Quiz — série-complexe-1.pdf

Question 1 sur 10 20:00

[{"id":45463,"question":"Quelle est la forme algébrique de z = 3(cos(π\/4) + i sin(π\/4)) ?","option_a":"3 + 3i","option_b":"3√2\/2 + 3i√2\/2","option_c":"3\/2 + 3i√3\/2","option_d":"√3\/2 + i\/2","option_e":"","option_f":"","bonne_reponse":"b","explication":"En développant cos(π\/4) = sin(π\/4) = √2\/2, on obtient z = 3(√2\/2 + i√2\/2) = 3√2\/2 + 3i√2\/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\": \"3 + 3i\", \"b\": \"3√2\/2 + 3i√2\/2\", \"c\": \"3\/2 + 3i√3\/2\", \"d\": \"","_debug_options_count":4},{"id":45464,"question":"Le module d'un nombre complexe est toujours un nombre réel positif.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module est défini comme la racine carrée d'une somme de carrés, donc toujours positif ou nul.","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":45465,"question":"Quelle est la solution de l'équation z² = -4 dans ℂ ?","option_a":"z = 2i ou z = -2i","option_b":"z = 4i","option_c":"z = 2 ou z = -2","option_d":"z = -4","option_e":"","option_f":"","bonne_reponse":"a","explication":"Les solutions sont z = ±√(-4) = ±2i, car (2i)² = 4i² = -4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"z = 2i ou z = -2i\", \"b\": \"z = 4i\", \"c\": \"z = 2 ou z = -2\", \"d\": \"","_debug_options_count":4},{"id":45466,"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], donc peut être négatif.","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":45467,"question":"Quelle est la forme exponentielle de z = -1 + i√3 ?","option_a":"2e^(iπ\/3)","option_b":"2e^(i2π\/3)","option_c":"√2 e^(i3π\/4)","option_d":"2e^(iπ\/4)","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le module est |z| = √((-1)² + (√3)²) = 2. L'argument θ vérifie cosθ = -1\/2 et sinθ = √3\/2, donc θ = 2π\/3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"2e^(iπ\/3)\", \"b\": \"2e^(i2π\/3)\", \"c\": \"√2 e^(i3π\/4)\", \"d\": \"2e","_debug_options_count":4},{"id":45468,"question":"Le conjugué d'un nombre complexe z = a + bi est égal à -a - bi.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le conjugué est z̄ = a - bi, pas -a - bi.","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":45469,"question":"Quelle est la partie réelle de z = (3 + 2i)(1 - i) ?","option_a":"1","option_b":"5","option_c":"-1","option_d":"3","option_e":"","option_f":"","bonne_reponse":"c","explication":"En développant z = 3(1) + 3(-i) + 2i(1) + 2i(-i) = 3 - 3i + 2i + 2 = 5 - i. La partie réelle est 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"1\", \"b\": \"5\", \"c\": \"-1\", \"d\": \"3\"}}","_debug_options_count":4},{"id":45470,"question":"L'équation z + z̄ = 4 admet-elle des solutions dans ℂ ?","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 = 2a = 4 ⇒ a = 2. Les solutions sont tous les nombres complexes de partie réelle 2 (b ∈ ℝ).","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":45471,"question":"Quelle est la valeur de i^100 ?","option_a":"1","option_b":"-1","option_c":"i","option_d":"-i","option_e":"","option_f":"","bonne_reponse":"a","explication":"i^100 = (i^4)^25 = 1^25 = 1, car i^4 = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"1\", \"b\": \"-1\", \"c\": \"i\", \"d\": \"-i\"}}","_debug_options_count":4},{"id":45472,"question":"Le produit de deux nombres complexes de module 1 a toujours un module égal à 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si |z1| = |z2| = 1, alors |z1 × z2| = |z1| × |z2| = 1 × 1 = 1.","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}]