Quiz — Correction Compiègne.pdf

Question 1 sur 10 20:00

[{"id":106006,"question":"Quelle est la limite de la suite définie par uₙ = (3n² + 2n - 1)\/(2n² + 5) quand n tend vers +∞ ?","option_a":"A. 0","option_b":"B. 3\/2","option_c":"C. +∞","option_d":"D. 1","option_e":"","option_f":"","bonne_reponse":"b","explication":"La limite d'une suite rationnelle est égale au rapport des termes de plus haut degré. Ici, 3n²\/2n² = 3\/2.","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. 3\/2\", \"c\": \"C. +∞\", \"d\": \"D. 1\"}}","_debug_options_count":4},{"id":106007,"question":"La fonction f(x) = x³ - 3x² + 4 est croissante sur l'intervalle [-1, 3].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f'(x) = 3x² - 6x s'annule en x=0 et x=2. Le signe de f' montre que la fonction est décroissante sur [0,2].","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},{"id":106008,"question":"Quelle est la dérivée de la fonction f(x) = ln(2x + 1) ?","option_a":"A. 1\/(2x + 1)","option_b":"B. 2\/(2x + 1)","option_c":"C. 1\/(x + 1)","option_d":"D. 2x\/(2x + 1)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de ln(u) est u'\/u. Ici, u = 2x + 1 donc u' = 2. Ainsi, f'(x) = 2\/(2x + 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. 1\/(2x + 1)\", \"b\": \"B. 2\/(2x + 1)\", \"c\": \"C. 1\/(x + 1)\", \"d\": \"","_debug_options_count":4},{"id":106009,"question":"Soit X une variable aléatoire suivant une loi normale N(0,1). Quelle est la probabilité P(-1 ≤ X ≤ 1) ?","option_a":"A. 0.3413","option_b":"B. 0.6826","option_c":"C. 0.9544","option_d":"D. 0.9974","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour une loi normale centrée réduite, environ 68% des valeurs sont dans l'intervalle [-1,1].","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. 0.3413\", \"b\": \"B. 0.6826\", \"c\": \"C. 0.9544\", \"d\": \"D. 0.9974\"}","_debug_options_count":4},{"id":106010,"question":"L'équation différentielle y' + 2y = 0 a pour solution générale y = Ce^(-2x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution générale est y = Ce^(-2x) seulement si la condition initiale est y(0) = C. La solution générale correcte est y = Ce^(-2x).","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":106011,"question":"Quelle est l'intégrale ∫(3x² + 2x)dx ?","option_a":"A. x³ + x² + C","option_b":"B. x³ + x + C","option_c":"C. 6x + 2 + C","option_d":"D. x³ + x²","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale de 3x² est x³ et celle de 2x est x². La constante d'intégration C est indispensable.","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. x³ + x² + C\", \"b\": \"B. x³ + x + C\", \"c\": \"C. 6x + 2 + C\", \"","_debug_options_count":4},{"id":106012,"question":"Dans une classe de 30 élèves, 18 sont des filles. On choisit un élève au hasard. Quelle est la probabilité que ce soit une fille ?","option_a":"A. 0.4","option_b":"B. 0.6","option_c":"C. 0.5","option_d":"D. 0.3","option_e":"","option_f":"","bonne_reponse":"b","explication":"La probabilité est le nombre de cas favorables divisé par le nombre total de cas : 18\/30 = 0.6.","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. 0.4\", \"b\": \"B. 0.6\", \"c\": \"C. 0.5\", \"d\": \"D. 0.3\"}}","_debug_options_count":4},{"id":106013,"question":"La fonction f(x) = e^(-x²) est paire.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une fonction paire vérifie f(-x) = f(x). Ici, f(-x) = e^(-(-x)²) = e^(-x²) = f(x). La fonction est donc paire.","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":106014,"question":"Quelle est la solution de l'équation différentielle y'' + 4y = 0 avec y(0) = 1 et y'(0) = 0 ?","option_a":"A. y = cos(2x)","option_b":"B. y = sin(2x)","option_c":"C. y = e^(2x)","option_d":"D. y = 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation caractéristique est r² + 4 = 0, soit r = ±2i. La solution générale est y = A cos(2x) + B sin(2x). Avec les conditions initiales, on obtient y = cos(2x).","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 = cos(2x)\", \"b\": \"B. y = sin(2x)\", \"c\": \"C. y = e^(2x)\", \"d\"","_debug_options_count":4},{"id":106015,"question":"Soit une série statistique de moyenne 50 et d'écart-type 10. Quel est l'intervalle [μ - σ, μ + σ] ?","option_a":"A. [40, 60]","option_b":"B. [30, 70]","option_c":"C. [20, 80]","option_d":"D. [10, 90]","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intervalle est [50 - 10, 50 + 10] = [40, 60]. Cet intervalle contient environ 68% des données pour une distribution normale.","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. [40, 60]\", \"b\": \"B. [30, 70]\", \"c\": \"C. [20, 80]\", \"d\": \"D. [1","_debug_options_count":4}]