Maîtrisez les nombres complexes : module, argument et applications

Question 1 sur 10 10:00

[{"id":27838,"question":"Quel est le module du nombre complexe z = 3 - 4i ?","option_a":"5","option_b":"7","option_c":"√7","option_d":"√13","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le module se calcule par |z| = √(a²+b²) = √(3²+(-4)²) = √25 = 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27839,"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 défini dans l'intervalle ]-π, π], donc il peut être négatif.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27840,"question":"Quelle est la forme trigonométrique de z = -1 + i√3 ?","option_a":"2(cos(2π\/3) + i sin(2π\/3))","option_b":"2(cos(π\/3) + i sin(π\/3))","option_c":"√3(cos(π\/6) + i sin(π\/6))","option_d":"2(cos(5π\/6) + i sin(5π\/6))","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le module est 2, et l'argument est 2π\/3 (car z est dans le 2ème quadrant).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27841,"question":"L'équation z² = -4 admet pour solutions dans ℂ :","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 = ±2i, car (2i)² = 4i² = -4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27842,"question":"Le conjugué d'un nombre complexe 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 est z̄ = a - bi, qui n'est réel que si b = 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27843,"question":"Quelle est la forme exponentielle de z = 1 - i ?","option_a":"√2 e^(-iπ\/4)","option_b":"√2 e^(iπ\/4)","option_c":"2 e^(-iπ\/4)","option_d":"e^(-iπ\/4)","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le module est √2 et l'argument est -π\/4, donc z = √2 e^(-iπ\/4).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27844,"question":"Dans le plan complexe, la multiplication par i correspond à une rotation de :","option_a":"90° dans le sens horaire","option_b":"90° dans le sens anti-horaire","option_c":"180°","option_d":"45°","option_e":"","option_f":"","bonne_reponse":"B","explication":"Multiplier par i revient à multiplier par e^(iπ\/2), soit une rotation de 90° anti-horaire.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27845,"question":"L'équation z³ = 8 admet exactement trois solutions distinctes dans ℂ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"C'est un polynôme de degré 3, donc il admet exactement 3 solutions (théorème de d'Alembert).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27846,"question":"Quel est le résultat de (1 + i)^4 ?","option_a":"-4","option_b":"4","option_c":"0","option_d":"-4i","option_e":"","option_f":"","bonne_reponse":"A","explication":"(1 + i)^2 = 2i, donc (1 + i)^4 = (2i)^2 = -4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27847,"question":"La distance entre deux points M(z₁) et N(z₂) dans le plan complexe est égale à :","option_a":"|z₁ + z₂|","option_b":"|z₁ - z₂|","option_c":"|z₁| + |z₂|","option_d":"|z₁| - |z₂|","option_e":"","option_f":"","bonne_reponse":"B","explication":"La distance est donnée par la norme de la différence des affixes : |z₁ - z₂|.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]