Quiz — 6357baf716fae_EX10.pdf

Question 1 sur 10 20:00

[{"id":58152,"question":"Quelle est la limite de la fonction f(x) = (3x² - 2x + 1)\/(x² + 1) lorsque x tend vers l'infini ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 3","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"La limite est obtenue en divisant le numérateur et le dénominateur par x², ce qui donne 3\/1 = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. 3\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":58153,"question":"La fonction f(x) = x³ - 3x² + 2x est croissante sur l'intervalle [0 ; 2].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 3x² - 6x + 2 est positive sur [0 ; 2] (car Δ \u003C 0 et coefficient dominant positif), donc la fonction est croissante.","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":58154,"question":"Résolvez l'équation ln(x² - 5x + 6) = 0. Quelle est la somme des solutions ?","option_a":"A. 5","option_b":"B. 6","option_c":"C. 1","option_d":"D. 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"ln(x² - 5x + 6) = 0 ⇒ x² - 5x + 6 = 1 ⇒ x² - 5x + 5 = 0. La somme des racines est 5 (formule -b\/a).","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. 5\", \"b\": \"B. 6\", \"c\": \"C. 1\", \"d\": \"D. 0\"}}","_debug_options_count":4},{"id":58155,"question":"La dérivée de la fonction f(x) = e^(2x+1) est f'(x) = 2e^(2x+1).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En appliquant la formule de dérivation des fonctions exponentielles, f'(x) = 2e^(2x+1) est correcte.","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":58156,"question":"Quel est le domaine de définition de la fonction f(x) = √(x² - 4x + 3) ?","option_a":"A. ]-∞ ; 1] ∪ [3 ; +∞[","option_b":"B. [1 ; 3]","option_c":"C. ]-∞ ; +∞[","option_d":"D. ]-∞ ; 0] ∪ [4 ; +∞[","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le discriminant de x² - 4x + 3 est Δ = 4 \u003E 0, les racines sont 1 et 3. Le domaine est donc ]-∞ ; 1] ∪ [3 ; +∞[.","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] ∪ [3 ; +∞[\", \"b\": \"B. [1 ; 3]\", \"c\": \"C. ]-∞ ","_debug_options_count":4},{"id":58157,"question":"Soit une variable aléatoire X suivant une loi normale N(0 ; 1). Que vaut 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, P(-1 ≤ X ≤ 1) ≈ 0,6826 (règle des 68-95-99,7).","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":58158,"question":"La fonction f(x) = |x² - 4| est dérivable en x = 2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction n'est pas dérivable en x = 2 car la valeur absolue crée un point anguleux (dérivée à gauche ≠ dérivée à droite).","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":58159,"question":"Calculez l'intégrale ∫(2x + 1)dx de 0 à 1. Le résultat est :","option_a":"A. 1","option_b":"B. 2","option_c":"C. 1,5","option_d":"D. 3","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'intégrale de 2x + 1 est x² + x. Évaluée entre 0 et 1, on obtient (1 + 1) - (0 + 0) = 2. Mais ∫(2x + 1)dx = x² + x + C, donc entre 0 et 1 : (1² + 1) - (0² + 0) = 2. Correction : la bonne réponse est B (2).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"A. 1\", \"b\": \"B. 2\", \"c\": \"C. 1,5\", \"d\": \"D. 3\"}}","_debug_options_count":4},{"id":58160,"question":"Soit une suite définie par u₀ = 2 et uₙ₊₁ = 3uₙ - 1. Cette suite est arithmético-géométrique.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une suite arithmético-géométrique est de la forme uₙ₊₁ = a·uₙ + b, ce qui correspond ici à a = 3 et b = -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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":58161,"question":"Quelle est la valeur de la somme S = 1 + 2 + 4 + ... + 2ⁿ pour n = 5 ?","option_a":"A. 31","option_b":"B. 63","option_c":"C. 127","option_d":"D. 255","option_e":"","option_f":"","bonne_reponse":"b","explication":"Il s'agit d'une suite géométrique de raison 2. La somme des n+1 premiers termes est S = 2⁶ - 1 = 63.","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. 31\", \"b\": \"B. 63\", \"c\": \"C. 127\", \"d\": \"D. 255\"}}","_debug_options_count":4}]