Quiz — 05_Complexes_Corrige.pdf

Question 1 sur 10 20:00

[{"id":22629,"question":"Quel est le module du nombre complexe z = 3 + 4i ?","option_a":"5","option_b":"7","option_c":"12","option_d":"25","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module est calculé par |z| = √(3² + 4²) = √(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\": \"12\", \"d\": \"25\"}}","_debug_options_count":4},{"id":22630,"question":"L'argument d'un nombre complexe z = a + bi est toujours défini dans l'intervalle [0, π].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'argument est généralement défini dans l'intervalle ]-π, π] ou [0, 2π[ selon les conventions.","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":22631,"question":"Si z = 2(cos(π\/3) + i sin(π\/3)), quelle est la forme algébrique de z ?","option_a":"1 + i√3","option_b":"√3 + i","option_c":"2 + 2i√3","option_d":"1 + i","option_e":"","option_f":"","bonne_reponse":"a","explication":"En développant z = 2(cos(π\/3) + i sin(π\/3)) = 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 + 2i√3\", \"d\": \"1 + i\"}}","_debug_options_count":4},{"id":22632,"question":"Le conjugué d'un nombre complexe z = a + bi est donné par :","option_a":"a - bi","option_b":"b + ai","option_c":"-a + bi","option_d":"a + bi","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le conjugué de z = a + bi est défini comme z̄ = a - bi.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"a - bi\", \"b\": \"b + ai\", \"c\": \"-a + bi\", \"d\": \"a + bi\"}}","_debug_options_count":4},{"id":22633,"question":"L'équation z² + 4 = 0 admet deux solutions complexes.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Les solutions sont z = 2i et z = -2i, qui sont bien des nombres complexes.","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":22634,"question":"Quel est l'argument principal du nombre complexe z = -1 - i ?","option_a":"-3π\/4","option_b":"π\/4","option_c":"5π\/4","option_d":"3π\/4","option_e":"","option_f":"","bonne_reponse":"a","explication":"z = -1 - i est dans le 3ème quadrant. Son argument est θ = arctan(1\/1) + π = π\/4 + π = 5π\/4, mais l'argument principal est -3π\/4 (ou 5π\/4 - 2π = -3π\/4).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"-3π\/4\", \"b\": \"π\/4\", \"c\": \"5π\/4\", \"d\": \"3π\/4\"}}","_debug_options_count":4},{"id":22635,"question":"Si z₁ = 2 + 3i et z₂ = 1 - i, quel est le produit z₁ × z₂ ?","option_a":"5 + i","option_b":"-1 + 5i","option_c":"2 + 3i - i + 3","option_d":"1 + 5i","option_e":"","option_f":"","bonne_reponse":"b","explication":"z₁ × z₂ = (2 + 3i)(1 - i) = 2 - 2i + 3i - 3i² = 2 + i + 3 = 5 + i (car i² = -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\": \"5 + i\", \"b\": \"-1 + 5i\", \"c\": \"2 + 3i - i + 3\", \"d\": \"1 + 5i\"}}","_debug_options_count":4},{"id":22636,"question":"La forme trigonométrique d'un nombre complexe est toujours unique.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La forme trigonométrique n'est pas unique : on peut ajouter 2π à l'argument sans changer le nombre complexe.","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":22637,"question":"Quel est le résultat de (1 + i)³ ?","option_a":"2 + 2i","option_b":"-2 + 2i","option_c":"1 + 3i","option_d":"0","option_e":"","option_f":"","bonne_reponse":"b","explication":"(1 + i)³ = (1 + i)(1 + 2i + i²) = (1 + i)(2i) = 2i + 2i² = 2i - 2 = -2 + 2i.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"2 + 2i\", \"b\": \"-2 + 2i\", \"c\": \"1 + 3i\", \"d\": \"0\"}}","_debug_options_count":4},{"id":22638,"question":"Si |z| = 5 et arg(z) = π\/2, quelle est la forme algébrique de z ?","option_a":"5i","option_b":"5","option_c":"-5i","option_d":"-5","option_e":"","option_f":"","bonne_reponse":"a","explication":"z = |z|(cos(arg(z)) + i sin(arg(z))) = 5(cos(π\/2) + i sin(π\/2)) = 5(0 + i) = 5i.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"5i\", \"b\": \"5\", \"c\": \"-5i\", \"d\": \"-5\"}}","_debug_options_count":4}]