Year 12 Mathematics

Students will explore the fundamental concepts of mathematical proof, distinguishing between conjecture and proven statements.

Mastering formal methods of proving mathematical statements through deduction, exhaustion, and counter-example.

Students will learn to construct proofs by contradiction and effectively use counter-examples to disprove statements.

Review and extend skills in manipulating algebraic expressions, including fractions and surds.

Deep dive into quadratic functions, including completing the square, the quadratic formula, and discriminant analysis.

Students will learn polynomial division and apply the factor and remainder theorems to solve polynomial equations.

Analyzing the properties of higher degree polynomials and the relationship between algebraic factors and graphical intercepts.

Introduction to different types of functions, domain, range, and inverse functions.

Exploring the composition of functions and understanding their domains and ranges.

Investigating translations, reflections, and stretches of functions and their impact on graphs.

Review of straight-line equations, parallel and perpendicular lines, and distance/midpoint formulas.

Extending linear geometry to circular paths and exploring the properties of tangents and normals.

Solving simultaneous equations involving lines and curves, including circles and parabolas.

Understanding the definition and properties of the modulus function and solving equations/inequalities involving it.

Decomposing rational expressions into simpler fractions for integration and other applications.

Solving linear and quadratic inequalities, including those involving rational expressions and graphs.

Developing the concept of the derivative as a limit and its application in finding gradients of curves.

Understanding the formal definition of the derivative using limits.

Applying standard rules for differentiating polynomials, powers, and sums/differences of functions.

Finding equations of tangents and normals to curves at specific points.

Identifying and classifying stationary points (maxima, minima, points of inflection) using first and second derivatives.

Applying differentiation to solve real-world problems involving maximizing or minimizing quantities.

Solving problems involving rates of change in various contexts, including related rates.

Understanding integration as the inverse of differentiation and its use in calculating areas under curves.

Applying standard rules for indefinite integration of polynomials and powers.

Calculating definite integrals and using them to find the area under a curve and between curves.

Applying definite integration to calculate areas in more complex scenarios, including areas below the x-axis.

Using the trapezium rule to estimate the area under a curve when analytical integration is not possible.

Applying differentiation rules to sine, cosine, and tangent functions.

Applying vectors to describe positions of points and displacements between them.

Using vectors to prove geometric properties and solve problems in geometry.

Understanding the scalar product and its applications, including finding angles between vectors and perpendicularity.

Generalizing trigonometry beyond right-angled triangles using the unit circle and introducing radian measure.

Analyzing the properties of sine, cosine, and tangent graphs, including amplitude, period, and phase shift.

Deriving and applying identities to simplify expressions and solve trigonometric equations.

Solving trigonometric equations within a given range using identities and inverse functions.

Deriving and applying formulae for sin(A±B), cos(A±B), and tan(A±B).

Deriving and applying formulae for sin(2A), cos(2A), and tan(2A).

Expressing Acosθ + Bsinθ in the form Rcos(θ±α) or Rsin(θ±α) and its applications.

Investigating the function e^x and its inverse, the natural logarithm.

Applying the laws of logarithms to simplify expressions and solve equations.

Solving equations involving exponential and logarithmic functions.

Using trigonometric functions to model periodic phenomena in real-world contexts.

Applying differentiation rules to functions involving e^x and ln(x).

Applying integration rules to functions involving e^x and 1/x.

Evaluating different sampling techniques and their impact on the validity of statistical conclusions.

Using various graphical methods to represent data and drawing conclusions from them.

Evaluating different sampling techniques and their impact on the validity of statistical conclusions.

Modeling scenarios with two possible outcomes and calculating probabilities of success over multiple trials.

Using probability distributions to make decisions about the validity of a null hypothesis.

Understanding basic probability rules, Venn diagrams, and conditional probability.

Defining and working with discrete random variables, probability distributions, and expected values.

Deriving and applying the equations of motion for particles moving in a straight line.

Applying SUVAT equations to objects moving under constant gravitational acceleration.

Investigating the relationship between force, mass, and acceleration using vector diagrams.

Resolving forces into perpendicular components and applying Newton's laws to inclined planes.

Understanding the concept of friction and its role in motion, including static and dynamic friction.