Les nombres complexes : quiz interactif pour Terminale Sciences

Question 1 sur 10 10:00

[{"id":9523,"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 se calcule par |z| = √(a² + b²) = √(3² + (-4)²) = √(9 + 16) = √25 = 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":9524,"question":"L'argument d'un nombre complexe 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 θ est généralement défini modulo 2π et peut prendre n'importe quelle valeur réelle, pas seulement entre 0 et π.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":9525,"question":"Quelle est la forme trigonométrique de z = 1 + i√3 ?","option_a":"2(cos(π\/3) + i sin(π\/3))","option_b":"2(cos(π\/6) + i sin(π\/6))","option_c":"√3(cos(π\/3) + i sin(π\/3))","option_d":"√2(cos(π\/4) + i sin(π\/4))","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le module est √(1² + (√3)²) = 2. L'argument est arctan(√3\/1) = π\/3. D'où z = 2(cos(π\/3) + i sin(π\/3)).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":9526,"question":"Le conjugué de z = 2 - 5i est égal à :","option_a":"2 + 5i","option_b":"-2 + 5i","option_c":"5 + 2i","option_d":"-5 + 2i","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le conjugué d'un nombre complexe z = a + bi est défini par z̄ = a - bi. Ici, z̄ = 2 + 5i.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":9527,"question":"L'équation z² + 4z + 13 = 0 admet deux solutions complexes conjuguées.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le discriminant Δ = 16 - 52 = -36 \u003C 0, donc les solutions sont complexes conjuguées : z = -2 ± 3i.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":9528,"question":"Quel est le résultat de (1 + i)³ ?","option_a":"2 + 2i","option_b":"1 + 3i","option_c":"2i","option_d":"-2 + 2i","option_e":"","option_f":"","bonne_reponse":"D","explication":"(1 + i)³ = 1 + 3i + 3i² + i³ = 1 + 3i - 3 - i = -2 + 2i (car i² = -1 et i³ = -i).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":9529,"question":"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 un nombre réel.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":9530,"question":"Quelle est la forme exponentielle de z = -√2\/2 + i√2\/2 ?","option_a":"e^(iπ\/4)","option_b":"e^(i3π\/4)","option_c":"e^(iπ\/2)","option_d":"e^(iπ)","option_e":"","option_f":"","bonne_reponse":"B","explication":"Le module est 1. L'argument est 3π\/4 (car le point est dans le 2ème quadrant). D'où z = e^(i3π\/4).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":9531,"question":"L'équation z⁴ = 16 admet exactement 4 solutions complexes.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Les solutions sont les racines quatrièmes de 16, soit z = 2, z = 2i, z = -2, z = -2i. Il y a bien 4 solutions.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":9532,"question":"Quel est le résultat de (3 - 2i)(1 + i) ?","option_a":"5 + i","option_b":"1 + i","option_c":"5 - i","option_d":"1 - i","option_e":"","option_f":"","bonne_reponse":"A","explication":"(3 - 2i)(1 + i) = 3×1 + 3×i - 2i×1 - 2i×i = 3 + 3i - 2i - 2i² = 3 + i + 2 = 5 + i (car i² = -1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]