← Hub
Edexcel · Mathematics
● Ready
These notes are AI-assisted study material. Always cross-check against the official Pearson Edexcel spec or your teacher before relying on them in an exam.

P1 — Pure Mathematics 1

Unit WMA11 · Edexcel International A Level Mathematics (2018)

1Algebra & Functions

Indices, surds, expansion, factorisation.

Laws of indices

For any real numbers $a>0$ and rationals $m,n$: $a^m a^n = a^{m+n}$, $\dfrac{a^m}{a^n} = a^{m-n}$, $(a^m)^n = a^{mn}$, $a^0 = 1$, $a^{-n} = \dfrac{1}{a^n}$, $a^{1/n} = \sqrt[n]{a}$.

Surds

A surd is an irrational root such as $\sqrt{2}$. Simplify by extracting square factors: $\sqrt{50} = 5\sqrt{2}$. Rationalise the denominator by multiplying top and bottom by the conjugate: $\dfrac{1}{a+\sqrt{b}} = \dfrac{a-\sqrt{b}}{a^2 - b}$.

Expanding brackets and factorising

Expand using distributivity; factorise by pulling out common factors or grouping. Recognise: $a^2 - b^2 = (a-b)(a+b)$; $a^2 \pm 2ab + b^2 = (a \pm b)^2$.

2Quadratics

Solving, completing the square, discriminant, sketching.

Quadratic formula

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

Completing the square

Rewrite $ax^2 + bx + c$ as $a(x + p)^2 + q$ where $p = \dfrac{b}{2a}$ and $q = c - ap^2$. Useful for finding the vertex $(-p,\; q)$ and solving without the formula.

Discriminant

$\Delta = b^2 - 4ac$. If $\Delta > 0$: two real roots. If $\Delta = 0$: one repeated root. If $\Delta < 0$: no real roots (curve doesn't cross the x-axis).

3Equations & Inequalities

Simultaneous equations, linear and quadratic inequalities.

Simultaneous equations

Two linear equations — solve by elimination or substitution. One linear, one quadratic — substitute the linear into the quadratic and solve.

Linear inequalities

Solve like equations, but flip the inequality sign when multiplying/dividing by a negative number.

Quadratic inequalities

Factorise, sketch the parabola, and read off which region satisfies the inequality. E.g. $x^2 - 5x + 6 > 0 \;\Rightarrow\; (x-2)(x-3) > 0 \;\Rightarrow\; x < 2$ or $x > 3$.

4Graphs & Transformations

Sketching cubics, reciprocals, and applying transformations.

Standard shapes

Cubic $y = ax^3 + \ldots$: 1-2 turning points, opposite end behaviours. Reciprocal $y = \dfrac{k}{x}$: two branches, asymptotes at $x=0$ and $y=0$. Modulus $y = |f(x)|$: reflect any negative $y$-values in the x-axis.

Transformations of $y=f(x)$

FormEffect
$y = f(x) + a$translate by $\binom{0}{a}$
$y = f(x + a)$translate by $\binom{-a}{0}$
$y = af(x)$stretch parallel to y-axis by factor $a$
$y = f(ax)$stretch parallel to x-axis by factor $\tfrac{1}{a}$
$y = -f(x)$reflect in x-axis
$y = f(-x)$reflect in y-axis

5Coordinate Geometry

Straight lines: gradient, equation, distance, midpoint.

Gradient and equation

Gradient between $(x_1,y_1)$ and $(x_2,y_2)$: $m = \dfrac{y_2 - y_1}{x_2 - x_1}$. Point-gradient form: $y - y_1 = m(x - x_1)$. Standard form: $ax + by + c = 0$.

Parallel and perpendicular

Two lines are parallel iff their gradients are equal. Perpendicular iff $m_1 m_2 = -1$.

Distance and midpoint

Distance: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. Midpoint: $\left( \dfrac{x_1 + x_2}{2},\; \dfrac{y_1 + y_2}{2} \right)$.

6Circles

Equation of a circle, tangent, chord.

Equation

Circle centre $(a,b)$ radius $r$$$(x - a)^2 + (y - b)^2 = r^2$$

Expanded form $x^2 + y^2 + 2gx + 2fy + c = 0$ has centre $(-g, -f)$ and radius $\sqrt{g^2 + f^2 - c}$.

Tangent and chord properties

  • The radius drawn to a tangent point is perpendicular to the tangent.
  • The perpendicular from the centre to a chord bisects the chord.
  • An angle in a semicircle is a right angle (Thales).

7Algebraic Methods

Polynomial division, factor theorem, remainder theorem.

Polynomial (long) division

Divide $P(x) = (\text{divisor})(\text{quotient}) + \text{remainder}$. Set up like numeric long division; keep like powers aligned.

Factor theorem

If $P(a) = 0$ then $(x - a)$ is a factor of $P(x)$. Use to find one root, then divide down.

Remainder theorem

The remainder when $P(x)$ is divided by $(x - a)$ is $P(a)$.

8Binomial Expansion

Expanding $(a + b)^n$ for positive integer $n$.
Binomial theorem$$(a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^{r} \quad \text{where} \quad \binom{n}{r} = \dfrac{n!}{r!(n-r)!}$$

The coefficients come from Pascal's triangle. For $(1 + x)^n$: $1 + nx + \dfrac{n(n-1)}{2!}x^2 + \dfrac{n(n-1)(n-2)}{3!}x^3 + \ldots$

9Trigonometry

Ratios, graphs, identities, equations.

Definitions and identities

In a right triangle: $\sin\theta = \dfrac{\text{opp}}{\text{hyp}}$, $\cos\theta = \dfrac{\text{adj}}{\text{hyp}}$, $\tan\theta = \dfrac{\text{opp}}{\text{adj}}$.

Core identities$$\sin^2\theta + \cos^2\theta = 1 \qquad \tan\theta = \dfrac{\sin\theta}{\cos\theta}$$

Sine and cosine rules

$$\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C} \qquad a^2 = b^2 + c^2 - 2bc\cos A \qquad \text{Area} = \tfrac{1}{2}ab\sin C$$

Solving trig equations

Find all solutions in the given interval by using the principal value then adding/subtracting periods ($2\pi$ or $\pi$) or using symmetry (CAST diagram).

10Differentiation

Derivatives from first principles, power rule, tangents and normals.

First principles

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

Power rule

$\dfrac{d}{dx}(x^n) = nx^{n-1}$. Linearity: $\dfrac{d}{dx}(af + bg) = af' + bg'$.

Tangents, normals, stationary points

Gradient of curve at point = value of derivative at that point. Tangent has same gradient; normal has gradient $-1/f'(x_0)$. Stationary point: $f'(x) = 0$; second derivative or sign analysis classifies max/min/inflection.

11Integration

Indefinite and definite integrals of polynomials.

Reversing differentiation

$$\int x^n \, dx = \dfrac{x^{n+1}}{n+1} + C \qquad (n \ne -1)$$

Definite integrals and area

$\displaystyle \int_a^b f(x)\,dx = F(b) - F(a)$, the (signed) area between the curve and x-axis from $x=a$ to $x=b$. Area below the axis counts as negative.

12Vectors (2D)

Component form, magnitude, unit vectors, position vectors.

A 2D vector $\mathbf{a} = \binom{a_1}{a_2}$ has magnitude $|\mathbf{a}| = \sqrt{a_1^2 + a_2^2}$. Unit vector: $\hat{\mathbf{a}} = \dfrac{1}{|\mathbf{a}|}\mathbf{a}$. Position vector of $P$ from origin: $\overrightarrow{OP} = \mathbf{p}$. Vector from $A$ to $B$: $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$.

These notes are AI-assisted study material. Always cross-check against the official Pearson Edexcel spec or your teacher before relying on them in an exam.

P2 — Pure Mathematics 2

Unit WMA12 · Edexcel International A Level Mathematics (2018)

1Algebra & Functions

Partial fractions, algebraic division extended.

Partial fractions

Split a rational function into simpler pieces. For distinct linear factors: $\dfrac{P(x)}{(x-a)(x-b)} = \dfrac{A}{x-a} + \dfrac{B}{x-b}$. For repeated: include $\dfrac{C}{(x-a)^2}$. Multiply and equate coefficients (or substitute convenient values) to find $A,B,C$.

Improper fractions

If degree of numerator $\ge$ degree of denominator, divide first to get a polynomial plus a proper fractional remainder before splitting.

2Coordinate Geometry

Extending straight-line and circle work.

Locus problems: describe the set of points satisfying a geometric condition, e.g. equidistance from two points $\Rightarrow$ perpendicular bisector. Use $x, y$ algebra to derive the locus equation.

3Sequences & Series

Arithmetic and geometric.

Arithmetic progression

$$u_n = a + (n-1)d \qquad S_n = \dfrac{n}{2}[2a + (n-1)d]$$

Geometric progression

$$u_n = ar^{n-1} \qquad S_n = \dfrac{a(1 - r^n)}{1 - r} \qquad S_\infty = \dfrac{a}{1 - r} \; (|r| < 1)$$

Recurrence relations: $u_{n+1} = f(u_n)$. Iterate to find terms.

4Binomial Expansion

Beyond positive integers.

Still positive-integer expansions in P2. Extended to any rational $n$ (with $|x|<1$) in P4.

5Trigonometry

Radians, arc length, sector area, general solutions.

Radian measure

Convert $\theta_\text{rad} = \theta_\text{deg} \times \dfrac{\pi}{180}$. Arc length $s = r\theta$; sector area $A = \tfrac{1}{2}r^2\theta$ (with $\theta$ in radians).

Solving equations over given intervals

Rewrite using identities where needed, find all solutions in the interval; check for extraneous ones after squaring or dividing by trig functions.

6Exponentials & Logarithms

Growth models, laws of logs, natural log $\ln$ and $e^x$.

Laws of logs

$$\log_a xy = \log_a x + \log_a y \quad \log_a \tfrac{x}{y} = \log_a x - \log_a y \quad \log_a x^n = n\log_a x$$

Change of base: $\log_a x = \dfrac{\log_b x}{\log_b a}$. Special: $\log_a 1 = 0$, $\log_a a = 1$.

Solving exponential equations

Take logs both sides, apply the power law. E.g. $3^x = 20 \Rightarrow x = \dfrac{\ln 20}{\ln 3}$.

7Differentiation

Chain, product, quotient rules; trig, exponential, log.
Standard derivatives$$\dfrac{d}{dx}(e^x) = e^x \qquad \dfrac{d}{dx}(\ln x) = \dfrac{1}{x} \qquad \dfrac{d}{dx}(\sin x) = \cos x \qquad \dfrac{d}{dx}(\cos x) = -\sin x \qquad \dfrac{d}{dx}(\tan x) = \sec^2 x$$
Rules$$\text{Product: } (uv)' = u'v + uv' \qquad \text{Quotient: } \left( \dfrac{u}{v} \right)' = \dfrac{u'v - uv'}{v^2} \qquad \text{Chain: } \dfrac{dy}{dx} = \dfrac{dy}{du} \cdot \dfrac{du}{dx}$$

8Integration

Standard integrals, substitution basics.
Standard integrals$$\int e^x \, dx = e^x + C \qquad \int \dfrac{1}{x} \, dx = \ln|x| + C \qquad \int \sin x \, dx = -\cos x + C \qquad \int \cos x \, dx = \sin x + C$$

9Numerical Methods

Locating roots, iteration, trapezium rule.

Change of sign

If $f$ is continuous and $f(a)$, $f(b)$ have opposite signs, a root lies in $[a,b]$.

Iteration

Rearrange $f(x)=0$ as $x = g(x)$ then iterate $x_{n+1} = g(x_n)$. Converges if $|g'(\alpha)| < 1$ near the root $\alpha$.

Trapezium rule

$$\int_a^b f(x)\,dx \approx \dfrac{h}{2}[y_0 + y_n + 2(y_1 + y_2 + \ldots + y_{n-1})], \quad h = \dfrac{b-a}{n}$$
These notes are AI-assisted study material. Always cross-check against the official Pearson Edexcel spec or your teacher before relying on them in an exam.

P3 — Pure Mathematics 3

Unit WMA13 · Edexcel International A Level Mathematics (2018)

1Algebraic Fractions

Simplifying, adding, multiplying, dividing.

Factorise everything first, then cancel or find a common denominator. Watch domain restrictions when cancelling.

2Functions

Domain, range, composite, inverse, modulus.

Notation $f:x \mapsto \ldots$ or $f(x) = \ldots$. Composite: $(g \circ f)(x) = g(f(x))$; order matters. Inverse $f^{-1}$: swap $x$ and $y$ and solve; domain of $f^{-1}$ = range of $f$. Modulus function $|x|$: distance from 0.

Sketch $y = |f(x)|$: reflect any part below x-axis. Sketch $y = f(|x|)$: keep the right side and reflect it in the y-axis.

3Trigonometry

Reciprocal ratios, further identities.
Reciprocals and Pythagorean forms$$\sec\theta = \dfrac{1}{\cos\theta}, \; \csc\theta = \dfrac{1}{\sin\theta}, \; \cot\theta = \dfrac{1}{\tan\theta}$$ $$1 + \tan^2\theta = \sec^2\theta \qquad 1 + \cot^2\theta = \csc^2\theta$$
Compound and double angles$$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$$ $$\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$$ $$\sin 2A = 2\sin A\cos A \qquad \cos 2A = \cos^2 A - \sin^2 A = 1 - 2\sin^2 A = 2\cos^2 A - 1$$
Harmonic form$$a\sin\theta + b\cos\theta = R\sin(\theta + \alpha) \text{ with } R = \sqrt{a^2 + b^2}, \; \tan\alpha = \tfrac{b}{a}$$

4Exponentials & Logarithms

Modelling exponential growth/decay, linearising by logs.

To reduce $y = a b^x$ to linear form: take logs — $\log y = \log a + x \log b$. Plot $\log y$ vs $x$; gradient is $\log b$, intercept is $\log a$.

5Differentiation

Implicit, parametric, second derivatives.

Implicit differentiation

Differentiate both sides of an equation in $x, y$ with respect to $x$, treating $y$ as a function of $x$ (chain rule adds a $\tfrac{dy}{dx}$ factor for $y$-terms). Rearrange to isolate $\tfrac{dy}{dx}$.

Second derivative and inflections

$f'' > 0$: concave up (curve holds water). $f'' < 0$: concave down. Point of inflection: $f'' = 0$ and sign of $f''$ changes.

6Integration

Standard results extended.
Useful forms$$\int \dfrac{f'(x)}{f(x)} \, dx = \ln|f(x)| + C \qquad \int f'(x) [f(x)]^n \, dx = \dfrac{[f(x)]^{n+1}}{n+1} + C$$

7Numerical Methods

Newton–Raphson.
Iteration formula$$x_{n+1} = x_n - \dfrac{f(x_n)}{f'(x_n)}$$

Converges quadratically near a simple root when $f'$ is non-zero and $f$ is smooth. Fails if $f'(x_n)=0$ or stationary point is near.

These notes are AI-assisted study material. Always cross-check against the official Pearson Edexcel spec or your teacher before relying on them in an exam.

P4 — Pure Mathematics 4

Unit WMA14 · Edexcel International A Level Mathematics (2018)

1Series — Binomial Extended

Expansion of $(1+x)^n$ for any rational $n$, valid for $|x| < 1$.
$$(1 + x)^n = 1 + nx + \dfrac{n(n-1)}{2!}x^2 + \dfrac{n(n-1)(n-2)}{3!}x^3 + \ldots \quad |x| < 1$$

For $(a + bx)^n$ factor out $a^n$: $(a + bx)^n = a^n\!\left(1 + \tfrac{bx}{a}\right)^n$, valid for $\left|\tfrac{bx}{a}\right| < 1$.

2Parametric Equations

$x = f(t),\; y = g(t)$; sketching, converting, differentiating.

Eliminate the parameter to get a Cartesian equation (often by expressing $t$ in one and substituting). Chain rule for slope: $\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}$.

3Trigonometry (further)

Half-angle formulas, sum-to-product.

Useful half-angle identities: $\sin^2 A = \tfrac{1 - \cos 2A}{2}$, $\cos^2 A = \tfrac{1 + \cos 2A}{2}$ — often needed to integrate $\sin^2, \cos^2$.

4Differentiation (further)

Inverse trig, related rates.
$$\dfrac{d}{dx}(\sin^{-1}x) = \dfrac{1}{\sqrt{1 - x^2}} \qquad \dfrac{d}{dx}(\tan^{-1}x) = \dfrac{1}{1 + x^2}$$

Related rates: connect $\tfrac{dV}{dt}$ to $\tfrac{dr}{dt}$ via $\tfrac{dV}{dr}$ (chain rule).

5Integration by Substitution

Change of variable $u = g(x)$.

Choose $u$ so $du = g'(x) \, dx$ appears (up to a constant) in the integrand. Change limits when definite. Trigonometric substitutions: $x = a\sin\theta$ for $\sqrt{a^2 - x^2}$; $x = a\tan\theta$ for $a^2 + x^2$.

6Integration by Parts

Product rule in reverse.
$$\int u \dfrac{dv}{dx} \, dx = uv - \int v \dfrac{du}{dx} \, dx$$

Choose $u$ using the LIATE heuristic (Log, Inverse trig, Algebraic, Trig, Exponential — pick the first one on the list). Some problems reduce to themselves (recursive) and can be solved algebraically.

7Differential Equations

Separable first-order.

If $\dfrac{dy}{dx} = f(x)g(y)$, separate: $\dfrac{dy}{g(y)} = f(x)\,dx$; integrate both sides. Apply initial condition to find the constant of integration.

8Vectors (3D)

Dot product, angle between vectors, vector equation of a line.
Dot product$$\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 = |\mathbf{a}||\mathbf{b}|\cos\theta$$

Perpendicular iff $\mathbf{a} \cdot \mathbf{b} = 0$. Vector equation of a line: $\mathbf{r} = \mathbf{a} + t\mathbf{d}$, position of a point plus a scalar multiple of the direction.

These notes are AI-assisted study material. Always cross-check against the official Pearson Edexcel spec or your teacher before relying on them in an exam.

S1 — Statistics 1

Unit WST01 · Edexcel International A Level Mathematics (2018)

1Mathematical Models

Model-building cycle in statistics.

Simplification → apply mathematics → interpret → compare with reality → refine. A model isn't reality; it's a useful approximation.

2Data Representation

Histograms, stem-and-leaf, box plots.

Histogram: area of each bar equals the frequency; frequency density = frequency / class width. Use for continuous data with unequal class widths. Box plot: min, Q1, median, Q3, max — shows spread and skew. Outliers commonly flagged beyond $Q_3 + 1.5(Q_3 - Q_1)$ or below $Q_1 - 1.5(Q_3 - Q_1)$.

3Location & Spread

Mean, median, mode, variance, standard deviation.
Mean and variance$$\bar{x} = \dfrac{\sum x}{n} \qquad \sigma^2 = \dfrac{\sum (x - \bar{x})^2}{n} = \dfrac{\sum x^2}{n} - \bar{x}^2$$

Standard deviation is the (positive) square root of variance. Use the sample formula ($n-1$ denominator) only when explicitly required.

4Probability

Events, Venn diagrams, conditional probability.
$$P(A \cup B) = P(A) + P(B) - P(A \cap B) \qquad P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$$

Independence: $P(A \cap B) = P(A)P(B)$. Mutually exclusive: $P(A \cap B) = 0$.

5Correlation & Regression

PMCC and least-squares line.
Sums of squares$$S_{xx} = \sum x^2 - \dfrac{(\sum x)^2}{n}, \quad S_{yy} = \sum y^2 - \dfrac{(\sum y)^2}{n}, \quad S_{xy} = \sum xy - \dfrac{(\sum x)(\sum y)}{n}$$
PMCC and regression$$r = \dfrac{S_{xy}}{\sqrt{S_{xx}S_{yy}}} \qquad \hat{y} = a + bx \text{ with } b = \dfrac{S_{xy}}{S_{xx}},\; a = \bar{y} - b\bar{x}$$

$r$ ranges from $-1$ to $+1$. Regression $y$ on $x$ is used to predict $y$; don't extrapolate outside the data range.

6Discrete Random Variables

Probability distribution, expectation, variance.
$$E(X) = \sum x P(X=x) \qquad \text{Var}(X) = E(X^2) - [E(X)]^2$$

Linearity: $E(aX + b) = aE(X) + b$; $\text{Var}(aX + b) = a^2 \text{Var}(X)$.

7Normal Distribution

Standardising, using tables.

If $X \sim N(\mu, \sigma^2)$ then $Z = \dfrac{X - \mu}{\sigma} \sim N(0, 1)$. Look up $\Phi(z) = P(Z \le z)$ in the standard normal table. Symmetry: $\Phi(-z) = 1 - \Phi(z)$.

These notes are AI-assisted study material. Always cross-check against the official Pearson Edexcel spec or your teacher before relying on them in an exam.

M1 — Mechanics 1

Unit WME01 · Edexcel International A Level Mathematics (2018)

1Mathematical Models

Common simplifying assumptions.
  • Particle: object treated as a point; no rotation or size.
  • Light string/rod: mass negligible.
  • Inextensible string: doesn't stretch.
  • Smooth surface/pulley: no friction.
  • Rigid body: shape doesn't deform.

2Vectors (2D)

Position, velocity, acceleration as vectors.

Add/subtract componentwise; scalar multiply to scale; magnitude via Pythagoras. Unit vectors $\mathbf{i}, \mathbf{j}$ along x, y axes.

3Kinematics (straight line)

Constant acceleration and calculus.
SUVAT (constant $a$)$$v = u + at \quad s = ut + \tfrac{1}{2}at^2 \quad s = \tfrac{1}{2}(u+v)t \quad v^2 = u^2 + 2as$$

Variable acceleration: $v = \tfrac{ds}{dt}$, $a = \tfrac{dv}{dt}$; $s = \int v\,dt$, $v = \int a\,dt$.

4Dynamics of a Particle

Newton's laws, weight, friction.
$$\mathbf{F} = m\mathbf{a} \qquad W = mg \qquad F_{\max} = \mu R$$

Weight acts downward with magnitude $mg$ (take $g = 9.8\,\text{m s}^{-2}$ unless told). Friction: static friction adjusts to prevent motion up to $F_{\max} = \mu R$; kinetic friction is $\mu R$ opposite the motion.

5Connected Particles

Ropes, pulleys, coupled motion.

Two particles joined by a taut, inextensible string share magnitude of acceleration. Apply $F = ma$ to each separately along the direction of motion; solve simultaneously. For a smooth pulley: tension is the same throughout the string.

6Statics

Equilibrium of a particle.

Resolve forces in perpendicular directions; both totals equal zero. On an inclined plane: resolve along and perpendicular to the slope for a cleaner solution.

7Moments

Turning effect of a force.
$$M = Fd$$

Moment about a point = force $\times$ perpendicular distance from the point to the line of action. In equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments about any point.

Formula Sheet

Curated formulas for Edexcel IAL Mathematics and Further Mathematics. Cross-reference against the official Pearson formula booklet before your exam.

Algebra

Quadratic formula$$x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$$
Binomial (positive integer $n$)$$(a+b)^n = \sum_{r=0}^n \binom{n}{r} a^{n-r}b^r$$
Binomial series (rational $n$, $|x|<1$)$$(1+x)^n = 1 + nx + \dfrac{n(n-1)}{2!}x^2 + \dfrac{n(n-1)(n-2)}{3!}x^3 + \ldots$$
Logarithms$$\log_a xy = \log_a x + \log_a y \quad \log_a \tfrac{x}{y} = \log_a x - \log_a y \quad \log_a x^n = n\log_a x$$

Series

Arithmetic$$u_n = a + (n-1)d \qquad S_n = \tfrac{n}{2}[2a + (n-1)d]$$
Geometric$$u_n = ar^{n-1} \qquad S_n = \dfrac{a(1-r^n)}{1-r} \qquad S_\infty = \dfrac{a}{1-r} \; (|r|<1)$$
Sums (Further)$$\sum_{r=1}^n r = \tfrac{n(n+1)}{2} \quad \sum_{r=1}^n r^2 = \tfrac{n(n+1)(2n+1)}{6} \quad \sum_{r=1}^n r^3 = \left[ \tfrac{n(n+1)}{2} \right]^2$$

Trigonometry

Identities$$\sin^2\theta + \cos^2\theta = 1 \quad 1 + \tan^2\theta = \sec^2\theta \quad 1 + \cot^2\theta = \csc^2\theta$$
Compound angles$$\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$$$$\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$$$$\tan(A \pm B) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$$
Double angles$$\sin 2A = 2\sin A\cos A$$$$\cos 2A = \cos^2 A - \sin^2 A = 1 - 2\sin^2 A = 2\cos^2 A - 1$$$$\tan 2A = \dfrac{2\tan A}{1 - \tan^2 A}$$
Sine, cosine, area$$\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C} \qquad a^2 = b^2 + c^2 - 2bc\cos A \qquad \text{Area} = \tfrac{1}{2}ab\sin C$$
Radians$$s = r\theta \qquad A = \tfrac{1}{2}r^2\theta$$
Harmonic form$$a\sin\theta + b\cos\theta = R\sin(\theta+\alpha) \text{ where } R = \sqrt{a^2+b^2},\; \tan\alpha = \tfrac{b}{a}$$

Differentiation

Standard$$\dfrac{d}{dx}(x^n) = nx^{n-1} \quad \dfrac{d}{dx}(e^x)=e^x \quad \dfrac{d}{dx}(\ln x)=\tfrac{1}{x}$$$$\dfrac{d}{dx}(\sin x) = \cos x \quad \dfrac{d}{dx}(\cos x) = -\sin x \quad \dfrac{d}{dx}(\tan x) = \sec^2 x$$
Inverse trig$$\dfrac{d}{dx}(\sin^{-1} x) = \dfrac{1}{\sqrt{1-x^2}} \qquad \dfrac{d}{dx}(\tan^{-1} x) = \dfrac{1}{1+x^2}$$
Rules$$(uv)' = u'v + uv' \quad \left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2} \quad \dfrac{dy}{dx} = \dfrac{dy}{du}\cdot\dfrac{du}{dx}$$
Hyperbolic (Further)$$\dfrac{d}{dx}(\sinh x) = \cosh x \quad \dfrac{d}{dx}(\cosh x) = \sinh x \quad \dfrac{d}{dx}(\tanh x) = \text{sech}^2 x$$

Integration

Standard$$\int x^n \, dx = \dfrac{x^{n+1}}{n+1} + C \; (n \ne -1) \qquad \int \tfrac{1}{x} \, dx = \ln|x| + C$$$$\int e^x \, dx = e^x + C \qquad \int \sin x \, dx = -\cos x + C \qquad \int \cos x \, dx = \sin x + C$$
Useful forms$$\int \dfrac{f'(x)}{f(x)}\,dx = \ln|f(x)| + C \qquad \int f'(x)[f(x)]^n\,dx = \dfrac{[f(x)]^{n+1}}{n+1} + C$$
By parts$$\int u \dfrac{dv}{dx}\,dx = uv - \int v \dfrac{du}{dx}\,dx$$
Trapezium rule$$\int_a^b y\,dx \approx \tfrac{h}{2}[y_0 + y_n + 2(y_1 + \ldots + y_{n-1})], \; h = \tfrac{b-a}{n}$$

Numerical Methods

Newton–Raphson$$x_{n+1} = x_n - \dfrac{f(x_n)}{f'(x_n)}$$

Vectors

Magnitude, unit vector$$|\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2} \qquad \hat{\mathbf{a}} = \dfrac{\mathbf{a}}{|\mathbf{a}|}$$
Dot product$$\mathbf{a}\cdot\mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 = |\mathbf{a}||\mathbf{b}|\cos\theta$$
Line and plane (Further)$$\mathbf{r} = \mathbf{a} + t\mathbf{d} \qquad \mathbf{r}\cdot\mathbf{n} = d$$
Cross product (Further)$$\mathbf{a}\times\mathbf{b} = \begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix},\quad |\mathbf{a}\times\mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta$$

Statistics

Mean and variance$$\bar{x} = \dfrac{\sum x}{n} \qquad \sigma^2 = \dfrac{\sum x^2}{n} - \bar{x}^2$$
PMCC and regression$$r = \dfrac{S_{xy}}{\sqrt{S_{xx}S_{yy}}} \qquad b = \dfrac{S_{xy}}{S_{xx}},\; a = \bar{y} - b\bar{x}$$
Discrete RV$$E(X) = \sum x P(X=x) \qquad \text{Var}(X) = E(X^2) - [E(X)]^2$$
Normal, standardising$$Z = \dfrac{X - \mu}{\sigma}$$
Binomial (Further)$$X \sim B(n,p): \; P(X=x) = \binom{n}{x} p^x (1-p)^{n-x},\; E(X) = np,\; \text{Var}(X) = np(1-p)$$
Poisson (Further)$$X \sim \text{Po}(\lambda): \; P(X=x) = \dfrac{e^{-\lambda}\lambda^x}{x!},\; E(X) = \text{Var}(X) = \lambda$$

Mechanics

SUVAT$$v = u + at \quad s = ut + \tfrac{1}{2}at^2 \quad s = \tfrac{1}{2}(u+v)t \quad v^2 = u^2 + 2as$$
Newton, friction$$\mathbf{F} = m\mathbf{a} \qquad W = mg \qquad F_{\max} = \mu R$$
Momentum, impulse$$p = mv \qquad I = Ft = mv - mu$$
Work, energy, power (M2)$$W = Fs\cos\theta \quad \text{KE} = \tfrac{1}{2}mv^2 \quad \text{GPE} = mgh \quad P = Fv$$
Moments$$M = Fd$$

Further Pure

Complex numbers$$z = a + bi = r(\cos\theta + i\sin\theta) = re^{i\theta}, \; |z| = r, \; \arg z = \theta$$$$z_1 z_2 = r_1 r_2 [\cos(\theta_1+\theta_2) + i\sin(\theta_1+\theta_2)]$$
De Moivre$$[r(\cos\theta + i\sin\theta)]^n = r^n(\cos n\theta + i\sin n\theta)$$
Maclaurin$$f(x) = f(0) + f'(0)x + \dfrac{f''(0)}{2!}x^2 + \dfrac{f'''(0)}{3!}x^3 + \ldots$$
Polar area$$A = \tfrac{1}{2}\int_\alpha^\beta r^2 \, d\theta$$
2×2 matrix inverse$$\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)^{-1} = \dfrac{1}{ad-bc}\left(\begin{smallmatrix}d&-b\\-c&a\end{smallmatrix}\right)$$

Mathematics AI Tutor