Checklist

Revision checklist

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Jump to: Proof · Algebra and functions · Coordinate geometry · Sequences and series · Trigonometry · Exponentials and logarithms · Differentiation · Integration · Numerical methods · Vectors · Statistics · Mechanics · Further proof · Complex numbers · Matrices · Further algebra and series · Further vectors · Further calculus · Hyperbolic functions · Polar coordinates · Differential equations · Further Pure 1 · Further Statistics 1 · Further Pure 2 · Further Statistics 2 · Further Mechanics 1 · Further Mechanics 2 · Decision Mathematics 1 · Decision Mathematics 2

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Proof

The structure of proof: deduction and exhaustion Edexcel 9MA0 1.1

Set a proof out properly, from stated assumptions through checkable steps to a conclusion.
Prove statements by deduction and by exhaustion, and choose the method that suits the claim.

Disproof and proof by contradiction Edexcel 9MA0 1.1

Disprove a universal claim with a single counter example, sought where the claim is most likely to fail.
Reproduce the contradiction proof that √2 is irrational.
Reproduce the contradiction proof that the primes never run out.
Apply contradiction to a statement you have never seen before.

Algebra and functions

Indices and surds Edexcel 9MA0 2.1, 2.2

Use the index laws for every rational exponent, negative and fractional included.
Simplify a surd by its largest square factor and do surd arithmetic exactly.
Rationalise a denominator, using the conjugate when the denominator has two terms.

Quadratic functions Edexcel 9MA0 2.3

Read roots, vertex and intercept off the three written forms of a quadratic.
Complete the square when the leading coefficient is not 1.
Use the discriminant to count real roots.
Find the values of a parameter that force real, repeated or no roots.
Solve a quadratic in a function of the unknown by naming the substitution.

Simultaneous equations and inequalities Edexcel 9MA0 2.4, 2.5

Solve one linear and one quadratic equation together by substitution.
Read the discriminant of the substituted equation as the number of intersections.
Solve linear, quadratic and fractional inequalities, and write the answers in set notation.
Shade a region of the plane with the right boundary convention.

Polynomials and the factor theorem Edexcel 9MA0 2.6

Divide a polynomial by a linear expression of the form (ax ± b), quotient and remainder included.
Use the factor theorem to find factors, fix unknown coefficients, and factorise a cubic completely.
Simplify rational expressions by factorise-and-cancel and by algebraic division.

Graphs, proportion and transformations Edexcel 9MA0 2.7, 2.9

Sketch a cubic or quartic from its factors, with crossings and touches in the right places.
Sketch y = a/x and y = a/x² and state their asymptotes.
Set up direct, inverse and inverse-square proportion, and fit the constant from one data point.
Apply the four transformations of y = f(x), singly and in combination.

Functions, inverses and the modulus Edexcel 9MA0 2.7, 2.8, 2.9

Distinguish a one-to-one mapping from a many-to-one mapping, and state a function's domain and range.
Build composite functions in the right order.
Find an inverse and give its domain and range.
Sketch modulus graphs, and solve modulus equations and inequalities from them.

Partial fractions Edexcel 9MA0 2.10

Split a fraction whose denominator is two or three distinct linear brackets.
Handle a repeated linear factor with the two-fraction template.

Functions in modelling Edexcel 9MA0 2.11

Match a situation's behaviour to the function family that describes it.
Fit and interpret a model's constants in context, and state clearly where the model fails.

Coordinate geometry

Straight lines Edexcel 9MA0 3.1

Find a line's equation from a point and a gradient, or from two points, in either standard form.
Use the gradient conditions for parallel and perpendicular lines, and show the working that earns the mark.
Build and interpret a straight-line model, with the gradient as a rate and the intercept as a starting value.

Circles Edexcel 9MA0 3.2

Move between the centre-radius form and the general form by completing the square.
Find the tangent at a point from the gradient of the radius.
Get a chord length from the half-chord right triangle without solving anything.
Find the circle through three given points.

Parametric equations Edexcel 9MA0 3.3, 3.4

Trace and sketch a curve given parametrically, treating the parameter as a clock.
Convert between parametric and Cartesian forms, and say which part of the Cartesian curve you actually get.
Use parametric equations as models, projectile motion included.

Sequences and series

The binomial expansion Edexcel 9MA0 4.1

Compute binomial coefficients from Pascal's triangle or from the nCr formula.
Expand (a + bx)ⁿ for positive integer n, and pull out one coefficient without expanding the rest.
Truncate an expansion to estimate a power of a number close to 1.

The general binomial expansion Edexcel 9MA0 4.1

Expand (1 + x)ⁿ for negative and fractional n, as far as a stated power.
State the range of validity, and say why the series is silent outside it.
Handle (a + bx)ⁿ by taking aⁿ out of the bracket first.
Combine expansions with partial fractions, and choose the right window for the sum.

Sequences and sigma notation Edexcel 9MA0 4.2, 4.3

Generate a sequence from an nth-term formula or from a recurrence, and classify how it behaves.
Read and write sums in sigma notation.

Arithmetic series Edexcel 9MA0 4.4, 4.6

Use u_n = a + (n − 1)d fluently, including to recover a and d from two given terms.
Prove the sum formula, and use it in both of its forms.
Apply arithmetic series to modelling, saving schemes included.

Geometric series Edexcel 9MA0 4.5, 4.6

Use u_n = arⁿ⁻¹, and recover a and r from two given terms.
Prove the finite sum formula by the subtraction trick, and use it.
Say when a series converges, and find the sum to infinity.
Answer how-many-terms and how-many-years questions with logarithms.

Trigonometry

Triangles and the sine and cosine rules Edexcel 9MA0 5.1

Use the sine rule to find sides and angles, pairing each side with the angle facing it.
Use the cosine rule from two sides and the included angle, or from all three sides.
Handle the ambiguous case, and find areas from two sides and the included angle.

Trigonometric graphs and equations Edexcel 9MA0 5.3, 5.5, 5.7

Sketch sine, cosine and tangent, and use their symmetry and period.
Quote exact values at the standard angles, and extend them past 90° by symmetry.
Use tan θ = sin θ/cos θ and sin²θ + cos²θ = 1 to reshape an equation.
Find every solution in a stated interval, multiple angles and hidden quadratics included.

Radians, arcs and small angles Edexcel 9MA0 5.1, 5.2

Convert between degrees and radians, and know the standard angles in both.
Use s = rθ and A = ½r²θ for arcs and sectors.
Find a segment area as sector minus triangle.
Apply the small-angle approximations, in radians only.

Reciprocal and inverse trigonometric functions Edexcel 9MA0 5.4, 5.5

Work with sec, cosec and cot, including their graphs, asymptotes and ranges.
Derive and use sec² = 1 + tan² and cosec² = 1 + cot².
Solve equations that are quadratics in a reciprocal ratio.
Use arcsin, arccos and arctan with their restricted domains and ranges.

Compound angles and the harmonic form Edexcel 9MA0 5.6, 5.8

Use the compound angle formulae, forwards and backwards.
Derive the double angle formulae by setting B = A, and choose between the three forms of cos 2A.
Rearrange cos 2A into the half-angle forms that integration later depends on.
Construct identity proofs of the kind the specification names.
Write a sin θ + b cos θ in harmonic form, and use it for equations and extremes.

Trigonometric modelling Edexcel 9MA0 5.9

Build and read wheel-style models of the form k − R cos ωt, in radians.
Extract centre, amplitude, period and phase from any periodic model.
Use the harmonic form to analyse a model carrying two trig terms.

Exponentials and logarithms

Exponential functions and e Edexcel 9MA0 6.1, 6.2

Sketch y = a^x for any positive base except 1, growth and decay alike, with the right asymptote.
Use the gradient property of e^kx, and transform e-curves into the form e^ax+b + c.
Recognise when a situation calls for an exponential model, and read its constants.

Logarithms and their laws Edexcel 9MA0 6.3, 6.4, 6.5

Convert between exponential and logarithmic statements, in any base and in base e.
Solve e^ax+b = p and ln (ax + b) = q by applying the right inverse to the whole side.
Use the three log laws to combine, split and simplify expressions.
Solve a^x = b, and anything else with the unknown in an exponent.

Log graphs and exponential models Edexcel 9MA0 6.6, 6.7

Linearise y = axⁿ and y = kb^x with the right choice of log plot.
Read the constants of a model off a log graph's gradient and intercept.
Interpret, use and criticise exponential growth and decay models.

Differentiation

The derivative from first principles Edexcel 9MA0 7.1

Explain the derivative as the limit of chord gradients, and use f'(x) and dy/dx notation.
Differentiate small powers of x from first principles.
Sketch a gradient function from the graph of a curve.
Read the second derivative as the rate of change of the gradient.

Differentiating powers of x Edexcel 9MA0 7.2

Differentiate xⁿ for any rational n, with sums, differences and constant multiples.
Rewrite roots, reciprocals, products and quotients as powers before differentiating.

Tangents, turning points and curve behaviour Edexcel 9MA0 7.3, 7.1

Find equations of tangents and normals at a point on a curve.
Locate stationary points and classify them with the second derivative.
State the intervals on which a curve is increasing or decreasing.
Use convex, concave and point of inflection correctly, and know what f'' = 0 does and does not prove.
Solve practical maximisation and minimisation problems, domain included.

Differentiating trig, exponentials and logs Edexcel 9MA0 7.1, 7.2

Differentiate sin kx, cos kx, tan kx, e^kx and ln x fluently.
Handle a^kx, whose derivative carries a factor of ln a.
Differentiate sin x and cos x from first principles using the small-angle approximations.
Find gradients and tangents on trig, exponential and log curves.

The product, quotient and chain rules Edexcel 9MA0 7.4

Differentiate composite functions with the chain rule.
Differentiate products and quotients with their rules, laid out cleanly.
Use the booklet derivatives of sec x, cosec x and cot x.
Combine the rules on nested expressions like cos² x, tan² 2x and e^3x/x.

Implicit and parametric differentiation Edexcel 9MA0 7.5

Differentiate implicit relations term by term, with the chain rule supplying dy/dx.
Use dy/dx = 1/(dx/dy) for relations given as x in terms of y.
Find gradients, tangents and normals on parametric curves by dividing rates.

Rates of change and building differential equations Edexcel 9MA0 7.4, 7.6

Connect rates through the chain rule and evaluate them at an instant.
Translate rate sentences into differential equations, signs and constants included.

Integration

Integration as antidifferentiation Edexcel 9MA0 8.1, 8.2

Integrate powers of x by reversing the power rule, with the constant of integration.
Rewrite expressions into powers before integrating, and check answers by differentiating.
Recover a curve from its gradient function and one known point.

Definite integrals and areas Edexcel 9MA0 8.3

Evaluate definite integrals with the square-bracket routine.
Find areas under curves and between a curve and a line.
Handle regions below the axis by splitting at the roots.

Integrating standard functions Edexcel 9MA0 8.2

Integrate e^kx, 1/x, sin kx, cos kx and sec² kx, dividing by k throughout.
Use ∫(1/x) dx = ln|x| + c, the case the power rule could not reach.
Reshape sin² x, cos² kx and tan² x by identity before integrating.

Integration by substitution and by parts Edexcel 9MA0 8.5

Spot reverse chain rule patterns, f'/f and f' times a power of f, at sight.
Run full substitutions, converting integrand, dx and limits together.
Integrate by parts, choosing which factor to differentiate, including ∫ln x dx.
Apply parts twice where one pass is not enough.

Integrating rational functions Edexcel 9MA0 8.6

Split a rational integrand into partial fractions and integrate each piece.
Recognise f'/f numerators and negative-power forms that need no split at all.
Evaluate definite integrals of rational functions, compressing the answer with log laws.

Areas, parametric curves and the limit of a sum Edexcel 9MA0 8.3, 8.4

Find the area between two curves with one integral of top minus bottom.
Find areas under parametric curves using ∫y (dx/dt) dt, with a Cartesian check.
Read a definite integral as the limit of a sum of strips of width δx.

Solving differential equations Edexcel 9MA0 8.7, 8.8

Solve separable equations, factorising first where separation needs it.
Turn a general solution into a particular one with a known condition.
Interpret solutions in context and state the limits of what the model can say.

Numerical methods

Locating roots and iteration Edexcel 9MA0 9.1, 9.2

Trap a root by a change of sign, with continuity stated.
Prove an accuracy claim by choosing the right pair of test values.
Name the two ways the sign change test fails.
Iterate x_n+1 = g(x_n) and read convergence off a staircase or cobweb diagram.

Newton-Raphson and the trapezium rule Edexcel 9MA0 9.3, 9.4, 9.5

Run Newton-Raphson, reading each step as a tangent sliding to the axis.
Explain the method's failure when the tangent is horizontal or nearly so.
Estimate definite integrals by the trapezium rule with a stated number of strips.
Decide from the bend of the curve whether the estimate is too big or too small.

Vectors

Vectors in two dimensions Edexcel 9MA0 10.1, 10.2, 10.3, 10.4, 10.5

Work in column and i-j notation, converting between components and magnitude-direction form.
Add vectors, scale them, and recognise parallel vectors.
Use position vectors, $\overrightarrow{AB}$ = b − a, and distances to solve geometric problems.
Find the unit vector in the direction of a given vector, and scale it to any length asked for.

Vectors in three dimensions Edexcel 9MA0 10.1, 10.2, 10.3, 10.4, 10.5

Work with i, j, k components, magnitudes and distances in three dimensions.
Solve geometric problems in space: collinearity, midpoints, and triangle shapes by distance alone.

Statistics

Sampling and the large data set Edexcel 9MA0 P3 1.1

Define population, census, sample and sampling frame, and say when a census is impractical.
Describe simple random, systematic, stratified, quota and opportunity sampling, and give an advantage and a limitation of each in a context.
Criticise someone else's sampling method by naming the group it under-represents.
Know the shape of the large data set: eight stations, two years, and the columns that are not numbers.

Measures of location and spread Edexcel 9MA0 P3 2.3

Find the median and quartiles of a listed data set using Edexcel's position rules.
Estimate the median, quartiles and percentiles of grouped data by linear interpolation.
Calculate variance and standard deviation from raw data or from the sums Σx and Σx².
Undo coding to recover the mean and standard deviation of the original variable.

Representing and interpreting data Edexcel 9MA0 P3 2.1, 2.4

Construct and read histograms using frequency density, including recovering frequencies from areas.
Draw and compare box plots, and identify outliers with the quartile fence rule.
Describe skewness from quartiles or from the mean against the median.
Interpret cumulative frequency diagrams, and clean data before summarising it.
Choose or criticise a diagram for the job the question is doing.

Correlation and regression Edexcel 9MA0 P3 2.2, 5.1

Describe correlation from a scatter diagram and interpret a product moment correlation coefficient r between −1 and 1.
Use a regression line y = a + bx to make predictions, interpreting a and b in context.
Say when a prediction is trustworthy, and name the lurking variable when correlation is being mistaken for cause.
Take logs to straighten y = axⁿ and y = kb^x, then read the constants off the regression line.

Probability and Venn diagrams Edexcel 9MA0 P3 3.1, 3.2

Represent events on a Venn diagram, including three sets, and read probabilities from its regions.
Use P(A ∪ B) = P(A) + P(B) − P(A ∩ B) in both directions.
Test events for mutual exclusivity and for independence, and keep the two ideas apart.

Conditional probability Edexcel 9MA0 P3 3.2, 3.3

Calculate conditional probabilities from the formula, from Venn diagrams and from two-way tables.
Build tree diagrams for successive events, including sampling without replacement.
Use P(A|B) = P(A) as a test of independence.
Criticise the assumptions behind a probability model and say which way the answer would move.

The binomial distribution Edexcel 9MA0 P3 4.1, 4.3

Use a discrete probability distribution given as a table or as a formula, including finding an unknown constant.
Recognise and name the discrete uniform distribution, and say why a given situation is or is not one.
State and check the four conditions for X ~ B(n, p) to model a situation.
Calculate P(X = x) from the formula and cumulative probabilities from a calculator.
Convert “more than”, “at least” and “fewer than” into the ≤ form a calculator will accept.

The normal distribution Edexcel 9MA0 P3 4.2, 4.3

Use the shape and symmetry of X ~ N(μ, σ²), including the points of inflection at μ ± σ.
Find probabilities and inverse-normal values with a calculator.
Standardise with Z = (X − μ)/σ to find an unknown μ or σ from given probabilities.
Approximate a binomial by a normal when n is large and p is near 0.5, applying a continuity correction.
Say when neither model suits the context.

Hypothesis testing with the binomial Edexcel 9MA0 P3 5.1, 5.2

Set up H₀ and H₁ for a claim about a binomial probability p.
Carry out one- and two-tailed tests by comparing a tail probability with the significance level, concluding in context.
Find a critical region and state the actual significance level of a test.

Hypothesis testing: correlation and the normal Edexcel 9MA0 P3 5.1, 5.3

Test H₀: ρ = 0 against a critical value read from the table, using the right column for the number of tails.
Use the fact that the mean of n observations of N(μ, σ²) is N(μ, σ²/n).
Carry out a hypothesis test for the mean of a normal distribution with known σ, concluding in context.

Mechanics

Modelling, quantities and units Edexcel 9MA0 P3 6.1

Use the standard modelling words and say what each assumption removes.
Work in SI units, converting into them before any calculation.
Distinguish scalars from vectors, and signed values from magnitudes.

Kinematics with constant acceleration Edexcel 9MA0 P3 7.1, 7.2, 7.3

Read velocity from a displacement-time graph, and acceleration and displacement from a velocity-time graph.
Select and use the five suvat equations for constant acceleration.
Split a multi-stage motion at the moments the acceleration changes.
Model vertical motion under gravity, using the symmetry of up-and-down flight.

Kinematics with variable acceleration Edexcel 9MA0 P3 7.4

Differentiate x(t) to get velocity and acceleration, and integrate a(t) back with constants found from stated conditions.
Find times and positions where a particle is at rest or changes direction.
Distinguish displacement from total distance when the motion reverses.
Differentiate and integrate a position vector in i and j notation, and find speed as the magnitude of the velocity.

Forces and Newton's laws Edexcel 9MA0 P3 8.1, 8.2, 8.3, 8.5

Draw free body diagrams with weight, normal reaction, tension, thrust and friction correctly placed.
Apply F = ma along a chosen direction, and the equilibrium condition when a = 0.
Resolve a force at an angle into components, and see what that does to the normal reaction.
Work with forces written in i and j notation, including the magnitude and direction of a resultant.
Combine F = ma with suvat when a constant force produces constant acceleration.

Connected particles and pulleys Edexcel 9MA0 P3 8.4

Model connected objects sharing one acceleration through a light inextensible string.
Choose between whole-system and single-particle equations deliberately.
Analyse smooth-pulley systems, including a mass on a table, finding acceleration and tension.

Projectiles Edexcel 9MA0 P3 7.5

Resolve a launch velocity into horizontal and vertical components.
Apply constant-velocity motion horizontally and suvat with g vertically, linked by time.
Find time of flight, range, greatest height, and the velocity at any instant.
Derive and use the equation of the path.

Friction and inclined planes Edexcel 9MA0 P3 8.2, 8.6

Use F ≤ μR, with equality only in limiting equilibrium or during sliding.
Work out R properly when the applied force has a vertical component.
Resolve forces along and perpendicular to an inclined plane.
Decide whether an object on a rough slope slips, and find its acceleration when it does.

Statics of a particle Edexcel 9MA0 P3 8.4, 8.5, 8.6

Resolve a set of coplanar forces in two perpendicular directions and set each sum to zero.
Solve string-and-weight equilibrium problems for unknown tensions.
Handle equilibrium on rough surfaces, including the least and greatest force that keeps an object still.

Moments Edexcel 9MA0 P3 9.1

Calculate moments as force times perpendicular distance, with a consistent sense of rotation.
Use both equilibrium conditions for beams: forces balance, and moments about any point balance.
Model uniform and non-uniform beams, and use tilting conditions where a reaction vanishes.

Further proof

Proof by induction: sums and series Edexcel 9FM0 CP1 1.1

Prove a sum formula by induction, showing plainly where the assumption is used.
Close a proof with the conclusion sentence that earns the final mark.

Induction: divisibility and matrices Edexcel 9FM0 CP1 1.1

Prove divisibility statements by induction using the subtract-a-multiple rearrangement.
Prove a formula for the nth power of a matrix by induction, shown entry by entry.

Complex numbers

Complex arithmetic and the Argand diagram Edexcel 9FM0 CP1 2.1, 2.2, 2.3, 2.4

Solve any quadratic, reading complex roots from a negative discriminant.
Add, subtract and multiply complex numbers without special rules.
Divide by using the conjugate to clear the denominator.
Plot numbers and their conjugates on the Argand diagram, and read addition as vectors.

Modulus, argument and loci Edexcel 9FM0 CP1 2.2, 2.5, 2.6, 2.7

Find the modulus and argument of a complex number in every quadrant.
Convert between x + yi and modulus-argument form, and multiply and divide in it.
Sketch and describe circles, perpendicular bisectors and half-lines.
Shade the region satisfying an inequality in z.

De Moivre's theorem and trigonometric identities Edexcel 9FM0 CP2 2.6, 2.8, 2.9

Use De Moivre's theorem to evaluate powers of complex numbers.
Write numbers in exponential form and use it for products and powers.
Derive multiple-angle identities such as cos 3θ in terms of cos θ.

Roots of unity and complex roots Edexcel 9FM0 CP2 2.10, 2.11

Find all nth roots of unity and place them on the unit circle.
Solve z to the n = w for any non-zero complex w, spacing the roots by 2π/n, and know that w = 0 gives z = 0 with multiplicity n.
Use the geometry, including regular polygons and roots summing to zero.

Matrices

Matrix algebra and transformations Edexcel 9FM0 CP1 3.1, 3.2, 3.3

Add and multiply matrices, respecting the size rules and the order of factors.
Use the zero and identity matrices, and say why AB = 0 does not force A or B to be zero.
Read a transformation matrix from the images of the unit vectors.
Identify and construct rotations, reflections, stretches and enlargements in the plane.
Write down the standard 3 × 3 reflections in coordinate planes and rotations about axes.

Determinants and inverses Edexcel 9FM0 CP1 3.5, 3.6

Evaluate 2 × 2 and 3 × 3 determinants and interpret them as scale factors.
Recognise singular matrices and describe what a zero determinant does to space.
Find and verify inverses of 2 × 2 and 3 × 3 matrices, and use the reversal rule for products.

Systems of equations and invariance Edexcel 9FM0 CP1 3.4, 3.7, 3.8

Write simultaneous equations as a matrix equation and solve with an inverse.
Interpret consistent and inconsistent three-plane systems geometrically.
Find invariant points and invariant lines of a matrix transformation.

Further algebra and series

Roots of polynomials Edexcel 9FM0 CP1 2.1, 2.3, 4.1, 4.2

Read the sum and product of roots straight from a polynomial's coefficients.
Evaluate symmetric functions such as α² + β² + γ² without finding any root.
Build a new polynomial whose roots are a transformation of the old ones.
Solve a cubic or quartic with real coefficients when one complex root is given.

Summing series and the method of differences Edexcel 9FM0 CP1 4.3, 4.4

Quote and combine the standard results for Σr, Σr² and Σr³.
Split a term into a difference and telescope the sum.

Maclaurin series Edexcel 9FM0 CP2 4.5, 4.6

Derive a Maclaurin series from repeated differentiation at zero.
Quote the standard series and state where ln(1 + x) and arctan x are valid.
Adapt standard series to composites, products and quotients.

Further vectors

Lines and planes in three dimensions Edexcel 9FM0 CP1 6.1, 6.2, 6.3, 6.4

Write a line as r = a + λb and convert to cartesian form and back.
Write a plane as r·n = d and as ax + by + cz = e, and switch between them.
Find a plane's normal from two directions lying in it.
Decide whether two lines meet, are parallel or are skew.

Intersections, angles and distances Edexcel 9FM0 CP1 6.3, 6.5

Substitute a parametric line into a plane's equation to find where they meet.
Compute angles between a line and a plane, and between two planes.
Apply the perpendicular distance formula from a point to a plane.
Find the distance from a point to a line, and the shortest distance between two skew lines.

Further calculus

Volumes of revolution Edexcel 9FM0 CP1 5.1

Use V = π∫y² dx for rotation about the x-axis, with the right limits.
Swap to V = π∫x² dy for rotation about the y-axis.
Subtract solids for a region between two curves, and handle parametric curves.
Check answers against known solids such as cones.

Mean values and improper integrals Edexcel 9FM0 CP2 5.2, 5.3

Compute the mean value of a function as the integral divided by the width.
Evaluate improper integrals as limits, stating convergence or divergence.
Split an integral at an interior singularity and test each one-sided limit separately.

Calculus with inverse trigonometric functions Edexcel 9FM0 CP2 5.4, 5.5, 5.6

Differentiate arcsin x, arccos x and arctan x from scratch via implicit differentiation.
Integrate 1/√(a² − x²) and 1/(a² + x²) by recognising the patterns.
Complete the square and use partial fractions to reach the same two patterns.
Choose and carry out the substitutions x = a sin θ and x = a tan θ for the associated forms.

Hyperbolic functions

Hyperbolic functions and identities Edexcel 9FM0 CP2 8.1, 8.3, 8.4

Define sinh, cosh, tanh and their reciprocals from exponentials, and sketch the graphs.
Prove and use cosh²x − sinh²x = 1 and its relatives.
Apply Osborn's rule to convert a trig identity into its hyperbolic twin.
Solve hyperbolic equations exactly via the logarithmic forms of the inverses.

Calculus with hyperbolic functions Edexcel 9FM0 CP2 8.2, 8.5

Differentiate sinh, cosh and tanh directly from the exponential definitions.
Integrate 1/√(x² + a²) and 1/√(x² − a²) via arsinh and arcosh.
Choose a hyperbolic substitution when recognition alone will not do.

Polar coordinates

Polar curves Edexcel 9FM0 CP2 7.1, 7.2

Plot points from (r, θ) and convert both ways with x = r cos θ, y = r sin θ.
Convert polar equations to cartesian form and recognise the result.
Sketch the standard families, including circles, half-lines, cardioids and roses.
Locate tangents parallel and perpendicular to the initial line.

Areas with polar coordinates Edexcel 9FM0 CP2 7.3

Apply A = ½∫r² dθ with limits that trace the region exactly once.
Square and simplify r with double angle identities before integrating.
Find the area of a region bounded by two polar curves.

Differential equations

First order equations and integrating factors Edexcel 9FM0 CP2 9.1, 9.2

Recognise the linear form dy/dx + P(x)y = Q(x) and rearrange into it.
Build the integrating factor and collapse the left side to one derivative.
Sketch members of the family of solution curves a general solution describes.
Fix the constant from a boundary condition and check by substituting back.

Second order equations Edexcel 9FM0 CP2 9.2, 9.4, 9.5, 9.6

Reduce ay'' + by' + cy = 0 to its auxiliary quadratic and classify the roots.
Write the general solution in each of the three root cases.
Solve ay'' + by' + cy = f(x) as complementary function plus particular integral.
Use two initial conditions to pin both constants, applied to the full solution.

Modelling with differential equations Edexcel 9FM0 CP2 9.3, 9.7, 9.8, 9.9

Recognise simple harmonic motion from its differential equation and state period and amplitude.
Model a damped oscillator and interpret each term physically.
Classify light, critical and heavy damping via the discriminant.
Reduce a coupled pair of first order equations to one second order equation.

Further Pure 1

The t-formulae Edexcel 9FM0 FP1 1.1, 1.2, 1.3

Derive sin θ, cos θ and tan θ in terms of t = tan(θ/2), and quote them without hesitating.
Prove trig identities by converting every term to t.
Turn a cos x + b sin x = c into a quadratic in t and solve it.
Test x = π by hand, since the substitution cannot reach it.

Taylor series Edexcel 9FM0 FP1 2.1

Quote the Taylor series of f about x = a in powers of (x − a), and assemble it from a derivative table.
Choose the anchor closest to the point being approximated.

Limits and L'Hospital's rule Edexcel 9FM0 FP1 2.2, 2.4

Recognise the indeterminate forms 0/0 and ∞/∞, and say why they carry no verdict.
Evaluate limits by substituting series expansions and cancelling.
Apply L'Hospital's rule, repeatedly where needed, and check its conditions first.
Convert products, differences and power forms into a quotient before starting.

Leibnitz's theorem and the Weierstrass substitution Edexcel 9FM0 FP1 2.3, 2.5

Apply Leibnitz's theorem to find high derivatives of products directly.
Convert trig integrands to rational functions of t = tan(x/2).
Transform the limits of a definite integral along with the variable.

Series solutions of differential equations Edexcel 9FM0 FP1 3.1, 3.2

Extract successive derivative values at the starting point directly from the equation.
Assemble the Taylor series of the solution as far as a stated power.
Apply a given substitution to reduce an equation to a solvable type.

Conic sections Edexcel 9FM0 FP1 4.1, 4.2

Find the cartesian and parametric forms of the four standard conics in the booklet and use them.
Compute eccentricity, foci and directrices from a curve's equation.
Use the focus-directrix property to solve distance problems.
Write down the asymptotes of a hyperbola and use them when sketching.
Handle an ellipse whose major axis runs vertically.

Tangents, normals and loci of conics Edexcel 9FM0 FP1 4.3, 4.4

Find tangents and normals at a general parametric point of any of the four conics.
Quote the standard tangent forms, and derive them when the question demands it.
Use and verify the tangency condition for y = mx + c.
Eliminate the parameter to find the locus of a moving point.

The vector product and the scalar triple product Edexcel 9FM0 FP1 5.1, 5.2, 5.3

Compute a × b from components and check it is perpendicular to both factors.
Use |a × b| as an area and |a·(b × c)| as a volume, with the tetrahedron's sixth.
Write a line as (r − a) × b = 0 and read off direction ratios and cosines.
Find distances in three dimensions using the vector and triple products.

Numerical methods for differential equations Edexcel 9FM0 FP1 6.1, 6.2

Quote and apply the forward and central difference formulae for dy/dx.
Step a first or second order equation forward with a stated step length h.
Apply Simpson's rule with an even number of strips and judge its accuracy.

Inequalities and inequations Edexcel 9FM0 FP1 7.1

Solve rational inequalities by collecting on one side and using a sign diagram.
Handle modulus inequalities by squaring or by splitting into cases.
State solution sets correctly, excluding the values that break a denominator.
Use a sketch as a check on an algebraic answer.

Further Statistics 1

Discrete random variables and expectation Edexcel 9FM0 FS1 1.1

Compute E(X) and Var(X) from a probability distribution table.
Evaluate E(g(X)) for functions such as X² and aX + b.
Use the mean and variance to judge whether a proposed model fits observed data.

The Poisson distribution Edexcel 9FM0 FS1 2.1, 2.2, 2.3

Model random events with Po(λ) and compute probabilities on a calculator.
State the conditions a Poisson model requires and check them against a context.
Use E(X) = Var(X) = λ, and the additive property for independent counts.
Approximate B(n, p) by Po(np) when n is large and p is small.

Geometric and negative binomial distributions Edexcel 9FM0 FS1 3.1, 3.2, 3.3

Model the trial of a first success with the geometric distribution.
Extend to the rth success with the negative binomial distribution.
Quote and apply the means and variances of both.

Hypothesis tests for Poisson and geometric models Edexcel 9FM0 FS1 4.1, 4.2

State hypotheses about λ for a Poisson model or p for a geometric one.
Find the tail probability of the observed value and compare with the level.
Locate a critical region, find its actual size, and report conclusions in context.

The Central Limit Theorem Edexcel 9FM0 FS1 5.1

State the Central Limit Theorem and the distribution it gives the sample mean.
Compute probabilities for a sample mean or a sample total from a parent distribution meeting the theorem's conditions, for large enough n.
Say what the theorem does not claim.

Goodness-of-fit tests Edexcel 9FM0 FS1 6.1

Compute expected frequencies from a proposed model and form the χ² statistic.
Count degrees of freedom, subtracting one for each estimated parameter.
Pool classes with expected frequency below 5 and recount the classes.
Fit and test a Poisson or binomial model from start to finish.

Contingency tables Edexcel 9FM0 FS1 6.1

Compute expected frequencies as row total × column total ÷ grand total.
Use (rows − 1)(columns − 1) degrees of freedom.
State the hypotheses as independence and association, and conclude in context.

Probability generating functions Edexcel 9FM0 FS1 7.1, 7.2, 7.3

Define G(t) = E(t^X) and derive it for standard distributions.
Find the mean from G'(1) and the variance from G''(1) + G'(1) − [G'(1)]².
Match a generating function to its distribution using the booklet's table.
Use the product rule for generating functions of independent sums.

The quality of tests Edexcel 9FM0 FS1 8.1

Distinguish Type I and Type II errors and compute their probabilities.
Find the size of a test from its critical region.
Evaluate and interpret the power function at particular alternatives.

Further Pure 2

Groups and their axioms Edexcel 9FM0 FP2 1.1, 1.2, 1.3

State the four group axioms and test whether a set and operation satisfy them.
Build and read a Cayley table, and spot the identity and the inverses in it.
Find the order of a group and the order of an element.
Recognise cyclic groups and name a generator.

Subgroups, Lagrange's theorem and isomorphism Edexcel 9FM0 FP2 1.3, 1.4, 1.5

Test whether a subset is a subgroup, and list the subgroups of a small group.
Apply Lagrange's theorem to rule out impossible subgroup orders.
Decide whether two groups of the same order are isomorphic.

Reduction formulae Edexcel 9FM0 FP2 2.1

Derive a reduction formula by integrating by parts and rearranging.
Apply the recurrence down to a base case you can integrate directly.
Recognise when a formula drops by one step and when it drops by two.

Arc length and surface area Edexcel 9FM0 FP2 2.2

Apply the arc length formula in cartesian, parametric and polar form.
Compute the area of a surface of revolution as 2π times the integral of (radius) ds.
Choose the correct radius when the axis of rotation changes.
Check answers against circles, cones and spheres.

Eigenvalues and eigenvectors Edexcel 9FM0 FP2 3.1

Form and solve the characteristic equation of a 2 × 2 or 3 × 3 matrix.
Find eigenvectors for each eigenvalue, and normalise them when asked.
Interpret eigenvectors as the directions a transformation leaves alone.
Say what repeated and complex eigenvalues mean for the transformation.

Diagonalisation and the Cayley-Hamilton theorem Edexcel 9FM0 FP2 3.2, 3.3

Build P from eigenvectors and D from eigenvalues so that P⁻¹MP = D.
Use diagonalisation to compute high powers of a matrix.
Diagonalise a symmetric matrix orthogonally, with P⁻¹ equal to the transpose.
Apply the Cayley-Hamilton theorem to find powers and inverses.

Further loci and regions in the Argand diagram Edexcel 9FM0 FP2 4.1

Identify |z − a| = k|z − b| as a circle when k ≠ 1, and find its centre and radius.
Recognise a constant argument of a quotient as an arc of a circle.
Shade regions defined by combined inequalities.
Mark clearly which boundaries and endpoints belong to a locus.

Transformations of the complex plane Edexcel 9FM0 FP2 4.2

Find the image of a line or circle under w = z² by eliminating between real and imaginary parts.
Invert a Möbius transformation to express z in terms of w before substituting.
Recognise that w = 1/z sends lines not through the origin to circles through it.

The Euclidean algorithm and Bezout's identity Edexcel 9FM0 FP2 5.1, 5.2

State the division theorem and run the Euclidean algorithm to a highest common factor.
Back substitute to express the highest common factor as an integer combination.
Recognise when two numbers are coprime, and say what that permits.

Modular arithmetic and Fermat's little theorem Edexcel 9FM0 FP2 5.3, 5.4, 5.5, 5.6

Use the congruence laws for addition, subtraction, multiplication and powers.
Apply divisibility tests and justify them with congruences.
Solve linear congruence equations by inverting the coefficient.
Use Fermat's little theorem to cut a large exponent down to size.

Combinatorics Edexcel 9FM0 FP2 5.7

Apply the multiplicative principle, and the addition and subtraction principles.
Choose between permutations and combinations by asking whether order matters.
Count subsets, and count by complement when the direct count is awkward.

Recurrence relations Edexcel 9FM0 FP2 6.1, 6.2, 6.3; 9FM0 D2 6.1, 6.2

Solve first order recurrence relations with a complementary function and a particular solution.
Handle second order relations through their auxiliary equation.
Set up a recurrence from a modelling context and interpret the closed form back in that context.
Fit the constants to the given initial terms.
Prove a supplied closed form by induction.

Further Statistics 2

Least squares regression and residuals Edexcel 9FM0 FS2 1.1, 1.2

Calculate regression coefficients from summary statistics and write the line down.
Compute residuals and use them to judge a fit and spot outliers.
Find the residual sum of squares and say what it measures.

Continuous random variables: density and distribution functions Edexcel 9FM0 FS2 2.1, 2.2

Check that a proposed density is valid and use it to find probabilities.
Move between the density and the cumulative distribution function in both directions.
Use the distribution function to find the median and other percentiles.

Mean, variance and skewness of continuous variables Edexcel 9FM0 FS2 2.3

Find the mean, variance and E(g(X)) for a continuous variable by integration.
Locate the mode, median and percentiles and distinguish them.
Describe skewness from the ordering of the three averages and justify it.
Apply the coding results E(aX + b) and Var(aX + b) in the continuous case.

The continuous uniform distribution Edexcel 9FM0 FS2 2.4

Write down the density and distribution function of U(a, b) and read probabilities off as lengths.
Derive its mean and variance from the general integrals.

Correlation coefficients: product moment and Spearman Edexcel 9FM0 FS2 3.1, 3.2

Calculate the product moment correlation coefficient from summary statistics.
State the conditions under which it is the appropriate measure.
Predict the effect of coding on it without recalculating.
Rank data, handle ties, and calculate Spearman's coefficient.

Testing a correlation coefficient Edexcel 9FM0 FS2 3.3

State hypotheses about a population correlation coefficient correctly.
Read a critical value from the tables and reach a conclusion in context.
Explain the condition the product moment test needs and why the rank test avoids it.

Combinations of normal random variables Edexcel 9FM0 FS2 4.1

Write down the distribution of aX ± bY for independent normal X and Y.
Distinguish the sum of n independent copies from n times a single one.
Use the combined distribution to answer a probability question.

Estimators, standard error and confidence intervals Edexcel 9FM0 FS2 5.1, 5.2, 5.3

Explain what an unbiased estimator is and check whether a given one qualifies.
Compute an unbiased estimate of a population variance from summary statistics.
Calculate a standard error and say what it measures.
Construct and interpret a confidence interval for a normal mean.

Comparing two normal means Edexcel 9FM0 FS2 5.4, 5.5

Find the standard error of a difference of sample means.
Carry out a two-sample z test and state the conclusion in context.
Build a confidence interval for the difference and link it to the test.

Testing variances: chi-squared and the F-distribution Edexcel 9FM0 FS2 6.1, 6.2

Test a claimed variance using the chi-squared statistic and the right degrees of freedom.
Build a confidence interval for a variance from the two chi-squared tails.
Test two variances for equality with an F ratio.

Confidence intervals and tests with the t-distribution Edexcel 9FM0 FS2 7.1, 7.2, 7.3

Carry out a one-sample t test and build the matching confidence interval.
Recognise paired data and reduce it to a single sample of differences.
Pool two sample variances and run a two-sample t test.
State the assumptions each version of the test requires.

Further Mechanics 1

Momentum and impulse Edexcel 9FM0 FM1 1.1

Use impulse equals change in momentum, including for a rebound.
Find an average force from an impulse and a contact time.
Apply conservation of momentum to a direct collision between two spheres.
Handle coalescence and explosion as special cases of the same principle.

Impulse and momentum as vectors Edexcel 9FM0 FM1 1.2

Apply the impulse-momentum principle in component form, and give a magnitude and direction at the end.
Conserve momentum in two dimensions by treating the components separately.

Work, energy and power Edexcel 9FM0 FM1 2.1

Calculate work done by a force at an angle, and by gravity and friction.
Apply the work-energy principle to motion on a slope.
Use conservation of mechanical energy where no resistance acts.
Use P = Fv to link engine power, driving force and acceleration.

Hooke's law and elastic strings Edexcel 9FM0 FM1 3.1

Find a tension, an extension or a modulus given the other two.
Solve equilibrium problems involving one or more elastic strings.

Elastic potential energy Edexcel 9FM0 FM1 3.2

Derive and use the formula for the energy stored in a stretched string.
Include elastic energy in a work-energy equation.
Solve problems where a particle is projected by a spring or oscillates on a string.

Direct impact and Newton's law of restitution Edexcel 9FM0 FM1 4.1

State Newton's law of restitution and the range of values e can take.
Solve a direct collision using conservation of momentum and restitution together.
Calculate the kinetic energy lost in an impact.
Recognise what happens at the two extreme values of e.

Successive impacts and impacts with a wall Edexcel 9FM0 FM1 4.2

Apply restitution to an impact with a fixed wall or floor.
Follow a chain of impacts and decide whether a further collision occurs.
Handle repeated bounces, including the total distance travelled.

Oblique impact and impact with a smooth surface Edexcel 9FM0 FM1 5.1, 5.2

Resolve a velocity into components along and perpendicular to a smooth surface.
Apply restitution to the perpendicular component only, and find the outgoing speed and direction.
Calculate the kinetic energy lost in an oblique impact.
Solve an oblique impact between two smooth spheres by resolving along the line of centres.
Follow a ball through successive oblique impacts with two surfaces.

Further Mechanics 2

Angular speed and horizontal circular motion Edexcel 9FM0 FM2 1.1

Convert between linear speed, angular speed and period.
Identify the force providing the radial acceleration in a given situation.
Solve conical pendulum and banked track problems.

Motion in a vertical circle Edexcel 9FM0 FM2 1.2

Combine conservation of energy with the radial equation of motion.
Find the condition for a particle on a string to complete a vertical circle.
Distinguish the string case from the rod and inside-surface cases.
Say what happens when the circle is not completed.

Centre of mass of a discrete distribution Edexcel 9FM0 FM2 2.1

Find the centre of mass of masses on a line and in a plane.
Work backwards from a given centre of mass to an unknown mass or position.

Centres of mass of plane figures and frameworks Edexcel 9FM0 FM2 2.2

Find the centre of mass of a composite lamina.
Handle a lamina with a piece removed, using a negative area.
Find the centre of mass of a framework of rods.
Use symmetry to avoid unnecessary calculation.

Centres of mass by integration Edexcel 9FM0 FM2 3.1

Find the centre of mass of a lamina bounded by a curve, by integration.
Find the centre of mass of a solid of revolution.
Handle a non-uniform body whose density varies with position.
Use symmetry to reduce the number of integrals needed.

Equilibrium, suspension, toppling and sliding Edexcel 9FM0 FM2 2.2, 3.2, 3.3

Find the angle at which a suspended lamina hangs.
Determine the angle at which a body on a rough slope topples.
Decide whether toppling or sliding happens first.

Newton's laws with a variable force Edexcel 9FM0 FM2 4.1

Choose between dv/dt and v dv/dx according to what the force depends on, then solve and apply the conditions.
Handle inverse square forces, including the work done against gravity.

Simple harmonic motion Edexcel 9FM0 FM2 4.2

Recognise and prove the simple harmonic condition from an equation of motion.
Use the standard formulae for displacement, speed and period.
Link speed to displacement without ever finding the time.
Solve problems mixing time, displacement and speed.

Oscillations on strings and springs Edexcel 9FM0 FM2 4.2

Prove that a mass on a spring moves with simple harmonic motion and find ω.
Locate the centre of the oscillation at the equilibrium position.
Recognise when a string goes slack and the motion stops being simple harmonic.

Further kinematics: acceleration as a function of x, t or v Edexcel 9FM0 FM2 5.1

Choose the form of the acceleration that separates for a given problem.
Solve the resulting equation and apply the conditions correctly.
Interpret limiting behaviour, including terminal speed and total distance.

Decision Mathematics 1

Algorithms, sorting and bin packing Edexcel 9FM0 D1 1.1, 1.2

Follow an algorithm given as text or a flow chart, and record the trace.
State the order of an algorithm and use it to predict running time.
Perform bubble sort and quick sort, showing every pass.
Apply the three bin packing algorithms and compare the result with a lower bound.

Graphs: order, Eulerian paths and planarity Edexcel 9FM0 D1 1.3, 1.4

Find the order of every node and use it to classify a graph.
Recognise complete, planar, isomorphic and Hamiltonian structures.
Apply the planarity algorithm to a Hamiltonian graph.

Minimum spanning trees: Prim and Kruskal Edexcel 9FM0 D1 2.1

Apply Prim's algorithm from a network and from a distance matrix, and Kruskal's from a sorted edge list.
State the number of edges in a spanning tree and check an answer against it.

Shortest paths: Dijkstra and Floyd Edexcel 9FM0 D1 2.2

Apply Dijkstra's algorithm with the standard boxed labels.
Read the shortest route back by working from the destination.
Carry out an iteration of Floyd's algorithm on a distance and a route matrix.

Route inspection Edexcel 9FM0 D1 3.1

Identify the odd nodes and list every possible pairing of them.
Find the shortest path between each pair and choose the cheapest pairing.
State the length of the shortest closed route and which edges are repeated.
Adapt the method when the route need not return to the start.

The travelling salesman problem Edexcel 9FM0 D1 3.2

Distinguish the practical and classical problems and convert between them.
Use the nearest neighbour algorithm to find an upper bound.
Use a minimum spanning tree, doubled and then short-cut, to find an upper bound.
Find a lower bound by deleting a node, and use both bounds to bracket the answer.

Critical path analysis Edexcel 9FM0 D1 4.1, 4.2, 4.3

Draw an activity network from a precedence table, using dummies where needed.
Carry out the forward and backward passes to find event times.
Identify the critical activities and the project duration.

Float, Gantt charts and scheduling Edexcel 9FM0 D1 4.4, 4.5, 4.6

Calculate the total float of every activity.
Draw a Gantt chart showing floats, and read simultaneous activities from it.
Draw a resource histogram, find the lower bound on workers, and schedule.

Linear programming: formulation and graphical solution Edexcel 9FM0 D1 5.1, 5.2

Define variables and write a problem as an objective and constraints.
Shade the feasible region and find the optimum by vertex or objective line.
Introduce slack, surplus and artificial variables correctly.
Deal with a problem whose variables must be whole numbers.

The Simplex algorithm Edexcel 9FM0 D1 5.3, 5.4

Set up the initial tableau from a formulated problem.
Choose the pivot column and row and carry out the row operations.
Read the solution from a final tableau and recognise when to stop.

Decision Mathematics 2

Transportation problems Edexcel 9FM0 D2 1.1

Balance a problem with a dummy source or destination.
Find an initial solution by the north-west corner method.
Count the cells used and recognise degeneracy.

The stepping-stone method Edexcel 9FM0 D2 1.2, 1.3

Find shadow costs from the cells in use.
Compute improvement indices and identify the entering cell.
Trace the stepping-stone route, find the exiting cell, and state the new cost.
Formulate a transportation problem as a linear program.

Allocation and the Hungarian algorithm Edexcel 9FM0 D2 2.1, 2.2

Reduce a cost matrix by rows and then by columns.
Use the minimum-lines test and augment the matrix when it fails.
Adapt the method for maximisation, dummies and forbidden allocations.

Flows in networks: cuts and capacity Edexcel 9FM0 D2 3.1, 3.4

Define a cut, calculate its capacity correctly, and explain why it bounds the flow.
Handle multiple sources, multiple sinks and restricted vertices.

Maximum flow and the labelling procedure Edexcel 9FM0 D2 3.2, 3.3, 3.5

Use the labelling procedure with excess and potential-backflow arrows.
Find flow-augmenting paths and state the extra flow each carries.
Prove a flow is maximal by exhibiting a cut of the same capacity.
Check flow conservation at every intermediate vertex.

Dynamic programming Edexcel 9FM0 D2 4.1

State Bellman's principle of optimality and explain what it licenses.
Set out a solution as a table with stage and state variables.
Handle minimax and maximin objectives as well as totals.

Game theory: play safe and stable solutions Edexcel 9FM0 D2 5.1, 5.2, 5.3

Read a pay-off matrix from the row player's point of view.
Find both play-safe strategies.
Test for a stable solution and state the value of the game.
Reduce a matrix using dominance arguments.

Mixed strategies Edexcel 9FM0 D2 5.4, 5.5

Set up the expected pay-off against each opposing choice as a function of p.
Solve a 2 by n or n by 2 game graphically and state the value.
Convert a larger game into a linear program for Simplex.

Decision analysis Edexcel 9FM0 D2 7.1, 7.2

Draw a decision tree and work expected monetary values back to the first decision.
Explain the limits of expected monetary value and where utility helps.

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