Quiz — serie n1 complexe exellent.pdf

Question 1 sur 10 20:00

[{"id":76836,"question":"Quel est le module du nombre complexe z = 3 - 4i ?","option_a":"5","option_b":"7","option_c":"25","option_d":"1","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module est calculé 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\": \"25\", \"d\": \"1\"}}","_debug_options_count":4},{"id":76837,"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 généralement compris entre -π et π (ou 0 et 2π selon les conventions).","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":76838,"question":"Quel est le conjugué du nombre complexe z = 2 + 5i ?","option_a":"2 - 5i","option_b":"-2 + 5i","option_c":"5 + 2i","option_d":"-2 - 5i","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le conjugué d'un nombre complexe a + bi est a - bi.","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 - 5i\", \"b\": \"-2 + 5i\", \"c\": \"5 + 2i\", \"d\": \"-2 - 5i\"}}","_debug_options_count":4},{"id":76839,"question":"Si z = 1 + i, alors z² est égal à :","option_a":"2i","option_b":"1 + 2i","option_c":"2","option_d":"0","option_e":"","option_f":"","bonne_reponse":"a","explication":"z² = (1 + i)² = 1 + 2i + i² = 1 + 2i - 1 = 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\": \"2i\", \"b\": \"1 + 2i\", \"c\": \"2\", \"d\": \"0\"}}","_debug_options_count":4},{"id":76840,"question":"L'équation z² = -4 admet deux solutions réelles.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Les solutions sont z = 2i et z = -2i, qui sont des nombres complexes non réels.","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":76841,"question":"Quel est l'argument principal du nombre complexe z = -1 - i ?","option_a":"-3π\/4","option_b":"π\/4","option_c":"5π\/4","option_d":"-π\/4","option_e":"","option_f":"","bonne_reponse":"a","explication":"z est dans le troisième quadrant. L'argument est calculé par atan(b\/a) = atan(1) = π\/4, mais comme a et b sont négatifs, on ajoute π : -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\": \"5π\/4\", \"d\": \"-π\/4\"}}","_debug_options_count":4},{"id":76842,"question":"La forme trigonométrique de z = 2(cos(π\/3) + i sin(π\/3)) est :","option_a":"2e^(iπ\/3)","option_b":"2(cos(π\/3) - i sin(π\/3))","option_c":"√3 + i","option_d":"1 + i√3","option_e":"","option_f":"","bonne_reponse":"a","explication":"La forme trigonométrique est z = r(cosθ + i sinθ), ici r = 2 et θ = π\/3, donc z = 2e^(iπ\/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\": \"2e^(iπ\/3)\", \"b\": \"2(cos(π\/3) - i sin(π\/3))\", \"c\": \"√3 + i\", ","_debug_options_count":4},{"id":76843,"question":"Si z = 3e^(iπ\/6), alors |z| est égal à :","option_a":"3","option_b":"π\/6","option_c":"√3","option_d":"1\/3","option_e":"","option_f":"","bonne_reponse":"a","explication":"Dans la forme trigonométrique z = re^(iθ), r est le module, donc |z| = 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\": \"3\", \"b\": \"π\/6\", \"c\": \"√3\", \"d\": \"1\/3\"}}","_debug_options_count":4},{"id":76844,"question":"L'équation z³ = 8 a pour solutions :","option_a":"2, 2e^(i2π\/3), 2e^(i4π\/3)","option_b":"2, -2, 2i","option_c":"1, i, -1","option_d":"√8, -√8, 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"Les solutions sont les racines cubiques de 8, soit z = 2, z = 2e^(i2π\/3) et z = 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\": \"2, 2e^(i2π\/3), 2e^(i4π\/3)\", \"b\": \"2, -2, 2i\", \"c\": \"1, i, -1\", ","_debug_options_count":4},{"id":76845,"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 + bi, alors z × z̄ = (a + bi)(a - bi) = a² + b², qui est un réel positif.","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}]