Quiz — Complex 2.pdf

Question 1 sur 10 20:00

[{"id":44320,"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 est calculé 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":44321,"question":"La forme trigonométrique d'un nombre complexe z = -1 + i√3 est :","option_a":"2(cos(2π\/3) + i sin(2π\/3))","option_b":"2(cos(π\/3) + i sin(π\/3))","option_c":"√2(cos(3π\/4) + i sin(3π\/4))","option_d":"√3(cos(π\/6) + i sin(π\/6))","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module est 2 et l'argument est 2π\/3 (120°), car z se situe dans le 2ème quadrant.","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(2π\/3) + i sin(2π\/3))\", \"b\": \"2(cos(π\/3) + i sin(π\/3))\",","_debug_options_count":4},{"id":44322,"question":"Vrai ou Faux ? Le conjugué d'un nombre complexe z = a + bi est donné par z̄ = -a + bi.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le conjugué est z̄ = a - bi, pas -a + bi. La partie réelle reste inchangée.","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":44323,"question":"Quelle est la solution de l'équation z² = -4 dans l'ensemble des nombres complexes ?","option_a":"z = 2i ou z = -2i","option_b":"z = 2 ou z = -2","option_c":"z = 4i ou z = -4i","option_d":"z = 1 + i ou z = -1 - i","option_e":"","option_f":"","bonne_reponse":"a","explication":"","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 = 2i ou z = -2i\", \"b\": \"z = 2 ou z = -2\", \"c\": \"z = 4i ou z = -","_debug_options_count":4},{"id":44324,"question":"L'argument principal d'un nombre complexe z = -√3 - i est :","option_a":"5π\/6","option_b":"7π\/6","option_c":"4π\/3","option_d":"π\/6","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le nombre se situe dans le 3ème quadrant. L'argument est calculé par θ = π + arctan(1\/√3) = 7π\/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\": \"5π\/6\", \"b\": \"7π\/6\", \"c\": \"4π\/3\", \"d\": \"π\/6\"}}","_debug_options_count":4},{"id":44325,"question":"Vrai ou Faux ? Le produit 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 z × z̄ = (a + bi)(a - bi) = a² + b², qui est bien un nombre réel.","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":44326,"question":"Quelle est la forme exponentielle de z = 1 + i ?","option_a":"√2 e^(iπ\/4)","option_b":"2 e^(iπ\/4)","option_c":"√2 e^(iπ\/2)","option_d":"e^(iπ\/4)","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module est √2 et l'argument est π\/4, donc z = √2 e^(iπ\/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 e^(iπ\/4)\", \"b\": \"2 e^(iπ\/4)\", \"c\": \"√2 e^(iπ\/2)\", \"d\": ","_debug_options_count":4},{"id":44327,"question":"L'équation z³ = 8 admet combien de solutions dans l'ensemble des nombres complexes ?","option_a":"1","option_b":"2","option_c":"3","option_d":"4","option_e":"","option_f":"","bonne_reponse":"c","explication":"Un nombre complexe non nul admet exactement n racines n-ièmes distinctes. Ici, n=3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"1\", \"b\": \"2\", \"c\": \"3\", \"d\": \"4\"}}","_debug_options_count":4},{"id":44328,"question":"Vrai ou Faux ? La partie réelle d'un nombre complexe est toujours positive.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La partie réelle peut être positive, négative ou nulle. Exemple : z = -3 + 2i a une partie réelle négative.","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":44329,"question":"Quel est le résultat de (1 + i)⁴ ?","option_a":"-4","option_b":"4","option_c":"-4i","option_d":"4i","option_e":"","option_f":"","bonne_reponse":"a","explication":"(1 + i)² = 2i, donc (1 + i)⁴ = (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\": \"-4\", \"b\": \"4\", \"c\": \"-4i\", \"d\": \"4i\"}}","_debug_options_count":4}]