Quiz — 6553c4878cf92_corrigé_femme-série5.pdf

Question 1 sur 10 20:00

[{"id":18669,"question":"Quelle est la limite de la suite définie par u_n = (3n² + 2n - 1)\/(n² + 5) lorsque n tend vers l'infini ?","option_a":"A. 3","option_b":"B. 0","option_c":"C. +∞","option_d":"D. 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La limite d'une suite rationnelle est donnée par le rapport des coefficients des termes de plus haut degré. Ici, 3n²\/n² = 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. 3\", \"b\": \"B. 0\", \"c\": \"C. +∞\", \"d\": \"D. 1\"}}","_debug_options_count":4},{"id":18670,"question":"La fonction f(x) = x³ - 3x² + 2x est dérivable sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Toute fonction polynôme est dérivable sur ℝ. La dérivée est f'(x) = 3x² - 6x + 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":18671,"question":"Quelle est la solution générale de l'équation différentielle y' + 2y = 0 ?","option_a":"A. y = Ce^{-2x}","option_b":"B. y = Ce^{2x}","option_c":"C. y = Cx","option_d":"D. y = C","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation est de la forme y' + ay = 0. La solution générale est y = Ce^{-ax}, soit ici y = Ce^{-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 = Ce^{-2x}\", \"b\": \"B. y = Ce^{2x}\", \"c\": \"C. y = Cx\", \"d\": \"","_debug_options_count":4},{"id":18672,"question":"Soit X une variable aléatoire suivant une loi normale N(0,1). Quelle est P(-1 ≤ X ≤ 1) ?","option_a":"A. 0,68","option_b":"B. 0,95","option_c":"C. 0,99","option_d":"D. 0,34","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une loi normale centrée réduite, environ 68% des valeurs sont dans l'intervalle [-1,1].","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. 0,68\", \"b\": \"B. 0,95\", \"c\": \"C. 0,99\", \"d\": \"D. 0,34\"}}","_debug_options_count":4},{"id":18673,"question":"La dérivée de la fonction f(x) = ln(2x + 1) est f'(x) = 2\/(2x + 1).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(u) est u'\/u. Ici, u = 2x + 1, donc u' = 2, et f'(x) = 2\/(2x + 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":18674,"question":"Quelle est la valeur de l'intégrale ∫₀¹ (3x² + 2x) dx ?","option_a":"A. 2","option_b":"B. 1","option_c":"C. 3","option_d":"D. 4","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale d'une somme est la somme des intégrales. ∫₀¹ 3x² dx = [x³]₀¹ = 1 et ∫₀¹ 2x dx = [x²]₀¹ = 1. Donc le résultat est 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. 1\", \"c\": \"C. 3\", \"d\": \"D. 4\"}}","_debug_options_count":4},{"id":18675,"question":"Soit une suite géométrique de premier terme u₀ = 2 et de raison q = -0,5. Quelle est la valeur de u₃ ?","option_a":"A. -0,25","option_b":"B. 0,25","option_c":"C. -0,5","option_d":"D. 0,5","option_e":"","option_f":"","bonne_reponse":"a","explication":"uₙ = u₀ * qⁿ. Donc u₃ = 2 * (-0,5)³ = 2 * (-0,125) = -0,25.","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. -0,25\", \"b\": \"B. 0,25\", \"c\": \"C. -0,5\", \"d\": \"D. 0,5\"}}","_debug_options_count":4},{"id":18676,"question":"La fonction f(x) = e^{2x} est croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de e^{2x} est 2e^{2x}, qui est toujours positive. Donc la fonction est croissante.","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":18677,"question":"Quelle est la probabilité d'obtenir exactement 2 fois le chiffre 6 en lançant 5 fois un dé équilibré ?","option_a":"A. 5\/324","option_b":"B. 25\/216","option_c":"C. 10\/243","option_d":"D. 5\/108","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est une loi binomiale de paramètres n=5 et p=1\/6. P(X=2) = C(5,2) * (1\/6)² * (5\/6)³ = 10 * (1\/36) * (125\/216) = 25\/216.","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. 5\/324\", \"b\": \"B. 25\/216\", \"c\": \"C. 10\/243\", \"d\": \"D. 5\/108\"}}","_debug_options_count":4},{"id":18678,"question":"L'équation différentielle y'' + y = 0 a pour solution générale y = A cos(x) + B sin(x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation caractéristique est r² + 1 = 0, donc r = ±i. La solution générale est y = A cos(x) + B sin(x).","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}]