Maîtrisez les nombres complexes : quiz interactif pour le Bac Sciences

Question 1 sur 10 10:00

[{"id":54861,"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 √(a² + b²). Ici √(3² + (-4)²) = √(9+16) = √25 = 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54862,"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] ou [0, 2π[ selon les conventions, mais pas toujours entre 0 et π.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54863,"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 satisfait tanθ = √3\/1 = √3, donc θ = π\/3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54864,"question":"Le produit de deux nombres complexes conjugués est toujours un nombre 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 = (a + bi)(a - bi) = a² + b² qui est bien réel.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54865,"question":"Quelle est la solution de l'équation z² = -4 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 = i ou z = -i","option_e":"","option_f":"","bonne_reponse":"A","explication":"Les solutions sont z = ±√(-4) = ±2i, car (2i)² = -4 et (-2i)² = -4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54866,"question":"Si z = 2e^(iπ\/4), quel est l'affixe du point image dans le plan complexe ?","option_a":"(√2, √2)","option_b":"(2, 2)","option_c":"(√2, 2√2)","option_d":"(2√2, 2√2)","option_e":"","option_f":"","bonne_reponse":"A","explication":"La forme trigonométrique z = r(cosθ + i sinθ) donne les coordonnées (r cosθ, r sinθ) = (2×√2\/2, 2×√2\/2) = (√2, √2).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54867,"question":"Le nombre complexe i est-il une racine de l'équation z⁴ = 1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Les racines 4èmes de l'unité sont 1, i, -1, -i. Donc i est bien une solution de z⁴ = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54868,"question":"Quelle est la valeur de (1 + i)^4 ?","option_a":"0","option_b":"4i","option_c":"-4","option_d":"4","option_e":"","option_f":"","bonne_reponse":"C","explication":"En utilisant la forme trigonométrique : 1 + i = √2 e^(iπ\/4), donc (1+i)^4 = (√2)^4 e^(iπ) = 4×(-1) = -4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54869,"question":"Si z₁ = 3e^(iπ\/6) et z₂ = 2e^(iπ\/3), quel est le module de z₁×z₂ ?","option_a":"5","option_b":"6","option_c":"√5","option_d":"3√2","option_e":"","option_f":"","bonne_reponse":"B","explication":"Le module du produit est le produit des modules : 3×2 = 6.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54870,"question":"L'équation z² + 2z + 5 = 0 admet-elle des solutions réelles ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Le discriminant Δ = 4 - 20 = -16 \u003C 0, donc les solutions sont complexes conjuguées : z = -1 ± 2i.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]