Quiz — corr serie passe bande 1.pdf

Question 1 sur 10 20:00

[{"id":66200,"question":"Quelle est la définition correcte d'un passe-bande pour une fonction f ?","option_a":"Un intervalle [a, b] où f est définie et continue","option_b":"Un ensemble de valeurs x pour lesquelles f(x) appartient à un intervalle donné","option_c":"Une fonction qui annule f(x) pour tout x","option_d":"Une suite qui converge vers une limite finie","option_e":"","option_f":"","bonne_reponse":"b","explication":"Un passe-bande pour une fonction f est un ensemble de valeurs x telles que f(x) appartient à un intervalle spécifique, souvent utilisé pour filtrer des données.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Un intervalle [a, b] où f est définie et continue\", \"b\": \"Un en","_debug_options_count":4},{"id":66201,"question":"Soit la fonction f(x) = ln(x). Quel est le passe-bande de f pour l'intervalle [0, 1] ?","option_a":"[1, e]","option_b":"[-∞, 0]","option_c":"[0, 1]","option_d":"[e, +∞]","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction ln(x) est définie pour x \u003E 0. Pour que ln(x) ∈ [0, 1], il faut que x ∈ [1, e], car ln(1) = 0 et ln(e) = 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\": \"[1, e]\", \"b\": \"[-∞, 0]\", \"c\": \"[0, 1]\", \"d\": \"[e, +∞]\"}}","_debug_options_count":4},{"id":66202,"question":"Vrai ou Faux ? Un passe-bande peut être vide.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Un passe-bande peut être vide si aucune valeur de la fonction ne satisfait la condition donnée, par exemple si l'intervalle cible est hors de l'ensemble des valeurs possibles de la fonction.","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":66203,"question":"Quelle est la passe-bande de la suite u_n = 2^n pour l'intervalle [4, 16] ?","option_a":"[2, 4]","option_b":"[0, 2]","option_c":"[n ∈ ℕ | 2 ≤ n ≤ 4]","option_d":"L'ensemble vide","option_e":"","option_f":"","bonne_reponse":"c","explication":"La suite u_n = 2^n prend les valeurs 2, 4, 8, 16, ... Pour que u_n ∈ [4, 16], il faut que n ∈ {2, 3, 4}.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"[2, 4]\", \"b\": \"[0, 2]\", \"c\": \"[n ∈ ℕ | 2 ≤ n ≤ 4]\", \"d\": ","_debug_options_count":4},{"id":66204,"question":"Vrai ou Faux ? La passe-bande d'une fonction constante f(x) = c est toujours ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La passe-bande de f(x) = c pour un intervalle [a, b] est ℝ seulement si c ∈ [a, b]. Sinon, la passe-bande est vide.","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":66205,"question":"Quelle est la passe-bande de la fonction f(x) = x² - 4x + 3 pour l'intervalle [0, 1] ?","option_a":"[-1, 0]","option_b":"[0, 1]","option_c":"[3, 4]","option_d":"L'ensemble vide","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction f(x) = x² - 4x + 3 a pour minimum -1 (en x=2) et f(0)=3, f(1)=0. Pour x ∈ [0,1], f(x) ∈ [-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\": \"[-1, 0]\", \"b\": \"[0, 1]\", \"c\": \"[3, 4]\", \"d\": \"L'ensemble vide\"}}","_debug_options_count":4},{"id":66206,"question":"Vrai ou Faux ? La passe-bande d'une fonction périodique est toujours un intervalle borné.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une fonction périodique comme sin(x) a une passe-bande non bornée pour tout intervalle contenant ses valeurs extrêmes (par exemple [-1, 1] pour sin(x)).","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":66207,"question":"Quelle est la passe-bande de la fonction f(x) = e^x pour l'intervalle [1, e] ?","option_a":"[0, 1]","option_b":"[1, e]","option_c":"[e, +∞]","option_d":"L'ensemble vide","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction e^x est croissante. Pour e^x ∈ [1, e], il faut que x ∈ [0, 1], car e^0 = 1 et e^1 = e.","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, 1]\", \"b\": \"[1, e]\", \"c\": \"[e, +∞]\", \"d\": \"L'ensemble vide\"}","_debug_options_count":4},{"id":66208,"question":"Vrai ou Faux ? La passe-bande d'une fonction affine f(x) = ax + b est toujours un intervalle borné.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une fonction affine est bijective sur ℝ. Sa passe-bande pour un intervalle [c, d] est donc un intervalle borné [ (c-b)\/a, (d-b)\/a ] si a ≠ 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":66209,"question":"Quelle est la passe-bande de la suite u_n = (-1)^n pour l'intervalle [-1, 1] ?","option_a":"[-1, 1]","option_b":"{ -1, 1 }","option_c":"L'ensemble vide","option_d":"ℕ","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite u_n = (-1)^n prend les valeurs -1 et 1. Pour u_n ∈ [-1, 1], tous les termes de la suite sont valides, mais la passe-bande est l'ensemble des indices n pour lesquels u_n ∈ [-1, 1], soit {n ∈ ℕ | n pair ou impair} = ℕ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"[-1, 1]\", \"b\": \"{ -1, 1 }\", \"c\": \"L'ensemble vide\", \"d\": \"ℕ\"}}","_debug_options_count":4}]