Quiz — 78611-magazine7nombres-complexes-enonce.pdf

Question 1 sur 10 20:00

[{"id":9993,"question":"Quel est le résultat 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 séparément les parties réelles et imaginaires : (3+1) + (2i-4i) = 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\": \"4 - 2i\", \"b\": \"4 + 6i\", \"c\": \"2 - 2i\", \"d\": \"3 - 2i\"}}","_debug_options_count":4},{"id":9994,"question":"Le nombre complexe 5i est-il un nombre réel ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Un nombre réel a une partie imaginaire nulle. 5i a une partie réelle nulle mais une partie imaginaire non nulle, donc ce n'est pas un nombre réel.","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":9995,"question":"Quel est le module du nombre complexe z = 3 - 4i ?","option_a":"5","option_b":"7","option_c":"1","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\": \"1\", \"d\": \"25\"}}","_debug_options_count":4},{"id":9996,"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 entre -π et π (ou 0 et 2π selon les conventions), mais pas systématiquement entre 0 et π.","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":9997,"question":"Quel est le produit (2 + i)(3 - 2i) ?","option_a":"8 - i","option_b":"6 - 2i","option_c":"4 + i","option_d":"8 + i","option_e":"","option_f":"","bonne_reponse":"a","explication":"On développe le produit : 2*3 + 2*(-2i) + i*3 + i*(-2i) = 6 - 4i + 3i - 2i² = 6 - i + 2 = 8 - i (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\": \"8 - i\", \"b\": \"6 - 2i\", \"c\": \"4 + i\", \"d\": \"8 + i\"}}","_debug_options_count":4},{"id":9998,"question":"Le conjugué d'un nombre complexe z = a + bi est défini par a - bi ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, le conjugué de z = a + bi est bien z̄ = a - bi.","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":9999,"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":"√2(cos(π\/4) + i sin(π\/4))","option_d":"2(cos(π\/2) + i sin(π\/2))","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module est √(1² + (√3)²) = 2. L'argument θ vérifie tan(θ) = √3\/1 = √3, donc θ = π\/3. D'où z = 2(cos(π\/3) + i sin(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":10000,"question":"L'équation z² = -4 admet-elle des solutions complexes ?","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, qui sont des nombres complexes.","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":10001,"question":"Quel est le résultat de (1 + i)² ?","option_a":"2i","option_b":"1 + 2i","option_c":"2","option_d":"1 + i","option_e":"","option_f":"","bonne_reponse":"a","explication":"On développe : (1 + i)² = 1 + 2i + i² = 1 + 2i - 1 = 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\": \"2i\", \"b\": \"1 + 2i\", \"c\": \"2\", \"d\": \"1 + i\"}}","_debug_options_count":4},{"id":10002,"question":"Le nombre complexe z = 0 est-il égal à son conjugué ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, car z = 0 + 0i et son conjugué z̄ = 0 - 0i = 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}]