Les nombres complexes : quiz interactif pour Terminale Mathématiques

Question 1 sur 10 10:00

[{"id":55201,"question":"Quel est le module du nombre complexe z = 3 - 4i ?","option_a":"5","option_b":"7","option_c":"√7","option_d":"√(3² + 4²)","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le module de z = a + ib est donné par |z| = √(a² + b²). Ici, |z| = √(3² + (-4)²) = √(9 + 16) = √25 = 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":55202,"question":"L'argument d'un nombre complexe réel positif est toujours égal à 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Un nombre complexe réel positif z = a (avec a \u003E 0) a pour argument θ = 0, car il se situe sur l'axe des réels positifs.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":55203,"question":"Quelle est la forme trigonométrique de z = -1 + i ?","option_a":"z = √2 (cos(3π\/4) + i sin(3π\/4))","option_b":"z = 2 (cos(π\/4) + i sin(π\/4))","option_c":"z = √2 (cos(π\/4) + i sin(π\/4))","option_d":"z = 1 (cos(π) + i sin(π))","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le module de z = -1 + i est |z| = √((-1)² + 1²) = √2. L'argument θ vérifie cosθ = -1\/√2 et sinθ = 1\/√2, donc θ = 3π\/4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":55204,"question":"Si z = 2e^(iπ\/3), alors z² est égal à :","option_a":"4e^(i2π\/3)","option_b":"2e^(i2π\/3)","option_c":"4e^(iπ\/6)","option_d":"8e^(iπ\/3)","option_e":"","option_f":"","bonne_reponse":"A","explication":"Pour z = re^(iθ), z² = r²e^(i2θ). Ici, z² = 2²e^(i2×π\/3) = 4e^(i2π\/3).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":55205,"question":"Le conjugué d'un nombre complexe z = a + ib est toujours un nombre réel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Le conjugué de z = a + ib est z̄ = a - ib. Il 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":55206,"question":"Quelle est la solution de l'équation z² + 4z + 13 = 0 dans ℂ ?","option_a":"z = -2 + 3i ou z = -2 - 3i","option_b":"z = 2 + 3i ou z = 2 - 3i","option_c":"z = -4 + 3i ou z = -4 - 3i","option_d":"z = 1 + 2i ou z = 1 - 2i","option_e":"","option_f":"","bonne_reponse":"A","explication":"En résolvant l'équation avec le discriminant Δ = 16 - 52 = -36, on obtient z = (-4 ± √(-36))\/2 = -2 ± 3i.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":55207,"question":"L'équation z³ = 8 a pour solutions :","option_a":"z = 2, z = -1 + i√3, z = -1 - i√3","option_b":"z = 2, z = 1 + i√3, z = 1 - i√3","option_c":"z = 2, z = 2e^(i2π\/3), z = 2e^(i4π\/3)","option_d":"z = 2, z = 2i, z = -2i","option_e":"","option_f":"","bonne_reponse":"A","explication":"Les solutions de z³ = 8 sont z = 2, z = 2e^(i2π\/3) = -1 + i√3, et z = 2e^(i4π\/3) = -1 - i√3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":55208,"question":"Pour tout nombre complexe z, |z × z̄| = |z|².","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le conjugué de z est z̄, et |z × z̄| = |z| × |z̄| = |z| × |z| = |z|², car |z̄| = |z|.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":55209,"question":"Quelle est la forme exponentielle de z = -√3 + i ?","option_a":"z = 2e^(i5π\/6)","option_b":"z = 2e^(iπ\/6)","option_c":"z = √3 e^(i5π\/6)","option_d":"z = 2e^(i7π\/6)","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le module de z = -√3 + i est |z| = √((-√3)² + 1²) = 2. L'argument θ vérifie cosθ = -√3\/2 et sinθ = 1\/2, donc θ = 5π\/6.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":55210,"question":"Si z₁ = 1 + i et z₂ = 1 - i, alors z₁ × z₂ est égal à :","option_a":"2","option_b":"0","option_c":"1 + i","option_d":"1 - i","option_e":"","option_f":"","bonne_reponse":"A","explication":"z₁ × z₂ = (1 + i)(1 - i) = 1 - i² = 1 - (-1) = 2. C'est aussi égal à |z₁|² = (√2)² = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]