Quiz — corrigé série 10.pdf

Question 1 sur 10 20:00

[{"id":30030,"question":"Quelle est la dérivée de la fonction f(x) = x³ - 3x² + 2x - 1 ?","option_a":"3x² - 6x + 2","option_b":"x² - 3x + 2","option_c":"3x² - 6x","option_d":"x³ - 3x + 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme se calcule terme à terme : (x³)' = 3x², (-3x²)' = -6x, (2x)' = 2, et (-1)' = 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\": \"3x² - 6x + 2\", \"b\": \"x² - 3x + 2\", \"c\": \"3x² - 6x\", \"d\": \"x³ ","_debug_options_count":4},{"id":30031,"question":"L'équation x² - 5x + 6 = 0 admet deux solutions réelles distinctes.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le discriminant Δ = 25 - 24 = 1 \u003E 0, donc l'équation admet deux solutions réelles distinctes : 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":30032,"question":"Quelle est la limite de la fonction f(x) = (2x² + 3x - 1)\/(x² - 1) lorsque x tend vers +∞ ?","option_a":"2","option_b":"0","option_c":"1","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour x → +∞, on divise numérateur et dénominateur par x² : f(x) ≈ (2 + 3\/x - 1\/x²)\/(1 - 1\/x²) → 2\/1 = 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\": \"2\", \"b\": \"0\", \"c\": \"1\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":30033,"question":"La fonction f(x) = x² - 4x + 3 est croissante sur l'intervalle [2 ; +∞[.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 2x - 4 est positive pour x ≥ 2, donc la fonction est croissante sur [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":30034,"question":"Résoudre l'inéquation : (x - 1)(x + 2) \u003E 0.","option_a":"x ∈ ]-∞ ; -2[ ∪ ]1 ; +∞[","option_b":"x ∈ ]-2 ; 1[","option_c":"x ∈ ]-∞ ; -1[ ∪ ]2 ; +∞[","option_d":"x ∈ ]-1 ; 2[","option_e":"","option_f":"","bonne_reponse":"a","explication":"Les racines sont x = -2 et x = 1. Le produit est positif à l'extérieur des racines (tableau de signes).","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[ ∪ ]1 ; +∞[\", \"b\": \"x ∈ ]-2 ; 1[\", \"c\": \"x","_debug_options_count":4},{"id":30035,"question":"La fonction f(x) = 1\/x est définie sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction 1\/x n'est pas définie en x = 0, donc son domaine de définition est ℝ*.","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":30036,"question":"Quelle est la solution de l'équation ln(x) = 2 ?","option_a":"x = e²","option_b":"x = 2","option_c":"x = e","option_d":"x = 1\/2","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition du logarithme, ln(x) = 2 ⇔ x = e² (car e² \u003E 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\": \"x = e²\", \"b\": \"x = 2\", \"c\": \"x = e\", \"d\": \"x = 1\/2\"}}","_debug_options_count":4},{"id":30037,"question":"La fonction f(x) = x³ est convexe sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée seconde f''(x) = 6x est positive pour x \u003E 0 et négative pour x \u003C 0, donc la fonction n'est pas convexe sur ℝ.","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":30038,"question":"Résoudre l'équation : 2^(x+1) = 8.","option_a":"x = 2","option_b":"x = 3","option_c":"x = 1","option_d":"x = 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"8 = 2³, donc 2^(x+1) = 2³ ⇒ x + 1 = 3 ⇒ 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\": \"x = 2\", \"b\": \"x = 3\", \"c\": \"x = 1\", \"d\": \"x = 0\"}}","_debug_options_count":4},{"id":30039,"question":"La fonction f(x) = √x est dérivable en x = 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f'(x) = 1\/(2√x) n'est pas définie en x = 0 (dérivée à droite infinie).","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}]