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Technique practice for A-level Maths and Physics.
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A-level Maths
Pure and applied techniques across the A-level Maths specification, from algebra to differentiation, integration and numerical methods.
Pure — Algebra and functions
- Algebra and functionsSolve a quadraticFactorise a quadratic with whole-number or simple fraction roots, and give both solutions.
- Algebra and functionsCompleting the squareWrite a quadratic as (x + p)² + q, or as a(x + p)² + q when x² has a coefficient.
- Algebra and functionsPartial fractionsExpress (ax + b)/((x + c)(px + d)) as A/(x + c) + B/(px + d) and find A and B.
- Algebra and functionsValue of a composite functionFind fg(a) or gf(a) for two functions: work out the inner function first, then the outer.
- Algebra and functionsValue of an inverse functionFind the inverse of a linear, rational or completed-square function and evaluate it.
Pure — Coordinate geometry
Pure — Sequences and series
- Sequences and seriesSum of an arithmetic seriesFind the first term and common difference, then use S = n/2 (2a + (n − 1)d).
- Sequences and seriesGeometric sum to infinityFind the common ratio, check |r| < 1, then use S = a/(1 − r) to give an exact value.
- Sequences and seriesBinomial expansion coefficientFind the coefficient of x^k in (a + bx)^n using the general term nCk a^(n−k) (bx)^k.
- Sequences and seriesBinomial expansion, any powerFind a coefficient in (1 + ax)^p for a negative or fractional power p, as an exact fraction.
Pure — Trigonometry
- TrigonometrySolve a trigonometric equationSolve sin, cos or tan equations for 0 ≤ x < 360°, with or without a shifted angle.
- TrigonometryArc length and sector areaUse s = rθ and A = ½r²θ for a sector, with the angle in radians.
- TrigonometryThe R-formulaWrite a sin x + b cos x as R sin(x + α) or R cos(x − α): R in exact form, α in degrees.
Pure — Exponentials and logarithms
- Exponentials and logarithmsExponential model: find the timeSolve an exponential growth or decay model for the time using natural logs.
- Exponentials and logarithmsExact solutions with logarithmsSolve e^(ax + b) = c and give x in exact form using ln.
- Exponentials and logarithmsSolve an exponential equationSolve a^x = b, a^(kx + m) = b or a^(x + m) = c^x with logarithms, to 3 s.f.
- Exponentials and logarithmsLaws of logarithmsCombine logarithms with the power, product and quotient laws to reach a single exact number.
Pure — Differentiation
- DifferentiationChain rule: gradient at a pointDifferentiate (ax + b)^n with the chain rule and find the gradient at a point.
- DifferentiationProduct rule: gradient at a pointDifferentiate x^n e^(kx) with the product rule and find the gradient at a point.
- DifferentiationChain rule: differentiateDifferentiate (ax + b)^n, 1/(ax + b)^n or e^(kx^m) with the chain rule and give dy/dx.
- DifferentiationStationary points of a cubicDifferentiate a cubic, set dy/dx = 0 and solve for the x-coordinates of the stationary points.
- DifferentiationTangent to a curveFind the gradient with dy/dx at a point and give the tangent as y = mx + c.
Pure — Integration
- IntegrationIntegrating a linear powerIntegrate (ax + b)^n by reversing the chain rule, then evaluate between limits.
- IntegrationIndefinite integralsIntegrate m(ax + b)^n or m e^(kx) by reversing the chain rule. Give the answer with + c.
- IntegrationIntegration by partsIntegrate m x e^(kx) between 0 and a by parts, to 3 significant figures.
- IntegrationArea under a curveIntegrate a polynomial between limits to find an area, as an exact fraction.
Pure — Numerical methods
Pure — Vectors
Statistics — Sampling and data
Statistics — Probability
Statistics — Statistical distributions
Statistics — Hypothesis testing
Mechanics — Kinematics
- KinematicsConstant acceleration (suvat)Choose the suvat equation that fits the three known quantities and find the fourth.
- KinematicsProjectile motionResolve the launch velocity, then find the time of flight, range or greatest height.
- KinematicsVariable accelerationDifferentiate or integrate a polynomial in t to link displacement, velocity and acceleration.
Mechanics — Forces and Newton's laws
Mechanics — Moments
A-level Physics
Calculation techniques across the A-level Physics specification, from mechanics and electricity to fields, capacitors, oscillations and nuclear physics.
Measurements and errors
Particles and radiation
- PhotonsPhoton energy in eVFind a photon's energy in eV from its wavelength or frequency, or its wavelength from eV.
- Photoelectric effectPhotoelectric maximum KEUse hf = φ + E_k(max) with the work function in eV to find the most energetic electron's KE.
- Photoelectric effectThreshold frequencyLink the work function to the threshold frequency or wavelength using φ = hf₀.
- Wave-particle dualityde Broglie wavelengthFind λ = h/p for an electron accelerated through a pd, a moving electron or proton, or a ball.
Waves and optics
- InterferenceDouble-slit fringe spacingUse w = λD/s to find the fringe spacing, wavelength, slit separation or screen distance.
- Diffraction gratingsDiffraction grating equationUse d sin θ = nλ with lines per mm to find an angle, a wavelength or the lines per mm.
- Diffraction gratingsHighest grating orderFind the highest order of maximum a grating can give, or the total number of maxima.
- RefractionSnell's law: angle of refractionUse n₁ sin θ₁ = n₂ sin θ₂ to find the angle of refraction at a boundary.
- Total internal reflectionCritical angleUse sin C = n₂/n₁ for the critical angle at a boundary, or find n from a critical angle.
- Stationary wavesStationary waves on a stringUse f = (1/2L)√(T/μ) for a string's harmonics, or find the tension for a given note.
Mechanics and materials
- MaterialsYoung modulus of a wireFind E = FL/(AΔL) from a load, length, diameter in mm and extension, or find the extension.
- MaterialsElastic strain energyFind the energy stored in a spring or wire from ½kx² or ½FΔL, converting cm and mm first.
- MomentumMomentum in collisionsUse conservation of momentum, with directions as signs, for a collision or a recoil.
- Work, energy and powerPower = force × velocityUse P = Fv against a resistive force, at constant speed or while accelerating.
- ProjectilesHorizontal projectileFind the time of fall, range or impact speed of a body launched horizontally.
Electricity
- ResistivityResistivity of a wireUse R = ρL/A with the area found from a diameter in mm, to find R, ρ or the length.
- emf and internal resistanceemf and internal resistanceUse ε = I(R + r) to find the terminal pd or current, or the internal resistance from readings.
- Potential dividersPotential divider outputFind V_out = V_in R₂/(R₁ + R₂), with or without a load, or the R₂ needed for a given output.
- DC circuitsSeries-parallel resistanceFind the total resistance of a network described in words, combining series and parallel parts.
Thermal physics
- Specific heat capacitySpecific heat capacityUse Q = mcΔθ for an energy, a final temperature with a heater, or the capacity itself.
- Specific latent heatSpecific latent heatUse Q = mL for an energy, the mass changed by a heater, or the latent heat from an experiment.
- Ideal gasesIdeal gas equationUse pV = nRT to find p, V, n or T, converting °C, kPa and dm³ to SI first.
- Kinetic theoryKinetic energy of a moleculeUse ½m⟨c²⟩ = 3kT/2 for a molecule's mean kinetic energy, the temperature or the rms speed.
Capacitors
- CapacitorsCapacitor dischargeFind the potential difference on a discharging capacitor, converting kΩ and μF first.
- CapacitorsCapacitor chargingUse V = V₀(1 − e^(−t/RC)) for the pd or charge while charging, or the time to reach a pd.
- CapacitorsEnergy stored in a capacitorFind the energy stored from ½CV², ½QV or Q²/2C, converting μF and μC to SI first.
Further mechanics
- Circular motionCentripetal forceFind the centripetal force from the speed, or from the period using ω = 2π/T.
- OscillationsSHM: maximum speedFind the maximum speed of an oscillator from its frequency or a mass on a spring.
- OscillationsSHM: displacement, accelerationUse x = A cos ωt (radians) for the displacement at a time, or a = −ω²x for the acceleration.
Fields
- Gravitational fieldsGravitational field strengthFind g at a height above a planet's surface using g = GM/r² with r = R + h.
- Gravitational fieldsOrbital periodUse T² = 4π²r³/GM for a satellite or planet's period, or the orbit radius for a given period.
- Gravitational fieldsGravitational potentialUse V = −GM/r, remembering the minus sign, and the work done moving a mass between two radii.
- Electric fieldsCoulomb's lawUse F = Q₁Q₂/4πε₀r² for point charges in nC or μC, charges in a nucleus, or the separation.
- Electric fieldsForce in a uniform fieldUse E = V/d between parallel plates, then F = eE and a = F/m for an electron.
- Magnetic fieldsForce on a current in a fieldUse F = BIL sin θ, with B in mT and L in cm, to find the force, the field or the current.
- Magnetic fieldsPath of a charged particleUse r = mv/BQ for an electron or proton moving at right angles to a field, or after a pd.
- Electromagnetic inductionFaraday's law: induced emfUse ε = NΔΦ/Δt with the area in cm² and the time in ms, or from a change of flux linkage.
- TransformersTransformer turns and efficiencyUse Vs/Vp = Ns/Np for voltage or turns, and efficiency = (IsVs)/(IpVp) for a real transformer.
Nuclear physics
- RadioactivityRadioactive decay activityTurn a half-life into a decay constant, then find the activity after a given time.
- Radioactive decayHalf-life from measurementsFind T½ from two activities and the time between them, with a background correction if needed.
- Binding energyBinding energy per nucleonTurn a mass defect in u into MeV with 1 u = 931.5 MeV, then divide by the nucleon number.
- Gamma radiationInverse-square law for gammaUse the inverse-square law for a count rate or a distance, with a background correction.
Astrophysics and cosmology
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