Maîtrisez les nombres complexes — Quiz interactif pour la Terminale

Question 1 sur 10 10:00

[{"id":40215,"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 de z = a + bi est donné par √(a² + b²). Ici, √(3² + 4²) = √(9 + 16) = √25 = 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":40216,"question":"L'argument d'un nombre complexe est toujours défini à 2π près.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'argument est défini à 2π près car les angles sont périodiques de période 2π.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":40217,"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":"√2(cos(π\/4) + i sin(π\/4))","option_d":"√3(cos(π\/6) + i sin(π\/6))","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le module est √((-1)² + (√3)²) = 2. L'argument est π - π\/3 = 2π\/3. La forme trigonométrique est donc 2(cos(2π\/3) + i sin(2π\/3)).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":40218,"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 (partie réelle nulle).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":40219,"question":"Quel est l'argument principal de z = 1 - i ?","option_a":"-π\/4","option_b":"π\/4","option_c":"3π\/4","option_d":"7π\/4","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le point (1, -1) est dans le 4ème quadrant. L'argument est donc -π\/4 (ou 7π\/4).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":40220,"question":"La multiplication de deux nombres complexes correspond à une addition de leurs arguments.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La multiplication de deux nombres complexes correspond à une addition de leurs arguments (et une multiplication de leurs modules).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":40221,"question":"Quelle est la solution de l'équation z³ = 8i ?","option_a":"2, 2e^(iπ\/3), 2e^(i5π\/3)","option_b":"2i, -2i, 2","option_c":"√8 e^(iπ\/6), √8 e^(i5π\/6), √8 e^(i3π\/2)","option_d":"2, -1 + i√3, -1 - i√3","option_e":"","option_f":"","bonne_reponse":"C","explication":"Les solutions sont les racines cubiques de 8i, soit 2 e^(i(π\/6 + 2kπ\/3)) pour k = 0, 1, 2. Cela donne 2, -1 + i√3, -1 - i√3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":40222,"question":"Le conjugué d'un nombre complexe z = a + bi est toujours de module inférieur ou égal à celui de z.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Le conjugué a le même module que z, car |z| = |conjugué(z)| = √(a² + b²).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":40223,"question":"Quelle est la forme exponentielle de z = -2√3 - 2i ?","option_a":"4 e^(i7π\/6)","option_b":"4 e^(i5π\/6)","option_c":"2√3 e^(iπ\/6)","option_d":"2 e^(i7π\/6)","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le module est √((-2√3)² + (-2)²) = 4. L'argument est π + π\/6 = 7π\/6. La forme exponentielle est donc 4 e^(i7π\/6).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":40224,"question":"L'ensemble des nombres complexes est un corps commutatif.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Les nombres complexes forment un corps commutatif, car ils sont stables pour l'addition et la multiplication, avec des éléments neutres et des inverses.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]