Mathematical Formulae - A-Levels Maths

5d. Mathematical formulae and identities

Learners must be able to use the following formulae and identities for A Level mathematics, without these formulae and identities being provided, either in these forms or in equivalent forms. These formulae and identities may only be provided where they are the starting point for a proof or as a result to be proved.

Pure Mathematics

Quadratic Equations

Formula
$ax^2 + bx + c = 0$ has roots $\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Laws of Indices

Formula
$a^x a^y = a^{x+y}$
$a^x \div a^y = a^{x-y}$
$(a^x)^y = a^{xy}$

Laws of Logarithms

Formula
$x = a^y \Leftrightarrow y = \log_a x$ , for $a > 0$ and $x > 0$
$\log_a x + \log_a y = \log_a(xy)$
$\log_a x - \log_a y = \log_a\left(\frac{x}{y}\right)$
$k\log_a x = \log_a(x^k)$

Coordinate Geometry

Formula
A straight line graph, gradient $m$ passing through $(x_1, y_1)$ has equation $y - y_1 = m(x - x_1)$
Straight lines with gradients $m_1$ and $m_2$ are perpendicular when $m_1m_2 = -1$

Sequences

Formula
General term of an arithmetic progression: $u_n = a + (n - 1)d$
General term of a geometric progression: $u_n = ar^{n-1}$

Trigonometry

Formula
In the triangle $ABC$
Sine rule: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
Cosine rule: $a^2 = b^2 + c^2 - 2bc\cos A$
Area $= \frac{1}{2}ab\sin C$
$\cos^2 A + \sin^2 A = 1$
$\sec^2 A = 1 + \tan^2 A$
$\cosec^2 A = 1 + \cot^2 A$
$\sin 2A = 2\sin A\cos A$
$\cos 2A = \cos^2 A - \sin^2 A$
$\tan 2A = \frac{2\tan A}{1 - \tan^2 A}$

Mensuration

Formula
Circumference and Area of circle, radius $r$ and diameter $d$ : $C = 2\pi r = \pi d$ , $A = \pi r^2$
Pythagoras' Theorem: In any right-angled triangle where $a$ , $b$ and $c$ are the lengths of the sides and $c$ is the hypotenuse: $c^2 = a^2 + b^2$
Area of a trapezium $= \frac{1}{2}(a + b)h$ , where $a$ and $b$ are the lengths of the parallel sides and $h$ is their perpendicular separation.
Volume of a prism $=$ area of cross section $\times$ length
For a circle of radius $r$ , where an angle at the centre of $\theta$ radians subtends an arc of length $s$ and encloses an associated sector of area $A$ : $s = r\theta$ , $A = \frac{1}{2}r^2\theta$

Differentiation

Function	Derivative
$x^n$	$nx^{n-1}$
$\sin kx$	$k\cos kx$
$\cos kx$	$-k\sin kx$
$e^{kx}$	$ke^{kx}$
$\ln x$	$\frac{1}{x}$
$f(x) + g(x)$	$f'(x) + g'(x)$
$f(x)g(x)$	$f'(x)g(x) + f(x)g'(x)$
$f(g(x))$	$f'(g(x))g'(x)$

Integration

Function	Integral
$x^n$	$\frac{1}{n + 1}x^{n+1} + c$ , $n \ne -1$
$\cos kx$	$\frac{1}{k}\sin kx + c$
$\sin kx$	$-\frac{1}{k}\cos kx + c$
$e^{kx}$	$\frac{1}{k}e^{kx} + c$
$\frac{1}{x}$	$\ln\|x\| + c$ , $x \ne 0$
$f'(x) + g'(x)$	$f(x) + g(x) + c$
$f'(g(x))g'(x)$	$f(g(x)) + c$

Area Under a Curve

Formula
Area under a curve $= \int_a^b y\,dx$ $(y \ge 0)$

Vectors

Formula
$\|xi + yj\| = \sqrt{x^2 + y^2}$
$\|xi + yj + zk\| = \sqrt{x^2 + y^2 + z^2}$

Mechanics - Forces and Equilibrium

Formula
Weight $=$ mass $\times\,g$
Friction: $F \le \mu R$
Newton's second law in the form: $F = ma$

Mechanics - Kinematics

Formula
For motion in a straight line with variable acceleration:
$\mathbf{v} = \frac{d\mathbf{r}}{dt}$ , $\mathbf{a} = \frac{d\mathbf{v}}{dt} = \frac{d^2\mathbf{r}}{dt^2}$
$\mathbf{r} = \int \mathbf{v}\,dt$ , $\mathbf{v} = \int \mathbf{a}\,dt$
$v = \frac{ds}{dt}$ , $a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$
$s = \int v\,dt$ , $v = \int a\,dt$

Statistics

Formula
The mean of a set of data: $\bar{x} = \frac{\sum x}{n} = \frac{\sum fx}{\sum f}$
The standard Normal variable: $Z = \frac{X - \mu}{\sigma}$ where $X \sim N(\mu, \sigma^2)$

Formulae A Level Mathematics A (H240)

Learners will be given the following formulae sheet in each question paper.

Arithmetic Series

Formula
$S_n = \frac{1}{2}n(a + l) = \frac{1}{2}n\{2a + (n - 1)d\}$

Geometric Series

Formula
$S_n = \frac{a(1 - r^n)}{1 - r}$
$S_\infty = \frac{a}{1 - r}$ for $\|r\| < 1$

Binomial Series

Formula
$(a + b)^n = a^n + {^nC_1}a^{n-1}b + {^nC_2}a^{n-2}b^2 + ... + {^nC_r}a^{n-r}b^r + ... + b^n$ $(n \in \mathbb{N})$
where $^nC_r = {}_nC_r = \binom{n}{r} = \frac{n!}{r!(n-r)!}$
$(1 + x)^n = 1 + nx + \frac{n(n - 1)}{2!}x^2 + ... + \frac{n(n - 1)...(n - r + 1)}{r!}x^r + ...$ $(\|x\| < 1,\ n \in \mathbb{R})$

Differentiation

$f(x)$	$f'(x)$
$\tan kx$	$k\sec^2 kx$
$\sec x$	$\sec x \tan x$
$\cot x$	$-\cosec^2 x$
$\cosec x$	$-\cosec x \cot x$

Quotient Rule

Formula
$y = \frac{u}{v}$ , $\frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$

Differentiation from First Principles

Formula
$f'(x) = \lim_{h \to 0}\frac{f(x + h) - f(x)}{h}$

Integration

Formula
$\int \frac{f'(x)}{f(x)}\,dx = \ln\|f(x)\| + c$
$\int f'(x)(f(x))^n\,dx = \frac{1}{n + 1}(f(x))^{n+1} + c$
Integration by parts: $\int u\frac{dv}{dx}\,dx = uv - \int v\frac{du}{dx}\,dx$

Small Angle Approximations

Formula
$\sin\theta \approx \theta$ , $\cos\theta \approx 1 - \frac{1}{2}\theta^2$ , $\tan\theta \approx \theta$ where $\theta$ is measured in radians

Trigonometric Identities

Formula
$\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$
$\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$
$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$ $(A \pm B \ne (k + \frac{1}{2})\pi)$

Numerical Methods

Formula
Trapezium rule: $\int_a^b y\,dx \approx \frac{1}{2}h\{(y_0 + y_n) + 2(y_1 + y_2 + ... + y_{n-1})\}$ , where $h = \frac{b - a}{n}$
The Newton-Raphson iteration for solving $f(x) = 0$ : $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

Probability

Formula
$\mathrm{P}(A \cup B) = \mathrm{P}(A) + \mathrm{P}(B) - \mathrm{P}(A \cap B)$
$\mathrm{P}(A \cap B) = \mathrm{P}(A)\mathrm{P}(B\|A) = \mathrm{P}(B)\mathrm{P}(A\|B)$ or $\mathrm{P}(A\|B) = \frac{\mathrm{P}(A \cap B)}{\mathrm{P}(B)}$

Standard Deviation

Formula
$\sqrt{\frac{\sum (x - \bar{x})^2}{n}} = \sqrt{\frac{\sum x^2}{n} - \bar{x}^2}$ or $\sqrt{\frac{\sum f(x - \bar{x})^2}{\sum f}} = \sqrt{\frac{\sum fx^2}{\sum f} - \bar{x}^2}$

The Binomial Distribution

Formula
If $X \sim B(n, p)$ then $\mathrm{P}(X = x) = \binom{n}{x}p^x(1 - p)^{n-x}$ , Mean of $X$ is $np$ , Variance of $X$ is $np(1-p)$

Hypothesis Test for the Mean of a Normal Distribution

Formula
If $X \sim N(\mu, \sigma^2)$ then $\bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right)$ and $\frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \sim N(0, 1)$

Percentage Points of the Normal Distribution

$p$	0.75	0.90	0.95	0.975	0.99	0.995	0.9975	0.999	0.9995
$z$	0.674	1.282	1.645	1.960	2.326	2.576	2.807	3.090	3.291

Kinematics

Motion in a straight line	Motion in two dimensions
$v = u + at$	$\mathbf{v} = \mathbf{u} + \mathbf{a}t$
$s = ut + \frac{1}{2}at^2$	$\mathbf{s} = \mathbf{u}t + \frac{1}{2}\mathbf{a}t^2$
$s = \frac{1}{2}(u + v)t$	$\mathbf{s} = \frac{1}{2}(\mathbf{u} + \mathbf{v})t$
$v^2 = u^2 + 2as$
$s = vt - \frac{1}{2}at^2$	$\mathbf{s} = \mathbf{v}t - \frac{1}{2}\mathbf{a}t^2$