Free IGCSE, A Level and IB notes made to be understood, not just memorised.

Night-before sheets 2

Formulas, definitions, practical skills and examiner traps, condensed for the last days before the exam.

• Formula sheetMaths Formula & Method SheetEvery formula you must be able to write down from memory, drawn where a picture helps, plus the first move for the most common question types. One sheet for the night before.ReadPDF
• Exam trapsMaths Exam Traps & Last-Night ChecklistThe 30 mistakes that lose the most marks in IGCSE Maths, grouped by topic, plus command words, calculator habits and a checklist for the night before.ReadPDF

Chapter notes 12

1. 01Number toolkitTypes of number · Primes, HCF & LCM · Sets · Fractions · Rounding & bounds · Indices · Standard form · Surds20 minPDF
2. 02Ratio, percentages & moneyRatio · Proportion · Rates & speed · Percentages · Reverse percentage · Interest · Growth & decay21 minPDF
3. 03Algebraic manipulationExpanding · Factorising · Algebraic fractions · Rearranging formulae · Substitution18 minPDF
4. 04Equations & inequalitiesLinear equations · Simultaneous equations · Quadratic equations · Quadratic formula · Completing the square · Inequalities21 minPDF
5. 05Sequences, functions & graphsnth term · Functions & inverse · Straight-line & curve graphs · Gradient of curves · Differentiation18 minPDF
6. 06Coordinate geometryGradient · y = mx + c · Midpoint & length · Parallel & perpendicular lines25 minPDF
7. 07Geometry & circle theoremsAngle facts · Polygons · Similarity & congruence · Symmetry · Circle theorems31 minPDF
8. 08Mensuration — area & volumeUnits · Perimeter & area · Circles, arcs & sectors · Surface area · Volume of prisms, cylinders, cones, spheres28 minPDF
9. 09Pythagoras & trigonometryPythagoras · SOHCAHTOA · Bearings · Angles of elevation · Sine & cosine rules · Area ½ab sin C · 3D · Trig graphs34 minPDF
10. 10Transformations & vectorsReflection · Rotation · Translation · Enlargement · Vectors · Magnitude · Vector geometry29 minPDF
11. 11ProbabilityProbability scale · Relative & expected frequency · Combined events · Tree diagrams · Conditional probability21 minPDF
12. 12StatisticsAverages & range · Frequency tables · Charts · Scatter diagrams · Cumulative frequency · Histograms34 minPDF

Night-before sheets 2

Formulas, definitions, practical skills and examiner traps, condensed for the last days before the exam.

• Formula sheetIGCSE Physics Formula SheetEvery equation with formula triangles, units and worked examples, Core + Extended.Download PDF
• ATP shortcutsPhysics ATP Shortcuts (Paper 6)Precautions, sources of error, control variables and the 7-mark plan structure, practical by practical.Download PDF

Chapter notes 12

1. 01Measurement & motionMeasuring instruments · Scalars & vectors · Speed & velocity · Acceleration · Distance–time & speed–time graphs · Free fall & terminal velocity10 minPDF
2. 02Mass, weight & densityMass vs weight · Gravitational field strength · Density · Measuring density · Floating & sinking7 minPDF
3. 03Forces, moments & momentumResultant force · Springs & Hooke's law · Friction & drag · F = ma · Circular motion · Moments · Centre of gravity · Momentum & impulse10 minPDF
4. 04Energy, work & powerEnergy stores & transfers · Conservation of energy · Kinetic & gravitational energy · Work done · Power · Efficiency & Sankey diagrams · Energy resources11 minPDF
5. 05PressurePressure = force ÷ area · Everyday examples · Liquid pressure, depth & density · Δp = ρgΔh7 minPDF
6. 06Thermal physicsParticle model · Changes of state · Brownian motion · Absolute zero · Gas pressure & pV = constant · Thermal expansion · Specific heat capacity · Evaporation · Conduction, convection, radiation13 minPDF
7. 07Waves & soundWave terms · Transverse & longitudinal waves · v = fλ · Reflection, refraction, diffraction · Sound · Speed of sound · Echoes & ultrasound10 minPDF
8. 08Light & the EM spectrumPlane mirrors · Refraction & critical angle · Optical fibres · Dispersion · Converging lenses · Sight correction · Electromagnetic spectrum · Digital & analogue signals11 minPDF
9. 09Electricity & circuitsStatic charge · Current & charge · e.m.f. & p.d. · Resistance & V = IR · I–V graphs · Series & parallel · Potential divider · Power & kWh · Electrical safety13 minPDF
10. 10Magnetism & electromagnetismMagnetic poles & materials · Fields · Electromagnets · Relays & loudspeakers · Force on a current · d.c. motor · Induction · a.c. generator · Transformers · Power transmission11 minPDF
11. 11Nuclear physicsAtomic structure · Nuclide notation & isotopes · Background radiation · Alpha, beta, gamma · Decay equations · Half-life · Uses & safety · Fission & fusion11 minPDF
12. 12Space physicsEarth, Sun & Moon · Seasons & phases · Solar System · Orbits · Stars & life cycle · Light-years · Redshift · Big Bang & Hubble constant12 minPDF

Night-before sheets 2

Formulas, definitions, practical skills and examiner traps, condensed for the last days before the exam.

• Night-before sheetChemistry Facts, Equations & TestsEvery test and colour, the mole calculations, key equations, industrial processes and the reactivity series on one sheet. Learn the tests word for word, because examiners mark the exact observation.ReadPDF
• ATP shortcutsChemistry ATP Shortcuts (Paper 6)The practical skills sheet: apparatus and readings, titrations, collecting gases, rates and energy, chromatography, planning and evaluating. Written for the practical and alternative-to-practical papers.ReadPDF

Chapter notes 12

1. 01States of matterSolids, liquids, gases · Changes of state · Heating curves · Gas pressure · Diffusion12 minPDF
2. 02Atoms, elements, compounds & bondingAtomic structure · Isotopes · Electronic configuration · Ionic bonding · Covalent bonding · Giant structures · Metallic bonding18 minPDF
3. 03Stoichiometry — formulae & molesFormulae · Balancing equations · Mr · The mole · Reacting masses · Gas volumes · Concentration · Empirical formula · % yield19 minPDF
4. 04ElectrochemistryElectrolysis · Products at electrodes · Electroplating · Ionic half-equations · Fuel cells15 minPDF
5. 05Chemical energeticsExothermic & endothermic · Energy profiles · ΔH · Activation energy · Bond energies9 minPDF
6. 06Rates, equilibrium & redoxPhysical vs chemical change · Rate factors · Collision theory · Catalysts · Reversible reactions · Haber & Contact processes · Redox17 minPDF
7. 07Acids, bases & saltspH & indicators · Reactions of acids · Oxides · Making salts · Solubility rules14 minPDF
8. 08The Periodic TableGroups & periods · Group I · Group VII · Transition elements · Noble gases · Trends12 minPDF
9. 09MetalsProperties · Alloys · Reactivity series · Corrosion · Extraction of iron & aluminium14 minPDF
10. 10Chemistry of the environmentWater tests & treatment · Fertilisers · Air composition · Pollutants · Climate change · Acid rain12 minPDF
11. 11Organic chemistryHomologous series · Alkanes · Alkenes · Isomers · Alcohols · Carboxylic acids · Esters · Polymers · Fuels & crude oil14 minPDF
12. 12Experimental techniques & chemical analysisApparatus · Separation methods · Chromatography · Purity · Tests for ions & gases15 minPDF

Night-before sheets 2

Formulas, definitions, practical skills and examiner traps, condensed for the last days before the exam.

• Night-before sheetBiology Definitions & Key FactsThe syllabus definitions in exam wording, plus the equations, tests, processes and numbers that come up every year. Learn the definitions word for word.ReadPDF
• ATP shortcutsBiology ATP Shortcuts (Paper 6)The practical skills sheet: biological drawings, magnification, variables, tables and graphs, the classic practicals and how to plan and evaluate. Written for the practical and alternative-to-practical papers.ReadPDF

Chapter notes 15

1. 01Characteristics & classificationMRS GREN · Species & binomial names · Dichotomous keys · Kingdoms · Vertebrates & arthropods · Viruses12 minPDF
2. 02Cells & organisationAnimal & plant cells · Bacteria · Specialised cells · Tissues & organs · Magnification9 minPDF
3. 03Movement into & out of cellsDiffusion · Osmosis · Water potential · Turgor · Active transport12 minPDF
4. 04Biological molecules & enzymesCarbohydrates, fats, proteins · Food tests · DNA · Enzymes · Temperature & pH11 minPDF
5. 05Plant nutritionPhotosynthesis · Limiting factors · Leaf structure · Mineral ions11 minPDF
6. 06Human nutritionBalanced diet · Digestive system · Digestion & enzymes · Absorption · Teeth10 minPDF
7. 07Transport in plantsXylem & phloem · Water uptake · Transpiration · Factors · Translocation11 minPDF
8. 08Transport in animalsDouble circulation · Heart · Coronary heart disease · Blood vessels · Blood components12 minPDF
9. 09Diseases, immunity & drugsPathogens · Transmission · Body defences · Antibodies · Vaccination · Antibiotics · Drug misuse14 minPDF
10. 10Gas exchange, respiration & excretionLungs & alveoli · Breathing · Aerobic & anaerobic respiration · Kidneys · Liver19 minPDF
11. 11Coordination & responseNervous system · Reflex arc · Eye · Hormones · Homeostasis · Tropisms17 minPDF
12. 12ReproductionAsexual & sexual · Flowers & pollination · Germination · Human reproductive systems · Menstrual cycle · STIs21 minPDF
13. 13Inheritance, variation & selectionChromosomes & genes · Mitosis & meiosis · Monohybrid crosses · Variation · Mutation · Natural & artificial selection24 minPDF
14. 14Ecology & human influencesFood chains & webs · Energy pyramids · Carbon & nitrogen cycles · Populations · Food supply · Pollution · Conservation30 minPDF
15. 15Biotechnology & genetic modificationBacteria & yeast · Enzymes in industry · Fermenters · Genetic modification18 minPDF

Exam specials 1

Built from the Cambridge examiner reports and mark schemes.

• Exam guideA Level Maths 9709: Exam Technique and the MF19 Formula ListHow 9709 is examined: the paper routes, timings and marks, the accuracy rules (3 s.f., angles to 1 d.p.), what the MF19 formula list gives you and what you must learn, calculator rules, and the errors the examiner reports flag most.ReadPDF

Topic notes 32

1. 1.1QuadraticsCompleting the square · Vertex & range · Discriminant · Solving quadratics · Quadratic inequalities · Line meets curve · Quadratics in disguiseP113 minPDF
2. 1.2Functions and transformationsDomain & range · One-one functions · Inverse functions · Composite functions · Translations · Stretches · Reflections · Combined transformationsP111 minPDF
3. 1.3Coordinate geometry: lines and circlesGradient & line equations · Parallel & perpendicular · Midpoint & distance · Perpendicular bisector · Circle equation · Centre & radius · Tangents · Line meets circleP18 minPDF
4. 1.4Circular measure: radians, arcs and sectorsThe radian · Degrees ↔ radians · Arc length s = rθ · Sector area ½r²θ · Segment area · Composite shapes · Exact answersP16 minPDF
5. 1.5Trigonometry: graphs, identities and equationsTrig graphs & transformations · Exact values · CAST & related angles · Principal values · tan θ = sin θ/cos θ · sin²θ + cos²θ = 1 · Proving identities · Solving equationsP110 minPDF
6. 1.6Series: binomial expansion, AP and GPBinomial expansion · ⁿCᵣ & Pascal · Specific coefficients · Term independent of x · Arithmetic progressions · Geometric progressions · Sum to infinity · Linked AP & GPP18 minPDF
7. 1.7DifferentiationGradient as a limit · Power rule · Chain rule · Tangents & normals · Increasing & decreasing · Stationary points · Second derivative · Optimisation · Connected rates of changeP19 minPDF
8. 1.8IntegrationReverse differentiation · ∫(ax + b)ⁿ dx · Constant of integration · Definite integrals · Improper integrals · Area under a curve · Area between curves · Volumes of revolutionP18 minPDF
9. 3.1Algebra: modulus, polynomials and partial fractionsModulus graphs · Modulus equations & inequalities · Polynomial division · Factor theorem · Remainder theorem · Partial fractions · Binomial expansion (rational n) · Range of validityP311 minPDF
10. 3.2Logarithms and exponential functionsLogs & indices · Laws of logarithms · eˣ and ln x · Graphs of e^{kx} · Index equations · Inequalities · Hidden quadratics · Linearising with logsP38 minPDF
11. 3.3Further trigonometrysec, cosec, cot · Pythagorean identities · Addition formulae · Double-angle formulae · Exact values · Proving identities · R sin(θ + α) · Solving equations · Max & minP38 minPDF
12. 3.4Further differentiationDerivatives of eˣ & ln x · Trig derivatives · tan⁻¹x · Chain rule · Product rule · Quotient rule · Implicit differentiation · Parametric differentiation · Tangents & normalsP36 minPDF
13. 3.5Further integrationStandard integrals · Trig identities · Partial fractions · f′(x)/f(x) · Integration by parts · Substitution · Trapezium rule · Over/under-estimatesP38 minPDF
14. 3.6Numerical solution of equationsChange of sign · Sketching to count roots · Rearranging to x = F(x) · Iteration · Staircase & cobweb diagrams · Convergence · Stating accuracyP36 minPDF
15. 3.7VectorsPosition & direction vectors · Magnitude · Unit vectors · Scalar product · Angle between vectors · Vector equation of a line · Parallel, intersecting, skew · Foot of perpendicular · Shortest distanceP36 minPDF
16. 3.8Differential equationsForming equations · Proportionality · Separating variables · General solution · Particular solution · Exponential growth & decay · Partial fractions in DEs · Interpreting solutionsP35 minPDF
17. 3.9Complex numbersi and x + iy · Conjugate · Division · Modulus & argument · Argand diagram · Square roots · Conjugate-pair roots · Polar & exponential form · Multiplying & dividing · Loci & regionsP36 minPDF
18. 4.1Forces and equilibriumTypes of force · Force diagrams · Resolving · Resultant force · Equilibrium of a particle · Friction & μ · Limiting equilibrium · Inclined planes · Smooth & rough contactMech6 minPDF
19. 4.2Kinematics of motion in a straight lineDisplacement vs distance · Velocity vs speed · s–t and v–t graphs · Area & gradient · suvat equations · Vertical motion · Variable acceleration · Differentiating & integratingMech5 minPDF
20. 4.3MomentumMomentum · Conservation of momentum · Direct collisions · Coalescing particles · Signs & directions · Loss of kinetic energy · Further collisionsMech4 minPDF
21. 4.4Newton's laws of motionNewton's three laws · Resultant force = ma · Weight mg · Lifts · Inclined planes with friction · Car and trailer · Pulleys · Connected particles · After the string breaksMech5 minPDF
22. 4.5Energy, work and powerWork done · Kinetic energy · Potential energy · Work–energy principle · Work against friction · Conservation of energy · Power P = Fv · Maximum speed · HillsMech5 minPDF
23. 5.1Representation of dataStem-and-leaf · Box plots · Histograms & frequency density · Cumulative frequency · Median & quartiles · Mean & standard deviation · Coded data Σ(x − a) · Comparing distributionsS15 minPDF
24. 5.2Permutations and combinationsFactorials · Arrangements · Repeated letters · Together & apart · Permutations ⁿPᵣ · Combinations ⁿCᵣ · Restrictions · "At least" · Probability by countingS14 minPDF
25. 5.3ProbabilitySample spaces · Addition rule · Mutually exclusive · Independent events · Conditional probability · Tree diagrams · Without replacement · Reversed conditioning · Venn diagramsS14 minPDF
26. 5.4Discrete random variablesProbability distributions · Σp = 1 · E(X) · Var(X) · Binomial B(n, p) · Conditions for binomial · Geometric Geo(p) · Mean & variance formulae · "At least" questionsS14 minPDF
27. 5.5The normal distributionNormal curve · Standardising · Φ(z) tables · Inverse normal · Finding μ and σ · Symmetry · Normal approximation to binomial · Continuity correctionS14 minPDF
28. 6.1The Poisson distributionPo(λ) · Conditions · P(X = r) · Scaling λ · Mean = variance · Poisson approximation to binomial · Normal approximation to Poisson · Continuity correctionS24 minPDF
29. 6.2Linear combinations of random variablesE(aX + b) · Var(aX + b) · E(X ± Y) · Var(X ± Y) · 3X vs X₁ + X₂ + X₃ · Sums of normals · Sums of Poissons · Comparing two quantitiesS24 minPDF
30. 6.3Continuous random variablesProbability density function · Total area = 1 · Probabilities as areas · Sketching pdfs · E(X) · Var(X) · Median · Percentiles · ModeS23 minPDF
31. 6.4Sampling and estimationPopulations & samples · Random sampling · Distribution of X̄ · Central Limit Theorem · Unbiased estimates · s² with n − 1 · Confidence intervals for μ · Confidence intervals for pS24 minPDF
32. 6.5Hypothesis testsH₀ and H₁ · Significance level · One- & two-tailed · Critical regions · Binomial test · Poisson test · Normal mean test · Type I error · Type II error · Conclusions in contextS25 minPDF

Exam specials 2

Built from the Cambridge examiner reports and mark schemes.

• DefinitionsA Level Physics 9702: Definitions You Must KnowEvery definition 9702 asks for, AS and A Level, in one place: worded precisely enough to earn the mark, with the key words examiners look for and the near-misses that score zero.ReadPDF
• Practical guideA Level Physics 9702: Paper 3 Practical and Paper 5 Planning GuideHow to score on the practical papers: tables and significant figures, graphs and gradients, uncertainties, the limitations-and-improvements question on Paper 3, and planning, log graphs, error bars and worst-fit lines on Paper 5.ReadPDF

Topic notes 25

1. 01Physical quantities and unitsSI base units · Derived units · Homogeneity · Prefixes · Estimates · Systematic & random errors · Precision & accuracy · Uncertainties · Scalars & vectors · Resolving vectorsAS9 minPDF
2. 02KinematicsDistance & displacement · Speed & velocity · Acceleration · Motion graphs · Deriving suvat · Free fall · Measuring g · ProjectilesAS9 minPDF
3. 03DynamicsMass & inertia · Newton's laws · F = ma · Momentum · F = Δp/Δt · Weight · Drag & terminal velocity · Conservation of momentum · Elastic & inelastic collisions · 2D collisionsAS9 minPDF
4. 04Forces, density and pressureCentre of gravity · Moment of a force · Couple & torque · Principle of moments · Equilibrium · Vector triangles · Density · Pressure · Δp = ρgΔh · UpthrustAS8 minPDF
5. 05Work, energy and powerWork done · Conservation of energy · Efficiency · Power · P = Fv · ΔEp = mgΔh · Ek = ½mv² · Energy with friction · Maximum speedAS7 minPDF
6. 06Deformation of solidsLoad & extension · Hooke's law · Limit of proportionality · Stress & strain · Young modulus · Measuring E · Elastic & plastic · Elastic potential energyAS7 minPDF
7. 07WavesProgressive waves · Amplitude, phase, period · Oscilloscope · v = fλ · Intensity ∝ amplitude² · Transverse & longitudinal · Doppler effect · EM spectrum · Polarisation · Malus's lawAS8 minPDF
8. 08SuperpositionSuperposition · Stationary waves · Nodes & antinodes · Microwaves, strings, air columns · Diffraction · Interference & coherence · Double slit λ = ax/D · Diffraction gratingAS9 minPDF
9. 09ElectricityCharge carriers · Q = It · I = Anvq · Potential difference · Power · Resistance · Ohm's law · I–V characteristics · Resistivity · LDRs & thermistorsAS7 minPDF
10. 10D.C. circuitsCircuit symbols · E.m.f. vs p.d. · Internal resistance · Kirchhoff's first law · Kirchhoff's second law · Series & parallel · Potential dividers · Sensors · Potentiometer · Null methodAS9 minPDF
11. 11Particle physicsα-particle scattering · Nuclear model · Nucleon & proton number · Isotopes · α, β⁻, β⁺, γ · Antiparticles · Neutrinos · Decay equations · Quarks · Hadrons & leptonsAS7 minPDF
12. 12Motion in a circleRadians · Angular displacement · Angular speed ω · v = rω · Centripetal acceleration · Centripetal force · Vertical circles · Banked & flat bendsA25 minPDF
13. 13Gravitational fieldsField lines · Field strength g · Newton's law F = GMm/r² · g = GM/r² · Potential φ = −GM/r · Potential energy · Orbits · T² ∝ r³ · Geostationary orbitsA25 minPDF
14. 14TemperatureThermal equilibrium · Thermodynamic scale · Absolute zero · Kelvin & Celsius · Specific heat capacity · Specific latent heat · Electrical method · Heat lossesA25 minPDF
15. 15Ideal gasesAmount of substance · Avogadro constant · Ideal gas equation · Boltzmann constant · Kinetic theory assumptions · pV = ⅓Nm⟨c²⟩ · Mean kinetic energy · r.m.s. speedA24 minPDF
16. 16ThermodynamicsInternal energy · Random kinetic & potential energy · Temperature and internal energy · Work done W = pΔV · First law ΔU = q + W · Sign convention · Gas processesA24 minPDF
17. 17OscillationsAmplitude, period, frequency, phase · SHM definition a = −ω²x · x = x₀ sin ωt · v = ±ω√(x₀² − x²) · Energy in SHM · Light, critical & heavy damping · Forced oscillations · ResonanceA25 minPDF
18. 18Electric fieldsElectric field lines · Field strength E = F/q · Uniform field E = ΔV/Δd · Charged particles between plates · Coulomb's law · Field of a point charge · Electric potential · Potential energyA24 minPDF
19. 19CapacitanceCapacitance C = Q/V · Isolated sphere 4πε₀r · Series & parallel · Energy W = ½QV = ½CV² · Area under Q–V graph · Discharge Q = Q₀e^{−t/RC} · Time constant τ = RCA24 minPDF
20. 20Magnetic fieldsField patterns · F = BIL sin θ · Tesla · F = BQv · Circular orbits r = mv/BQ · Velocity selector · Hall voltage · Forces between wires · Magnetic flux · Faraday’s law · Lenz’s lawA26 minPDF
21. 21Alternating currentsPeriod, frequency, peak value · I = I₀ sin ωt · r.m.s. values · Mean power = ½I₀²R · Half-wave rectification · Full-wave bridge rectifier · Smoothing capacitor · RippleA23 minPDF
22. 22Quantum physicsPhoton energy E = hf · Photon momentum · Electronvolt · Photoelectric effect · Work function & threshold frequency · hf = Φ + ½mv²max · Electron diffraction · de Broglie λ = h/p · Energy levels · Line spectraA25 minPDF
23. 23Nuclear physicsE = mc² · Mass defect · Binding energy · Binding energy per nucleon · Fission & fusion · Random & spontaneous decay · Activity A = λN · Exponential decay · Half-life t½ = ln 2/λA25 minPDF
24. 24Medical physicsPiezoelectric transducers · Acoustic impedance Z = ρc · Intensity reflection coefficient · Coupling gel · Attenuation I = I₀e^{−μx} · X-ray production · Image sharpness & contrast · CT scanning · PET & annihilationA25 minPDF
25. 25Astronomy and cosmologyLuminosity · Radiant flux intensity F = L/4πd² · Standard candles · Wien's law · Stefan–Boltzmann law · Stellar radius · Redshift Δλ/λ = v/c · Hubble's law · Expanding universe · Big BangA24 minPDF

Exam specials 7

Built from the IB examiners' subject reports: grade boundaries, traps, paper strategy, booklets and the IA.

• Grade boundariesIB Maths AA Grade Boundaries (May 2025 & Nov 2024)Every grade boundary for IB Maths AA from the IB subject reports: May 2025 time zones 1–3 and November 2024, SL and HL, overall and paper by paper, and what it takes to get a 7.ReadPDF
• Exam trapsIB Maths AA Exam Traps: 57 Mistakes Examiners Flagged57 mistakes IB examiners flagged in the November 2024 and May 2025 Maths AA subject reports, grouped by topic, each with the fix, plus a last-week checklist.ReadPDF
• IA guideIB Maths IA Guide: How to Score Highly on the Exploration (AA & AI)A practical guide to the IB Maths exploration for AA and AI, SL and HL: what each criterion rewards, how to choose a topic with enough mathematics, a structure that works and the mistakes moderators flagged in the 2024–25 reports.ReadPDF
• Command terms & marksIB Maths Command Terms and Markscheme Codes (M1, A1, R1, AG, FT)What every IB Maths command term asks for, and how examiners award marks: M1, A1, R1, AG, FT, implied marks, misreads, "hence" vs "hence or otherwise", accuracy and the presentation rules that cost students marks.ReadPDF
• Paper 3 strategyIB Maths AA HL Paper 3: Strategy for the Problem-Solving PaperHow to approach the IB Maths AA HL Paper 3 investigations: the structure of the paper, timing, a step-by-step method for unfamiliar problems, a worked mini-investigation, and what the 2024–25 examiners said.ReadPDF
• Paper 1 guideIB Maths AA Paper 1: Non-Calculator Skills and Exact ValuesThe skills IB Maths AA Paper 1 tests without a calculator: exact trig values and the unit circle, logs, surds and indices, arithmetic shortcuts, sketching by hand, solving trig equations and the slips examiners see every session.ReadPDF
• Formula booklet guideIB Maths AA Formula Booklet: What Is Given and What to MemoriseWhat the IB Maths AA formula booklet (SL and HL) contains, topic by topic, and the formulas, values and facts it leaves out that you must memorise, plus how to use the booklet quickly in the exam.ReadPDF

Topic notes 16

1. T1Sequences, series and the binomial theoremArithmetic sequences · Geometric sequences · Sigma notation · Sum to infinity · Binomial theorem · ⁿCᵣ · Binomial series (HL)SL+HL8 minPDF
2. T1Exponents and logarithmsLaws of exponents · Logarithms · Natural log & e · Laws of logs · Change of base · Exponential equations · Exponential & log graphs · Growth & decay modelsSL7 minPDF
3. T1Proof, counting and partial fractionsDeductive proof · LHS = RHS · Proof by induction (HL) · Proof by contradiction (HL) · Permutations & combinations (HL) · Partial fractions (HL) · Systems of equations (HL)SL+HL8 minPDF
4. T1Complex numbersi² = −1 · Cartesian form · Argand diagram · Modulus & argument · Polar & Euler form · Conjugate roots · De Moivre · nth rootsHL6 minPDF
5. T2Functions, inverses and compositesStraight lines · Domain & range · Key features of graphs · Composite functions · Inverse functions · Odd & even functions (HL) · Self-inverse (HL)SL+HL6 minPDF
6. T2Quadratics and the discriminantStandard, factorised & vertex form · Completing the square · Axis of symmetry · Quadratic formula · Discriminant · Inequalities · Hidden quadraticsSL6 minPDF
7. T2Transformations of graphsTranslations · Reflections · Stretches · Combined transformations · Mapping points · |f(x)| & f(|x|) (HL) · 1/f(x) (HL) · [f(x)]² (HL)SL+HL6 minPDF
8. T2Polynomials, rational functions and inequalitiesReciprocal function · Rational functions & asymptotes · Factor theorem (HL) · Remainder theorem (HL) · Sum & product of roots (HL) · Oblique asymptotes (HL) · Inequalities (HL)SL+HL6 minPDF
9. T3Trigonometry: triangles, radians, arcs and sectors3D distance & midpoint · Volume & surface area · Angles in 3D · Sine rule · Ambiguous case · Cosine rule · Area ½ab sin C · Bearings · Radians · Arcs & sectorsSL7 minPDF
10. T3Trig identities, equations and graphsUnit circle · Exact values · sin² + cos² = 1 · Double angle identities · Trig graphs · Amplitude & period · Trig models · Trig equations · sec, cosec, cot (HL) · arcsin, arccos, arctan (HL) · Compound angles (HL)SL+HL9 minPDF
11. T3Vectors, lines and planesVector notation · Magnitude & unit vectors · Position vectors · Scalar (dot) product · Vector equation of a line · Skew lines · Vector (cross) product · Equations of a plane · Line–plane & plane–plane intersections · AnglesHL10 minPDF
12. T4Statistics and probabilitySampling & bias · Mean, median, mode · Quartiles & IQR · Standard deviation · Outliers · Box plots · Cumulative frequency · Venn diagrams · Tree diagrams · Conditional probability · Independence · Bayes’ theorem (HL)SL+HL9 minPDF
13. T4Random variables, binomial and normal distributionsDiscrete random variables · Expected value · Fair games · Binomial distribution · Normal distribution · z-scores · Inverse normal · Variance & linear transformations (HL) · Continuous random variables (HL) · Median & mode of a pdf (HL)SL+HL8 minPDF
14. T5DifferentiationFirst principles · Chain, product & quotient rules · Tangents & normals · Stationary points & inflection · Optimisation · Implicit differentiation · Related rates · L'Hôpital's ruleSL+HL20 minPDF
15. T5Integration and kinematicsIndefinite integrals · Boundary conditions · Definite integrals · Area under a curve · Area between curves · eˣ, 1/x, sin, cos · Reverse chain rule · Kinematics · Distance vs displacement · Substitution (HL) · By parts (HL) · Volumes of revolution (HL)SL+HL9 minPDF
16. T5Differential equations and Maclaurin seriesDifferential equations · Separating variables · Slope fields · Homogeneous equations (y = vx) · Integrating factor · Exponential models · Newton’s law of cooling · Maclaurin series · Combining series · Series from a DE · Limits with seriesHL9 minPDF

Exam specials 7

Built from the IB examiners' subject reports: grade boundaries, traps, paper strategy, booklets and the IA.

• Grade boundariesIB Maths AI Grade Boundaries (May 2025 & Nov 2024)Every grade boundary for IB Maths AI from the IB subject reports: May 2025 time zones 1–3 and November 2024, SL and HL, overall and paper by paper, and what it takes to get a 7.ReadPDF
• Exam trapsIB Maths AI Exam Traps: Mistakes Examiners FlaggedThe mistakes IB examiners flagged in the November 2024 and May 2025 Maths AI subject reports, grouped by topic, each with the fix, plus a last-week checklist.ReadPDF
• GDC guideIB Maths GDC Guide: TI-84 Plus CE and Casio fx-CG50The calculator skills IB examiners say students under-use, with menu paths for the TI-84 Plus CE and the Casio fx-CG50: solving, graphs, calculus, statistics, distributions, finance, tests, matrices and complex numbers.ReadPDF
• Answer templatesIB Maths AI Answer Templates: Interpret, Justify, ConcludeFill-in-the-blank model answers for the IB Maths AI questions where examiners say students lose reasoning marks: hypothesis tests, interpreting parameters, regression, comparing data, Voronoi, finance and model limitations.ReadPDF
• IA guideIB Maths IA Guide: How to Score Highly on the Exploration (AA & AI)A practical guide to the IB Maths exploration for AA and AI, SL and HL: what each criterion rewards, how to choose a topic with enough mathematics, a structure that works and the mistakes moderators flagged in the 2024–25 reports.ReadPDF
• Command terms & marksIB Maths Command Terms and Markscheme Codes (M1, A1, R1, AG, FT)What every IB Maths command term asks for, and how examiners award marks: M1, A1, R1, AG, FT, implied marks, misreads, "hence" vs "hence or otherwise", accuracy and the presentation rules that cost students marks.ReadPDF
• Formula booklet guideIB Maths AI Formula Booklet: What Is Given and What to MemoriseWhat the IB Maths AI formula booklet (SL and HL) contains and the facts it leaves out, from Voronoi diagrams, sinusoidal models and outliers to χ² degrees of freedom, eigenvalues and graph theory, plus tips for using it in the exam.ReadPDF

Topic notes 18

1. T1Approximation, standard form and errorsStandard form a × 10ᵏ · GDC E-notation · Decimal places & significant figures · Upper and lower bounds · Bounds of calculations · Estimation · Percentage errorSL6 minPDF
2. T1Sequences and financial mathematicsArithmetic sequences & series · Geometric sequences & series · Sigma notation · Compound interest · Depreciation · Inflation & real value · Loans & amortisation · Annuities · TVM solver · Sum to infinity (HL)SL+HL8 minPDF
3. T1Exponents, logarithms and equationsLaws of exponents · Logarithms base 10 and e · Solving aˣ = b · pH and decibel scales · Systems of linear equations · Polynomial equations · Laws of logarithms (HL) · Rational exponents (HL)SL+HL6 minPDF
4. T1Complex numbersi² = −1 · Cartesian form a + bi · Argand diagram · Modulus & argument · Polar & Euler form · Multiplying & dividing · Powers · Complex roots of quadratics · Adding sinusoids (phasors)HL6 minPDF
5. T1Matrices and eigenvaluesMatrix order · Adding & multiplying matrices · Identity matrix · Determinant · Inverse matrix · Solving AX = B · Eigenvalues · Eigenvectors · Characteristic polynomial · Diagonalisation · Powers of a matrixHL6 minPDF
6. T2Lines, functions and graphsGradient · y = mx + c and other forms · Parallel & perpendicular lines · Functions, domain & range · Inverse functions · Key features of graphs · GDC zero, max, min & intersect · AsymptotesSL6 minPDF
7. T2Modelling with functionsModelling cycle · Linear & piecewise models · Quadratic models · Exponential growth & decay · Direct & inverse variation · Cubic models · Sinusoidal models · Fitting parameters · Logistic models (HL) · Logarithmic models (HL)SL+HL8 minPDF
8. T2Composites, transformations and linearising dataComposite functions · Inverse functions & restricted domains · Translations · Stretches · Reflections · Combined transformations · Log scales · Semi-log graphs (exponential models) · Log-log graphs (power models)HL7 minPDF
9. T3Trigonometry, 3D geometry and Voronoi diagrams3D distance & midpoint · Volume & surface area · Angles of elevation & depression · Sine rule · Cosine rule · Area ½ab sin C · Bearings · Arcs & sectors · Perpendicular bisectors · Voronoi diagrams · Nearest neighbour interpolation · Toxic waste dump problemSL8 minPDF
10. T3Radians, vectors and matrix transformationsRadians · Arc length & sector area · Unit circle · sin²θ + cos²θ = 1 · Trig equations · Matrix transformations · Composite transformations · Determinant as area factor · Vectors · Scalar product · Vector equation of a line · Vector kinematics · Closest approach · Vector productHL8 minPDF
11. T3Graph theoryVertices, edges & degree · Simple, complete, weighted & directed graphs · Trees · Eulerian trails & circuits · Hamiltonian paths & cycles · Adjacency matrix · Counting walks · Kruskal & Prim · Minimum spanning tree · Chinese postman · Travelling salesman · Nearest neighbour & deleted vertexHL8 minPDF
12. T4Statistics, correlation and regressionPopulation & sample · Sampling methods & bias · Histograms · Cumulative frequency · Box plots · Mean, median, mode · Quartiles & IQR · Standard deviation · Outliers · Scatter diagrams · Pearson’s r · Regression line · Spearman’s rank · Data collection design (HL) · R² & non-linear regression (HL)SL+HL8 minPDF
13. T4Probability, binomial and normal distributionsProbability rules · Expected number of outcomes · Venn diagrams · Tree diagrams · Sample spaces · Conditional probability · Independence · Discrete random variables · Expected value & fair games · Binomial distribution · Normal distribution · Inverse normalSL6 minPDF
14. T4Chi-squared and t-testsNull & alternative hypotheses · Significance level · p-values · Critical values · Contingency tables · Expected frequencies · Degrees of freedom · χ² test for independence · χ² goodness of fit · t-test for two means · One-tailed & two-tailed tests · Conclusions in contextSL6 minPDF
15. T4Estimation, Poisson, hypothesis tests and Markov chainsE(aX + b) & Var(aX + b) · Linear combinations · Unbiased estimators · Central limit theorem · Confidence intervals (z and t) · Poisson distribution · Binomial hypothesis tests · Critical regions · Type I & Type II errors · Tests for a normal mean · Markov chains · Transition matrices · Steady stateHL8 minPDF
16. T5Differentiation and optimisationDerivative as rate of change · Limits & gradients · Differentiating axⁿ · Increasing & decreasing · Tangents & normals · Stationary points · Local max & min · Optimisation · Trig, exp & log derivatives (HL) · Chain, product & quotient rules (HL) · Related rates (HL) · Second derivative (HL)SL+HL8 minPDF
17. T5Integration, areas and kinematicsAnti-differentiation · + C and boundary conditions · Definite integrals · Area under a curve · Trapezoidal rule · Integrating sin, cos, eˣ, 1/x (HL) · Integration by inspection (HL) · Area with the y-axis (HL) · Volumes of revolution (HL) · Kinematics: displacement & distance (HL)SL+HL7 minPDF
18. T5Differential equations and phase portraitsSetting up DEs · Exponential growth & decay models · Slope fields · Separation of variables · Euler’s method · Coupled systems · Eigenvalue solutions · Phase portraits · Saddles, nodes, spirals & centres · Equilibrium & stability · Second-order DEsHL8 minPDF

Exam specials 7

Built from the IB examiners' subject reports: grade boundaries, traps, paper strategy, booklets and the IA.

• Grade boundariesIB Physics Grade Boundaries (May 2025, new syllabus)Every grade boundary for IB Physics from the IB subject reports: May 2025 time zones 1–3 (first new-syllabus session) and November 2024, SL and HL, overall and paper by paper.ReadPDF
• Paper 1A strategyIB Physics Paper 1A: Multiple-Choice Strategies and TrapsHow to attack IB Physics Paper 1A multiple choice: timing with Paper 1B, the shortcuts examiners recommend, worked examples and a bank of the misconceptions flagged in the 2024–25 subject reports.ReadPDF
• Paper 1B skillsIB Physics Paper 1B: Data, Uncertainties and Graph SkillsEverything Paper 1B tests in IB Physics: recording uncertainties, propagating them, max and min gradient lines, linearising graphs, units of gradients and evaluating experiments, with the errors examiners flagged.ReadPDF
• Paper 2 answersIB Physics Paper 2: How to Write Full-Mark AnswersHow to write IB Physics Paper 2 answers that earn every mark: setting out calculations, "show that" questions, model answers for the "explain" questions examiners see go wrong, and how to tackle the long multi-concept questions.ReadPDF
• IA guideIB Physics IA Guide: Scientific Investigation (New Syllabus)A practical guide to the new-syllabus IB Physics internal assessment: what each of the four criteria rewards, how to write a focused research question, the uncertainty analysis moderators expect and how to write a conclusion and evaluation that score.ReadPDF
• Data booklet guideIB Physics Data Booklet: Equations Not Given and How to Use ItWhat the IB Physics data booklet (first assessment 2025) contains, the equations and facts it leaves out that you must know or derive, and how to use the booklet quickly in Papers 1 and 2.ReadPDF
• Command terms & marksIB Physics Command Terms and How Markschemes Award MarksWhat every IB Physics command term asks for, and how Paper 2 markschemes award marks: marking points, "max" totals, alternative answers, essential words, OWTTE, error carried forward and the rules on units and significant figures.ReadPDF

Topic notes 24

1. A.1KinematicsDisplacement & velocity · Motion graphs · suvat equations · Free fall · Projectile motion · Fluid resistance · Measuring g & uncertaintiesSL+HL15 minPDF
2. A.2Forces and momentumNewton's laws · Free-body diagrams · Friction & Hooke's law · Drag & buoyancy · Momentum & impulse · Collisions · Circular motionSL+HL12 minPDF
3. A.3Work, energy and powerWork done · Kinetic & potential energy · Conservation of energy · Sankey diagrams · Power · Efficiency · Energy densitySL+HL7 minPDF
4. A.4Rigid body mechanicsTorque · Rotational equilibrium · Angular kinematics · Moment of inertia · τ = Iα · Angular momentum · Rotational kinetic energyHL9 minPDF
5. A.5Galilean and special relativityReference frames · Galilean transformation · Postulates · Lorentz transformation · Time dilation · Length contraction · Spacetime diagrams · Muon decayHL8 minPDF
6. B.1Thermal energy transfersParticle model · Internal energy · Specific heat & latent heat · Conduction · Convection · Black-body radiation · Luminosity & brightness · Wien's lawSL+HL8 minPDF
7. B.2Greenhouse effectEmissivity · Albedo · Solar constant & S/4 · Energy balance · Greenhouse gases · Resonance & molecular energy levels · Enhanced greenhouse effectSL+HL6 minPDF
8. B.3Gas lawsPressure · Moles & Avogadro · Boyle, Charles & pressure laws · PV = nRT · Kinetic model · P = ⅓ρv² · Internal energy · Real gasesSL+HL7 minPDF
9. B.4ThermodynamicsFirst law · W = PΔV · P–V diagrams · Adiabatic processes · Cycles & heat engines · Carnot efficiency · Entropy · Second lawHL8 minPDF
10. B.5Current and circuitsCurrent · Potential difference · Resistance & resistivity · Ohm's law & I–V graphs · Power · Series & parallel · Potential dividers · emf & internal resistanceSL+HL8 minPDF
11. C.1Simple harmonic motionConditions for SHM · a = −ω²x · Period & angular frequency · Mass–spring · Pendulum · Energy in SHM · Phase angle (HL)SL+HL6 minPDF
12. C.2Wave modelTransverse & longitudinal · Wavelength, frequency & speed · Wave graphs · Sound · Electromagnetic spectrum · Mechanical vs EM wavesSL+HL6 minPDF
13. C.3Wave phenomenaWavefronts & rays · Snell's law · Total internal reflection · Diffraction · Interference · Double slit s = λD/d · Single slit (HL) · Gratings (HL)SL+HL7 minPDF
14. C.4Standing waves and resonanceStanding waves · Nodes & antinodes · Harmonics on strings · Open & closed pipes · Resonance · DampingSL+HL5 minPDF
15. C.5Doppler effectDoppler effect · Wavefront diagrams · Redshift & blueshift · Δf/f ≈ v/c · Moving source (HL) · Moving observer (HL)SL+HL5 minPDF
16. D.1Gravitational fieldsKepler's laws · Newton's law of gravitation · Field strength g · Field lines · Orbits · Potential (HL) · Escape & orbital speed (HL)SL+HL7 minPDF
17. D.2Electric and magnetic fieldsCharge & charging · Coulomb's law · Electric field strength · Field lines · Parallel plates · Millikan · Magnetic fields · Electric potential (HL)SL+HL7 minPDF
18. D.3Motion in electromagnetic fieldsCharges in E fields · F = qvB sin θ · Circular motion in B · Crossed fields · F = BIL sin θ · Parallel wiresSL+HL6 minPDF
19. D.4InductionMagnetic flux · Faraday's law · ε = BvL · Lenz's law · AC generators · Effect of rotation speedHL6 minPDF
20. E.1Structure of the atomRutherford scattering · Nuclear notation · Emission & absorption spectra · Energy levels · Photons E = hf · Nuclear radius (HL) · Bohr model (HL)SL+HL6 minPDF
21. E.2Quantum physicsPhotons · Photoelectric effect · Work function · Stopping potential · de Broglie wavelength · Electron diffraction · Compton scatteringHL6 minPDF
22. E.3Radioactive decayNuclear stability · Mass defect & binding energy · BE per nucleon · α, β, γ decay · Neutrinos · Half-life · Activity · Decay constant (HL)SL+HL7 minPDF
23. E.4FissionFission · Energy released · Chain reaction · Critical mass · Moderator & control rods · Heat exchanger · Shielding · Nuclear wasteSL+HL5 minPDF
24. E.5Fusion and starsNuclear fusion · Conditions for fusion · Stellar equilibrium · Parallax & the parsec · HR diagram · Stellar evolution · White dwarfs, neutron stars, black holesSL+HL6 minPDF