Quiz — 6705627cb58b4_corrigé_Trigonométrie et suites.pdf

Question 1 sur 10 20:00

[{"id":5626,"question":"Quelle est la valeur de sin(π\/6) ?","option_a":"A. 0","option_b":"B. 1\/2","option_c":"C. √2\/2","option_d":"D. 1","option_e":"","option_f":"","bonne_reponse":"b","explication":"La valeur de sin(π\/6) est 1\/2, car π\/6 correspond à 30° dans le cercle trigonométrique.","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. 0\", \"b\": \"B. 1\/2\", \"c\": \"C. √2\/2\", \"d\": \"D. 1\"}}","_debug_options_count":4},{"id":5627,"question":"La suite définie par uₙ = n² - 5n + 1 est-elle croissante à partir d'un certain rang ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La suite est croissante car sa dérivée (2n - 5) devient positive pour n ≥ 3.","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":5628,"question":"Quelle identité trigonométrique est toujours vraie pour tout réel x ?","option_a":"A. sin²x + cos²x = 1","option_b":"B. sinx + cosx = 1","option_c":"C. tanx = sinx\/cosx","option_d":"D. A et C","option_e":"","option_f":"","bonne_reponse":"d","explication":"Les identités A et C sont toujours vraies. L'identité B n'est vraie que pour x = π\/4 + kπ.","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. sin²x + cos²x = 1\", \"b\": \"B. sinx + cosx = 1\", \"c\": \"C. tanx","_debug_options_count":4},{"id":5629,"question":"La suite 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 oscille entre -1 et 1 et n'a pas de limite, donc 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":5630,"question":"Si sin(x) = 3\/5 et x ∈ [0, π\/2], quelle est la valeur de cos(x) ?","option_a":"A. 4\/5","option_b":"B. -4\/5","option_c":"C. 1\/5","option_d":"D. √34\/5","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant l'identité sin²x + cos²x = 1, on trouve cos(x) = 4\/5 (car x est dans le premier quadrant).","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. 4\/5\", \"b\": \"B. -4\/5\", \"c\": \"C. 1\/5\", \"d\": \"D. √34\/5\"}}","_debug_options_count":4},{"id":5631,"question":"La suite uₙ = (n+1)\/n est-elle bornée ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La suite est bornée car elle est décroissante et converge vers 1 (limite inférieure et supérieure).","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":5632,"question":"Quelle est la période de la fonction f(x) = sin(2x) ?","option_a":"A. π\/2","option_b":"B. π","option_c":"C. 2π","option_d":"D. 4π","option_e":"","option_f":"","bonne_reponse":"a","explication":"La période de sin(2x) est π car la période de sin(x) est 2π, et le coefficient 2 divise cette période par 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\": \"A. π\/2\", \"b\": \"B. π\", \"c\": \"C. 2π\", \"d\": \"D. 4π\"}}","_debug_options_count":4},{"id":5633,"question":"La suite uₙ = n\/(n+1) est-elle convergente ? Si oui, vers quelle limite ?","option_a":"A. Oui, vers 0","option_b":"B. Oui, vers 1","option_c":"C. Non, elle diverge","option_d":"D. Oui, vers 1\/2","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite converge vers 1 car lim(n→∞) n\/(n+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\": \"A. Oui, vers 0\", \"b\": \"B. Oui, vers 1\", \"c\": \"C. Non, elle diverg","_debug_options_count":4},{"id":5634,"question":"L'équation sin(x) = 1 admet-elle une solution dans l'intervalle [0, 2π] ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation admet une solution unique dans [0, 2π], qui est x = 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":5635,"question":"Quelle est la limite de la suite uₙ = (3n² + 2n - 1)\/(n² + 5) quand n tend vers l'infini ?","option_a":"A. 0","option_b":"B. 3","option_c":"C. 1","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par n², on obtient lim(uₙ) = 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\": \"A. 0\", \"b\": \"B. 3\", \"c\": \"C. 1\", \"d\": \"D. +∞\"}}","_debug_options_count":4}]