Surds and exact answers
Keep a square root exact until the question asks for a decimal.
Further Maths topics
Choose a topic for an explanation, a worked example and practice questions. Select your course to filter the practice.
Algebra & functions
Show 21 topics
Fractional and negative indices
A negative power takes a reciprocal; a fractional power describes a root.
Expanding and factorising
Expansion distributes multiplication; factorisation reverses it.
Solving quadratic equations
Solve a quadratic by factorising, completing the square or using the quadratic formula.
Completing the square
Complete the square to find a quadratic’s turning point.
Algebraic fractions
Cancel factors, and retain the values excluded by the original denominator.
The factor theorem
A polynomial has factor x − a exactly when its value at a is zero.
Quadratic inequalities
Locate the roots, then determine the sign in each interval.
Domain and range
The domain lists allowed inputs; the range lists outputs actually attained.
Composite functions
In fg(x), apply g first and then apply f.
Inverse functions
An inverse reverses the input-output relationship on a one-to-one domain.
Sketching functions
A useful sketch records structure: intercepts, turning points, restrictions and asymptotes.
Function notation and evaluation
Replace every occurrence of the input variable by the complete new input.
Simultaneous linear and quadratic equations
Substitute one equation into the other, then recover every ordered pair.
Partial fractions
Split a rational expression into simpler fractions before integrating or analysing it.
Rearranging formulae
Collect every occurrence of the new subject before dividing.
Three simultaneous linear equations
Eliminate one variable twice to obtain two equations in two variables.
Piecewise graphs and endpoints
Choose the rule using the input, then mark whether each boundary belongs to the graph.
Exponential graphs and parameters
Equal steps in the input multiply the output by a constant factor.
Choosing an inverse branch
A restricted domain determines which root gives the inverse.
Inequalities and interval notation
Record both the permitted regions and whether their endpoints are included.
Geometry & trigonometry
Show 19 topics
Straight lines and gradients
Use a gradient and one known point to determine a nonvertical line.
Circles and tangents
Complete the square to read off a circle’s centre and radius.
Scalar products and angles
The scalar product relates coordinates to lengths and angles.
Trigonometric equations in intervals
Find every permitted angle, then check the interval endpoints.
Trigonometric identities
An identity holds for every angle in its stated domain.
Area, volume and angle facts
Identify the perpendicular height and distinguish a surface area from a volume.
Circle theorems and angle reasoning
Name the chord, arc or tangent that makes a circle theorem applicable.
Pythagoras in three dimensions
Find a face diagonal before using it in a second right-angled triangle.
Sine rule, cosine rule and triangle area
Match each side to its opposite angle before choosing a rule.
Angles between lines and planes
Use the perpendicular projection of the line onto the plane to identify the angle.
Sketching sine, cosine and tangent
Mark a full period and the key values before repeating the curve.
Trigonometric ratios in each quadrant
Use the coordinates on a unit circle to determine the signs.
Exact trigonometric values
Derive the special values from two right-angled triangles.
Midpoints and section ratios
Move the required fraction of the way from the first endpoint to the second.
Exact distances between coordinates
Use the horizontal and vertical displacements as the perpendicular sides of a right triangle.
Proving equal tangent lengths
The perpendicular radii create two right-angled triangles that can be compared.
The two possible sine-rule angles
A sine value may give two angles inside a triangle; test both against the other information.
Angles between two planes
Use a cross-section perpendicular to the line where the planes meet.
Identity proofs and permitted angles
An identity proof preserves the domain of every expression it uses.
Calculus & change
Show 14 topics
Differentiating powers
Differentiation finds an instantaneous rate of change.
Tangents and normals
The derivative gives a tangent’s gradient at the chosen point.
Stationary points
A stationary point has derivative zero, but it need not be a maximum or minimum.
Integration and exact area
An antiderivative reverses differentiation; a definite integral measures signed accumulation.
Areas in polar coordinates
A polar curve requires both a radial equation and an angular interval.
Hyperbolic functions
The exponential definitions explain the signs in hyperbolic identities.
First order differential equations
Solving a differential equation determines a family of functions until a condition fixes the constant.
Newton-Raphson iteration
Use the tangent at a current estimate to produce a new estimate of a root.
Bisection and root brackets
A sign change brackets a root when the function is continuous on the interval.
Maclaurin series
A Maclaurin expansion uses derivatives at zero to approximate a function near zero.
Optimisation with a stated domain
Write the quantity in terms of one variable. Check stationary points and any permitted endpoints.
Area estimates using the trapezium rule
Approximate each strip by a trapezium between its endpoint heights.
Increasing and decreasing intervals
The sign of the derivative describes how the function changes between critical inputs.
Sketching a curve from its derivative
Combine intercepts, stationary points and derivative signs into one consistent sketch.
Matrices & complex numbers
Show 11 topics
Matrix multiplication
Multiply a row of the first matrix by a column of the second, then add the products.
Matrix transformations
The columns show where the two coordinate unit vectors move.
Determinants and singular matrices
A zero determinant identifies a square matrix with no inverse.
Inverse matrices
An inverse reverses an invertible linear transformation.
Complex arithmetic
Use i² = −1 and collect real and imaginary parts.
Modulus and argument
An Argand diagram turns a complex number into a point and a vector.
De Moivre and complex roots
Use De Moivre’s theorem to find powers and all the roots of a complex number.
Complex loci
Modulus statements describe distances on an Argand diagram.
The identity matrix
The identity matrix leaves every column vector unchanged.
Eigenvalues and eigenvectors
An eigenvector keeps its direction under a linear transformation, up to sign and scale.
Combining matrix transformations
For column vectors, the transformation performed first is on the right of the product.
Sequences & proof
Show 11 topics
Quadratic sequences
Constant second differences identify a quadratic nth term.
Arithmetic series
An arithmetic sequence adds a constant difference; a series adds its terms.
Geometric series
A geometric sequence multiplies by a constant ratio.
Proof by induction
Prove the first case, then show that if a case is true, the next case must be true.
Algebraic proof and counterexamples
Use a general expression to prove a statement about every permitted value.
Linear sequence nth terms
Use the constant difference to find the coefficient of n.
Counting with restrictions
Count the choices at each position after allowing for the restrictions.
Sequence terms and limiting values
A limiting value describes what the terms approach as the index increases.
Writing a geometric proof
State a reason for each angle or length equality, then connect them to the required result.
Sums of integer powers
Expand the summand and apply each finite-sum formula over the correct index range.
Proving divisibility with consecutive integers
Write a general integer as n and use it to prove the statement for every permitted value.
Applied mathematics
Show 5 topics
Binomial probabilities
A binomial model counts successes in a fixed number of independent trials.
Shortest paths with Dijkstra’s algorithm
Dijkstra’s algorithm settles vertices in order of their current distance from the start.
Momentum and impulse
Momentum is a vector; choose and state a positive direction.
Expectation of a discrete random variable
Multiply each possible value by its probability and add the products.
Linear programming on a feasible region
Find the feasible vertices, then compare the objective function at those points.