A-level and IB · Mathematics

A-level Maths: pure mathematics

A-level pure maths flashcards: proof, algebra, functions, series, trigonometry, logs, differentiation, integration and numerical methods, with worked examples.

45 cards7 topicsAbout 20 minutes a roundUK A-level Mathematics pure content (England), comparable to Grade 12 calculus plus early first-year university algebra

Independent study material written by StudyBrix, not produced or reviewed by any exam board or by the International Baccalaureate Organization. Check your own specification or subject guide.

Print cards to cut out (PDF)Print a study list (PDF)

About this deck

This deck is for students revising the pure half of A-level Mathematics, and for Canadian or international students who want to see how far the English course goes before first-year calculus. It covers proof, surds and indices, quadratics, polynomials and partial fractions, functions and graph transformations, coordinate geometry and circles, sequences and series, binomial expansion, radians and trigonometric identities, exponentials and logarithms, differentiation (including the chain, product and quotient rules and implicit differentiation), integration, numerical methods and vectors.

The content standard for A-level Maths in England is set nationally, but the way each exam board (AQA, Edexcel, OCR) arranges papers, notation and formula booklets differs. Check your own board's specification and formula booklet before an exam, especially for which formulae you must memorize.

You should already be fluent with GCSE higher algebra: expanding, factoring, rearranging formulae and solving simultaneous equations. Study each card by covering the example and working it on paper first, then compare. Mix tags once you are confident, because exam questions often combine two topics, such as a trigonometric identity inside an integral.

Common mistakes: forgetting the plus or minus when taking a square root; using degrees in calculus when radians are required; writing ln(a + b) as ln a + ln b; and losing the constant of integration or the limits when substituting. Also state the validity range when you expand a binomial with a negative or fractional power.

Before you start: GCSE higher tier maths (or Grade 11 pre-calculus): algebraic manipulation, quadratics, basic trigonometry and coordinate geometry.

All 45 cards

Cover the definitions with a sheet of paper and test yourself, or use the study modes above.

Proof and algebra

Proof by contradiction: Assume the statement you want to prove is false, then reason step by step until you reach something impossible; the assumption must therefore be wrong.Example: Suppose sqrt(2) = a/b in lowest terms. Then a^2 = 2b^2, so a is even, a = 2k, giving b^2 = 2k^2, so b is even too. Both even contradicts lowest terms.
Proof by exhaustion: Split the claim into a finite set of cases that together cover every possibility, and check each case separately.Example: Any integer n leaves remainder 0, 1 or 2 when divided by 3; n^2 then leaves 0, 1 or 1. So a square never leaves remainder 2 on division by 3.
Disproof by counterexample: To show a general claim is false, it is enough to find one specific case where it fails.Example: Claim: 2^n - 1 is prime whenever n is prime. For n = 11, 2^11 - 1 = 2047 = 23 * 89, so the claim is false.
Rationalizing a denominator: Rewriting a fraction so no surd remains on the bottom, by multiplying top and bottom by the denominator's conjugate.Example: 6/(3 - sqrt(3)) = 6(3 + sqrt(3))/(9 - 3) = 3 + sqrt(3).
Discriminant: The value b^2 - 4ac for ax^2 + bx + c = 0. Positive: two real roots; zero: one repeated root; negative: no real roots.Example: kx^2 + 4x + 1 = 0 has equal roots when 16 - 4k = 0, so k = 4.
Completing the square: Rewriting ax^2 + bx + c as a(x + p)^2 + q, which shows the turning point (-p, q) and helps solve or sketch quadratics.Example: x^2 - 6x + 11 = (x - 3)^2 + 2, so the minimum point is (3, 2) and the graph never meets the x-axis.
Factor theorem: For a polynomial f, (x - a) is a factor exactly when f(a) = 0. Used with division to factor cubics fully.Example: f(x) = x^3 - 2x^2 - 5x + 6: f(1) = 0, so f(x) = (x - 1)(x^2 - x - 6) = (x - 1)(x - 3)(x + 2).
Partial fractions: Splitting a rational expression with a factored denominator into a sum of simpler fractions, usually so it can be integrated or expanded.Example: (5x + 1)/((x + 1)(x - 1)) = 2/(x + 1) + 3/(x - 1). Substitute x = 1 and x = -1 into 5x + 1 = A(x - 1) + B(x + 1).

Functions and graphs

Composite function fg(x): Apply g first, then apply f to the result. Order matters, so fg and gf are usually different.Example: f(x) = 2x + 1, g(x) = x^2: fg(3) = f(9) = 19, while gf(3) = g(7) = 49.
Inverse function: A function f^-1 that undoes f, so f^-1(f(x)) = x. It exists only if f is one-to-one; its graph is the reflection of f in y = x.Example: f(x) = 3x - 4 gives f^-1(x) = (x + 4)/3. The domain of f^-1 is the range of f.
Graph transformations: y = f(x) + a moves the graph up a; y = f(x + a) moves it left a; y = af(x) stretches vertically by a; y = f(ax) stretches horizontally by 1/a.Example: y = (x - 2)^2 + 3 is y = x^2 translated 2 right and 3 up, so its vertex is (2, 3).
Parametric equations: A curve described by x = f(t) and y = g(t) using a parameter t. The gradient is dy/dx = (dy/dt)/(dx/dt).Example: x = t^2, y = 2t: dy/dx = 2/(2t) = 1/t. Eliminating t gives the Cartesian form y^2 = 4x.

Coordinate geometry and series

Equation of a circle: (x - a)^2 + (y - b)^2 = r^2 has centre (a, b) and radius r. Complete the square in x and y to read these off an expanded form.Example: x^2 + y^2 - 4x + 6y - 12 = 0 becomes (x - 2)^2 + (y + 3)^2 = 25: centre (2, -3), radius 5.
Perpendicular gradients: Two non-vertical lines meet at right angles when the product of their gradients is -1, so each slope is the negative reciprocal of the other.Example: The line through (4, 1) at right angles to y = 2x + 1 has gradient -1/2: y = -x/2 + 3.
Arithmetic series: A sum whose terms rise by a fixed difference d. The sum of n terms is S = n/2 (2a + (n - 1)d), where a is the first term.Example: 3 + 7 + 11 + ... to 20 terms: S = 10(6 + 19 * 4) = 10 * 82 = 820.
Sum to infinity of a geometric series: When the common ratio satisfies |r| < 1, the partial sums approach a/(1 - r), where a is the first term.Example: 12 + 6 + 3 + ... has a = 12, r = 1/2, so the sum to infinity is 12/(1/2) = 24.
Binomial expansion for a positive integer power: (a + b)^n expands into n + 1 terms with coefficients nCr, the powers of a falling and of b rising.Example: (1 + 2x)^5 = 1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5.
Binomial expansion for any rational power: (1 + x)^n = 1 + nx + n(n - 1)x^2/2! + ... is an infinite series when n is negative or fractional, valid only for |x| < 1.Example: (1 + x)^(1/2) is about 1 + x/2 - x^2/8. With x = 0.02: about 1.00995, very close to sqrt(1.02) = 1.0099505.
Recurrence relation: A rule that defines each term from the one before, written u(n+1) = f(u(n)), plus a starting value.Example: u1 = 2 and u(n+1) = 3u(n) - 1 give 2, 5, 14, 41, ...

Trigonometry

Radian measure: An angle of one radian subtends an arc equal to the radius. Arc length is r times theta and sector area is half of r^2 times theta, with theta in radians.Example: r = 6, theta = pi/3: arc = 2pi (about 6.28) and sector area = 6pi (about 18.85).
Small angle approximations: For small x in radians: sin x is about x, tan x is about x, and cos x is about 1 - x^2/2.Example: cos(0.1) is about 1 - 0.005 = 0.995; the true value is 0.99500 to 5 d.p.
Pythagorean identities: sin^2 x + cos^2 x = 1, and dividing through gives 1 + tan^2 x = sec^2 x and 1 + cot^2 x = cosec^2 x.Example: If tan x = 3/4 with x acute, sec^2 x = 1 + 9/16 = 25/16, so sec x = 5/4.
Double angle formulae: sin 2A = 2 sin A cos A; cos 2A = cos^2 A - sin^2 A = 1 - 2sin^2 A = 2cos^2 A - 1; tan 2A = 2tan A/(1 - tan^2 A).Example: sin A = 3/5, A acute: cos A = 4/5, so sin 2A = 24/25 and cos 2A = 16/25 - 9/25 = 7/25.
Compound angle formulae: sin(A + B) = sin A cos B + cos A sin B and cos(A + B) = cos A cos B - sin A sin B; change signs for A - B.Example: sin 75 degrees = sin 45 cos 30 + cos 45 sin 30 = (sqrt(6) + sqrt(2))/4, about 0.9659.
Harmonic form R cos(x - alpha): a cos x + b sin x can be written as one wave R cos(x - alpha), with R = sqrt(a^2 + b^2) and tan alpha = b/a.Example: 3cos x + 4 sin x = 5 cos(x - 53.13 degrees), so its greatest value is 5.
Solving trig equations in an interval: Find the principal value, use the symmetry of the graph (or CAST diagram) for the other solutions, then add whole periods to stay in the interval.Example: 2 sin x = 1 for 0 <= x < 360 degrees: sin x = 1/2, so x = 30 or 150 degrees.

Exponentials and logarithms

Laws of logarithms: log a + log b = log ab, log a - log b = log(a/b), and k log a = log a^k, for logs with the same base.Example: Solve 3^x = 20: x ln 3 = ln 20, so x = ln 20/ln 3, about 2.727.
Exponential growth model: A quantity growing at a rate proportional to its size follows N = N0 e^(kt), where N0 is the starting amount and k > 0 is the growth constant.Example: 500 bacteria doubling every 3 hours: k = ln 2/3, about 0.231. After 7 hours N = 500 * 2^(7/3), about 2520.
Reducing a power law to a straight line: If y = ax^n then log y = n log x + log a, so plotting log y against log x gives gradient n and intercept log a.Example: A log10 plot with gradient 1.5 and intercept 0.3 gives n = 1.5 and a = 10^0.3, about 2.00.

Differentiation

Differentiation from first principles: Finding f'(x) as the limit, as h tends to 0, of (f(x + h) - f(x))/h.Example: For f(x) = x^2: ((x + h)^2 - x^2)/h = 2x + h, which tends to 2x.
Chain rule: For a function of a function y = f(u) with u = g(x), dy/dx = dy/du * du/dx.Example: y = (3x^2 + 1)^4: dy/dx = 4(3x^2 + 1)^3 * 6x = 24x(3x^2 + 1)^3.
Product rule: For y = uv, dy/dx = u dv/dx + v du/dx.Example: y = x^2 sin x gives dy/dx = 2x sin x + x^2 cos x.
Quotient rule: For y = u/v, dy/dx = (v du/dx - u dv/dx)/v^2.Example: y = x/(x + 1): dy/dx = ((x + 1) - x)/(x + 1)^2 = 1/(x + 1)^2.
Stationary points: Points where dy/dx = 0. The second derivative classifies them: positive means a minimum, negative a maximum, zero needs more checking.Example: y = x^3 - 3x: dy/dx = 3x^2 - 3 = 0 at x = 1 and -1. y'' = 6x gives a minimum at (1, -2) and a maximum at (-1, 2).
Implicit differentiation: Differentiate every term with respect to x when y is not isolated, multiplying each derivative of a y term by dy/dx.Example: x^2 + y^2 = 25 gives 2x + 2y dy/dx = 0, so dy/dx = -x/y. At (3, 4) the gradient is -3/4.
Connected rates of change: Link rates with the chain rule, for example dV/dt = dV/dr * dr/dt.Example: Sphere with dr/dt = 0.1 cm/s at r = 5 cm: dV/dt = 4 pi r^2 * 0.1 = 10 pi, about 31.4 cm^3/s.
Point of inflection: A point where the curve changes concavity, so the second derivative changes sign there.Example: For y = x^3, y'' = 6x changes from negative to positive at x = 0, so (0, 0) is a point of inflection.

Integration and numerical methods

Definite integral as area: The integral of f from a to b gives the signed area between the curve and the x-axis; regions below the axis count as negative.Example: The integral of 3x^2 from 0 to 2 is [x^3] = 8 - 0 = 8.
Integration by substitution: Replace part of the integrand with u, rewrite dx in terms of du, integrate, then substitute back (or change the limits).Example: For 2x(x^2 + 1)^3, let u = x^2 + 1, du = 2x dx: the integral is u^4/4 + c = (x^2 + 1)^4/4 + c.
Integration by parts: The integral of u dv/dx equals uv minus the integral of v du/dx. Choose u so that it gets simpler when differentiated.Example: Integral of x e^x: u = x, dv/dx = e^x gives x e^x - e^x + c.
Separable differential equation: An equation dy/dx = f(x)g(y) that can be rearranged so all y terms are with dy and all x terms with dx, then both sides integrated.Example: dy/dx = 2y with y(0) = 3: ln y = 2x + c, so y = 3e^(2x).
Trapezium rule: Estimates an integral using strips of width h: (h/2)(first y + last y + 2 times the sum of the middle ys).Example: Integral of x^2 from 0 to 1 with h = 0.5: 0.25(0 + 1 + 2 * 0.25) = 0.375, an overestimate of 1/3 because the curve is convex.
Change of sign method: If f is continuous and f(a), f(b) have opposite signs, a root of f(x) = 0 lies between a and b.Example: f(x) = x^3 - x - 1: f(1) = -1 and f(2) = 5, so a root lies in (1, 2).
Newton-Raphson method: An iteration for roots: x(n+1) = x(n) - f(x(n))/f'(x(n)). It fails or jumps if f'(x) is near zero.Example: f(x) = x^2 - 2 from x0 = 1.5: x1 = 1.5 - 0.25/3 = 1.41667, already close to sqrt(2).
Magnitude of a 3D vector: For a vector with components (x, y, z), the length is sqrt(x^2 + y^2 + z^2); dividing by it gives a unit vector.Example: (2, -3, 6) has magnitude sqrt(4 + 9 + 36) = 7, so its unit vector is (2/7, -3/7, 6/7).