1.1 Arithmetic sequences and series; sum of finite arithmetic series; geometric sequences and series; sum of finite and infinite geometric series Sigma notation
1.2 Exponents and logarithms Laws of exponents; laws of logarithms Change of base
1.3 Counting principles, including permutations and combinations The binomial theorem: expansion of (a + b)^n, n∈ N
1.4 Proof by mathematical induction Forming conjectures to be proved by mathematical induction
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
1.6 Sums, products and quotients of complex numbers
1.7 De Moivre’s theorem Powers and roots of a complex number
1.8 Conjugate roots of polynomial equations with real coefficients
Topic 2: Functions and equations
2.1 Concept of function f: x → f(x): domain, range; image (value) Composite functions f ◦ g; identity function Inverse function f^(-1)
2.2 The graph of a function; its equation y = f(x) Function graphing skills: use of a GDC to graph a variety of functions, investigation of key features of graphs, solutions of equations graphically
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|)
2.4 The reciprocal function x → 1/x, x ≠ 0: its graph; its self-inverse nature
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
2.6 The solution of ax^2 + bx + c = 0, a ≠ 0 The quadratic formula Use of the discriminant Δ = b^2 - 4ac
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
2.8 The exponential function x → e^x The logarithmic function x → lnx, x > 0
2.9 Inequalities in one variable, using their graphical representation Solution of g(x) ≥ f(x), where f, g are linear or quadratic
2.10 Polynomial functions The factor and remainder theorems, with application to the solution of polynomial equations and inequalities
Topic 3: Circular functions and trigonometry
3.1 The circle: radian measure of angles; length of an arc; area of a sector
3.2 Definition of cosθ and sinθ in terms of the unit circle Definition of tanθ as sinθ/cosθ Definition of secθ, cscθ and cotθ Pythagorean identities: cos^2 θ + sin^2 θ = 1; 1 + tan^2 θ = sec^2 θ; 1 + cot^2 θ = csc^2 θ
3.4 The circular functions sinx , cosx and tanx; their domains, ranges, periodic nature and graphs Composite functions of the form f(x) = asin(b(x + c)) + d The inverse functions x → arcsinx, x → arccosx, x → arctanx; their domains, ranges and graphs
3.5 Solution of trigonometric equations in a finite interval Use of trigonometric identies and factorization to transform equations
3.6 Solution of triangles The cosine rule The sine rule Area of a triangle as 1/2 absinC
Topic 4: Matrices
4.1 Definition of a matrix: the terms element, row, column and order
4.2 Algebra of matrices: equality; addition; subtraction; multiplication by a scalar Multiplication of matrices Identity and zero matrices
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
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
Topic 5: Vectors
5.1 Vectors in the plane and in three dimensions Components of a vector The sum and difference of two vectors; the zero vector, the opposite vector, multiplication by a scalar, magnitude of a vector; unit vectors; base vectors i, j, k; position vector
5.2 The scalar product of two vectors Algebraic properties of the scalar product Perpendicular vectors; parallel vectors The angle between two vectors
5.3 Vector equation of a line r = a + λb The angle between two lines
5.4 Coincident, parallel, intersecting and distinguishing between these cases Points of intersection
5.5 The vector product of two vectors, v x w The determinant representation Geometric interpretation of |v x w|
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
5.7 Intersections of: a line with a plane; two planes; three planes Angle between: a line and a plane; two planes
Topic 6: Statistics and probability
6.1 Concepts of population, sample, random sample and frequency distribution of discrete and continuous data
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
6.4 Cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles
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);
6.6 Combined events, the formula: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) P(A ∩ B) = 0 for mutually exclusive events
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
6.8 Use of Venn diagrams, tree diagrams and tables of outcomes to solve problems
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
6.10 Binomial distribution, its mean and variance Poisson distribution, its mean and variance
6.11 Normal distribution Properties of the normal distribution Standardization of normal variables
Topic 7: Calculus
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
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
7.3 Local maximum and minimum points Use of the first and second derivative in optimization problems
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
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
7.6 Kinematic problems involving displacement: s, velocity: v, and acceleration: a
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
7.8 Implicit differentiation
7.9 Further integration: integration by substitution; integration by parts
7.10 Solution of first order differential equations by separation of variables
Option
Option A: Statistics and probability
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
A2 Cumulative distribution functions Discrete distributions: uniform, Bernoulli, binomial, negative binomial, Poisson, geometric, hypergeometric Continuous distributions: uniform, exponential, normal
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
A4 Finding confidence intervals for the mean of a population Finding confidence intervals for the proportion of successes in a population
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
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
Option B: Sets, relations and groups
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)
B2 Ordered pairs: the Cartesian product of two sets Relations; equivalence relations; equivalence classes
B3 Functions: injections; surjections; bijections Composition of functions and inverse functions
B5 Binary operations with associative, distributive and commutative properties
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
B7 The axioms of a group {G, *} Abelian groups
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
B9 Finite and infinite groups The order of a group element and the order of a group
B10 Cyclic groups Proof that all cyclic groups are Abelian
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)
B12 Isomorphism of groups Proof of isomorphism properties for identities and inverses
Option C: Series and differential equations
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
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
C3 Series that converge absolutely Series that converge conditionally Alternating series
C4 Power series: radius of convergence and interval of convergence Determination of the radius of convergence by the ratio test
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
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
Option D: Discrete mathematics
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
D2 Representation of integers in different bases
D3 Linear diophantine equations ax + by = c
D4 Modular arithmetic Linear congruences Chinese remainder theorem
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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