Quiz — Nombres complex.pdf

Question 1 sur 10 20:00

[{"id":7485,"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 se calcule par |z| = √(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":7486,"question":"L'argument principal 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 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":7487,"question":"Quelle est la forme trigonométrique de z = 1 + i ?","option_a":"√2(cos(π\/4) + i sin(π\/4))","option_b":"2(cos(π\/4) + i sin(π\/4))","option_c":"√2(cos(π\/2) + i sin(π\/2))","option_d":"1(cos(π\/4) + i sin(π\/4))","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module est √(1² + 1²) = √2 et l'argument est π\/4, d'où la forme √2(cos(π\/4) + i sin(π\/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\": \"√2(cos(π\/4) + i sin(π\/4))\", \"b\": \"2(cos(π\/4) + i sin(π\/4))\"","_debug_options_count":4},{"id":7488,"question":"L'équation z² + 4 = 0 a-t-elle des solutions dans ℂ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, les solutions sont z = 2i et z = -2i, car (2i)² = -4 et (-2i)² = -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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":7489,"question":"Quel est le résultat de (1 + i)³ ?","option_a":"2 + 2i","option_b":"-2 + 2i","option_c":"2 - 2i","option_d":"-2 - 2i","option_e":"","option_f":"","bonne_reponse":"b","explication":"En développant (1 + i)³ = 1 + 3i + 3i² + i³ = 1 + 3i - 3 - i = -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\": \"2 - 2i\", \"d\": \"-2 - 2i\"}}","_debug_options_count":4},{"id":7490,"question":"Le produit de deux nombres complexes a toujours un module égal au produit de leurs modules.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une propriété fondamentale : |z1 * z2| = |z1| * |z2|.","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":7491,"question":"Quelle est la forme algébrique de z = 2e^(iπ\/3) ?","option_a":"1 + i√3","option_b":"√3 + i","option_c":"1 + i","option_d":"√3 + i√3","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la formule d'Euler, 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\": \"1 + i\", \"d\": \"√3 + i√3\"}}","_debug_options_count":4},{"id":7492,"question":"L'équation z² = -1 a-t-elle une solution réelle ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Non, car le carré d'un nombre réel est toujours positif ou nul.","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":7493,"question":"Quel est l'argument principal de z = -1 + i√3 ?","option_a":"2π\/3","option_b":"π\/3","option_c":"4π\/3","option_d":"-2π\/3","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le point (-1, √3) est dans le 2ème quadrant. L'angle est π - π\/3 = 2π\/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\": \"2π\/3\", \"b\": \"π\/3\", \"c\": \"4π\/3\", \"d\": \"-2π\/3\"}}","_debug_options_count":4},{"id":7494,"question":"La somme de deux nombres complexes conjugués est toujours un nombre réel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si z = a + bi, alors son conjugué est z̄ = a - bi. Leur somme est 2a, qui est bien réelle.","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}]