integration and applications

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Power Rule for Integration $\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C (n \neq = - 1)$ Practise
Constant Multiple Rule for Integration $\int k f (x) d x = k \int f (x) d x$ Practise
Sum Rule for Integration $\int [f (x) + g (x)] d x = \int f (x) d x + \int g (x) d x$ Practise
Integration by Substitution $\int_{a}^{b} f (g (x)) g^{'} (x) d x = \int_{g (a)}^{g (b)} f (u) d u$ Practise
Definite Integral Linear Property $\int_{a}^{b} [f (x) + g (x)] d x = \int_{a}^{b} f (x) d x + \int_{a}^{b} g (x) d x$ Practise
Integral of sin x $\int sin x d x = - cos x + C$ Practise
Integral of cos x $\int cos x d x = sin x + C$ Practise
Integral of sec^2 x $\int sec^{2} x d x = tan x + C$ Practise
Integral of e^x $\int e^{x} d x = e^{x} + C$ Practise
Integral of 1/x $\int \frac{1}{x} d x = ln ∣ x ∣ + C (x \neq = 0)$ Practise
Chain Rule $\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) \cdot g^{'} (x)$ Practise
Product Rule $\frac{d}{d x} [f (x) g (x)] = f^{'} (x) g (x) + f (x) g^{'} (x)$ Practise
Quotient Rule $\frac{d}{d x} [\frac{f ( x )}{g ( x )}] = \frac{f ^{'} ( x ) g ( x ) - f ( x ) g ^{'} ( x )}{[ g ( x ) ] ^{2}}$ Practise
d/dx(sin x) = cos x $\frac{d}{d x} (sin x) = cos x$ Practise
d/dx(cos x) = -sin x $\frac{d}{d x} (cos x) = - sin x$ Practise
d/dx(tan x) = sec^2(x) $\frac{d}{d x} (tan x) = sec^{2} x$ Practise
d/dx(e^x) = e^x $\frac{d}{d x} (e^{x}) = e^{x}$ Practise

Not yet covered

Integration by parts (no formula in corpus)
Integration using partial fractions (no formula in corpus)
Arc length of a curve (L = int sqrt(1 + (dy/dx)^2) dx — no formula in corpus)
Volume of revolution around both axes (V = pi int [f(x)]^2 dx — no formula in corpus)
Second derivatives and applications in dynamics (no specific formula)

Statistical inference

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Margin of Error $ME = z^{*} \frac{σ}{n}$ Practise
CI for Mean (Known sigma) $\overset{x}{ˉ} \pm z^{*} \frac{σ}{n}$ Practise
CI for Mean (Unknown sigma) $\overset{x}{ˉ} \pm t^{*} \frac{s}{n}$ Practise
CI for Proportion $\overset{p}{^} \pm z^{*} \frac{p ^ ( 1 - p ^ )}{n}$ Practise
Sample Size Formula $n = (\frac{z ^{*} σ}{ME})^{2}$ Practise
Normal Distribution PDF $f (x) = \frac{1}{σ 2 π} e^{- \frac{( x - μ ) ^{2}}{2 σ ^{2}}}$ Practise
Z-Score $z = \frac{x - μ}{σ}$ Practise
Expected Value $E (X) = \sum_{x} x \cdot P (X = x)$ Practise
Variance $Var (X) = E (X^{2}) - [E (X)]^{2}$ Practise
Standard Deviation $σ = Var (X)$ Practise

Not yet covered

t-distribution confidence intervals for unknown population standard deviation (corpus has mean-unknown-sigma, but no explicit t-critical values)
Hypothesis testing — null and alternative hypotheses, p-values, Type I and Type II errors (conceptual, no formulas in corpus)
Chi-squared goodness-of-fit test (no formula in corpus)
Central Limit Theorem for sample means (conceptual, no formula)