Lessons · A/L Combined Mathematics

Unit 1 · Real Numbers

• Number systems: N, Z, Q and R
• Proving a number is irrational
• Surds and rationalising the denominator
• Sets and intervals
• Properties of real numbers

Unit 2 · Functions

• Relations and functions: domain, codomain and range
• One-one, onto and bijective functions
• Composite functions
• Inverse functions
• Graphs of basic functions and transformations
• Even and odd functions, and piecewise functions

Unit 3 · Quadratic Functions and Quadratic Equations

• Completing the square, vertex form and the graph
• Roots of a quadratic equation and the discriminant
• Relations between roots and coefficients
• Sign of a quadratic expression and quadratic inequalities
• Common roots of two quadratic equations

Unit 4 · Polynomials

• Polynomials and long division
• Remainder theorem, factor theorem and cubics
• Identities and comparing coefficients

Unit 5 · Rational Functions and Partial Fractions

• Partial fractions with linear factors
• Partial fractions with quadratic factors and improper fractions

Unit 6 · Index and Logarithm Laws

• Laws of indices
• Logarithms and the laws of logarithms
• Changing the base and solving exponential equations

Unit 7 · Inequalities

• Inequalities: properties, linear and quadratic
• Rational inequalities and sign tables
• The modulus and modulus equations
• Modulus inequalities
• Proving inequalities and the AM-GM inequality

Unit 8 · Angular Measure

• Radian measure, circular functions and their graphs

Unit 9 · Circular Functions

• Solving trigonometric equations: particular and general solutions

Unit 10 · Trigonometric Identities

• Trigonometric identities, compound and multiple angles
• Sum-to-product formulae and the form R cos(x − α)

Unit 11 · Sine Rule and Cosine Rule

• The sine rule, the cosine rule and solving triangles

Unit 12 · Inverse Trigonometric Functions

• Inverse trigonometric functions and principal values

Unit 13 · Limits

• The idea of a limit
• Limit theorems and the sandwich theorem
• Standard limits: $\dfrac{x^n - a^n}{x - a}$ and $\dfrac{\sin x}{x}$
• One-sided limits and continuity
• Limits at infinity and asymptotes

Unit 14 · Derivatives

• The derivative from first principles and the tangent
• Product and quotient rules and the derivatives of trigonometric functions
• The chain rule and the derivative of an inverse function
• Derivatives of exponential, logarithmic and inverse trigonometric functions
• Implicit and parametric differentiation
• Higher derivatives

Unit 15 · Applications of Derivatives

• Increasing and decreasing functions and stationary points
• Maxima and minima: the first and second derivative tests
• Concavity and points of inflexion
• Curve sketching with asymptotes
• Optimisation problems

Unit 16 · Integration

• Integration as the reverse of differentiation: antiderivatives and standard integrals
• Integrating rational functions and trigonometric expressions
• Integration by substitution
• Integration by parts
• Definite integrals and the fundamental theorem of calculus
• Areas and volumes of revolution

Unit 17 · The Rectangular Cartesian System

• Distance, section formula and area of a triangle

Unit 18 · The Straight Line

• Gradient and the equation of a straight line
• Angle between lines, parallel and perpendicular lines, distance from a point
• Intersection of lines, concurrency and the family of lines $l_1 + \lambda l_2 = 0$
• Angle bisectors and simple loci

Unit 19 · Principle of Mathematical Induction

• The principle of mathematical induction
• Proving summation formulae by induction
• Divisibility results by induction
• Inequalities and other results by induction

Unit 20 · Finite Series

• Sequences, series and sigma notation
• Arithmetic progressions
• Geometric progressions
• The sums $\sum r$, $\sum r^2$ and $\sum r^3$

Unit 21 · Infinite Series

• Summing series by differences and partial fractions
• Proving series results by mathematical induction
• Infinite series: convergence and the sum to infinity

Unit 22 · Binomial Expansion

• The binomial theorem for a positive integer index
• The general term and finding a required term
• Particular terms: constant term, middle terms and products
• Relations between binomial coefficients

Unit 23 · Complex Numbers

• Complex numbers: algebra and the conjugate
• Modulus, the Argand diagram and polar form
• Operations on the Argand diagram and the triangle inequality
• De Moivre's theorem and its uses
• Loci in the Argand diagram

Unit 24 · Permutations and Combinations

• Counting principles and factorials
• Permutations and arrangements
• Combinations and selections
• Selections with conditions

Unit 25 · Matrices

• Matrices: notation, order, types, equality, addition and scalar multiplication
• Matrix multiplication and its properties
• The transpose and symmetric matrices
• The determinant and inverse of a 2 × 2 matrix
• Solving linear systems and matrix equations

Unit 26 · The Circle: Equations

• The equation of a circle

Unit 27 · The Circle: Geometric Properties

• A point or a line and a circle
• Tangents, normals and the chord of contact
• Two circles: position, common chord and touching
• Orthogonal circles and families of circles

Unit 28 · Vectors (Combined Maths II)

• Vectors, addition and components
• Position vectors and displacement
• The ratio theorem
• Parallel vectors, collinear points and vector proofs
• The scalar (dot) product
• Projections and geometry with the dot product

Unit 29 · Coplanar Force Systems

• Forces as vectors: resultant and resolution
• Equilibrium of a particle: triangle of forces and Lami's theorem
• Moment of a force about a point
• Parallel forces and couples
• Reduction of a force system and equilibrium of a rigid body
• Friction: limiting equilibrium on rough surfaces and ladders
• Jointed rods: reactions at smooth hinges
• Frameworks of light rods: method of joints and Bow's notation
• Centre of mass of particles, laminas and composite bodies
• Centre of mass by integration: arcs, sectors, cones and hemispheres
• Suspended bodies, toppling and stability

Unit 30 · Motion

• Motion in a straight line: displacement, velocity, acceleration and graphs
• Uniform acceleration and motion under gravity
• Motion in a plane: position, velocity and acceleration vectors
• Relative motion: relative velocity, closest approach and interception
• Projectile motion
• Newton's laws of motion: connected particles, inclines and lifts
• Work, energy and power
• Impulse, momentum and direct collisions
• Elastic strings and springs: Hooke's law and elastic energy
• Motion in a horizontal circle and the conical pendulum
• Motion in a vertical circle
• Simple harmonic motion

Unit 31 · Probability

• Random experiments, sample space and events
• Probability, the axioms and the addition rule
• Conditional probability, the multiplication rule and tree diagrams
• Independent events
• Total probability and Bayes' theorem

Unit 32 · Statistics

• Data, frequency tables and the mean, median, mode and harmonic mean
• Grouped data: mean, median, mode and the coded mean
• Quartiles, percentiles and the box plot
• Measures of dispersion: range, variance, standard deviation and the z-score
• Skewness: the shape of a distribution