P1 · Pure 1
Algebra · Coordinate · Trig · Diff · Int
Algebra
- Quadratic: $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
- Discriminant: $\Delta = b^2 - 4ac$
- Surds: $\sqrt{a}\,\sqrt{b} = \sqrt{ab}$; $\dfrac{1}{\sqrt{a}} = \dfrac{\sqrt{a}}{a}$
Coordinate Geometry
- Gradient: $m = \dfrac{y_2 - y_1}{x_2 - x_1}$
- Distance: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
- Midpoint: $\left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2}\right)$
- Perp: $m_1 m_2 = -1$
Trigonometry
- Sine rule: $\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$
- Cosine rule: $a^2 = b^2 + c^2 - 2bc\cos A$
- Area: $\dfrac{1}{2}ab\sin C$
Diff & Int
- Power: $\dfrac{d}{dx}x^n = nx^{n-1}$
- Power (int): $\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C$ ($n \neq -1$)
P2 · Pure 2
Sequences · Logs · Circles · Trig
Sequences & Series
- AP $S_n$: $S_n = \dfrac{n}{2}[2a + (n-1)d]$
- GP $S_n$: $S_n = \dfrac{a(1 - r^n)}{1 - r}$
- GP $S_\infty$: $\dfrac{a}{1 - r}$ for $|r| < 1$
Logs & Exponentials
- Product: $\log(xy) = \log x + \log y$
- Quotient: $\log\left(\dfrac{x}{y}\right) = \log x - \log y$
- Power: $\log(x^n) = n\log x$
- Change base: $\log_a x = \dfrac{\log_b x}{\log_b a}$
Circles
- Std form: $(x-a)^2 + (y-b)^2 = r^2$
- Tangent: perpendicular to the radius at point of contact
- Arc: $s = r\theta$ · Sector: $A = \tfrac{1}{2}r^2\theta$ ($\theta$ in rad)
Trig identities
- $\sin^2\theta + \cos^2\theta = 1$
- $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$
P3 · Pure 3
Functions · Numerical · Further Diff/Int
Numerical Methods
- Newton–Raphson: $x_{n+1} = x_n - \dfrac{f(x_n)}{f'(x_n)}$
- Sign change: If $f$ continuous and $f(a)f(b) < 0$, then a root lies in $(a, b)$.
Standard derivatives
- $\dfrac{d}{dx}e^x = e^x$ · $\dfrac{d}{dx}\ln x = \dfrac{1}{x}$
- $\dfrac{d}{dx}\sin x = \cos x$ · $\dfrac{d}{dx}\cos x = -\sin x$ · $\dfrac{d}{dx}\tan x = \sec^2 x$
Chain / Product / Quotient
- Chain: $\dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx}$
- Product: $(uv)' = u'v + uv'$
- Quotient: $\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}$
Standard integrals
- $\int \dfrac{1}{x}\,dx = \ln|x| + C$ · $\int e^x\,dx = e^x + C$
- By parts: $\int u\,dv = uv - \int v\,du$
P4 · Pure 4
Vectors · Partial · Parametric · DEs
Vectors
- Magnitude: $|\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}$
- Dot product: $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta$
- Line eqn: $\mathbf{r} = \mathbf{a} + t\mathbf{d}$
Partial fractions
- $\dfrac{f(x)}{(x-a)(x-b)} = \dfrac{A}{x-a} + \dfrac{B}{x-b}$
- Repeated linear factor: $\dfrac{A}{x-a} + \dfrac{B}{(x-a)^2}$
- Irreducible quadratic: $\dfrac{Bx+C}{x^2+px+q}$
Parametric
- $\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}$
Differential Equations
- Separable: $\int \dfrac{1}{f(y)}\,dy = \int g(x)\,dx$
S1 · Statistics 1
Data · Prob · Corr · DRV · Normal
Descriptive Statistics
- Mean: $\bar{x} = \dfrac{\sum x_i}{n}$
- SD: $\sigma = \sqrt{\dfrac{\sum(x_i - \bar{x})^2}{n}}$
- Q1–Q3: IQR = $Q_3 - Q_1$
Probability
- $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
- Indep: $P(A \cap B) = P(A)P(B)$
- Cond: $P(A|B) = \dfrac{P(A \cap B)}{P(B)}$
Correlation & Regression
- PMCC: $r = \dfrac{S_{xy}}{\sqrt{S_{xx} S_{yy}}}$
- Line: $y = a + bx$, $b = \dfrac{S_{xy}}{S_{xx}}$, $a = \bar{y} - b\bar{x}$
Random Variables
- $E(X)$: $\sum x P(X = x)$
- $\text{Var}(X)$: $E(X^2) - [E(X)]^2$
- Normal: $X \sim N(\mu, \sigma^2)$; $Z = \dfrac{X - \mu}{\sigma}$
M1 · Mechanics 1
Kinematics · Newton · Moments · Vec
Kinematics (SUVAT)
- $v = u + at$
- $s = ut + \tfrac{1}{2}at^2$
- $v^2 = u^2 + 2as$
- $s = \tfrac{1}{2}(u + v)t$
Newton's Laws
- Second law: $F = ma$ (net force)
- Weight: $W = mg$, $g = 9.8\text{ m s}^{-2}$
- Friction: $F \le \mu R$, equality on point of slipping
- Inclined plane: along-slope $= mg\sin\theta$; normal $= mg\cos\theta$
Moments
- Moment: $M = F \times d$ (perpendicular distance)
- Equilibrium: $\sum F = 0$ and $\sum M = 0$
- Uniform rod: weight acts at midpoint
UK · Extra Integration Forms
Pure Y2 · used across Papers 1 & 2
Logarithmic integrals
- $\displaystyle\int \dfrac{f'(x)}{f(x)}\,dx = \ln|f(x)| + C$
- $\displaystyle\int \tan x\,dx = -\ln|\cos x| + C = \ln|\sec x| + C$
- $\displaystyle\int \sec x\,dx = \ln|\sec x + \tan x| + C$
- $\displaystyle\int \cot x\,dx = \ln|\sin x| + C$
Trig identities (Y2)
- $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$
- $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$
- $\sin 2A = 2\sin A \cos A$; $\cos 2A = \cos^2 A - \sin^2 A$
- $R\sin(x + \alpha) = a\sin x + b\cos x$ with $R = \sqrt{a^2 + b^2}$, $\tan\alpha = b/a$
Hypothesis Testing
- Binomial test: $X \sim B(n, p)$; $H_0: p = p_0$ vs $H_1: p \neq p_0$ / $> p_0$ / $< p_0$
- Type I: $P(\text{reject } H_0 \mid H_0 \text{ true})$
- Normal approx: $X \sim B(n,p) \approx N(np, np(1-p))$ (with continuity correction $\pm 0.5$)
UK · Proof Techniques
Y1 · often 1st question on Paper 1
Common structures
- Direct: assume premise, derive conclusion via known facts.
- Contradiction: assume the conclusion is false; derive a contradiction.
- Contrapositive: to prove $P \Rightarrow Q$, prove $\neg Q \Rightarrow \neg P$.
- Counterexample: one instance where a claim fails is enough to disprove.
Useful facts
- $n$ even $\Leftrightarrow n = 2k$ for some integer $k$; $n$ odd $\Leftrightarrow n = 2k+1$.
- Product of any $n$ consecutive integers is divisible by $n!$.
- Every integer $> 1$ has a unique prime factorisation (FTA).