Les nombres complexes : module, argument et applications

Question 1 sur 10 10:00

[{"id":75292,"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 √(3² + (-4)²) = √(9 + 16) = √25 = 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":75293,"question":"L'argument d'un nombre complexe négatif est toujours π.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"L'argument est π uniquement si le complexe est réel négatif (ex: z = -2). Pour z = -2i, l'argument est -π\/2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":75294,"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 tanθ = √3\/1 ⇒ θ = π\/3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":75295,"question":"Le produit de deux complexes de module 2 a toujours un module de 4.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le module d'un produit est le produit des modules (propriété multiplicative). Ici, 2 × 2 = 4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":75296,"question":"Si z = 2e^(iπ\/4), quelle est sa forme algébrique ?","option_a":"√2 + i√2","option_b":"2 + i2","option_c":"√2 + i2","option_d":"2√2 + i2√2","option_e":"","option_f":"","bonne_reponse":"A","explication":"z = 2(cos(π\/4) + i sin(π\/4)) = 2(√2\/2 + i√2\/2) = √2 + i√2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":75297,"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":"z = 1 + i√3 ou z = -1 - i√3","option_e":"","option_f":"","bonne_reponse":"A","explication":"Les solutions sont z = ±√(-4) = ±2i. On vérifie (2i)² = -4 et (-2i)² = -4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":75298,"question":"Le conjugué de z = 5 - 3i est 5 + 3i.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le conjugué change le signe de la partie imaginaire : si z = a + ib, alors \u0015z = a - ib.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":75299,"question":"Quelle transformation géométrique correspond à la multiplication par i ?","option_a":"Translation de vecteur (1,0)","option_b":"Rotation de π\/2 autour de l'origine","option_c":"Homothetie de rapport 2","option_d":"Symétrie par rapport à l'axe des réels","option_e":"","option_f":"","bonne_reponse":"B","explication":"Multiplier par i revient à une rotation de 90° (π\/2 radians) dans le sens direct autour de l'origine.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":75300,"question":"Si |z| = 3 et arg(z) = π\/3, alors z = :","option_a":"3(cos(π\/3) + i sin(π\/3))","option_b":"3(cos(π\/6) + i sin(π\/6))","option_c":"(3\/2) + i(3√3\/2)","option_d":"3 + i3√3","option_e":"","option_f":"","bonne_reponse":"C","explication":"z = 3(cos(π\/3) + i sin(π\/3)) = 3(1\/2 + i√3\/2) = (3\/2) + i(3√3\/2).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":75301,"question":"L'ensemble ℂ est stable par addition et multiplication.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"ℂ est un corps : il est stable par addition, multiplication, et contient les inverses pour la multiplication (sauf 0).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]