Quiz — 66fc175a40065_série 2 -suites réelles et trigonométie.pdf

Question 1 sur 10 20:00

[{"id":13964,"question":"Quelle est la limite de la suite définie par uₙ = (3n + 2)\/(2n - 1) lorsque n tend vers l'infini ?","option_a":"A. 1","option_b":"B. 3\/2","option_c":"C. 2\/3","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"La limite d'une suite rationnelle s'obtient en comparant les termes de plus haut degré : (3n)\/(2n) = 3\/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\": \"A. 1\", \"b\": \"B. 3\/2\", \"c\": \"C. 2\/3\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":13965,"question":"L'équation sin(x) = √2\/2 admet-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 sin(x) = √2\/2 dans [0, π] sont x = π\/4 et x = 3π\/4.","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":13966,"question":"Soit la suite définie par u₀ = 1 et uₙ₊₁ = 2uₙ + 3. Quelle est l'expression de uₙ en fonction de n ?","option_a":"A. uₙ = 2ⁿ - 3","option_b":"B. uₙ = 4ⁿ - 3","option_c":"C. uₙ = 2ⁿ⁺¹ - 3","option_d":"D. uₙ = 3·2ⁿ - 3","option_e":"","option_f":"","bonne_reponse":"d","explication":"La solution générale d'une suite arithmético-géométrique uₙ₊₁ = a·uₙ + b est uₙ = A·aⁿ - b\/(1-a). Ici, A = 4 et uₙ = 4·2ⁿ - 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"A. uₙ = 2ⁿ - 3\", \"b\": \"B. uₙ = 4ⁿ - 3\", \"c\": \"C. uₙ = 2","_debug_options_count":4},{"id":13967,"question":"La fonction f(x) = sin(2x) est-elle périodique de période π ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction sin(2x) a une période de π car sin(2(x + π)) = sin(2x + 2π) = sin(2x).","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":13968,"question":"Quelle est la valeur de cos(π\/3) + sin(π\/6) ?","option_a":"A. 1","option_b":"B. √3\/2","option_c":"C. 3\/2","option_d":"D. 1\/2","option_e":"","option_f":"","bonne_reponse":"c","explication":"cos(π\/3) = 1\/2 et sin(π\/6) = 1\/2, donc la somme est 1\/2 + 1\/2 = 1. Réponse C : 3\/2 est incorrecte.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"A. 1\", \"b\": \"B. √3\/2\", \"c\": \"C. 3\/2\", \"d\": \"D. 1\/2\"}}","_debug_options_count":4},{"id":13969,"question":"Soit la suite uₙ définie par uₙ = (-1)ⁿ. Cette suite 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 finie : elle est divergente.","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":13970,"question":"Quelle est la solution générale de l'équation trigonométrique cos(x) = -1\/2 ?","option_a":"A. x = 2π\/3 + 2kπ ou x = 4π\/3 + 2kπ, k ∈ ℤ","option_b":"B. x = π\/3 + 2kπ ou x = 5π\/3 + 2kπ, k ∈ ℤ","option_c":"C. x = π\/2 + kπ, k ∈ ℤ","option_d":"D. x = π + 2kπ, k ∈ ℤ","option_e":"","option_f":"","bonne_reponse":"a","explication":"Les solutions de cos(x) = -1\/2 sont x = 2π\/3 + 2kπ et x = 4π\/3 + 2kπ pour tout entier k.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. x = 2π\/3 + 2kπ ou x = 4π\/3 + 2kπ, k ∈ ℤ\", \"b\": \"B. x =","_debug_options_count":4},{"id":13971,"question":"La suite uₙ = n²\/(n + 1) est-elle majorée ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite uₙ = n²\/(n + 1) tend vers +∞ lorsque n tend vers l'infini, donc elle n'est pas majorée.","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":13972,"question":"Quelle identité trigonométrique permet d'écrire sin²(x) + cos²(x) = 1 ?","option_a":"A. Identité de Pythagore","option_b":"B. Formule d'addition","option_c":"C. Formule de duplication","option_d":"D. Identité fondamentale","option_e":"","option_f":"","bonne_reponse":"d","explication":"L'identité sin²(x) + cos²(x) = 1 est appelée l'identité fondamentale de la trigonométrie.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"A. Identité de Pythagore\", \"b\": \"B. Formule d'addition\", \"c\": \"C","_debug_options_count":4},{"id":13973,"question":"Soit la suite définie par u₀ = 0 et uₙ₊₁ = uₙ + 2n + 1. Quelle est la valeur de u₅ ?","option_a":"A. 15","option_b":"B. 25","option_c":"C. 30","option_d":"D. 35","option_e":"","option_f":"","bonne_reponse":"b","explication":"En calculant les termes : u₁ = 1, u₂ = 4, u₃ = 9, u₄ = 16, u₅ = 25. La suite correspond à uₙ = n².","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A. 15\", \"b\": \"B. 25\", \"c\": \"C. 30\", \"d\": \"D. 35\"}}","_debug_options_count":4}]