Nombres complexes et géométrie : Testez vos connaissances !

Question 1 sur 10 10:00

[{"id":46016,"question":"Quel est le module du nombre complexe z = 3 + 4i ?","option_a":"5","option_b":"7","option_c":"12","option_d":"25","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le module de z = a + bi est donné par √(a² + b²). Ici, √(3² + 4²) = 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":46017,"question":"L'affixe d'un point M dans le plan complexe est z = 2 - i. Quel est son image par la rotation de centre O et d'angle π\/2 ?","option_a":"1 + 2i","option_b":"-1 + 2i","option_c":"2 + i","option_d":"-2 + i","option_e":"","option_f":"","bonne_reponse":"B","explication":"La rotation d'angle π\/2 multiplie z par i : (2 - i) * i = 2i - i² = 1 + 2i. L'image est donc 1 + 2i.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":46018,"question":"Deux droites sont parallèles si et seulement si leurs affixes vérifient une certaine condition. Vrai ou Faux ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Faux. Deux droites sont parallèles si leurs vecteurs directeurs ont des affixes proportionnelles (même argument).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":46019,"question":"Quel est l'argument principal du nombre complexe z = -1 + i√3 ?","option_a":"π\/3","option_b":"2π\/3","option_c":"4π\/3","option_d":"5π\/3","option_e":"","option_f":"","bonne_reponse":"B","explication":"z = -1 + i√3 a pour module 2 et pour argument 2π\/3 (dans le 2ème quadrant).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":46020,"question":"Si z1 et z2 sont deux nombres complexes non nuls, alors |z1 * z2| = |z1| * |z2|. Vrai ou Faux ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Vrai. La propriété du module d'un produit est |z1 * z2| = |z1| * |z2|.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":46021,"question":"Quelle transformation géométrique est associée à la multiplication par un nombre complexe de module 1 ?","option_a":"Translation","option_b":"Rotation","option_c":"Homothetie","option_d":"Symétrie axiale","option_e":"","option_f":"","bonne_reponse":"B","explication":"Multiplier par un nombre complexe de module 1 (e^(iθ)) correspond à une rotation d'angle θ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":46022,"question":"L'ensemble des points M d'affixe z tels que |z - 2| = 3 est :","option_a":"Une droite","option_b":"Un cercle de centre 2 et de rayon 3","option_c":"Une demi-droite","option_d":"Un segment","option_e":"","option_f":"","bonne_reponse":"B","explication":"|z - a| = r représente un cercle de centre a et de rayon r. Ici, c'est un cercle de centre 2 et de rayon 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":46023,"question":"Si z = 1 + i, alors z² est égal à :","option_a":"2i","option_b":"1 + 2i","option_c":"2 + 2i","option_d":"0","option_e":"","option_f":"","bonne_reponse":"A","explication":"z² = (1 + i)² = 1 + 2i + i² = 1 + 2i - 1 = 2i.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":46024,"question":"Deux nombres complexes sont égaux si et seulement si leurs parties réelles et imaginaires sont égales. Vrai ou Faux ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Vrai. Deux nombres complexes z1 = a + bi et z2 = c + di sont égaux si et seulement si a = c et b = d.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":46025,"question":"Quel est le conjugué du nombre complexe z = 5 - 3i ?","option_a":"5 + 3i","option_b":"-5 + 3i","option_c":"-5 - 3i","option_d":"3 - 5i","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le conjugué de z = a + bi est z̄ = a - bi. Ici, z̄ = 5 + 3i.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]