Singapore A-Level H2 Mathematics

syllabus 9758

97 of 102 skills covered

Applications, worked type by type: rational functions · scatter plots and correlation · probability with several events · vectors in the plane · planes and the vector product · sets and counting · probability distributions · sampling · statistical inference · correlation and regression · complex arithmetic · the complex plane · using differentiation · techniques of integration · polar curves · sequences and series

✓ covered · ◐ partial · ○ not yet · — excluded

H2 Mathematics (9758)

	skill	lessons
1	Function, domain and range	Function Notation, Domain and Range	✓ covered
2	Inverse and composite functions	Inverse Functions, Composite Functions	✓ covered
3	Conditions for existence of inverse/composite	Inverse Functions, Restricting the Domain for an Inverse, Composite Functions, When a Composite Function Exists	✓ covered
4	Domain restriction to obtain an inverse	Restricting the Domain for an Inverse	✓ covered
5	Graph of a one-to-one function and its inverse	Inverse Functions, Restricting the Domain for an Inverse	✓ covered
6	Graphing a function with a graphing calculatorgraphing-calculator content — deferred until the planned graphing tool ships (owner ruling 2026-08-25; these rows then join the exam-item batches)		○ deferred
7	Characteristics of conics and rational functions	Zeros and Vertical Asymptotes, Holes Where a Factor Cancels, Horizontal Asymptotes by Degree, Slant Asymptotes by Long Division, The Equation of an Ellipse, The Equation of a Hyperbola, Values a Rational Function Cannot Take	✓ covered
8	Equations of asymptotes, axes of symmetry, value restrictions	Zeros and Vertical Asymptotes, Horizontal Asymptotes by Degree, Slant Asymptotes by Long Division, The Equation of an Ellipse, The Equation of a Hyperbola, Values a Rational Function Cannot Take	✓ covered
9	Transformations af(x), f(x)+a, f(x+a), f(ax) and combinations	Transforming Graphs, Stretching a Graph Vertically, Stretching a Graph Horizontally, Combining Graph Transformations	✓ covered
10	y = \\|f(x)\\|, y = f(\\|x\\|), y = 1/f(x) related to y = f(x)	Taking the Modulus of a Whole Expression, Replacing x with a Modulus, The Graph of y = \|x\|, Reciprocal Graphs, The Graph of y = 1/f(x)	✓ covered
11	Simple parametric equations and their graphs	Parametric Equations	✓ covered
12	Formulating equations/systems/inequalities from a problem	Forming Equations, Writing Formulas from Words	✓ covered
13	Solving an equation with a graphing calculatorgraphing-calculator content — deferred until the planned graphing tool ships (owner ruling 2026-08-25; these rows then join the exam-item batches)		○ deferred
14	Solving a linear system with a graphing calculatorgraphing-calculator content — deferred until the planned graphing tool ships (owner ruling 2026-08-25; these rows then join the exam-item batches)		○ deferred
15	Inequalities f(x)/g(x) > 0	Solving a Rational Inequality, Solving a Quadratic Inequality	✓ covered
16	\\|x\\| and \\|x−a\\| < b ⟺ a−b < x < a+b	Solving Absolute Value Inequalities, Absolute Value, Solving Absolute Value Equations	✓ covered
17	Solving inequalities graphically	Intersections of Graphs, Solving Inequalities from Graphs	✓ covered
18	Sequence and series, finite and infinite	Term-to-Term Rules, Sigma Notation, The Sum to Infinity	✓ covered
19	Sequence as y = f(n)	The nth Term, Term-to-Term Rules	✓ covered
20	Relationship between uₙ and Sₙ	The Sum of an Arithmetic Series, The Sum of a Geometric Series, The nth Term from the Sum	✓ covered
21	nth-term formula and uₙ₊₁ = f(uₙ) recurrences	The nth Term, Recurrence Relations	✓ covered
22	Sum and difference of two series	Sigma Notation	✓ covered
23	Convergence of a series and sum to infinity	The Sum to Infinity, Two Infinite Sums, Opposite Fates	✓ covered
24	nth term and sum of a finite arithmetic series	The Sum of an Arithmetic Series	✓ covered
25	nth term and sum of a finite geometric series	Geometric Sequences, The Sum of a Geometric Series	✓ covered
26	Convergence condition for an infinite GP; sum to infinity	The Sum to Infinity	✓ covered
27	Vector addition, scalar multiplication, geometry	Adding Vectors, Subtracting Vectors, Scalar Multiplication	✓ covered
28	Position, displacement and direction vectors	Position Vectors, Vectors	✓ covered
29	Magnitude; unit vectors	Magnitude of a Vector, Unit Vectors	✓ covered
30	Distance between two points; collinearity	The Distance Between Two Points, Collinear Vectors	✓ covered
31	Ratio theorem	Vector Proofs with Midpoints, The Ratio Theorem	✓ covered
32	Scalar and vector product and their properties	The Dot Product, The Vector Product, Expanding Products of Vectors	✓ covered
33	Angle between two vectors	The Angle Between Vectors	✓ covered
34	Geometrical meanings of \\|a·n̂\\| and \\|a × n̂\\|	The Distance from a Point to a Plane, Parallelogram Area from the Vector Product, The Distance from a Point to a Line	✓ covered
35	Vector and cartesian equations of lines and planes	The Vector Equation of a Line, The Vector Equation of a Plane, The Cartesian Equation of a Plane	✓ covered
36	Foot of perpendicular; distance from a point to a line/plane	The Distance from a Point to a Plane, The Distance from a Point to a Line	✓ covered
37	Angle between two lines / line and plane / two planes	The Angle Between Vectors, The Angle Between a Line and a Plane, The Angle Between Two Planes	✓ covered
38	Relationships between lines, lines and planes, planes	Intersecting and Skew Lines, The Intersection of a Line and a Plane, The Intersection of Two Planes	✓ covered
39	Extension from real to complex numbers	Roots of Negative Numbers, The Imaginary Unit	✓ covered
40	Complex roots of quadratic equations	Complex Roots of a Quadratic	✓ covered
41	Modulus, argument, conjugate	The Modulus of a Complex Number, The Argument of a Complex Number, Complex Conjugates	✓ covered
42	Four operations; equality of complex numbers	Adding Complex Numbers, Multiplying Complex Numbers, Dividing Complex Numbers	✓ covered
43	Conjugate roots of real-coefficient polynomial equations	Complex Roots of a Quadratic, Complex Roots of a Cubic	✓ covered
44	Representation on the Argand diagram	The Argand Diagram	✓ covered
45	Geometrical effects of conjugation, negation, ±, × i	The Imaginary Unit, Complex Conjugates, Adding Complex Numbers, Moving Points on the Argand Diagram	✓ covered
46	Graphical interpretation of f′ and f″ signs	Increasing and Decreasing Functions, Points of Inflection	✓ covered
47	Relating the graph of y = f′(x) to y = f(x)	Curve Sketching with Derivatives, From the Derivative Back to the Function	✓ covered
48	Implicit and parametric differentiation	Implicit Differentiation, Parametric Differentiation	✓ covered
49	Nature of stationary points analytically	The First Derivative Test, The Second Derivative Test	✓ covered
50	Locating max/min and derivative values with a GCgraphing-calculator content — deferred until the planned graphing tool ships (owner ruling 2026-08-25; these rows then join the exam-item batches)		○ deferred
51	Tangents and normals, incl. implicit/parametric curves	Tangents and Normals to Curves, Implicit Differentiation, Parametric Differentiation	✓ covered
52	Local maxima and minima problems	Optimization Problems	✓ covered
53	Connected rates of change	Related Rates	✓ covered
54	Standard Maclaurin expansions	Maclaurin Series	✓ covered
55	Deriving the first terms by repeated differentiation	Maclaurin Series	✓ covered
56	Range of x for which a standard series converges	Binomial Series for a Negative or Fractional Index, Expanding (a + bx) to a Rational Power, The Sum to Infinity, Maclaurin Series, Where a Series Is Valid	✓ covered
57	Maclaurin series as an approximation	Maclaurin Series	✓ covered
58	Small-angle approximations	Small-Angle Approximations, The Limit of sin x over x	✓ covered
59	∫ f′(x)[f(x)]ⁿ and ∫ f′(x)e^f(x)	Integrals of f′ over f, Integration by Substitution	✓ covered
60	∫ sin²x, cos²x, tan²x	Integrating sin² and cos², Integrating sec²x, sec x tan x and tan x	✓ covered
61	∫ 1/(a²+x²), 1/√(a²−x²), 1/(a²−x²), 1/(x²−a²)	Integrals That Give tan⁻¹ and sin⁻¹, Integration by Partial Fractions	✓ covered
62	Integration by a given substitution	Integration by Substitution	✓ covered
63	Integration by parts	Integration by Parts	✓ covered
64	Definite integral as a limit of sum and as area	The Definite Integral, Area Under a Curve	✓ covered
65	Areas incl. between two curves and below the axis	Area Between Two Curves, Area Below the Axis	✓ covered
66	Volume of revolution about x- or y-axis	Volumes of Revolution	✓ covered
67	Approximate definite integral with a graphing calculatorgraphing-calculator content — deferred until the planned graphing tool ships (owner ruling 2026-08-25; these rows then join the exam-item batches)		○ deferred
68	dy/dx = f(x)g(y), incl. reduction by a given substitution	Separable Differential Equations, Solving a Differential Equation by Substitution	✓ covered
69	Formulating a differential equation from a situation	Forming Differential Equations	✓ covered
70	Interpreting a DE and its solution in context	Forming Differential Equations, Separable Differential Equations	✓ covered
71	Addition and multiplication principles	Adding and Multiplying Choices	✓ covered
72	Permutations and combinations	Permutations, Combinations	✓ covered
73	Arrangements in a line or in a circle, with repetition/restriction	Permutations, Repeats and Restrictions, Arrangements in a Circle	✓ covered
74	Addition and multiplication of probabilities	The Addition Rule for Overlapping Events, Independent Events	✓ covered
75	Mutually exclusive and independent events	Mutually Exclusive Events, Independent Events	✓ covered
76	Tables, Venn, tree diagrams and P&C for probabilities	Possibility Diagrams, Venn Diagrams, Tree Diagrams	✓ covered
77	Conditional probabilities	Conditional Probability, Conditional Probability from Tables	✓ covered
78	P(A′), P(A∪B), P(A\\|B) formulae	Complementary Events, The Addition Rule for Overlapping Events, Conditional Probability Notation	✓ covered
79	Discrete RVs: distributions, expectations, variances	Random Variables, Probability Distributions, Expected Value, Variance	✓ covered
80	Binomial distribution and conditions for suitability	The Binomial Distribution, Modeling with a Binomial	✓ covered
81	Mean and variance of the binomial	Binomial Mean and Variance	✓ covered
82	Continuous RVs; normal model; N(μ, σ²)	Continuous Variables, The Normal Distribution	✓ covered
83	Standard normal distribution	Standardizing, Reading the Z-Table	✓ covered
84	P(X < x₁) given x₁, μ, σ	Reading Normal Probabilities	✓ covered
85	Symmetry of the normal curve	The Normal Distribution	✓ covered
86	Relationship between x₁, μ, σ from a given probability	Inverse Normal	✓ covered
87	E(aX + b), Var(aX + b)	Transforming a Random Variable	✓ covered
88	E(aX + bY), Var(aX + bY)	Combining Normals	✓ covered
89	Population and simple random sample	Populations and Samples, Sampling Methods	✓ covered
90	E(X̄) = μ, Var(X̄) = σ²/n	Mean and Variance of X-bar, Standard Error	✓ covered
91	Distribution of X̄ from a normal population	The Sampling Distribution	✓ covered
92	Central Limit Theorem	The Central Limit Theorem, Applying the Theorem	✓ covered
93	Unbiased estimates, incl. from summarised data	Point Estimates, Unbiased Sample Variance, Estimates from Summarized Data	✓ covered
94	H₀/H₁, test statistic, critical region, level, p-value	Null and Alternative Hypotheses, Critical Regions, P-Values	✓ covered
95	Hypothesis test for a population mean	The Z-Test	✓ covered
96	1-tail and 2-tail tests; interpretation	One-Tailed and Two-Tailed, P-Values	✓ covered
97	Scatter diagram to judge a linear relationship	Scatter Plots, Correlation	✓ covered
98	Correlation coefficient; interpreting r	Pearson’s Correlation Coefficient, Spearman’s Rank Correlation	✓ covered
99	Linear regression; least squares; the regression equation	The Least-Squares Regression Line	✓ covered
100	Interpolation and extrapolation	Interpolation and Extrapolation, Predicting from a Regression Line	✓ covered
101	Regression line for prediction, with model commentary	Predicting from a Regression Line, Correlation Is Not Causation	✓ covered
102	Square/reciprocal/log transformation to achieve linearity	Straightening Growth with Logarithms, Straightening a Power Law, Straightening with a Square or a Reciprocal	✓ covered

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Other syllabuses: Singapore PSLE Mathematics · Singapore O-Level Mathematics · SAT Math · GCSE Mathematics (Higher)