Quiz — serie6 (Ben Amor).pdf

Question 1 sur 10 20:00

[{"id":37200,"question":"Quelle est la solution de l'équation différentielle y' = 2y avec y(0) = 3 ?","option_a":"A) y = 3e^(2x)","option_b":"B) y = 3e^(-2x)","option_c":"C) y = 2e^(3x)","option_d":"D) y = 3x + 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation différentielle y' = 2y a pour solution générale y = Ce^(2x). Avec y(0) = 3, on trouve C = 3, d'où y = 3e^(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\": \"A) y = 3e^(2x)\", \"b\": \"B) y = 3e^(-2x)\", \"c\": \"C) y = 2e^(3x)\", \"","_debug_options_count":4},{"id":37201,"question":"La fonction f(x) = x^3 - 3x^2 + 2x est croissante sur l'intervalle [0, 1].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f'(x) = 3x^2 - 6x + 2 s'annule en x = 1 ± √(1\/3). Sur [0,1], f'(x) est négative (ex: f'(0.5) = -0.25), donc f 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":37202,"question":"Quel est le déterminant de la matrice A = [[2, 1], [3, 4]] ?","option_a":"A) 5","option_b":"B) -5","option_c":"C) 11","option_d":"D) 8","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le déterminant se calcule par ad - bc = (2*4) - (1*3) = 8 - 3 = 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) 5\", \"b\": \"B) -5\", \"c\": \"C) 11\", \"d\": \"D) 8\"}}","_debug_options_count":4},{"id":37203,"question":"L'intégrale ∫(de 0 à 1) (2x + 1) dx vaut :","option_a":"A) 2","option_b":"B) 1.5","option_c":"C) 3","option_d":"D) 0.5","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégrale se calcule comme [x^2 + x] de 0 à 1 = (1 + 1) - (0 + 0) = 2. Erreur dans les options : la bonne réponse est 2 (A).","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) 2\", \"b\": \"B) 1.5\", \"c\": \"C) 3\", \"d\": \"D) 0.5\"}}","_debug_options_count":4},{"id":37204,"question":"La suite définie par u_n = (-1)^n \/ n converge vers 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par le théorème des gendarmes, |u_n| = 1\/n → 0, donc u_n → 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":37205,"question":"Quelle est la solution du système { 2x + y = 5 ; x - y = 1 } ?","option_a":"A) (2, 1)","option_b":"B) (1, 2)","option_c":"C) (3, -1)","option_d":"D) (0, 5)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En additionnant les deux équations, on obtient 3x = 6 → x = 2. En substituant, y = 1. Solution : (2, 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) (2, 1)\", \"b\": \"B) (1, 2)\", \"c\": \"C) (3, -1)\", \"d\": \"D) (0, 5)\"","_debug_options_count":4},{"id":37206,"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 ln(x) est définie uniquement pour x \u003E 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":37207,"question":"Le produit scalaire de deux vecteurs orthogonaux est toujours égal à :","option_a":"A) 1","option_b":"B) 0","option_c":"C) -1","option_d":"D) Leur norme","option_e":"","option_f":"","bonne_reponse":"b","explication":"Par définition, deux vecteurs orthogonaux ont un produit scalaire nul.","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) 1\", \"b\": \"B) 0\", \"c\": \"C) -1\", \"d\": \"D) Leur norme\"}}","_debug_options_count":4},{"id":37208,"question":"La limite de (sin(x)\/x) quand x tend vers 0 est égale à :","option_a":"A) 0","option_b":"B) 1","option_c":"C) ∞","option_d":"D) -1","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est un résultat fondamental du cours : 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\": \"A) 0\", \"b\": \"B) 1\", \"c\": \"C) ∞\", \"d\": \"D) -1\"}}","_debug_options_count":4},{"id":37209,"question":"Quelle est la dérivée de la fonction f(x) = e^(2x) ?","option_a":"A) 2e^(2x)","option_b":"B) e^(2x)","option_c":"C) 2x e^(2x)","option_d":"D) e^(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de e^(u(x)) est u'(x)e^(u(x)). Ici u(x) = 2x, donc u'(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) 2e^(2x)\", \"b\": \"B) e^(2x)\", \"c\": \"C) 2x e^(2x)\", \"d\": \"D) e^(x","_debug_options_count":4}]