Quiz — Fonctions ArcSin et ArcCos, I.pdf

Question 1 sur 10 20:00

[{"id":65156,"question":"Quelle est la valeur de ArcSin(1\/2) ?","option_a":"π\/6","option_b":"π\/3","option_c":"π\/4","option_d":"π\/2","option_e":"","option_f":"","bonne_reponse":"a","explication":"ArcSin(1\/2) = π\/6 car sin(π\/6) = 1\/2 et π\/6 ∈ [-π\/2, π\/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\": \"π\/6\", \"b\": \"π\/3\", \"c\": \"π\/4\", \"d\": \"π\/2\"}}","_debug_options_count":4},{"id":65157,"question":"La fonction ArcCos est-elle paire ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"ArcCos n'est pas paire car ArcCos(-x) ≠ ArcCos(x). Par exemple, ArcCos(-1\/2) = 2π\/3 ≠ ArcCos(1\/2) = π\/3.","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":65158,"question":"Quelle est la dérivée de ArcCos(x) ?","option_a":"1\/√(1 - x²)","option_b":"-1\/√(1 - x²)","option_c":"√(1 - x²)","option_d":"-√(1 - x²)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de ArcCos(x) est -1\/√(1 - x²), car ArcCos(x) = π\/2 - ArcSin(x).","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\/√(1 - x²)\", \"b\": \"-1\/√(1 - x²)\", \"c\": \"√(1 - x²)\", \"d\"","_debug_options_count":4},{"id":65159,"question":"Pour x ∈ [-1, 1], quelle relation lie ArcSin(x) et ArcCos(x) ?","option_a":"ArcSin(x) + ArcCos(x) = π\/2","option_b":"ArcSin(x) = ArcCos(x)","option_c":"ArcSin(x) - ArcCos(x) = π","option_d":"ArcSin(x) * ArcCos(x) = 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour tout x ∈ [-1, 1], ArcSin(x) + ArcCos(x) = π\/2, ce qui est une propriété fondamentale de ces fonctions.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"ArcSin(x) + ArcCos(x) = π\/2\", \"b\": \"ArcSin(x) = ArcCos(x)\", \"c\":","_debug_options_count":4},{"id":65160,"question":"La fonction ArcSin est-elle définie pour x = 2 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Non, ArcSin n'est pas défini pour x = 2 car 2 ∉ [-1, 1]. Le domaine de définition de ArcSin est [-1, 1].","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":65161,"question":"Quelle est la valeur de ArcCos(-1) ?","option_a":"0","option_b":"π\/2","option_c":"π","option_d":"3π\/2","option_e":"","option_f":"","bonne_reponse":"c","explication":"ArcCos(-1) = π car cos(π) = -1 et π ∈ [0, π], le domaine de définition de ArcCos.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"0\", \"b\": \"π\/2\", \"c\": \"π\", \"d\": \"3π\/2\"}}","_debug_options_count":4},{"id":65162,"question":"Si f(x) = ArcSin(x), alors f'(x) = ?","option_a":"1\/√(1 - x)","option_b":"1\/√(1 - x²)","option_c":"√(1 - x²)","option_d":"-1\/√(1 - x²)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de ArcSin(x) est 1\/√(1 - x²), obtenue par dérivation de la fonction inverse.","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\/√(1 - x)\", \"b\": \"1\/√(1 - x²)\", \"c\": \"√(1 - x²)\", \"d\": \"","_debug_options_count":4},{"id":65163,"question":"La fonction ArcCos est-elle croissante sur son domaine ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, ArcCos est décroissante sur [-1, 1] car sa dérivée -1\/√(1 - x²) est négative sur cet intervalle.","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":65164,"question":"Résoudre l'équation ArcSin(x) = π\/4.","option_a":"x = √2\/2","option_b":"x = 1\/2","option_c":"x = 1","option_d":"x = 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"ArcSin(x) = π\/4 ⇒ x = sin(π\/4) = √2\/2, car ArcSin est la fonction inverse de sin sur [-π\/2, π\/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\": \"x = √2\/2\", \"b\": \"x = 1\/2\", \"c\": \"x = 1\", \"d\": \"x = 0\"}}","_debug_options_count":4},{"id":65165,"question":"Quelle est la limite de ArcSin(x) lorsque x tend vers 1 par la gauche ?","option_a":"0","option_b":"π\/2","option_c":"π","option_d":"+∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"Lorsque x → 1⁻, ArcSin(x) → π\/2, car sin(π\/2) = 1 et ArcSin est continue sur [-1, 1].","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0\", \"b\": \"π\/2\", \"c\": \"π\", \"d\": \"+∞\"}}","_debug_options_count":4}]