Singapore · SEABSyllabus
Maths syllabus, dot point by dot point
Every dot point in the Singapore 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.
Calculus
Module overview →- How do we locate and classify stationary points and solve optimisation problems?Find and classify stationary points, determine increasing and decreasing intervals and concavity, and solve optimisation problems in context10 min answer →
- How does the definite integral measure area, and how do we handle areas below the axis or between curves?Evaluate definite integrals, use them to find the area under a curve and between curves, and apply the fundamental theorem of calculus10 min answer →
- How do we solve first-order differential equations and interpret their solutions?Solve first-order differential equations by direct integration and by separating variables, find particular solutions from conditions, and interpret solutions in context10 min answer →
- How do the product, quotient and chain rules let us differentiate any combination of standard functions?Differentiate standard functions and use the product, quotient and chain rules to differentiate products, quotients and composite functions9 min answer →
- How do we differentiate when y is defined implicitly or through a parameter?Differentiate relations defined implicitly and curves defined parametrically, and find gradients, tangents and second derivatives in each case10 min answer →
- How do substitution, integration by parts and partial fractions extend what we can integrate?Integrate standard functions and use substitution, integration by parts and partial fractions to evaluate a wide range of integrals10 min answer →
- How does the Maclaurin series approximate a function as a power series, and how do we use the standard expansions?Derive and use the Maclaurin series of a function, apply the standard series for common functions, and use series to obtain approximations10 min answer →
- How do derivatives give tangent and normal lines and connect related rates of change?Find equations of tangents and normals to curves, and solve connected rates of change problems using the chain rule9 min answer →
- How does integration give the volume of a solid formed by rotating a region about an axis?Find volumes of revolution generated by rotating a region about the x-axis or y-axis, including the volume between two curves9 min answer →
Functions and Graphs
Module overview →- What are the key features of a curve, and how do asymptotes and symmetry guide a sketch?Identify and use the key features of a curve - intercepts, turning points, asymptotes, symmetry and behaviour at infinity - to produce and interpret graph sketches9 min answer →
- How do we combine functions and reverse them, and when does an inverse exist?Form and find the domain of composite functions, determine when a composite is defined, find inverse functions and their domains, and use the graphical relationship between a function and its inverse10 min answer →
- How do we recognise standard conic graphs and work with curves defined parametrically?Recognise and sketch the standard conics (circle, ellipse, parabola, hyperbola) from their equations, and sketch and analyse curves defined parametrically9 min answer →
- What makes a relation a function, and how do its domain and range determine its behaviour?Define a function and its domain and range, decide whether a relation is a function or one-to-one, and find the range of a given function over a stated domain9 min answer →
- How do we sketch a rational function by finding its intercepts, asymptotes and turning points?Sketch graphs of rational functions of the form a linear over linear and a quadratic over linear, finding intercepts, asymptotes, stationary points and the regions where the curve lies10 min answer →
- How do we solve polynomial, rational and modulus inequalities reliably?Solve quadratic, polynomial and rational inequalities algebraically and graphically, using a sign analysis and respecting the sign of any denominator9 min answer →
- How does the modulus function behave, and how do we solve equations and sketch graphs that involve it?Define the modulus function, sketch graphs involving the modulus of a function, and solve equations and inequalities involving the modulus9 min answer →
- How do translations, stretches and reflections change the graph and the equation of a function?Relate the graph of y equals a f(b(x + c)) + d to the graph of y equals f(x) through translations, stretches and reflections, and apply combined transformations in the correct order9 min answer →
Probability and Statistics
Module overview →- When do the binomial and Poisson distributions apply, and how do we compute their probabilities?Model situations with the binomial and Poisson distributions, state the conditions for each, and compute probabilities, means and variances10 min answer →
- How does conditioning on information change a probability, and what does independence mean?Calculate conditional probabilities, test for independence, and apply the conditional probability formula and the law of total probability9 min answer →
- How do we measure the strength of a linear relationship and fit a line for prediction?Compute and interpret the product moment correlation coefficient, find the least squares regression line, and use it for prediction within the data range9 min answer →
- How do we describe a discrete random variable and compute its expectation and variance?Construct probability distributions for discrete random variables and compute the expectation and variance, including for functions of the variable9 min answer →
- How do we test a claim about a population mean using sample evidence?Carry out a hypothesis test for a population mean, stating hypotheses, computing a test statistic or p-value, and interpreting the conclusion in context10 min answer →
- When can the binomial or Poisson be approximated by the normal or Poisson, and how is the continuity correction applied?Approximate the binomial by the Poisson or the normal, and the Poisson by the normal, under stated conditions, applying a continuity correction where appropriate10 min answer →
- How do we compute probabilities for a normally distributed variable using standardisation?Model continuous data with the normal distribution, standardise to the Z-distribution to find probabilities, and find values from given probabilities10 min answer →
- How do we count arrangements and selections, and when does order matter?Use the addition and multiplication principles, permutations and combinations to count arrangements and selections, including cases with restrictions9 min answer →
- How do we calculate probabilities of combined events using the basic rules?Use the probability rules for the complement, union and intersection of events, and apply Venn diagrams and tree diagrams to combined events9 min answer →
- How does the distribution of a sample mean behave, and what does the Central Limit Theorem guarantee?Describe the distribution of the sample mean, use the Central Limit Theorem, and find unbiased estimates of the population mean and variance from a sample9 min answer →
Sequences and Series
Module overview →- How do arithmetic progressions grow, and how do we sum them?Use the formulae for the nth term and the sum of the first n terms of an arithmetic progression, and solve problems involving arithmetic sequences and series8 min answer →
- How do we expand a binomial raised to a rational power, and when is the expansion valid?Expand (1 + x) to the power n for rational n as a series, state the range of validity, and use the expansion to obtain approximations9 min answer →
- What does it mean for a series to converge, and how do we reason about the behaviour of a sequence as n grows?Describe the behaviour of a sequence as n tends to infinity, determine the convergence of a geometric series, and interpret the limit of a sequence or partial sum9 min answer →
- How do geometric progressions grow, and when does their sum converge?Use the formulae for the nth term and the sum of a geometric progression, determine convergence, and find the sum to infinity of a convergent geometric series9 min answer →
- How does proof by mathematical induction establish a result for every positive integer?Use the principle of mathematical induction to prove statements about sums of series, divisibility and other results indexed by the positive integers9 min answer →
- How does a telescoping sum collapse, and how do we exploit it to evaluate a series?Use the method of differences, including the use of partial fractions, to find the sum of a series whose terms telescope, and deduce the sum to infinity where it exists10 min answer →
- How do recurrence relations define a sequence, and how do we find or verify a closed form?Use recurrence relations to generate sequences, find and verify a conjectured formula for the nth term, and analyse long-term behaviour9 min answer →
- How does sigma notation express a sum, and which standard results let us evaluate it?Use sigma notation and the standard results for the sums of integers, squares and cubes, and the linearity of summation, to evaluate finite series9 min answer →
Vectors and Complex Numbers
Module overview →- How do complex numbers describe geometry in the Argand diagram, and what loci do conditions on modulus and argument define?Represent complex numbers on an Argand diagram and identify and sketch loci defined by conditions on the modulus and argument10 min answer →
- How do we do arithmetic with complex numbers and solve polynomial equations using them?Perform arithmetic with complex numbers in Cartesian form, use the conjugate, and solve polynomial equations including the use of conjugate root pairs9 min answer →
- How do modulus-argument and exponential forms simplify complex multiplication and powers?Express complex numbers in modulus-argument and exponential form, convert between forms, and use them to multiply, divide and take powers via de Moivre's theorem10 min answer →
- How do we describe a line in space and decide how two lines relate?Write the vector and Cartesian equations of a line in three dimensions, find the intersection of two lines, and classify lines as parallel, intersecting or skew10 min answer →
- How do we describe a plane, and how do lines and planes meet?Write the scalar product and Cartesian equations of a plane, find the intersection of a line with a plane and of two planes, and compute distances and angles involving planes10 min answer →
- How do we find all the nth roots of a complex number and the roots of higher polynomial equations?Find the nth roots of a complex number using de Moivre's theorem, and solve polynomial equations with complex roots, interpreting the roots geometrically10 min answer →
- How does the scalar product measure the angle between vectors and project one onto another?Define and compute the scalar (dot) product, use it to find angles between vectors, test for perpendicularity, and find the projection of one vector onto another9 min answer →
- How does the vector product produce a perpendicular vector and measure area?Define and compute the vector (cross) product, use it to find a vector perpendicular to two given vectors, the area of a triangle or parallelogram, and the sine of the angle between vectors9 min answer →
- How do we represent and manipulate vectors in two and three dimensions?Represent vectors in component and position form, add and scale them, find magnitudes and unit vectors, and use the ratio theorem for points dividing a line segment9 min answer →