Quiz — serie 26 p1.pdf

Question 1 sur 10 20:00

[{"id":7525,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"6x² + 2","option_c":"3x + 2","option_d":"6x - 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme se calcule terme par terme. La dérivée de 3x² est 6x, celle de 2x est 2, et celle de -5 est 0. Donc f'(x) = 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\": \"6x + 2\", \"b\": \"6x² + 2\", \"c\": \"3x + 2\", \"d\": \"6x - 5\"}}","_debug_options_count":4},{"id":7526,"question":"La limite de la fonction f(x) = (x² - 1)\/(x - 1) quand x tend vers 1 est égale à 2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En factorisant le numérateur, on obtient (x - 1)(x + 1)\/(x - 1). Pour x ≠ 1, on simplifie par (x - 1), ce qui donne x + 1. La limite quand x tend vers 1 est donc 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":7527,"question":"Quelle est la solution de l'équation différentielle y' + 2y = 0 ?","option_a":"y = Ce^(-2x)","option_b":"y = Ce^(2x)","option_c":"y = C + e^(-2x)","option_d":"y = Cx^(-2)","option_e":"","option_f":"","bonne_reponse":"a","explication":"Cette équation est de la forme y' + ay = 0. La solution générale est y = Ce^(-ax), ici a = 2, donc 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\": \"y = Ce^(-2x)\", \"b\": \"y = Ce^(2x)\", \"c\": \"y = C + e^(-2x)\", \"d\": \"","_debug_options_count":4},{"id":7528,"question":"Soit une variable aléatoire X suivant une loi normale centrée réduite. Quelle est la probabilité P(-1 ≤ X ≤ 1) ?","option_a":"0,68","option_b":"0,95","option_c":"0,34","option_d":"0,50","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une loi normale centrée réduite, environ 68% des valeurs se situent entre -1 et 1. Donc P(-1 ≤ X ≤ 1) ≈ 0,68.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"0,68\", \"b\": \"0,95\", \"c\": \"0,34\", \"d\": \"0,50\"}}","_debug_options_count":4},{"id":7529,"question":"La suite définie par uₙ = (n² + 1)\/n est bornée.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"En simplifiant, uₙ = n + 1\/n. Quand n tend vers l'infini, uₙ tend vers l'infini, donc la suite n'est pas bornée.","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":7530,"question":"Quelle est la valeur de la somme S = 1 + 2 + 3 + ... + n ?","option_a":"n(n+1)\/2","option_b":"n²","option_c":"(n+1)²\/2","option_d":"n(n-1)\/2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La somme des n premiers entiers est donnée par la formule S = n(n+1)\/2, démontrée par Gauss.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"n(n+1)\/2\", \"b\": \"n²\", \"c\": \"(n+1)²\/2\", \"d\": \"n(n-1)\/2\"}}","_debug_options_count":4},{"id":7531,"question":"Soit f une fonction continue sur [a, b]. Le théorème des valeurs intermédiaires garantit que f prend toutes les valeurs entre f(a) et f(b).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le théorème des valeurs intermédiaires stipule que si f est continue sur [a, b], alors elle prend toutes les valeurs intermédiaires entre f(a) et f(b).","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":7532,"question":"Quelle est l'intégrale de f(x) = 2x + 1 entre 0 et 2 ?","option_a":"6","option_b":"5","option_c":"4","option_d":"3","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale de 2x est x² et celle de 1 est x. Donc ∫(2x + 1)dx = x² + x. Évaluée entre 0 et 2, on obtient (4 + 2) - (0 + 0) = 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\": \"6\", \"b\": \"5\", \"c\": \"4\", \"d\": \"3\"}}","_debug_options_count":4},{"id":7533,"question":"Soit une suite géométrique de premier terme u₀ = 3 et de raison q = 2. Quel est le terme u₅ ?","option_a":"96","option_b":"48","option_c":"192","option_d":"64","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le terme général d'une suite géométrique est uₙ = u₀ * qⁿ. Donc u₅ = 3 * 2⁵ = 3 * 32 = 96.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"96\", \"b\": \"48\", \"c\": \"192\", \"d\": \"64\"}}","_debug_options_count":4},{"id":7534,"question":"La fonction exponentielle est toujours positive sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle, notée exp(x) ou eˣ, est définie pour tout x réel et prend uniquement des valeurs strictement positives.","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}]