01
Number & Algebra
Indices & Surds
▼
Key Rules to Memorise
\(a^m \times a^n = a^{m+n}\)
\(a^m \div a^n = a^{m-n}\)
\((a^m)^n = a^{mn}\)
\(a^0 = 1\), \(a^{-n} = \dfrac{1}{a^n}\), \(a^{1/n} = \sqrt[n]{a}\)
Surd rule: \(\sqrt{ab} = \sqrt{a}\cdot\sqrt{b}\); rationalise denominator by multiplying by conjugate.
Worked Example
Simplify \(\dfrac{6\sqrt{3}}{\sqrt{3}-1}\).
Multiply top and bottom by \((\sqrt{3}+1)\): \(\dfrac{6\sqrt{3}(\sqrt{3}+1)}{3-1} = \dfrac{6(3+\sqrt{3})}{2} = 9+3\sqrt{3}\).
Arithmetic & Geometric Sequences
▼
Formulae
Arithmetic: \(u_n = u_1 + (n-1)d\) — \(d\) = common difference
Arithmetic sum: \(S_n = \dfrac{n}{2}(2u_1 + (n-1)d)\)
Geometric: \(u_n = u_1 \cdot r^{n-1}\) — \(r\) = common ratio
Geometric sum: \(S_n = \dfrac{u_1(r^n - 1)}{r-1}\)
Worked Example
Find the 15th term of the AP: 7, 11, 15, …
\(d=4,\; u_{15} = 7 + 14 \times 4 = 63\)
Quadratics & Functions
▼
Core Facts
Quadratic formula: \(x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\)
Discriminant \(\Delta = b^2-4ac\): >0 two real roots, =0 one, <0 no real roots
Vertex form: \(f(x) = a(x-h)^2+k\), vertex at \((h,k)\)
Axis of symmetry: \(x = -\dfrac{b}{2a}\)
Worked Example
Solve \(2x^2 - 5x - 3 = 0\).
\(\Delta = 25+24=49\). \(x = \dfrac{5\pm7}{4}\). So \(x=3\) or \(x=-\tfrac{1}{2}\).
01
Indices
Simplify \(2^5 \times 2^{-3} \div 2^2\). Give your answer as an integer.
[1 mark]
02
Surds
Rationalise the denominator and simplify: \(\dfrac{10}{\sqrt{5}}\).
[2 marks]
03
Sequences
The first term of a geometric sequence is 3 and the common ratio is 2. Find the sum of the first 6 terms.
[2 marks]
04
Quadratics
Find the \(x\)-coordinates of the turning point of \(f(x) = 3x^2 - 12x + 7\).
[2 marks]
05
Functions
Given \(f(x) = 2x + 1\) and \(g(x) = x^2 - 3\), find \(f(g(2))\).
[2 marks]
06
Algebra — Discriminant
For the equation \(kx^2 - 6x + 3 = 0\) to have exactly one real solution, find the value of \(k\).
[2 marks]
02
Geometry & Trigonometry
Trigonometry & 3D Pythagoras
▼
Essential Formulae
SOH CAH TOA: \(\sin\theta = \dfrac{opp}{hyp}\), \(\cos\theta = \dfrac{adj}{hyp}\), \(\tan\theta = \dfrac{opp}{adj}\)
Sine rule: \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B}\)
Cosine rule: \(a^2 = b^2 + c^2 - 2bc\cos A\)
3D diagonal of cuboid: \(d = \sqrt{l^2+w^2+h^2}\)
Circle area: \(\pi r^2\); Arc length: \(\dfrac{\theta}{360}\times 2\pi r\)
Worked Example
In △ABC, \(a=7\), \(b=9\), \(C=60°\). Find side \(c\).
\(c^2 = 49+81-2(7)(9)\cos60° = 130-63=67\). \(c=\sqrt{67}\approx 8.19\)
Circle Theorems
▼
Theorems to Know
Angle at centre = 2 × angle at circumference (same arc)
Angles in same segment are equal
Opposite angles in a cyclic quadrilateral sum to 180°
Tangent ⊥ radius at point of contact
Tangents from an external point are equal in length
Worked Example
An inscribed angle subtends an arc. The central angle on the same arc is 110°. Find the inscribed angle.
Inscribed angle = 110° ÷ 2 = 55°.
07
Trigonometry
A ladder leans against a wall, making an angle of 70° with the ground. The foot of the ladder is 1.5 m from the wall. Find the length of the ladder, correct to 2 decimal places (in metres).
[2 marks]
08
3D Pythagoras
A cuboid has dimensions 3 cm × 4 cm × 12 cm. Find the length of the main diagonal. Give an exact answer.
[2 marks]
09
Circle Theorems
O is the centre of a circle. Points A, B, and C lie on the circle. Angle AOC = 148°. Find angle ABC.
[2 marks]
10
Similarity
Two similar triangles have corresponding sides in the ratio 3 : 5. If the area of the smaller triangle is 27 cm², find the area of the larger triangle.
[2 marks]
11
Sine Rule
In triangle PQR, angle P = 35°, angle Q = 80°, and side \(p = 8\) cm. Find the length of side \(q\), correct to 1 decimal place.
[2 marks]
03
Statistics & Probability
Descriptive Statistics
▼
Formulae
Mean: \(\bar{x} = \dfrac{\sum x}{n}\)
Population std dev: \(\sigma = \sqrt{\dfrac{\sum(x-\bar{x})^2}{n}}\)
IQR = Q3 − Q1; Outlier if > Q3 + 1.5×IQR or < Q1 − 1.5×IQR
Correlation: strong if |r| ≥ 0.8, weak if |r| ≤ 0.4
Worked Example
Data: 4, 7, 7, 8, 10. Find the mean and range.
Mean = 36/5 = 7.2. Range = 10 − 4 = 6.
Probability
▼
Rules
P(A∪B) = P(A) + P(B) − P(A∩B)
Conditional: \(P(A|B) = \dfrac{P(A\cap B)}{P(B)}\)
Independent events: P(A∩B) = P(A)·P(B)
Complementary: P(A') = 1 − P(A)
Worked Example
P(A)=0.4, P(B)=0.3, independent. Find P(A∩B).
P(A∩B) = 0.4 × 0.3 = 0.12.
12
Mean & Standard Deviation
The data set is: 2, 4, 4, 4, 5, 5, 7, 9. Find the mean.
[1 mark]
13
Probability — Combined
A bag contains 4 red balls and 6 blue balls. Two balls are drawn without replacement. Find the probability that both are red. Express as a simplified fraction.
[2 marks]
14
Conditional Probability
In a class, 60% of students study French (F) and 50% study German (G). 30% study both. A student is chosen at random. Given the student studies German, what is the probability they also study French?
[2 marks]
15
Statistics — Outlier
For the data set {12, 15, 15, 17, 18, 19, 22, 45}, Q1 = 15 and Q3 = 20.5. Determine whether 45 is an outlier. Show your reasoning.
[2 marks]
04
Calculus & Mathematical Modelling
Derivatives & Rate of Change
▼
Differentiation Rules
Power rule: \(\dfrac{d}{dx}(x^n) = nx^{n-1}\)
Constant: \(\dfrac{d}{dx}(c) = 0\)
Gradient of tangent at \(x=a\) is \(f'(a)\)
Stationary point: \(f'(x)=0\); min if \(f''(x)>0\), max if \(f''(x)<0\)
Worked Example
Find \(f'(x)\) for \(f(x) = 4x^3 - 6x + 2\).
\(f'(x) = 12x^2 - 6\)
Exponential & Linear Models
▼
Model Types
Linear: \(y = mx + c\); constant rate of change
Exponential growth: \(y = A \cdot b^t\) where \(b > 1\)
Exponential decay: \(y = A \cdot b^t\) where \(0 < b < 1\)
Half-life: time for quantity to halve (\(y = A(0.5)^{t/h}\))
Worked Example
A population starts at 500 and doubles every 4 years. Write a model for the population \(P\) after \(t\) years.
\(P(t) = 500 \times 2^{t/4}\)
16
Differentiation
Find the gradient of \(f(x) = 3x^2 - 4x + 1\) at the point where \(x = 2\).
[2 marks]
17
Stationary Points
Find the \(x\)-coordinate of the stationary point of \(g(x) = x^3 - 6x^2 + 9x + 2\), and state whether it is a local maximum or minimum at \(x = 1\).
[3 marks]
18
Exponential Model
A radioactive substance decays according to \(M(t) = 200 \times (0.5)^{t/3}\) grams, where \(t\) is in years. Find the mass after 9 years.
[2 marks]
19
Linear Modelling
A car hire company charges a fixed fee of $25 plus $0.30 per kilometre. Write a linear model for cost \(C\) in terms of distance \(d\). How much does it cost to travel 120 km?
[2 marks]
20
Optimisation
A farmer has 120 m of fencing to enclose a rectangular field against a straight wall (no fencing needed along the wall). Let the width perpendicular to the wall be \(x\) m. Find the value of \(x\) that maximises the area, and state the maximum area.
[3 marks]