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Core (226)
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Topic 1: Algebra
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1.1 Arithmetic sequences and series; sum of finite arithmetic series; geometric sequences and series; sum of finite and infinite geometric series Sigma notation
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1.2 Exponents and logarithms Laws of exponents; laws of logarithms Change of base
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1.3 Counting principles, including permutations and combinations The binomial theorem: expansion of (a + b)^n, n∈ N
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1.4 Proof by mathematical induction Forming conjectures to be proved by mathematical induction
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1.5 Complex numbers: the number i; the terms real part, imaginary part, conjugate, modulus and argument Cartesian form z = a + ib Modulus-argument form z = r(cosθ + isinθ) The complex plane
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1.6 Sums, products and quotients of complex numbers
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1.7 De Moivre’s theorem Powers and roots of a complex number
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1.8 Conjugate roots of polynomial equations with real coefficients
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2.1 Concept of function f: x → f(x): domain, range; image (value) Composite functions f ◦ g; identity function Inverse function f^(-1)
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2.3 Transformations of graphs: translations; stretches; reflections in the axes The graph of y = f^(-1) (x) as the reflection in the line y = x of the graph of y = f(x) The graph of 1/f(x) from y = f(x) The graphs of y = |f(x)| and y = f(|x|)
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2.4 The reciprocal function x → 1/x, x ≠ 0: its graph; its self-inverse nature
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2.5 The quadratic function and its graph Axis of symmetry of the quadratic function The vertex and the factor form of the quadratic function
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2.6 The solution of ax^2 + bx + c = 0, a ≠ 0 The quadratic formula Use of the discriminant Δ = b^2 - 4ac
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2.7 The function: x → a^x, a > 0 The inverse function x → loga(x), x > 0 Graphs of y = a^x and y = loga(x) Solution of a^x = b using logarithms
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2.8 The exponential function x → e^x The logarithmic function x → lnx, x > 0
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2.9 Inequalities in one variable, using their graphical representation Solution of g(x) ≥ f(x), where f, g are linear or quadratic
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2.10 Polynomial functions The factor and remainder theorems, with application to the solution of polynomial equations and inequalities
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Topic 4: Matrices
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4.1 Definition of a matrix: the terms element, row, column and order
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4.2 Algebra of matrices: equality; addition; subtraction; multiplication by a scalar Multiplication of matrices Identity and zero matrices
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4.3 Determinant of a square matrix Calculation of 2 x 2 and 3 x 3 determinants Inverse of a matrix: conditions for its existence
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4.4 Solution of systems of linear equations (a maximum of three equations in three unknowns) Conditions for the existence of a unique solution, no solution and an infinity of solutions
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5.3 Vector equation of a line r = a + λb The angle between two lines
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5.4 Coincident, parallel, intersecting and distinguishing between these cases Points of intersection
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5.5 The vector product of two vectors, v x w The determinant representation Geometric interpretation of |v x w|
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5.6 Vector equation of a plane r = a + λb + µc Use of normal vector to obtain the form r ∙ n = a ∙ n Cartesian equation of a plane ax + by + cz = d
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5.7 Intersections of: a line with a plane; two planes; three planes Angle between: a line and a plane; two planes
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Topic 6: Statistics and probability
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6.1 Concepts of population, sample, random sample and frequency distribution of discrete and continuous data
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6.2 Presentation of data: frequency tables and diagrams, box and whisker plots Grouped data: mid-interval values, interval width, upper and lower interval boundaries, frequency histograms
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6.3 Mean, median, mode; quartiles, percentiles Range; interquartile range; variance, standard deviation
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6.4 Cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles
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6.5 Concepts of trial, outcome, equally likely outcomes, sample space (U) and event The probability of an event A as P(A) = n(A)/n(U) The complementary events A and A′ (not A);
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6.6 Combined events, the formula: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) P(A ∩ B) = 0 for mutually exclusive events
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6.7 Conditional probability; the definition: P(A | B) = P(A ∩ B)/P(B) Independent events; the definition: P(A | B) = P(A) = P(A | B') Use of Bayes’ theorem for two events
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6.8 Use of Venn diagrams, tree diagrams and tables of outcomes to solve problems
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6.9 Concept of discrete and continuous random variables and their probability distributions Definition and use of probability density functions Expected value (mean), mode, median, variance and standard deviation
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6.10 Binomial distribution, its mean and variance Poisson distribution, its mean and variance
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6.11 Normal distribution Properties of the normal distribution Standardization of normal variables
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Topic 7: Calculus
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7.1 Informal ideas of limit and convergence Definition of derivative Derivative of x^n, sinx, cosx, tanx, e^x, lnx, a^x , loga(x), cotx, secx, cscx, arcsinx, arccosx, arctanx Derivative interpreted as a gradient function and as rate of change
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7.2 Differentiation of a sum and a real multiple of the functions x^n, sinx, cosx, tanx, e^x, ln x, a^x, loga(x), arcsinx, arccosx, arctanx, cotx, secx, cscx The chain rule for composite functions The product and quotient rules The second derivative
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7.3 Local maximum and minimum points Use of the first and second derivative in optimization problems
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7.4 Indefinite integration as anti-differentiation Indefinite integral of x^n (n ≠ -1), sinx, cosx, e^x, 1/x The composites of any of these with the linear function ax + b
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7.5 Anti-differentiation with a boundary condition to determine the constant term Definite integrals Area between a curve and the x-axis or y-axis in a given interval, areas between curves Volumes of revolution
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7.6 Kinematic problems involving displacement: s, velocity: v, and acceleration: a
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7.7 Graphical behaviour of functions: tangents and normals, behaviour for large |x|; asymptotes The significance of the second derivative; distinction between maximum and minimum points Points of inflexion with zero and non-zero gradients
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7.8 Implicit differentiation
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7.9 Further integration: integration by substitution; integration by parts
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7.10 Solution of first order differential equations by separation of variables
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Option
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Option A: Statistics and probability
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A1 Expectation algebra Linear transformation of a single random variable Mean and variance of linear combinations of two independent random variables Extension to linear combinations of n independent random variables
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A2 Cumulative distribution functions Discrete distributions: uniform, Bernoulli, binomial, negative binomial, Poisson, geometric, hypergeometric Continuous distributions: uniform, exponential, normal
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A3 Distribution of the sample mean The distribution of linear combinations of independent normal random variables The central limit theorem The approximate normality of the proportion of successes in a large sample
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A4 Finding confidence intervals for the mean of a population Finding confidence intervals for the proportion of successes in a population
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A5 Significance testing for a mean Significance testing for a proportion Null and alternative hypotheses H0 and H1 Type I and Type II errors Significance levels; critical region, critical values, p-values; one-tailed and two-tailed tests
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A6 The chi-squared distribution: degrees of freedom, υ The χ^2 statistic The χ^2 goodness of fit test Contingency tables: the χ^2 test for the independence of two variables
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Option B: Sets, relations and groups
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B1 Finite and infinite sets Subsets Operations on sets: union; intersection; complement, set difference, symmetric difference De Morgan’s laws; distributive, associative and commutative laws (for union and intersection)
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B2 Ordered pairs: the Cartesian product of two sets Relations; equivalence relations; equivalence classes
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B3 Functions: injections; surjections; bijections Composition of functions and inverse functions
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B4 Binary operations Operation tables (Cayley tables)
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B5 Binary operations with associative, distributive and commutative properties
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B6 The identity element e The inverse a^(−1) of an element a Proof that left-cancellation and right-cancellation by an element a hold, provided that a has an inverse Proofs of the uniqueness of the identity and inverse elements
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B7 The axioms of a group {G, *} Abelian groups
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B8 R, Q, Z, C under addition; matrices of the same order under addition; invertible matrices under multiplication; symmetries of a triangle, rectangle; invertible functions under composition of functions; permutations under composition of permutations
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B9 Finite and infinite groups The order of a group element and the order of a group
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B10 Cyclic groups Proof that all cyclic groups are Abelian
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B11 Subgroups, proper subgroups Use and proof of subgroup tests Lagrange’s theorem Use and proof of the result that the order of a finite group is divisible by the order of any element (Corollary to Lagrange’s theorem)
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B12 Isomorphism of groups Proof of isomorphism properties for identities and inverses
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Option C: Series and differential equations
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C1 Infinite sequences of real numbers Limit theorems as n approaches infinity Limit of a sequence Improper integrals The integral as a limit of a sum; lower sum and upper sum
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C2 Convergence of infinite series Partial fractions and telescoping series (method of differences) Tests for convergence: comparison test; limit comparison test; ratio test; integral test The p-series Use of integrals to estimate sums of series
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C3 Series that converge absolutely Series that converge conditionally Alternating series
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C4 Power series: radius of convergence and interval of convergence Determination of the radius of convergence by the ratio test
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C5 Taylor polynomials and series, including the error term Maclaurin series for e^x, sinx, cosx, arctanx, ln(1 + x), (1 + x)^p Use of substitution to obtain other series The evaluation of limits using l’Hôpital’s Rule and/or the Taylor series
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C6 First order differential equations; numerical solution of dy/dx = f(x, y) using Euler’s method Homogeneous differential equation dy/dx = f(y/x) using the substitution y = vx Solution of y' + P(x)y = Q(x) using the integrating factor
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Option D: Discrete mathematics
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D1 Division and Euclidean algorithms The greatest common divisor, gcd(a, b), and the least common multiple, lcm(a, b), of integers a and b Relatively prime numbers; prime numbers and the fundamental theorem of arithmetic
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D2 Representation of integers in different bases
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D3 Linear diophantine equations ax + by = c
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D4 Modular arithmetic Linear congruences Chinese remainder theorem
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D5 Fermat’s little theorem
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D6 Graphs, vertices, edges Adjacent vertices, adjacent edges Simple graphs; connected graphs; complete graphs; bipartite graphs; planar graphs, trees, weighted graphs Subgraphs; complements of graphs Graph isomorphism
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D7 Walks, trails, paths, circuits, cycles Hamiltonian paths and cycles; Eulerian trails and circuits
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D8 Adjacency matrix Cost adjacency matrix
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D9 Graph algorithms: Prim’s; Kruskal’s; Dijkstra’s
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D10 “Chinese postman” problem (“route inspection”) “Travelling salesman” problem Algorithms for determining upper and lower bounds of the travelling salesman problem
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angle
