Maîtrisez la géométrie des nombres complexes en 10 questions !

Question 1 sur 10 10:00

[{"id":21918,"question":"Quel est le module du nombre complexe z = 3 + 4i ?","option_a":"5","option_b":"7","option_c":"12","option_d":"25","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le module est calculé par |z| = √(3² + 4²) = √(9 + 16) = √25 = 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":21919,"question":"L'argument d'un nombre complexe négatif est toujours compris entre -π et π.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'argument d'un nombre complexe négatif est bien compris entre -π et π, car il correspond à l'angle entre le vecteur et l'axe des réels.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":21920,"question":"Si z = 1 + i, quel est le résultat de z × z̄ (z̄ étant le conjugué de z) ?","option_a":"0","option_b":"1","option_c":"2","option_d":"i","option_e":"","option_f":"","bonne_reponse":"C","explication":"z × z̄ = (1 + i)(1 - i) = 1 - i² = 1 - (-1) = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":21921,"question":"La multiplication de deux nombres complexes correspond à :","option_a":"Une translation","option_b":"Une rotation suivie d'une homothétie","option_c":"Une symétrie par rapport à l'axe des réels","option_d":"Une inversion","option_e":"","option_f":"","bonne_reponse":"B","explication":"La multiplication de deux nombres complexes correspond à une rotation (par l'argument) suivie d'une homothétie (par le module).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":21922,"question":"L'équation z² = -1 admet pour solutions :","option_a":"z = 1 et z = -1","option_b":"z = i et z = -i","option_c":"z = 0 et z = 1","option_d":"Aucune solution réelle","option_e":"","option_f":"","bonne_reponse":"B","explication":"Les solutions de z² = -1 sont z = i et z = -i, car i² = -1 et (-i)² = -1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":21923,"question":"Le conjugué d'un nombre complexe z = a + ib est toujours réel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Le conjugué z̄ = a - ib n'est réel que si b = 0, c'est-à-dire si z est réel.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":21924,"question":"Si z = 2(cos(π\/3) + i sin(π\/3)), quel est son module ?","option_a":"1","option_b":"2","option_c":"π\/3","option_d":"√2","option_e":"","option_f":"","bonne_reponse":"B","explication":"Le module d'un nombre complexe en forme trigonométrique z = r(cosθ + i sinθ) est r. Ici, r = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":21925,"question":"La transformation géométrique associée à la multiplication par i est :","option_a":"Une rotation de 90° dans le sens horaire","option_b":"Une rotation de 90° dans le sens anti-horaire","option_c":"Une homothétie de rapport 2","option_d":"Une symétrie par rapport à l'origine","option_e":"","option_f":"","bonne_reponse":"B","explication":"Multiplier par i correspond à une rotation de 90° dans le sens anti-horaire, car i = e^(iπ\/2).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":21926,"question":"L'équation z³ = 8 admet trois solutions complexes distinctes.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'équation z³ = 8 admet une solution réelle (z = 2) et deux solutions complexes conjuguées (z = -1 + i√3 et z = -1 - i√3).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":21927,"question":"Si z = 1 + √3i, quel est son argument principal (en radians) ?","option_a":"π\/6","option_b":"π\/3","option_c":"π\/2","option_d":"2π\/3","option_e":"","option_f":"","bonne_reponse":"B","explication":"L'argument θ est donné par tanθ = √3\/1 = √3, donc θ = π\/3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]