Quiz — simil.pdf

Question 1 sur 10 20:00

[{"id":72665,"question":"Quelle transformation géométrique est une similitude directe ?","option_a":"Une translation","option_b":"Une rotation suivie d'une homothétie","option_c":"Une réflexion","option_d":"Une projection orthogonale","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une similitude directe combine une rotation et une homothétie (éventuellement une translation), ce qui conserve les angles et les rapports de distances.","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 translation\", \"b\": \"Une rotation suivie d'une homothétie\", \"","_debug_options_count":4},{"id":72666,"question":"Dans le plan complexe, une similitude est représentée par l'application z ↦ (2 + 3i)z + 1. Quel est le rapport de cette similitude ?","option_a":"2","option_b":"3","option_c":"√13","option_d":"5","option_e":"","option_f":"","bonne_reponse":"c","explication":"Le rapport d'une similitude est donné par le module du coefficient multiplicatif (ici |2 + 3i| = √(2² + 3²) = √13).","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\", \"b\": \"3\", \"c\": \"√13\", \"d\": \"5\"}}","_debug_options_count":4},{"id":72667,"question":"Une similitude indirecte conserve l'orientation du plan.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une similitude indirecte renverse l'orientation (exemple : une réflexion), contrairement à une similitude directe qui la conserve.","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":72668,"question":"Quelle est l'image d'un point d'affixe z = 1 + i par la similitude z ↦ (1 - i)z + 2i ?","option_a":"3 + i","option_b":"1 + 3i","option_c":"2 + 2i","option_d":"0 + 2i","option_e":"","option_f":"","bonne_reponse":"a","explication":"Calcul : (1 - i)(1 + i) + 2i = (1 - i²) + 2i = (1 + 1) + 2i = 2 + 2i. L'image est donc 2 + 2i.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"3 + i\", \"b\": \"1 + 3i\", \"c\": \"2 + 2i\", \"d\": \"0 + 2i\"}}","_debug_options_count":4},{"id":72669,"question":"Toute similitude est une bijection du plan.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une similitude est une transformation bijective car elle admet une application réciproque (inverse de la similitude).","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":72670,"question":"Quel est l'angle de rotation d'une similitude directe de rapport 2 et d'argument π\/4 ?","option_a":"π\/4","option_b":"π\/2","option_c":"π","option_d":"2π","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'argument du coefficient multiplicatif (ici π\/4) donne directement l'angle de rotation de la similitude.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"π\/4\", \"b\": \"π\/2\", \"c\": \"π\", \"d\": \"2π\"}}","_debug_options_count":4},{"id":72671,"question":"Une homothétie est-elle une similitude ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une homothétie est un cas particulier de similitude où l'angle de rotation est nul (argument du coefficient multiplicatif = 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":72672,"question":"Quelle est la nature de la transformation z ↦ -z ?","option_a":"Rotation de π","option_b":"Homothétie de rapport -1","option_c":"Symétrie centrale","option_d":"Translation","option_e":"","option_f":"","bonne_reponse":"a","explication":"La transformation z ↦ -z est une rotation de π radians (180°) autour de l'origine, donc une similitude directe de rapport 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\": \"Rotation de π\", \"b\": \"Homothétie de rapport -1\", \"c\": \"Symétri","_debug_options_count":4},{"id":72673,"question":"Le produit de deux similitudes est-il toujours une similitude ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le produit de deux similitudes (composées) est une similitude, car la composition de transformations bijectives est bijective et conserve les rapports de 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},{"id":72674,"question":"Quelle similitude transforme le point A(1, 0) en B(0, 1) et conserve l'orientation ?","option_a":"Rotation de π\/2 autour de l'origine","option_b":"Homothétie de centre O et rapport 1","option_c":"Translation de vecteur (0,1)","option_d":"Réflexion d'axe y = x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La rotation de π\/2 autour de l'origine transforme (1,0) en (0,1) et conserve l'orientation (similitude directe).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Rotation de π\/2 autour de l'origine\", \"b\": \"Homothétie de centr","_debug_options_count":4}]