Singapore · SEABSyllabus
Further Maths syllabus, dot point by dot point
Every dot point in the Singapore Further Mathssyllabus, with a focused answer for each one. Click any dot point for a worked explainer, past exam questions, and links to related dot points. Written by Claude Opus 4.8, Anthropic's latest AI.
Complex Numbers and Polynomials
Module overview →- How do we represent complex numbers in Cartesian, polar and exponential form, and what does the Argand diagram show?Represent complex numbers in Cartesian, polar and exponential form, perform arithmetic, and interpret them on the Argand diagram11 min answer →
- How does de Moivre's theorem let us take powers of complex numbers and derive trigonometric identities?State and apply de Moivre's theorem to find powers of complex numbers and to derive multiple-angle and power-reduction trigonometric identities11 min answer →
- How do equations and inequalities involving the modulus and argument define loci and regions on the Argand diagram?Sketch loci and regions in the Argand diagram defined by conditions on the modulus and argument of a complex number11 min answer →
- How do the roots of a polynomial relate to its coefficients, and how do complex roots occur in conjugate pairs?Use the relationships between the roots and coefficients of a polynomial and apply the conjugate root theorem for real polynomials11 min answer →
- What are the nth roots of unity and of a general complex number, and how are they arranged geometrically?Find the nth roots of unity and the nth roots of a general complex number, and describe their geometric arrangement on the Argand diagram11 min answer →
Differential Equations
Module overview →- How do we solve first-order differential equations by separating variables and by the integrating factor method?Solve first-order differential equations by separation of variables and by the integrating factor method, applying initial conditions11 min answer →
- How do we set up a differential equation to model a real situation and interpret its solution?Formulate differential equations from descriptions of rates of change and interpret the solutions in context, including long-term behaviour11 min answer →
- How do we solve a non-homogeneous second-order linear differential equation using a complementary function and a particular integral?Solve non-homogeneous second-order linear differential equations by finding the complementary function and a particular integral for standard forcing terms12 min answer →
- How do we solve a homogeneous second-order linear differential equation with constant coefficients using the auxiliary equation?Solve homogeneous second-order linear differential equations with constant coefficients using the auxiliary equation, covering real, repeated and complex roots11 min answer →
- How do we solve a coupled system of first-order linear differential equations by reducing it to a single second-order equation?Solve coupled systems of first-order linear differential equations by reduction to a single second-order equation11 min answer →
Further Calculus
Module overview →- How do we compute the length of a curve and the area of a surface of revolution by integration?Calculate the arc length of a curve and the area of a surface of revolution for curves given in Cartesian or parametric form11 min answer →
- How do trigonometric and hyperbolic substitutions and standard inverse-function integrals extend our integration toolkit?Integrate using trigonometric and hyperbolic substitutions and recognise standard integrals giving inverse trigonometric and logarithmic forms11 min answer →
- What is an improper integral, and how do we decide whether one converges and find its value?Evaluate improper integrals with infinite limits or integrands with a singularity, determining convergence by means of limits10 min answer →
- How do we derive a Maclaurin series, including by repeated implicit differentiation, and use it for limits and approximations?Derive Maclaurin series including by repeated implicit differentiation and use series to evaluate limits and approximations11 min answer →
- How do we derive and use a reduction formula to integrate a family of integrals indexed by an integer?Derive reduction formulae using integration by parts and apply them to evaluate families of integrals11 min answer →
Further Probability and Statistics
Module overview →- How does a probability density function describe a continuous random variable, and how do we find its mean, variance and median?Work with continuous random variables defined by a probability density function, finding probabilities, the cumulative distribution function, expectation, variance and median11 min answer →
- How do we describe a discrete random variable and compute its expectation and variance?Work with discrete random variables, their probability distributions, expectation, variance, and the expectation and variance of linear functions11 min answer →
- How do we estimate a population parameter from a sample, and what does a confidence interval mean?Compute unbiased estimates of a population mean and variance and construct and interpret confidence intervals for a population mean11 min answer →
- How do Type I and Type II errors and the power of a test describe what a hypothesis test can get wrong?Carry out hypothesis tests and analyse Type I and Type II errors and the power of a test11 min answer →
- How do non-parametric tests such as the sign test and Wilcoxon tests work when we cannot assume a normal distribution?Apply non-parametric tests including the sign test and the Wilcoxon signed-rank test, and know when they are appropriate11 min answer →
- What are the geometric and negative binomial distributions, and when does each model a counting situation?Recognise and apply the geometric and negative binomial distributions, including their probabilities, expectations and variances11 min answer →
Mathematical Induction, Inequalities and Recurrences
Module overview →- How do we prove and solve inequalities rigorously, and which standard inequalities are worth knowing?Prove and apply inequalities including the use of the discriminant, completing the square, and standard results such as the AM-GM inequality10 min answer →
- What are the standard methods of mathematical proof, and how do we structure a rigorous argument?Construct rigorous mathematical arguments using direct proof, proof by contradiction, proof by contrapositive, and disproof by counterexample10 min answer →
- How does proof by mathematical induction establish a statement for all positive integers, and how do we write a watertight induction argument?Prove statements about sums, divisibility and inequalities for all positive integers using the principle of mathematical induction11 min answer →
- How do we solve linear recurrence relations to find a closed form for the nth term?Solve first- and second-order linear recurrence relations with constant coefficients and find closed-form expressions for the nth term11 min answer →
- How do we sum a finite series in closed form using the method of differences and the standard power sums?Sum finite series using the method of differences, standard results for powers of integers, and partial fractions10 min answer →
Matrices and Linear Spaces
Module overview →- How do we diagonalise a matrix, and why does diagonalisation make computing powers of a matrix easy?Diagonalise a matrix using its eigenvalues and eigenvectors and use the diagonal form to compute powers of the matrix11 min answer →
- What are the eigenvalues and eigenvectors of a matrix, and how do we find them?Find the eigenvalues and eigenvectors of 2x2 and 3x3 matrices using the characteristic equation11 min answer →
- How do we invert a matrix and use it to solve a system of linear equations?Find the inverse of a non-singular matrix and use matrices to solve systems of linear equations, recognising consistent, inconsistent and dependent cases11 min answer →
- What is a vector space, and how do the ideas of linear independence, basis, dimension and rank organise it?Use the concepts of vector spaces and subspaces, linear independence, spanning sets, basis, dimension and the rank of a matrix11 min answer →
- How do we perform matrix arithmetic and compute determinants, and what does a determinant tell us?Carry out matrix addition and multiplication and evaluate the determinant of 2x2 and 3x3 matrices, interpreting its geometric meaning11 min answer →
Numerical Methods
Module overview →- How does rearranging an equation into the form x = g(x) give an iterative method, and what controls whether it converges?Solve an equation by fixed-point iteration of the form x = g(x), and use the derivative condition to decide convergence11 min answer →
- How does the Newton-Raphson method find a root of an equation, and when does it succeed or fail?Apply the Newton-Raphson method to find a root of an equation numerically and discuss its convergence and failure cases11 min answer →
- How do the trapezium rule and Simpson's rule approximate a definite integral, and how accurate are they?Approximate a definite integral using the trapezium rule and Simpson's rule and comment on the accuracy of the estimate11 min answer →
- How does Euler's method step a differential equation forward numerically, and what limits its accuracy?Use Euler's method and the improved Euler (midpoint) method to obtain a numerical solution of a first-order differential equation11 min answer →
Vectors and the Geometry of Three Dimensions
Module overview →- How do we find where lines and planes intersect, and how do we compute the shortest distance between geometric objects?Find the intersection of lines and planes and compute shortest distances from a point to a line or plane and between two skew lines12 min answer →
- How do we describe a line in three dimensions, and how do we test whether two lines intersect, are parallel or are skew?Write the vector, parametric and Cartesian equations of a line in three dimensions and classify the relationship between two lines11 min answer →
- How do we describe a plane in three dimensions using a normal vector, and how do we move between its forms?Write the vector, scalar product and Cartesian equations of a plane using a normal vector and find the angle between planes and between a line and a plane11 min answer →
- What do the scalar and vector products compute, and how do we use them to find angles, areas and volumes?Use the scalar and vector products and the scalar triple product to find angles, areas and volumes in three dimensions11 min answer →
- How do we use vector methods to prove geometric results and find reflections, foot of perpendicular and other constructions?Apply vector methods to geometric problems including the foot of the perpendicular, reflections of points, and proofs of geometric properties11 min answer →