Quiz — Série 6 Complexes

Question 1 sur 5 10:00

[{"id":2479,"question":"Quel est le module du nombre complexe z = 3 - 4i ?","option_a":"5","option_b":"7","option_c":"√7","option_d":"√(3² + 4²)","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module de 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\": \"√(3² + 4²)\"}}","_debug_options_count":4},{"id":2480,"question":"Si z = 2(cos(π\/3) + i sin(π\/3)), quelle est sa forme algébrique ?","option_a":"1 + i√3","option_b":"1 - i√3","option_c":"√3 + i","option_d":"2 + i√3","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant les formules cos(π\/3) = 1\/2 et sin(π\/3) = √3\/2, on obtient 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\": \"1 - i√3\", \"c\": \"√3 + i\", \"d\": \"2 + i√3\"}}","_debug_options_count":4},{"id":2481,"question":"Quelle transformation géométrique correspond à la multiplication par i dans le plan complexe ?","option_a":"Une translation de vecteur (1, 0)","option_b":"Une rotation de 90° dans le sens direct","option_c":"Une homothétie de rapport 2","option_d":"Une symétrie par rapport à l'axe des réels","option_e":"","option_f":"","bonne_reponse":"b","explication":"Multiplier un nombre complexe par i revient à le multiplier par e^(iπ\/2), ce qui correspond à une rotation de 90° dans le sens direct autour de l'origine.","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 de vecteur (1, 0)\", \"b\": \"Une rotation de 90° da","_debug_options_count":4},{"id":2482,"question":"Résoudre dans ℂ l'équation z² = -16.","option_a":"z = 4i ou z = -4i","option_b":"z = 4 ou z = -4","option_c":"z = 2i ou z = -2i","option_d":"z = 8i ou z = -8i","option_e":"","option_f":"","bonne_reponse":"a","explication":"Les solutions de z² = -16 sont z = ±√(-16) = ±4i, car i² = -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\": \"z = 4i ou z = -4i\", \"b\": \"z = 4 ou z = -4\", \"c\": \"z = 2i ou z = -","_debug_options_count":4},{"id":2483,"question":"Quel est l'argument principal du nombre complexe z = -1 - i ?","option_a":"-π\/4","option_b":"3π\/4","option_c":"5π\/4","option_d":"7π\/4","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'argument θ d'un nombre complexe z = a + bi est donné par θ = arctan(b\/a) + kπ (selon le quadrant). Ici, z = -1 - i est dans le 3ème quadrant, donc θ = arctan(1) + π = π\/4 + π = 5π\/4. L'argument principal est 5π\/4 - 2π = -3π\/4, mais on le ramène souvent à l'intervalle ]-π, π], soit 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\": \"-π\/4\", \"b\": \"3π\/4\", \"c\": \"5π\/4\", \"d\": \"7π\/4\"}}","_debug_options_count":4}]