Mathematical formulae and identities

Based on Section 5d of the OCR MEI B (H640) specification.

Formulae you must know (not provided in exams)

Quadratic equations

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

Laws of indices

Product
$a^x a^y = a^{x+y}$
Quotient
$a^x \div a^y = a^{x-y}$
Power of a power
$(a^x)^y = a^{xy}$

Laws of logarithms

Definition
$x = a^n \iff n = \log_a x$
$a>0, x>0$
Product
$\log_a x + \log_a y = \log_a(xy)$
Quotient
$\log_a x - \log_a y = \log_a\left(\frac{x}{y}\right)$
Power
$k\log_a x = \log_a(x^k)$

Coordinate geometry

Line through $(x_1, y_1)$ with gradient $m$
$y - y_1 = m(x - x_1)$
Perpendicular lines
$m_1 m_2 = -1$

Sequences

Arithmetic nth term
$u_n = a + (n-1)d$
Geometric nth term
$u_n = a r^{n-1}$

Trigonometry

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 of triangle
$\text{Area} = \tfrac{1}{2}ab\sin C$
Pythagorean identity
$\cos^2 A + \sin^2 A \equiv 1$
Secant identity
$\sec^2 A \equiv 1 + \tan^2 A$
Cosecant identity
$\operatorname{cosec}^2 A \equiv 1 + \cot^2 A$
Double angle (sin)
$\sin 2A \equiv 2\sin A\cos A$
Double angle (cos)
$\cos 2A \equiv \cos^2 A - \sin^2 A$
Double angle (tan)
$\tan 2A \equiv \frac{2\tan A}{1 - \tan^2 A}$

Mechanics

Weight
$W = mg$
Friction
$F \le \mu R$
Newton's Second Law
$F = ma$
Variable acceleration
$v = \frac{dr}{dt},\ a = \frac{dv}{dt} = \frac{d^2r}{dt^2}$
Integrate
$r = \int v\,dt,\ v = \int a\,dt$

Statistics

Mean
$\bar{x} = \frac{\sum x}{n} = \frac{\sum fx}{\sum f}$
Standard Normal
$Z = \frac{X - \mu}{\sigma}$ , $X \sim N(\mu, \sigma^2)$

Formulae provided in exams

Arithmetic series

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

Geometric series

Sum
$S_n = \frac{a(1-r^n)}{1-r}$
Sum to infinity
$S_\infty = \frac{a}{1-r}$
$|r| < 1$

Binomial series

Expansion
$(a+b)^n = a^n + {}^nC_1 a^{n-1}b + \dots + {}^nC_r a^{n-r}b^r + \dots + b^n$
$n \in \mathbb{N}$
General power
$(1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \dots$
$|x| < 1$ , $n \in \mathbb{R}$

Differentiation

Quotient rule
$y = \frac{u}{v},\ \frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$
First principles
$f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$

Integration

Log form
$\int\frac{f'(x)}{f(x)}\,dx = \ln|f(x)| + c$
Power form
$\int f'(x)(f(x))^n\,dx = \frac{(f(x))^{n+1}}{n+1} + c$

Small angles

Approximations (radians)
$\sin\theta \approx \theta,\ \cos\theta \approx 1 - \tfrac{1}{2}\theta^2,\ \tan\theta \approx \theta$

Compound angles

sin
$\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$
cos
$\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$
tan
$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$

Numerical methods

Trapezium rule
$\int_a^b y\,dx \approx \tfrac{1}{2}h[(y_0+y_n) + 2(y_1+\dots+y_{n-1})]$
$h = (b-a)/n$
Newton–Raphson
$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

Probability

Addition
$P(A \cup B) = P(A) + P(B) - P(A \cap B)$
Multiplication
$P(A \cap B) = P(A)P(B|A)$
Conditional
$P(A|B) = \frac{P(A \cap B)}{P(B)}$

Sample variance

Variance
$s^2 = \frac{1}{n-1}S_{xx}$
$S_{xx} = \sum(x_i - \bar{x})^2$

Binomial distribution

Probability
$P(X=r) = {}^nC_r p^r q^{n-r}$
$q = 1-p$
Mean
$E(X) = np$

Normal sample mean

Distribution
$\bar{X} \sim N\!\left(\mu, \frac{\sigma^2}{n}\right)$
Standardised
$\frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \sim N(0,1)$

Kinematics

SUVAT 1D
$v = u + at,\ s = ut + \tfrac{1}{2}at^2$
SUVAT 1D
$s = \tfrac{1}{2}(u+v)t,\ v^2 = u^2 + 2as$
SUVAT 1D
$s = vt - \tfrac{1}{2}at^2$
SUVAT 2D
$\mathbf{v} = \mathbf{u} + \mathbf{a}t,\ \mathbf{s} = \mathbf{u}t + \tfrac{1}{2}\mathbf{a}t^2$