Topic 1 · Algebra
Arithmetic & Geometric Sequences
$$u_n = u_1 + (n-1)d \qquad S_n = \frac{n}{2}(2u_1 + (n-1)d)$$
$$u_n = u_1 \cdot r^{n-1} \qquad S_n = \frac{u_1(r^n - 1)}{r-1}, \; r \neq 1 \qquad S_\infty = \frac{u_1}{1-r}, \; |r|<1$$
Example
AP: 3, 7, 11, … → d = 4, u₁₀ = 3 + 9·4 = 39
GP: 2, 6, 18, … → r = 3, u₅ = 2·3⁴ = 162
Key: identify whether common difference (AP) or common ratio (GP). For S∞, require |r| < 1.
Topic 2 · Functions
Transformations & Inverse Functions
$$y = af(b(x-h)) + k$$
a = vertical stretch, b = horizontal compression, h = horizontal shift, k = vertical shift.
$$f^{-1}: \text{swap } x \text{ and } y, \text{ then solve for } y$$
Example
If f(x) = 2x + 3, then f⁻¹(x): x = 2y + 3 → y = (x−3)/2, so f⁻¹(x) = (x−3)/2
Topic 3 · Trigonometry
Exact Values, Identities & Sine/Cosine Rules
$$\sin^2\theta + \cos^2\theta = 1 \qquad \tan\theta = \frac{\sin\theta}{\cos\theta}$$
$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \qquad c^2 = a^2 + b^2 - 2ab\cos C$$
Exact Values to Memorise
sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3
sin 45° = cos 45° = 1/√2, tan 45° = 1
sin 60° = √3/2, cos 60° = ½, tan 60° = √3
Topic 2 · Functions
Exponential & Logarithm Laws
$$\log_b(xy) = \log_b x + \log_b y \qquad \log_b\!\left(\frac{x}{y}\right) = \log_b x - \log_b y$$
$$\log_b(x^n) = n\log_b x \qquad b^x = e^{x\ln b} \qquad \log_b x = \frac{\ln x}{\ln b}$$
Example
Solve 3ˣ = 81 → 3ˣ = 3⁴ → x = 4
Solve log₂(x) = 5 → x = 2⁵ = 32
Topic 5 · Calculus
Differentiation Rules
$$\frac{d}{dx}(x^n) = nx^{n-1} \qquad \frac{d}{dx}(e^x) = e^x \qquad \frac{d}{dx}(\ln x) = \frac{1}{x}$$
$$\text{Chain: } \frac{d}{dx}[f(g(x))] = f'(g(x))\cdot g'(x)$$
$$\text{Product: } (uv)' = u'v + uv' \qquad \text{Quotient: } \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$$
Example
f(x) = x³ − 4x → f'(x) = 3x² − 4 → stationary pts: 3x² = 4 → x = ±2/√3
Topic 5 · Calculus
Integration
$$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1) \qquad \int e^x\,dx = e^x + C$$
$$\int_a^b f(x)\,dx = [F(x)]_a^b = F(b) - F(a)$$
Example
∫(3x² + 2) dx = x³ + 2x + C
∫₀² x² dx = [x³/3]₀² = 8/3 − 0 = 8/3
Topic 4 · Statistics & Probability
Probability & Distributions
$$P(A\cup B) = P(A) + P(B) - P(A\cap B)$$
$$X \sim B(n,p): \quad P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}$$
$$X \sim N(\mu, \sigma^2): \quad Z = \frac{X - \mu}{\sigma}$$
Example
X ~ B(5, 0.4): P(X=2) = C(5,2)·0.4²·0.6³ = 10·0.16·0.216 = 0.3456
Topic 3 · Geometry & Trigonometry
Vectors
$$\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_1b_1 + a_2b_2 + a_3b_3$$
$$|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2} \qquad \text{Line: } \vec{r} = \vec{a} + t\vec{b}$$
Example
a = (2, 1, −1), b = (1, 3, 2): a·b = 2+3−2 = 3
|a| = √(4+1+1) = √6; cos θ = 3/(√6·√14)