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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.

FP1 — Further Pure 1

Unit WFM01 · Edexcel International A Level Further Mathematics (2018)

1Complex Numbers

Cartesian and Argand diagram, modulus-argument.

$z = a + bi$ with $i^2 = -1$. Real part $\Re(z) = a$, imaginary part $\Im(z) = b$. Complex conjugate $\bar{z} = a - bi$. Modulus $|z| = \sqrt{a^2+b^2}$; argument $\arg z$ is the angle from positive real axis, $-\pi < \arg z \le \pi$.

Arithmetic: add componentwise; multiply as $(a+bi)(c+di) = (ac-bd) + (ad+bc)i$; divide by multiplying by conjugate of denominator.

An Argand diagram plots $z$ as $(a, b)$. Loci: $|z - z_0| = r$ is a circle; $\arg(z - z_0) = \theta$ is a half-line.

2Roots of Quadratic Equations

Vieta's formulas.

If $\alpha, \beta$ are roots of $ax^2 + bx + c = 0$: $\alpha + \beta = -\dfrac{b}{a}$, $\alpha\beta = \dfrac{c}{a}$. Given transformed roots (e.g. $\alpha^2, \beta^2$ or $\tfrac{1}{\alpha}, \tfrac{1}{\beta}$), compute their sum and product and reconstruct the new quadratic.

3Numerical Solutions

Interval bisection, linear interpolation.

If $f$ continuous and $f(a) \cdot f(b) < 0$, a root lies in $(a, b)$. Bisection: repeatedly test the midpoint. Linear interpolation: estimate root as $x = a - \dfrac{f(a)(b-a)}{f(b) - f(a)}$.

4Coordinate Systems

Parabola and rectangular hyperbola.

Parabola $y^2 = 4ax$: focus $(a, 0)$, directrix $x = -a$. Parametric form $(at^2, 2at)$.

Rectangular hyperbola $xy = c^2$: parametric form $(ct, c/t)$. Tangent at parameter $t$: $x + t^2 y = 2ct$.

5Matrix Algebra

Addition, multiplication, determinant, inverse.

Matrix multiplication is associative but not commutative. $(AB)C = A(BC)$; usually $AB \ne BA$.

2×2 inverse$$\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)^{-1} = \dfrac{1}{ad - bc}\left(\begin{smallmatrix}d&-b\\-c&a\end{smallmatrix}\right), \; \det = ad - bc$$

Singular matrix: $\det = 0$; no inverse exists.

6Transformations

Linear transformations of the plane represented by matrices.

The columns of a $2 \times 2$ matrix are the images of $\binom{1}{0}$ and $\binom{0}{1}$. Composition of transformations $=$ matrix multiplication in reverse order: transformation by $A$ then $B$ gives matrix $BA$.

Common: rotation about origin by $\theta$: $\left(\begin{smallmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{smallmatrix}\right)$. Enlargement scale $k$: $kI$. Reflection in line $y=x\tan\theta$: standard matrix.

7Series (sums of powers)

Closed forms for sums of natural numbers, squares, cubes.
$$\sum_{r=1}^n r = \tfrac{n(n+1)}{2} \qquad \sum_{r=1}^n r^2 = \tfrac{n(n+1)(2n+1)}{6} \qquad \sum_{r=1}^n r^3 = \left[\tfrac{n(n+1)}{2}\right]^2$$

Use these with linearity: $\sum (ar^2 + br + c) = a\sum r^2 + b\sum r + cn$.

8Proof by Induction

Standard technique for statements about integers.

Base case: verify statement for $n = 1$ (or smallest relevant integer). Inductive step: assume true for $n = k$, prove true for $n = k+1$. Conclusion: since base holds and the step is valid, statement holds for all $n \ge 1$.

Common applications: summation identities, divisibility statements, matrix powers.

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.

FP2 — Further Pure 2

Unit WFM02 · Edexcel International A Level Further Mathematics (2018)

1Inequalities

Algebraic manipulation, modulus, rational inequalities.

Multiply through by a squared expression to avoid sign issues; multiply by the square of the denominator when solving rational inequalities. Solutions to $|f(x)| < a$: $-a < f(x) < a$. Solutions to $|f(x)| > a$: $f(x) < -a$ or $f(x) > a$.

2Series — Method of Differences

Telescoping sums.

Rewrite $u_r = f(r) - f(r+1)$ (or similar) so that consecutive terms cancel. The sum $\sum_{r=1}^n u_r$ collapses to endpoints: $f(1) - f(n+1)$. Often used after partial fractions.

3Further Complex Numbers

De Moivre's theorem, roots of unity, exponential form.
Exponential and De Moivre$$e^{i\theta} = \cos\theta + i\sin\theta \qquad (r\, e^{i\theta})^n = r^n\, e^{in\theta}$$

$n$th roots of $z = re^{i\theta}$: $r^{1/n}\, e^{i(\theta + 2k\pi)/n}$ for $k = 0, 1, \ldots, n-1$. On the Argand diagram they lie on a circle at equal angular spacing.

Applications: expressing $\cos n\theta$, $\sin n\theta$ in terms of $\cos\theta$, $\sin\theta$; simplifying trig sums.

4First-order Differential Equations

Integrating factor.

For $\dfrac{dy}{dx} + P(x)y = Q(x)$: multiply both sides by integrating factor $\mu = e^{\int P\,dx}$. LHS becomes $\dfrac{d}{dx}(\mu y)$; integrate both sides then divide.

5Second-order Differential Equations

Linear with constant coefficients.

Solve $ay'' + by' + cy = f(x)$ in two steps: general solution = complementary function (from auxiliary $am^2 + bm + c = 0$) + particular integral (guessed form based on $f$).

  • Two distinct real roots $m_1, m_2$: $y_{cf} = Ae^{m_1 x} + Be^{m_2 x}$.
  • Repeated root $m$: $y_{cf} = (A + Bx)e^{mx}$.
  • Complex $m = \alpha \pm i\beta$: $y_{cf} = e^{\alpha x}(A\cos\beta x + B\sin\beta x)$.

6Maclaurin & Taylor Series

Power series expansion of a function.
Maclaurin$$f(x) = f(0) + f'(0)x + \dfrac{f''(0)}{2!}x^2 + \dfrac{f'''(0)}{3!}x^3 + \ldots$$

Standard: $e^x = 1 + x + \tfrac{x^2}{2!} + \ldots$, $\sin x = x - \tfrac{x^3}{3!} + \tfrac{x^5}{5!} - \ldots$, $\cos x = 1 - \tfrac{x^2}{2!} + \tfrac{x^4}{4!} - \ldots$, $\ln(1+x) = x - \tfrac{x^2}{2} + \tfrac{x^3}{3} - \ldots$ (for $|x|<1$).

7Polar Coordinates

$(r, \theta)$, conversion, area.

Conversion: $x = r\cos\theta$, $y = r\sin\theta$; $r = \sqrt{x^2+y^2}$, $\theta = \arctan(y/x)$ (with quadrant care).

Area of a sector$$A = \tfrac{1}{2}\int_\alpha^\beta r^2 \, d\theta$$

Common curves: circles $r = a$, cardioids $r = a(1 + \cos\theta)$, roses $r = a\sin n\theta$.

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.

FP3 — Further Pure 3

Unit WFM03 · Edexcel International A Level Further Mathematics (2018)

1Hyperbolic Functions

$\sinh, \cosh, \tanh$ and their inverses.
$$\sinh x = \dfrac{e^x - e^{-x}}{2} \qquad \cosh x = \dfrac{e^x + e^{-x}}{2} \qquad \tanh x = \dfrac{\sinh x}{\cosh x}$$

Osborn's rule: to translate a trig identity to hyperbolic, replace each $\sin$ with $\sinh$ and each $\cos$ with $\cosh$, but flip the sign of a product of two $\sinh$'s.

Identity, derivatives$$\cosh^2 x - \sinh^2 x = 1$$$$\dfrac{d}{dx}\sinh x = \cosh x, \; \dfrac{d}{dx}\cosh x = \sinh x, \; \dfrac{d}{dx}\tanh x = \text{sech}^2 x$$

Inverses in log form: $\sinh^{-1}x = \ln(x + \sqrt{x^2+1})$, $\cosh^{-1}x = \ln(x + \sqrt{x^2-1})$ for $x \ge 1$, $\tanh^{-1}x = \tfrac{1}{2}\ln\tfrac{1+x}{1-x}$ for $|x|<1$.

2Further Coordinate Systems

Ellipse and hyperbola.

Ellipse $\tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} = 1$: foci $(\pm ae, 0)$ where $b^2 = a^2(1-e^2)$; directrices $x = \pm a/e$. Parametric $(a\cos t, b\sin t)$.

Hyperbola $\tfrac{x^2}{a^2} - \tfrac{y^2}{b^2} = 1$: foci $(\pm ae, 0)$ with $b^2 = a^2(e^2-1)$; asymptotes $y = \pm \tfrac{b}{a}x$. Parametric $(a\sec t, b\tan t)$ or hyperbolic $(a\cosh t, b\sinh t)$.

3Differentiation (further)

Inverse trig and hyperbolic.
$$\dfrac{d}{dx}(\sinh^{-1}x) = \dfrac{1}{\sqrt{1+x^2}} \qquad \dfrac{d}{dx}(\cosh^{-1}x) = \dfrac{1}{\sqrt{x^2-1}} \qquad \dfrac{d}{dx}(\tanh^{-1}x) = \dfrac{1}{1-x^2}$$

4Integration (further)

Reduction formulae, standard forms.
Standard forms$$\int \dfrac{1}{\sqrt{a^2-x^2}}\,dx = \sin^{-1}\tfrac{x}{a} + C$$$$\int \dfrac{1}{a^2+x^2}\,dx = \tfrac{1}{a}\tan^{-1}\tfrac{x}{a} + C$$$$\int \dfrac{1}{\sqrt{x^2+a^2}}\,dx = \sinh^{-1}\tfrac{x}{a} + C$$$$\int \dfrac{1}{\sqrt{x^2-a^2}}\,dx = \cosh^{-1}\tfrac{x}{a} + C$$

Reduction formulae: expressions of the form $I_n = \int \ldots$ recursively in terms of $I_{n-1}$ or $I_{n-2}$; use integration by parts to derive them, then evaluate iteratively.

5Vectors (3D)

Cross product, vector/scalar equations of a line and plane.
Cross product$$\mathbf{a} \times \mathbf{b} = \left|\begin{smallmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\a_1&a_2&a_3\\b_1&b_2&b_3\end{smallmatrix}\right|, \qquad |\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta$$

$\mathbf{a} \times \mathbf{b}$ is perpendicular to both $\mathbf{a}$ and $\mathbf{b}$ (right-hand rule). Its magnitude equals the area of the parallelogram they span.

Plane$$\mathbf{r}\cdot \mathbf{n} = \mathbf{a}\cdot \mathbf{n} \quad \Leftrightarrow \quad n_1 x + n_2 y + n_3 z = d$$

Distance from point $(x_0,y_0,z_0)$ to plane $n_1 x + n_2 y + n_3 z = d$: $\dfrac{|n_1 x_0 + n_2 y_0 + n_3 z_0 - d|}{|\mathbf{n}|}$.

6Further Matrix Algebra

3×3 matrices, eigenvalues, eigenvectors.

Eigenvalue $\lambda$ satisfies $\det(A - \lambda I) = 0$. Corresponding eigenvector $\mathbf{v}$ solves $(A - \lambda I)\mathbf{v} = \mathbf{0}$.

For 3×3 the characteristic equation is a cubic; factorise to find the three eigenvalues. Diagonalisation: $A = PDP^{-1}$ where $P$'s columns are eigenvectors and $D$ has eigenvalues on the diagonal.

7Further Trigonometry

Products of trig and hyperbolic; identities via complex exponentials.

Using $e^{i\theta} = \cos\theta + i\sin\theta$ and $\sin\theta = \tfrac{e^{i\theta} - e^{-i\theta}}{2i}$, $\cos\theta = \tfrac{e^{i\theta} + e^{-i\theta}}{2}$, derive expressions for $\sin^n\theta$, $\cos^n\theta$ in terms of multiple angles — useful for integration and summation.

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.

M2 — Mechanics 2

Unit WME02 · Edexcel International A Level Further Mathematics (2018)

1Kinematics (variable acceleration)

Calculus-based motion; projectiles.

$\mathbf{v} = \tfrac{d\mathbf{r}}{dt}$, $\mathbf{a} = \tfrac{d\mathbf{v}}{dt}$. Sometimes $a = f(v)$ or $a = f(x)$ — use $a = v\tfrac{dv}{dx}$ to change variable.

Projectiles: split velocity into horizontal (constant) and vertical (under gravity) components. Time of flight, range, max height derived from SUVAT in each direction.

2Centres of Mass

Systems of particles, laminas, uniform bodies.
Discrete system$$\bar{x} = \dfrac{\sum m_i x_i}{\sum m_i}, \; \bar{y} = \dfrac{\sum m_i y_i}{\sum m_i}$$

Composite laminas: treat each piece as a particle at its own centroid. For a body suspended from a point, the centre of mass lies vertically below the point.

3Work, Energy, Power

Energy conservation and non-conservative forces.
$$W = Fs\cos\theta \qquad \text{KE} = \tfrac{1}{2}mv^2 \qquad \text{GPE} = mgh \qquad P = Fv$$

Work–energy theorem: total work done = change in KE. When only conservative forces act, KE + GPE is constant.

4Collisions

Impulse, momentum, coefficient of restitution.

Momentum $p = mv$; impulse $I = Ft = \Delta p$. In collisions momentum is conserved if no external impulse: $m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$.

Coefficient of restitution $e = \dfrac{v_2 - v_1}{u_1 - u_2}$ ($0 \le e \le 1$). Perfectly elastic: $e = 1$. Perfectly inelastic: $e = 0$ (particles stick).

5Statics of Rigid Bodies

Equilibrium under multiple forces and moments.

Rigid body equilibrium: sum of forces = 0 in each direction AND sum of moments about any point = 0. Common exam context: ladder against a wall — resolve horizontally and vertically, take moments about the foot.

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.

S2 — Statistics 2

Unit WST02 · Edexcel International A Level Further Mathematics (2018)

1Binomial Distribution

Discrete count of successes.

$X \sim B(n, p)$ if there are $n$ independent trials, each with success probability $p$. $P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}$. Mean $np$, variance $np(1-p)$.

2Poisson Distribution

Rare events over a fixed interval.
$$X \sim \text{Po}(\lambda): \; P(X = x) = \dfrac{e^{-\lambda}\lambda^x}{x!}, \; E(X) = \text{Var}(X) = \lambda$$

Approximations: If $n$ large and $p$ small, $B(n, p) \approx \text{Po}(np)$. Sum of independent Poissons is Poisson with sum of means.

3Continuous Random Variables

Probability density and cumulative distribution.

$f(x)$ is a valid pdf if $f(x) \ge 0$ and $\int_{-\infty}^{\infty} f(x)\,dx = 1$. CDF: $F(x) = \int_{-\infty}^x f(t)\,dt$; $P(a < X < b) = F(b) - F(a) = \int_a^b f\,dx$.

$E(X) = \int x f(x)\,dx$, $E(X^2) = \int x^2 f(x)\,dx$, $\text{Var}(X) = E(X^2) - [E(X)]^2$.

4Continuous Distributions

Uniform, extending Normal.

Continuous uniform on $[a, b]$: $f(x) = \tfrac{1}{b-a}$; mean $\tfrac{a+b}{2}$, variance $\tfrac{(b-a)^2}{12}$.

Normal approximation to Binomial: valid for large $n$ and $p$ near $\tfrac{1}{2}$: $B(n,p) \approx N(np, np(1-p))$. Apply continuity correction ($\pm 0.5$).

5Hypothesis Testing

Framework.
  1. State $H_0$ (null) and $H_1$ (alternative); pick a significance level $\alpha$ (e.g. 5%).
  2. Under $H_0$ compute the probability of observed (or more extreme) result.
  3. If p-value $\le \alpha$: reject $H_0$. Else insufficient evidence to reject.
  4. State conclusion in context — never say "prove".
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.

D1 — Decision Mathematics 1

Unit WDM11 · Edexcel International A Level Further Mathematics (2018)

1Algorithms

Sorting and packing.
  • Bubble sort: pass through list swapping adjacent out-of-order pairs; repeat until no swaps. $O(n^2)$.
  • Quick sort: pick a pivot; partition into $< $ pivot and $\ge$ pivot; recurse. Average $O(n \log n)$.
  • Bin packing: first-fit, first-fit decreasing, full-bin — greedy heuristics; not guaranteed optimal.

2Algorithms on Graphs

Minimum spanning tree and shortest path.

Kruskal's (MST): sort edges by weight; add smallest unless it creates a cycle; stop when $n-1$ edges included.

Prim's (MST): start at any vertex; repeatedly add the smallest edge that connects a visited vertex to an unvisited one.

Dijkstra's (shortest path): initialise source label 0, others $\infty$; permanently label the smallest unlabelled; update neighbours' temporary labels; repeat.

3Route Inspection

Chinese Postman Problem.

Find the shortest closed route that traverses every edge. If graph is Eulerian (all vertex degrees even), total weight = sum of edges. If some vertices are odd: pair them up; find the pairing whose sum of shortest paths is minimum; add these repeats to the total.

4Critical Path Analysis

Activity networks; earliest/latest event times.

Represent activities on arcs (or nodes). Forward pass: compute earliest event times. Backward pass: compute latest event times. Float of an activity = latest finish − earliest start − duration. Critical path: activities with zero float; their durations sum to the project duration.

5Linear Programming

Graphical and simplex.

Graphical: plot constraints, identify feasible region, evaluate the objective at each vertex — optimum is at a vertex (or along an edge).

Simplex: introduce slack variables; construct initial tableau; pivot on the most negative bottom-row entry to improve; repeat until no negatives left in the objective row.

6Matchings

Bipartite graphs.

A matching pairs elements of one set with elements of another. To improve a partial matching, find an alternating path starting and ending at an unmatched vertex; toggle the edges along it.

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$$\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)^{-1} = \dfrac{1}{ad-bc}\left(\begin{smallmatrix}d&-b\\-c&a\end{smallmatrix}\right)$$
Hyperbolic identity$$\cosh^2 x - \sinh^2 x = 1$$

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