Quiz — 66f801321b999_Corrigé Trigonomètrie et suites.pdf

Question 1 sur 10 20:00

[{"id":5646,"question":"Quelle est la valeur de cos(π\/3) ?","option_a":"1\/2","option_b":"-1\/2","option_c":"√2\/2","option_d":"√3\/2","option_e":"","option_f":"","bonne_reponse":"a","explication":"cos(π\/3) = 1\/2, valeur fondamentale à connaître pour les calculs trigonométriques.","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\": \"-1\/2\", \"c\": \"√2\/2\", \"d\": \"√3\/2\"}}","_debug_options_count":4},{"id":5647,"question":"Une suite est définie par uₙ₊₁ = 2uₙ + 1 avec u₀ = 0. Cette suite est-elle arithmétique ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une suite arithmétique a une différence constante entre deux termes consécutifs. Ici, u₁ = 1, u₂ = 3, donc la différence n'est pas constante.","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":5648,"question":"Quelle est la solution générale de l'équation sin(x) = √2\/2 ?","option_a":"x = π\/4 + 2kπ ou x = 3π\/4 + 2kπ, k ∈ ℤ","option_b":"x = π\/4 + kπ, k ∈ ℤ","option_c":"x = π\/2 + kπ, k ∈ ℤ","option_d":"x = π\/3 + kπ, k ∈ ℤ","option_e":"","option_f":"","bonne_reponse":"a","explication":"Les solutions de sin(x) = √2\/2 sont x = π\/4 + 2kπ ou x = 3π\/4 + 2kπ, car sin(π\/4) = sin(3π\/4) = √2\/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 = π\/4 + 2kπ ou x = 3π\/4 + 2kπ, k ∈ ℤ\", \"b\": \"x = π\/4 +","_debug_options_count":4},{"id":5649,"question":"La suite définie par uₙ = (-1)ⁿ est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite uₙ = (-1)ⁿ alterne entre -1 et 1, donc elle n'a pas de limite et n'est pas convergente.","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":5650,"question":"Quelle est la formule de la somme des n premiers termes d'une suite géométrique de raison q ≠ 1 ?","option_a":"Sₙ = u₀(1 - qⁿ)\/(1 - q)","option_b":"Sₙ = u₀(1 + qⁿ)\/(1 - q)","option_c":"Sₙ = u₀(1 - qⁿ)\/(1 + q)","option_d":"Sₙ = u₀(qⁿ - 1)\/(1 - q)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La somme des n premiers termes d'une suite géométrique est donnée par Sₙ = u₀(1 - qⁿ)\/(1 - q).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Sₙ = u₀(1 - qⁿ)\/(1 - q)\", \"b\": \"Sₙ = u₀(1 + qⁿ)\/(1 - ","_debug_options_count":4},{"id":5651,"question":"L'équation cos(2x) = 1\/2 a-t-elle des solutions dans l'intervalle [0, π] ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Les solutions de cos(2x) = 1\/2 sont 2x = π\/3 + 2kπ ou 2x = 5π\/3 + 2kπ. Dans [0, π], x = π\/6 et x = 5π\/6 sont solutions.","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":5652,"question":"Une suite est définie par uₙ₊₁ = uₙ + 3 avec u₀ = 2. Quel est le terme u₅ ?","option_a":"17","option_b":"14","option_c":"20","option_d":"11","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une suite arithmétique de raison 3. u₅ = u₀ + 5×3 = 2 + 15 = 17.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"17\", \"b\": \"14\", \"c\": \"20\", \"d\": \"11\"}}","_debug_options_count":4},{"id":5653,"question":"La fonction f(x) = sin(x) est-elle périodique ? Si oui, quelle est sa période ?","option_a":"Vrai, période = π","option_b":"Vrai, période = 2π","option_c":"Faux","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction sin(x) est périodique de période 2π, car sin(x + 2π) = sin(x) pour tout 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\": \"Vrai, période = π\", \"b\": \"Vrai, période = 2π\", \"c\": \"Faux\", \"","_debug_options_count":4},{"id":5654,"question":"Quelle est la limite de la suite uₙ = (1 + 1\/n)ⁿ lorsque n tend vers l'infini ?","option_a":"0","option_b":"1","option_c":"e","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"La suite uₙ = (1 + 1\/n)ⁿ tend vers e (nombre d'Euler) lorsque n tend vers l'infini.","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\": \"1\", \"c\": \"e\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":5655,"question":"L'équation tan(x) = 1 a-t-elle des solutions dans l'intervalle [0, π\/2] ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"tan(x) = 1 a pour solution x = π\/4 dans l'intervalle [0, π\/2], car tan(π\/4) = 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}]