Quiz — Corrigé-série-2-5-Nombres complexes 26-9.pdf

Question 1 sur 10 20:00

[{"id":74015,"question":"Quelle est la forme algébrique de (3 + 2i) + (1 - 4i) ?","option_a":"4 - 2i","option_b":"4 + 6i","option_c":"2 - 2i","option_d":"3 - 2i","option_e":"","option_f":"","bonne_reponse":"a","explication":"On additionne les parties réelles (3+1=4) et les parties imaginaires (2i-4i=-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\": \"4 - 2i\", \"b\": \"4 + 6i\", \"c\": \"2 - 2i\", \"d\": \"3 - 2i\"}}","_debug_options_count":4},{"id":74016,"question":"Le module de z = 5 - 12i est égal à :","option_a":"13","option_b":"√119","option_c":"√(-119)","option_d":"17","option_e":"","option_f":"","bonne_reponse":"a","explication":"|z| = √(5² + (-12)²) = √(25 + 144) = √169 = 13.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"13\", \"b\": \"√119\", \"c\": \"√(-119)\", \"d\": \"17\"}}","_debug_options_count":4},{"id":74017,"question":"L'argument principal de z = -1 - i est :","option_a":"3π\/4","option_b":"5π\/4","option_c":"π\/4","option_d":"-π\/4","option_e":"","option_f":"","bonne_reponse":"b","explication":"z est dans le 3ème quadrant. θ = arctan(1) + π = π\/4 + π = 5π\/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\": \"3π\/4\", \"b\": \"5π\/4\", \"c\": \"π\/4\", \"d\": \"-π\/4\"}}","_debug_options_count":4},{"id":74018,"question":"Le conjugué de z = 2 + 3i est :","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le conjugué de z = a + bi est a - bi, donc ici 2 - 3i.","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":74019,"question":"Quelle est la forme trigonométrique de z = 1 + i√3 ?","option_a":"2(cos(π\/3) + i sin(π\/3))","option_b":"2(cos(π\/6) + i sin(π\/6))","option_c":"√3(cos(π\/3) + i sin(π\/3))","option_d":"2(cos(π\/4) + i sin(π\/4))","option_e":"","option_f":"","bonne_reponse":"a","explication":"Module r = √(1² + (√3)²) = 2. Argument θ = arctan(√3\/1) = π\/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\": \"2(cos(π\/3) + i sin(π\/3))\", \"b\": \"2(cos(π\/6) + i sin(π\/6))\", \"","_debug_options_count":4},{"id":74020,"question":"L'équation z² + 4 = 0 admet pour solutions :","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":"z² = -4 ⇒ z = ±√(-4) = ±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":74021,"question":"Le produit (1 + i)(1 - i) est égal à :","option_a":"2","option_b":"0","option_c":"1 + i²","option_d":"2i","option_e":"","option_f":"","bonne_reponse":"a","explication":"(1 + i)(1 - i) = 1 - i² = 1 - (-1) = 2.","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\", \"b\": \"0\", \"c\": \"1 + i²\", \"d\": \"2i\"}}","_debug_options_count":4},{"id":74022,"question":"L'argument d'un nombre complexe est toujours compris entre 0 et π.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'argument principal est généralement compris entre -π et π (ou 0 et 2π).","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":74023,"question":"Quelle est la forme exponentielle de z = -√3 + i ?","option_a":"2e^(i5π\/6)","option_b":"2e^(iπ\/6)","option_c":"e^(i5π\/6)","option_d":"2e^(iπ\/3)","option_e":"","option_f":"","bonne_reponse":"a","explication":"Module r = 2, argument θ = 5π\/6 (3ème quadrant).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"2e^(i5π\/6)\", \"b\": \"2e^(iπ\/6)\", \"c\": \"e^(i5π\/6)\", \"d\": \"2e^(iπ","_debug_options_count":4},{"id":74024,"question":"Si z = 3(cos(π\/4) + i sin(π\/4)), alors z³ = :","option_a":"27(cos(3π\/4) + i sin(3π\/4))","option_b":"9(cos(3π\/4) + i sin(3π\/4))","option_c":"27(cos(π\/4) + i sin(π\/4))","option_d":"3(cos(3π\/4) + i sin(3π\/4))","option_e":"","option_f":"","bonne_reponse":"a","explication":"Formule de Moivre : zⁿ = rⁿ(cos(nθ) + i sin(nθ)). Ici, 3³=27 et 3×π\/4=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\": \"27(cos(3π\/4) + i sin(3π\/4))\", \"b\": \"9(cos(3π\/4) + i sin(3π\/4)","_debug_options_count":4}]