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EnglandMaths

Pure mathematics

10 dot points across 10 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do you manipulate, factorise and transform algebraic expressions and functions accurately?

Indices, surds, quadratics, simultaneous equations, inequalities, polynomials, the factor theorem, partial fractions, graphs of functions, composite and inverse functions, the modulus function and graph transformations.
A focused answer to the AQA A-Level Mathematics algebra and functions content, covering indices, surds, quadratics, simultaneous equations and inequalities, polynomials and the factor theorem, partial fractions, modulus and the transformations of graphs.
11 min answer →

How do you describe straight lines, circles and curves using coordinates and equations?

Equations of straight lines, gradients, parallel and perpendicular lines, the equation of a circle, tangents and chords, and parametric equations of curves.
A focused answer to the AQA A-Level Mathematics coordinate geometry content, covering the equation of a straight line, gradient conditions for parallel and perpendicular lines, the equation of a circle and its key properties, and parametric equations of curves.
10 min answer →

How do you find the rate at which a quantity changes, and how do you use that to analyse curves and solve optimisation problems?

Differentiation from first principles, the rules for powers, the chain, product and quotient rules, derivatives of standard functions, stationary points and their nature, and connected rates of change.
A focused answer to the AQA A-Level Mathematics differentiation content, covering first principles, the chain, product and quotient rules, derivatives of standard functions, stationary points and their nature, and applications to optimisation.
11 min answer →

How do exponential and logarithmic functions describe growth and decay, and how do you manipulate and solve equations involving them?

The exponential function and its derivative, the natural logarithm, the laws of logarithms, solving exponential and logarithmic equations, and using logarithms to linearise data and model exponential growth and decay.
A focused answer to the AQA A-Level Mathematics exponentials and logarithms content, covering the function with base e, the laws of logarithms, solving exponential equations, linearising data with logs and modelling growth and decay.
11 min answer →

How do you reverse differentiation to find areas, and what techniques let you integrate more complicated functions?

Integration as the reverse of differentiation, indefinite and definite integrals, the area under a curve, integration of standard functions, integration by substitution and by parts, and using partial fractions to integrate rational functions.
A focused answer to the AQA A-Level Mathematics integration content, covering indefinite and definite integrals, area under a curve, standard integrals, integration by substitution and by parts, and integrating with partial fractions.
11 min answer →

When you cannot solve an equation or integrate a function exactly, how do you find a reliable approximate answer?

Locating roots by sign change, iterative methods including fixed point iteration and the Newton-Raphson method, the conditions under which they succeed or fail, and the trapezium rule for approximating definite integrals.
A focused answer to the AQA A-Level Mathematics numerical methods content, covering locating roots by sign change, fixed point iteration, the Newton-Raphson method and its failures, and the trapezium rule for definite integrals.
11 min answer →

How do you prove a mathematical statement is always true, and how do you disprove one?

Methods of proof including proof by deduction, proof by exhaustion, disproof by counter-example and proof by contradiction, applied to statements about numbers and inequalities.
A focused answer to the AQA A-Level Mathematics proof content, covering proof by deduction, proof by exhaustion, disproof by counter-example and proof by contradiction, with the irrationality of root 2 and the infinitude of primes.
10 min answer →

How do you describe, sum and expand sequences and series?

Arithmetic and geometric sequences and series, sigma notation, the conditions for convergence of a geometric series, the binomial expansion for positive integer and rational powers, and recurrence relations.
A focused answer to the AQA A-Level Mathematics sequences and series content, covering arithmetic and geometric progressions, sigma notation, the sum to infinity of a convergent geometric series, the binomial expansion for positive integer and rational powers, and recurrence relations.
10 min answer →

How do the trigonometric functions, their identities and inverse functions let you model and solve problems involving angles?

Radian measure, arc length and sector area, the trigonometric ratios and their graphs, exact values, identities, the reciprocal and inverse functions, the addition and double angle formulae, and solving trigonometric equations.
A focused answer to the AQA A-Level Mathematics trigonometry content, covering radians, arc and sector formulae, exact values, the Pythagorean and addition formulae, reciprocal and inverse functions, and solving trigonometric equations.
11 min answer →

How do vectors represent position and movement in two and three dimensions, and how do you calculate with them?

Vectors in two and three dimensions, magnitude and direction, addition and scalar multiplication, unit vectors and components, position vectors, and using vectors to solve geometric problems.
A focused answer to the AQA A-Level Mathematics vectors content, covering vectors in two and three dimensions, magnitude and direction, addition and scalar multiplication, unit vectors, position vectors and geometric applications.
10 min answer →