Quiz — 65ef676aec433_corrigé série 16.pdf

Question 1 sur 10 20:00

[{"id":38260,"question":"Quelle est la limite de la suite définie par u_{n+1} = 2u_n + 1 avec u_0 = 0 ?","option_a":"A. 0","option_b":"B. -1","option_c":"C. +∞","option_d":"D. 1","option_e":"","option_f":"","bonne_reponse":"c","explication":"La suite est croissante et non majorée, donc sa limite est +∞. La formule de récurrence montre que u_n tend vers l'infini.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"A. 0\", \"b\": \"B. -1\", \"c\": \"C. +∞\", \"d\": \"D. 1\"}}","_debug_options_count":4},{"id":38261,"question":"La fonction f(x) = e^{2x} - 3e^x + 2 admet-elle un minimum ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 2e^{2x} - 3e^x s'annule en x = ln(3\/2), et f''(x) \u003E 0, donc f admet un minimum en ce point.","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":38262,"question":"Résoudre l'équation différentielle y' + 2y = e^{-x} :","option_a":"A. y = Ce^{-2x} + e^{-x}","option_b":"B. y = Ce^{-x} - e^{-2x}","option_c":"C. y = Ce^{-2x} - e^{-x}","option_d":"D. y = Ce^{-x} + e^{-2x}","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution générale est y = Ce^{-2x} + e^{-x}, où C est une constante réelle. La solution particulière est obtenue par variation de la constante.","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 = Ce^{-2x} + e^{-x}\", \"b\": \"B. y = Ce^{-x} - e^{-2x}\", \"c\": ","_debug_options_count":4},{"id":38263,"question":"Soit X une variable aléatoire suivant une loi normale N(0,1). P(X ≤ -1) est égal à :","option_a":"A. 0.1587","option_b":"B. 0.3413","option_c":"C. 0.6826","option_d":"D. 0.8413","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par symétrie de la loi normale centrée réduite, P(X ≤ -1) = 1 - P(X ≤ 1) ≈ 0.1587.","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. 0.1587\", \"b\": \"B. 0.3413\", \"c\": \"C. 0.6826\", \"d\": \"D. 0.8413\"}","_debug_options_count":4},{"id":38264,"question":"Le nombre complexe z = (1 + i)^4 est égal à :","option_a":"A. -4","option_b":"B. 4i","option_c":"C. -4i","option_d":"D. 4","option_e":"","option_f":"","bonne_reponse":"a","explication":"En développant (1 + i)^2 = 2i, puis (2i)^2 = -4. Donc z = -4.","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. -4\", \"b\": \"B. 4i\", \"c\": \"C. -4i\", \"d\": \"D. 4\"}}","_debug_options_count":4},{"id":38265,"question":"L'intégrale ∫_{0}^{1} x e^{x^2} dx vaut :","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"En posant u = x^2, du = 2x dx, l'intégrale devient (1\/2)∫_{0}^{1} e^u du = (1\/2)(e - 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":38266,"question":"Soit f(x) = ln(x) - x + 1. Le nombre de solutions de l'équation f(x) = 0 est :","option_a":"A. 0","option_b":"B. 1","option_c":"C. 2","option_d":"D. Infini","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction f admet un maximum en x = 1 où f(1) = 0. Elle est strictement croissante avant 1 et décroissante après, donc une seule solution.","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. 2\", \"d\": \"D. Infini\"}}","_debug_options_count":4},{"id":38267,"question":"Dans un triangle ABC, si AB = 5, AC = 7 et BC = 8, alors cos(B) vaut :","option_a":"A. 0.5","option_b":"B. 0.6","option_c":"C. 0.7","option_d":"D. 0.8","option_e":"","option_f":"","bonne_reponse":"b","explication":"Par la loi des cosinus : cos(B) = (AB² + BC² - AC²)\/(2·AB·BC) = (25 + 64 - 49)\/80 = 40\/80 = 0.5.","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. 0.5\", \"b\": \"B. 0.6\", \"c\": \"C. 0.7\", \"d\": \"D. 0.8\"}}","_debug_options_count":4},{"id":38268,"question":"La série ∑_{n=1}^{∞} 1\/n^2 converge-t-elle ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La série de Riemann ∑ 1\/n^α converge si α \u003E 1. Ici α = 2 \u003E 1, donc la série converge.","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":38269,"question":"Soit f(x) = x^3 - 3x + 1. Le nombre de points critiques de f est :","option_a":"A. 0","option_b":"B. 1","option_c":"C. 2","option_d":"D. 3","option_e":"","option_f":"","bonne_reponse":"c","explication":"La dérivée f'(x) = 3x² - 3 s'annule en x = 1 et x = -1. Donc deux points critiques.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. 2\", \"d\": \"D. 3\"}}","_debug_options_count":4}]