Key ideas — A-level maths

Key ideas

The two sentences worth carrying out of every topic, taken straight from the notes. Open a unit to skim it, or search to find the one idea you are missing.

388 ideas

ProofYears 12-132 topics

The structure of proof: deduction and exhaustion

A proof runs from stated assumptions through checkable steps to a conclusion, and settles every case in one go.
Deduce when the cases never run out. Exhaust when they do.

Disproof and proof by contradiction

One counter example is enough to disprove a universal claim.
Assume the opposite, reason correctly until you reach an impossibility, and the original statement is proved.

Algebra and functionsYears 12-138 topics

Indices and surds

Three laws rule every power, and a fraction in the exponent is a root.
Take out the largest square factor; rationalise with the conjugate when the denominator has two terms.

Quadratic functions

The completed square names the vertex, and the leftover constant is the extreme value.
The discriminant counts the roots for you. Two, one, or none.

Simultaneous equations and inequalities

Substitute the linear equation into the quadratic. The roots are the crossings and the discriminant counts them.
Solve a quadratic inequality from a sketch. Roots, shape, sign.

Polynomials and the factor theorem

Divide f(x) by (x − a) and the remainder is f(a). A remainder of zero names a factor.
One found root, one division, one quadratic you already know how to finish.

Graphs, proportion and transformations

The factors are the sketch. Simple roots cross, squared roots touch, and an asymptote is what the curve settles onto, which is not the same as a line it never touches.
Outside the bracket acts on y and does what it says. Inside acts on x and does the opposite.

Functions, inverses and the modulus

fg runs g first. An inverse undoes, reflects in y = x, and swaps domain with range.
Only one-one functions invert.

Partial fractions

One fraction per distinct bracket, and a squared bracket takes both the bracket and its square.
Substitute the roots to harvest constants, then compare a coefficient for whatever is left.

Functions in modelling

Behaviour picks the family. Repetition is trig, a rate proportional to the amount is exponential, a constant product is reciprocal.
Fit the constants, read them in units, and say where the model breaks.

Coordinate geometryYears 12-133 topics

Straight lines

One point and one gradient determine a line, and y − y₁ = m(x − x₁) turns them into its equation.
Parallel lines share m. Perpendicular gradients multiply to −1.

Circles

A circle is Pythagoras about a fixed centre, and completing the square recovers the centre and radius from any form.
Radius meets tangent at a right angle, the perpendicular from the centre bisects a chord, and a right angle on the circle names a diameter.

Parametric equations

A parametric curve is a point with a clock, and the trail it leaves is the graph.
Eliminate t by substitution or by identity, then say which part of the curve the parameter really visits.

Sequences and seriesYears 12-135 topics

The binomial expansion

Row n of Pascal's triangle is nCr, and every term of (a + b)ⁿ keeps total degree n.
For small x the early terms carry nearly everything; truncate, then say what you dropped.

The general binomial expansion

A negative or non-integer exponent makes the expansion infinite; a positive whole number is rational too, and terminates. |bx/a| < 1 is where the infinite one converges, with the endpoints needing their own check.
Take out the a, expand the rest, and never leave the validity unsaid.

Sequences and sigma notation

A formula serves any term on demand; a recurrence builds the list step by step from a stated start.
Increasing, decreasing or periodic with its order, and Σ carries a counter, a recipe and a range.

Arithmetic series

The nth term is a plus n minus 1 steps of d; count steps, not positions.
Forwards plus backwards makes n columns of a + l; halve for the sum.

Geometric series

Each term is r times the last; the sum telescopes when you subtract r times itself.
|r| < 1 gives convergence and a over 1 minus r; logs answer the how-long questions.

TrigonometryYears 12-136 topics

Triangles and the sine and cosine rules

Sine rule for opposite pairs, cosine rule for included angles or three sides.
Inverse sine gives two possible angles; the geometry of the triangle determines how many are valid.

Trigonometric graphs and equations

The unit circle defines sin and cos for every angle; the graphs repeat and the table of exact values covers the rest.
Widen the interval with the substituted angle, find every crossing in it, then translate back.

Radians, arcs and small angles

A radian is one radius of arc; π of them make 180°.
Arc rθ, sector ½r²θ, and for tiny radian angles sine and tan are θ itself.

Reciprocal and inverse trigonometric functions

sec, cosec and cot are reciprocals that blow up where their parents vanish.
Divide the Pythagorean identity for two new ones; restrict before inverting, and the range picks the angle.

Compound angles and the harmonic form

Angles add through the compound formulae, never through the functions themselves.
a sin θ + b cos θ is one wave of amplitude √(a² + b²); fold first, then read off extremes and roots.

Trigonometric modelling

Centre line, amplitude, period: three dials, three separate homes in the formula.
Fold two-term models with the harmonic form, then read the extremes straight off R.

Exponentials and logarithmsYear 123 topics

Exponential functions and e

Every base's curve passes through (0, 1) and hugs an asymptote it never touches.
e is the base whose gradient equals its height; e to the kx grows at k times itself.

Logarithms and their laws

A logarithm is an exponent: a to the n equals x and log base a of x equals n are one statement.
Logs turn multiplication into addition and bring exponents down as multipliers.

Log graphs and exponential models

Power laws straighten on log-log axes, exponentials on log-linear; the straightening plot is the diagnosis.
In Ae to the kt, A is the start, k the proportional rate, and ln 2 over k the halving or doubling time.

DifferentiationYears 12-137 topics

The derivative from first principles

The derivative is the limit of chord gradients as the chord shrinks to a point.
Cancel h first, then let it vanish; what survives is the gradient function.

Differentiating powers of x

Multiply down by the exponent, then knock the exponent down by one: nx to the n minus 1.
The rule sees only powers, so rewrite products, roots and fractions until powers are all there is.

Tangents, turning points and curve behaviour

Tangents take f'(a); normals take its negative reciprocal when f'(a) ≠ 0, and the vertical x = a when f'(a) = 0; flat points solve f'(x) = 0.
The second derivative's sign classifies; only a sign change makes an inflection.

Differentiating trig, exponentials and logs

Sine to cosine, cosine to minus sine, e to itself, ln to the reciprocal: the shelf, in radians.
Every k multiplies out front; every base other than e pays a factor of its log.

The product, quotient and chain rules

Chains multiply rates, products share the differentiation, quotients keep strict order over v squared.
Pick the rule from the shape, name your u, and tidy the answer by factorising.

Implicit and parametric differentiation

Differentiate equations as they stand; y-terms carry dy/dx through the chain rule.
Parametric gradients divide the two rates; x-in-terms-of-y flips its derivative.

Rates of change and building differential equations

Linked quantities chain their rates: build the link, differentiate it, multiply.
Rate sentences translate word by word: derivative, k, and a sign that tells the truth.

IntegrationYears 12-137 topics

Integration as antidifferentiation

Raise the exponent by one, divide by it, and never leave without the + c.
A gradient function names a family; one known point picks the member.

Definite integrals and areas

Antidifferentiate, bracket, substitute both limits and subtract; the c never survives.
Integrals are signed; areas are not. Sketch, split at the roots, and add sizes.

Integrating standard functions

Reverse the shelf and divide by every k; the missing power-rule case is ln|x|.
No entry for a squared trig function exists; an identity trades it for terms that have one.

Integration by substitution and by parts

A function beside its own derivative is a reversed chain; relabel or just read it off.
Parts trades one integral for another: differentiate what simplifies, and run it twice when once will not do.

Integrating rational functions

Split a rational integrand by partial fractions; every linear piece becomes a logarithm.
Logs come from f'/f and nowhere else; other bottoms are powers written in fraction notation.

Areas, parametric curves and the limit of a sum

The area between curves is one integral of top minus bottom, wherever the region sits.
Parametric areas run in t via ∫y (dx/dt) dt, and every integral is secretly a limit of strip sums.

Solving differential equations

Separate y-things from x-things, integrate both sides, and let one constant cover it.
The constant boards at integration time, and in context the solution comes with a validity window.

Numerical methodsYear 132 topics

Locating roots and iteration

A sign change plus continuity traps a root; tighter traps prove accuracy claims.
Iterate x = g(x) from a start value; a shallow crossing pulls in sequences that start near it, a steep one throws them out.

Newton-Raphson and the trapezium rule

Newton-Raphson slides tangents to the axis and doubles its correct digits, until a flat tangent makes it fail.
The trapezium rule adds straight-topped strips, ends once and middles twice, and the curve's bend determines the sign of the error.

VectorsYears 12-132 topics

Vectors in two dimensions

Components add slot by slot, scalars stretch, and parallel means scalar multiple.
$\overrightarrow{AB}$ = b − a, destination minus start, and its magnitude |AB| is the distance.

Vectors in three dimensions

A third slot, the same rules: magnitudes by three-square Pythagoras, AB = b − a as ever.
Space geometry reduces to distances and scalar multiples, checked component by component.

StatisticsYears 12-1310 topics

Sampling and the large data set

A sample represents its population only if chance, not convenience, chose it.
Stratify when you know the structure. The same fraction from every stratum keeps the sample's proportions matched to the population's.

Measures of location and spread

Location by position: order the data, find the position, then read the value.
Spread by squares. σ² is the mean of the squares minus the square of the mean, and only stretching the data, never sliding it, changes σ.

Representing and interpreting data

In a histogram, area is frequency and the vertical axis is frequency density.
Fences stand one and a half IQRs beyond the quartiles, and whatever is beyond the fence is plotted alone.

Correlation and regression

r scores the direction and strength of a linear link, never cause.
Regress y on x, predict only inside the data, and read a and b in the question's own units.

Probability and Venn diagrams

Draw the diagram, fill the overlap first, and make everything total one.
Exclusive means the overlap is zero. Independent means the overlap equals the product. The two are almost never both true.

Conditional probability

Condition means divide: the overlap, measured against the new, smaller world.
On a tree, multiply along and add across, and without replacement the second denominators shrink.

The binomial distribution

Fixed n, two outcomes, constant p, independent trials. Only then is it binomial.
Point probabilities by the formula, everything else through ≤ and the cumulative function.

The normal distribution

The bell is symmetric about μ, with inflection points one σ out and 95% of it within two.
Standardise with Z = (X − μ)/σ whenever a parameter is unknown, and the calculator handles everything else.

Hypothesis testing with the binomial

Assume H₀, measure how extreme the data is, compare with the agreed level.
The critical region is decided before the data, and its probability under H₀ is the actual significance level.

Hypothesis testing: correlation and the normal

Correlation: hypotheses in ρ, and compare the sample r against the table value for the right number of tails.
A mean of n readings lives on N(μ, σ²/n), so standardise with σ/√n and test as usual.

MechanicsYears 12-139 topics

Modelling, quantities and units

Model first. Every standard word deletes one complication, and you should know which one.
SI units in, signed one-dimensional vectors throughout, magnitudes only at the end.

Kinematics with constant acceleration

List s, u, v, a, t, then pick the equation missing the letter you do not need.
Gravity is a constant −9.8 with up positive, all flight long, peak included.

Kinematics with variable acceleration

Differentiate x to v to a, integrate back, and let the conditions fix the constants.
Solve v = 0 to find the turning moments, and add distance leg by leg.

Forces and Newton's laws

Diagram first. Every force on the object, nothing it exerts on anything else.
Resolve, sum with signs, apply F = ma. Equilibrium is the a = 0 special case.

Connected particles and pulleys

System for the acceleration, single particle for the tension.
One equation per mass, own direction positive, then add to eliminate T.

Projectiles

Resolve once. u cos θ across for ever, u sin θ up into gravity's hands.
Solve whichever direction knows the answer, then carry t across to the other.

Friction and inclined planes

Friction matches the applied force up to μR, and only sliding or the limiting case makes that an equality.
On a slope, mg sin θ along and mg cos θ into it, and compare tan θ with μ to see whether it holds.

Statics of a particle

Equilibrium means the components sum to zero in two chosen directions.
Choose axes that flatten the geometry, and let friction fill the gap up to μR.

Moments

Moment = force × perpendicular distance, with a declared sense of rotation.
Beams balance twice over: forces to zero, and moments about your best pivot to zero.

Further proofFurther Maths2 topics

Proof by induction: sums and series

Verify P(1), then show P(k) forces P(k + 1). Both, every time.
For sums, the step is always the same. Add the next term to the assumed total, then tidy.
The conclusion sentence is a mark. Learn it and write it out.

Induction: divisibility and matrices

Divisibility steps rewrite the k + 1 case as (multiple of the k case) + (visible multiple).
Matrix steps are one multiplication, A to the k times A, worked entry by entry.

Complex numbersFurther Maths4 topics

Complex arithmetic and the Argand diagram

i squared is minus one. Everything else is ordinary algebra.
To divide, multiply top and bottom by the conjugate of the denominator.
Conjugation reflects in the real axis, and real quadratics have conjugate root pairs.

Modulus, argument and loci

Modulus is distance from the origin; argument is the angle in (−π, π], read from a sketch.
Products multiply moduli and add arguments; quotients divide and subtract.
|z − a| = r is a circle, |z − a| = |z − b| a bisector, arg(z − a) = θ a half-line.

De Moivre's theorem and trigonometric identities

Powers in modulus-argument form: raise the modulus, multiply the argument.
z = r e^iθ turns De Moivre into an index law.
For identities, expand (cos θ + i sin θ)ⁿ binomially and equate parts with cos nθ + i sin nθ.

Roots of unity and complex roots

nth roots of a non-zero number: root the modulus, divide the argument by n, then space by 2π/n.
The n roots of unity form a regular n-gon on the unit circle and, for n ≥ 2, sum to zero.
Every root of zⁿ = w is one root times a root of unity, and zⁿ = 0 has only z = 0.

MatricesFurther Maths3 topics

Matrix algebra and transformations

A matrix's columns are the images of the unit vectors, so a stretch parallel to the axes reads as rows (a, 0) and (0, b) and the size of the determinant gives the area scale factor.
Doing B then A is the product AB, with the first-applied matrix nearest the vector.
In 3-D, reflections in coordinate planes flip one sign; rotations put a 2 × 2 rotation into the other two coordinates.
AB = 0 does not make A or B zero, so cancelling a matrix off both sides needs that matrix to be invertible.

Determinants and inverses

det = ad − bc gives the area scale factor as its size, with the sign recording orientation, and a 3 × 3 determinant does the same for volume.
Zero determinant means the plane is flattened onto a line or a point, so there is no inverse.
2 × 2 inverse: swap the diagonal, negate the off-diagonal, divide by the determinant.
3 × 3 inverse: minors, signs, transpose, divide.

Systems of equations and invariance

det ≠ 0 solves Mv = b as v = M⁻¹b; det = 0 means the planes miss or share a line.
Invariant lines survive the matrix as sets; lines of fixed points survive point by point.

Further algebra and seriesFurther Maths3 topics

Roots of polynomials

Expand a(z − α)(z − β)…: root sums and products sit in the coefficients, signs alternating.
For transformed roots, substitute z in terms of w, or rebuild from the new sum and product.
Real coefficients force complex roots into conjugate pairs.

Summing series and the method of differences

Polynomial series split into Σr³, Σr², Σr. Two of those are in the booklet and Σr is not.
Telescopes split each term into a difference. Write both ends and keep the survivors.

Maclaurin series

Maclaurin: the coefficient of x^r is the rth derivative at zero over r factorial.
Substitute into standard series for composites, and carry ln's and arctan's windows through the substitution.
Simplify each derivative before evaluating it at zero.

Further vectorsFurther Maths2 topics

Lines and planes in three dimensions

Line: r = a + λb, one anchor point plus multiples of a direction.
Plane: r·n = d, every point whose displacement from the anchor is perpendicular to n.

Intersections, angles and distances

Line into plane: substitute the parametric coordinates and solve one equation in λ.
Line-plane angles use sine with the normal; plane-plane angles use cosine.
Shortest distances come from making the joining vector perpendicular to every direction in sight.

Further calculusFurther Maths3 topics

Volumes of revolution

About the x-axis, V = π∫y² dx between x-limits. Square first, integrate second.
About the y-axis, V = π∫x² dy between y-limits, with x² rewritten in terms of y.
Parametric curves: substitute for y² and dx, and change the limits to t.

Mean values and improper integrals

Mean value is the integral divided by the interval width, the level line of equal area.
Improper integrals: integrate to a letter, take the limit, then name the verdict.
A singularity inside the interval splits the integral in two, and it converges only if both halves do.

Calculus with inverse trigonometric functions

Differentiate inverses implicitly. sin y = x gives dy/dx = 1/√(1 − x²), with the trig gone.
∫1/√(a² − x²) = arcsin(x/a) and ∫1/(a² + x²) = (1/a) arctan(x/a).
Complete the square, split numerators, and give quadratic factors a Bx + C numerator.
Beyond the patterns, substitute: x = a sin θ collapses a² − x², x = a tan θ collapses a² + x².

Hyperbolic functionsFurther Maths2 topics

Hyperbolic functions and identities

cosh and sinh are the even and odd halves of e^x, and cosh² − sinh² = 1 follows in two lines.
Osborn's rule converts trig identities by flipping the sign on products of two sinhs.
arsinh x = ln(x + √(x² + 1)); arcosh x = ln(x + √(x² − 1)) for x ≥ 1; artanh x = ½ln((1 + x)/(1 − x)) for −1 < x < 1.

Calculus with hyperbolic functions

sinh' = cosh and cosh' = sinh, with no minus sign anywhere; tanh' = sech²x.
√(x² + a²) integrals go to arsinh and √(x² − a²) to arcosh.
Substitute x = a sinh u or x = a cosh u when the root will not yield to recognition.

Polar coordinatesFurther Maths2 topics

Polar curves

x = r cos θ and y = r sin θ. Build r² and r cos θ to convert equations.
Sketch polar curves from r at the four compass angles, marking any pass through the pole.
Tangents parallel to the initial line come from dy/dθ = 0 and perpendicular ones from dx/dθ = 0, each with the other derivative non-zero.

Areas with polar coordinates

Polar area is ½∫r² dθ, so the radius comes in squared.
Square out, deploy cos²θ = (1 + cos 2θ)/2, and set limits between zeros of r or at intersections.

Differential equationsFurther Maths3 topics

First order equations and integrating factors

Linear first order: multiply by e^{∫P dx} and the left side becomes d(IF × y)/dx.
Constant P splits solutions into steady state plus a transient that dies away.
A general solution is a family of curves, one per value of the constant; a boundary condition picks out the one particular member.

Second order equations

Substitute e^mx: the equation becomes am² + bm + c = 0, and the roots classify the motion.
Real roots decay or grow, repeated roots need (A + Bx), complex p ± qi oscillate at frequency q inside e^px.
With a right-hand side, the answer is complementary function plus particular integral.
If the trial already solves the homogeneous equation, multiply it by x.

Modelling with differential equations

SHM is ẍ = −ω²x, with period 2π/ω and amplitude fixed by the starting conditions.
Damping is read from b² − 4c: negative oscillates in an envelope, zero settles fastest, positive creeps.
Collapse coupled pairs by differentiating one equation and substituting the other.

Further Pure 1Further Maths10 topics

The t-formulae

With t = tan(θ/2): sin θ = 2t/(1 + t²), cos θ = (1 − t²)/(1 + t²), tan θ = 2t/(1 − t²).
a cos x + b sin x = c becomes a quadratic in t. Solve it, then take x = 2 arctan t.
The substitution never reaches x = π, so test that value by hand.

Taylor series

Taylor about a: the coefficient of (x − a)^r is the rth derivative at a over r factorial.
Anchor where the derivatives are exact and the target is near. Maclaurin is the a = 0 special case.

Limits and L'Hospital's rule

0/0 says nothing by itself. Expand in series and compare the lowest surviving powers.
L'Hospital replaces f/g by f'/g' while the form stays indeterminate, one named round at a time.
Products, differences and power forms must be forced into a quotient before the rule applies.

Leibnitz's theorem and the Weierstrass substitution

Leibnitz: the nth derivative of fg is the binomial-weighted sum of split derivatives.
t = tan(x/2) with dx = 2dt/(1 + t²) turns a rational trig integrand into a rational function of t; transform the limits too.

Series solutions of differential equations

Solve for the highest derivative, then differentiate and substitute repeatedly; each round yields the next Taylor coefficient.
A given substitution turns the equation into a standard type. Transform the derivatives, solve, translate back.

Conic sections

Parabola (at², 2at); ellipse (a cos t, b sin t); hyperbola (a sec t, b tan t); rectangular hyperbola (ct, c/t).
Distance to focus = e × distance to directrix, with e < 1 an ellipse, e = 1 a parabola, e > 1 a hyperbola.
Foci sit at (±ae, 0) and directrices at x = ±a/e when the major axis is horizontal.
The hyperbola x²/a² − y²/b² = 1 has asymptotes y = ±(b/a)x.

Tangents, normals and loci of conics

Tangent to y² = 4ax at (at², 2at) is ty = x + at²; to xy = c² at (ct, c/t) it is x + t²y = 2ct.
A repeated root in the substituted quadratic means tangency, and a zero discriminant is how the conditions are derived.
For a locus, parametrise the moving point and eliminate the parameter; constants built from the roots survive.

The vector product and the scalar triple product

a × b is perpendicular to both factors, with |a × b| = |a||b| sin θ, the parallelogram's area.
|a·(b × c)| is the parallelepiped's volume, the tetrahedron takes a sixth, and zero means coplanar.

Numerical methods for differential equations

The forward difference steps ahead; the central difference straddles the point and is the more accurate.
Second order equations use (y_n+1 − 2y_n + y_n-1)/h² and need two starting values.
Simpson: h/3 with weights 1, 4, 2, …, 4, 1, and an even number of strips or no rule at all.

Inequalities and inequations

Collect on one side, factorise, and read a sign diagram; multiplying by an unknown sign loses intervals.
Square a modulus inequality when both sides are non-negative, otherwise split into cases and sketch to check.

Further Statistics 1Further Maths9 topics

Discrete random variables and expectation

E(X) = Σ x P(X = x) is the balance point, and Var(X) = E(X²) − μ² measures the scatter round it.
E(aX + b) = aE(X) + b while Var(aX + b) = a²Var(X). Shifting moves the centre; scaling squares into the spread.

The Poisson distribution

Poisson counts events occurring singly, independently and at a constant average rate, with E(X) = Var(X) = λ.
Independent Poissons add and windows scale λ, and B(n, p) becomes Po(np) for large n and small p.

Geometric and negative binomial distributions

Geometric: P(X = x) = p(1 − p)^x-1, mean 1/p, variance (1 − p)/p².
Negative binomial for the rth success: the combination arranges the earlier successes, and the mean is r/p.

Hypothesis tests for Poisson and geometric models

Poisson tests fix hypotheses on λ, and you compare the tail probability of the observed count under H₀ with the level.
Discreteness drops the actual significance level below the nominal one, and geometric upper tails are (1 − p) to the power x − 1.

The Central Limit Theorem

For large n from a random sample, the sample mean is approximately N(μ, σ²/n) and the total approximately N(nμ, nσ²), whatever the parent.
The parent itself never turns normal, and the standard error σ/√n needs four times the data for half the spread.

Goodness-of-fit tests

χ² sums (O − E)²/E, so each squared gap is weighted down by how large its class was expected to be.
Degrees of freedom are classes minus 1, minus one more for each parameter estimated from the data, counted after pooling.

Contingency tables

Expected = row total × column total ÷ grand total, which is independence written as arithmetic.
Degrees of freedom are (rows − 1)(columns − 1), because the margins have already used up the rest.

Probability generating functions

G(t) = E(t^X) stores every probability as a coefficient, G(1) = 1 always, and the mean is G'(1).
Variance is G''(1) + G'(1) − [G'(1)]², and independent sums multiply their generating functions.

The quality of tests

Type I rejects a true H₀, with probability equal to the test's size; Type II misses a real change.
Power = 1 − P(Type II error), rising as the alternative moves away from H₀, and with n fixed size and power trade off.

Further Pure 2Further Maths12 topics

Groups and their axioms

A group needs four axioms: closure, associativity, an identity, and an inverse for every element.
The order of a group is its size; the order of an element is the least power reaching the identity.
A group generated by one element is cyclic, and the integers modulo n under addition always are.

Subgroups, Lagrange's theorem and isomorphism

A subgroup needs the identity, closure and inverses; associativity comes free from the parent group.
Lagrange: every subgroup's order divides the group's order, and so does every element's order.

Reduction formulae

Integrate by parts once, rearrange into a recurrence, then walk it down to a base case you can integrate directly.

Arc length and surface area

Arc length integrates the hypotenuse: √(1 + (dy/dx)²) dx, or the parametric and polar equivalents.
A surface of revolution is 2π∫(radius) ds, with the same ds as the arc length integral.
Check every answer against a circle, a cone or a sphere before moving on.

Eigenvalues and eigenvectors

Eigenvectors satisfy Mv = λv; solve det(M − λI) = 0 for the eigenvalues, then substitute back.
Trace is the sum of the eigenvalues and determinant their product; symmetric matrices have perpendicular eigenvectors whenever the eigenvalues differ.

Diagonalisation and the Cayley-Hamilton theorem

P from eigenvectors, D from eigenvalues: P⁻¹MP = D, and Mⁿ = PDⁿP⁻¹ makes powers cheap.
For a symmetric matrix, an orthonormal set of eigenvectors makes P orthogonal and P⁻¹ is simply the transpose.
Cayley-Hamilton: a matrix satisfies its own characteristic equation, which yields inverses and collapses powers.

Further loci and regions in the Argand diagram

|z − a| = k|z − b| is a circle unless k = 1, when it degenerates to the perpendicular bisector.
A constant argument of a quotient traces an arc through a and b, with those two points excluded.
Sketch every boundary before shading, and record which ones the region includes.

Transformations of the complex plane

Under w = z², separate u = x² − y² and v = 2xy, then eliminate to find the image curve.
For a Möbius map, invert to z in terms of w first; w = 1/z turns lines missing the origin into circles through it.

The Euclidean algorithm and Bezout's identity

Euclid: divide, keep the remainder, repeat; the last non-zero remainder is the highest common factor.
Bezout: back substitution writes that factor as ax + by, and when it is 1 the coefficient x inverts a modulo b.

Modular arithmetic and Fermat's little theorem

Congruences add, subtract, multiply and take powers; reduce at every step to keep the numbers small.
Cancelling is only safe when the common factor is coprime to the modulus.
Fermat: for prime p and a not divisible by p, a to the power p − 1 is 1 modulo p, so exponents reduce mod p − 1.

Combinatorics

Multiply choices made in sequence, add alternatives, and subtract from the total when 'at least one' appears.
Permutations order the selection and combinations do not, so they differ by exactly r factorial.

Recurrence relations

Recurrences solve like differential equations: complementary function plus particular solution, constants fitted to the initial terms.
Second order relations use an auxiliary equation in m, and a repeated root needs the extra factor of n.
Modelled situations that grow by a factor and change by a fixed amount each step give u sub n+1 = au sub n + b.

Further Statistics 2Further Maths11 topics

Least squares regression and residuals

Least squares minimises the sum of the squared vertical residuals, giving b = S(xy)/S(xx), with the line through the mean point.
Residuals sum to zero, their pattern tests the model, and RSS = S(yy) − S(xy)²/S(xx) measures what is left unexplained.

Continuous random variables: density and distribution functions

For a continuous variable, probability is area under the density: P(a < X ≤ b) is the integral of f from a to b, and P(X = c) = 0.
F is the integral of f from the lower end, climbing from 0 to 1; differentiating F gives f back, and F(m) = 0.5 gives the median.

Mean, variance and skewness of continuous variables

Replace sums with integrals: E(X) is the integral of xf(x), E(g(X)) the integral of g(x)f(x), and Var(X) = E(X²) − [E(X)]².
The mode is where f peaks, endpoints included, any percentile solves F(x) = p/100, and the order of the three averages names the skew.

The continuous uniform distribution

U(a, b) has density 1/(b − a) on the interval, and a distribution function in three branches: 0 below a, (x − a)/(b − a) on it, 1 above b, so probability is proportion of length.
Its mean is the midpoint (a + b)/2 and its variance is (b − a)²/12, a standard deviation of about 29% of the range.

Correlation coefficients: product moment and Spearman

The product moment coefficient r = S(xy)/√(S(xx)S(yy)) measures linear association only, and linear coding leaves its size unchanged.
Spearman's rs = 1 − 6Σd²/[n(n² − 1)] measures agreement of orderings, with tied values sharing the average rank.

Testing a correlation coefficient

Hypotheses are about the population coefficient ρ or ρs, tested against a critical value that depends on n, the tail and the level.
Critical values fall as n grows; the product moment test assumes a bivariate normal population, while the rank test assumes nothing about shape.

Combinations of normal random variables

For independent normals, aX ± bY is normal with mean aμx ± bμy and variance a²σx² + b²σy².
Variances add for a difference as well as a sum, and n independent copies give variance nσ² while n times one gives n²σ².

Estimators, standard error and confidence intervals

An estimator is unbiased when its expected value is the parameter, and among unbiased ones you prefer the smaller variance.
The standard error σ/√n sets the width: a 95% interval is the sample mean ± 1.96 standard errors, and 95% of such intervals capture μ.

Comparing two normal means

The difference of two independent sample means is normal, with the two variance-over-n terms added under the square root.
With large samples the sample variances may replace the population ones, and an interval missing zero matches a significant two-tailed test.

Testing variances: chi-squared and the F-distribution

(n − 1)S²/σ² is chi-squared on n − 1 degrees of freedom, and the distribution is skewed, so a two-tailed test needs two different critical values.
The ratio of two sample variances is F on the two degrees of freedom, numerator first; the alternative hypothesis picks the numerator, and larger-over-smaller is the two-tailed procedure at half the level.

Confidence intervals and tests with the t-distribution

With σ estimated from the sample, use t on n − 1 degrees of freedom: heavier tails than the normal, so wider intervals.
Paired data becomes one sample of differences; independent samples with a common variance use the pooled s² on n₁ + n₂ − 2 degrees of freedom.

Further Mechanics 1Further Maths8 topics

Momentum and impulse

Impulse is force times time and equals the change in momentum, so a rebound needs an impulse equal to the sum of the two speeds.
In a collision the internal forces are equal and opposite, so when the external impulse is negligible the total momentum before equals the total momentum after.
Divide an impulse by the contact time to get the average force.

Impulse and momentum as vectors

In vector form I = mv − mu, so the i components and the j components each obey the principle on their own.
Take magnitudes only after the vector is known. A change of direction at constant speed still needs an impulse.

Work, energy and power

Work is Fd cos θ, and the work-energy principle says the total work done by all forces equals the change in kinetic energy.
With no resistance, kinetic plus gravitational potential energy stays constant.
Power is P = Fv, so the driving force falls as speed rises, and maximum speed is where it has dropped to equal the resistance.

Hooke's law and elastic strings

Hooke's law is T = λx/l, where x is the extension, l the natural length and λ the modulus in newtons.
A string pulls only and has zero tension when slack. A spring also pushes, with the compression in place of the extension.

Elastic potential energy

The energy stored is λx²/(2l), the area of the triangle under the Hooke's law line, so it grows with the square of the extension.
Add it to kinetic and gravitational potential energy, and set the total change equal to any work done against friction.

Direct impact and Newton's law of restitution

Newton's law of restitution says the separation speed is e times the approach speed, with 0 ≤ e ≤ 1.
Use it with conservation of momentum to get two equations in the two unknown velocities.
Kinetic energy is lost in every impact except e = 1; at e = 0 the spheres coalesce.

Successive impacts and impacts with a wall

Against a fixed surface only restitution applies: the rebound speed is e times the approach speed, and momentum is not conserved for the ball alone.
In a chain of impacts, carry signed velocities forward; a further collision happens exactly when the bodies are still approaching.

Oblique impact and impact with a smooth surface

A smooth surface leaves the parallel component untouched and multiplies the perpendicular component by −e.
The outgoing path therefore always hugs the surface more closely than the incoming one.
All the energy lost comes from the perpendicular component, which keeps only e² of its share.
For two smooth spheres the impulse acts along the line of centres, so resolve there and leave both perpendicular components alone.

Further Mechanics 2Further Maths10 topics

Angular speed and horizontal circular motion

Circular motion at constant speed still accelerates, towards the centre, with magnitude rω² or v²/r.
Some real force must supply that acceleration, so name it before writing an equation.
On a banked track the reaction alone suffices at the design speed given by tan θ = v²/rg.

Motion in a vertical circle

Energy relates speed to height, since the tension does no work; the radial equation then gives the tension at any point.
A string or an inside track can act inwards only, so it needs v² ≥ gr at the top; a rod or a wire can also push, and needs only v > 0.
From the lowest point, u² ≥ 5gr completes the circle and u² ≤ 2gr oscillates; anything between leaves the circle above the horizontal.

Centre of mass of a discrete distribution

The centre of mass is the mass-weighted average of the positions, taken one coordinate at a time, and it follows from moments, so an object balances when supported there and hangs with it below any point of suspension.

Centres of mass of plane figures and frameworks

For a uniform lamina, weight each piece by its area at its own centre.
The booklet gives the centres of seven standard uniform bodies, so read them off instead of memorising them.
Anything removed goes in with a negative area at its own centre.
For a framework, weight each rod by its length at its own midpoint, so the wire and the lamina of the same shape generally differ.

Centres of mass by integration

Cut into strips: x_G is ∫xy dx over ∫y dx, but y_G is ∫½y² dx over ∫y dx, because each strip acts at its own halfway height.
For a solid of revolution use y² in place of y and take the centre on the axis by symmetry; for a non-uniform body put ρ inside both integrals.

Equilibrium, suspension, toppling and sliding

A freely suspended body hangs with its centre of mass vertically below the point of suspension.
On a slope, sliding needs tan θ = μ and toppling needs tan θ = a/h, with a the base half-width and h the height of G.
Whichever of those two angles is smaller is the one that happens.

Newton's laws with a variable force

Use v dv/dx when the force depends on position and dv/dt when it depends on time; SUVAT is unavailable either way.
For gravitation, write GM as gR² so the inverse square force is mgR²/x², and integrate it for the work done.

Simple harmonic motion

Simple harmonic motion is the equation acceleration = −ω²x, with solutions a sin ωt or a cos ωt.
The period is 2π/ω whatever the amplitude.
Speed and displacement are linked by v² = ω²(a² − x²): fastest at the centre, at rest at the ends.

Oscillations on strings and springs

Measured from equilibrium the weight cancels, leaving acceleration = −(λ/ml)x: simple harmonic with ω² = λ/ml.
A spring stays simple harmonic throughout; a string goes slack above its natural length, so compare the amplitude with the equilibrium extension.

Further kinematics: acceleration as a function of x, t or v

Match the form of the acceleration to what it depends on: dv/dt for t or v, v dv/dx for x or v.
Separate, integrate, then apply the conditions at once.
A terminal speed is found by setting the acceleration to zero, not by taking a limit.

Decision Mathematics 1Further Maths10 topics

Algorithms, sorting and bin packing

An algorithm is a precise finite recipe, and its order says how the work grows with the size of the problem.
First fit, first fit decreasing and full bin packing are all quick and none is guaranteed optimal.
Always compare a packing with the lower bound of total size over bin size, rounded up.

Graphs: order, Eulerian paths and planarity

The orders always total twice the number of edges, so the number of odd nodes is always even.
For a connected graph, no odd nodes means Eulerian, exactly two means semi-Eulerian, and more than two means neither.
The planarity algorithm needs a Hamiltonian cycle: draw it as a circle and try to split the chords between inside and outside.

Minimum spanning trees: Prim and Kruskal

A spanning tree of n nodes has exactly n − 1 edges and no cycle; the minimum one has least total weight.
Prim grows outwards from a node and Kruskal works down the sorted edges rejecting cycles, and both always give the same total.

Shortest paths: Dijkstra and Floyd

Dijkstra makes permanent the smallest temporary label anywhere in the network, not the cheapest edge from where you are.
Backtrack from the destination through matching labels, since the match alone only names an edge on some shortest route; Floyd instead does every pair at once, one intermediate node per iteration.

Route inspection

Odd nodes force repetition, so pair them up and repeat a shortest path between each pair.
The answer is the total weight of the network plus the cost of the cheapest pairing, and the repeated edges must be named.

The travelling salesman problem

Any tour at all is an upper bound, and nearest neighbour finds one quickly, rarely the best one.
Doubling a minimum spanning tree also gives an upper bound, which short cuts then improve.
A lower bound comes from deleting a node, finding a minimum spanning tree on the rest, and adding back the two shortest edges from it.

Critical path analysis

The forward pass takes the largest incoming finish and the backward pass the smallest outgoing start.
An activity is critical when its earliest and latest starts agree; the critical activities form one path whose durations total the project duration.

Float, Gantt charts and scheduling

Total float is latest finish minus earliest start minus duration.
Float is shared along a chain, so two activities on the same path cannot both use it all.
The lower bound on workers is total work divided by project duration, rounded up; levelling shifts non-critical activities within their float to reach it.

Linear programming: formulation and graphical solution

The optimum of a linear program sits at a vertex of the feasible region, though not necessarily the furthest one from the origin.
A ≤ constraint gains a slack variable; a ≥ constraint loses a surplus variable and gains an artificial one.

The Simplex algorithm

Pivot on the most negative entry in the objective row.
Choose the row by the smallest non-negative ratio of value to pivot-column entry, skipping negative and zero entries.
Stop when no negative entries remain in the objective row; ≥ constraints need the two-stage or big-M method to get started.

Decision Mathematics 2Further Maths9 topics

Transportation problems

Balance the problem with a zero-cost dummy first, then fill from the north-west corner without looking at any cost.
A solution to an m by n problem uses m + n − 1 cells; fewer means degeneracy, fixed by recording a zero allocation.

The stepping-stone method

Shadow costs satisfy R + K = cost for every cell in use, and the improvement index of an unused cell is cost − R − K.
The most negative index enters; θ is the smallest minus-cell allocation, that cell exits, and the cost falls by θ times the index.
As a linear program it is minimise Σ c x subject to one equation per source and one per destination.

Allocation and the Hungarian algorithm

Reducing by rows then columns removes the same amount from every complete allocation, so the cheapest one is unchanged.
If the minimum covering lines number fewer than the matrix size, augment using the smallest uncovered entry and test again.

Flows in networks: cuts and capacity

A cut's capacity is the total of the arcs crossing from the source side to the sink side only.
Every flow is bounded by every cut, so a small cut is a proof of an upper bound; restricted vertices are split into two joined by one arc.

Maximum flow and the labelling procedure

Label spare capacity forwards and existing flow backwards; an augmenting path carries the smallest of those numbers along it.
The maximum flow equals the minimum cut, so a flow is proved maximal by exhibiting a cut of the same capacity.
The minimum cut separates the vertices still reachable from the source from those that are not.

Dynamic programming

Bellman's principle says any part of an optimal path is optimal, which licenses working backwards from the end.
Tabulate stage against state, quoting values already found.
For minimax or maximin replace the addition with a maximum or a minimum along the route.

Game theory: play safe and stable solutions

The matrix is from the row player's view, so they maximise the row minima and the column player minimises the column maxima.
The game is stable exactly when maximin equals minimax, and that common number is the value.
Delete any strictly dominated row or column before doing anything else.

Mixed strategies

Plot the expected pay-off against each opposing choice as a line in p; the optimum is the highest point of the lower boundary, which is a crossing or an end.
Equate the pair meeting there for p and the value, test p = 0 and p = 1 too, check against the other player's mix, and send larger games to Simplex.

Decision analysis

Chance nodes take the probability-weighted average and decision nodes take the best available, worked back to the first decision.
Expected monetary value is an average over repetitions, so for a one-off decision with large stakes utility may favour the safer option.

Each topic links to its full lesson. See also the equation index, key definitions and the revision checklist.