Quiz — 33_CourbesEnPolaires.pdf

Question 1 sur 10 20:00

[{"id":31210,"question":"Quelle est l'équation polaire d'un cercle de centre O et de rayon 3 ?","option_a":"r = 3","option_b":"r = 3θ","option_c":"r = 3 sin(θ)","option_d":"r = 3 cos(θ)","option_e":"","option_f":"","bonne_reponse":"a","explication":"Un cercle centré à l'origine a une équation polaire de la forme r = constante, ici r = 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\": \"r = 3\", \"b\": \"r = 3θ\", \"c\": \"r = 3 sin(θ)\", \"d\": \"r = 3 cos(θ)","_debug_options_count":4},{"id":31211,"question":"La courbe d'équation polaire r = 2 + 2 cos(θ) est une...","option_a":"Spirale","option_b":"Cardioïde","option_c":"Lemniscate","option_d":"Rosace","option_e":"","option_f":"","bonne_reponse":"b","explication":"Cette équation correspond à une cardioïde, reconnaissable à sa forme de cœur.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Spirale\", \"b\": \"Cardioïde\", \"c\": \"Lemniscate\", \"d\": \"Rosace\"}}","_debug_options_count":4},{"id":31212,"question":"Vrai ou Faux ? Une courbe polaire peut avoir une asymptote verticale.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Les asymptotes des courbes polaires sont généralement obliques ou horizontales, jamais verticales.","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":31213,"question":"Quelle propriété géométrique vérifie la courbe r = sin(2θ) ?","option_a":"Symétrie par rapport à l'axe polaire","option_b":"Symétrie par rapport à l'axe θ = π\/4","option_c":"Période π\/2","option_d":"Branche infinie","option_e":"","option_f":"","bonne_reponse":"b","explication":"La courbe r = sin(2θ) est une rosace à 4 pétales, symétrique par rapport aux axes θ = π\/4 et θ = 3π\/4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Symétrie par rapport à l'axe polaire\", \"b\": \"Symétrie par rapp","_debug_options_count":4},{"id":31214,"question":"Vrai ou Faux ? La dérivée dr\/dθ permet de déterminer la pente de la tangente en un point de la courbe polaire.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La pente de la tangente est donnée par dy\/dx = (dr\/dθ sin(θ) + r cos(θ)) \/ (dr\/dθ cos(θ) - r sin(θ)).","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":31215,"question":"Quelle est l'aire balayée par un rayon vecteur pour θ variant de 0 à π\/2 pour la courbe r = 2 cos(θ) ?","option_a":"π","option_b":"π\/2","option_c":"2π","option_d":"4","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'aire est donnée par (1\/2) ∫[0, π\/2] r² dθ = (1\/2) ∫[0, π\/2] 4 cos²(θ) dθ = π\/2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"π\", \"b\": \"π\/2\", \"c\": \"2π\", \"d\": \"4\"}}","_debug_options_count":4},{"id":31216,"question":"La courbe r = 1\/θ est une...","option_a":"Spirale d'Archimède","option_b":"Spirale logarithmique","option_c":"Spirale hyperbolique","option_d":"Rosace","option_e":"","option_f":"","bonne_reponse":"c","explication":"Cette équation définit une spirale hyperbolique, car r est inversement proportionnel à θ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"Spirale d'Archimède\", \"b\": \"Spirale logarithmique\", \"c\": \"Spiral","_debug_options_count":4},{"id":31217,"question":"Vrai ou Faux ? La lemniscate de Bernoulli a pour équation polaire r² = a² cos(2θ).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation correcte est r² = a² cos(2θ), donc l'affirmation est vraie.","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":31218,"question":"Quelle est la période de la fonction r = sin(3θ) ?","option_a":"π\/3","option_b":"2π\/3","option_c":"π","option_d":"2π","option_e":"","option_f":"","bonne_reponse":"b","explication":"La période de sin(3θ) est 2π\/3, car 3θ varie de 0 à 2π pour θ de 0 à 2π\/3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"π\/3\", \"b\": \"2π\/3\", \"c\": \"π\", \"d\": \"2π\"}}","_debug_options_count":4},{"id":31219,"question":"Quel est le nombre de pétales de la rosace r = 3 sin(4θ) ?","option_a":"4","option_b":"8","option_c":"12","option_d":"16","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour r = a sin(nθ), le nombre de pétales est 2n si n est pair, ici n = 4 donc 8 pétales.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"4\", \"b\": \"8\", \"c\": \"12\", \"d\": \"16\"}}","_debug_options_count":4}]