Quiz : Maîtrisez les nombres complexes en Terminale Scientifique

Question 1 sur 10 10:00

[{"id":30748,"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 est calculé par √(a² + b²) = √(3² + (-4)²) = √(9 + 16) = √25 = 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":30749,"question":"L'argument principal 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 principal est généralement défini dans l'intervalle ]-π, π] ou [0, 2π[ selon les conventions.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":30750,"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\/4) + i sin(π\/4))","option_c":"2(cos(π\/6) + i sin(π\/6))","option_d":"√3(cos(π\/3) + i sin(π\/3))","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le module est √(1² + (√3)²) = 2. L'argument est π\/3 car tan(θ) = √3\/1 = √3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":30751,"question":"Si z = 2e^(iπ\/4), alors z² = ?","option_a":"4e^(iπ\/2)","option_b":"4e^(iπ\/4)","option_c":"2e^(iπ\/2)","option_d":"8e^(iπ\/4)","option_e":"","option_f":"","bonne_reponse":"A","explication":"En utilisant la formule (re^(iθ))^n = r^n e^(inθ), on obtient 2² e^(i2*π\/4) = 4e^(iπ\/2).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":30752,"question":"Le conjugué de z = a + bi 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 + bi est a - bi, qui est réel seulement si b = 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":30753,"question":"Quelle est la solution de l'équation z² + 4z + 13 = 0 dans ℂ ?","option_a":"z = -2 ± 3i","option_b":"z = 2 ± 3i","option_c":"z = -2 ± i√13","option_d":"z = 2 ± i√13","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le discriminant est Δ = 16 - 52 = -36. Les solutions sont z = (-4 ± √(-36))\/2 = -2 ± 3i.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":30754,"question":"Si z = 5(cos(π\/6) + i sin(π\/6)), alors Re(z) = ?","option_a":"5","option_b":"5√3\/2","option_c":"5\/2","option_d":"5√2\/2","option_e":"","option_f":"","bonne_reponse":"B","explication":"La partie réelle est r cos(θ) = 5 * cos(π\/6) = 5 * (√3\/2) = 5√3\/2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":30755,"question":"La multiplication de deux nombres complexes correspond à une rotation dans le plan complexe.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La multiplication de deux nombres complexes correspond à une rotation et une homothétie (dilatation).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":30756,"question":"Quelle est la valeur de i^100 ?","option_a":"1","option_b":"-1","option_c":"i","option_d":"-i","option_e":"","option_f":"","bonne_reponse":"A","explication":"i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1. Comme 100 est divisible par 4, i^100 = (i^4)^25 = 1^25 = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":30757,"question":"Si z = 3 - 4i, alors z * z̄ = ?","option_a":"25","option_b":"7","option_c":"9 + 16i","option_d":"9 - 16i","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le produit d'un nombre complexe par son conjugué est égal au carré de son module : z * z̄ = |z|² = 3² + (-4)² = 25.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]