Quiz — Corrigé Exam S1 2010-2011.doc

Question 1 sur 10 20:00

[{"id":19409,"question":"Quelle est la dérivée de la fonction f(x) = x² * ln(x) ?","option_a":"A: 2x * ln(x) + x","option_b":"B: 2x * ln(x) + 1","option_c":"C: x * ln(x) + 2x","option_d":"D: x * ln(x) + 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de x² * ln(x) s'obtient par la formule de dérivation d'un produit : (u*v)' = u'v + uv'. Ici, u = x² et v = ln(x), donc u' = 2x et v' = 1\/x. Ainsi, f'(x) = 2x * ln(x) + x² * (1\/x) = 2x * ln(x) + x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A: 2x * ln(x) + x\", \"b\": \"B: 2x * ln(x) + 1\", \"c\": \"C: x * ln(x) ","_debug_options_count":4},{"id":19410,"question":"L'intégrale de 0 à 1 de e^(-x) dx est égale à :","option_a":"A: 1 - 1\/e","option_b":"B: e - 1","option_c":"C: 1\/e - 1","option_d":"D: 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale de e^(-x) est -e^(-x). En évaluant entre 0 et 1, on obtient [-e^(-1)] - [-e^(0)] = -1\/e + 1 = 1 - 1\/e.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A: 1 - 1\/e\", \"b\": \"B: e - 1\", \"c\": \"C: 1\/e - 1\", \"d\": \"D: 1\"}}","_debug_options_count":4},{"id":19411,"question":"La fonction f(x) = x³ - 3x² + 4 admet un point d'inflexion en x = :","option_a":"A: 0","option_b":"B: 1","option_c":"C: 2","option_d":"D: 3","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le point d'inflexion correspond à un changement de concavité, donc où la dérivée seconde s'annule. f''(x) = 6x - 6. f''(x) = 0 donne x = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A: 0\", \"b\": \"B: 1\", \"c\": \"C: 2\", \"d\": \"D: 3\"}}","_debug_options_count":4},{"id":19412,"question":"Vrai ou Faux ? La probabilité d'un événement impossible est toujours égale à 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, un événement impossible a une probabilité de 0, car il ne peut pas se produire.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":19413,"question":"Soit X une variable aléatoire suivant une loi normale N(0,1). Quelle est la probabilité P(X ≤ 1) ?","option_a":"A: 0.6826","option_b":"B: 0.8413","option_c":"C: 0.9544","option_d":"D: 0.9772","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour une loi normale centrée réduite, P(X ≤ 1) ≈ 0.8413 (valeur lue dans la table de la loi normale).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A: 0.6826\", \"b\": \"B: 0.8413\", \"c\": \"C: 0.9544\", \"d\": \"D: 0.9772\"}","_debug_options_count":4},{"id":19414,"question":"L'équation de la tangente à la courbe y = x² + 2x en x = 1 est :","option_a":"A: y = 4x + 1","option_b":"B: y = 4x - 1","option_c":"C: y = 2x + 1","option_d":"D: y = 2x - 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La tangente en x = 1 a pour équation y = f'(1)(x - 1) + f(1). f'(x) = 2x + 2, donc f'(1) = 4. f(1) = 1 + 2 = 3. Ainsi, y = 4(x - 1) + 3 = 4x - 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A: y = 4x + 1\", \"b\": \"B: y = 4x - 1\", \"c\": \"C: y = 2x + 1\", \"d\": ","_debug_options_count":4},{"id":19415,"question":"Vrai ou Faux ? Si deux événements A et B sont indépendants, alors P(A ∩ B) = P(A) * P(B).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est la définition de l'indépendance entre deux événements : P(A ∩ B) = P(A) * P(B).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":19416,"question":"Quelle est la limite de (sin(x) - x)\/x³ quand x tend vers 0 ?","option_a":"A: 0","option_b":"B: -1\/6","option_c":"C: 1\/6","option_d":"D: ∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"En utilisant le développement limité de sin(x) = x - x³\/6 + o(x³), on obtient (sin(x) - x)\/x³ ≈ (-x³\/6)\/x³ = -1\/6.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A: 0\", \"b\": \"B: -1\/6\", \"c\": \"C: 1\/6\", \"d\": \"D: ∞\"}}","_debug_options_count":4},{"id":19417,"question":"Soit f une fonction dérivable sur R. Si f'(x) \u003E 0 pour tout x ∈ R, alors :","option_a":"A: f est décroissante sur R","option_b":"B: f est croissante sur R","option_c":"C: f est constante sur R","option_d":"D: f est périodique","option_e":"","option_f":"","bonne_reponse":"b","explication":"Si la dérivée est strictement positive sur un intervalle, la fonction est strictement croissante sur cet intervalle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A: f est décroissante sur R\", \"b\": \"B: f est croissante sur R\", ","_debug_options_count":4},{"id":19418,"question":"Vrai ou Faux ? L'espérance d'une variable aléatoire discrète est toujours un nombre entier.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'espérance est une moyenne pondérée des valeurs possibles, elle peut être un nombre réel non entier.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]