Mathematics (Edexcel)

Indices and Surds

be able to use and manipulate surds, including rationalising the denominator · understand and use the laws of indices for all rational exponents · be able to evaluate expressions involving surds and rational indices without a calculator

Quadratic Functions

be able to solve quadratic equations by factorising, by completing the square and by using the quadratic formula · be able to complete the square of a quadratic expression and use the result to sketch the graph and identify the vertex and line of symmetry · understand and use the discriminant b^2 - 4ac to determine the nature of the roots of a quadratic equation · be able to solve equations that can be reduced to a quadratic in a function of x (e.g. quadratic in x^2 or in sqrt(x))

Simultaneous Equations and Inequalities

be able to solve simultaneous equations in two unknowns (linear-linear and linear-quadratic) algebraically and graphically · be able to solve linear and quadratic inequalities, expressing solution sets using inequality and interval notation · understand how to represent linear and quadratic inequalities graphically, including regions defined by multiple inequalities

Polynomials and Algebraic Manipulation

be able to manipulate polynomials algebraically, including expanding brackets, collecting like terms and simplifying rational expressions · be able to factorise quadratic and cubic polynomials, recognising standard forms such as the difference of two squares

Graphs and Transformations

be able to sketch graphs of quadratic, cubic, reciprocal and simple rational functions, identifying key features (intercepts, asymptotes, turning points) · understand the effect of the transformations y = f(x) + a, y = f(x + a), y = af(x) and y = f(ax) on the graph of y = f(x), and apply these to sketch transformed curves

Straight Lines

understand and use the equation of a straight line in the forms y = mx + c, y - y1 = m(x - x1) and ax + by + c = 0 · be able to calculate the gradient of a line through two points and the distance between two points · be able to find the midpoint of a line segment and the coordinates of the point dividing it in a given ratio

Parallel and Perpendicular Lines

understand the conditions for two lines to be parallel (equal gradients) or perpendicular (product of gradients = -1) · be able to find the equation of a line parallel or perpendicular to a given line through a specified point

Trigonometric Ratios and the Unit Circle

understand and use the definitions of sine, cosine and tangent for any angle, using the unit circle to interpret signs of trigonometric ratios in each quadrant · know and use exact values of sin, cos and tan of 0°, 30°, 45°, 60°, 90° and related angles · be able to sketch the graphs of y = sin x, y = cos x and y = tan x and recognise their periodicity and symmetries

Sine and Cosine Rules

be able to apply the sine rule and the cosine rule to solve problems involving triangles, including the ambiguous case of the sine rule · be able to use the formula area = 1/2 ab sin C to find the area of a triangle

Trigonometric Identities and Equations

understand and use the identities sin^2 θ + cos^2 θ = 1 and tan θ = sin θ / cos θ · be able to solve trigonometric equations of the form sin(kθ + α) = c, cos(kθ + α) = c and tan(kθ + α) = c within a given interval

The Derivative

understand the derivative of f(x) as the gradient of the tangent to the graph of y = f(x) at a general point, and use the notations dy/dx and f'(x) · be able to differentiate x^n for any rational n, including sums and constant multiples of such terms · be able to differentiate from first principles for simple polynomial functions

Applications of Differentiation

be able to find equations of tangents and normals to a curve at a given point · be able to locate stationary points of a curve and determine their nature using the second derivative or sign of the first derivative · be able to apply differentiation to problems involving rates of change, optimisation and curve sketching

Indefinite Integration

understand integration as the reverse of differentiation and use the notation ∫ f(x) dx, including the role of the constant of integration · be able to integrate x^n for n ≠ -1, including sums and constant multiples of such terms, and find a curve from its gradient function given a point

Definite Integration and Area

be able to evaluate definite integrals and interpret them as signed areas between a curve and the x-axis · be able to find the area between a curve and the x-axis, including regions that cross the axis (treating sign appropriately)

Polynomial Division and the Factor Theorem

be able to divide a polynomial by a linear or quadratic divisor, expressing the result as quotient plus remainder · understand and apply the factor theorem and the remainder theorem to factorise cubic and quartic polynomials

Algebraic Fractions and Identities

be able to simplify rational expressions by factorising and cancelling, and add or subtract algebraic fractions with linear or quadratic denominators

Arithmetic Sequences and Series

understand and use the formula for the nth term of an arithmetic sequence: u_n = a + (n - 1)d · be able to use the formula for the sum of the first n terms of an arithmetic series: S_n = n/2 [2a + (n - 1)d]

Geometric Sequences and Series

understand and use the formula for the nth term of a geometric sequence: u_n = a r^(n-1) · be able to use the formula for the sum of the first n terms of a geometric series and the sum to infinity for |r| < 1

Sigma Notation

understand and use sigma notation to represent and evaluate sums of arithmetic and geometric series

Pascal's Triangle and the Binomial Theorem

understand and use Pascal's triangle to expand (a + b)^n for small positive integer n · be able to use the binomial theorem to expand (a + b)^n for positive integer n, using the notation nCr (or (n r))

Coefficient Extraction

be able to identify and calculate specific terms or coefficients in a binomial expansion without expanding fully

Equations of Circles

understand and use the equation of a circle (x - a)^2 + (y - b)^2 = r^2, including completing the square to find the centre and radius from a general second-degree form

Geometric Properties of Circles

be able to find equations of tangents and chords to a circle and solve problems using the perpendicularity of a tangent to the radius at the point of contact · understand and use the standard circle theorems: angle in a semicircle, perpendicular from centre bisects chord, tangent perpendicular to radius

Radian Measure

understand and use the definition of a radian, and convert between degrees and radians · be able to use the formulae for arc length s = rθ and area of a sector A = 1/2 r^2 θ where θ is in radians

Trigonometric Identities and Equations

be able to use the identities sin^2 θ + cos^2 θ = 1 and tan θ = sin θ / cos θ to prove identities and solve equations in radians · be able to solve trigonometric equations involving multiple angles or compound expressions within a given interval expressed in radians

Exponential Functions

understand the function y = a^x and sketch its graph for a > 1 and 0 < a < 1 · understand the function y = e^x as the exponential function and recognise its special role in calculus

Logarithms

understand and use the laws of logarithms: log(ab), log(a/b), log(a^n), change of base · be able to solve equations of the form a^x = b using logarithms, including problems modelled by exponential growth and decay · be able to use logarithms to linearise relationships of the form y = a x^n and y = a b^x, and interpret the gradient and intercept of the linearised graph

Stationary Points and Curve Sketching

be able to find and classify stationary points (maxima, minima, points of inflection) using the second derivative test · be able to apply differentiation to optimisation problems involving maxima and minima in geometric and applied contexts

Area Between Curves

be able to find areas between two curves, between a curve and a line, or between a curve and the y-axis using definite integration

Locating Roots

be able to locate roots of f(x) = 0 by using a change of sign in f(x) over a suitable interval, recognising the conditions under which this method may fail

Iterative Methods

be able to solve equations approximately using an iteration of the form x_(n+1) = g(x_n), and understand the conditions for convergence

Partial Fractions

be able to express a proper rational function as a sum of partial fractions when the denominator factorises into distinct linear factors, repeated linear factors or a linear and a quadratic factor · be able to reduce an improper rational function to a polynomial plus a proper rational function before decomposing into partial fractions

The Modulus Function

understand the definition |x| of the modulus function and be able to sketch graphs of y = |f(x)| and y = f(|x|) · be able to solve equations and inequalities involving the modulus function algebraically and graphically

Composite and Inverse Functions

understand the definitions of one-one, many-one and inverse functions, and the domain and range of a function · be able to form composite functions fg(x) and inverse functions f^(-1)(x), and sketch the graph of an inverse as the reflection of the graph of f in the line y = x

Reciprocal Trigonometric Functions

understand and use the reciprocal functions sec θ = 1/cos θ, cosec θ = 1/sin θ and cot θ = 1/tan θ, including their graphs and domains · be able to use the identities 1 + tan^2 θ = sec^2 θ and 1 + cot^2 θ = cosec^2 θ to prove identities and solve equations

Compound and Double-Angle Formulae

be able to use the compound-angle formulae for sin(A ± B), cos(A ± B) and tan(A ± B) · be able to use the double-angle formulae sin 2A = 2 sin A cos A, cos 2A = 1 - 2 sin^2 A = 2 cos^2 A - 1 and tan 2A = 2 tan A / (1 - tan^2 A) · be able to express a sin θ + b cos θ in the form R sin(θ + α) or R cos(θ - α) and use this form to find maxima, minima and to solve equations

The Exponential and Natural Logarithm Functions

understand the function y = e^(kx) and its derivative ky' = y, including modelling applications of exponential growth and decay · understand and use the natural logarithm function y = ln x as the inverse of y = e^x, including sketching its graph and noting its domain

Differentiation of Standard Functions

be able to differentiate e^x, ln x, sin x, cos x, tan x and their constant multiples and sums

Chain, Product and Quotient Rules

be able to apply the chain rule to differentiate composite functions · be able to apply the product rule and the quotient rule to differentiate products and quotients of functions

Implicit and Parametric Differentiation

be able to differentiate implicitly to find dy/dx for relations between x and y · be able to use parametric differentiation: if x = x(t), y = y(t), then dy/dx = (dy/dt)/(dx/dt)

Integration of Standard Functions

be able to integrate e^x, 1/x, sin x, cos x, sec^2 x and related forms involving linear substitutions

Integration Techniques

be able to integrate by substitution, including using a given substitution and choosing an appropriate one · be able to integrate by parts using the formula ∫ u dv = uv - ∫ v du, including repeated application · be able to integrate rational functions by first decomposing into partial fractions

The Newton-Raphson Method

be able to use the Newton-Raphson iteration x_(n+1) = x_n - f(x_n)/f'(x_n) to find roots of f(x) = 0, and identify situations where the method may fail to converge

Proof by Contradiction

understand and use proof by contradiction for simple results, including the irrationality of sqrt(2) and the infinitude of primes

The General Binomial Expansion

be able to expand (1 + x)^n for any rational n, stating the range of values of x for which the expansion is valid · be able to adapt the binomial series to expand (a + bx)^n by first factorising out a^n

Curves Defined Parametrically

understand and use parametric equations of curves, including converting between parametric and Cartesian forms · be able to find points of intersection, tangents and areas under curves defined parametrically

Related Rates and Connected Variables

be able to use the chain rule to solve related-rates problems where one variable is changing as a known function of time and another depends on it

Volumes of Revolution

be able to find the volume of revolution generated by rotating a curve about the x-axis or the y-axis through 2π radians, using V = π ∫ y^2 dx or V = π ∫ x^2 dy

First-Order Separable Differential Equations

be able to formulate and solve first-order differential equations with separable variables, including those arising from contextual modelling problems · be able to find particular solutions of separable differential equations using given boundary conditions and interpret solutions in context

Vectors in 3D and the Scalar Product

understand and use vectors in three dimensions, including position vectors, magnitude, unit vectors and the i, j, k basis · understand and use the scalar (dot) product a · b = |a||b| cos θ to find angles between vectors and to test for perpendicularity · be able to find the vector equation of a line in 3D, determine whether two lines are parallel, intersect or are skew, and find the angle between two lines

Mathematical Models and Assumptions

understand the role of mathematical modelling in mechanics and the meaning of standard modelling assumptions (particle, light, smooth, rough, inextensible, rigid, uniform) · understand the distinctions between scalars and vectors, and recognise common quantities (position, velocity, acceleration, force) and their vector or scalar nature

Vector Operations and Resolution

be able to add and subtract vectors, multiply a vector by a scalar, and resolve a vector into components parallel and perpendicular to a given direction · be able to find the magnitude and direction of a vector expressed in component form, and write a vector in i, j component form given its magnitude and direction

Constant Acceleration Equations

be able to use the equations for uniformly accelerated motion in one dimension: v = u + at, s = (u + v)t/2, s = ut + 1/2 at^2 and v^2 = u^2 + 2as · be able to apply these equations to problems involving vertical motion under gravity, taking g = 9.8 m s^(-2) unless otherwise specified

Motion Graphs

be able to draw and interpret displacement-time and velocity-time graphs, including the use of gradients to find velocity and acceleration, and areas to find displacement and change in velocity

Newton's Laws of Motion

understand and apply Newton's first, second and third laws of motion to particles moving in one dimension, including using F = ma · be able to draw force diagrams and write equations of motion for particles experiencing weight, normal reaction, tension, thrust and friction

Friction

understand the model of friction and use F ≤ μR for a rough contact, recognising that at the point of slipping F = μR

Connected Particles

be able to apply Newton's laws to systems of connected particles, including particles linked by a light inextensible string passing over a smooth pulley

Equilibrium of a Particle

be able to solve problems involving the equilibrium of a particle under the action of two or more coplanar forces by resolving in two perpendicular directions

Moments of a Force

understand and use the moment of a force about a point: moment = F × d, where d is the perpendicular distance from the point to the line of action of the force · be able to apply the principle of moments to a rigid body in equilibrium under coplanar forces, including problems involving uniform and non-uniform rods and reactions at supports

Projectile Motion

be able to analyse projectile motion by resolving the initial velocity into horizontal and vertical components and treating each independently · be able to derive and use formulae for the range, maximum height and time of flight of a projectile launched from a level surface

Variable Acceleration via Calculus

understand the relationships v = dx/dt and a = dv/dt = d^2 x/dt^2, and use calculus to solve kinematics problems with variable acceleration in one dimension

Centres of Mass of Discrete and Composite Systems

be able to find the centre of mass of a system of particles in 1D and 2D using x̄ = Σ m_i x_i / Σ m_i and the corresponding formula for ȳ

Centres of Mass of Uniform Laminae

be able to find the centre of mass of standard uniform laminae (rectangle, triangle, semicircle, circular sector) and of composite laminae formed by combining these shapes

Work and Kinetic Energy

understand and use the definitions of work done by a force, kinetic energy KE = 1/2 m v^2 and gravitational potential energy PE = m g h · be able to apply the work-energy principle and the principle of conservation of mechanical energy to motion under gravity, friction and other forces

Power

understand and use the definition of power as P = Fv for a vehicle of constant driving force, and apply it to problems of motion on level ground and on an inclined plane

Impulse and Momentum

understand the definition of momentum p = mv and impulse J = F t = Δp, and apply the principle of conservation of linear momentum to direct collisions between particles

Direct Collisions and the Coefficient of Restitution

understand and use Newton's experimental law of restitution: e = (separation speed) / (approach speed) for two particles in direct collision, where 0 ≤ e ≤ 1 · be able to apply impulse-momentum and the coefficient of restitution to solve direct-collision problems, including collisions with a fixed wall

Equilibrium of a Rigid Body in 2D

be able to solve problems on the equilibrium of a rigid body under coplanar forces, including a ladder against a wall and rods on inclined planes, using both force equations and moment equations

Variable Acceleration as a Function of Displacement or Velocity

be able to use the relation a = v dv/dx (and equivalent forms) to solve kinematics problems where acceleration is given as a function of displacement or velocity

Hooke's Law and Elastic Potential Energy

understand and use Hooke's law T = λ x / l for an elastic string or spring, where λ is the modulus of elasticity, x is the extension and l the natural length · be able to use the formula EPE = λ x^2 / (2 l) for the elastic potential energy stored in a stretched string or spring, and apply energy conservation to problems involving elastic strings and springs

Variable Forces and Work-Energy Methods

be able to apply the work-energy principle and Newton's laws to problems involving variable forces, including forces that depend on displacement or velocity

Uniform Circular Motion

understand and use the formulae for the acceleration of a particle moving in a circle at constant speed: a = v^2 / r = r ω^2, where ω is the angular speed · be able to apply Newton's second law to circular motion in horizontal circles, including the conical pendulum and a particle on a banked track

Non-Uniform Circular Motion in a Vertical Plane

be able to use energy conservation together with Newton's second law to analyse motion in a vertical circle, including the bucket-overhead and ball-on-string cases · understand the difference between a string (which can only pull) and a rigid track/wire (which can push or pull), and derive the critical condition for completing a vertical loop

Toppling and Sliding

be able to determine whether a rigid body on a rough inclined surface will slide first or topple first as the angle of inclination increases (slides when tan θ = μ; topples when tan θ = a/h — does whichever happens at the smaller angle) · be able to determine whether a rigid body on a rough horizontal surface will slide first or topple first as an applied horizontal force increases (slides when F = μW; topples when F·h = W·a — does whichever happens at the smaller F)

The Modelling Process

understand the role of a statistical model — a simplified mathematical description of a real-world population or process — and the necessary trade-off between simplicity and fidelity · understand the four-stage modelling cycle (formulate → test → refine → use), and the limitations and assumptions that every model carries

Diagrams for Data

be able to construct and interpret stem-and-leaf diagrams and histograms (including histograms with unequal class widths, using frequency density = frequency / class width) · be able to construct and interpret cumulative-frequency diagrams and box plots, identify outliers using the 1.5 × IQR rule, and comment on skewness from a diagram

Measures of Location and Spread

be able to calculate the mean, median, mode, range, and interquartile range (IQR) for raw and grouped data, and choose an appropriate measure of location and spread · be able to calculate the variance and standard deviation of raw and grouped data using Σ(x − x̄)²/n = Σx²/n − x̄²; understand coding (linear transformation) and its effect on mean and standard deviation

Events and Probability Rules

understand the language of probability: sample space, outcomes, mutually exclusive events, exhaustive events, complementary events · be able to use the addition rule P(A ∪ B) = P(A) + P(B) - P(A ∩ B) and the multiplication rule for independent events P(A ∩ B) = P(A) P(B)

Conditional Probability

be able to use conditional probability P(A | B) = P(A ∩ B) / P(B), interpret the result and assess independence by comparing P(A | B) with P(A) · be able to draw and interpret tree diagrams and Venn diagrams to solve probability problems

Correlation

be able to calculate the product moment correlation coefficient (PMCC) r from summary statistics, interpret its value in context and comment on the existence and strength of any linear relationship

Linear Regression

be able to find the equation of the least-squares regression line y on x from summary statistics, use it to estimate values of y for given x, and comment on the appropriateness of interpolation versus extrapolation

Probability Distributions

understand the concept of a discrete random variable, its probability distribution and the requirement that probabilities sum to 1 · be able to calculate the expectation E(X), variance Var(X) and standard deviation of a discrete random variable, including E(aX + b) and Var(aX + b)

The Discrete Uniform Distribution

understand and use the discrete uniform distribution as a model, and find its expectation and variance

Properties and Standardisation

understand the properties of the normal distribution N(μ, σ^2), including symmetry about μ and the inflection points at μ ± σ · be able to use standardisation Z = (X - μ) / σ and tables (or a calculator) of the standard normal distribution to find probabilities and inverse values · be able to solve problems where μ and/or σ are unknown by setting up and solving simultaneous equations from given probabilities

Modelling with the Binomial Distribution

understand the conditions under which a random variable can be modelled by B(n, p): a fixed number of independent trials with two outcomes and constant probability of success · be able to calculate probabilities, expectation E(X) = np and variance Var(X) = np(1 - p) for a binomially distributed variable

Modelling with the Poisson Distribution

understand the conditions under which a random variable can be modelled by Po(λ): events occurring independently at a constant average rate λ in a fixed interval · be able to calculate Poisson probabilities, use E(X) = Var(X) = λ, and use the Poisson distribution as an approximation to the binomial when n is large and p is small

Probability Density Functions

understand the concept of a continuous random variable defined by a probability density function f(x) and use ∫ f(x) dx = 1 over the support · be able to find probabilities, the cumulative distribution function, the median and other percentiles, and the expectation and variance of a continuous random variable

The Continuous Uniform Distribution

understand and use the continuous uniform (rectangular) distribution on an interval [a, b], including its mean (a + b)/2 and variance (b - a)^2 / 12

Hypothesis Tests for the Binomial Distribution

understand the language of hypothesis testing: null hypothesis, alternative hypothesis, test statistic, significance level, critical region, p-value, and one- and two-tailed tests · be able to carry out a hypothesis test for the parameter p of a binomial distribution, identify the critical region and state conclusions in context

Hypothesis Tests for the Poisson Distribution

be able to carry out a hypothesis test for the parameter λ of a Poisson distribution, identify the critical region and state conclusions in context

Linear Combinations of Independent Variables

understand and use the results E(aX + bY) = a E(X) + b E(Y) and, for independent X and Y, Var(aX + bY) = a^2 Var(X) + b^2 Var(Y) · understand that linear combinations of independent normal variables are themselves normally distributed, and use this to find probabilities

Sampling Methods

understand the purpose of sampling and recognise common sampling methods (simple random, stratified, systematic, quota, cluster, opportunity), including their advantages and disadvantages

The Distribution of the Sample Mean

understand and use the result that if X has mean μ and variance σ^2, then the sample mean of n independent observations has mean μ and variance σ^2 / n, and is normally distributed when X is normal or n is large (central limit theorem)

Confidence Intervals for the Mean

be able to construct and interpret confidence intervals for a population mean when σ is known (using the normal distribution) and when σ must be estimated (using the t distribution)

Tests for a Single Mean and for the Difference of Two Means

be able to carry out hypothesis tests for the mean of a normal distribution and for the difference between two means using independent samples, applying both z- and t-tests where appropriate

Chi-Squared Tests

be able to carry out a chi-squared goodness-of-fit test for a hypothesised distribution, calculating expected frequencies, the test statistic Σ (O - E)^2 / E and identifying degrees of freedom · be able to carry out a chi-squared test for association in a contingency table and interpret the conclusion in context

Spearman's Rank Correlation Coefficient

be able to calculate Spearman's rank correlation coefficient r_s from ranked data and carry out an associated hypothesis test for the existence of a monotonic relationship

Algorithms and Sorting

understand what is meant by an algorithm and follow algorithms expressed in words or pseudocode · be able to apply standard sorting and searching algorithms (bubble sort, quick sort, binary search) and compare their efficiency informally

Bin-Packing Algorithms

be able to apply the first-fit, first-fit decreasing and full-bin algorithms to bin-packing problems, and comment on optimality

Graph Terminology

understand and use graph terminology: vertex, edge, degree, path, cycle, simple graph, connected graph, tree, complete graph and bipartite graph · understand the Eulerian and Hamiltonian properties of graphs and the conditions under which a graph is traversable

Kruskal's and Prim's Algorithms

be able to apply Kruskal's algorithm to find a minimum spanning tree of a weighted connected graph, expressing the result both as a list of selected edges and as a diagram · be able to apply Prim's algorithm (both on a graph and from a network distance matrix) to find a minimum spanning tree

Dijkstra's Algorithm

be able to apply Dijkstra's algorithm to find the shortest path between two vertices of a weighted graph with non-negative edge weights, recording working values at each vertex

The Chinese Postman Problem

be able to solve the route inspection problem for a connected graph by identifying odd-degree vertices, pairing them optimally and adding repeated edges to produce a minimum-weight closed walk traversing every edge

Activity Networks and the Critical Path

be able to construct activity networks from a precedence table, calculate earliest start and latest finish times for each activity and identify the critical path · be able to calculate total float for non-critical activities and produce Gantt (cascade) charts and resource histograms

Formulation and Graphical Solution

be able to formulate a linear programming problem in two variables by identifying the decision variables, the objective function and the constraints · be able to solve a two-variable linear programming problem graphically by drawing the feasible region and using the objective-line (ruler) or vertex methods to find optimal solutions, including integer solutions where appropriate

Matchings in Bipartite Graphs

be able to find a maximum matching in a bipartite graph using the matching improvement algorithm based on alternating paths