Quiz — Lycée Pilote Sousse - DS 1 2018 (+Corrigé).pdf

Question 1 sur 10 20:00

[{"id":41495,"question":"Quelle est la solution de l'équation x² - 5x + 6 = 0 ?","option_a":"A) x = 2 ou x = 3","option_b":"B) x = -2 ou x = -3","option_c":"C) x = 1 ou x = 6","option_d":"D) x = 0 ou x = 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation x² - 5x + 6 = 0 se factorise en (x-2)(x-3) = 0, donc les solutions sont x = 2 et x = 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) x = 2 ou x = 3\", \"b\": \"B) x = -2 ou x = -3\", \"c\": \"C) x = 1 ou","_debug_options_count":4},{"id":41496,"question":"La fonction f(x) = (x² + 1)\/x est-elle définie en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction f(x) = (x² + 1)\/x n'est pas définie en x = 0 car le dénominateur s'annule. C'est une fonction rationnelle avec une discontinuité en 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":41497,"question":"Quelle est la dérivée de la fonction f(x) = x³ - 2x² + 5x - 7 ?","option_a":"A) f'(x) = 3x² - 4x + 5","option_b":"B) f'(x) = 3x² - 4x + 5x","option_c":"C) f'(x) = 3x³ - 4x² + 5x","option_d":"D) f'(x) = x² - 2x + 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme est obtenue en appliquant la règle de dérivation terme à terme : f'(x) = 3x² - 4x + 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) f'(x) = 3x² - 4x + 5\", \"b\": \"B) f'(x) = 3x² - 4x + 5x\", \"c\":","_debug_options_count":4},{"id":41498,"question":"Si lim(x→2) f(x) = 3 et lim(x→2) g(x) = 5, alors lim(x→2) [f(x) + g(x)] = ?","option_a":"A) 8","option_b":"B) 2","option_c":"C) 15","option_d":"D) -2","option_e":"","option_f":"","bonne_reponse":"a","explication":"D'après la propriété des limites, lim(x→2) [f(x) + g(x)] = lim(x→2) f(x) + lim(x→2) g(x) = 3 + 5 = 8.","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) 8\", \"b\": \"B) 2\", \"c\": \"C) 15\", \"d\": \"D) -2\"}}","_debug_options_count":4},{"id":41499,"question":"La fonction f(x) = x² est-elle dérivable en x = 1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction f(x) = x² est dérivable en tout point de son domaine, y compris en x = 1. Sa dérivée est f'(x) = 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":41500,"question":"Quelle est la solution de l'inéquation x² - 4x + 3 \u003E 0 ?","option_a":"A) x ∈ ]-∞; 1[ ∪ ]3; +∞[","option_b":"B) x ∈ ]1; 3[","option_c":"C) x ∈ ]-∞; +∞[","option_d":"D) x ∈ {1, 3}","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'inéquation x² - 4x + 3 \u003E 0 se résout en étudiant le signe du polynôme. Les racines sont x = 1 et x = 3, et le polynôme est positif à l'extérieur des racines.","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) x ∈ ]-∞; 1[ ∪ ]3; +∞[\", \"b\": \"B) x ∈ ]1; 3[\", \"c\": \"","_debug_options_count":4},{"id":41501,"question":"La fonction f(x) = 1\/x est-elle continue sur son domaine de définition ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction f(x) = 1\/x est continue sur son domaine de définition, c'est-à-dire pour tout x ≠ 0. Elle présente une discontinuité en x = 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":41502,"question":"Quelle est la limite de la fonction f(x) = (x² - 1)\/(x - 1) lorsque x tend vers 1 ?","option_a":"A) 2","option_b":"B) 0","option_c":"C) 1","option_d":"D) +∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction f(x) = (x² - 1)\/(x - 1) se simplifie en f(x) = x + 1 pour x ≠ 1. Donc lim(x→1) f(x) = 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) 1\", \"d\": \"D) +∞\"}}","_debug_options_count":4},{"id":41503,"question":"Si f est une fonction dérivable sur ℝ et f'(x) = 0 pour tout x, alors f est :","option_a":"A) Constante","option_b":"B) Croissante","option_c":"C) Décroissante","option_d":"D) Périodique","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si la dérivée d'une fonction est nulle sur tout son domaine, alors la fonction est constante (théorème des fonctions constantes).","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) Constante\", \"b\": \"B) Croissante\", \"c\": \"C) Décroissante\", \"d\"","_debug_options_count":4},{"id":41504,"question":"La fonction f(x) = x³ - 3x² + 2x est-elle 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 + 2 s'annule en x = 1 ± √(1\/3). Sur [0; 2], la fonction n'est pas monotone (elle décroît puis croît).","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}]