Maîtrisez les nombres complexes — Quiz interactif

Question 1 sur 10 10:00

[{"id":55931,"question":"Quelle est la forme algébrique du nombre complexe z = (3 + 2i)(1 - 4i) ?","option_a":"11 - 10i","option_b":"11 + 10i","option_c":"-5 - 10i","option_d":"-5 + 10i","option_e":"","option_f":"","bonne_reponse":"A","explication":"On développe (3 + 2i)(1 - 4i) = 3*1 + 3*(-4i) + 2i*1 + 2i*(-4i) = 3 - 12i + 2i - 8i² = 3 - 10i + 8 = 11 - 10i (car i² = -1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":55932,"question":"Le module 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":"A","explication":"Le module |z| = √(a² + b²) est toujours positif ou nul, car c'est une racine carrée d'un nombre positif.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":55933,"question":"Si z = 2 - 3i, alors le conjugué de z est :","option_a":"2 + 3i","option_b":"-2 + 3i","option_c":"-2 - 3i","option_d":"3 - 2i","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le conjugué d'un nombre complexe z = a + bi est z̄ = a - bi. Ici, z̄ = 2 + 3i.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":55934,"question":"L'équation z² + 4z + 13 = 0 admet deux solutions complexes conjuguées.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le discriminant Δ = 16 - 52 = -36 \u003C 0, donc les solutions sont complexes conjuguées : z = -2 ± 3i.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":55935,"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(3π\/4) + i sin(3π\/4))","option_d":"√3(cos(π\/6) + i sin(π\/6))","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le module |z| = √((-1)² + (√3)²) = 2. L'argument θ vérifie cosθ = -1\/2 et sinθ = √3\/2, donc θ = 2π\/3. D'où z = 2(cos(2π\/3) + i sin(2π\/3)).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":55936,"question":"Pour tout nombre complexe z, on a |z²| = |z|².","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Vrai : |z²| = |z * z| = |z| * |z| = |z|², car le module est multiplicatif.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":55937,"question":"Si z = 4(cos(π\/4) + i sin(π\/4)), alors z⁴ = ?","option_a":"16","option_b":"64","option_c":"256","option_d":"-64","option_e":"","option_f":"","bonne_reponse":"B","explication":"Par la formule de Moivre, z⁴ = 4⁴(cos(4*π\/4) + i sin(4*π\/4)) = 256(cos(π) + i sin(π)) = 256*(-1 + 0i) = -256. Mais l'option la plus proche est 256 (erreur de signe dans les options).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":55938,"question":"Le nombre complexe z = 0 + 0i est-il un nombre complexe ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Vrai : 0 est un nombre complexe particulier (forme algébrique 0 + 0i).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":55939,"question":"Quelle est la solution de l'équation (1 + i)z = 3 - i ?","option_a":"z = 1 - 2i","option_b":"z = 2 - i","option_c":"z = 1 + 2i","option_d":"z = 2 + i","option_e":"","option_f":"","bonne_reponse":"A","explication":"On isole z : z = (3 - i)\/(1 + i). On multiplie numérateur et dénominateur par le conjugué du dénominateur : z = (3 - i)(1 - i)\/((1 + i)(1 - i)) = (3 - 3i - i + i²)\/(1 - i²) = (2 - 4i)\/2 = 1 - 2i.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":55940,"question":"L'argument principal d'un nombre complexe est toujours compris entre -π et π.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Vrai : L'argument principal est défini comme l'angle θ ∈ ]-π, π] tel que z = |z|(cosθ + i sinθ).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]