Les nombres complexes : module, argument et équations — Terminale Math

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[{"id":18068,"question":"Quel est le module du nombre complexe z = 3 - 4i ?","option_a":"5","option_b":"7","option_c":"√7","option_d":"√34","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le module se calcule par √(3² + (-4)²) = √(9 + 16) = √25 = 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":18069,"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[. Par exemple, l'argument de -1 est π (ou -π selon la convention).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":18070,"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 θ vérifie tan(θ) = √3\/1 = √3, donc θ = π\/3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":18071,"question":"L'équation z² = -4 a pour solutions :","option_a":"z = 2i ou z = -2i","option_b":"z = 2 ou z = -2","option_c":"z = 4i ou z = -4i","option_d":"Aucune solution dans ℂ","option_e":"","option_f":"","bonne_reponse":"A","explication":"Les solutions sont z = 2i et z = -2i car (2i)² = 4i² = -4 et (-2i)² = 4i² = -4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":18072,"question":"Le produit de deux nombres complexes a pour module la somme de leurs modules.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Le module du produit est le produit des modules. Par exemple, |z1 * z2| = |z1| * |z2|.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":18073,"question":"La forme exponentielle d'un complexe z = r(cosθ + i sinθ) est :","option_a":"z = re^(iθ)","option_b":"z = r(cosθ + i sinθ)","option_c":"z = e^(rθ)","option_d":"z = r + iθ","option_e":"","option_f":"","bonne_reponse":"A","explication":"La formule d'Euler donne z = re^(iθ) = r(cosθ + i sinθ).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":18074,"question":"Quel est l'argument principal de z = -1 - i ?","option_a":"-3π\/4","option_b":"5π\/4","option_c":"π\/4","option_d":"-π\/4","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le point M(-1, -1) est dans le 3ème quadrant. L'angle est π + π\/4 = 5π\/4, mais l'argument principal est -3π\/4 (ou 5π\/4 selon la convention).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":18075,"question":"L'équation z³ = 8 a trois solutions dans ℂ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'équation z³ - 8 = 0 se factorise en (z - 2)(z² + 2z + 4) = 0. Les solutions sont z = 2, z = -1 + i√3 et z = -1 - i√3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":18076,"question":"Le conjugué d'un nombre complexe z = a + ib est :","option_a":"a - ib","option_b":"-a + ib","option_c":"a + ib","option_d":"√(a² + b²)","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le conjugué de z = a + ib est noté \u0015z = a - ib.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":18077,"question":"La distance entre deux points M(z1) et N(z2) dans le plan complexe est |z1 - z2|.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"En effet, |z1 - z2| représente la norme du vecteur MN, donc la distance entre M et N.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]