Year 11: Singapore H2 Mathematics 9758

Listed Mathematics sub-topics at Higher 2 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 2 Mathematics syllabus (9758) for the 2026 examination. H1 and H2 are each a single syllabus covering both years of junior college, examined at the end of Year 12, so this page lists all of it. The practice beside each sub-topic is the Year 11 practice; the Year 12 page maps the same sub-topics onto the Year 12 skills. Topics 1 to 5 are the Pure Mathematics section, examined in Paper 1 and in Section A of Paper 2; topic 6 is Probability and Statistics, examined in Section B of Paper 2. Knowledge of O-Level Mathematics is assumed.

19sub-topics
148skills 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

1.1FunctionsInclude: • concepts of function, domain and range • inverse functions and composite functions • conditions for the existence of inverse functions and composite functions • domain restriction to obtain an inverse function • relationship between graphs of a one-to-one function and its inverse
Excluded by syllabusthe use of the relation (fg)⁻¹ = g⁻¹f⁻¹, and restriction of domain to obtain a composite function.
1.2Graphs and transformationsInclude: • use of a graphing calculator or a graphing software to graph a given function • important characteristics of graphs such as symmetry, intersections with the axes, turning points and asymptotes of the following: y² = ax; x² = by; x²/a² + y²/b² = 1; x²/a² − y²/b² = 1; y²/b² − x²/a² = 1; y = (ax + b)/(cx + d); y = (ax² + bx + c)/(dx + e) • equations of asymptotes, axes of symmetry, and restrictions on the possible values of x and/or y • effect of transformations on the graph of y = f(x) as represented by y = af(x), y = f(x) + a, y = f(x + a) and y = f(ax) and combinations of these transformations • relating the graphs of y = |f(x)|, y = f(|x|), and y = 1/f(x) to the graph of y = f(x) • simple parametric equations and their graphs
1.3Equations and inequalitiesInclude: • formulating an equation, a system of linear equations, or inequalities from a problem situation • solving an equation exactly or approximately using a graphing calculator or a graphing software • solving a system of linear equations using a graphing calculator or a graphing software • solving inequalities of the form f(x)/g(x) > 0 where f(x) and g(x) are linear expressions or quadratic expressions • concept of |x|, and use of relations |x − a| < b ⇔ a − b < x < a + b and |x − a| > b ⇔ x < a − b or x > a + b • solving inequalities by graphical methods

2 Sequences and series

2.1Sequences and seriesInclude: • concepts of sequence and series for finite and infinite cases • sequence as function y = f(n) where n is a positive integer • relationship between uₙ (the nth term) and Sₙ (the sum to n terms) • sequence given by a formula for the nth term • sequence generated by the relation uₙ₊₁ = f(uₙ), including the use of a graphing calculator or a computer to generate the sequence • sum and difference of two series • convergence of a series and the sum to infinity • formula for the nth term and the sum of a finite arithmetic series • formula for the nth term and the sum of a finite geometric series • condition for convergence of an infinite geometric series • formula for the sum to infinity of a convergent geometric series

3 Vectors

3.1Basic properties of vectors in two and three dimensionsInclude: • addition and subtraction of vectors, multiplication of a vector by a scalar, and their geometrical interpretations • position vectors, displacement vectors and direction vectors • magnitude of a vector • unit vectors • distance between two points • collinearity • use of the ratio theorem in geometrical applications
3.2Scalar and vector products in vectorsInclude: • concepts of scalar product and vector product of vectors and their properties • angle between two vectors • geometrical meanings of |a · n̂| and |a × n̂|, where n̂ is a unit vector
Excluded by syllabustriple products a × b · c and a × b × c.
- Vector dot product
3.3Three-dimensional vector geometryInclude: • vector and cartesian equations of lines and planes • foot of the perpendicular and distance from a point to a line or to a plane • angle between two lines, between a line and a plane, or between two planes • relationships between (i) two lines (coplanar or skew) (ii) a line and a plane (iii) two planes
Excluded by syllabus• shortest distance between two skew lines • common perpendicular to two skew lines

4 Introduction to Complex numbers

4.1Complex numbers expressed in cartesian form and Argand diagramsInclude: • extension of the number system from real numbers to complex numbers • complex roots of quadratic equations • modulus, argument and conjugate of a complex number • four operations of complex numbers • equality of complex numbers • conjugate roots of a polynomial equation with real coefficients • representation of complex numbers in the Argand diagram • geometrical effects of conjugation, negation, addition, subtraction, and multiplication by i
Excluded by syllabuscomplex numbers expressed in polar (or modulus-argument) form and exponential form.

5 Calculus

5.1DifferentiationInclude: • graphical interpretation of (i) f′(x) > 0, f′(x) = 0 and f′(x) < 0 (ii) f″(x) > 0 and f″(x) < 0 • relating the graph of y = f′(x) to the graph of y = f(x) • differentiation of simple functions defined implicitly or parametrically • determining the nature of the stationary points (local maximum and minimum points and points of inflexion) analytically, in simple cases, using the first derivative test or the second derivative test • 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 and normals to curves, including cases where the curve is defined implicitly or parametrically • local maxima and minima problems • connected rates of change problems
Excluded by syllabusnon-stationary points of inflexion and finding second derivative of functions defined parametrically.
5.2Maclaurin seriesInclude: • standard series expansion of (1 + x)ⁿ for any rational n, eˣ, sin x, cos x and ln(1 + x) • derivation of the first few terms of the Maclaurin series by − repeated differentiation, e.g. sec x − repeated implicit differentiation, e.g. y³ + y² + y = x² − 2x − using standard series, e.g. eˣ cos 2x, ln((1 + x)/(1 − x)) • range of values of x for which a standard series converges • concept of Maclaurin’s series as an approximation of a function • small angle approximations: sin x ≈ x, cos x ≈ 1 − ½x², tan x ≈ x
Excluded by syllabusproblems involving derivation of the general term of a series.
5.3Integration techniquesInclude: • integration of f′(x)[f(x)]ⁿ (including n = −1), f′(x)e^f(x), sin²x, cos²x, tan²x, 1/(a² + x²), 1/√(a² − x²), 1/(a² − x²) and 1/(x² − a²) • integration by a given substitution • integration by parts
Excluded by syllabusreduction formulae.
- Indefinite integration of polynomials
5.4Definite integralsInclude: • concept of definite integral as a limit of sum • definite integral as the area under a curve • evaluation of definite integrals • 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 • area below the x-axis • volume of revolution about the x- or y-axis • finding the approximate value of a definite integral using a graphing calculator or a graphing software
Excluded by syllabusarea and volume of revolution about the x-axis or y-axis where curve is defined parametrically.
- Definite integration
- Application of integration for polynomial equations
5.5Differential equationsInclude: • solving for the general solutions and particular solutions of differential equations of the form dy/dx = f(x)g(y), including reducing a given differential equation to this form by means of a given substitution • formulating a differential equation from a problem situation • interpreting a differential equation and its solution in terms of a problem situation

6 Probability and Statistics

6.1ProbabilityInclude: • addition and multiplication principles for counting • concepts of permutation (ⁿPᵣ) and combination (ⁿCᵣ) • arrangements of objects in a line or in a circle, including cases involving repetition and 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)
6.2Discrete random variablesInclude: • concept of discrete random variables, probability distributions, expectations and variances • concept of binomial distribution B(n, p) as an example of a discrete probability distribution and use of B(n, p) as a probability model, including conditions under which the binomial distribution is a suitable model • use of mean and variance of binomial distribution (without proof)
Excluded by syllabusfinding cumulative distribution function of a discrete random variable.
6.3Normal 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.)
6.4SamplingInclude: • 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
6.5Hypothesis 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.
6.6Correlation 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 • use of a square, reciprocal or logarithmic transformation to achieve linearity
Excluded by syllabus• problems involving derivation of formulae • relationship r² = b₁b₂, where b₁ and b₂ are regression coefficients • hypothesis tests
- Understanding scatter plots

Other Year 11 skills

Year 11 skills not currently linked to an entry on this page. These may include revision, extension practice, or skills still awaiting a curriculum mapping.