Maîtrisez la géométrie complexe : quiz interactif sur les nombres complexes

Question 1 sur 10 10:00

[{"id":60857,"question":"Quel est le module du nombre complexe z = 3 + 4i ?","option_a":"5","option_b":"7","option_c":"√7","option_d":"1","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le module est calculé par |z| = √(a² + b²) = √(3² + 4²) = 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":60858,"question":"L'argument d'un nombre complexe négatif est toujours compris entre 0 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 2π (ou -π et 0 selon la convention).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":60859,"question":"Quelle transformation géométrique correspond à la multiplication par i ?","option_a":"Translation","option_b":"Rotation de 90°","option_c":"Homothétie de rapport 2","option_d":"Symétrie axiale","option_e":"","option_f":"","bonne_reponse":"B","explication":"Multiplier par i correspond à une rotation de 90° (π\/2 radians) dans le sens direct.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":60860,"question":"Si z = 2(cos(π\/3) + i sin(π\/3)), quelle est la forme algébrique de z ?","option_a":"1 + i√3","option_b":"√3 + i","option_c":"2 + 2i√3","option_d":"1 + i","option_e":"","option_f":"","bonne_reponse":"A","explication":"En développant : z = 2cos(π\/3) + 2i sin(π\/3) = 2*(1\/2) + 2i*(√3\/2) = 1 + i√3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":60861,"question":"Le conjugué d'un nombre complexe z = a + bi est toujours égal à -z.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Le conjugué est a - bi, tandis que -z = -a - bi. Ils ne sont égaux que si a = 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":60862,"question":"Quelle est la condition pour que deux nombres complexes soient égaux ?","option_a":"Mêmes parties réelles et imaginaires","option_b":"Mêmes modules","option_c":"Mêmes arguments","option_d":"Mêmes conjugués","option_e":"","option_f":"","bonne_reponse":"A","explication":"Deux nombres complexes sont égaux si et seulement si leurs parties réelles et imaginaires sont égales.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":60863,"question":"Si z₁ = 1 + i et z₂ = 1 - i, quel est le produit z₁ × z₂ ?","option_a":"0","option_b":"2","option_c":"1 + i²","option_d":"2i","option_e":"","option_f":"","bonne_reponse":"B","explication":"Le produit est (1+i)(1-i) = 1 - i² = 1 - (-1) = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":60864,"question":"Une homothétie de rapport k \u003E 0 et de centre O est représentée par la multiplication par k.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Une homothétie de rapport k est représentée par la multiplication par k (si k \u003E 0) ou par -k (si k \u003C 0 pour une inversion).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":60865,"question":"Quel est l'argument principal du nombre complexe z = -1 - i√3 ?","option_a":"π\/3","option_b":"2π\/3","option_c":"4π\/3","option_d":"5π\/3","option_e":"","option_f":"","bonne_reponse":"C","explication":"Le point est dans le 3ème quadrant. L'angle est π + arctan(√3\/1) = π + π\/3 = 4π\/3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":60866,"question":"Si |z| = 2 et arg(z) = π\/4, quelle est la forme trigonométrique de z ?","option_a":"2(cos(π\/4) + i sin(π\/4))","option_b":"2(cos(π\/4) - i sin(π\/4))","option_c":"√2 + i√2","option_d":"1 + i","option_e":"","option_f":"","bonne_reponse":"A","explication":"La forme trigonométrique est |z|(cos(arg(z)) + i sin(arg(z))) = 2(cos(π\/4) + i sin(π\/4)).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]