Quiz — révision_complexe.pdf

Question 1 sur 10 20:00

[{"id":70089,"question":"Quel est le module du nombre complexe z = 3 - 4i ?","option_a":"5","option_b":"7","option_c":"1","option_d":"12","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\": \"12\"}}","_debug_options_count":4},{"id":70090,"question":"L'argument d'un nombre complexe négatif est toujours égal à π.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Un nombre complexe négatif a un argument égal à π + 2kπ (k entier), mais pas toujours exactement π selon sa position dans le plan complexe.","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":70091,"question":"Quelle est la forme trigonométrique de z = -1 + i√3 ?","option_a":"2(cos(2π\/3) + i sin(2π\/3))","option_b":"2(cos(π\/3) + i sin(π\/3))","option_c":"√3(cos(π\/6) + i sin(π\/6))","option_d":"1(cos(π) + i sin(π))","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module est √((-1)² + (√3)²) = 2. L'argument θ vérifie cosθ = -1\/2 et sinθ = √3\/2, donc θ = 2π\/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(2π\/3) + i sin(2π\/3))\", \"b\": \"2(cos(π\/3) + i sin(π\/3))\",","_debug_options_count":4},{"id":70092,"question":"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 + ib, alors z × z̄ = (a + ib)(a - ib) = 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":70093,"question":"Quelle est la solution de l'équation z² = -4 dans ℂ ?","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√3 ou z = -1 - i√3","option_e":"","option_f":"","bonne_reponse":"a","explication":"Les solutions sont z = √(-4) = 2i et z = -2i, car (2i)² = 4i² = -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\": \"z = 2i ou z = -2i\", \"b\": \"z = 2 ou z = -2\", \"c\": \"z = 4i ou z = -","_debug_options_count":4},{"id":70094,"question":"Le nombre complexe z = 5(cos(π\/4) + i sin(π\/4)) a pour forme algébrique :","option_a":"5\/√2 + i5\/√2","option_b":"5√2\/2 + i5√2\/2","option_c":"5 + i5","option_d":"5\/2 + i5\/2","option_e":"","option_f":"","bonne_reponse":"b","explication":"cos(π\/4) = sin(π\/4) = √2\/2, donc z = 5(√2\/2 + i√2\/2) = 5√2\/2 + i5√2\/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\": \"5\/√2 + i5\/√2\", \"b\": \"5√2\/2 + i5√2\/2\", \"c\": \"5 + i5\", \"d\":","_debug_options_count":4},{"id":70095,"question":"Dans le plan complexe, l'ensemble des points M d'affixe z tels que |z - 2| = 3 est :","option_a":"Une droite","option_b":"Un cercle de centre O et de rayon 3","option_c":"Un cercle de centre 2 et de rayon 3","option_d":"Une demi-droite","option_e":"","option_f":"","bonne_reponse":"c","explication":"|z - 2| représente la distance entre M(z) et le point A(2). L'ensemble des points à distance 3 de A est un cercle de centre A et de rayon 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\": \"Une droite\", \"b\": \"Un cercle de centre O et de rayon 3\", \"c\": \"Un","_debug_options_count":4},{"id":70096,"question":"Le conjugué d'un nombre complexe est toujours différent de lui-même.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Un nombre complexe est égal à son conjugué si et seulement si sa partie imaginaire est nulle (z 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":70097,"question":"Quelle est la valeur de (1 + i)⁴ ?","option_a":"-4","option_b":"4i","option_c":"4","option_d":"-4i","option_e":"","option_f":"","bonne_reponse":"a","explication":"(1 + i)² = 1 + 2i + i² = 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\": \"4i\", \"c\": \"4\", \"d\": \"-4i\"}}","_debug_options_count":4},{"id":70098,"question":"L'équation z³ = 8 a trois solutions complexes distinctes.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Les solutions sont z = 2, z = 2e^(2iπ\/3) et z = 2e^(4iπ\/3), qui sont distinctes 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}]