Quiz — serie 24 p2.pdf

Question 1 sur 10 20:00

[{"id":35090,"question":"Soit f une fonction dérivable sur ℝ telle que f'(x) = 2x + 1. Quelle est la dérivée de la fonction g(x) = f(x²) ?","option_a":"A) 2x(2x + 1)","option_b":"B) 4x² + 2x","option_c":"C) 2x + 1","option_d":"D) 4x + 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"On utilise la formule de dérivation des fonctions composées : g'(x) = f'(x²) * (x²)' = (2x² + 1) * 2x = 4x³ + 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) 2x(2x + 1)\", \"b\": \"B) 4x² + 2x\", \"c\": \"C) 2x + 1\", \"d\": \"D) 4","_debug_options_count":4},{"id":35091,"question":"La fonction f(x) = x³ - 3x² + 2 est croissante sur l'intervalle [-1, 1].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f'(x) = 3x² - 6x = 3x(x - 2). Sur [-1, 1], f'(x) ≤ 0, donc la fonction est décroissante 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":35092,"question":"Soit (Un) une suite géométrique de raison q = 2 et de premier terme U0 = 3. Quelle est la valeur de U3 ?","option_a":"A) 6","option_b":"B) 12","option_c":"C) 24","option_d":"D) 48","option_e":"","option_f":"","bonne_reponse":"c","explication":"U3 = U0 * q³ = 3 * 2³ = 3 * 8 = 24.","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) 6\", \"b\": \"B) 12\", \"c\": \"C) 24\", \"d\": \"D) 48\"}}","_debug_options_count":4},{"id":35093,"question":"Dans une classe de 30 élèves, 12 sont des filles. On choisit un élève au hasard. Quelle est la probabilité que ce soit une fille ?","option_a":"A) 0.3","option_b":"B) 0.4","option_c":"C) 0.5","option_d":"D) 0.6","option_e":"","option_f":"","bonne_reponse":"b","explication":"La probabilité est le nombre de filles divisé par le nombre total d'élèves : 12\/30 = 0.4.","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) 0.3\", \"b\": \"B) 0.4\", \"c\": \"C) 0.5\", \"d\": \"D) 0.6\"}}","_debug_options_count":4},{"id":35094,"question":"Une équation différentielle de la forme y' + 2y = 0 a pour solution générale y(x) = Ce^(-2x), où C est une constante.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution générale de y' + 2y = 0 est bien y(x) = Ce^(-2x), avec C une constante réelle.","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":35095,"question":"Soit A et B deux événements indépendants tels que P(A) = 0.4 et P(B) = 0.5. Quelle est la probabilité de P(A ∩ B) ?","option_a":"A) 0.2","option_b":"B) 0.4","option_c":"C) 0.5","option_d":"D) 0.9","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour des événements indépendants, P(A ∩ B) = P(A) * P(B) = 0.4 * 0.5 = 0.2.","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) 0.2\", \"b\": \"B) 0.4\", \"c\": \"C) 0.5\", \"d\": \"D) 0.9\"}}","_debug_options_count":4},{"id":35096,"question":"La fonction f(x) = e^(2x) a pour dérivée f'(x) = 2e^(2x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de e^(u(x)) est u'(x)e^(u(x)). Ici, u(x) = 2x, donc u'(x) = 2. Ainsi, f'(x) = 2e^(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":35097,"question":"Soit (Un) une suite arithmétique de premier terme U0 = 5 et de raison r = -2. Quelle est la valeur de U5 ?","option_a":"A) -5","option_b":"B) -3","option_c":"C) 1","option_d":"D) 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"U5 = U0 + 5r = 5 + 5*(-2) = 5 - 10 = -5.","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) -3\", \"c\": \"C) 1\", \"d\": \"D) 5\"}}","_debug_options_count":4},{"id":35098,"question":"Soit f une fonction continue sur [a, b] et dérivable sur ]a, b[. Si f(a) = f(b), alors il existe c ∈ ]a, b[ tel que f'(c) = 0 (Théorème de Rolle).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le théorème de Rolle stipule exactement cela : si f est continue sur [a, b], dérivable sur ]a, b[ et f(a) = f(b), alors il existe c ∈ ]a, b[ tel que f'(c) = 0.","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":35099,"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, P(-1 ≤ X ≤ 1) ≈ 0.6826 (environ 68% des valeurs sont dans 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) 0.3413\", \"b\": \"B) 0.6826\", \"c\": \"C) 0.9544\", \"d\": \"D) 0.9974\"}","_debug_options_count":4}]