Fourteen lessons covering core Further Mathematics content (proof, complex numbers, matrices, series, calculus, hyperbolic functions, differential equations, vectors, polar coordinates) and mechanics, statistics and discrete options, with interactive explorers.
A study guide to A-level Further Mathematics, not a full course. The paper structure and topic list follow the AQA 7367 pages; other boards (Edexcel, OCR, WJEC) organize options differently. Method and formulae are standard A-level content, and you should check your board's specification and formula booklet. Lab numbers are practice values.
['A-level Mathematics (taken at the same time or before)']
Course outline
Proof by induction
Prove statements for all positive integers using induction.
Complex numbers: arithmetic and the Argand diagram
Calculate with complex numbers and use modulus and argument.
De Moivre's theorem and roots of unity
Raise complex numbers to powers and find nth roots.
Matrices and transformations
Use 2 by 2 and 3 by 3 matrices, determinants, inverses and invariant lines.
Roots of polynomials and series
Use relations between roots and coefficients and sum series with standard results.
Maclaurin series and further calculus
Build series for functions and evaluate improper integrals, means and volumes.
Hyperbolic functions
Use sinh, cosh and tanh, their identities, derivatives and inverses.
Differential equations and Euler's method
Solve first and second order equations and approximate with steps.
Further vectors: dot product, cross product and planes
Use scalar and vector products to find angles, areas and planes.
Polar coordinates
Sketch curves in polar form and find areas.
Mechanics option: momentum and circular motion
Use momentum, impulse and circular motion formulas.
Statistics option: the Poisson distribution
Model rare random events and combine Poisson variables.
Discrete mathematics option: graphs and algorithms
Use graph theory, networks and algorithm counts.
Exam skills for Further Mathematics
Plan the three papers and the optional content.
Sources and curriculum note
Reviewed October 7, 2026. Confirm the specification for your board and exam year.
Read every lesson below. The interactive reader above contains the same explanations, with visual tools and quizzes.
1. Proof by induction
Learning goal: Prove statements for all positive integers using induction.
Proof by induction shows a statement P(n) is true for every positive integer n. There are four parts. First the basis: show P(1) is true. Then assume P(k) is true for some positive integer k. Then show that P(k + 1) must be true. Finally write the conclusion: P(1) is true and P(k) implies P(k + 1), so P(n) is true for all positive integers n.
A standard example is the sum of the first n integers: 1 + 2 + ... + n = n(n + 1)/2. For n = 1 both sides equal 1. Assume the result for k, then 1 + ... + k + (k + 1) = k(k + 1)/2 + (k + 1) = (k + 1)(k + 2)/2, which is the formula with n = k + 1.
Induction also proves divisibility results. To show 7^n - 1 is divisible by 6, write 7^(k+1) - 1 = 7(7^k - 1) + 6, which is a multiple of 6 when 7^k - 1 is. It proves results for matrices, such as a power of a matrix, and for recurrence relations.
Markers want the full structure and the final sentence. Say clearly where the assumption is used, and write what you are trying to show for k + 1 before you start.
Worked example
Write the proof.
State P(n)
Basis
Step
Conclusion
Practice problem and solution
What is 1^3 + 2^3 + 3^3 + 4^3? Enter a number.
1 + 8 + 27 + 64 = 100, which equals (4 x 5 / 2)^2.
Mental model: Basis, assumption, step, conclusion. All four parts are required.
Common trap: Leaving out the basis case or the final sentence.
2. Complex numbers: arithmetic and the Argand diagram
Learning goal: Calculate with complex numbers and use modulus and argument.
The imaginary unit i satisfies i^2 = -1. A complex number is z = x + iy with real part x and imaginary part y. Add and subtract by parts. Multiply by expanding brackets and replacing i^2 by -1. To divide, multiply the top and bottom by the conjugate of the denominator.
The conjugate of x + iy is x - iy. The product z times its conjugate equals x^2 + y^2, which is the square of the modulus. The modulus |z| is the distance from the origin on the Argand diagram, the square root of x^2 + y^2. The argument is the angle from the positive real axis, taken in the range from minus pi to pi for the principal value.
A quadratic or polynomial with real coefficients has complex roots in conjugate pairs. If 2 + 3i is a root then 2 - 3i is a root as well. Loci such as |z - a| = r describe a circle with centre a and radius r on the Argand diagram, and |z - a| = |z - b| is the perpendicular bisector of the points a and b.
Draw an Argand diagram whenever a question mentions modulus or argument. Check the quadrant before you report an argument.
Worked example
Sketch a locus.
Identify the form
Find centre or line
Check the inequality
Shade
Practice problem and solution
What is the real part of (1 + 2i)(3 - i)? Enter a number.
3 - i + 6i - 2i^2 = 3 + 5i + 2 = 5 + 5i.
Mental model: Use the conjugate to divide, and picture modulus and argument on the Argand diagram.
Common trap: Forgetting i^2 = -1 when expanding.
3. De Moivre's theorem and roots of unity
Learning goal: Raise complex numbers to powers and find nth roots.
Polar form writes z = r(cos theta + i sin theta). Multiplying two complex numbers multiplies their moduli and adds their arguments. Dividing divides the moduli and subtracts the arguments. In exponential form z = r e^(i theta), which makes these rules look like the index laws.
De Moivre's theorem says (cos theta + i sin theta)^n = cos n theta + i sin n theta. It gives powers quickly and gives multiple-angle identities when you expand both sides and compare real and imaginary parts.
The nth roots of 1 are at the angles 2 pi k / n for k = 0, 1, ..., n - 1. They lie at equal angles around the unit circle and form a regular polygon. Their sum is 0 for n at least 2. For the nth roots of a general number w with modulus R, use R^(1/n) and add the equal angle steps to the root angle.
Always give n roots for an nth-root question, and say what range your arguments lie in.
Worked example
Find cube roots.
Write in polar form
Take the cube root of the modulus
Divide angle by 3
Add 120 degrees
Practice problem and solution
What is the sum of the five fifth roots of unity? Enter a number.
They form a regular pentagon centred at the origin, so they cancel.
Mental model: De Moivre gives powers and roots; roots of unity are equally spaced and sum to zero.
Common trap: Giving only one root of an nth-root question.
4. Matrices and transformations
Learning goal: Use 2 by 2 and 3 by 3 matrices, determinants, inverses and invariant lines.
Matrix multiplication combines rows of the first matrix with columns of the second, and is not commutative: AB is usually not BA. A 2 by 2 matrix transforms the plane. Its columns are the images of the points (1, 0) and (0, 1). Rotation, reflection, enlargement, stretch and shear all have matrix forms.
The determinant of [[a, b], [c, d]] is ad - bc. Its absolute value is the area scale factor, and its sign shows whether orientation is preserved. If the determinant is 0 the matrix is singular and has no inverse. The inverse is (1/det) times [[d, -b], [-c, a]]. The inverse of a product reverses the order: (AB)^(-1) = B^(-1) A^(-1).
Eigenvectors are non-zero vectors that only change by a scale factor under the matrix: Av = lambda v. The eigenvalues solve det(A - lambda I) = 0. For a 2 by 2 matrix this is lambda^2 - (trace) lambda + det = 0, so the sum of the eigenvalues is the trace and their product is the determinant. Invariant lines and diagonalization use these.
Simultaneous equations can be written as Ax = b. If det A is not zero there is a unique solution x = A^(-1) b; if it is zero there are no solutions or infinitely many.
Worked example
Find eigenvalues.
Form A - lambda I
Set determinant to zero
Solve the quadratic
Find eigenvectors
Practice problem and solution
What is the determinant of [[2, 1], [3, 4]]? Enter a number.
2 x 4 - 1 x 3 = 5.
Mental model: Columns are the images of the axes; determinant is area scale; eigenvalues sum to the trace.
Common trap: Multiplying in the wrong order.
5. Roots of polynomials and series
Learning goal: Use relations between roots and coefficients and sum series with standard results.
For a quadratic ax^2 + bx + c = 0 with roots alpha and beta, alpha + beta = -b/a and alpha beta = c/a. For a cubic ax^3 + bx^2 + cx + d = 0 the sum of the roots is -b/a, the sum of the products in pairs is c/a and the product of all three is -d/a. You can find expressions like alpha^2 + beta^2 from these without solving.
To form a new equation with transformed roots, such as 2 alpha and 2 beta, substitute x = y/2 into the original equation, or use the new sum and product. Check with a simple case.
Standard sums are the sum of r from 1 to n = n(n + 1)/2, the sum of r^2 = n(n + 1)(2n + 1)/6 and the sum of r^3 = (n(n + 1)/2)^2. Use these to sum polynomial series. The method of differences writes each term as f(r) - f(r + 1) so that most terms cancel and only the first and last remain.
Further algebra also includes partial fractions, which help with series that telescope, and inequalities involving rational functions.
Worked example
Use root relations.
Write sum and product
Express the target
Substitute
Calculate
Practice problem and solution
What is the sum of r^2 for r = 1 to 5? Enter a number.
5 x 6 x 11 / 6 = 55.
Mental model: Roots link to coefficients; standard sums and differences evaluate series.
Common trap: Sign errors in the root relations.
6. Maclaurin series and further calculus
Learning goal: Build series for functions and evaluate improper integrals, means and volumes.
The Maclaurin series of f(x) is f(0) + f'(0) x + f''(0) x^2 / 2! + f'''(0) x^3 / 3! + ... It approximates a function near x = 0 using derivatives. Standard series are e^x = 1 + x + x^2/2! + x^3/3! + ..., sin x = x - x^3/3! + x^5/5! - ..., cos x = 1 - x^2/2! + x^4/4! - ..., and ln(1 + x) = x - x^2/2 + x^3/3 - ... for x between -1 and 1 (including 1).
Series let you approximate values and evaluate limits. Compare the first few terms to see how closely a polynomial fits near zero, and note that the fit gets worse far from zero. You can combine series by substitution and multiplication.
An improper integral has an infinite limit or an infinite integrand. Evaluate it as a limit. For example the integral of 1/x^2 from 1 to infinity equals 1 but the integral of 1/x diverges. The mean value of f on [a, b] is (1/(b - a)) times the integral of f. The volume of revolution about the x-axis is pi times the integral of y^2 dx.
Always state whether an integral converges. Show the limit step in improper integrals; that is where marks are awarded.
Worked example
Approximate a value.
Pick the series
Use a few terms
Substitute x
Compare with the true value
Practice problem and solution
What is the mean value of x^2 on the interval 0 to 3? Enter a number.
(1/3) x [x^3/3] from 0 to 3 = (1/3) x 9 = 3.
Mental model: Maclaurin series approximate near zero; improper integrals need limits; volumes use pi times the integral of y squared.
Common trap: Using a series far from the centre.
7. Hyperbolic functions
Learning goal: Use sinh, cosh and tanh, their identities, derivatives and inverses.
The hyperbolic functions are defined with exponentials: sinh x = (e^x - e^(-x))/2, cosh x = (e^x + e^(-x))/2 and tanh x = sinh x / cosh x. Cosh is even and sinh is odd. Cosh x is always at least 1. The graph of cosh describes a hanging chain, called a catenary.
The key identity is cosh^2 x - sinh^2 x = 1, which parallels cos^2 + sin^2 = 1 for circles. It comes from squaring the definitions. Other identities mirror trigonometric ones, with sign changes where products of sinh appear (Osborn's rule).
The derivative of sinh x is cosh x and the derivative of cosh x is sinh x. The inverse functions have logarithmic forms: arsinh x = ln(x + sqrt(x^2 + 1)) and arcosh x = ln(x + sqrt(x^2 - 1)) for x at least 1. These help to integrate expressions with sqrt(x^2 + a^2).
Check definitions first. Most proofs rewrite sinh and cosh as exponentials and simplify.
Worked example
Use an identity.
Pick the identity
Substitute
Simplify
Check at x = 0
Practice problem and solution
What is sinh(0)? Enter a number.
(1 - 1)/2 = 0.
Mental model: Hyperbolic functions are built from e^x; cosh^2 - sinh^2 = 1; derivatives swap sinh and cosh.
Common trap: Using the trigonometric sign in the identity.
8. Differential equations and Euler's method
Learning goal: Solve first and second order equations and approximate with steps.
A differential equation relates a function to its derivatives. A separable equation dy/dx = f(x) g(y) is solved by writing dy/g(y) = f(x) dx and integrating both sides. A first-order linear equation dy/dx + P y = Q uses an integrating factor e^(integral of P dx): multiply through and the left side becomes the derivative of a product.
A second-order equation with constant coefficients a y'' + b y' + c y = 0 has the auxiliary equation a m^2 + b m + c = 0. Real distinct roots m1 and m2 give y = A e^(m1 x) + B e^(m2 x). A repeated root gives y = (A + Bx) e^(mx). Complex roots p plus or minus iq give e^(px)(A cos qx + B sin qx). For a non-zero right-hand side add a particular integral to the complementary function.
Euler's method approximates a solution step by step: y(n+1) = y(n) + h f(x(n), y(n)). Smaller steps reduce the error but need more calculation. For dy/dx = y with y(0) = 1, n steps from 0 to 1 give (1 + 1/n)^n, which approaches e as n grows.
Always apply initial conditions to find the constants, and say whether the solution models the situation, for example growth that cannot continue forever.
Worked example
Solve a second order equation.
Auxiliary equation
Find roots
Write general solution
Apply conditions
Practice problem and solution
What is the larger root of m^2 - 3m + 2 = 0? Enter a number.
(m - 1)(m - 2) = 0, so m = 1 or 2.
Mental model: Choose the method by the form; apply initial conditions; Euler approximates step by step.
Common trap: Forgetting the second constant.
9. Further vectors: dot product, cross product and planes
Learning goal: Use scalar and vector products to find angles, areas and planes.
The scalar product a.b = a1 b1 + a2 b2 + a3 b3 = |a||b| cos theta. It gives the angle between vectors and shows perpendicularity: a.b = 0 means the vectors are perpendicular. The vector product a x b is perpendicular to both a and b and has magnitude |a||b| sin theta, equal to the area of the parallelogram they span.
A line has vector equation r = a + t d, where a is a point on it and d is the direction. Two lines in three dimensions may intersect, be parallel or be skew. To check, set components equal and solve; if there is no consistent solution and the directions differ, they are skew.
A plane has equation r.n = k, or in Cartesian form ax + by + cz = d with normal vector (a, b, c). The distance from the point (x0, y0, z0) to the plane is |a x0 + b y0 + c z0 - d| / sqrt(a^2 + b^2 + c^2). The angle between planes is the angle between their normals. The angle between a line and a plane uses the angle between the direction and the normal.
Draw a sketch. Check your final vector is perpendicular to the ones you started with by taking a dot product.
Worked example
Find a plane.
Two direction vectors
Cross product
Normal
Use a point
Practice problem and solution
What is a.b for a = (1, 2, 3) and b = (4, -5, 6)? Enter a number.
4 - 10 + 18 = 12.
Mental model: Dot product for angles and perpendicularity; cross product for normals and areas; planes use a normal vector.
Common trap: Mixing up the dot and cross product.
10. Polar coordinates
Learning goal: Sketch curves in polar form and find areas.
In polar coordinates a point is given by its distance r from the origin and the angle theta from the positive x-axis. Convert with x = r cos theta, y = r sin theta, and r^2 = x^2 + y^2, tan theta = y/x (check the quadrant).
Sketch polar curves by listing r for key values of theta, noting where r is zero and where it is largest, and watching for negative r, which reflects the point through the origin. Standard curves include circles through the origin, cardioids and rose curves.
The area enclosed by a polar curve between two angles alpha and beta is half the integral of r^2 d theta. For r = a theta from 0 to theta, the area is a^2 theta^3 / 6. Find where a curve passes through the origin by solving r = 0 to get the tangents at the pole.
Give angles in radians unless told otherwise, and make clear which interval of theta you used.
Worked example
Convert an equation.
Substitute x and y
Simplify
Use r^2
Identify the curve
Practice problem and solution
A spiral r = theta (a = 1) is swept from 0 to 3 radians. What is the area? Enter a number.
theta^3 / 6 = 27 / 6 = 4.5.
Mental model: Polar form uses distance and angle; area is half the integral of r squared.
Common trap: Forgetting the half in the area formula.
11. Mechanics option: momentum and circular motion
Learning goal: Use momentum, impulse and circular motion formulas.
Momentum is mass times velocity. Impulse is force times time and equals the change in momentum. In a collision with no external force, total momentum is conserved. The coefficient of restitution e is the ratio of speed of separation to speed of approach along the line of impact, from 0 for perfectly inelastic to 1 for elastic.
For motion in a circle at constant speed v and radius r, the acceleration is v^2 / r towards the centre, and the angular speed omega = v / r. The resultant force towards the centre is m v^2 / r. Examples are tension in a string, friction on a turning car and the normal reaction on a banked track.
Energy methods use work done = force times distance moved in the direction of the force, kinetic energy 1/2 m v^2 and potential energy m g h. For vertical circular motion, speed changes with height, so combine the energy equation with F = ma towards the centre. Elastic strings and springs store energy of 1/2 k x^2 for extension x.
Define a direction as positive and keep signs consistent in collision problems. Check units for each quantity.
Worked example
Solve a collision.
Positive direction
Conserve momentum
Use e
Solve both
Practice problem and solution
A particle moves at 6 m/s on a circle of radius 3 m. What is its acceleration in m/s^2? Enter a number.
v^2 / r = 36 / 3 = 12.
Mental model: Momentum is conserved; impulse is change in momentum; circular acceleration is v squared over r.
Common trap: Dropping the sign of velocity in a collision.
12. Statistics option: the Poisson distribution
Learning goal: Model rare random events and combine Poisson variables.
The Poisson distribution models the number of events in a fixed interval when they occur independently at a constant average rate. If X follows Poisson with mean lambda, then P(X = r) = e^(-lambda) lambda^r / r! for r = 0, 1, 2, ... Typical contexts are calls to a help line per hour, defects per meter of cloth and radioactive decays per minute.
For a Poisson distribution the mean and the variance are both lambda. A quick check on real data is that the sample variance should be close to the sample mean. If the variance is much larger, events may be clustered and a Poisson model would not fit.
The sum of independent Poisson variables with means lambda and mu is Poisson with mean lambda + mu. Rates scale with the interval: a rate of 3 per hour is 1.5 per half hour. The Poisson also approximates a binomial when n is large and p is small, with lambda = n p.
In a hypothesis test, state H0 as a value of lambda, find the probability of a result at least as extreme, and compare with the significance level.
Worked example
Check the model.
Mean
Variance
Compare
Conclude
Practice problem and solution
A Poisson variable has mean 4. What is its variance? Enter a number.
For a Poisson distribution the variance equals the mean.
Mental model: Poisson models independent events at a constant rate; mean equals variance; rates scale with the interval.
Common trap: Using a Poisson when events cluster.
13. Discrete mathematics option: graphs and algorithms
Learning goal: Use graph theory, networks and algorithm counts.
A graph has vertices joined by edges. The degree of a vertex is the number of edges at it, and the sum of all degrees is twice the number of edges. A complete graph K_n has every pair of vertices joined, so it has n(n - 1)/2 edges. A tree is connected with no cycles and has n - 1 edges for n vertices.
In a network, edges carry weights such as distance or cost. A minimum spanning tree connects all vertices at least total weight and is found by Kruskal's or Prim's algorithm. Dijkstra's algorithm finds shortest paths from one vertex to all others when weights are not negative. Eulerian trails use every edge once and Hamiltonian cycles visit every vertex once.
Algorithms are step-by-step procedures. Bubble sort compares adjacent items and may make n(n - 1)/2 comparisons for a list of n items in the worst case. This quadratic growth means doubling the list roughly quadruples the work. Binary search halves the remaining items each time, taking roughly log base 2 of n steps on a sorted list.
Critical path analysis uses activity networks to find the minimum project time and the activities with no float. Linear programming sets up constraints and an objective, then finds the best point on the feasible region.
Worked example
Count work.
Define the list size
Count comparisons
Find the pattern
State the order
Practice problem and solution
How many edges does the complete graph K5 have? Enter a number.
5 x 4 / 2 = 10.
Mental model: Graph counts, spanning trees, shortest paths and algorithm growth form the discrete option.
Common trap: Confusing a tree with a complete graph.
14. Exam skills for Further Mathematics
Learning goal: Plan the three papers and the optional content.
AQA A-level Further Mathematics (7367) has three written papers, each 2 hours and 100 marks, each one third of the A-level. Papers 1 and 2 may assess the core content: proof, complex numbers, matrices, further algebra, further calculus, further vectors, polar coordinates, hyperbolic functions, differential equations and numerical methods. Paper 3 covers two optional applications from mechanics, statistics and discrete mathematics, with an answer booklet for each.
Further Mathematics is taken alongside Mathematics, so students need the full A-level Mathematics content. Choose your options with your teacher, since the available combinations depend on the school. Other boards organize the content differently, so check your own specification.
Rigor matters more than in A-level Mathematics. In proof questions state each logical step. In 'show that' questions show every line. Give exact values such as surds, fractions and multiples of pi unless asked for decimals.
Allow about one minute per mark. Use the formula booklet for the formulas it provides and learn those it does not. Practice with past papers under timing and use mark schemes to check how method marks are given.
Worked example
Plan revision.
List topics
Rank weak areas
Schedule
Test
Practice problem and solution
Three papers each have 100 marks. What is the total number of marks? Enter a number.
3 x 100 = 300.
Mental model: Three papers of 100 marks; learn core plus options; show rigorous method.
Common trap: Giving decimals when exact values are expected.