Quiz — 65faac1ca3350_corrigé_Série 32.pdf

Question 1 sur 10 20:00

[{"id":41715,"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 d'un polynôme de même degré au numérateur et au dénominateur est égale au rapport des coefficients dominants. Ici, 3x²\/x² = 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":41716,"question":"La fonction f(x) = x³ - 3x² + 2 est strictement croissante sur l'intervalle [0, 2].","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. Sur ]0,2[, f'(x) \u003C 0, donc la fonction est décroissante.","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":41717,"question":"Soit la suite définie par u₀ = 2 et uₙ₊₁ = 2uₙ - 1. Quelle est la limite de cette suite ?","option_a":"A) 0","option_b":"B) 1","option_c":"C) 2","option_d":"D) +∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite est arithmético-géométrique. En résolvant l'équation l = 2l - 1, on trouve l = 1. La suite converge vers 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) +∞\"}}","_debug_options_count":4},{"id":41718,"question":"Quelle est la valeur de la dérivée de f(x) = ln(2x + 1) en x = 0 ?","option_a":"A) 0","option_b":"B) 1","option_c":"C) 2","option_d":"D) 4","option_e":"","option_f":"","bonne_reponse":"c","explication":"La dérivée de ln(u) est u'\/u. Ici, u = 2x + 1, donc u' = 2. En x=0, f'(0) = 2\/(2*0 + 1) = 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) 0\", \"b\": \"B) 1\", \"c\": \"C) 2\", \"d\": \"D) 4\"}}","_debug_options_count":4},{"id":41719,"question":"L'équation x² - 4x + 5 = 0 admet deux solutions réelles distinctes.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le discriminant Δ = (-4)² - 4*1*5 = 16 - 20 = -4 \u003C 0. L'équation n'a pas de solutions réelles.","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":41720,"question":"Quelle est la probabilité d'obtenir un nombre pair en lançant un dé équilibré ?","option_a":"A) 1\/6","option_b":"B) 1\/3","option_c":"C) 1\/2","option_d":"D) 2\/3","option_e":"","option_f":"","bonne_reponse":"c","explication":"Un dé a 6 faces. Les nombres pairs sont 2, 4 et 6, soit 3 cas favorables sur 6 possibles. La probabilité est donc 3\/6 = 1\/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\/6\", \"b\": \"B) 1\/3\", \"c\": \"C) 1\/2\", \"d\": \"D) 2\/3\"}}","_debug_options_count":4},{"id":41721,"question":"Soit A = (1 2; 3 4) une matrice carrée. Quel est son déterminant ?","option_a":"A) -2","option_b":"B) 0","option_c":"C) 2","option_d":"D) 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le déterminant d'une matrice 2x2 (a b; c d) est ad - bc. Ici, (1*4) - (2*3) = 4 - 6 = -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) -2\", \"b\": \"B) 0\", \"c\": \"C) 2\", \"d\": \"D) 5\"}}","_debug_options_count":4},{"id":41722,"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 f est paire si 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":41723,"question":"Quelle est l'équation de la tangente à la courbe y = x² - 3x + 2 au point d'abscisse x = 1 ?","option_a":"A) y = -x + 1","option_b":"B) y = x - 1","option_c":"C) y = -x - 1","option_d":"D) y = x + 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La tangente a pour équation y = f'(a)(x - a) + f(a). Ici, f'(x) = 2x - 3, f(1) = 0, f'(1) = -1. Donc y = -1(x - 1) + 0 = -x + 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 = -x + 1\", \"b\": \"B) y = x - 1\", \"c\": \"C) y = -x - 1\", \"d\": \"","_debug_options_count":4},{"id":41724,"question":"Soit une variable aléatoire X 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 (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}]