A Level Maths formulas, unit by unit

P1 · Pure 1

Algebra · Coordinate · Trig · Diff · Int · no calculator

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: $\dfrac{1}{2}ab\sin C$
• Radians: $\pi \text{ rad} = 180^\circ$
• Arc: $s = r\theta$  ·  Sector: $A = \tfrac{1}{2}r^2\theta$ ($\theta$ in rad)

Diff & Int

• Power: $\dfrac{d}{dx}x^n = nx^{n-1}$
• Power (int): $\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C$ ($n \neq -1$)
• Normal: gradient $-\dfrac{1}{f'(x_0)}$

P2 · Pure 2

Algebra · Sequences · Binomial · Logs · Circles · Trig · Calculus

Algebra

• Factor theorem: $f(a) = 0 \Leftrightarrow (x - a)$ is a factor
• Remainder theorem: remainder on dividing by $(x - a)$ is $f(a)$

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 $|r| < 1$
• Binomial ($n \in \mathbb{Z}^+$): $(a + b)^n = \sum_{r=0}^{n} {}^nC_r\, a^{n-r}b^r$,   ${}^nC_r = \dfrac{n!}{r!(n-r)!}$

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

Trig identities

• $\sin^2\theta + \cos^2\theta = 1$
• $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$

Calculus

• Stationary point: $f'(x) = 0$; $f''(x) > 0$ min, $f''(x) < 0$ max
• Area: $\int_a^b y\,dx$
• Trapezium rule: $\int_a^b y\,dx \approx \tfrac{h}{2}\big[y_0 + y_n + 2(y_1 + \cdots + y_{n-1})\big]$, $h = \dfrac{b-a}{n}$

P3 · Pure 3

Functions · Trig · Exp/Log · Diff/Int · Numerical

Trigonometry

• $\sec^2\theta = 1 + \tan^2\theta$  ·  $\operatorname{cosec}^2\theta = 1 + \cot^2\theta$
• $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$
• $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$
• $\tan(A \pm B) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$
• $\sin 2A = 2\sin A \cos A$; $\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A$
• $R$-form: $a\sin x + b\cos x = R\sin(x + \alpha)$, $R = \sqrt{a^2 + b^2}$, $\tan\alpha = \dfrac{b}{a}$

Numerical Methods

• Sign change: If $f$ continuous and $f(a)f(b) < 0$, then a root lies in $(a, b)$.
• Iteration: $x_{n+1} = g(x_n)$ from a rearrangement $x = g(x)$

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

• $\int \dfrac{1}{x}\,dx = \ln|x| + C$  ·  $\int e^x\,dx = e^x + C$
• $\int \sin x\,dx = -\cos x + C$  ·  $\int \cos x\,dx = \sin x + C$
• $f(ax+b)$: $\int f(ax + b)\,dx = \dfrac{1}{a}F(ax + b) + C$

P4 · Pure 4

Proof · Partial · Binomial · Parametric · Calculus · DEs · Vectors

Proof & Binomial

• Contradiction: assume the statement is false and derive a contradiction
• Binomial ($n$ rational): $(1 + x)^n = 1 + nx + \dfrac{n(n-1)}{2!}x^2 + \cdots$,   $|x| < 1$

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: $\dfrac{A}{x-a} + \dfrac{B}{(x-a)^2}$
• Improper fraction: divide first

Parametric & implicit

• $\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}$
• Implicit: $\dfrac{d}{dx}(y^n) = ny^{n-1}\dfrac{dy}{dx}$

Integration

• By parts: $\int u\dfrac{dv}{dx}\,dx = uv - \int v\dfrac{du}{dx}\,dx$
• $\int \dfrac{f'(x)}{f(x)}\,dx = \ln|f(x)| + C$
• Volume ($x$-axis): $V = \pi\int_a^b y^2\,dx$

Differential Equations

• Separable: $\int \dfrac{1}{f(y)}\,dy = \int g(x)\,dx$