Quiz — 63385dc94eb9b_Enoncé-Maths-Nombres Complexes (suite).pdf

Question 1 sur 10 20:00

[{"id":87513,"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 + ib est donné par |z| = √(a² + b²). Ici, |z| = √(3² + (-4)²) = √(9 + 16) = √25 = 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"5\", \"b\": \"7\", \"c\": \"√7\", \"d\": \"√13\"}}","_debug_options_count":4},{"id":87514,"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":"Par définition, l'argument principal θ vérifie -π \u003C θ ≤ π, donc l'affirmation est vraie.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":87515,"question":"Quelle est la solution de l'équation z² = -1 dans l'ensemble des nombres complexes ?","option_a":"z = i","option_b":"z = -i","option_c":"z = i ou z = -i","option_d":"z = 1 + i","option_e":"","option_f":"","bonne_reponse":"c","explication":"Les solutions de z² = -1 sont les racines carrées de -1, soit z = i et z = -i.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"z = i\", \"b\": \"z = -i\", \"c\": \"z = i ou z = -i\", \"d\": \"z = 1 + i\"}}","_debug_options_count":4},{"id":87516,"question":"Si z = 2(cos(π\/3) + i sin(π\/3)), quel est son conjugué ?","option_a":"z̄ = 2(cos(π\/3) - i sin(π\/3))","option_b":"z̄ = 2(cos(-π\/3) + i sin(-π\/3))","option_c":"z̄ = -2(cos(π\/3) + i sin(π\/3))","option_d":"z̄ = 2(cos(2π\/3) + i sin(2π\/3))","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le conjugué d'un nombre complexe z = a + ib est z̄ = a - ib. En forme trigonométrique, cela donne z̄ = r(cosθ - i sinθ).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"z̄ = 2(cos(π\/3) - i sin(π\/3))\", \"b\": \"z̄ = 2(cos(-π\/3) + i s","_debug_options_count":4},{"id":87517,"question":"L'équation z³ = 8 admet trois solutions complexes distinctes.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation z³ = 8 admet trois solutions complexes distinctes : 2, 2e^(i2π\/3) et 2e^(i4π\/3).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":87518,"question":"Quel est l'argument principal du nombre complexe z = -1 - i√3 ?","option_a":"-2π\/3","option_b":"2π\/3","option_c":"π\/3","option_d":"-π\/3","option_e":"","option_f":"","bonne_reponse":"a","explication":"z = -1 - i√3 correspond à un point dans le 3ème quadrant. Son argument principal est θ = arctan(√3\/1) - π = π\/3 - π = -2π\/3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"-2π\/3\", \"b\": \"2π\/3\", \"c\": \"π\/3\", \"d\": \"-π\/3\"}}","_debug_options_count":4},{"id":87519,"question":"Si z₁ = 1 + i et z₂ = 1 - i, quel est le produit z₁ × z₂ ?","option_a":"0","option_b":"2","option_c":"1 + i","option_d":"2i","option_e":"","option_f":"","bonne_reponse":"b","explication":"z₁ × z₂ = (1 + i)(1 - i) = 1 - i² = 1 - (-1) = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0\", \"b\": \"2\", \"c\": \"1 + i\", \"d\": \"2i\"}}","_debug_options_count":4},{"id":87520,"question":"L'ensemble des nombres complexes est un corps algébriquement clos.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, tout polynôme non constant à coefficients complexes admet au moins une racine complexe (théorème de d'Alembert-Gauss).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":87521,"question":"Quel est le module du nombre complexe z = (1 + i)⁴ ?","option_a":"2","option_b":"4","option_c":"√2","option_d":"8","option_e":"","option_f":"","bonne_reponse":"b","explication":"|z| = |1 + i|⁴ = (√(1² + 1²))⁴ = (√2)⁴ = 4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"2\", \"b\": \"4\", \"c\": \"√2\", \"d\": \"8\"}}","_debug_options_count":4},{"id":87522,"question":"Si z = 4e^(iπ\/4), quelle est sa forme algébrique ?","option_a":"2 + 2i","option_b":"2√2 + 2√2i","option_c":"4 + 4i","option_d":"√2 + √2i","option_e":"","option_f":"","bonne_reponse":"b","explication":"z = 4(cos(π\/4) + i sin(π\/4)) = 4(√2\/2 + i√2\/2) = 2√2 + 2√2i.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"2 + 2i\", \"b\": \"2√2 + 2√2i\", \"c\": \"4 + 4i\", \"d\": \"√2 + √2i","_debug_options_count":4}]