← Maths syllabus

EnglandMaths

Pure mathematics: foundations

7 dot points across 7 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do you solve quadratics, simultaneous equations and inequalities, and read information from the discriminant?

Quadratic functions, completing the square, the quadratic formula and the discriminant, simultaneous equations (linear and quadratic), and linear and quadratic inequalities.
A focused answer to the OCR A-Level Mathematics A algebra content, covering solving quadratics by factorising, completing the square and the formula, the discriminant and the nature of roots, simultaneous linear and quadratic equations, and solving linear and quadratic inequalities.
11 min answer →

How do you find equations of straight lines and circles, and use the geometry of tangents and chords?

Straight lines, gradients, parallel and perpendicular conditions, the equation of a circle, the relationship between a tangent and the radius, and parametric equations of curves.
A focused answer to the OCR A-Level Mathematics A coordinate geometry content, covering the equation of a straight line, gradient conditions for parallel and perpendicular lines, the equation of a circle, tangent and chord properties, and parametric equations of curves.
11 min answer →

How do you sketch standard curves and predict the effect of translating, stretching or reflecting a graph?

Sketching curves including polynomials, the reciprocal function and its variations, intersections of graphs, and the transformations y equals f(x) plus a, f(x plus a), f(ax) and af(x).
A focused answer to the OCR A-Level Mathematics A graphs and transformations content, covering sketching polynomial and reciprocal curves, asymptotes, points of intersection, and the four standard graph transformations of translation, stretch and reflection.
10 min answer →

How do you manipulate powers and irrational roots exactly, without a calculator?

Laws of indices for all rational exponents, surd manipulation and rationalising denominators, and the meaning of negative and fractional indices.
A focused answer to the OCR A-Level Mathematics A indices and surds content, covering the laws of indices for all rational exponents, negative and fractional powers, simplifying surds, and rationalising denominators including those of the form a plus root b.
9 min answer →

How do you factorise and divide polynomials, and how do you expand a bracket raised to a power?

Polynomial manipulation, the factor theorem and algebraic division, and the binomial expansion of (a plus b) to the power n for positive integer n using binomial coefficients.
A focused answer to the OCR A-Level Mathematics A polynomials and binomial theorem content, covering polynomial addition and multiplication, algebraic division, the factor theorem for finding roots, and the binomial expansion of a bracket raised to a positive integer power using binomial coefficients.
10 min answer →

How do you prove a mathematical statement is always true, and how do you disprove one that is not?

Methods of proof: proof by deduction, proof by exhaustion, disproof by counter-example, and proof by contradiction, including the irrationality of root 2 and the infinitude of primes.
A focused answer to the OCR A-Level Mathematics A proof content, covering proof by deduction, proof by exhaustion, disproof by counter-example and proof by contradiction, with the standard results that root 2 is irrational and that there are infinitely many primes.
10 min answer →

How do you describe and sum arithmetic and geometric sequences, and when does an infinite series converge?

Arithmetic and geometric sequences and series, sigma notation, sum formulae, recurrence relations, increasing, decreasing and periodic sequences, and the sum to infinity of a convergent geometric series.
A focused answer to the OCR A-Level Mathematics A sequences and series content, covering arithmetic and geometric sequences, the sum formulae, sigma notation, recurrence relations, increasing, decreasing and periodic sequences, and the sum to infinity of a convergent geometric series.
11 min answer →