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Mathematics HL
1 · Number and algebra
Sequences and series
Recover an arithmetic sequence • Geometric growth and accumulated change • Partial sums and mixed progressions
Exponents and logarithms
Change base and combine • Substitute to expose algebra • Growth, decay and comparison
Counting
Selection or arrangement • Together and separated • Complement and disjoint cases
Binomial expansions
Target a power or constant term • Compare unknown coefficients • Products and approximations
Mathematical induction
Induction for sums and recurrences • Divisibility and inequalities • Direct proof, contradiction and counterexamples
Complex numbers: Cartesian form
Simplify a quotient • Unknown components and realness • Polynomial roots and loci
Complex numbers: polar form
Convert and calculate powers • Find every nth root • Geometry and real powers
Linear systems
Solve a numerical system • Find exceptional parameter values • Describe infinitely many solutions
Mathematics HL
2 · Functions
Functions and inverses
Form a composition • Find the inverse and select a branch • Domain and range through operations
Quadratic functions
Root number and tangency • Root signs and transformed roots • Quadratic substitutions and global positivity
Polynomial functions
Factors and remainders • Repeated roots and graph behaviour • Root relationships without explicit roots
Rational functions
Sketch from structure • Solve inequalities safely • Inverse, reciprocal and range
Graph transformations
Map points through scales and shifts • Absolute-value transformations • Reciprocals and intersections
Inequalities
Polynomial and rational sign charts • Modulus inequalities • Substitution and domain intersection
Mathematics HL
3 · Geometry and trigonometry
Radians and sectors
A sector from area, arc or perimeter • Segments and overlaps • Composite boundaries and folded sectors
Trigonometric graphs
Recover a sinusoidal model • Starting at the bottom or crossing a level • Period, tangent and transformed graphs
Trigonometric identities and equations
Reduce to a quadratic • Solve on a transformed interval • Prove identities and combine sine/cosine
Inverse trigonometric functions
Principal values and graphs • Sum and difference of inverse angles • Equations and exact triangle values
Triangle trigonometry
Choose sine or cosine rule • Ambiguous triangles • Area, exact values and bearings
Vector products
Dot product for angles and perpendicularity • Cross product for area and normals • Division, projection and shortest distance
Vector lines
Construct and intersect lines • Distances and common perpendiculars • Motion and closest approach
Vector planes
Construct a plane and intersect it • Angles and common lines • Perpendicular distance and reflection
Mathematics HL
4 · Statistics and probability
Descriptive statistics
Means, variance and missing values • Cumulative data, quartiles and outliers • Regression and prediction
Probability
Venn regions and independence • Trees and reversed conditions • Changing counts and repeated games
Discrete random variables
Complete a distribution • Moments and winnings • Build a maximum or waiting-time distribution
Binomial distribution
Translate the tail • Recover parameters • Condition or find the minimum trial count
Normal distribution
Direct and conditional probabilities • Inverse quantiles and unknown parameters • A new experiment based on a threshold
Continuous random variables
Normalise the density • Choose the correct statistic • Conditional events and new scores
Mathematics HL
5 · Calculus
Limits and supplied models
An indeterminate quotient • A leading-term limit • First-principles differentiation
Differentiation rules
Chain rules and nested functions • Products, quotients and tables • Inverse derivatives and identities
Implicit differentiation
Collect all derivative terms • Horizontal and vertical tangents • Normal, branch or second derivative
Tangents and normals
Known point or prescribed slope • A line through an external point • Common tangents and angles
Stationary points and curve features
Stationary points and monotonic intervals • Inflection, asymptotes and range • Parameters and numbers of solutions
Optimisation
Eliminate a dimension • Distance, time or cost • Prove the optimum
Related rates
One dimension determines the shape • Several dimensions change • Distance and angular motion
Integration methods
Reverse chain rule and substitution • Integration by parts • Constants, conditions and exact evaluation
Areas and definite integrals
Area between curves • Area versus signed integral • Accumulation and unknown boundaries
Volumes of revolution
Discs about a coordinate axis • Washers and holes • Inverse or composite solids
Calculus of motion
Move between position, velocity and acceleration • Distance and maximum speed • Velocity given as a function of position
Differential equations
Separation and initial values • Homogeneous or linear forms • Euler approximation and solution checks
Maclaurin series
Coefficients from derivatives • Products and compositions • Approximate integrals and leading-term limits
Mathematics HL
Extra · Legacy and extension
Poisson distribution: legacy extension
Scale the parameter • Recover the rate • Condition and count qualifying intervals
Legacy matrices and current linear systems
Current linear systems by elimination • Legacy inverse and singularity methods • Legacy transformations and transferable coordinates
Legacy Statistics option
Sampling means and confidence intervals • Compare means with a test • Goodness-of-fit and independence
Legacy Sets, Relations and Groups option
Sets and equivalence relations • Verify a group operation • Orders, subgroups and maps
Legacy Discrete Mathematics option
Euclidean algorithm and integer equations • Choose the correct graph task • Solve a linear recurrence
Legacy Analysis and Approximation option
Numerical iteration • Quadrature with a guaranteed bound • Series convergence and remainders
Legacy Geometry option
Build a locus from distances or midpoints • Tangent and normal to a conic • Ratios, concurrency and circle reasoning
Physics HL
A · Space, time and motion
Kinematics
Separate a projectile into two motions • Read a graph as a physical operation • Choose the constant-acceleration relationship
Forces and momentum
Build a force balance for the correct object • Keep collision momentum separate from energy • Supply the required inward resultant
Work, energy and power
Account for work and dissipative transfer • Turn energy into useful and input power • Use flow energy and graph areas carefully
Rigid body mechanics (HL)
Choose the axis and use perpendicular lever arms • Conserve angular momentum over the correct interval • Share energy between rolling and translation
Galilean and special relativity (HL)
Identify proper time and proper length • Transform events instead of guessing simultaneity • Add velocities with a defined sign convention
Physics HL
B · Particulate nature of matter
Thermal energy transfers
Split a heating or cooling process into stages • Compare conduction paths and heat-loss biases • Use spectrum and radiated-power information together
Greenhouse effect
Write the energy balance for the stated model • Explain infrared absorption as a causal chain • Track a change in albedo or emissivity
Gas laws
Use the gas law with a controlled amount • Turn a linear graph into a physical constant • Explain pressure and internal energy microscopically
Thermodynamics (HL)
Apply the first law one process at a time • Read path shapes and cycle area • Check efficiency and total entropy
Current and circuits
Reduce the network by nodes, then recover currents • Include internal resistance and operating conditions • Connect wire geometry, heating and uncertainty
Physics HL
C · Wave behaviour
Simple harmonic motion
Read the oscillator from its graph or equation • Use energy fractions without losing the reference point • Establish the restoring law before using the period
Wave model
Extract wavelength, period and speed from axes • Infer particle motion from a travelling profile • Link the wave model to particle SHM and energy
Wave phenomena
Use the correct angles at a boundary • Convert path difference into interference conditions • Read a diffraction pattern and count orders
Standing waves and resonance
Fit allowed modes to the boundaries • Explain formation, phase and imperfect nodes • Read damping and resonance as energy behaviour
Doppler effect
Separate moving-source and moving-observer effects • Extract radial speed from a spectral line • Treat reflection and changing motion in stages
Physics HL
D · Fields
Gravitational fields
Add fields and potentials in different ways • Derive orbit relationships from the inward force • Use escape energy and scaling arguments
Electric and magnetic fields
Superpose electric fields and potentials • Connect charge, potential and conductor behaviour • Use magnetic direction rules with conventional current
Motion in electromagnetic fields
Turn potential energy into motion in an electric field • Use magnetic curvature and its direction • Balance crossed fields or forces on currents
Induction (HL)
Differentiate flux and distinguish angle conventions • Use Lenz’s law to determine current and force • Connect motional emf to energy and braking
Physics HL
E · Nuclear and quantum physics
Structure of the atom
Link scattering evidence to nuclear structure • Choose a transition from an energy-level diagram • Apply nuclear size and simple atomic-model scaling
Quantum physics (HL)
Use the photoelectric energy balance • Calculate matter wavelength from momentum • Account for photon energy and momentum in scattering
Radioactive decay
Balance a nuclear decay equation • Use exponential decay after correcting the measurement • Calculate nuclear energy with consistent masses
Fission
Balance the reaction and identify the released energy • Distinguish moderation from control • Connect a steady chain reaction to power and fuel
Fusion and stars
Build distance and luminosity from observations • Read the HR diagram as luminosity and temperature • Calculate fusion yield and interpret stellar lifetimes
Physics HL