Quiz — Fractions continues.pdf

Question 1 sur 10 20:00

[{"id":98392,"question":"Quelle est la forme générale d'une fraction continue simple ?","option_a":"a₀ + 1\/(a₁ + 1\/(a₂ + ...))","option_b":"a₀ × (a₁ + 1\/(a₂ + ...))","option_c":"a₀ + a₁ + a₂ + ...","option_d":"a₀ \/ (a₁ + a₂ + ...)","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une fraction continue simple s'écrit sous la forme a₀ + 1\/(a₁ + 1\/(a₂ + ...)), où chaque aᵢ est un entier.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"a₀ + 1\/(a₁ + 1\/(a₂ + ...))\", \"b\": \"a₀ × (a₁ + 1\/(a₂ ","_debug_options_count":4},{"id":98393,"question":"L'algorithme d'Euclide est-il utilisé pour calculer les fractions continues ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. L'algorithme d'Euclide permet de trouver les coefficients aᵢ en répétant les divisions euclidiennes.","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":98394,"question":"Quelle fraction continue représente le nombre d'or φ ?","option_a":"[1; 1, 1, 1, ...]","option_b":"[1; 2, 2, 2, ...]","option_c":"[2; 1, 1, 1, ...]","option_d":"[1; 3, 3, 3, ...]","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le nombre d'or φ = (1 + √5)\/2 a pour développement en fraction continue [1; 1, 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\": \"[1; 1, 1, 1, ...]\", \"b\": \"[1; 2, 2, 2, ...]\", \"c\": \"[2; 1, 1, 1, ","_debug_options_count":4},{"id":98395,"question":"Une fraction continue finie représente toujours un nombre rationnel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. Une fraction continue finie se termine par un entier, donc elle représente un nombre rationnel.","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":98396,"question":"Quel est le premier coefficient a₀ dans le développement de √2 en fraction continue ?","option_a":"0","option_b":"1","option_c":"2","option_d":"√2","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le développement de √2 commence par [1; 2, 2, 2, ...], donc a₀ = 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\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"√2\"}}","_debug_options_count":4},{"id":98397,"question":"Les réduites d'une fraction continue sont-elles toujours des approximations par excès ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. Les réduites alternent entre approximations par excès et par défaut selon leur rang.","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":98398,"question":"Quelle est la réduite d'ordre 2 pour la fraction continue [1; 2, 3] ?","option_a":"1 + 1\/2","option_b":"1 + 1\/(2 + 1\/3)","option_c":"1 + 1\/3","option_d":"2 + 1\/3","option_e":"","option_f":"","bonne_reponse":"b","explication":"La réduite d'ordre 2 est 1 + 1\/(2 + 1\/3) = 1 + 1\/(7\/3) = 1 + 3\/7 = 10\/7.","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\/2\", \"b\": \"1 + 1\/(2 + 1\/3)\", \"c\": \"1 + 1\/3\", \"d\": \"2 + 1\/3\"}","_debug_options_count":4},{"id":98399,"question":"Une fraction continue infinie peut-elle représenter un nombre rationnel ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. Une fraction continue infinie représente toujours un nombre irrationnel.","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":98400,"question":"Quel algorithme permet de trouver le développement en fraction continue d'un nombre réel ?","option_a":"Algorithme de Newton","option_b":"Algorithme d'Euclide","option_c":"Méthode de Horner","option_d":"Algorithme de Gauss","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'algorithme d'Euclide est utilisé pour calculer les coefficients d'une fraction continue.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Algorithme de Newton\", \"b\": \"Algorithme d'Euclide\", \"c\": \"Méthod","_debug_options_count":4},{"id":98401,"question":"Les fractions continues sont-elles utilisées en cryptographie ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. Elles sont utilisées dans certains algorithmes de cryptographie, comme le RSA, pour la factorisation ou la génération de clés.","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}]