Quiz — 4. Complexes 3. maths corrections..pdf

Question 1 sur 10 20:00

[{"id":21059,"question":"Quel est le module du nombre complexe z = 3 + 4i ?","option_a":"5","option_b":"7","option_c":"12","option_d":"√7","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module d'un nombre complexe z = a + bi est donné par |z| = √(a² + b²). Ici, |3 + 4i| = √(3² + 4²) = √(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\": \"12\", \"d\": \"√7\"}}","_debug_options_count":4},{"id":21060,"question":"L'équation x² + 4 = 0 admet-elle des solutions complexes ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, les solutions sont x = 2i et x = -2i, car x² = -4 ⇒ x = ±√(-4) = ±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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":21061,"question":"Quelle est la forme trigonométrique de z = 1 - i ?","option_a":"√2 (cos(π\/4) + i sin(π\/4))","option_b":"√2 (cos(-π\/4) + i sin(-π\/4))","option_c":"2 (cos(π\/4) + i sin(π\/4))","option_d":"√2 (cos(3π\/4) + i sin(3π\/4))","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le module de z = 1 - i est √(1² + (-1)²) = √2. L'argument θ vérifie cos(θ) = 1\/√2 et sin(θ) = -1\/√2, donc θ = -π\/4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"√2 (cos(π\/4) + i sin(π\/4))\", \"b\": \"√2 (cos(-π\/4) + i sin(-","_debug_options_count":4},{"id":21062,"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.","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":21063,"question":"La formule de Moivre s'applique-t-elle aux nombres complexes ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, la formule de Moivre (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ) est valable pour tout nombre complexe sous forme trigonométrique.","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":21064,"question":"Quel est le résultat de (1 + i)² ?","option_a":"2i","option_b":"1 + 2i","option_c":"2 + 2i","option_d":"0","option_e":"","option_f":"","bonne_reponse":"a","explication":"(1 + i)² = 1² + 2*1*i + 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 + 2i\", \"d\": \"0\"}}","_debug_options_count":4},{"id":21065,"question":"L'argument d'un nombre complexe est-il 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 θ est défini à 2π près et peut être négatif ou supérieur à π selon le quadrant du plan complexe.","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":21066,"question":"Quelle est la solution de l'équation z² = -9 dans ℂ ?","option_a":"z = 3i ou z = -3i","option_b":"z = 3 ou z = -3","option_c":"z = 9i ou z = -9i","option_d":"Pas de solution","option_e":"","option_f":"","bonne_reponse":"a","explication":"z² = -9 ⇒ z = ±√(-9) = ±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\": \"z = 3i ou z = -3i\", \"b\": \"z = 3 ou z = -3\", \"c\": \"z = 9i ou z = -","_debug_options_count":4},{"id":21067,"question":"Quel est le produit de z₁ = 2 + 3i et z₂ = 1 - i ?","option_a":"5 + i","option_b":"2 - 3i","option_c":"5 - i","option_d":"-1 + i","option_e":"","option_f":"","bonne_reponse":"c","explication":"z₁ * z₂ = (2 + 3i)(1 - i) = 2*1 + 2*(-i) + 3i*1 + 3i*(-i) = 2 - 2i + 3i - 3i² = 2 + i + 3 = 5 + i.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"5 + i\", \"b\": \"2 - 3i\", \"c\": \"5 - i\", \"d\": \"-1 + i\"}}","_debug_options_count":4},{"id":21068,"question":"La forme algébrique d'un nombre complexe est-elle unique ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, la forme algébrique z = a + bi est unique pour un nombre complexe donné, où a et b sont des réels.","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}]