Quiz — td_complexes.pdf

Question 1 sur 10 20:00

[{"id":91337,"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| = √(a² + b²) = √(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":91338,"question":"La forme trigonométrique d'un nombre complexe z = -1 + i√3 est :","option_a":"2(cos(π\/3) + i sin(π\/3))","option_b":"2(cos(2π\/3) + i sin(2π\/3))","option_c":"√3(cos(π\/6) + i sin(π\/6))","option_d":"2(cos(5π\/6) + i sin(5π\/6))","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le module est 2 et l'argument est 2π\/3 (120°), car z se situe dans le deuxième quadrant.","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(cos(π\/3) + i sin(π\/3))\", \"b\": \"2(cos(2π\/3) + i sin(2π\/3))\",","_debug_options_count":4},{"id":91339,"question":"Vrai ou Faux : Le conjugué d'un nombre complexe z = a + bi est défini par z̄ = a - bi.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une définition fondamentale : le conjugué change le signe de la partie imaginaire.","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":91340,"question":"Quelle est la solution de l'équation z² + 4z + 13 = 0 dans ℂ ?","option_a":"z = -2 ± 3i","option_b":"z = 2 ± 3i","option_c":"z = -2 ± i√3","option_d":"z = 2 ± i√13","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la formule des racines, Δ = 16 - 52 = -36, donc z = (-4 ± √(-36))\/2 = -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\": \"z = -2 ± 3i\", \"b\": \"z = 2 ± 3i\", \"c\": \"z = -2 ± i√3\", \"d\": \"","_debug_options_count":4},{"id":91341,"question":"Vrai ou Faux : Deux nombres complexes sont égaux si et seulement si leurs parties réelles et imaginaires sont égales.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une propriété fondamentale de l'égalité dans ℂ : z1 = z2 ⇔ Re(z1)=Re(z2) et Im(z1)=Im(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":91342,"question":"Quel est l'argument principal (dans ]-π, π]) du nombre complexe z = -√3 - i ?","option_a":"-π\/6","option_b":"-5π\/6","option_c":"π\/6","option_d":"7π\/6","option_e":"","option_f":"","bonne_reponse":"b","explication":"z se situe dans le troisième quadrant. L'angle est π + π\/6 = 7π\/6, mais l'argument principal est -5π\/6.","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\/6\", \"b\": \"-5π\/6\", \"c\": \"π\/6\", \"d\": \"7π\/6\"}}","_debug_options_count":4},{"id":91343,"question":"La multiplication de deux nombres complexes z1 = 2(cos(π\/4) + i sin(π\/4)) et z2 = 3(cos(π\/3) + i sin(π\/3)) donne :","option_a":"6(cos(7π\/12) + i sin(7π\/12))","option_b":"5(cos(7π\/12) + i sin(7π\/12))","option_c":"6(cos(π\/7) + i sin(π\/7))","option_d":"5(cos(π\/12) + i sin(π\/12))","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module est 2×3=6 et l'argument est π\/4 + π\/3 = 7π\/12 (formule de multiplication en forme trigonométrique).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"6(cos(7π\/12) + i sin(7π\/12))\", \"b\": \"5(cos(7π\/12) + i sin(7π\/","_debug_options_count":4},{"id":91344,"question":"Vrai ou Faux : Le nombre complexe i est une racine carrée de -1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, i² = -1, donc i est bien une racine carrée de -1 dans ℂ.","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":91345,"question":"Quelle est la forme algébrique du nombre complexe z = 4e^(iπ\/3) ?","option_a":"2 + 2i√3","option_b":"2√3 + 2i","option_c":"4\/2 + 4i√3\/2","option_d":"2 + i√3","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la formule d'Euler : z = 4(cos(π\/3) + i sin(π\/3)) = 4(1\/2 + i√3\/2) = 2 + 2i√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 + 2i√3\", \"b\": \"2√3 + 2i\", \"c\": \"4\/2 + 4i√3\/2\", \"d\": \"2 + ","_debug_options_count":4},{"id":91346,"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":"a","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\": \"a\", \"options\": {\"a\": \"-2 + 2i\", \"b\": \"2 - 2i\", \"c\": \"-2 - 2i\", \"d\": \"2 + 2i\"}}","_debug_options_count":4}]