Quiz — 6339a1eaa180d_Fonction réciproque et nombres complexes.pdf

Question 1 sur 10 20:00

[{"id":29000,"question":"Soit f une fonction bijective définie par f(x) = 2x + 3. Quelle est sa fonction réciproque f⁻¹ ?","option_a":"f⁻¹(x) = (x - 3)\/2","option_b":"f⁻¹(x) = 2x - 3","option_c":"f⁻¹(x) = (x + 3)\/2","option_d":"f⁻¹(x) = x\/2 - 3","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour trouver f⁻¹, on résout y = 2x + 3 en x : x = (y - 3)\/2. Ainsi, f⁻¹(x) = (x - 3)\/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\": \"f⁻¹(x) = (x - 3)\/2\", \"b\": \"f⁻¹(x) = 2x - 3\", \"c\": \"f⁻¹(x","_debug_options_count":4},{"id":29001,"question":"Le module d'un nombre complexe z = a + ib est donné par :","option_a":"|z| = a² + b²","option_b":"|z| = √(a² + b²)","option_c":"|z| = a + b","option_d":"|z| = |a| + |b|","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le module d'un nombre complexe z = a + ib est défini par |z| = √(a² + b²), représentant la distance entre l'origine et le point (a,b) dans le 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\": \"|z| = a² + b²\", \"b\": \"|z| = √(a² + b²)\", \"c\": \"|z| = a + b\"","_debug_options_count":4},{"id":29002,"question":"La fonction réciproque d'une fonction strictement croissante est toujours strictement croissante.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si f est strictement croissante, alors f⁻¹ l'est également, car la croissance est préservée par la réciproque.","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":29003,"question":"Quel est le résultat de (3 + 4i) × (1 - 2i) ?","option_a":"11 - 2i","option_b":"11 + 2i","option_c":"-5 + 10i","option_d":"3 - 8i","option_e":"","option_f":"","bonne_reponse":"a","explication":"En développant : (3 + 4i)(1 - 2i) = 3×1 + 3×(-2i) + 4i×1 + 4i×(-2i) = 3 - 6i + 4i - 8i² = 3 - 2i + 8 = 11 - 2i (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\": \"11 - 2i\", \"b\": \"11 + 2i\", \"c\": \"-5 + 10i\", \"d\": \"3 - 8i\"}}","_debug_options_count":4},{"id":29004,"question":"Si f est une fonction paire, alors sa réciproque f⁻¹ est aussi paire.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une fonction paire vérifie f(-x) = f(x). Sa réciproque n'est pas nécessairement paire, car la parité n'est pas préservée par la réciproque.","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":29005,"question":"Quel est le conjugué du nombre complexe z = 5 - 3i ?","option_a":"5 + 3i","option_b":"-5 + 3i","option_c":"5 - 3i","option_d":"-5 - 3i","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le conjugué d'un nombre complexe z = a + ib est défini par z̄ = a - ib. Ainsi, le conjugué de 5 - 3i est 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\": \"5 - 3i\", \"d\": \"-5 - 3i\"}}","_debug_options_count":4},{"id":29006,"question":"La fonction f(x) = x² n'a pas de fonction réciproque sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction f(x) = x² n'est pas bijective sur ℝ (elle n'est pas injective). Cependant, si on restreint son domaine à ℝ⁺, elle devient bijective et admet une réciproque : f⁻¹(x) = √x.","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":29007,"question":"Quel est le résultat de (2 - i)² ?","option_a":"3 - 4i","option_b":"4 - 4i + i²","option_c":"3 - 2i","option_d":"5 - 4i","option_e":"","option_f":"","bonne_reponse":"b","explication":"En développant : (2 - i)² = 2² - 2×2×i + i² = 4 - 4i - 1 = 3 - 4i (car i² = -1). La réponse '4 - 4i + i²' est équivalente mais moins simplifiée.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"3 - 4i\", \"b\": \"4 - 4i + i²\", \"c\": \"3 - 2i\", \"d\": \"5 - 4i\"}}","_debug_options_count":4},{"id":29008,"question":"Si f est une fonction strictement décroissante, alors f⁻¹ est strictement décroissante.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si f est strictement décroissante, alors f⁻¹ l'est également, car la décroissance est préservée par la réciproque.","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":29009,"question":"Quel est l'argument principal du nombre complexe z = -1 + i ?","option_a":"π\/4","option_b":"3π\/4","option_c":"5π\/4","option_d":"-π\/4","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le nombre complexe z = -1 + i se situe dans le 2ème quadrant. Son argument θ vérifie tan(θ) = -1\/1 = -1, donc θ = 3π\/4 (car arctan(-1) = -π\/4, mais l'argument principal est dans [0, 2π[).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"π\/4\", \"b\": \"3π\/4\", \"c\": \"5π\/4\", \"d\": \"-π\/4\"}}","_debug_options_count":4}]