Edexcel IGCSE Further Pure Mathematics
Last Accessed
Recently visited lessons will appear here for quick access.Start exploring your first lesson today!
Number
Exact numbers
1C - Working with surdsA square root can be an exact answer even when its decimal expansion never ends. You will simplify square roots, combine like surds and multiply them without losing accuracy, ready to use exact lengths and solve equations later in the course.
1D - Rationalising denominatorsThe same exact number can have a surd in its denominator or only in its numerator. You will change between these forms, use conjugates to remove two-term denominators and choose an efficient method for simple surd equations.
Powers and exponential graphs
1B - Integer and fractional indicesPowers do more than count repeated multiplication: they can also describe reciprocals and roots. You will use the index laws to simplify exact expressions and solve equations by writing both sides as powers of the same base.
1A - Exponential functionsAn exponential function changes by a constant multiplier over equal input steps. You will build its values, recognise growth and decay graphs, and read the effect of multiplying or shifting an exponential expression.
Logarithms
1A, 1B - Logarithms as exponentsWhat power of 2 gives 8, and what power gives 1/8? A logarithm answers this kind of question by reversing exponentiation. In 1A you will connect logarithms to exponential graphs; in 1B you will find exact logarithms from known powers.
1B - Combining logarithmsCan adding two logarithms turn a difficult calculation into an exact power? In 1B you will use the index laws to explain the logarithm laws, then expand and combine expressions. The same-base and positive-argument conditions will keep every rearrangement valid.
1B - Solving logarithmic equationsWhy can algebra produce two candidates when a logarithmic equation has only one solution? In 1B you will isolate or combine logarithms, reverse them into powers and check the answers in the original equation. The positive-argument conditions will decide which algebraic candidates really work.
1B - Changing base and finding exponentsHow can you find the exponent in 3^x=20 when 20 is not a familiar power of 3? In 1B you will change logarithm bases, use reciprocal logarithms and calculate unknown exponents. You will also interpret those exponents as times in growth and decay models.
Algebra
Quadratics
2A - Factorising quadratic expressionsCan you turn a sum of three terms into a product of two brackets? You will use the connections between a quadratic's coefficients and its factors, then decide when integer factors are possible and when exact surds are needed.
2A - Completing the squareHow small can a quadratic expression become? You will rewrite quadratics as a multiple of one square plus a constant, making their minimum or maximum visible without plotting lots of points.
2B, 2C - Quadratic roots and the discriminantA quadratic can have two different real solutions, a repeated real solution or no real solutions. In 2B you will use the discriminant to distinguish these cases and determine conditions on coefficients; in 2C you will find the roots exactly with the quadratic formula, ready to use expressions inv...
2C - Sums and products of rootsThe coefficients of a quadratic reveal the sum and product of its roots, even when the roots themselves are awkward. You will use these links to calculate expressions in the roots without solving the equation, then work backwards to form a quadratic from a known sum and product.
2C - Quadratics with related rootsYou can form a quadratic for a changed pair of roots without solving the original equation. You will calculate the new sum and product for shifts, multiples, reciprocals, squares and simple combinations, then turn them into a complete equation.
Polynomial equations
3A - Dividing polynomialsCan a cubic divided by a linear expression give an exact quadratic answer? You will divide polynomials by monic and non-monic linear expressions, keep track of missing powers and describe any remainder. Multiplying back will give you a reliable way to check every division.
3B - Remainders and factorsCan one substitution replace a whole polynomial division? You will use the remainder theorem to find what is left over, recognise linear factors and turn factor or remainder information into equations for unknown coefficients. The link comes directly from the division identity.
3B, 3C - Factorising and solving cubicsHow can one known root unlock a cubic equation? In 3B you will use a supplied factor and divide to factorise a cubic; in 3C you will find a simple rational root when needed and solve for every real root. You will also see why a cubic need not have three different real solutions.
3C - A line and a quadratic togetherCan two equations describe the same point in more than one way? You will substitute a linear relation into a quadratic equation, find every matching coordinate pair and check both equations. You will then connect two, one or no real pairs to intersections and tangency.
Inequalities
3D - Solving linear inequalitiesAn inequality can have infinitely many solutions, so finding one successful value is only a start. You will rearrange linear inequalities, describe their full solution sets and combine conditions while keeping track of which endpoints are allowed.
3D - Solving quadratic inequalitiesThe roots of a quadratic mark where its value is zero, but an inequality asks where its value has the required sign. You will find those boundaries, choose intervals using signs or a sketch, and combine the result with another condition without rounding exact endpoints.
3E - Regions and linear optimisationWith two variables, an inequality describes a region of possible pairs rather than an interval of numbers. You will draw its boundaries, combine constraints into a feasible region and find the greatest or least value of a linear expression, including situations where the variables must be integers.
Function graphs
4A - Sketching polynomial graphsHow much of a curve can you predict before plotting any points? Factors reveal where a polynomial meets an axis, while the highest power tells you where its ends go. You will combine these clues to make a labelled sketch without finding exact turning points.
4A - Rational functions and asymptotesDoes every forbidden input produce a graph that shoots upwards or downwards? You will distinguish a missing point from a vertical asymptote, find axis-parallel asymptotes and use them with intercepts to sketch rational functions with linear denominators.
4B - Solving equations with graphsWhat can you do when an equation has no convenient exact solution? Graphs turn equality into a visible meeting point. You will choose graphs that represent an equation, use a suitable straight line with logarithmic or exponential curves, and report solutions at an accuracy the graph supports.
Series
Arithmetic sequences and sums
5A, 5B - Arithmetic sequencesHow can you find a distant term without writing every term before it? In 5B, you will use constant differences to describe arithmetic sequences and recover their rules from known terms. In 5A, sigma notation gives you a compact way to describe a total of selected terms.
5A, 5B - Arithmetic seriesAn arithmetic sequence is easy to continue, but adding many terms one by one is slow. In 5B, you will calculate arithmetic totals, recover an unknown number of terms, and find when a total reaches a target. The sigma notation from 5A lets you specify exactly which consecutive terms to add.
Geometric sequences and sums
5B - Geometric sequencesRepeated multiplication produces a different kind of pattern from repeated addition. You will find geometric terms, recover possible ratios from term conditions, and use logarithms to locate the first term beyond a growth or decay threshold.
5A, 5B - Finite geometric seriesRepeated multiplication can make a long total surprisingly compact. In 5B, you will add finite geometric series and recover their parameters from sums. In 5A, you will use sigma bounds to identify the first included term and the number of terms being added.
5B - Convergent geometric seriesCan infinitely many nonzero terms have a finite total? You will interpret a sum to infinity through partial sums, use the geometric convergence condition, and combine finite and infinite information to recover parameters or reach a target total.
Binomial expansions
6A - Binomial expansions with integer powersRepeated brackets soon become slow to multiply out. You will see where binomial coefficients come from, use them to expand positive integer powers, and find one required term without calculating the rest.
6A - Binomial series with rational powersA square root cannot be expanded by multiplying half a bracket. The binomial series extends the coefficient pattern to fractional and negative powers; you will use it carefully, keep any constant factor, and state where the resulting series is valid.
6A - Using a binomial approximationA short binomial expansion can turn a root or reciprocal into ordinary arithmetic. You will choose a useful small variable, calculate an estimate from a prescribed number of terms, and combine short expansions without losing contributions to the powers you need.
Geometry
Coordinate geometry
8A - Distances between pointsCoordinates describe position, but they also let you measure a length exactly. You will build the distance formula from a right-angled triangle, find missing coordinates from length conditions and use exact lengths to investigate geometric figures.
8B - Dividing a line in a ratioHow can you locate a point one-third of the way along a sloping segment without measuring the diagram? You will find midpoints, divide segments in a stated ratio and work backwards to find a missing endpoint or ratio.
8C - Gradients of straight linesA line can rise gently, fall steeply or stand vertically. You will describe its direction using coordinate changes, distinguish zero gradient from undefined gradient and decide whether three points lie on one straight line.
8D - Equations of straight linesA point fixes a position; a gradient fixes a direction. You will combine them to describe a whole line, recognise equivalent equation forms and find where lines meet by solving their equations together.
8E - Parallel and perpendicular linesMatching directions and right angles can be expressed using gradients. You will find parallel and perpendicular lines, handle horizontal and vertical directions, and combine these ideas with midpoints and intersections to solve exact coordinate problems.
Vectors
7A, 7B - Vectors and componentsIf you walk east and then north, what single movement takes you from your starting point to your finishing point? Vectors describe that overall movement with both size and direction. You will add, subtract and scale vectors, then express them using horizontal and vertical components. In 7A, you l...
7C, 7E - Vector lengths and unit vectorsA displacement has components 3 units east and 4 units north: how long is the direct journey, and how can we keep its direction while changing its length? You will calculate vector magnitudes, build unit vectors and construct vectors of a specified magnitude. In 7C, you find a vector’s ma...
7D, 7F - Position vectors and division pointsTwo places can be described by arrows from one origin, but how do we find the movement from one place to the other? You will use position vectors to join points, locate a point dividing a segment in a given ratio and calculate the same vector by different routes through a figure. In 7D, you descr...
7A, 7F - Parallel lines and collinearityIf two arrows have parallel directions, what extra evidence proves that three points lie on one line? You will compare coefficients in nonparallel reference directions, prove that directions are parallel and identify the extra connection needed to prove that points are collinear. In 7A, you compa...
7F - Vector intersections and concurrencyHow can you prove that three lines meet at one point when a sketch is too imprecise to decide? You will describe that shared point, solve for its position and ratios, then test whether a third line passes through exactly the same point.
Trigonometry
Angles and graphs
10A - Radians, arcs and sectorsMeasure an angle using the circle itself, then use radians to calculate curved lengths and areas. You will distinguish an arc from a sector perimeter and a sector from a segment, keeping exact answers where possible.
10B - Trigonometric ratios beyond acute anglesExtend sine, cosine and tangent from right-angled triangles to rotations of any size. You will use coordinates to choose signs, derive the standard exact values and find related values in degrees or radians.
10B - Sine, cosine and tangent graphsBuild the three trigonometric graphs from unit-circle values, recognise their repeating patterns and sketch simple transformations. You will use intersections to locate all values in a stated interval, checking its endpoints and the gaps in tangent.
Triangles and space
10D - The sine rule and triangle areaCan the same two side lengths and angle describe two different triangles? You will use opposite side–angle pairs to find lengths, decide how many triangles are possible, and calculate area from two sides and their included angle.
10D - The cosine ruleTwo paths have the same lengths but meet at different angles. How much can the direct distance between their endpoints change? You will use the cosine rule to find sides and angles, then choose a suitable triangle method in exact calculations and two-dimensional journeys.
10C - Angles and lengths in three dimensionsWhich angle measures how steeply a line climbs out of a flat surface? You will extract triangles from solids, use perpendicular projections to measure line–plane angles, and construct cross-sections for angles between planes. Exact lengths will let you combine these ideas without accumulating rou...
Identities and equations
10E, 10F - Using trigonometric identitiesKnowing one trigonometric ratio often lets you find the others without finding the angle. In 10E, you will connect sine and cosine through the unit circle; in 10F, you will write tangent as their quotient. Together, these identities let you calculate exact values and prove that different expressi...
10G - Compound-angle formulaeCan you find 15° exactly using only familiar angles? You will use the supplied sine, cosine and tangent formulae for sums and differences of angles. These connect known ratios to new exact values and let you establish short identities by expanding the angles carefully.
10E, 10G - Double-angle identitiesDoubling an angle does not simply double its sine or cosine. In 10G, you will set the two angles equal in the supplied addition formulae; in 10E, you will use the Pythagorean identity to rewrite the resulting cosine expression. These connections give several useful forms for exact values and shor...
10H - Trigonometric equations on an intervalA calculator gives one angle when you enter an inverse trigonometric function. How can you know that you have found every angle an equation allows? You will use symmetry, periods and the stated interval to solve equations in degrees and radians, including multiplied and shifted angles.
10E, 10F, 10H - Trigonometric equations using identitiesHow can an equation involving both sine and cosine turn into an ordinary quadratic? You will use the Pythagorean identity (10E) and tangent quotient identity (10F) to solve the interval equations of 10H, recovering every permitted angle. You will also keep zero-factor solutions, both signs of a s...
Calculus
Differentiation rules
9A - Gradients and the power ruleA straight line has one gradient, but a curve becomes steeper or flatter as you move along it. How can we assign a gradient at one point? You will connect nearby-point gradients to a tangent, learn to differentiate powers and their sums, and evaluate derivatives to describe a curve locally.
9A - Differentiating sine, cosine and exponentialsThe sine curve rises through the origin, flattens at its peak and then falls. Which function records those changing gradients? You will differentiate sine, cosine and the exponential function with constant multiples of the input, then combine these rules with powers and evaluate exact gradients.
9B - The chain ruleHow can you differentiate a bracket raised to a power without expanding it? You will separate an inner function from an outer function, then use the chain rule to differentiate powers, trigonometric functions and exponentials of simple expressions.
9B - The product ruleIf two factors both change, how does their product change? You will account for both contributions, differentiate products of familiar functions, and combine the product and chain rules without losing factors or signs.
9B - The quotient ruleCan you differentiate a fraction by dividing the derivatives of its numerator and denominator? You will use the quotient rule, choose when algebra gives a shorter route, and evaluate derivatives while keeping the original function's domain.
Curves and optimisation
9F - Tangents and normalsMany straight lines pass through one point on a curve. You will use a derivative to choose the tangent, find its perpendicular normal, and locate points where a tangent or normal satisfies a given slope condition.
9D, 9E - Stationary and turning pointsA horizontal tangent can mark the top of a hill, the bottom of a valley, or a place where a curve briefly flattens and carries on. You will find stationary points, justify their nature and distinguish a turning point from a stationary point without a turn.
9E - Maximum and minimum problemsA fixed amount of material can make a tall narrow container or a short wide one. You will turn a practical constraint into a function of one variable, use calculus to find its best value and check that the answer is possible in the original problem.
Integration
9A - Antiderivatives and integration rulesKnowing every gradient of a curve still leaves its height undetermined. You will reverse differentiation to find families of functions, integrate powers and standard trigonometric and exponential forms, and use a point to select one function from the family.
9A, 9C - Definite integralsAn indefinite integral gives a family of functions; a definite integral gives a number for a specified interval. You will evaluate it exactly, interpret its sign and work backwards to find a missing limit or parameter, ready for later area and motion applications.
9C - Areas bounded by curvesHow can two regions with positive areas give an integral of zero? You will turn signed integrals into geometric areas, find the region between two curves, and choose between vertical and horizontal strips when one direction makes the calculation simpler.
9C - Volumes of revolutionWhat solid appears when a flat curved region spins around an axis? You will build its volume from thin circular discs, identify the correct radius, and integrate with respect to the coordinate running along the axis of rotation.
Motion and change
9C - Displacement, velocity and accelerationCan a moving particle have zero acceleration, or pass through rest without changing direction? You will use differentiation and integration to connect position, velocity and acceleration, then distinguish the distance travelled from the displacement between two times.
9G - Connected rates of changeIf water enters a vessel at a constant volume rate, must its depth rise at a constant speed? You will connect rates through a shared changing quantity, use geometric relationships at one instant, and interpret signs and time units in growing and shrinking objects.
9G - Estimating small changesHow can a derivative estimate a nearby square root or the effect of a tiny change in radius? You will use a tangent to estimate changes and new values, then compare estimates with exact calculations so that their accuracy is stated meaningfully.