Quiz — Complex 3.pdf

Question 1 sur 10 20:00

[{"id":7495,"question":"Quel est le module du nombre complexe z = 3 - 4i ?","option_a":"5","option_b":"7","option_c":"√7","option_d":"25","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module de z = a + bi est √(a² + b²). Ici, √(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\": \"25\"}}","_debug_options_count":4},{"id":7496,"question":"L'argument d'un nombre complexe est toujours mesuré en degrés.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'argument peut être exprimé en degrés ou en radians, selon le contexte. Les deux unités sont valides.","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":7497,"question":"Si z = 2(cos(π\/3) + i sin(π\/3)), quelle est la forme algébrique de z ?","option_a":"1 + i√3","option_b":"2 + i2√3","option_c":"√3 + i","option_d":"1 + i","option_e":"","option_f":"","bonne_reponse":"a","explication":"En développant : z = 2cos(π\/3) + i 2sin(π\/3) = 2*(1\/2) + i 2*(√3\/2) = 1 + i√3. Mais ici, le module est 2, donc z = 2*(1\/2) + i 2*(√3\/2) = 1 + i√3. L'option correcte est donc 1 + i√3, mais avec un module de 2, la forme algébrique est 1 + i√3. L'option 2 + i2√3 est incorrecte car le module serait alors 4. La bonne réponse est donc 1 + i√3 (option 0). Correction : l'option correcte est 1 + i√3 (module 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\": \"1 + i√3\", \"b\": \"2 + i2√3\", \"c\": \"√3 + i\", \"d\": \"1 + i\"}}","_debug_options_count":4},{"id":7498,"question":"Quel est le produit de (1 + i) et (2 - i) ?","option_a":"3 + i","option_b":"3 - i","option_c":"1 + 3i","option_d":"2 + i","option_e":"","option_f":"","bonne_reponse":"a","explication":"On développe : (1 + i)(2 - i) = 1*2 + 1*(-i) + i*2 + i*(-i) = 2 - i + 2i - i² = 2 + i + 1 = 3 + i (car i² = -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\": \"3 + i\", \"b\": \"3 - i\", \"c\": \"1 + 3i\", \"d\": \"2 + i\"}}","_debug_options_count":4},{"id":7499,"question":"L'équation z² = -4 a-t-elle des solutions dans ℂ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, les solutions sont z = 2i et z = -2i, car (2i)² = 4i² = -4 et (-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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":7500,"question":"Quel est le conjugué du nombre complexe z = 5 - 3i ?","option_a":"5 + 3i","option_b":"-5 - 3i","option_c":"3 - 5i","option_d":"5 - 3i","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le conjugué d'un nombre complexe z = a + bi est z̄ = a - bi. Ici, z̄ = 5 + 3i.","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 + 3i\", \"b\": \"-5 - 3i\", \"c\": \"3 - 5i\", \"d\": \"5 - 3i\"}}","_debug_options_count":4},{"id":7501,"question":"Si z = 1 + i, quel est son argument principal (en radians) ?","option_a":"π\/4","option_b":"π\/2","option_c":"π\/3","option_d":"0","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le point (1,1) dans le plan complexe est situé à 45° de l'axe réel, soit π\/4 radians.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"π\/4\", \"b\": \"π\/2\", \"c\": \"π\/3\", \"d\": \"0\"}}","_debug_options_count":4},{"id":7502,"question":"L'égalité |z₁ * z₂| = |z₁| * |z₂| est-elle toujours vraie ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, c'est une propriété fondamentale des modules : le module d'un produit est égal au produit des modules.","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":7503,"question":"Quel est le résultat de (3 + 2i) \/ (1 - i) ?","option_a":"(1 + 5i)\/2","option_b":"(5 + i)\/2","option_c":"1 + 5i","option_d":"5 + i","option_e":"","option_f":"","bonne_reponse":"b","explication":"On multiplie par le conjugué du dénominateur : (3 + 2i)(1 + i) \/ (1 - i)(1 + i) = (3 + 3i + 2i + 2i²) \/ (1 + 1) = (3 + 5i - 2)\/2 = (1 + 5i)\/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\": \"(1 + 5i)\/2\", \"b\": \"(5 + i)\/2\", \"c\": \"1 + 5i\", \"d\": \"5 + i\"}}","_debug_options_count":4},{"id":7504,"question":"Un nombre complexe de module 1 est-il nécessairement sur le cercle unité ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, par définition, le cercle unité est l'ensemble des nombres complexes de module 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}]