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A-level Maths 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.
Read and write the implication and equivalence arrows, and tell a statement apart from its converse.
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 there are infinitely many primes.
Apply proof by contradiction to an unfamiliar statement.

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.
Expand a product of polynomial brackets and collect like terms, and use an expansion to check a 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.
Sketch y = 1/f(x) from a given curve, trading zeros for asymptotes and turning points for their reciprocals.
Diagnose a rational curve from its coefficients, oblique asymptotes included, and find where it meets a straight line.
Find the range and the stationary points of a rational function from a discriminant rather than a derivative.
Count the solutions of f(x) = k from where the line y = k crosses the curve.

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.
Use the two modulus relations: equal moduli mean equal squares, and a modulus held under b is a two-sided interval.

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

Name the four stages of the problem solving cycle, and say which stage a piece of work belongs to.
Specify a modelling question precisely enough to answer, and plan what to measure and how often.
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, showing the gradient product where perpendicularity is claimed.
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 where a line meets a circle by substituting it in, and read the discriminant as two crossings, a tangent or a miss.
Use the angle in a semicircle in both directions: a diameter subtends a right angle, and a right angle names a diameter.
Find the circle through three given points.
Decide how two circles lie by comparing the distance between their centres with the sum and the difference of the radii, and find the crossings when there are two.

Parametric equations Edexcel 9MA0 3.3, 3.4

Trace and sketch a curve given parametrically, using values of the parameter.
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 why the series does not represent the function outside it.
Handle (a + bx)ⁿ by taking aⁿ out of the bracket first.
Combine expansions with partial fractions, and state the interval of validity of the combined expansion.

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.
Total a stated number of terms of a sequence, by listing the terms or by pairing them where the sequence repeats.

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.
Apply geometric series to modelling, compound growth included.

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.
Prove sin (A + B) and cos (A + B) from the two-triangle construction, and say what that figure assumes.
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.
Solve a model for the times at which it takes a given value, in a stated interval.

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 rational n on a domain where the function is differentiable, 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.
Find tangents and normals on implicit curves from dy/dx at the point.

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.
Verify that a given function satisfies a differential equation by differentiating and comparing.

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 the conditions that pin its constants.
Explain why the constant of integration is required, and why two antiderivatives of the same function differ only by a constant.

Definite integrals and areas Edexcel 9MA0 8.3

Evaluate definite integrals with the square-bracket routine.
Find geometric areas under curves and between a curve and a line.
Split the interval at the axis crossings and add the absolute values where part of the region lies below the axis.
Use the properties of a definite integral: adjacent intervals add, reversed limits change the sign, and an odd integrand over a symmetric interval gives zero.

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.
Choose a substitution from the shape of the integrand, including x = a sin θ for a root of a difference of two squares.
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.
Compare the degrees before choosing a method, and divide an improper integrand down before splitting what is left.
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 first-order separable equations, factorising first where separation requires it, and retain the constant solutions lost by division.
Turn a general solution into a particular one using a given condition.
Interpret a solution in context, stating the interval on which it is valid and its behaviour for large values.
Verify that a proposed solution satisfies the equation and its condition, by differentiating it and comparing both sides.

Numerical methods

Locating roots and iteration Edexcel 9MA0 9.1, 9.2

Show that a root lies in an interval from a change of sign, with continuity stated.
Prove an accuracy claim by choosing the pair of test values the rounding requires.
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.
Rearrange f(x) = 0 into x = g(x), and say why a fixed point of g is a root of f.

The Newton-Raphson method Edexcel 9MA0 9.3, 9.5

Run Newton-Raphson, reading each step as the tangent meeting the x-axis.
State why no next iterate exists when f'(x_n) = 0, and what a nearly horizontal tangent does to the sequence.
Turn a question in context into an equation, and give the root back with its units.
State how fast the iterates converge near a simple root, where the correct digits roughly double each step.

The trapezium rule Edexcel 9MA0 9.4, 9.5

Estimate definite integrals by the trapezium rule with a stated number of strips.
Decide from the sign of f'' whether the estimate is an overestimate or an underestimate.
Count ordinates against strips, and state how an estimate can be improved.
Bound an integral between two rectangle sums, and state the condition each bound requires.
Turn a record of readings into a quantity, with its units and the assumptions it rests on.

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.
State the structure of the Pearson Edexcel large data set: eight stations, two years, and the columns that are not numerical.
Say why a conclusion drawn from the large data set may be unsafe, missing entries included.

Measures of location and spread Edexcel 9MA0 P3 2.3

Find the mean, median and mode of a data set, and say which of them best represents it when one extreme value is present.
Find the median and quartiles of a listed data set, or of one already ordered by a display, 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

Read a stem-and-leaf diagram, a dot plot and a vertical line chart, and say which kind of data each one suits.
Construct and read histograms using frequency density, including recovering frequencies from areas.
Draw and compare box plots, and identify outliers using the rule stated in the question.
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.
Read an unfamiliar diagram through its axes, its key, what one mark represents and a total that can be checked.

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.
State the limitations of a prediction, and identify a possible confounding variable where correlation is being treated as evidence of 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 state the direction in which the answer would change.

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.
State 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.
Define the p-value as the probability, if H₀ is true, of a result at least as extreme as the one observed, and say what it is not.

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.
Extension: write the dimensions of a quantity in M, L and T from its units, and recognise a dimensionless quantity.
Extension: check an equation for dimensional consistency, and say what a pass and a fail each establish.
Extension: find unknown indices by matching powers, to predict the shape of a formula or an equation of motion.

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.
Derive the five suvat equations from the velocity-time graph, and obtain each of the last three from the first two.
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.
Read the formulae as vector equations, in column or i and j notation, and recover the two resolved equations from them.

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.
State Newton's first and third laws, and name the third-law partner of a force, which acts on the other body.
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.
Write the launch velocity and the acceleration as vectors, with a = (0, −g) or −gj, and read the two component equations off the vector one.
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.
Treat the normal and frictional forces as the two components of one contact force, and find its magnitude.
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.
Fix the direction of friction before writing an equation, and reverse it when the sliding it opposes reverses.
Solve the standard arrangement of a particle on a plane joined by a string over a pulley at the top of it to a particle hanging freely.

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.
Take the weight of a rectangular lamina at its point of symmetry, and find the force that puts it on the point of toppling.
Solve for the forces on a ladder resting on rough ground against a smooth vertical wall, with one declared sense of rotation.
Decide whether a body slides or topples first by comparing the sliding threshold with the toppling force, and find the coefficient of friction at which the answer changes.

Further proof

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

Say why the base case and the inductive step are both required, and what a proof missing either one establishes.
Prove a sum formula by induction, showing plainly where the assumption is used.
Conjecture a formula from the first few cases, then prove the conjecture by induction.
Close a proof with the required conclusion sentence.

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.
Prove an inequality by induction, chaining a second inequality onto the assumed one.
Find the value of n from which an inequality holds, and put the base case there.

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.
Find the two square roots of a complex number by squaring x + yi and comparing real and imaginary parts.
Plot numbers and their conjugates on the Argand diagram, and read addition as vectors.
Solve a cubic or quartic with real coefficients from one given root, by pairing it with its conjugate.
Equate real and imaginary parts to turn one complex equation into two real ones.

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, read the shorthands [r, θ] and r cis θ, and multiply and divide in that form.
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.
Derive z + 1/z = 2 cos θ and z − 1/z = 2i sin θ, and use them to rewrite a power of cos θ or sin θ as multiple angles ready to integrate.
Sum a series of cosines or sines by summing one geometric series with a complex ratio, quoting the condition it needs.

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 conformable 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, shears and enlargements in the plane.
Find the equation of a curve's image under a single or composite transformation.
Write down the standard 3 × 3 reflections in coordinate planes and rotations about axes.
Read the area scale factor and the change of orientation off the determinant.

Determinants and inverses Edexcel 9FM0 CP1 3.5, 3.6

Evaluate 2 × 2 and 3 × 3 determinants and interpret them as scale factors.
Use row and column operations to simplify a determinant and return it in factorised form.
Recognise singular matrices and describe the lower-dimensional image a zero determinant gives.
Find and verify inverses of 2 × 2 and 3 × 3 matrices, and use det(AB) = det A det B and 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.
Use Σr to count the comparisons an algorithm makes, read its order off the dominant term, and say what doubling the list does to the work.
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.
Estimate the value of a function by truncating its series, and comment on the accuracy of the estimate.

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 in two dimensions or three, and convert to cartesian form and back.
Write a plane as r·n = d, as ax + by + cz = e and as r = a + λb + μc, 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 signed integral divided by the interval width.
Evaluate improper integrals as limits, stating convergence or divergence.
Settle an exponential or a logarithmic tail with the two standard limits.
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.
Divide an improper fraction down first, whether its denominator is linear or quadratic, and split what is left.
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.
State the domain and range of all six hyperbolic functions and of the three inverses.
Prove and use cosh²x − sinh²x = 1 and its relatives.
Apply Osborn's rule to convert a trigonometric identity into the corresponding hyperbolic identity.
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.
Derive the derivatives of arsinh, arcosh and artanh by implicit differentiation, with the ranges deciding each sign.
Integrate 1/√(x² + a²) and 1/√(x² − a²) via arsinh and arcosh.
Choose a hyperbolic substitution when recognition alone will not do.
Integrate sinh x, cosh x and sech² x by reading their derivatives backwards.

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.
Read a value or the long-term behaviour off the particular solution.

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.
Multiply the trial function by x when it duplicates a term of the complementary function.
Use two initial conditions to pin both constants, applied to the full solution.
Carry the same three root cases across to a second order recurrence relation.

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 suitable linear coupled pair of first order equations to one second order equation.
State the stability condition on the roots before reading a particular integral as the long-term behaviour.

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.
Check every odd multiple of π in the stated interval by hand, since the substitution does not reach them.

Taylor series Edexcel 9FM0 FP1 2.1

Quote the Taylor polynomial of f about x = a in powers of (x − a), and assemble it from a derivative table.
Expand about a point close to the value being approximated, and say why the choice reduces the error.
State the remainder condition that justifies an accuracy claim, and distinguish a polynomial from an infinite series.

Limits and L'Hôpital'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'Hôpital's rule, repeatedly where needed, stating its hypotheses first.
Convert products, differences and power forms into a quotient before starting.
Quote the two standard limits, of x^ke^−x at infinity and of x^kln x at the origin.

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

Apply Leibniz'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 polynomial of the solution as far as a stated power.
Apply a given substitution to reduce an equation to a solvable type, and list any solution the substitution excludes.
Compare a truncated series with an exact solution, near the expansion point and far from it.

Conic sections Edexcel 9FM0 FP1 4.1, 4.2

Write down the cartesian and parametric forms of the four standard conics, with the restriction each parametrisation carries, 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, and state every restriction on it.

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.
Settle the product's direction from the right-handed triple, and expand a product of sums using the distributive rule.
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.
Use |a × b| = |a||b| sin θ, and read a zero product as parallel vectors.

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 compare its estimate with an exact value.
Apply the mid-ordinate rule, and say which way its error runs against that of the trapezium rule, with the assumptions that comparison needs.

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

Check that a distribution is valid, and find an unknown constant in one by making the probabilities total 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.
Locate the median of a discrete distribution from its running total.
Derive the mean and variance of a discrete uniform distribution.
Use Var(aX + b) = a²Var(X), where the shift contributes nothing.

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, scale λ with the length of the window, and add 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 whose expected frequency falls below the stated condition, then recount the classes.
Fit and test a Poisson or binomial model from start to finish.
Build expected frequencies for a continuous model from the probability of each class interval.
State the hypotheses, compare χ² with the critical value, and conclude in the language of evidence.

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.
Merge rows or columns whose expected frequency falls below the stated condition, and recount the degrees of freedom from the merged table.
State the hypotheses as independence and association, and conclude in context.
Apply Yates' correction to a two by two table where the specification requires it, and say why it is used.
Form the chi-squared statistic from the cell contributions (O − E)²/E, and read the largest contributions and their signs when interpreting the result.

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 table of standard discrete distributions.
Use the product rule for generating functions of independent sums.

Type I and Type II errors, test size and power 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.
Prove or refute associativity for an operation defined by a formula.
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, a quantity some specifications call its period.
Recognise cyclic groups and name a generator, and generate a non-cyclic group from two elements.
Work with an infinite group, and say which of the finite results stop applying to it.

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

Test whether a subset is a subgroup, and name the proper and non-trivial ones.
Apply Lagrange's theorem to rule out impossible subgroup orders.
Prove a property of an arbitrary group from the axioms, with no table and no named elements.
Decide whether two groups of the same order are isomorphic.
Quote how many structures exist at each order up to seven, and use it to identify a small group.

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 that integrates directly.
Recognise when a formula drops by one step and when it drops by two.
State the values of n for which a reduction formula and its consequences hold.

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 the non-negative distance from the axis with respect to arc length.
Choose the correct radius when the axis of rotation changes.
Check answers against circles, cones and spheres.

Functions of two variables

Read z = f(x, y) as a surface, and move between that form and an implicit g(x, y, z) = c.
Carry the same algebra to more than two variables, and the same reading to trigonometric, logarithmic and exponential rules.
Sketch sections z = f(a, y) and z = f(x, b), and contours c = f(x, y), and relate both to the surface.
Find first, second and mixed partial derivatives in both notations, and state the condition under which the mixed derivatives agree.
Locate stationary points by solving f_x = 0 and f_y = 0 together, and describe maxima, minima and saddle points.
Classify a stationary point by the sign of the Hessian determinant, and say when the test gives no verdict.
Write down the tangent plane to a surface at a point where the function is differentiable.
Use the tangent plane as a linear approximation to the function near the point of contact, and compare the estimate with the exact value.

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 real eigenvectors as the directions whose line the transformation maps to itself.
Say what repeated and non-real eigenvalues mean for the transformation, and over which field the calculation is being made.

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

Test whether an n × n matrix has n independent eigenvectors, and build P and D when it has.
Use diagonalisation to compute high powers of a matrix.
Diagonalise a real symmetric matrix orthogonally, with P⁻¹ equal to the transpose.
Apply the Cayley-Hamilton theorem to reduce powers, and to find an inverse when the matrix is non-singular.

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

Identify |z − a| = k|z − b| as a circle when a ≠ b and k is positive and not 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.
Identify |z − a| = |z − b| as the perpendicular bisector of the segment joining a and b, and find its cartesian equation.

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, and state the condition ad − bc ≠ 0.
Recognise that on the extended plane w = 1/z sends a line not through the origin to a circle through the origin, and say which point an ordinary Argand diagram omits.

The Euclidean algorithm and Bézout'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, and write down the whole family of integer solutions.
Recognise when two numbers are coprime, and say what that permits.
Use the fact that a common divisor divides every integer combination, and apply Euclid's lemma.
State the fundamental theorem of arithmetic and say what its uniqueness rests on.
Write an integer in a base other than ten, and convert between bases.

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 the standard divisibility tests, including for composite divisors, and justify them with congruences.
Solve linear congruence equations by inverting the coefficient, and systems of them by substitution.
Decide which numbers are squares to a given modulus, and solve a quadratic congruence or prove it insoluble.
Use both forms of Fermat's little theorem, and say why passing its test does not make a number prime.
Reduce a sum raised to a prime power using the binomial coefficients that the prime divides.

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.
Count arrangements in a line of objects that are not all distinct, and of only some of them.
Handle a restriction by gluing objects into a block or by subtracting the arrangements that break it.
Use the pigeonhole principle to settle that a repeat is there.
Count the partitions of a set into a given number of parts, and total them to the Bell numbers.
Count derangements by an ad hoc argument, and count the selections a graph of forbidden pairs allows.
Turn a count of arrangements into a probability.
Count arrangements round a circle by fixing one object, so that n distinct objects sit round a table in (n − 1)! ways.

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, adding a particular solution when the relation has a right-hand side.
Multiply the trial solution by n when it duplicates a term of the complementary function.
Describe a sequence as monotonic, convergent, divergent, oscillating or periodic, naming the root that decides it and the condition that root has to meet.
Work with the Fibonacci and Lucas numbers, and use the algebraic properties of the golden ratio.
Set up a recurrence from a modelling context, integer part function included, 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.
Recover the line in the original variables from one fitted to coded data.
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.
Find the distribution of a function of X by rewriting its event and differentiating back.
Work with a variable that is part discrete and part continuous.
Write down the exponential distribution as the wait between Poisson events and use it.
Prove the memoryless property from the exponential tail, and say which situations it suits and which it does not.

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.
Prove that an exponential wait has mean 1/λ and variance 1/λ², and read off its median.

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.
Say what each coefficient measures and the values it can take.

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.
Say what makes a test non-parametric, and weigh the assumption it drops against the power it costs.

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 from how it was collected 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.

Non-parametric tests

Say what makes a test non-parametric and when one is worth its loss of power.
Test a claimed median with a sign test and with a Wilcoxon signed-rank test.
Run the paired forms of both, and the rank-sum test for two independent samples.
Choose the test from how the data were collected and what the readings carry.
Use the normal approximations for T and W, with a continuity correction.

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.
Find the impulse of a force given as a function of time, by integrating it.
Apply conservation of momentum to a direct collision between two spheres.
Handle coalescence and explosion as special cases of the same principle.
Work out the jerk when a slack inextensible string becomes taut.
Find the kinetic energy lost in a coalescence or a jerk, or released in an explosion, by totalling ½mv² before and after.

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.
Handle a particle jerked at an angle to a string by resolving along the string and across it.

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.
Find a kinetic energy from a velocity vector as ½m v·v.
Use v·v = u·u + 2a·s for constant acceleration in a plane, and recognise it as the work-energy principle in vector form.

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.
State Hooke's law, naming each symbol and the newton as the unit of the modulus.

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.
Say why the impulses in an impact cancel so momentum is conserved while kinetic energy is not.

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.
Obtain v = rω and the radial acceleration by differentiating a position vector.
Identify the force providing the radial acceleration in a given situation.
Solve conical pendulum and banked track problems.
Handle a conical pendulum held by two strings, and test whether both stay taut.
Apply F = mrω² to a horizontal circle in other contexts, such as an elastic string or a string through a hole.
Use a = rω² = v²/r numerically, from the speed or the angular speed.

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.
Read the centre of mass physically: a body balances when supported there, and hangs with it directly below the point of suspension.

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.
Quote the booklet's standard centre of mass for a single uniform figure.

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.
Find the centre of mass of a composite solid from standard results.

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

Find the moment of a couple, and use the fact that it is the same about any point.
Reduce a system of coplanar forces to a resultant force and a couple.
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.
Use the two conditions for the equilibrium of a rigid body: no resultant force and no resultant moment.

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.
Integrate a force given against time for its impulse, and read a speed off the change of momentum.

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.
Use the kinetic, gravitational and elastic energy of an oscillation as one constant total, and find a speed from it.

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

Sort a problem into the kinds existence, construction, enumeration and optimisation.
Follow an algorithm given as text, as a flow chart or as pseudo-code, and record the trace.
Use the INT and ABS functions inside an algorithm, and alter an algorithm to a new purpose.
Perform bubble sort, shuttle sort and quick sort, showing every pass.
Count the comparisons and the swaps two methods make on the same list.
Write a worst-case run-time as a function of n, read its order off the dominant term, and rank the orders.
Apply the four bin packing algorithms and compare the result with a lower bound.
Say what makes a method an algorithm: an input, an output, determinism and termination.

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.
Test a graph against Ore's condition, and name a Hamiltonian graph that fails it.
Apply the planarity algorithm to a Hamiltonian graph.
Split a non-planar graph into planar layers and give its thickness.
Distinguish walk, trail, path and cycle, and say what makes a graph simple.
Subdivide an edge and contract one, naming the contracted node in bracket notation.
Colour a graph to test whether it is bipartite, and write down a complement.
Count the regions of a plane drawing and use Euler's formula and the edge bound it gives.

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.
Quote the cubic order of both algorithms and use it to scale a running time.
Read a graph from its adjacency matrix, and check the row sums against the orders.
Update a minimum spanning tree when one arc is cheapened or removed, by the cycle it closes or the split it leaves, without rerunning the algorithm.

Shortest paths: Dijkstra and Floyd Edexcel 9FM0 D1 2.2

Apply Dijkstra's algorithm, with non-negative edge weights, using 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.
Quote the quadratic order of the implementation used here and use it to scale a running time.

Route inspection Edexcel 9FM0 D1 3.1

Identify the odd nodes and list every possible pairing of them.
Count the pairings of 2n odd nodes as the product of the odd numbers up to 2n − 1.
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.
Say what a network model assumed, and rework the answer when a road closes or a weight is revised.

Further packing and the knapsack problem

Say what a knapsack problem asks for, and how the question it puts differs from a bin packing one.
Write a knapsack problem as an integer programme with a nought-or-one variable for each item.
Rank items by profit per unit mass, load greedily, and compare the result with the best load.
Bound the best profit above by allowing fractions of items to be taken.
Build the dynamic programming table and read the chosen items back out of it.
Pack rectangles onto a sheet by shelves and compare the length used with the area lower bound.
Stack boxes in a container by layers, and compare the height with the volume bound.
Compare the cost of a complete search with the size of the table, and say what the unit of the limit does to it.

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, and name the merges and bursts they turn on.
Identify the critical activities and the project duration.
Draw and complete an activity-on-node network, filling in the six numbers each box carries.
Translate a project between the two conventions and check that the answers agree.

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

Calculate the total float of every activity.
Find the latest start and earliest finish of an activity, and split its float into the independent and interfering parts.
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.
Say over what range of a coefficient the optimal vertex survives, and what a relaxed constraint is worth.
Deal with a problem whose variables must be whole numbers, by branch and bound where the method is named.

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.
Read what each constraint is worth off the objective row, and say what a change to a coefficient would do.
Say why a ≥ constraint leaves no starting basis, and how the two-stage and big-M methods manufacture one from artificial variables.

Decision Mathematics 2

Transportation problems and the north-west corner method 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.
Add a zero allocation when too few cells are in use for the shadow costs.

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.
Read a complete allocation from the zeros and cost it in the original matrix.
Formulate an allocation problem as a linear program, defining the decision variables.

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.
Value a cut in a network whose arcs carry lower capacities as well as upper ones.
Say when a flow is feasible under lower capacities, and why one may not exist.

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.
Establish a feasible flow and augment it when arcs carry lower capacities as well as upper ones.

Dynamic programming and Bellman's principle 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.
Convert a constant-sum game to zero-sum form before solving it.
Find both play-safe strategies.
Test for a saddle point, a pure-strategy Nash equilibrium, and state the value of the game.
Reduce a matrix using dominance arguments.

Mixed strategies in game theory 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, and formulate a larger game as a linear programme.
Convert a larger game into a linear program for Simplex.

Decision trees, expected monetary value and utility 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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