Checklist

Revision checklist

Every objective in the Pure course, taken straight from the lessons. Tap the dot beside each one to rate how confident you feel. Ratings are saved in this browser only — nothing is sent anywhere.

Jump to: Proof · Algebra and functions · Coordinate geometry · Sequences and series · Trigonometry · Exponentials and logarithms · Differentiation · Integration · Numerical methods · Vectors

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Proof

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

Describe the structure of a proof: assumptions, logical steps, conclusion.
Prove statements by deduction, using algebra to cover every case at once.
Prove statements by exhaustion, checking a complete list of cases.

Disproof and proof by contradiction Edexcel 9MA0 1.1

Disprove statements by finding a single counter example.
Reproduce the contradiction proofs of the irrationality of √2 and the infinity of primes.
Apply proof by contradiction to unfamiliar statements.

Algebra and functions

Indices and surds Edexcel 9MA0 2.1, 2.2

Use the laws of indices for all rational exponents, including negative and fractional powers.
Simplify surds using the multiplication rule and exact arithmetic.
Rationalise denominators, including with the conjugate.

Quadratic functions Edexcel 9MA0 2.3

Read a quadratic's graph from its factorised, completed-square and expanded forms.
Use the discriminant to count real roots, and solve for unknowns that control it.
Solve quadratic equations, including quadratics in a function of the unknown.

Simultaneous equations and inequalities Edexcel 9MA0 2.4, 2.5

Solve simultaneous equations by substitution and elimination, including linear with quadratic.
Solve linear and quadratic inequalities, including with brackets and fractions, and write the answers in set notation.
Represent inequalities graphically: regions, shading, dotted and solid boundaries.

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 cubics completely.
Simplify rational expressions by factorise-and-cancel and by algebraic division.

Graphs, proportion and transformations Edexcel 9MA0 2.7, 2.9

Sketch cubics and quartics from factorised form, reading crossings and touches from root multiplicity.
Sketch y = a/x and y = a/x² with their asymptotes, and handle direct and inverse proportion.
Apply the four standard transformations of y = f(x), singly and in combination.

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

Work with domain, range, and one-one against many-one mappings.
Build composite functions in the right order, and find inverses with their domains.
Sketch modulus graphs and solve modulus equations and inequalities.

Partial fractions Edexcel 9MA0 2.10

Split fractions with distinct linear factors, two or three of them.
Handle a repeated linear factor with the two-fraction template.
Choose substitution values that make constants fall out one at a time.

Functions in modelling Edexcel 9MA0 2.11

Match a situation's behaviour to the right function family.
Fit and interpret a model's constants in context, with units.
State limitations honestly and suggest refinements.

Coordinate geometry

Straight lines Edexcel 9MA0 3.1

Find the equation of a line from a point and a gradient, or from two points, in either standard form.
Use gradient conditions for parallel and perpendicular lines.
Build and interpret straight-line models, gradient as rate and intercept as starting value.

Circles Edexcel 9MA0 3.2

Convert between centre-radius and expanded forms of a circle's equation by completing the square.
Use the tangent, chord and semicircle properties as coordinate calculations.
Find tangent equations at a point, and the circle through three given points.

Parametric equations Edexcel 9MA0 3.3, 3.4

Trace and sketch curves given parametrically, treating the parameter as a clock.
Convert between parametric and Cartesian forms, watching the parameter's domain.
Use parametric equations as models, including projectile-style motion.

Sequences and series

The binomial expansion Edexcel 9MA0 4.1

Compute binomial coefficients from Pascal's triangle or the nCr formula.
Expand (a + bx)ⁿ for positive integer n, and pick out single coefficients.
Truncate an expansion to approximate powers of numbers 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.
Handle (a + bx)ⁿ by factoring out aⁿ, and state the validity window.
Combine expansions with partial fractions, and use them for numerical approximation.

Sequences and sigma notation Edexcel 9MA0 4.2, 4.3

Generate sequences from nth-term formulae and from recurrence relations.
Classify sequences as increasing, decreasing or periodic, with justification.
Read and write sums in sigma notation.

Arithmetic series Edexcel 9MA0 4.4, 4.6

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

Geometric series Edexcel 9MA0 4.5, 4.6

Use the nth-term formula arⁿ⁻¹, and prove and use the finite sum formula.
Recognise convergence and use the sum to infinity when |r| < 1.
Solve how-many-terms questions with logarithms, in series and compound-growth models.

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 its opposite angle.
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 and read the sine, cosine and tangent graphs, with exact values at the standard angles.
Use tan θ = sin θ/cos θ and sin²θ + cos²θ = 1 to reshape and solve equations.
Find every solution in a stated interval, including for multiple angles and hidden quadratics.

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

Convert between degrees and radians, and know the exact-value angles in both.
Use s = rθ and A = ½r²θ for arcs, sectors and segments.
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 and undefined points.
Use sec² = 1 + tan² and cosec² = 1 + cot² in equations and identities.
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 and double angle formulae, forwards and backwards.
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 models with two trig terms.

Exponentials and logarithms

Exponential functions and e Edexcel 9MA0 6.1, 6.2

Sketch y = a^x for any positive base, growth and decay alike, with the right asymptote.
Use the gradient property of e^kx, and transform e-curves with the standard moves.
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.
Use the three log laws to combine, split and simplify expressions.
Solve equations with the unknown in the exponent, including a^x = b.

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 from 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 a curve, and meet the second derivative.

Differentiating powers of x Edexcel 9MA0 7.2

Differentiate xⁿ for any rational n, with sums, differences and constant multiples.
Rewrite roots, reciprocals and quotients as powers before differentiating.
Expand or simplify products and fractions that hide differentiable powers.

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.
Use increasing, decreasing, convex and concave language, and solve practical max/min problems.

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

Differentiate sin kx, cos kx, tan kx, e^kx, a^kx and ln x fluently.
Differentiate sin 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.
Combine the rules on expressions like cos² x 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 like x = sin y.
Find gradients and tangents 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.
Verify proposed solutions of a differential equation by differentiating.

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, integrating pieces separately.

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 resolution of the power rule's missing case.
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.

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, and justify accuracy claims with a tighter trap.
Know how the sign change test fails: discontinuities and evenly many roots.
Iterate x_n+1 = g(x_n), reading convergence from staircase and cobweb diagrams.

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

Run Newton-Raphson, understanding each step as a tangent sliding to the axis.
Know when the method fails: a horizontal tangent has no axis-cut to offer.
Estimate definite integrals by the trapezium rule and read the error's direction from a sketch.

Vectors

Vectors in two dimensions Edexcel 9MA0 10.1-10.5

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, AB = b − a, and distances to solve geometric problems.

Vectors in three dimensions Edexcel 9MA0 10.1-10.5

Work with i, j, k components and magnitudes via the three-term Pythagoras.
Find distances, midpoints and unit vectors in three dimensions.
Solve geometric problems: parallel and collinear checks, and triangle shapes by distances.

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