The Theory
Atlas.

Be able to simplify expressions involving indices, surds, algebraic fractions and brackets. Correct factorisation and use of algebraic identities are essential before solving equations.

Quadratic equations can be solved by factorisation, completing the square or the quadratic formula. The discriminant determines the nature of the roots.

• Δ > 0 gives two distinct real roots.
• Δ = 0 gives one repeated real root.
• Δ < 0 gives no real roots.

An arithmetic sequence has a constant difference between successive terms. A geometric sequence has a constant ratio between successive terms.

A straight-line equation can be determined from a point and gradient. Parallel lines have equal gradients while perpendicular lines have gradients whose product is −1.

When solving trigonometric equations, first find the principal solution and then use the symmetry and periodicity of the relevant trigonometric function to obtain all required solutions in the given interval.

Differentiation gives the gradient of a curve. At a stationary point, the first derivative is zero. The second derivative can be used to classify the stationary point.

• Second derivative positive → local minimum.
• Second derivative negative → local maximum.

Integration reverses differentiation. An arbitrary constant must be included for indefinite integrals. Definite integration can be used to calculate areas under curves.

A complete solution should establish the required equation, differentiate accurately, identify stationary points, classify them and interpret the result in the context of the question.

A logarithm is the inverse operation to exponentiation. Logarithm laws allow products, quotients and powers to be rewritten in simpler forms.

Exponential models describe quantities that grow or decay at a rate proportional to their current value. The natural exponential function has the special property that its derivative is itself.

The binomial theorem expands a power of a sum into a sequence of terms involving binomial coefficients.

The chain rule is used when one function is contained inside another. Differentiate the outer function with respect to the inner function and multiply by the derivative of the inner function.

When differentiating the product of two functions, differentiate one function while keeping the other unchanged, then reverse the roles and add the two results.

Identities are true for all permitted values of the variable. They can be used to transform complicated trigonometric expressions into a form that can be solved or differentiated.

A composite function applies one function to the output of another. The order matters, so fg(x) and gf(x) generally produce different results.

An inverse function reverses the effect of the original function. The original function must satisfy the necessary one-to-one condition on its domain for an inverse to exist.

Newton-Raphson is an iterative method for finding an approximate root of an equation. A starting value is required and repeated iterations move the estimate towards a root.

An equation can be rearranged into an iterative form x = g(x). Starting from an initial estimate, repeated substitution can converge to a solution.

When y cannot conveniently be expressed explicitly in terms of x, differentiate both sides with respect to x and treat y as a function of x.

Integration by parts is useful when the integrand is a product of functions and direct integration is not convenient.

Vectors have both magnitude and direction. They can represent displacement, velocity and other directed quantities.

The scalar product produces a scalar. It can be used to find the angle between vectors and determine whether two vectors are perpendicular.

Parametric equations express x and y in terms of a third parameter, usually t. They are particularly useful for curves that are difficult to describe directly as y=f(x).

A differential equation relates a function to one or more of its derivatives. Separation of variables allows the variables to be placed on opposite sides before integration.

After integration, use the supplied initial condition to determine the arbitrary constant. The final solution should satisfy both the differential equation and the initial condition.

A particle is treated as having mass but negligible dimensions. This model allows forces to be considered as acting at a single point.

Newton's laws describe how forces affect the motion of objects. The resultant force determines acceleration according to Newton's second law.

For connected particles, identify the forces acting on each particle separately. Use a consistent positive direction and apply Newton's second law to each particle.

Friction acts to oppose relative motion or the tendency for relative motion between surfaces. Limiting friction occurs when motion is about to begin.

When the external resultant impulse on a system is negligible, the total momentum of the system remains constant.

Energy cannot be created or destroyed. It can be transferred between kinetic, gravitational potential, elastic and other forms.

Resolve the initial velocity into horizontal and vertical components. Treat the horizontal motion independently from the vertical motion, where acceleration is due to gravity.

An object moving in a circle has acceleration directed towards the centre. The force responsible for this acceleration is the resultant radial force.

Momentum is conserved in an isolated collision. The coefficient of restitution describes the relative speed of separation compared with relative speed of approach.

The centre of mass is the point at which the mass of a system can be considered concentrated for translational motion.

For a body in equilibrium, both the resultant force and resultant moment must be zero.

A sample is a subset of a population. Sampling methods are used when studying the entire population is impractical.

• Simple random sampling
• Systematic sampling
• Stratified sampling
• Quota sampling
• Opportunity sampling

Data may be qualitative or quantitative. Quantitative data may be discrete or continuous.

Primary data are collected directly for the investigation. Secondary data were previously collected for another purpose or by another source.

Mean, median and mode describe the central location of data. The choice of measure depends on the distribution and the presence of extreme values.

Range, interquartile range and standard deviation describe how spread out data are. A larger measure of dispersion indicates greater variability.

An outlier is an observation that is unusually distant from the rest of the data. Outliers can have a substantial effect on the mean and standard deviation.

For grouped continuous data, histogram area represents frequency. Therefore class width must be considered when comparing bars.

Cumulative frequency is the running total of frequencies. A cumulative frequency graph can be used to estimate quartiles, median and percentages below a particular value.

A box plot displays the minimum, lower quartile, median, upper quartile and maximum. It allows distributions to be compared using location and spread.

Probability measures how likely an event is to occur. Probabilities lie between 0 and 1 inclusive.

Two events are independent when the occurrence of one does not change the probability of the other.

Correlation measures the strength and direction of a linear relationship between two variables. Correlation does not establish causation.

A regression line provides an equation for predicting one variable from another. Predictions should preferably be made within the range of the observed data.

Coding transforms every observation using a linear transformation. It can simplify calculations while preserving the underlying statistical information.

A strong statistical response should identify the correct method, perform the calculation accurately, interpret the result in context and give a conclusion that answers the question directly.

A discrete random variable takes a countable set of possible values, each associated with a probability.

A binomial model applies when there is a fixed number of independent trials, two possible outcomes and a constant probability of success.

The Poisson distribution models the number of occurrences of an event within a fixed interval when events occur with a constant average rate and independently.

The normal distribution is continuous, symmetric and bell-shaped. Its mean determines the centre and its standard deviation determines the spread.

A hypothesis test begins with a null hypothesis and alternative hypothesis. Evidence from a sample is used to determine whether the null hypothesis should be rejected at the stated significance level.

A scalar has magnitude only. A vector has both magnitude and direction.

Newton's laws explain the relationship between resultant force, mass and acceleration. In equilibrium the resultant force is zero.

For an isolated system, total momentum before an interaction equals total momentum after the interaction.

For a body in rotational equilibrium, the sum of clockwise moments about a point equals the sum of anticlockwise moments about that point.

Work is the transfer of energy by a force acting through a distance. Power is the rate of energy transfer.

For an elastic material below the limit of proportionality, extension is proportional to applied force.

Stress measures force per unit cross-sectional area. Strain measures fractional change in length. Young modulus describes the stiffness of a material in the elastic region.

For a small sphere moving slowly through a viscous fluid, viscous drag increases with velocity. As the sphere accelerates, drag increases until the resultant force becomes zero and terminal velocity is reached.

The initial straight-line region represents proportional behaviour. Beyond the elastic limit, permanent deformation may occur.

A progressive wave transfers energy and information from one location to another without transferring matter overall.

In a transverse wave, oscillations are perpendicular to the direction of energy transfer. In a longitudinal wave, oscillations are parallel to the direction of energy transfer.

Diffraction is the spreading of waves when they pass through a gap or around an obstacle. The effect is most significant when the gap size is comparable to the wavelength.

Interference occurs when waves overlap. Constructive interference occurs when the waves reinforce each other, while destructive interference occurs when they partially or completely cancel.

Refraction occurs when a wave changes speed as it enters another medium. A change in speed causes a change in wavelength; frequency remains constant.

Electric current is the rate of flow of charge through a conductor.

Resistance describes how strongly a component opposes the flow of electric charge. For an ohmic conductor at constant temperature, current is proportional to potential difference.

In series, the current is the same through each component. In parallel, the potential difference across each branch is the same.

The independent variable is deliberately changed. The dependent variable is measured. Control variables are kept constant to make the investigation fair.

Accuracy describes closeness to the accepted value. Precision describes how closely repeated measurements agree. Resolution is the smallest change that an instrument can detect.

Every measurement has an associated uncertainty. Repeated measurements can be used to estimate random uncertainty, while systematic errors tend to shift all measurements in the same direction.

A best-fit line should represent the overall trend of the data rather than simply joining adjacent points.

An object moving in a circle continuously changes direction, so it has centripetal acceleration directed towards the centre. The resultant radial force provides this acceleration.

A gravitational field is a region in which a mass experiences a gravitational force. Field strength is force per unit mass.

An electric field is a region where a charged particle experiences an electric force. Field direction is defined as the direction of force on a positive test charge.

Electric potential at a point is the work done per unit positive charge in bringing a small positive test charge from infinity to that point.

Particles can be classified into groups such as leptons, quarks, hadrons and exchange particles. Conservation laws constrain which particle interactions are possible.

Interactions are represented using particle diagrams and must obey conservation of charge, energy, momentum and other relevant quantum numbers.

Temperature is related to the average kinetic energy of particles. Heating transfers energy and can increase temperature or change the state of matter.

Simple harmonic motion occurs when acceleration is directly proportional to displacement from equilibrium and directed towards the equilibrium position.

Radioactive decay is spontaneous and random for individual nuclei. The activity of a sufficiently large sample follows a predictable statistical decay law.

Half-life is the time required for the number of undecayed nuclei, or the activity, to decrease to half its initial value.

Light from a receding astronomical object is observed at a longer wavelength than when emitted. The observed redshift provides evidence about the object's relative motion.

For sufficiently distant galaxies, recession speed is approximately proportional to distance. This relationship provides evidence for the expansion of the universe.

A good investigation identifies variables, selects suitable equipment, explains the method, controls relevant variables and considers safety.

Evaluation should identify limitations, distinguish random from systematic errors and propose realistic improvements that directly address the limitation.

Atoms contain protons and neutrons in the nucleus with electrons occupying shells or orbitals around the nucleus. Isotopes contain the same number of protons but different numbers of neutrons.

A mass spectrometer can be used to identify isotopes and determine relative isotopic abundance. Particles are ionised, accelerated, separated according to mass-to-charge ratio and detected.

Ionic bonding is the electrostatic attraction between oppositely charged ions formed following electron transfer.

A covalent bond is the electrostatic attraction between a shared pair of electrons and the nuclei of the bonded atoms.

Metallic bonding is the electrostatic attraction between positive metal ions and delocalised electrons.

Electronegativity is the ability of an atom to attract the bonding pair of electrons in a covalent bond.

A functional group determines many of the characteristic chemical reactions of an organic compound.

Structural isomers have the same molecular formula but different structural arrangements of atoms.

Enthalpy change measures the heat energy transferred during a reaction under constant pressure.

The enthalpy change for a reaction is independent of the route taken between reactants and products because enthalpy is a state function.

Average bond enthalpy is the average energy required to break one mole of a particular bond in gaseous molecules.

Down Group 2, atomic radius increases and the outer electrons are further from the nucleus. This affects ionisation energy and chemical reactivity.

Halogenoalkanes undergo reactions including nucleophilic substitution. Their reactivity is influenced by the strength of the carbon-halogen bond.

Alcohols can undergo oxidation, dehydration and reactions associated with the hydroxyl functional group.

A titration determines the concentration of an unknown solution by reacting it with a solution of known concentration.

Measured quantities contain uncertainty. The effect of uncertainty should be considered when reporting calculated results.

Chemical tests can identify ions and functional groups through characteristic observations such as colour changes, precipitates or gas formation.

For a reaction to occur, particles must collide with sufficient energy and suitable orientation. Increasing temperature increases the proportion of particles with energy above the activation energy.

A catalyst provides an alternative reaction pathway with a lower activation energy. It does not change the equilibrium position.

At dynamic equilibrium, the forward and reverse reactions continue at equal rates. Concentrations remain constant although particles continue to react.

When a change is imposed on a system at equilibrium, the system responds in the direction that tends to oppose the change.

An acid donates protons while a base accepts protons. Strong acids and bases ionise or dissociate extensively, whereas weak acids and bases establish equilibria.

Reaction mechanisms show how bonds break and form using curly arrows to represent electron movement.

Transition metals form ions with partially filled d orbitals and commonly exhibit variable oxidation states, coloured compounds and complex formation.

A complex ion consists of a central metal ion surrounded by ligands that donate lone pairs to form coordinate bonds.

A ligand is an ion or molecule that donates a lone pair of electrons to a central metal ion.

Amines are derivatives of ammonia in which one or more hydrogen atoms have been replaced by organic groups.

Amino acids contain both amine and carboxylic acid functional groups and can behave as both acids and bases.

A good chemical investigation identifies variables, controls conditions, selects appropriate apparatus and produces repeatable measurements.

An evaluation should identify specific limitations and explain how each limitation affects the result. Improvements should directly address the identified limitation.

Carbohydrates include monosaccharides, disaccharides and polysaccharides. Monosaccharides join through condensation reactions to form larger carbohydrates.

Proteins are polymers of amino acids. The sequence of amino acids determines the primary structure and influences higher levels of protein structure.

Triglycerides contain glycerol and three fatty acids. Phospholipids have hydrophilic phosphate-containing heads and hydrophobic fatty-acid tails.

Enzymes are biological catalysts. Their active site has a specific shape complementary to the substrate. Temperature, pH and substrate concentration can affect enzyme activity.

Cell membranes contain phospholipids and proteins arranged according to the fluid mosaic model. Membrane proteins perform functions including transport, recognition and reception.

Efficient gas exchange surfaces have a large surface area, short diffusion distance and suitable concentration gradients. Ventilation and blood flow help maintain concentration gradients.

Xylem transports water and mineral ions. Phloem transports assimilates such as sucrose between sources and sinks.

Eukaryotic cells contain membrane-bound organelles with specialised functions. Prokaryotic cells lack a membrane-bound nucleus and have a simpler internal organisation.

Mitosis produces genetically identical daughter cells. It is important for growth, repair and asexual reproduction.

Meiosis produces genetically different haploid cells. Variation arises through independent assortment and crossing over.

DNA replication is semi-conservative. Each original strand acts as a template for a new complementary strand.

Alleles are alternative forms of a gene. Genotype describes the alleles present while phenotype describes the observable characteristics.

Individuals within a population vary. Individuals with advantageous heritable characteristics are more likely to survive and reproduce, increasing the frequency of those alleles over generations.

Biodiversity refers to variation within and between species and ecosystems. Conservation aims to maintain biodiversity and prevent species loss.

Microscopes allow structures smaller than the unaided eye can resolve to be observed. Magnification and resolution are different properties.

Random sampling reduces selection bias. Systematic and stratified methods may be used where the population has known structure.

A biological investigation should clearly identify the independent and dependent variables and control relevant environmental factors.

A valid investigation measures what it is intended to measure. Reliability increases when repeated measurements produce consistent results.

Light energy is converted into chemical energy during photosynthesis. Light-dependent reactions generate ATP and reduced NADP, which support carbon fixation reactions.

Aerobic respiration transfers energy from glucose to ATP. Glycolysis occurs in the cytoplasm while later stages occur in mitochondria.

Energy enters many ecosystems through photosynthesis and passes between trophic levels. Energy is lost through respiration and other processes at each transfer.

Elements such as carbon and nitrogen cycle between organisms and the physical environment through biological and chemical processes.

Microorganisms include bacteria and fungi. They can be beneficial in biotechnology and decomposition but can also cause disease.

The immune system recognises foreign antigens and produces specific responses involving lymphocytes, antibodies and memory cells.

Vaccination exposes the immune system to an antigen or antigenic material so that memory cells are produced. A later exposure can therefore trigger a faster secondary immune response.

Homeostasis maintains a relatively stable internal environment despite external or internal changes. Negative feedback returns conditions towards a set point.

The nervous system coordinates rapid responses using electrical impulses transmitted along neurones.

At a chemical synapse, neurotransmitter is released from the presynaptic neurone, diffuses across the synaptic cleft and binds to receptors on the postsynaptic membrane.

Hormones are chemical messengers transported in the blood. They bind to specific receptors on target cells.

Gene expression involves transcription of DNA into RNA and translation of the genetic information into a polypeptide.

Gene technology involves techniques used to isolate, amplify, modify or transfer genetic material. Applications include medicine, agriculture and research.

The Hardy-Weinberg principle describes allele and genotype frequencies in an ideal population where evolutionary forces such as selection, mutation and migration are absent.

A strong investigation identifies the hypothesis, variables, controls, method, repeats, appropriate equipment and safety considerations.

Tables should include appropriate headings and units. Graphs should use suitable scales, labelled axes and appropriate plotting conventions.

Statistical tests are used to determine whether observed differences or relationships are likely to have arisen by chance.

An effective evaluation identifies specific limitations, explains their effect on the results and proposes practical improvements.