Year 12: Singapore H1 Mathematics 8865

Listed Mathematics sub-topics at Higher 1 based on the Singapore Examinations and Assessment Board (SEAB). Checked against the source on 2 October 2026.

Sub-topics from the Singapore–Cambridge GCE A-Level Higher 1 Mathematics syllabus (8865) for the 2026 examination. Year 12 is the second year of the course, at the end of which the papers are sat. There is one 3-hour paper: Section A is Pure Mathematics (40 marks) and Section B is Probability and Statistics (60 marks). H1 is the shorter course, and it leaves out a good deal that H2 covers – there is no trigonometry, no product or quotient rule, and no vectors or complex numbers – so the practice offered here is narrower than the full year list by design.

10sub-topics
80skills mapped
100%have practice

Coverage is the percentage of listed sub-topics with at least one linked practice skill; it does not measure how fully every part of an entry is practised. A skill can practise more than one sub-topic, so some appear more than once below.

1 Functions and Graphs

1.1Exponential and logarithmic functions and Graphing techniquesInclude: • concept of function as a rule or relationship where for every input there is only one output • functions eˣ and ln x and their graphs • exponential growth and decay • logarithmic growth • equivalence of y = eˣ and x = ln y • laws of logarithms • use of a graphing calculator or graphing software to graph a given function • characteristics of graphs such as symmetry, intersections with the axes, turning points and asymptotes (horizontal and vertical)
Excluded by syllabus• problems involving finding domain and range • use of notation f : x ↦ x² + 5 • change of base of logarithms
1.2Equations and inequalitiesInclude: • conditions for a quadratic equation to have (i) two real roots, (ii) two equal roots, and (iii) no real roots • conditions for ax² + bx + c to be always positive (or always negative) • solving simultaneous equations, one linear and one quadratic, by substitution • solving quadratic equations and inequalities in one unknown analytically • solving inequalities by graphical methods • formulating an equation or a system of linear equations from a problem situation • finding the approximate solution of an equation or a system of linear equations using a graphing calculator or a graphing software

2 Calculus

2.1DifferentiationInclude: • derivative of f(x) as the gradient of the tangent to the graph of y = f(x) at a point • derivatives of xⁿ for any rational n, eˣ, ln x, together with constant multiples, sums and differences • chain rule • graphical interpretation of f′(x) > 0, f′(x) = 0 and f′(x) < 0 • use of the first derivative test or second derivative test to determine the nature of the stationary points (local maximum and minimum points and points of inflexion) in simple cases • locating maximum and minimum points using a graphing calculator or a graphing software • finding the approximate value of a derivative at a given point using a graphing calculator or a graphing software • problems involving tangents to curves • local maxima and minima problems
Excluded by syllabus• finding derivatives from first principles • derivatives of products and quotients of functions • use of dy/dx = 1/(dx/dy) • differentiation of functions defined implicitly or parametrically • non-stationary points of inflexion • problems connecting the graph of y = f(x) to the graph of y = f′(x)
2.2IntegrationInclude: • integration as the reverse of differentiation • integration of xⁿ for any rational n, and eˣ, together with constant multiples, sums and differences • integration of (ax + b)ⁿ for any rational n, and e^(ax+b) • definite integral as the area under a curve • evaluation of definite integrals • finding the area of a region bounded by a curve and lines parallel to the coordinate axes, between a curve and a line, or between two curves • finding the approximate value of a definite integral using a graphing calculator or a graphing software
Excluded by syllabus• finding definite integral as a limit of sum • approximation of area under a curve using the trapezium rule • area below x-axis

3 Probability and Statistics

3.1ProbabilityInclude: • addition and multiplication principles for counting • concepts of permutation (ⁿPᵣ) and combination (ⁿCᵣ) • arrangements of distinct objects in a line including cases involving restriction • addition and multiplication of probabilities • mutually exclusive events and independent events • use of tables of outcomes, Venn diagrams, tree diagrams, and permutations and combinations techniques to calculate probabilities • calculation of conditional probabilities in simple cases • use of: P(A′) = 1 − P(A); P(A ∪ B) = P(A) + P(B) − P(A ∩ B); P(A|B) = P(A ∩ B)/P(B)
3.2Binomial distributionInclude: • concept of discrete random variables* • concept of binomial distribution B(n, p) and use of B(n, p) as a probability model, including conditions under which the binomial distribution is a suitable model • mean and variance of a binomial distribution (without proof) (* For teaching and learning only.)
3.3Normal distributionInclude: • concept of continuous random variables* • concept of a normal distribution as an example of a continuous probability model and its mean and variance; use of N(μ, σ²) as a probability model • standard normal distribution • finding the value of P(X < x₁) or a related probability, given the values of x₁, μ, σ • symmetry of the normal curve and its properties • finding a relationship between x₁, μ, σ given the value of P(X < x₁) or a related probability • solving problems involving the use of E(aX + b) and Var(aX + b) • solving problems involving the use of E(aX + bY) and Var(aX + bY), where X and Y are independent
Excluded by syllabusnormal approximation to binomial distribution. (* For teaching and learning only.)
3.4SamplingInclude: • concepts of population and simple random sample • concept of the sample mean X̄ as a random variable with E(X̄) = μ and Var(X̄) = σ²/n • distribution of sample mean from a normal population • use of the Central Limit Theorem to treat sample mean as having normal distribution when the sample size is sufficiently large (e.g. n ≥ 30) • use of unbiased estimates of the population mean and variance from a sample, including cases where the data are given in summarised form Σx and Σx², or Σ(x − a) and Σ(x − a)²
3.5Hypothesis testingInclude: • concepts of null hypothesis (H₀) and alternative hypotheses (H₁), test statistic, critical region, critical value, level of significance, and p-value • formulation of hypotheses and testing for a population mean based on − a sample from a normal population of known variance − a large sample from any population • 1-tail and 2-tail tests • interpretation of the results of a hypothesis test in the context of the problem
Excluded by syllabusthe use of the term ‘Type I error’, concept of Type II error and testing the difference between two population means.
3.6Correlation and Linear regressionInclude: • use of scatter diagram to judge if there is a plausible linear relationship between the two variables • correlation coefficient as a measure of the fit of a linear model to the scatter diagram • interpreting the product moment correlation coefficient (in particular, values close to −1, 0 and 1) • concepts of linear regression and method of least squares to find the equation of the regression line • concepts of interpolation and extrapolation • use of the appropriate regression line to make prediction or estimate a value in practical situations, including explaining how well the situation is modelled by the linear regression model
Excluded by syllabus• problems involving derivation of formulae • relationship r² = b₁b₂, where b₁ and b₂ are regression coefficients • hypothesis tests • transformation to achieve linearity of regression model

Other Year 12 skills

Year 12 skills not currently linked to an entry on this page. These may include revision, extension practice, or skills still awaiting a curriculum mapping.