Quiz — 12 - Arcos-Arcsin 1.pdf

Question 1 sur 10 20:00

[{"id":29100,"question":"Quelle est la valeur de arcsin(1\/2) ?","option_a":"π\/6","option_b":"π\/4","option_c":"π\/3","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\/4\", \"c\": \"π\/3\", \"d\": \"π\/2\"}}","_debug_options_count":4},{"id":29101,"question":"La fonction arccos(x) est-elle définie pour x = 2 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. Le domaine de arccos(x) est [-1, 1], donc x = 2 est hors domaine.","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":29102,"question":"Quelle est la dérivée de f(x) = arcsin(x²) ?","option_a":"2x\/√(1-x²)","option_b":"x\/√(1-x⁴)","option_c":"2x\/√(1-x⁴)","option_d":"x\/√(1-x²)","option_e":"","option_f":"","bonne_reponse":"c","explication":"En utilisant la règle de la chaîne : f'(x) = (1\/√(1-(x²)²)) * 2x = 2x\/√(1-x⁴).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"2x\/√(1-x²)\", \"b\": \"x\/√(1-x⁴)\", \"c\": \"2x\/√(1-x⁴)\", \"d\":","_debug_options_count":4},{"id":29103,"question":"Quelle identité relie arcsin(x) et arccos(x) ?","option_a":"arcsin(x) + arccos(x) = 0","option_b":"arcsin(x) + arccos(x) = π\/2","option_c":"arcsin(x) * arccos(x) = π\/2","option_d":"arcsin(x) - arccos(x) = π\/2","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour tout x ∈ [-1, 1], arcsin(x) + arccos(x) = π\/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\": \"arcsin(x) + arccos(x) = 0\", \"b\": \"arcsin(x) + arccos(x) = π\/2\", ","_debug_options_count":4},{"id":29104,"question":"La fonction arcsin(x) est-elle croissante sur son domaine ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. La dérivée de arcsin(x) est 1\/√(1-x²) \u003E 0 sur ]-1, 1[, donc la fonction est croissante.","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":29105,"question":"Quelle est l'intégrale de 1\/√(1-x²) ?","option_a":"arcsin(x) + C","option_b":"arccos(x) + C","option_c":"ln|x| + C","option_d":"√(1-x²) + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale de 1\/√(1-x²) est arcsin(x) + C, car c'est la dérivée de arcsin(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\": \"arcsin(x) + C\", \"b\": \"arccos(x) + C\", \"c\": \"ln|x| + C\", \"d\": \"√","_debug_options_count":4},{"id":29106,"question":"Pour x = 0, quelle est la valeur de arccos(x) ?","option_a":"0","option_b":"π\/2","option_c":"π","option_d":"π\/4","option_e":"","option_f":"","bonne_reponse":"b","explication":"arccos(0) = π\/2 car cos(π\/2) = 0 et π\/2 ∈ [0, π].","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\": \"π\/4\"}}","_debug_options_count":4},{"id":29107,"question":"La fonction arccos(x) est-elle paire ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. arccos(-x) = π - arccos(x), donc la fonction n'est pas paire.","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":29108,"question":"Quelle est la valeur de sin(arccos(√3\/2)) ?","option_a":"1\/2","option_b":"√3\/2","option_c":"1\/√2","option_d":"0","option_e":"","option_f":"","bonne_reponse":"a","explication":"Soit θ = arccos(√3\/2), alors cos(θ) = √3\/2. Donc sin(θ) = √(1 - (√3\/2)²) = 1\/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\/2\", \"b\": \"√3\/2\", \"c\": \"1\/√2\", \"d\": \"0\"}}","_debug_options_count":4},{"id":29109,"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":"1","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\": \"1\"}}","_debug_options_count":4}]