MSI Concours d'Admission cheat sheet

Quadratic formula: roots of ax²+bx+c=0 are x = (-b ± √(b²-4ac))/(2a); discriminant Δ=b²-4ac determines nature: Δ>0 two real, Δ=0 one real, Δ<0 two complex conjugates.

Derivative rules: (xⁿ)' = n xⁿ⁻¹; (uv)' = u'v+uv'; (u/v)' = (u'v−uv')/v²; (f∘g)' = f'(g(x))·g'(x); (eˣ)'=eˣ; (ln x)'=1/x; (sin x)'=cos x; (cos x)'=−sin x.

Integral: ∫xⁿ dx = xⁿ⁺¹/(n+1)+C for n≠−1; ∫dx/x = ln|x|+C; integration by parts ∫u dv = uv − ∫v du.

Limits: lim_{x→0} sin x/x = 1; lim_{x→0} (1−cos x)/x² = 1/2; lim_{x→∞} (1+1/x)ˣ = e; l'Hôpital applies to 0/0 or ∞/∞.

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

Combinatorics: number of permutations of n distinct objects = n!; arrangements of k from n = n!/(n−k)!; combinations C(n,k)=n!/(k!(n−k)!); binomial theorem (a+b)ⁿ=Σ C(n,k)aⁿ⁻ᵏbᵏ.

Arithmetic/geometric sequences: arithmetic uₙ=u₀+nr, sum Sₙ=(n+1)(u₀+uₙ)/2; geometric uₙ=u₀qⁿ, sum Sₙ=u₀(1−qⁿ⁺¹)/(1−q) for q≠1.

Complex numbers: i²=−1; modulus |a+bi|=√(a²+b²); argument θ=atan2(b,a); Euler form re^{iθ}=r(cosθ+i sinθ); de Moivre (cosθ+i sinθ)ⁿ=cos nθ+i sin nθ.

Newton's second law: ΣF = ma; weight W=mg with g≈9.81 m/s²; friction force f=μN where μ is coefficient of friction and N the normal force.

Kinematics: v=u+at; s=ut+½at²; v²=u²+2as; for uniform circular motion a=v²/r directed toward center.

Energy and momentum: kinetic energy KE=½mv²; gravitational potential near Earth PE=mgh; work W=F·d·cosθ; momentum p=mv; conservation of momentum in isolated systems.

Electricity: Ohm's law U=RI; power P=UI=I²R=U²/R; resistors in series R=R₁+R₂+…, in parallel 1/R=1/R₁+1/R₂+…; Coulomb force F=kq₁q₂/r² with k≈8.99×10⁹ N·m²/C².

Waves and optics: wave speed v=fλ; Snell's law n₁sinθ₁=n₂sinθ₂; refractive index n=c/v; total internal reflection when θ₁>arcsin(n₂/n₁).

Thermodynamics: ideal gas PV=nRT; first law ΔU=Q−W; Celsius to Kelvin T(K)=T(°C)+273.15; heat Q=mcΔT.

Mole concept: n=m/M; Avogadro number N_A≈6.022×10²³ mol⁻¹; molar volume of ideal gas at STP ≈22.4 L/mol; concentration C=n/V.

Acid-base: pH=−log[H₃O⁺]; pOH=−log[OH⁻]; pH+pOH=14 at 25°C; strong acid fully dissociates; buffer Henderson–Hasselbalch pH=pKa+log([A⁻]/[HA]).

Redox and electrochemistry: oxidation is loss of electrons, reduction is gain; oxidizing agent is reduced; cell potential E°cell=E°cathode−E°anode; Faraday's law m=(MIt)/(nF) with F≈96485 C/mol.

Cell theory: all living organisms are made of cells; cell is basic unit of life; cells arise from pre-existing cells; prokaryotes lack membrane-bound nucleus, eukaryotes have one.

DNA structure: double helix with antiparallel strands; adenine pairs with thymine (2 H-bonds), guanine with cytosine (3 H-bonds); in RNA thymine is replaced by uracil.

Central dogma: DNA → RNA (transcription) → protein (translation); genetic code is triplet, degenerate, non-overlapping; start codon AUG codes methionine; stop codons UAA, UAG, UGA.

Mitosis phases: prophase, metaphase, anaphase, telophase; produces two genetically identical diploid cells; meiosis produces four haploid gametes with crossing-over in prophase I.

Mendelian genetics: law of segregation and independent assortment; monohybrid cross F2 phenotypic ratio 3:1; dihybrid 9:3:3:1; test cross reveals genotype.

Photosynthesis: 6CO₂+6H₂O+light → C₆H₁₂O₆+6O₂; occurs in chloroplasts; light reactions produce ATP and NADPH; Calvin cycle fixes CO₂ via rubisco.

Cellular respiration: glycolysis (cytoplasm) yields 2 ATP and 2 pyruvate; Krebs cycle and oxidative phosphorylation in mitochondria; complete oxidation of glucose yields about 30–32 ATP.

Human physiology: blood has pH≈7.4; hemoglobin carries O₂; nephron is functional unit of kidney; neurons transmit action potentials; hormones are secreted by endocrine glands into blood.

Ecology: trophic levels, food webs, energy transfer ~10% between levels; carrying capacity K; exponential growth dN/dt=rN; logistic growth dN/dt=rN(1−N/K).

Accord du participe passé avec être: le participe s'accorde en genre et en nombre avec le sujet (elle est partie); avec avoir: pas d'accord avec le sujet, accord avec le COD antéposé (les lettres qu'il a écrites).

Temps composés: passé composé = auxiliaire avoir/être au présent + participe passé; plus-que-parfait = auxiliaire à l'imparfait + participe passé; futur antérieur = auxiliaire au futur simple + participe passé.

Subjonctif: utilisé après des verbes de volonté, doute, émotion (vouloir que, douter que, être content que) et après certaines conjonctions (bien que, pour que, avant que).

Figures de style: métaphore (comparaison implicite sans outil), comparaison (avec comme, tel que), hyperbole (exagération), métonymie (remplacer par un terme lié), oxymore (alliance de contraires), anaphore (répétition en début de phrase).

Genres littéraires: roman, nouvelle, conte, poésie (sonnet: 14 vers, deux quatrains et deux tercets), théâtre (tragédie, comédie; règle des trois unités au XVIIᵉ), essai.

Mouvements: humanisme (XVIᵉ), classicisme (XVIIᵉ, ordre et raison), Lumières (XVIIIᵉ, raison et progrès), romantisme (XIXᵉ, sentiment), réalisme/naturalisme (XIXᵉ, observation), surréalisme (XXᵉ, inconscient).

Auteurs majeurs: Molière (comédies), Racine (tragédies), Voltaire (contes philosophiques), Hugo (romantisme), Balzac (La Comédie humaine), Flaubert (réalisme), Zola (naturalisme), Proust (roman moderne).

Accord de l'adjectif: s'accorde en genre et en nombre avec le nom; adjectifs de couleur composés restent invariables (des yeux bleu clair); certains adjectifs restent invariables (marron, orange).

Connecteurs logiques: cause (parce que, puisque, car), conséquence (donc, ainsi, par conséquent), opposition (mais, cependant, néanmoins), concession (bien que, quoique), but (pour que, afin que).

Verb tenses: present perfect (have/has + past participle) links past to present; past simple for finished actions with definite time; past perfect for earlier past; future perfect (will have + past participle) for completion before a future point.

Conditionals: zero (if + present, present) for general truths; first (if + present, will) for real future; second (if + past, would) for unreal present; third (if + past perfect, would have + past participle) for unreal past.

Passive voice: be + past participle; only transitive verbs passivize; by-agent introduced with by; progressive passive be being + past participle.

Reported speech: backshift one tense (present→past, past→past perfect, will→would); pronouns and time expressions change (now→then, today→that day, tomorrow→the next day).

Relative clauses: defining clauses give essential information (no commas, can use that); non-defining clauses add extra information (commas, cannot use that); whose shows possession.

Modals: can (ability), may (permission/possibility), must (obligation), should (advice), might (possibility), could (past ability/polite), needn't (absence of obligation), mustn't (prohibition).

Word formation: prefixes (un-, dis-, re-, pre-, mis-) and suffixes (-tion, -ment, -ness, -able, -ful, -less, -ly) change meaning or part of speech; compound words and conversion (noun to verb) are common.

Common confusions: affect (verb) vs effect (noun); its (possessive) vs it's (it is); their/there/they're; lie (intransitive) vs lay (transitive); fewer (countable) vs less (uncountable).

French Revolution key dates: 1789 storming of the Bastille (14 July) and Declaration of the Rights of Man; 1792 proclamation of the First Republic; 1793–94 Reign of Terror; 1799 Napoleon's coup.

Napoleon: Emperor 1804–1814/15; Napoleonic Code (1804) influenced civil law; defeated at Waterloo 1815; Congress of Vienna 1814–15 redrew Europe.

World Wars: WWI 1914–1918, triggered by assassination of Archduke Franz Ferdinand; Treaty of Versailles 1919; WWII 1939–1945; Holocaust; United Nations founded 1945.

Cold War: 1947–1991; NATO (1949) vs Warsaw Pact (1955); Berlin Wall 1961–1989; Cuban Missile Crisis 1962; fall of USSR 1991.

Decolonization: India independent 1947; most African colonies independent 1958–1962; Algerian War 1954–1962; Bandung Conference 1955.

French Fifth Republic: founded 1958 by Charles de Gaulle; Constitution strengthened presidency; current institutions: President, Prime Minister, Parliament (Assemblée nationale and Sénat).

Geography of France: metropolitan France has 13 régions; major rivers Seine, Loire, Rhône, Garonne, Rhine; mountain ranges Alps, Pyrenees, Massif Central, Jura, Vosges; climate types oceanic, continental, Mediterranean, mountain.

Earth and maps: latitude/longitude; equator 0°, tropics at 23.5°, polar circles at 66.5°; time zones; map projections (Mercator distorts area, Peters preserves area); scale ratio.

Demography and urbanization: world population ~8 billion; demographic transition model stages; urbanization rate; megacities; push-pull migration factors.

Branches of philosophy: metaphysics (being, reality), epistemology (knowledge), ethics (morality), logic (reasoning), aesthetics (beauty), political philosophy.

Socrates: Socratic method (questioning to expose ignorance); Plato: theory of Forms, allegory of the cave, tripartite soul; Aristotle: four causes, virtue ethics as golden mean, logic syllogism.

Descartes: cogito ergo sum; mind-body dualism; methodical doubt. Kant: categorical imperative; transcendental idealism; distinction between phenomena and noumena.

Empiricism (Locke, Berkeley, Hume): knowledge from experience; rationalism (Descartes, Spinoza, Leibniz): reason as source of knowledge.

Nietzsche: will to power, Übermensch, critique of morality; existentialism (Sartre: existence precedes essence; freedom and responsibility); phenomenology (Husserl).

Political philosophy: Hobbes (social contract, absolute sovereignty), Locke (natural rights, limited government), Rousseau (general will, popular sovereignty), Montesquieu (separation of powers).

Art history periods: Antiquity, Middle Ages (Romanesque, Gothic), Renaissance (15th–16th c.), Baroque (17th c.), Neoclassicism, Romanticism, Realism, Impressionism, Cubism, Surrealism.

Music: Baroque (Bach, Vivaldi), Classical (Mozart, Haydn, Beethoven), Romantic (Chopin, Wagner), modern (Debussy, Stravinsky); basic forms: sonata, symphony, concerto, opera.

Science milestones: Copernicus heliocentrism; Galileo telescopic observations; Newton Principia (1687); Darwin On the Origin of Species (1859); Einstein relativity (1905, 1915); DNA double helix 1953.

Deductive validity: an argument is valid if the conclusion necessarily follows from the premises; soundness requires validity plus true premises; a valid argument can have false premises.

Common fallacies: ad hominem (attack person), straw man (misrepresent), false dilemma (only two options), slippery slope, appeal to authority, circular reasoning, post hoc ergo propter hoc.

Syllogism: categorical statements (All A are B; No A are B; Some A are B; Some A are not B); use Venn diagrams to test validity; converse of 'All A are B' is not necessarily true.

Conditional logic: 'If P then Q' is equivalent to 'If not Q then not P' (contrapositive); converse Q→P and inverse ¬P→¬Q are not equivalent; P is sufficient for Q, Q is necessary for P.

Quantifiers: negation of 'All A are B' is 'Some A are not B'; negation of 'Some A are B' is 'No A are B'; be careful with scope of 'only', 'unless', 'except'.

Sequences: arithmetic difference constant; geometric ratio constant; Fibonacci each term sum of previous two; look for alternating, square, prime, or combined patterns.

Problem-solving strategies: work backward, draw a diagram, make a table, find a pattern, simplify, use parity or extreme cases, check units and reasonableness of answer.

Ratios and proportions: cross-multiplication a/b=c/d ⇒ ad=bc; percentage change = (new−old)/old×100; average = sum/count; weighted average accounts for frequencies.