UB Concours d'Entrée cheat sheet

Quadratic formula: for ax² + bx + c = 0 (a ≠ 0), x = (−b ± √(b² − 4ac)) / 2a; the discriminant Δ = b² − 4ac gives two real roots if Δ > 0, one if Δ = 0, none real if Δ < 0.

Derivative rules: (xⁿ)' = nxⁿ⁻¹; (uv)' = u'v + uv'; (u/v)' = (u'v − uv')/v²; (f∘g)' = f'(g(x))·g'(x).

Integration by parts: ∫u dv = uv − ∫v du; choose u by the LIATE order (Logarithm, Inverse trig, Algebraic, Trigonometric, Exponential).

Arithmetic progression: uₙ = u₁ + (n−1)r, sum Sₙ = n(u₁ + uₙ)/2. Geometric: uₙ = u₁qⁿ⁻¹, sum Sₙ = u₁(1 − qⁿ)/(1 − q) for q ≠ 1.

Limit definition of the derivative: f'(a) = lim(h→0) [f(a+h) − f(a)]/h; a function is continuous at a only if this limit of f exists and equals f(a).

Probability: P(A∪B) = P(A) + P(B) − P(A∩B); independence means P(A∩B) = P(A)P(B); conditional P(A|B) = P(A∩B)/P(B).

Pythagorean and trigonometric identities: sin²x + cos²x = 1; cos(2x) = cos²x − sin²x; tan x = sin x / cos x.

Logarithm rules: ln(ab) = ln a + ln b; ln(a/b) = ln a − ln b; ln(aⁿ) = n ln a; ln 1 = 0, ln e = 1.

Vector dot product: u·v = |u||v|cos θ = x₁x₂ + y₁y₂; u ⟂ v exactly when u·v = 0.

Classical genres and registers: tragedy (Racine, Corneille), comedy (Molière), fable (La Fontaine), essay (Montaigne), novel; register is determined by lexicon, syntax and situation, not by topic alone.

Figures of speech: métaphore (implicit comparison, no tool word), comparaison (uses comme, tel que), métonymie (container for content), hyperbole (exaggeration), oxymore (contradictory pairing), anaphore (repetition at clause start).

Literary periods: Pléiade (16th c.), classicisme (17th c., rules of bienséance and vraisemblance), Lumières (18th c.), romantisme (early 19th c.), réalisme/naturalisme (later 19th c.), surréalisme (20th c.).

Narrative analysis: focalisation is interne (through a character), externe (observer outside), or omniscient/zero; the narrator is not the author.

French agreement rule: the past participle with avoir agrees with a preceding direct object only; with être (and pronominal verbs) it agrees with the subject in most cases.

Subjunctive is triggered after expressions of necessity, doubt, emotion and after conjunctions such as bien que, pour que, avant que, quoique.

Commentaire composé structure: introduction (situation of text, problem, plan), two or three developed axes with quotations analysed, conclusion.

Dissertation structure: introduction with problématique and announced plan, thesis/antithesis/synthesis or thematic parts, conclusion answering the problématique.

Socratic method: maieutics proceeds by question and refutation (elenchus), exposing contradiction in an interlocutor's definition to reach a better one.

Descartes' cogito: radical doubt strips away all that can be doubted, leaving 'I think, therefore I am' as the first certainty; mind and body are then treated as distinct substances.

Kant's three questions summarise his critical project: what can I know? what ought I do? what may I hope?; the categorical imperative tests maxims for universalisability.

Plato's allegory of the cave opposes opinion (doxa) to knowledge (epistemia); the sensible world is a shadow of the intelligible Ideas.

Aristotle: virtue lies as a mean between two vices (courage between cowardice and recklessness); the syllogism is the basic deductive form.

Hume's fork: relations of ideas (a priori, analytic) versus matters of fact (a posteriori, empirical); causal necessity is a habit of the mind, not observed in things.

Nietzsche announces the 'death of God' and critiques morality as historically produced; Sartre states existence precedes essence and defines bad faith (mauvaise foi) as refusing one's freedom.

Dissertation method: define terms, formulate a problem, argue a thesis with precise references, examine objections, conclude with a defended position rather than a summary.

French Revolution key dates: 1789 (États généraux, Bastille, Declaration of the Rights of Man), 1792 (Republic proclaimed), 1793–94 (Terror), 1799 (Brumaire coup).

World War I: 1914–1918; trench warfare on the Western Front; the Treaty of Versailles (1919) imposes reparations and territorial losses on Germany.

World War II: 1939–1945; 1940 armistice and Occupation in France; 1944 Liberation; the Cold War then divides Europe by the Iron Curtain.

Decolonisation: 1947 Indian independence, 1954–62 Algerian war ending in the Évian Accords (1962); 1960 marks independence for most French African colonies.

Fifth Republic: founded 1958 under de Gaulle; the president is elected by direct universal suffrage since 1962 (five-year term since 2000).

Demographic transition: a fall in birth and death rates as a society industrialises, passing through a phase of rapid natural increase.

Geography concepts: density = population / area; urbanisation rate = urban population / total; a gradient describes a spatial variation such as north–south or coast–inland.

Map reading: use the scale for distances, the legend for symbols, orientation for direction; relief is shown by contour lines, climate by isotherms and rainfall bands.

Newton's second law: ΣF = ma; weight is P = mg with g ≈ 9.81 m/s²; action and reaction forces are equal, opposite and act on different bodies.

Kinematics: v = v₀ + at, x = x₀ + v₀t + ½at², v² = v₀² + 2a(x − x₀) for constant acceleration.

Ohm's law U = RI; power P = UI = RI² = U²/R; resistors in series add, in parallel 1/R = 1/R₁ + 1/R₂.

Mole concept: n = m/M = N/N_A, with N_A ≈ 6.02 × 10²³ mol⁻¹; molar concentration C = n/V.

pH = −log[H₃O⁺]; a pH below 7 is acidic, above 7 basic, at 25 °C; strong acids dissociate completely in water.

Balancing equations conserves atoms and charge; stoichiometry uses mole ratios from the balanced coefficients.

Energy: E = hν for a photon; kinetic energy ½mv²; conservation of energy holds when friction is negligible, otherwise some energy dissipates as heat.

Period of a simple pendulum T = 2π√(L/g), independent of mass for small amplitudes.

Cell theory: all living organisms are made of cells; prokaryotes lack a nucleus, eukaryotes have membrane-bound organelles including mitochondria.

DNA structure: double helix of nucleotides paired A–T and G–C; replication is semi-conservative; transcription makes mRNA, translation at ribosomes makes protein.

Mendelian genetics: alleles segregate at meiosis; a dominant allele masks a recessive one; a monohybrid cross of heterozygotes gives a 3:1 phenotype ratio.

Photosynthesis: 6CO₂ + 6H₂O + light → C₆H₁₂O₆ + 6O₂, occurring in chloroplasts; respiration reverses it to release ATP.

Human circulation: double circulation, four chambers; oxygenated blood travels from lungs to left atrium, then left ventricle and aorta.

Homeostasis maintains internal stability through negative feedback (e.g. insulin lowers blood glucose); positive feedback amplifies a change.

Evolution: mutation generates variation, natural selection acts on phenotypes; Hardy–Weinberg equilibrium requires no selection, mutation, migration or drift and random mating.

Ecology: a food chain transfers energy between trophic levels, with roughly 10 % transferred; decomposers recycle nutrients.

English tenses: present perfect (have/has + past participle) links past to present; past simple marks a finished past time; continuous forms stress duration or activity.

Conditionals: type 1 (if + present, will + infinitive), type 2 (if + past, would + infinitive, unreal present), type 3 (if + had + past participle, would have + past participle, unreal past).

Passive voice: be + past participle; the object of the active clause becomes subject; use it when the agent is unknown or unimportant.

Reported speech shifts tense back one step (present to past, will to would) and changes time and place markers (now to then, here to there).

Modals express degrees of certainty and obligation: must (strong obligation), should (advice), might (possibility), can't (deduction of impossibility).

False friends to watch: actually means in fact; eventually means finally; sensible means reasonable; attend means go to.

Translation method: identify the register, keep proper nouns and institutions precise, then restructure the sentence rather than translating word for word.

Essay connectors: however, therefore, moreover, whereas, in spite of; each signals a precise logical relation and should not be swapped loosely.

French institutions: the president is head of state, the prime minister heads the government; the Assemblée nationale and Sénat form Parliament; the Conseil constitutionnel reviews laws.

European Union: 27 member states since Brexit (2020); the European Commission proposes legislation, the European Parliament and Council of the EU co-decide.

Major international organisations: UN (1945, New York), UNESCO (Paris), WHO (Geneva), IMF and World Bank (Washington), NATO (Brussels).

French symbols and values: motto Liberté, Égalité, Fraternité; national day 14 July; Marianne and the tricolour flag; laïcité separates church and state.

Climate frameworks: the 1997 Kyoto Protocol and the 2015 Paris Agreement, which aims to limit warming to well below 2 °C, ideally 1.5 °C.

Economic indicators: GDP measures output, inflation measures the change in the general price level, unemployment rate is unemployed persons divided by the labour force.

Scientific milestones: 1969 Moon landing, 2003 completion of the Human Genome Project, 2020 rapid development of mRNA vaccines.

Reading current affairs: check the source and date, distinguish fact from commentary, and place events in their institutional context before judging.

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