Quiz — 64f33781b8543_corrigé_Série 01-Complexe.pdf

Question 1 sur 10 20:00

[{"id":42735,"question":"Quel est le module du nombre complexe z = 3 - 4i ?","option_a":"5","option_b":"7","option_c":"√7","option_d":"√13","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\": \"√13\"}}","_debug_options_count":4},{"id":42736,"question":"Si z = 2(cos(π\/4) + i sin(π\/4)), quelle est sa forme algébrique ?","option_a":"√2 + i√2","option_b":"1 + i","option_c":"2 + 2i","option_d":"√2 + i","option_e":"","option_f":"","bonne_reponse":"a","explication":"En développant : z = 2(√2\/2 + i√2\/2) = √2 + i√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 + i√2\", \"b\": \"1 + i\", \"c\": \"2 + 2i\", \"d\": \"√2 + i\"}}","_debug_options_count":4},{"id":42737,"question":"Vrai ou Faux : Le conjugué d'un nombre complexe z = a + ib est toujours un nombre réel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le conjugué est z̄ = a - ib, qui n'est réel que si b = 0.","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":42738,"question":"Quelle est la solution de l'équation z² = -1 dans ℂ ?","option_a":"z = i","option_b":"z = -i","option_c":"z = 1 ou z = -1","option_d":"z = i ou z = -i","option_e":"","option_f":"","bonne_reponse":"d","explication":"Les solutions sont les racines carrées de -1, soit z = i et z = -i.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"z = i\", \"b\": \"z = -i\", \"c\": \"z = 1 ou z = -1\", \"d\": \"z = i ou z =","_debug_options_count":4},{"id":42739,"question":"Si z₁ = 1 + i et z₂ = 1 - i, quel est le produit z₁ × z₂ ?","option_a":"0","option_b":"1","option_c":"2","option_d":"i","option_e":"","option_f":"","bonne_reponse":"b","explication":"z₁ × z₂ = (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\": \"b\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"i\"}}","_debug_options_count":4},{"id":42740,"question":"Vrai ou Faux : L'argument d'un nombre complexe non nul est toujours un angle en radians.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'argument est bien défini en radians (ou degrés) pour tout nombre complexe non nul.","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":42741,"question":"Quel est l'argument principal de z = -1 - i ?","option_a":"-3π\/4","option_b":"π\/4","option_c":"3π\/4","option_d":"5π\/4","option_e":"","option_f":"","bonne_reponse":"a","explication":"z = -1 - i est dans le 3ème quadrant. Son argument est -3π\/4 (ou 5π\/4 en positif).","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\": \"3π\/4\", \"d\": \"5π\/4\"}}","_debug_options_count":4},{"id":42742,"question":"Si |z| = 5 et arg(z) = π\/3, quelle est la forme trigonométrique de z ?","option_a":"5(cos(π\/3) + i sin(π\/3))","option_b":"√5(cos(π\/3) + i sin(π\/3))","option_c":"5(cos(π\/6) + i sin(π\/6))","option_d":"5(cos(π\/3) - i sin(π\/3))","option_e":"","option_f":"","bonne_reponse":"a","explication":"La forme trigonométrique est z = |z|(cos(arg(z)) + i sin(arg(z))).","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(cos(π\/3) + i sin(π\/3))\", \"b\": \"√5(cos(π\/3) + i sin(π\/3))\"","_debug_options_count":4},{"id":42743,"question":"Vrai ou Faux : Le module d'un nombre complexe est toujours positif ou nul.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module est défini comme une racine carrée, donc toujours positif ou nul (|z| ≥ 0).","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":42744,"question":"Quelle transformation géométrique représente la multiplication par i dans le plan complexe ?","option_a":"Translation","option_b":"Rotation de π\/2 radians","option_c":"Symétrie par rapport à l'axe réel","option_d":"Homothetie de rapport 1","option_e":"","option_f":"","bonne_reponse":"b","explication":"Multiplier par i revient à une rotation de π\/2 radians (90°) dans le sens direct.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Translation\", \"b\": \"Rotation de π\/2 radians\", \"c\": \"Symétrie pa","_debug_options_count":4}]