Quiz — correction serie n1 copmlexe.pdf

Question 1 sur 5 10:00

[{"id":2560,"question":"Quel est le module du nombre complexe z = 3 - 4i ?","option_a":"5","option_b":"7","option_c":"√7","option_d":"25","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module d'un nombre complexe z = a + bi est donné par |z| = √(a² + b²). Ici, |3 - 4i| = √(3² + (-4)²) = √(9 + 16) = √25 = 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"5\", \"b\": \"7\", \"c\": \"√7\", \"d\": \"25\"}}","_debug_options_count":4},{"id":2561,"question":"Si z = 2(cos(π\/3) + i sin(π\/3)), quelle est sa forme algébrique ?","option_a":"1 + i√3","option_b":"√3 + i","option_c":"2 + i√3","option_d":"1 + i","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant les formules de trigonométrie : cos(π\/3) = 1\/2 et sin(π\/3) = √3\/2. Donc z = 2(1\/2 + i√3\/2) = 1 + i√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\": \"1 + i√3\", \"b\": \"√3 + i\", \"c\": \"2 + i√3\", \"d\": \"1 + i\"}}","_debug_options_count":4},{"id":2562,"question":"Résoudre l'équation complexe z² = -4 dans ℂ.","option_a":"z = 2i ou z = -2i","option_b":"z = 4i ou z = -4i","option_c":"z = 2 ou z = -2","option_d":"z = i ou z = -i","option_e":"","option_f":"","bonne_reponse":"a","explication":"Les solutions de z² = -4 sont les racines carrées de -4. Comme -4 = 4e^(iπ), les solutions sont z = 2e^(iπ\/2) = 2i et z = 2e^(i3π\/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\": \"z = 2i ou z = -2i\", \"b\": \"z = 4i ou z = -4i\", \"c\": \"z = 2 ou z = ","_debug_options_count":4},{"id":2563,"question":"Dans le plan complexe, quel est l'ensemble des points M d'affixe z telle que |z - 1| = 3 ?","option_a":"Un cercle de centre O et de rayon 3","option_b":"Un cercle de centre A(1) et de rayon 3","option_c":"Une droite passant par O","option_d":"Un disque de centre A(1) et de rayon 3","option_e":"","option_f":"","bonne_reponse":"b","explication":"|z - 1| = 3 représente tous les points M dont la distance au point A d'affixe 1 est égale à 3. C'est donc un cercle de centre A(1) et de rayon 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\": \"Un cercle de centre O et de rayon 3\", \"b\": \"Un cercle de centre A","_debug_options_count":4},{"id":2564,"question":"Quel est le conjugué du nombre complexe z = (2 + 3i)(1 - i) ?","option_a":"(2 - 3i)(1 + i)","option_b":"5 + i","option_c":"5 - i","option_d":"(2 + 3i)(1 + i)","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le conjugué d'un produit est le produit des conjugués : (2 + 3i)(1 - i) a pour conjugué (2 - 3i)(1 + i).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"(2 - 3i)(1 + i)\", \"b\": \"5 + i\", \"c\": \"5 - i\", \"d\": \"(2 + 3i)(1 + ","_debug_options_count":4}]