Quiz — Exercices sur les nombres complexes.pdf

Question 1 sur 10 20:00

[{"id":81969,"question":"Quel est le module du nombre complexe z = 3 - 4i ?","option_a":"5","option_b":"7","option_c":"√7","option_d":"√(3-4i)","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\": \"√(3-4i)\"}}","_debug_options_count":4},{"id":81970,"question":"L'affirmation suivante est-elle vraie ? Tout nombre complexe non nul admet exactement deux racines carrées.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une propriété fondamentale des nombres complexes : pour tout z ≠ 0, il existe exactement deux nombres complexes w tels que w² = z.","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":81971,"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 de z est |z| = √(1² + (√3)²) = 2. L'argument θ vérifie cosθ = 1\/2 et sinθ = √3\/2, donc θ = π\/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\": \"2(cos(π\/3) + i sin(π\/3))\", \"b\": \"2(cos(π\/6) + i sin(π\/6))\", \"","_debug_options_count":4},{"id":81972,"question":"L'équation z² + 4z + 13 = 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 = 16 - 52 = -36 \u003C 0, donc l'équation n'a pas de solutions réelles mais deux solutions complexes conjuguées.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":81973,"question":"Quel est le résultat de (1 + i)³ ?","option_a":"2 + 2i","option_b":"-2 + 2i","option_c":"2 - 2i","option_d":"-2 - 2i","option_e":"","option_f":"","bonne_reponse":"b","explication":"En développant (1 + i)³ = 1 + 3i + 3i² + i³ = 1 + 3i - 3 - i = -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 + 2i\", \"c\": \"2 - 2i\", \"d\": \"-2 - 2i\"}}","_debug_options_count":4},{"id":81974,"question":"La forme exponentielle d'un nombre complexe z = re^(iθ) est-elle équivalente à z = r(cosθ + i sinθ) ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est la formule d'Euler : e^(iθ) = cosθ + i sinθ, donc z = r(cosθ + i sinθ) = re^(iθ).","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":81975,"question":"Quel est l'argument principal du nombre complexe z = -1 + i ?","option_a":"π\/4","option_b":"3π\/4","option_c":"5π\/4","option_d":"-π\/4","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le point (-1, 1) est dans le deuxième quadrant. L'argument θ vérifie tanθ = -1, donc θ = 3π\/4 (argument principal).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"π\/4\", \"b\": \"3π\/4\", \"c\": \"5π\/4\", \"d\": \"-π\/4\"}}","_debug_options_count":4},{"id":81976,"question":"L'affirmation suivante est-elle vraie ? Le conjugué d'un nombre complexe z = a + ib est toujours différent de z.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le conjugué de z est z̄ = a - ib. z = z̄ si et seulement si b = 0, c'est-à-dire si z est réel.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":81977,"question":"Quelle est la solution de l'équation z³ = -8 dans ℂ ?","option_a":"z = 2","option_b":"z = -2","option_c":"z = 2e^(iπ\/3)","option_d":"z = 2e^(iπ\/3), 2e^(iπ), 2e^(i5π\/3)","option_e":"","option_f":"","bonne_reponse":"d","explication":"Les solutions sont les racines cubiques de -8, soit z = 2e^(i(π + 2kπ)\/3) pour k = 0, 1, 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"z = 2\", \"b\": \"z = -2\", \"c\": \"z = 2e^(iπ\/3)\", \"d\": \"z = 2e^(iπ\/3","_debug_options_count":4},{"id":81978,"question":"Quel est le résultat de (2e^(iπ\/4))² ?","option_a":"4e^(iπ\/2)","option_b":"2e^(iπ\/2)","option_c":"4e^(iπ\/4)","option_d":"4e^(iπ)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant (re^(iθ))^n = r^n e^(inθ), on obtient (2e^(iπ\/4))² = 4e^(iπ\/2).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"4e^(iπ\/2)\", \"b\": \"2e^(iπ\/2)\", \"c\": \"4e^(iπ\/4)\", \"d\": \"4e^(iπ)","_debug_options_count":4}]