Maîtrisez les nombres complexes — Quiz interactif pour Terminale Math

Question 1 sur 10 10:00

[{"id":10533,"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 est calculé par √(a² + b²) = √(3² + (-4)²) = √25 = 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10534,"question":"L'argument principal d'un nombre complexe négatif est toujours compris entre π\/2 et π.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"L'argument d'un nombre complexe négatif est compris entre π et 3π\/2 (ou -π et -π\/2).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10535,"question":"Si z = 2(cos(π\/3) + i sin(π\/3)), quelle est sa forme algébrique ?","option_a":"1 + i√3","option_b":"√3 + i","option_c":"1 + i","option_d":"2 + i√3","option_e":"","option_f":"","bonne_reponse":"A","explication":"En développant : z = 2cos(π\/3) + i 2sin(π\/3) = 2*(1\/2) + i 2*(√3\/2) = 1 + i√3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10536,"question":"Le produit de deux nombres complexes de module 1 est toujours de module 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le module d'un produit est le produit des modules : |z1 * z2| = |z1| * |z2| = 1 * 1 = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10537,"question":"Quelle est la solution de l'équation z² = -4 dans ℂ ?","option_a":"z = 2i ou z = -2i","option_b":"z = 4i","option_c":"z = 2 ou z = -2","option_d":"z = i√2 ou z = -i√2","option_e":"","option_f":"","bonne_reponse":"A","explication":"Les solutions sont z = 2i et z = -2i, car (2i)² = -4 et (-2i)² = -4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10538,"question":"Si z = 1 + i, quel est son argument principal ?","option_a":"π\/4","option_b":"π\/2","option_c":"π\/3","option_d":"3π\/4","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'argument est θ = arctan(b\/a) = arctan(1\/1) = π\/4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10539,"question":"La conjugaison complexe change le signe de l'argument d'un nombre complexe.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La conjugaison change le signe de la partie imaginaire, mais l'argument devient -θ (symétrie par rapport à l'axe réel).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10540,"question":"Quel est le résultat de (1 + i)³ ?","option_a":"2 + 2i","option_b":"1 + 3i","option_c":"-2 + 2i","option_d":"0","option_e":"","option_f":"","bonne_reponse":"C","explication":"En développant : (1 + i)³ = 1 + 3i + 3i² + i³ = 1 + 3i - 3 - i = -2 + 2i.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10541,"question":"Si |z| = 5 et arg(z) = π\/6, quelle est la forme trigonométrique de z ?","option_a":"5(cos(π\/6) + i sin(π\/6))","option_b":"5(cos(π\/3) + i sin(π\/3))","option_c":"3(cos(π\/6) + i sin(π\/6))","option_d":"5(cos(π\/2) + i sin(π\/2))","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},{"id":10542,"question":"L'équation z⁴ = 16 a-t-elle quatre solutions distinctes dans ℂ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Oui, les solutions sont z = 2, z = -2, z = 2i et z = -2i, soit quatre solutions distinctes.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]