Zhongkao - independent subject foundations cheat sheet

Parentheses can change the base and sign of a power.

Differentiate products of powers from sums of powers.

State nonzero conditions for zero/negative exponents and quotients.

Use structure, estimation and a direct arithmetic check.

Combine like terms only when variable parts match.

Distribute across every term and check a substitution.

Solve by preserving equality, including every denominator term.

Interpret rate and intercept with variables, units and model limits.

Use actual geometric conditions rather than visual appearance.

Central and inscribed angle relationships require the same intercepted arc.

Check centers, tangency and corresponding parts.

Length and area scale differently; keep units consistent.

Read the complete sentence and its time or logical relationship.

Check both grammatical form and meaning.

Learn collocations and complement patterns in context.

Preserve tense, qualification and meaning in a rewrite.

Use recognition practice alongside original writing and speaking.

Distinguish explicit detail from justified inference.

Read limiting words, negation and changes of plan carefully.

Check reference and logic across both sides of a text gap.

Distinguish narrator, speaker and author.

Support analysis with precise evidence and explained effects.

Resolve vectors before applying F = ma or the SUVAT equations

Momentum is a vector: conserve each component only when the corresponding net external impulse is zero

Equilibrium requires both zero resultant force and zero resultant moment

The centripetal component points to the centre; uniform circular motion has constant speed but changing velocity

SHM signature: a = −ω²x, giving T = 2π/ω

State g and sign conventions explicitly at the start of a solution

EMF is per unit charge from the source; terminal p.d. is E − Ir

Kirchhoff's laws follow from conservation of charge and energy

Capacitor energy is ½CV²; discharge is exponential with time constant RC

Magnetic-force magnitude is |q|vB sinθ and its vector is perpendicular to velocity, so magnetic force does no work

Relative atomic mass is an abundance-weighted mean of isotope masses, calculated from mass-spectrum data.

Across a period effective nuclear charge rises, so radius falls and ionisation energy and electronegativity rise.

Large jumps in successive ionisation energies reveal the number of electrons in the outer shell.

Cr and Cu adopt 3d5 4s1 and 3d10 4s1 configurations; 4s electrons are removed first on ionisation.

Transition metals require a partially filled d subshell in at least one common oxidation state.

VSEPR: count all electron domains, then apply lone pair–lone pair > lone pair–bond pair > bond pair–bond pair repulsion.

Molecular polarity requires both polar bonds and an asymmetric arrangement of those bonds.

Higher bond order gives a shorter, stronger bond; resonance spreads the order over equivalent positions.

Only period 3 elements and below can expand their octet; N, O, F and C cannot.

Ionic lattice strength increases with ion charge and decreases with ion radius.

Convert to K and m3 before using PV = nRT; R = 8.314 J mol−1 K−1.

Real gases deviate from ideality at high pressure and low temperature.

Prokaryotes lack a membrane-bound nucleus; bacterial peptidoglycan walls are not universal and archaeal walls differ.

Fluid mosaic membranes contain a lipid bilayer and proteins; sterol and carbohydrate composition depends on the cell.

Facilitated diffusion is passive; primary active transport may use ATP directly and secondary transport uses a gradient.

Water potential ψ = ψs + ψp; water moves from higher (less negative) to lower ψ.

Mitosis (P-M-A-T) after S-phase replication maintains chromosome number.

Cytokinesis: cleavage furrow in animals, cell plate in plants.

Condensation forms glycosidic, ester and peptide bonds; hydrolysis reverses them.

Reducing-sugar test interpretation requires suitable assay conditions and controls; hydrolysis can change sucrose test results.

Cellulose = β-1,4 glucose, straight H-bonded microfibrils; starch and glycogen = α-glucose.

Triglyceride = glycerol + 3 fatty acids; phospholipid = phosphate head + 2 tails, amphipathic.

Negative feedback reverses a deviation; positive feedback amplifies and must terminate.

Surface area to volume ratio drives the need for specialized exchange surfaces.

State resolution and uncertainty convention for each instrument

Random errors → repeats and means; systematic errors → correction

For first-order worst-case bounds, add relative contributions; independent standard uncertainties use quadrature

Gradient from two points on the best-fit line, with units

Average repeated timings and time many oscillations to cut reaction error

Never quote more significant figures than the data justify

Hypothesis = testable prediction; theory = broad supported explanation; law = consistent regularity.

Independent variable is manipulated; dependent is measured; controls are held constant.

Random assignment supports experimental causal inference; an observed association alone does not establish causation.

Accuracy = closeness to truth; precision = repeatability.

Replication and sample size reduce the influence of random chance.