Nombres complexes : maîtrisez les bases et les applications

Question 1 sur 10 10:00

[{"id":17788,"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":17789,"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":17790,"question":"Quel est le conjugué du nombre complexe z = 2 + 5i ?","option_a":"2 - 5i","option_b":"-2 + 5i","option_c":"5 + 2i","option_d":"-2 - 5i","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le conjugué d'un complexe z = a + ib est z̄ = a - ib.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17791,"question":"La forme trigonométrique de z = -1 + i√3 est :","option_a":"2(cos(2π\/3) + i sin(2π\/3))","option_b":"2(cos(π\/3) + i sin(π\/3))","option_c":"√2(cos(3π\/4) + i sin(3π\/4))","option_d":"1(cos(π) + i sin(π))","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le module est √((-1)² + (√3)²) = 2, et l'argument θ vérifie cosθ = -1\/2 et sinθ = √3\/2, donc θ = 2π\/3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17792,"question":"L'équation z² = -4 admet deux solutions complexes conjuguées.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Les solutions sont z = 2i et z = -2i, qui sont bien conjuguées.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17793,"question":"Quel est le résultat de (1 + i)² ?","option_a":"2i","option_b":"1 + 2i","option_c":"2 + 2i","option_d":"0","option_e":"","option_f":"","bonne_reponse":"A","explication":"(1 + i)² = 1 + 2i + i² = 1 + 2i - 1 = 2i.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17794,"question":"L'affixe d'un point M dans le plan complexe est z = -3 - 2i. Quelle est la distance OM où O est l'origine ?","option_a":"√13","option_b":"5","option_c":"√5","option_d":"13","option_e":"","option_f":"","bonne_reponse":"A","explication":"La distance OM est le module de z : |z| = √((-3)² + (-2)²) = √(9 + 4) = √13.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17795,"question":"Pour tout nombre complexe z, on a |z²| = |z|².","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"En effet, |z²| = |z|² car |z²| = |z * z| = |z| * |z| = |z|².","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17796,"question":"La forme exponentielle d'un nombre complexe z = r(cosθ + i sinθ) est :","option_a":"z = re^(iθ)","option_b":"z = r²e^(iθ)","option_c":"z = re^(-iθ)","option_d":"z = e^(irθ)","option_e":"","option_f":"","bonne_reponse":"A","explication":"La formule d'Euler donne z = r(cosθ + i sinθ) = re^(iθ).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17797,"question":"Si z₁ = 2e^(iπ\/4) et z₂ = 3e^(iπ\/6), quel est le module de z₁ * z₂ ?","option_a":"5","option_b":"6","option_c":"√6","option_d":"1","option_e":"","option_f":"","bonne_reponse":"B","explication":"Le module de z₁ * z₂ est |z₁| * |z₂| = 2 * 3 = 6.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]