Quiz — serie 2(corrigé).pdf

Question 1 sur 10 20:00

[{"id":17889,"question":"Quelle est la dérivée de la fonction f(x) = 3x² - 5x + 2 ?","option_a":"6x - 5","option_b":"3x - 5","option_c":"6x² - 5","option_d":"3x² - 5x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'un polynôme se calcule terme par terme : (3x²)' = 6x, (-5x)' = -5, et la dérivée de 2 est 0. Donc f'(x) = 6x - 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\": \"6x - 5\", \"b\": \"3x - 5\", \"c\": \"6x² - 5\", \"d\": \"3x² - 5x\"}}","_debug_options_count":4},{"id":17890,"question":"La fonction f(x) = x³ - 3x est croissante sur l'intervalle [-2, 2].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f'(x) = 3x² - 3 s'annule en x = ±1. La fonction est décroissante sur [-1, 1] et croissante ailleurs sur [-2, 2]. Donc l'affirmation est fausse.","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":17891,"question":"Quelle est la limite de (sin(x)\/x) lorsque x tend vers 0 ?","option_a":"0","option_b":"1","option_c":"∞","option_d":"n'existe pas","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est un cas classique de limite trigonométrique : lim(x→0) sin(x)\/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\": \"0\", \"b\": \"1\", \"c\": \"∞\", \"d\": \"n'existe pas\"}}","_debug_options_count":4},{"id":17892,"question":"Soit une suite définie par uₙ₊₁ = 0.5uₙ + 3 avec u₀ = 4. Cette suite est :","option_a":"Croissante","option_b":"Décroissante","option_c":"Constante","option_d":"Non monotone","option_e":"","option_f":"","bonne_reponse":"b","explication":"Calculons u₁ = 0.5*4 + 3 = 5, u₂ = 0.5*5 + 3 = 5.5, etc. La suite est strictement croissante car uₙ₊₁ \u003E uₙ pour tout n.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Croissante\", \"b\": \"Décroissante\", \"c\": \"Constante\", \"d\": \"Non mo","_debug_options_count":4},{"id":17893,"question":"L'équation eˣ = 2 admet une solution unique dans ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle est strictement croissante et bijective de ℝ vers ℝ⁺*, donc l'équation eˣ = 2 admet une solution unique (x = ln(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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":17894,"question":"Quelle est la forme trigonométrique du nombre complexe z = -1 + i√3 ?","option_a":"2(cos(π\/3) + i sin(π\/3))","option_b":"2(cos(2π\/3) + i sin(2π\/3))","option_c":"√2(cos(3π\/4) + i sin(3π\/4))","option_d":"2(cos(π\/6) + i sin(π\/6))","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le module de z est √((-1)² + (√3)²) = 2. L'argument θ vérifie cos(θ) = -1\/2 et sin(θ) = √3\/2, donc θ = 2π\/3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"2(cos(π\/3) + i sin(π\/3))\", \"b\": \"2(cos(2π\/3) + i sin(2π\/3))\",","_debug_options_count":4},{"id":17895,"question":"Soit X une variable aléatoire suivant une loi normale N(0,1). P(X ≤ 1.96) est approximativement égal à :","option_a":"0.95","option_b":"0.975","option_c":"0.99","option_d":"0.90","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour une loi normale centrée réduite, P(X ≤ 1.96) ≈ 0.975 (valeur issue des tables 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\": \"0.95\", \"b\": \"0.975\", \"c\": \"0.99\", \"d\": \"0.90\"}}","_debug_options_count":4},{"id":17896,"question":"La fonction f(x) = ln(x) est définie pour tout x réel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction logarithme est définie uniquement pour x \u003E 0. Elle n'est pas définie pour x ≤ 0.","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":17897,"question":"Quelle est la solution de l'inéquation 2ˣ \u003E 8 ?","option_a":"x \u003E 3","option_b":"x \u003C 3","option_c":"x \u003E 0","option_d":"x \u003C 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"2ˣ \u003E 8 ⇔ 2ˣ \u003E 2³ ⇔ x \u003E 3 (car la fonction exponentielle de base 2 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\": \"x \u003E 3\", \"b\": \"x \u003C 3\", \"c\": \"x \u003E 0\", \"d\": \"x \u003C 0\"}}","_debug_options_count":4},{"id":17898,"question":"Soit f une fonction continue sur [a,b] telle que f(a) \u003C 0 et f(b) \u003E 0. Alors f s'annule au moins une fois sur [a,b].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est le théorème des valeurs intermédiaires : si f est continue sur [a,b] et change de signe, alors elle s'annule au moins une fois sur cet intervalle.","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}]