Quiz — Serie complex 2.pdf

Question 1 sur 10 20:00

[{"id":47123,"question":"Quel est le module du nombre complexe z = 3 - 4i ?","option_a":"5","option_b":"7","option_c":"1","option_d":"25","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module se calcule par |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\": \"1\", \"d\": \"25\"}}","_debug_options_count":4},{"id":47124,"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 utilisées.","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":47125,"question":"Si z = 2(cos(π\/3) + i sin(π\/3)), quelle est sa forme algébrique ?","option_a":"1 + i√3","option_b":"1 - i√3","option_c":"2 + i√3","option_d":"2 - i√3","option_e":"","option_f":"","bonne_reponse":"a","explication":"En développant : z = 2cos(π\/3) + i2sin(π\/3) = 2*(1\/2) + i2*(√3\/2) = 1 + i√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\": \"1 + i√3\", \"b\": \"1 - i√3\", \"c\": \"2 + i√3\", \"d\": \"2 - i√3\"}","_debug_options_count":4},{"id":47126,"question":"Le conjugué d'un nombre complexe z = a + ib est donné par :","option_a":"a - ib","option_b":"b + ia","option_c":"a + ib","option_d":"-a - ib","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le conjugué de z = a + ib est défini comme z̄ = a - ib.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"a - ib\", \"b\": \"b + ia\", \"c\": \"a + ib\", \"d\": \"-a - ib\"}}","_debug_options_count":4},{"id":47127,"question":"L'équation z² + 4 = 0 admet pour solutions :","option_a":"z = 2i et z = -2i","option_b":"z = 2 et z = -2","option_c":"z = 4i et z = -4i","option_d":"z = 1 + i√3 et z = 1 - i√3","option_e":"","option_f":"","bonne_reponse":"a","explication":"En résolvant z² = -4, on obtient z = ±2i.","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 et z = -2i\", \"b\": \"z = 2 et z = -2\", \"c\": \"z = 4i et z = -","_debug_options_count":4},{"id":47128,"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 + ib, alors z * z̄ = (a + ib)(a - ib) = a² + b², qui est bien un nombre réel.","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":47129,"question":"Quel est l'argument du nombre complexe z = -1 - i ?","option_a":"-3π\/4","option_b":"π\/4","option_c":"3π\/4","option_d":"5π\/4","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le point (-1, -1) est dans le 3ème quadrant. arg(z) = arctan(1) + π = π\/4 + π = 5π\/4, mais l'argument principal est -3π\/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\": \"-3π\/4\", \"b\": \"π\/4\", \"c\": \"3π\/4\", \"d\": \"5π\/4\"}}","_debug_options_count":4},{"id":47130,"question":"La forme exponentielle d'un nombre complexe z = re^(iθ) est équivalente à :","option_a":"z = r(cosθ + i sinθ)","option_b":"z = r(cosθ - i sinθ)","option_c":"z = r sinθ + i r cosθ","option_d":"z = r²(cosθ + i sinθ)","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par la formule d'Euler, e^(iθ) = cosθ + i sinθ, donc z = 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\": \"z = r(cosθ + i sinθ)\", \"b\": \"z = r(cosθ - i sinθ)\", \"c\": \"z =","_debug_options_count":4},{"id":47131,"question":"Si z₁ = 2 + 3i et z₂ = 1 - i, alors z₁ + z₂ = :","option_a":"3 + 2i","option_b":"3 + 4i","option_c":"1 + 2i","option_d":"2 + 2i","option_e":"","option_f":"","bonne_reponse":"a","explication":"En additionnant les parties réelles et imaginaires : (2+1) + (3-1)i = 3 + 2i.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"3 + 2i\", \"b\": \"3 + 4i\", \"c\": \"1 + 2i\", \"d\": \"2 + 2i\"}}","_debug_options_count":4},{"id":47132,"question":"Le nombre complexe z = 0 est le seul nombre complexe dont le module est égal à 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module de z = 0 est bien 0, et c'est le seul nombre complexe avec cette propriété.","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}]