Quiz — Serie complex.pdf

Question 1 sur 10 20:00

[{"id":76966,"question":"Quelle est la forme trigonométrique du nombre complexe 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":"√2(cos(π\/4) + i sin(π\/4))","option_d":"√3(cos(π\/3) + i sin(π\/3))","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module de z = 1 + i√3 est √(1² + (√3)²) = 2. L'argument θ vérifie tanθ = √3\/1 = √3, donc θ = π\/3. La forme trigonométrique est donc 2(cos(π\/3) + i sin(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":76967,"question":"L'équation z² = -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":"Vrai. Les solutions sont z = 2i et z = -2i, car (2i)² = -4 et (-2i)² = -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":76968,"question":"Quel est le module du nombre complexe z = 3 - 4i ?","option_a":"5","option_b":"7","option_c":"1","option_d":"√7","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module de z = a + bi est √(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\": \"1\", \"d\": \"√7\"}}","_debug_options_count":4},{"id":76969,"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":"Faux. L'argument principal θ est généralement défini dans l'intervalle ]-π, π] ou [0, 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":76970,"question":"Quelle est la forme exponentielle de z = -1 + i ?","option_a":"√2 e^(i 3π\/4)","option_b":"√2 e^(i π\/4)","option_c":"2 e^(i 3π\/4)","option_d":"√2 e^(i 5π\/4)","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module de z = -1 + i est √((-1)² + 1²) = √2. L'argument θ vérifie cosθ = -1\/√2 et sinθ = 1\/√2, donc θ = 3π\/4. La forme exponentielle est √2 e^(i 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\": \"√2 e^(i 3π\/4)\", \"b\": \"√2 e^(i π\/4)\", \"c\": \"2 e^(i 3π\/4)\", ","_debug_options_count":4},{"id":76971,"question":"L'ensemble des solutions de l'équation z³ = 8 dans ℂ est un triangle équilatéral dans le plan complexe.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. Les solutions sont z = 2, z = 2e^(i 2π\/3) et z = 2e^(i 4π\/3), qui forment un triangle équilatéral centré à l'origine.","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":76972,"question":"Quel est le conjugué du nombre complexe z = 5 - 2i ?","option_a":"5 + 2i","option_b":"-5 + 2i","option_c":"5 - 2i","option_d":"-5 - 2i","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le conjugué d'un nombre complexe z = a + bi est z̄ = a - bi. Ici, z̄ = 5 - (-2i) = 5 + 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\": \"5 + 2i\", \"b\": \"-5 + 2i\", \"c\": \"5 - 2i\", \"d\": \"-5 - 2i\"}}","_debug_options_count":4},{"id":76973,"question":"L'équation z² + 2z + 5 = 0 admet-elle des solutions réelles ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. Le discriminant Δ = 4 - 20 = -16 \u003C 0. Les solutions sont complexes : z = -1 ± 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":76974,"question":"Quel est l'argument principal du nombre complexe z = -√3 - i ?","option_a":"-5π\/6","option_b":"7π\/6","option_c":"5π\/6","option_d":"-π\/6","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module de z = -√3 - i est 2. L'argument θ vérifie cosθ = -√3\/2 et sinθ = -1\/2, donc θ = -5π\/6 (ou 7π\/6 selon la convention).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"-5π\/6\", \"b\": \"7π\/6\", \"c\": \"5π\/6\", \"d\": \"-π\/6\"}}","_debug_options_count":4},{"id":76975,"question":"La multiplication de deux nombres complexes de module 1 donne toujours un nombre complexe de module 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. Si |z₁| = 1 et |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}]