Syllabus
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Unit 1: Combinatorics, vectors and proof
Topic 1: Combinatorics
The inclusion-exclusion principle for the union of two sets and three sets
Unit 1: Combinatorics, vectors and proof > Topic 1: Combinatorics > The inclusion-exclusion principle for the union of two sets and three sets
- Determine and use the formulas for finding the number of elements in the union of two and the union of three sets
- Use the multiplication principle
Permutations (ordered arrangements) and combinations (unordered selections)
Unit 1: Combinatorics, vectors and proof > Topic 1: Combinatorics > Permutations (ordered arrangements) and combinations (unordered selections)
- Solve problems involving permutations
- Use factorial notation
- Use the notation \( nPr \)
- Solve problems involving permutations with restrictions
- Solve problems involving combinations
- Use the notation \( nCr \)
- Derive and use simple identities associated with Pascal's triangle
- Solve problems involving combinations with restrictions
- Apply permutations and combinations to probability problems
The pigeon-hole principle
Unit 1: Combinatorics, vectors and proof > Topic 1: Combinatorics > The pigeon-hole principle
- Solve problems and prove results using the pigeon-hole principle
Topic 2: Vectors in the plane
Representing vectors in the plane by directed line segments
Unit 1: Combinatorics, vectors and proof > Topic 2: Vectors in the plane > Representing vectors in the plane by directed line segments
- Examine examples of vectors
- Understand the difference between a scalar and a vector
- Define and use the magnitude and direction of a vector
- Understand and use vector equality
- Understand and use both the Cartesian form and polar form of a vector
- Represent a scalar multiple of a vector
- Use the triangle rule to find the sum and difference of two vectors
Algebra of vectors in the plane
Unit 1: Combinatorics, vectors and proof > Topic 2: Vectors in the plane > Algebra of vectors in the plane
- Use ordered pair notation and column vector notation to represent a vector
- Understand and use vector notation: \( \overrightarrow{AB} \), \( \underset{\sim}{c} \), \( \mathbf{d} \), unit vector notation \( \hat{n} \)
- Convert between Cartesian form and polar form
- Determine a vector between two points
- Define and use unit vectors and the perpendicular unit vectors \(\vec{i}\) and \(\vec{j}\)
- Express a vector in component form using the unit vectors \(\vec{i}\) and \(\vec{j}\)
- Examine and use addition and subtraction of vectors in component form
- Define and use multiplication by a scalar of a vector in component form
- Define and use a vector representing the midpoint of a line segment
- Define and use scalar (dot) product
- Apply the scalar product to vectors expressed in component form
- Examine properties of parallel and perpendicular vectors and determine if two vectors are parallel or perpendicular
- Define and use projections of vectors
- Solve problems involving displacement, force, velocity, equilibrium and relative velocity involving the above concepts
Topic 3: Introduction to proof
The nature of proof
Unit 1: Combinatorics, vectors and proof > Topic 3: Introduction to proof > The nature of proof
- Use implication, contrapositive
- Use proof by contradiction
- Use the symbols for implication (\(\implies\)), equivalence (\(\iff\)), and equality (\(=\))
- Use the quantifiers 'for all' (\(\forall\)) and 'there exists' (\(\exists\))
- Use examples and counterexamples
Rational and irrational numbers
Unit 1: Combinatorics, vectors and proof > Topic 3: Introduction to proof > Rational and irrational numbers
- Prove simple results involving numbers
- Express rational numbers as terminating or eventually recurring decimals and vice versa
- Prove irrationality by contradiction
Circle properties and their proofs
Unit 1: Combinatorics, vectors and proof > Topic 3: Introduction to proof > Circle properties and their proofs
- Prove circle properties, such as: an angle in a semicircle is a right angle, the angle at the centre subtended by an arc of a circle is twice the angle at the circumference subtended by the same arc, angles at the circumference of a circle subtended by the same arc are equal, the opposite angles of a cyclic quadrilateral are supplementary, chords of equal length subtend equal angles at the centre and conversely chords subtending equal angles at the centre of a circle have the same length, a tangent drawn to a circle is perpendicular to the radius at the point of contact, the alternate segment theorem, when two chords of a circle intersect, the product of the lengths of the intervals on one chord equals the product of the lengths of the intervals on the other chord and its converse, when a secant (meeting the circle at A and B) and a tangent (meeting the circle at T) are drawn to a circle from an external point M, the square of the length of the tangent equals the product of the lengths to the circle on the secant; (\( AM \times BM = TM^2 \)) and its converse
- Solve problems finding unknown angles and lengths and prove further results using the circle properties listed above
Geometric proofs using vectors
Unit 1: Combinatorics, vectors and proof > Topic 3: Introduction to proof > Geometric proofs using vectors
- Prove the diagonals of a parallelogram meet at right angles if and only if it is a rhombus
- Prove midpoints of the sides of a quadrilateral join to form a parallelogram
- Prove the sum of the squares of the lengths of a parallelogram's diagonals is equal to the sum of the squares of the lengths of the sides
- Prove an angle in a semicircle is a right angle
Unit 2: Complex numbers, trigonometry, functions and matrices
Topic 1: Complex numbers 1
Complex numbers
Unit 2: Complex numbers, trigonometry, functions and matrices > Topic 1: Complex numbers 1 > Complex numbers
- Define the imaginary number \( i \) as a root of the equation \( x^2 = -1 \)
- Use complex numbers in the form \( a + bi \) where \( a \) and \( b \) are the real and imaginary parts
- Determine and use complex conjugates
- Perform complex-number arithmetic: addition, subtraction, multiplication and division
The complex plane (the Argand plane)
Unit 2: Complex numbers, trigonometry, functions and matrices > Topic 1: Complex numbers 1 > The complex plane (the Argand plane)
- Consider complex numbers as points in a plane with real and imaginary parts as Cartesian coordinates
- Examine and use addition of complex numbers as vector addition in the complex plane
- Understand and use location of complex conjugates in the complex plane
- Examine and use multiplication as a linear transformation in the complex plane
Complex arithmetic using polar form
Unit 2: Complex numbers, trigonometry, functions and matrices > Topic 1: Complex numbers 1 > Complex arithmetic using polar form
- Use the modulus \( |z| \) of a complex number \( z \) and the argument \( \text{Arg}(z) \) of a non-zero complex number \( z \)
- Convert between Cartesian form and polar form
- Define and use multiplication, division and powers of complex numbers in polar form and the geometric interpretation of these
Roots of equations
Unit 2: Complex numbers, trigonometry, functions and matrices > Topic 1: Complex numbers 1 > Roots of equations
- Use the general solution of real quadratic equations
- Determine complex conjugate solutions of real quadratic equations
- Determine linear factors of real quadratic polynomials
Topic 2: Trigonometry and functions
The basic trigonometric functions
Unit 2: Complex numbers, trigonometry, functions and matrices > Topic 2: Trigonometry and functions > The basic trigonometric functions
- Find all solutions of \( f(a(x - b)) = c \) where \( f(\theta) \) is one of \( \sin(\theta) \), \( \cos(\theta) \), or \( \tan(\theta) \)
- Sketch and graph functions with rules of the form \( y = f(a(x - b)) \) where \( f(\theta) \) is one of \( \sin(\theta) \), \( \cos(\theta) \), or \( \tan(\theta) \)
Sketching graphs
Unit 2: Complex numbers, trigonometry, functions and matrices > Topic 2: Trigonometry and functions > Sketching graphs
- Use and apply the notation \( |x| \) for the absolute value for the real number \( x \) and the graph of \( y = |x| \)
- Examine the relationship between the graph of \( y = f(x) \) and the graphs of \( y = \frac{1}{f(x)} \), \( y = |f(x)| \) and \( y = f(|x|) \)
- Sketch the graphs of simple rational functions where the numerator and denominator are polynomials of low degree
The reciprocal trigonometric functions, secant, cosecant and cotangent
Unit 2: Complex numbers, trigonometry, functions and matrices > Topic 2: Trigonometry and functions > The reciprocal trigonometric functions, secant, cosecant and cotangent
- Define the reciprocal trigonometric functions, sketch their graphs, and graph simple transformations of them
Trigonometric identities
Unit 2: Complex numbers, trigonometry, functions and matrices > Topic 2: Trigonometry and functions > Trigonometric identities
- Prove and apply the Pythagorean identities
- Prove and apply the angle sum, difference and double-angle identities for sines and cosines
- Prove and apply the identities for products of sines and cosines expressed as sums and differences
- Convert sums as \(\cos(x) + \sin(x)\) to \(R\cos(x + \alpha)\) or \(R\sin(x + \alpha)\) and apply these to sketch graphs, solve equations of the form \(a\cos(x) + b\sin(x) = c\) and solve real-world problems
- Use the binomial theorem to prove and apply multi-angle trigonometric identities up to \( \sin(4x) \) and \( \cos(4x) \)
Applications of trigonometric functions to model periodic phenomena
Unit 2: Complex numbers, trigonometry, functions and matrices > Topic 2: Trigonometry and functions > Applications of trigonometric functions to model periodic phenomena
- Model periodic motion using sine and cosine functions, and understand the relevance of the period and amplitude of these functions in the model
Topic 3: Matrices
Matrix arithmetic
Unit 2: Complex numbers, trigonometry, functions and matrices > Topic 3: Matrices > Matrix arithmetic
- Understand the matrix definition and notation
- Define and use addition and subtraction of matrices, scalar multiplication, matrix multiplication, multiplicative identity and multiplicative inverse
- Calculate the determinant and inverse of \(2 \times 2\) matrices algebraically and solve matrix equations of the form \(AX = B\), where \(A\) is a \(2 \times 2\) matrix and \(X\) and \(B\) are column vectors
- Calculate the determinant and inverse of higher order matrices and solve matrix equations using technology
Transformations in the plane
Unit 2: Complex numbers, trigonometry, functions and matrices > Topic 3: Matrices > Transformations in the plane
- Understand translations and their representation as column vectors
- Define and use basic linear transformations: dilations of the form \((x, y) \rightarrow (ax, by)\), rotations about the origin and reflection in a line that passes through the origin, and the representations of these transformations by 2 x 2 matrices
- Apply these transformations to points in the plane and geometric objects
- Define and use composition of linear transformations and the corresponding matrix products
- Define and use inverses of linear transformations and the relationship with the matrix inverse
- Examine the relationship between the determinant and the effect of a linear transformation on area
- Establish geometric results by matrix multiplications
Unit 3: Mathematical induction, and further vectors, matrices and complex numbers
view_agenda query_statsTopic 1: Proof by mathematical induction
view_agenda query_statsMathematical induction
view_agenda query_statsUnit 3: Mathematical induction, and further vectors, matrices and complex numbers > Topic 1: Proof by mathematical induction > Mathematical induction
- Understand the nature of inductive proof including the 'initial statement' and inductive step
- Prove results for sums for any positive integer \( n \)
- Prove divisibility results for any positive integer \( n \)
Topic 2: Vectors and matrices
view_agenda query_statsThe algebra of vectors in three dimensions
view_agenda query_statsUnit 3: Mathematical induction, and further vectors, matrices and complex numbers > Topic 2: Vectors and matrices > The algebra of vectors in three dimensions
- Review the concepts of vectors from Unit 1 and extend to three dimensions by introducing the unit vector \( \mathbf{k} \) and the altitude \( \phi \)
- Prove geometric results (review from the topic Geometric proofs using vectors) in the plane and construct simple proofs in three dimensions
Vector and Cartesian equations
view_agenda query_statsUnit 3: Mathematical induction, and further vectors, matrices and complex numbers > Topic 2: Vectors and matrices > Vector and Cartesian equations
- Introduce Cartesian coordinates for three-dimensional space, including plotting points and the equations of spheres
- Use vector equations of curves in two or three dimensions involving a parameter, and determine a 'corresponding' Cartesian equation in the two-dimensional case
- Determine a vector, parametric and Cartesian equation of a straight line and straight-line segment given the position of two points, or equivalent information, in both two and three dimensions
- Examine the position of two particles, each described as a vector function of time, and determine if their paths cross or if the particles meet
- Define and use the vector (cross) product to determine a vector normal to a given plane
- Use vector methods in applications, including areas of shapes and determining vector and Cartesian equations of a plane and of regions in a plane
Systems of linear equations
view_agenda query_statsUnit 3: Mathematical induction, and further vectors, matrices and complex numbers > Topic 2: Vectors and matrices > Systems of linear equations
- Recognise the general form of a system of linear equations in several variables and use Gaussian techniques of elimination to solve a system of linear equations
- Solve systems of linear equations using matrix algebra
- Examine the three cases for solutions of systems of equations; a unique solution, no solution and infinitely many solutions, and the geometric interpretation of a solution of a system of equations with three variables
Applications of matrices
view_agenda query_statsUnit 3: Mathematical induction, and further vectors, matrices and complex numbers > Topic 2: Vectors and matrices > Applications of matrices
- Model real-life situations using matrices, including Dominance and Leslie
- Investigate how matrices have been applied in other real-life situations, e.g. Leontief, Markov, area, cryptology, eigenvectors and eigenvalues - Note: The external examination may assess only Dominance and Leslie matrices
Vector calculus
view_agenda query_statsUnit 3: Mathematical induction, and further vectors, matrices and complex numbers > Topic 2: Vectors and matrices > Vector calculus
- Consider position of vectors as a function of time
- Derive the Cartesian equation of a path given as a vector equation in two dimensions, including circles, ellipses and hyperbolas
- Differentiate and integrate a vector function with respect to time
- Determine equations of motion of a particle travelling in a straight line with both constant and variable acceleration
- Apply vector calculus to motion in a plane, including projectile and circular motion
Topic 3: Complex numbers 2
view_agenda query_statsCartesian forms
view_agenda query_statsUnit 3: Mathematical induction, and further vectors, matrices and complex numbers > Topic 3: Complex numbers 2 > Cartesian forms
- Review real and imaginary parts \(\text{Re}(z)\) and \(\text{Im}(z)\) of a complex number \(z\)
- Review Cartesian form
- Review complex arithmetic using Cartesian form
Complex arithmetic using polar form
view_agenda query_statsUnit 3: Mathematical induction, and further vectors, matrices and complex numbers > Topic 3: Complex numbers 2 > Complex arithmetic using polar form
- Prove the identities involving modulus and argument
- Prove and use De Moivre's theorem for integral powers
The complex plane (the Argand plane)
view_agenda query_statsUnit 3: Mathematical induction, and further vectors, matrices and complex numbers > Topic 3: Complex numbers 2 > The complex plane (the Argand plane)
- Identify subsets of the complex plane determined by straight lines and circles
Roots of complex numbers
view_agenda query_statsUnit 3: Mathematical induction, and further vectors, matrices and complex numbers > Topic 3: Complex numbers 2 > Roots of complex numbers
- Determine and examine the nth roots of unity and their location on the unit circle
- Determine and examine the nth roots of complex numbers and their location in the complex plane
Factorisation of polynomials
view_agenda query_statsUnit 3: Mathematical induction, and further vectors, matrices and complex numbers > Topic 3: Complex numbers 2 > Factorisation of polynomials
- Prove and apply the factor theorem and the remainder theorem for polynomials
- Consider conjugate roots for polynomials with real coefficients
- Solve polynomial equations to order 4
Unit 4: Further calculus and statistical inference
view_agenda query_statsTopic 1: Integration and applications of integration
view_agenda query_statsIntegration techniques
view_agenda query_statsUnit 4: Further calculus and statistical inference > Topic 1: Integration and applications of integration > Integration techniques
- Integrate using the trigonometric identities \( \sin^2(x) = \frac{1}{2}(1 - \cos(2x)), \cos^2(x) = \frac{1}{2}(1 + \cos(2x)) \) and \( 1 + \tan^2(x) = \sec^2(x) \)
- Use substitution \( u = g(x) \) to integrate expressions of the form \( f(g(x)) \cdot g'(x) \)
- Establish and use the formula \( \int \frac{1}{x} dx = \ln |x| + c \), for \( x \neq 0 \)
- Find and use the inverse trigonometric functions: arcsine, arccosine and arctangent
- Find and use the derivative of the inverse trigonometric functions: arcsine, arccosine and arctangent
- Integrate expressions of the form \( \frac{1}{ax^2 + bx + c} \) and \( \frac{ax}{ax^2 + bx + c} \)
- Use partial fractions where necessary for integration in simple cases
- Integrate by parts
Applications of integral calculus
view_agenda query_statsUnit 4: Further calculus and statistical inference > Topic 1: Integration and applications of integration > Applications of integral calculus
- Calculate areas between curves determined by functions
- Determine volumes of solids of revolution about either axis
- Use the numerical integration method of Simpson's rule, using technology
- Use and apply the probability density function, \( f(x) = \lambda e^{-\lambda x} \) for \( t \geq 0 \), of the exponential random variable with parameter \( \lambda \), and use the exponential random variables and associated probabilities and quantiles to model data and solve practical problems
Topic 2: Rates of change and differential equations
view_agenda query_statsRates of change
view_agenda query_statsUnit 4: Further calculus and statistical inference > Topic 2: Rates of change and differential equations > Rates of change
- Use implicit differentiation to determine the gradient of curves whose equations are given in implicit form
- Use related rates as instances of the chain rule: \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \)
- Solve simple first-order differential equations of the form \( \frac{dy}{dx} = f(x) \), differential equations of the form \( \frac{dx}{dy} = g(y) \), and in general, differential equations of the form \( \frac{dx}{dy} = f(x)g(y) \) using separation of variables
- Examine slope (direction or gradient) fields of a first-order differential equation
- Formulate and use differential equations, including the logistic equation, e.g. examples in chemistry, biology and economics
Modelling motion
view_agenda query_statsUnit 4: Further calculus and statistical inference > Topic 2: Rates of change and differential equations > Modelling motion
- Examine momentum, force, resultant force, action and reaction
- Consider constant and non-constant force
- Understand motion of a body under concurrent forces
- Consider and solve problems involving motion in a straight line with both constant and non-constant acceleration, including simple harmonic motion and the use of expressions \( \frac{d^2x}{dt^2} \), \( \frac{dv}{dx} \) and \( \frac{d^2x}{dx^2} \) for acceleration
Topic 3: Statistical inference
view_agenda query_statsSample means
view_agenda query_statsUnit 4: Further calculus and statistical inference > Topic 3: Statistical inference > Sample means
- Examine the concept of the sample mean \( \bar{X} \) as a random variable whose value varies between samples where X is a random variable with mean \( \mu \) and the standard deviation \( \sigma \)
- Simulate repeated random sampling from a variety of distributions and a range of sample sizes to illustrate properties of the distribution of \( \bar{X} \) across samples of a fixed size n, including its mean \( \mu \), its standard deviation \( \frac{\sigma}{\sqrt{n}} \) (where \( \mu \) and \( \sigma \) are the mean and standard deviation of X) and its approximate normality if n is large
- Simulate repeated random sampling from a variety of distributions and a range of sample sizes to illustrate the approximate standard normality of \( \frac{\bar{X}-\mu}{(\sigma/\sqrt{n})} \) for large samples (\( n \geq 30 \)), where \( \sigma \) is the sample standard deviation
Confidence intervals for means
view_agenda query_statsUnit 4: Further calculus and statistical inference > Topic 3: Statistical inference > Confidence intervals for means
- Understand the concept of an interval estimate for a parameter associated with a random variable
- Examine the approximate confidence interval \( ( \bar{x} - z \frac{s}{\sqrt{n}}, \bar{x} + z \frac{s}{\sqrt{n}} ) \) as an interval estimate for \( \mu \), the population mean, where z is the appropriate quantile for the standard normal distribution
- Use simulation to illustrate variations in confidence intervals between samples and to show that most but not all confidence intervals contain \( \mu \)
- Use \( \bar{x} \) and s to estimate \( \mu \) and \( \sigma \), to obtain approximate intervals covering desired proportions of values of a normal random variable and compare with an approximate confidence interval for \( \mu \)
- Collect data and construct an approximate confidence interval to estimate a mean and to report on survey procedures and data quality