Nombres complexes : Maîtrisez les antidépresseurs de l'algèbre !

Question 1 sur 10 10:00

[{"id":27858,"question":"Quelle est la forme algébrique du nombre complexe z = 3 + 4i ?","option_a":"3 + 4i","option_b":"4 + 3i","option_c":"3i + 4","option_d":"4i + 3","option_e":"","option_f":"","bonne_reponse":"A","explication":"La forme algébrique d'un nombre complexe s'écrit sous la forme a + bi, où a et b sont des réels.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27859,"question":"Le nombre complexe z = 1 + i a pour module √2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le module de z = a + bi est √(a² + b²). Ici, √(1² + 1²) = √2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27860,"question":"Quelle est la partie réelle du nombre complexe z = 5 - 2i ?","option_a":"5","option_b":"-2","option_c":"5i","option_d":"-2i","option_e":"","option_f":"","bonne_reponse":"A","explication":"La partie réelle d'un nombre complexe a + bi est le réel a.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27861,"question":"L'argument d'un nombre complexe z = a + bi est toujours positif.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"L'argument est défini à 2π près et peut être négatif selon le quadrant du plan complexe.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27862,"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(π\/2) + i sin(π\/2))","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le module est √((-1)² + (√3)²) = 2. L'argument est π - π\/3 = 2π\/3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27863,"question":"Le produit de deux nombres complexes conjugués est toujours réel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Si z = a + bi, alors z × z̄ = a² + b², qui est un réel positif.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27864,"question":"Quelle est la solution de l'équation z² = -4 dans ℂ ?","option_a":"z = 2i ou z = -2i","option_b":"z = 4i ou z = -4i","option_c":"z = 2 ou z = -2","option_d":"z = 0","option_e":"","option_f":"","bonne_reponse":"A","explication":"Les solutions sont z = ±√(-4) = ±2i.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27865,"question":"Un nombre complexe non nul a toujours un argument défini.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'argument est défini pour tout nombre complexe non nul.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27866,"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.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27867,"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":"Dans ℂ, un polynôme de degré n a exactement n racines (théorème de d'Alembert-Gauss).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]