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- Complex numbers in polar formSuggested nextWrite complex numbers in polar and exponential form, use de Moivre’s theorem for powers, roots and identities, and sketch loci and regions in the Argand diagram.Shown in a check2 core lessonsLesson progress hasn’t loaded yet
- Polar coordinatesSuggested nextSketch simple polar curves and find the areas they enclose and their arc lengths.Shown in a check1 core lessonLesson progress hasn’t loaded yet
- Applications of integrationSuggested nextFind arc lengths, volumes by discs or shells, and surface areas of revolution.Shown in a check1 core lessonLesson progress hasn’t loaded yet
- Functions of two variablesSuggested nextUse partial derivatives, gradients, tangent planes and stationary points of surfaces.Shown in a check1 core lessonLesson progress hasn’t loaded yet
- Linear differential equationsSuggested nextSolve first- and second-order linear differential equations, use given substitutions, and analyse exponential and logistic growth with phase lines.Shown in a check2 core lessonsLesson progress hasn’t loaded yet
- Recurrence relationsSuggested nextSolve first- and second-order linear recurrences, including those reduced by a given substitution, analyse their long-term behaviour and model with them.Shown in a check2 core lessonsLesson progress hasn’t loaded yet
- Matrices and linear spacesSuggested nextSolve linear systems by row reduction, represent linear transformations, work with linear spaces, rank and nullity, and diagonalise matrices.Shown in a check3 core lessonsLesson progress hasn’t loaded yet
- Numerical methodsSuggested nextLocate and approximate roots, approximate integrals by the trapezium and Simpson's rules, and step through differential equations by Euler methods.Shown in a check2 core lessonsLesson progress hasn’t loaded yet
- Poisson and geometric distributionsSuggested nextModel counts and waiting times with Poisson and geometric distributions, including their conditions and properties.Shown in a check1 core lessonLesson progress hasn’t loaded yet
- Continuous random variablesSuggested nextWork with probability density and cumulative distribution functions, and uniform and exponential models.Shown in a check1 core lessonLesson progress hasn’t loaded yet
- Hypothesis tests and confidence intervalsSuggested nextCarry out t-tests and χ² tests and construct confidence intervals.Shown in a check3 core lessonsLesson progress hasn’t loaded yet
- Non-parametric testsSuggested nextUse sign tests and Wilcoxon matched-pair signed-rank tests.Shown in a check1 core lessonLesson progress hasn’t loaded yet
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Reference
Course info
- GCE A-Level H2 Further Mathematics 2027
- 12 topics
Course guidance
Lessons for H2 Further Mathematics cover every topic in its syllabus. Practice questions for this course are not available yet.
Learning goals
- Convert between Cartesian, modulus–argument and exponential forms of a complex number
- Multiply and divide complex numbers in polar form and interpret the rotation and scaling
- Represent simple complex equations and inequalities as loci and regions
- Use de Moivre’s theorem to find powers and every nth root of a complex number
- Derive multiple-angle and power identities using de Moivre’s theorem
- Sketch and interpret simple polar curves on a stated angle interval
- Calculate areas bounded by polar curves
- Calculate arc lengths of curves in polar form
- Calculate arc lengths of curves in Cartesian form
- Choose discs or shells to calculate a volume of revolution about either axis
- Calculate surface areas of revolution about either axis
- Calculate and interpret first- and second-order partial derivatives
- Find and classify maxima, minima and saddle points of a function of two variables
- Find tangent and normal planes and local linear and quadratic approximations
- Use the gradient vector and directional derivatives
- Solve first-order linear differential equations with an integrating factor
- Solve second-order homogeneous linear differential equations with constant coefficients
- Find a particular integral and the general solution of a non-homogeneous linear equation
- Reduce a differential equation to a standard linear form with a given substitution
- Interpret solution curves, phase lines and the stability of equilibrium points
- Model exponential and logistic growth, including harvesting
- Solve first-order linear recurrence relations with constant coefficients
- Solve second-order linear homogeneous recurrence relations with constant coefficients
- Analyse the long-term behaviour of a sequence defined by a recurrence
- Formulate and interpret recurrence-relation models
- Calculate determinants and inverses of 2 × 2 and 3 × 3 matrices
- Solve linear systems by row reduction and interpret the solution set geometrically
- Represent and interpret linear transformations by matrices
- Find real eigenvalues and their eigenvectors of 2 × 2 and 3 × 3 matrices
- Diagonalise a matrix as QDQ⁻¹ and use the form to find powers
- Test subspaces, linear independence and span, and find a basis and dimension
- Find rank, null space and column space and relate rank to nullity
- Locate a root by graphical or sign-change reasoning
- Approximate roots by linear interpolation and the Newton–Raphson method and explain failure
- Use fixed-point iteration and decide whether it converges
- Approximate integrals with the trapezium rule and Simpson’s rule
- Approximate solutions of first-order differential equations with Euler and improved Euler methods
- Choose and use Poisson and geometric models, including their means, variances and conditions
- Use probability density and cumulative distribution functions, expectations, medians and modes
- Use uniform and exponential models and relate exponential waiting times to Poisson counts
- Formulate and carry out t-tests, paired-sample tests and normal tests for a difference of means
- Carry out χ² goodness-of-fit and contingency-table independence tests
- Construct and interpret confidence intervals for a mean or a proportion
- Carry out sign and Wilcoxon matched-pair signed-rank tests and judge when to use them
Syllabus statements covered
- Include: complex numbers expressed in the form r(cos θ + i sin θ), re^(iθ), where r > 0 and −π < θ ⩽ π; multiplication and division of two complex numbers expressed in polar form and their geometrical interpretations; loci of simple equations and inequalities such as |z − c| ⩽ r, |z − a| = |z − b| and arg(z − a) = α (excluding loci of |z − a| = k|z − b|, where k ≠ 1 and arg(z − a) − arg(z − b) = α); use of de Moivre’s theorem to find the powers and nth roots of a complex number, and to derive trigonometric identities.
- Include: simple polar curves (for 0 ⩽ θ < 2π or −π < θ ⩽ π or a subset of either of these intervals; and where r is non-negative throughout the domain); use of formula A = ½∫_α^β r² dθ for the area of a sector; arc length of curves defined in polar form.
- Include: arc length of curves defined in cartesian form; volume of revolution about the x- or y-axis for curves defined in cartesian form using discs or shells as appropriate; surface area of revolution about the x- or y-axis for curves defined in cartesian form.
- Include: functions of two variables and surfaces z = f(x, y); first order and second order partial derivatives; minimum, maximum and saddle points; normal and tangent planes and local linearisation; directional derivatives and gradient; quadratic approximations; local maximum and minimum problems. Exclude Jacobian matrix, Lagrange multipliers and constrained optimisation.
- Include: analytical solution of first order and second order linear differential equations of the form: (i) dy/dx + p(x)y = q(x), using an integrating factor; (ii) d²y/dx² + a dy/dx + by = 0, where a, b ∈ ℝ; (iii) d²y/dx² + a dy/dx + by = f(x), where a, b ∈ ℝ and f(x) is a polynomial or pe^(kx) or p cos(kx) + q sin(kx), including those that can be reduced to the above by means of a given substitution; relationship between the solution of a non-homogenous equation and the associated homogenous equation; family of solution curves; phase lines and slope fields (for teaching and learning only); exponential growth model; logistic growth model, equilibrium points and their stability, and harvesting.
- Include: behaviour of a sequence, such as the limiting behaviour of a sequence; solution of (i) first order linear (homogeneous and non-homogeneous) recurrence relations with constant coefficients of the form uₙ = auₙ₋₁ + b, a, b ∈ ℝ, a ≠ 0; (ii) second order linear homogeneous recurrence relations with constant coefficients including those that can be transformed to the above by means of a given substitution; modelling with recurrence relations of the forms above.
- Include: use of matrices to represent a set of linear equations; operations on 3 × 3 matrices; determinant of a square matrix and inverse of a non-singular matrix (2 × 2 and 3 × 3 matrices only); use of matrices to solve a set of linear equations (including row reduction and echelon forms, and geometrical interpretation of the solution); linear transformations and matrices from ℝⁿ → ℝᵐ, where n, m ⩽ 3; eigenvalues and eigenvectors of square matrices (2 × 2 and 3 × 3 matrices, restricted to cases where the eigenvalues are real); diagonalisation of a square matrix M by expressing the matrix in the form QDQ⁻¹, where D is a diagonal matrix of eigenvalues and Q is a matrix whose columns are eigenvectors, and use of this expression such as to find the powers of M; linear spaces and subspaces, and the axioms (restricted to spaces of finite dimension over the field of real numbers only); linear independence and span; basis and dimension (in simple cases), including use of terms such as ‘column space’, ‘row space’, ‘range space’ and ‘null space’; rank of a square matrix and relation between rank, dimension of null space and order of the matrix.
- Include: location of roots of an equation by simple graphical or numerical methods; approximation of roots of equations using linear interpolation and Newton-Raphson method, with the aid of a computer, including cases where each method fails to converge to the required root; iterations involving recurrence relations of the form xₙ₊₁ = F(xₙ), including cases where the method fails to converge; approximation of integral of a function using the trapezium rule and Simpson’s rule, with the aid of a computer; approximation of solutions of first order differential equations using Euler method (including the use of the improved Euler formula), with the aid of a computer.
- Include: use of Poisson distribution Po(μ) and geometric distribution Geo(p) as probability models, including conditions under which each distribution is a suitable model; mean and variance for Poisson and geometric distributions; additive property of the Poisson distribution.
- Include: probability density function of a continuous random variable and its mean and variance (includes ‘piecewise’ probability density function); cumulative distribution function and its relationship with the probability density function; concepts of median and mode of a continuous random variable; use of the result E(g(X)) = ∫_−∞^∞ g(x) f(x) dx in simple cases, where f(x) is the probability density function of X and g(x) is a function of X; uniform distribution and exponential distribution as probability models; relationship between Poisson and exponential distributions.
- Include: formulation of hypotheses and testing for a population mean using a small sample drawn from a normal population of unknown variance using a t-test; formulation of hypotheses for the difference of population means, and apply, as appropriate: a paired sample t-test, a test using a normal distribution; contingency tables and χ²-tests of goodness of fit and independence (excluding Yates’ correction for continuity); connection between confidence interval and hypothesis test; confidence interval for the population mean based on: a random sample from a normal population of known variance, a small random sample drawn from a normal population of unknown variance, a large random sample from any population; confidence interval for population proportion (including concept of sample proportion) from a large random sample; interpretation of confidence intervals and the results of a hypothesis test in the context of the problem. Exclude the use of the term ‘Type I error’, concept of Type II error and power of a test.
- Include: formulation of hypotheses and testing for a population median using Sign test, and identical probability distributions for two sampled populations in a paired difference design using Wilcoxon matched-pair signed rank test; advantages and disadvantages of non-parametric tests. Exclude treatment of tied ranks.