This is the complete reference for WACE Year 12 Mathematics Specialist. Specialist is the highest-scaling subject in WA (typically +9 to +10 at the median in TISC scaling reports).
Vectors in 3D
- Magnitude: |v| = sqrt(v1^2 + v2^2 + v3^2)
- Unit vector: v_hat = v / |v|
- Dot product: u . v = u1 v1 + u2 v2 + u3 v3 = |u| |v| cos(theta)
- Cross product: |u x v| = |u| |v| sin(theta), perpendicular to both u and v
- Angle between vectors: cos(theta) = (u . v) / (|u| |v|)
- Vector projection of u onto v: proj_v(u) = ((u . v) / |v|^2) v
- Line through point A in direction d: r = a + t d, t scalar
- Plane through point A with normal n: n . (r - a) = 0
Complex numbers
- Cartesian: z = a + bi (a, b real)
- Polar: z = r(cos(theta) + i sin(theta)) = r cis(theta)
- Modulus: |z| = r = sqrt(a^2 + b^2)
- Argument: arg(z) = theta where tan(theta) = b/a (consider quadrant)
- Conjugate: z* = a - bi (reflects over real axis)
- Multiplication in polar: r1 cis(theta1) * r2 cis(theta2) = r1 r2 cis(theta1 + theta2)
- de Moivre's theorem: (r cis(theta))^n = r^n cis(n theta)
- nth roots of z: w_k = r^(1/n) cis((theta + 2 pi k) / n) for k = 0, 1, ..., n-1
- Euler's formula: e^(i theta) = cos(theta) + i sin(theta)
Functions and graphs
- Inverse trig functions domain/range: arcsin: [-1, 1] -> [-pi/2, pi/2]; arccos: [-1, 1] -> [0, pi]; arctan: R -> (-pi/2, pi/2).
- Reciprocal trig: sec(x) = 1/cos(x); csc(x) = 1/sin(x); cot(x) = 1/tan(x).
- Pythagorean identities: sin^2(x) + cos^2(x) = 1; 1 + tan^2(x) = sec^2(x); 1 + cot^2(x) = csc^2(x).
Calculus (advanced)
| Function | Derivative |
|---|---|
| arcsin(x) | 1 / sqrt(1 - x^2) |
| arccos(x) | -1 / sqrt(1 - x^2) |
| arctan(x) | 1 / (1 + x^2) |
Integration techniques
- Integration by substitution: let u = g(x), du = g'(x) dx, replace.
- Integration by parts: integral u dv = uv - integral v du.
- Trigonometric substitution: for sqrt(a^2 - x^2) try x = a sin(theta); for sqrt(a^2 + x^2) try x = a tan(theta).
- Partial fractions: decompose rational functions before integrating.
Differential equations
- Separable: dy/dx = f(x) g(y); separate as dy/g(y) = f(x) dx, integrate both sides.
- First-order linear: dy/dx + P(x) y = Q(x); use integrating factor e^(integral P(x) dx).
- Population model dN/dt = kN: N(t) = N_0 e^(kt); k > 0 growth, k < 0 decay.
- Logistic growth dN/dt = kN(1 - N/M): bounded by carrying capacity M.
Statistical inference
- Sample mean distribution: Xbar ~ N(mu, sigma^2 / n) for large n (Central Limit Theorem).
- Standard error of mean: SE = sigma / sqrt(n).
- Confidence interval for mean (large n): xbar +/- z* (sigma / sqrt(n)).
- Hypothesis test for mean: z = (xbar - mu_0) / (sigma / sqrt(n)).
- Critical values: 90% CI: z = 1.645; 95%: z = 1.96; 99%: z = 2.576.
Proof techniques
- Direct proof: assume hypothesis, derive conclusion through logical steps.
- Proof by contradiction: assume the negation, derive a contradiction, conclude original is true.
- Proof by induction (over integers):
- Base case: prove for n = 1.
- Inductive step: assume true for n = k, prove for n = k+1.
- Conclude: true for all positive integers.
- Counterexample: to disprove a "for all" statement, find one case where it fails.
Useful constants
| Constant | Value |
|---|---|
| e (Euler's number) | 2.71828... |
| pi | 3.14159... |
| i (imaginary unit) | sqrt(-1) |
Print and pin this
Print this page. Stick it in your Specialist folder. Specialist demands the kind of mathematical maturity that only comes from repeated exposure. Refer to this card daily.
For the strategic case for taking Specialist alongside Methods, see our Specialist vs Methods scaling post.
If you want a Specialist tutor who works through past paper proof and vector questions every week, book a free trial class.