Year 13 Mathematics

Mastering the logic of mathematical deduction to prove algebraic identities and properties for all real numbers.

Applying mathematical deduction to prove statements involving geometric properties and theorems.

Exploring proof by exhaustion for finite cases and disproving statements using counterexamples.

Mastering the technique of proof by contradiction to establish the truth of mathematical statements.

Decomposing rational expressions with distinct linear factors in the denominator to enable integration.

Extending partial fraction decomposition to include repeated linear and irreducible quadratic factors.

Expanding expressions of the form (a+bx)^n where n is a rational number, and determining validity.

Investigating the formation of composite functions and determining their valid domains and ranges.

Exploring the conditions for existence of inverse functions and their graphical relationship to original functions.

Solving equations and inequalities involving the modulus function and interpreting their graphical representations.

Analyzing and sketching graphs of various functions, including transformations (translations, stretches, reflections).

Using graphical methods to find approximate solutions to equations, including intersections of curves.

Exploring the properties, graphs, and applications of exponential and logarithmic functions.

Analyzing secant, cosecant, and cotangent functions, including their graphs and fundamental identities.

Understanding the definitions, domains, and ranges of arcsin, arccos, and arctan functions.

Calculating and applying the derivatives of secant, cosecant, and cotangent functions.

Finding and applying the derivatives of arcsin, arccos, and arctan functions.

Deriving and applying identities for sums and differences of angles (sin(A±B), cos(A±B), tan(A±B)).

Applying double angle identities and exploring their use in deriving half-angle identities and solving equations.

Differentiating equations where variables are linked indirectly through a parameter, using the chain rule.

Differentiating equations where variables cannot be easily isolated, such as circular or elliptical relations.

Calculating the second derivative for parametrically and implicitly defined functions to determine concavity.

Applying differentiation to solve problems involving rates of change in various contexts.

Developing strategies for integrating composite functions by changing the variable of integration.

Extending 2D vector concepts into the third dimension using i, j, k notation and performing basic operations.

Calculating the scalar product of two vectors and using it to find angles between vectors and test for perpendicularity.

Expressing lines in 3D using vector and parametric forms, understanding position and direction vectors.

Converting between vector/parametric and Cartesian forms of a line's equation in 3D.

Determining if two lines in 3D are parallel, intersecting, or skew, and finding intersection points.

Calculating the shortest distance from a point to a line in 3D using vector methods.

Determining the shortest distance between two non-parallel, non-intersecting lines in 3D space.

Representing planes in 3D space using vector equations, including normal vector and position vector.

Using tree diagrams and formulas to solve complex probability problems involving conditional events.

Using Venn diagrams to visualize and calculate conditional probabilities and test for independence.

Understanding the characteristics of the Normal distribution, including its parameters and symmetry.

Using the standard normal distribution (Z-distribution) and Z-scores for probability calculations and comparisons.

Understanding when and how to use the Normal distribution as an approximation for the Binomial distribution.

Calculating and interpreting the product moment correlation coefficient (PMCC) as a measure of linear association.

Modeling the path of objects moving under gravity in two dimensions, neglecting air resistance.

Solving complex projectile motion problems involving inclined planes or targets at different heights.

Analyzing forces on objects on rough horizontal surfaces, including static and kinetic friction.

Analyzing forces on objects on rough inclined planes, considering components of gravity and friction.

Defining complex numbers, the imaginary unit 'i', and performing basic arithmetic operations.

Representing complex numbers geometrically on the Argand diagram and converting to polar form.

Performing multiplication and division of complex numbers using their modulus-argument forms.

Calculating definite integrals to find the area bounded by curves and axes or between two curves.

Using integration to calculate the volume of solids formed by revolving a region around an axis.

Calculating the length of a curve and the surface area of a solid of revolution using integration.

Approximating definite integrals using the trapezium rule and understanding its accuracy.

Defining discrete random variables and their probability distributions, including probability mass functions.

Calculating the expectation (mean) and variance of discrete random variables.

Understanding the properties of the continuous uniform distribution and calculating probabilities.

Calculating work done by constant and variable forces, including work done against resistance.

Defining and calculating kinetic energy, gravitational potential energy, and elastic potential energy.