Quiz — corrigé série 8.pdf

Question 1 sur 10 20:00

[{"id":95469,"question":"Quelle est la solution de l'équation x² - 5x + 6 = 0 ?","option_a":"x = 2 ou x = 3","option_b":"x = -2 ou x = -3","option_c":"x = 1 ou x = 6","option_d":"x = 0 ou x = 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation 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\": \"x = 2 ou x = 3\", \"b\": \"x = -2 ou x = -3\", \"c\": \"x = 1 ou x = 6\", ","_debug_options_count":4},{"id":95470,"question":"La fonction f(x) = x³ - 3x est-elle croissante sur ℝ ?","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 n'est pas croissante sur tout ℝ (elle décroît entre -1 et 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":95471,"question":"Quelle est la dérivée de la fonction f(x) = e^(2x) ?","option_a":"f'(x) = 2e^(2x)","option_b":"f'(x) = e^(2x)","option_c":"f'(x) = 2xe^(2x)","option_d":"f'(x) = e^x","option_e":"","option_f":"","bonne_reponse":"a","explication":"En appliquant la règle de dérivation des fonctions composées, 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\": \"f'(x) = 2e^(2x)\", \"b\": \"f'(x) = e^(2x)\", \"c\": \"f'(x) = 2xe^(2x)\",","_debug_options_count":4},{"id":95472,"question":"L'intégrale de 0 à 1 de x² dx vaut :","option_a":"1\/3","option_b":"1","option_c":"0","option_d":"2\/3","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale de x² est [x³\/3] de 0 à 1, soit 1\/3 - 0 = 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\": \"1\/3\", \"b\": \"1\", \"c\": \"0\", \"d\": \"2\/3\"}}","_debug_options_count":4},{"id":95473,"question":"La limite de (sin x)\/x quand x tend vers 0 est égale à 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est un résultat fondamental en analyse : 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\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":95474,"question":"Quelle est la solution de l'inéquation 2x - 5 \u003E 3x + 1 ?","option_a":"x \u003C -6","option_b":"x \u003E -6","option_c":"x \u003E 6","option_d":"x \u003C 6","option_e":"","option_f":"","bonne_reponse":"a","explication":"En simplifiant, on obtient -x \u003E 6, soit x \u003C -6.","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 \u003C -6\", \"b\": \"x \u003E -6\", \"c\": \"x \u003E 6\", \"d\": \"x \u003C 6\"}}","_debug_options_count":4},{"id":95475,"question":"La fonction f(x) = ln(x) est définie pour :","option_a":"x \u003E 0","option_b":"x ≥ 0","option_c":"x \u003C 0","option_d":"x ∈ ℝ","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le logarithme naturel est défini uniquement pour les nombres strictement positifs.","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 0\", \"b\": \"x ≥ 0\", \"c\": \"x \u003C 0\", \"d\": \"x ∈ ℝ\"}}","_debug_options_count":4},{"id":95476,"question":"L'aire sous la courbe de f(x)=x² entre 0 et 2 est :","option_a":"8\/3","option_b":"4","option_c":"2","option_d":"16\/3","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale de x² de 0 à 2 vaut [x³\/3] de 0 à 2 = 8\/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\": \"8\/3\", \"b\": \"4\", \"c\": \"2\", \"d\": \"16\/3\"}}","_debug_options_count":4},{"id":95477,"question":"La dérivée de sin(2x) est :","option_a":"2cos(2x)","option_b":"cos(2x)","option_c":"2sin(2x)","option_d":"-2cos(2x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En appliquant la règle de dérivation des fonctions composées, f'(x) = 2cos(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\": \"2cos(2x)\", \"b\": \"cos(2x)\", \"c\": \"2sin(2x)\", \"d\": \"-2cos(2x)\"}}","_debug_options_count":4},{"id":95478,"question":"L'équation cos(x) = 0 admet pour solutions dans [0, π] :","option_a":"x = π\/2","option_b":"x = 0","option_c":"x = π","option_d":"x = π\/4","option_e":"","option_f":"","bonne_reponse":"a","explication":"cos(π\/2) = 0, et c'est la seule solution dans l'intervalle [0, π].","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 = π\/2\", \"b\": \"x = 0\", \"c\": \"x = π\", \"d\": \"x = π\/4\"}}","_debug_options_count":4}]