Nombres complexes : Maîtrisez les bases pour le Bac

Question 1 sur 10 10:00

[{"id":22588,"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},{"id":22589,"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},{"id":22590,"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},{"id":22591,"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},{"id":22592,"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},{"id":22593,"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},{"id":22594,"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},{"id":22595,"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},{"id":22596,"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},{"id":22597,"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}]