Quiz — corrigé_Séance -(Isométries-Bijections).pdf

Question 1 sur 10 20:00

[{"id":21839,"question":"Parmi les transformations suivantes, laquelle est une isométrie ?","option_a":"Une homothétie de rapport 2","option_b":"Une rotation d'angle 90°","option_c":"Une projection orthogonale","option_d":"Une translation de vecteur (0,0)","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une rotation conserve les distances et les angles, contrairement aux autres transformations proposées.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Une homothétie de rapport 2\", \"b\": \"Une rotation d'angle 90°\", ","_debug_options_count":4},{"id":21840,"question":"Une bijection est toujours inversible.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, une bijection est inversible car elle est à la fois injective et surjective.","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":21841,"question":"Quel est le noyau d'une application linéaire f définie par f(x,y) = (x+y, x-y) ?","option_a":"{(0,0)}","option_b":"{(1,-1)}","option_c":"L'ensemble des vecteurs (a,-a)","option_d":"L'ensemble des vecteurs (a,a)","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le noyau est l'ensemble des vecteurs (x,y) tels que f(x,y) = (0,0). La seule solution est (0,0).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"{(0,0)}\", \"b\": \"{(1,-1)}\", \"c\": \"L'ensemble des vecteurs (a,-a)\",","_debug_options_count":4},{"id":21842,"question":"La composition de deux symétries axiales est toujours une rotation.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La composition de deux symétries axiales est une rotation d'angle le double de l'angle entre les deux axes.","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":21843,"question":"Quelle propriété caractérise une application linéaire injective ?","option_a":"Son noyau est réduit à {0}","option_b":"Son image est l'espace entier","option_c":"Elle est surjective","option_d":"Son déterminant est non nul","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une application linéaire est injective si et seulement si son noyau est {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\": \"Son noyau est réduit à {0}\", \"b\": \"Son image est l'espace entie","_debug_options_count":4},{"id":21844,"question":"Soit f une bijection. Que peut-on dire de f⁻¹ ∘ f ?","option_a":"C'est l'identité","option_b":"C'est une constante","option_c":"C'est une projection","option_d":"C'est une symétrie","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, f⁻¹ ∘ f est l'application identité, car f⁻¹ annule l'effet de f.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"C'est l'identité\", \"b\": \"C'est une constante\", \"c\": \"C'est une p","_debug_options_count":4},{"id":21845,"question":"Une translation est une isométrie qui conserve les angles orientés.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une translation conserve les angles et les distances, donc les angles orientés.","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":21846,"question":"Quel est le déterminant d'une isométrie linéaire en dimension 2 ?","option_a":"1 ou -1","option_b":"0","option_c":"2","option_d":"Impossible à déterminer","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le déterminant d'une isométrie linéaire est toujours 1 (rotation) ou -1 (symétrie).","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 ou -1\", \"b\": \"0\", \"c\": \"2\", \"d\": \"Impossible à déterminer\"}}","_debug_options_count":4},{"id":21847,"question":"Soit f une application linéaire de noyau {0}. Que peut-on conclure sur f ?","option_a":"f est injective","option_b":"f est surjective","option_c":"f est bijective","option_d":"f est une homothétie","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si le noyau est {0}, alors f est injective. Si de plus l'espace d'arrivée est de même dimension, f est bijective.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"f est injective\", \"b\": \"f est surjective\", \"c\": \"f est bijective\"","_debug_options_count":4},{"id":21848,"question":"La composition d'une rotation et d'une translation est toujours une isométrie.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La composition d'une rotation et d'une translation est une isométrie car elle conserve les distances.","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}]