Quiz — Copie de تمارين-المتتاليات-في-البكالوريا 3 اداب من 2008 إلى 2024 مرتبة حسب الأفكار.pdf

Question 1 sur 10 20:00

[{"id":61257,"question":"Quelle est la nature d'une suite définie par u_{n+1} = 2u_n + 3 avec u_0 = 1 ?","option_a":"Arithmétique","option_b":"Géométrique","option_c":"Récurrente linéaire d'ordre 1","option_d":"Non définie","option_e":"","option_f":"","bonne_reponse":"c","explication":"La suite est récurrente linéaire d'ordre 1 car elle s'exprime sous la forme u_{n+1} = a*u_n + b. Ici, a=2 et b=3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"Arithmétique\", \"b\": \"Géométrique\", \"c\": \"Récurrente linéaire","_debug_options_count":4},{"id":61258,"question":"La suite (u_n) définie par u_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 prend alternativement les valeurs -1 et 1, donc elle est bornée par -1 et 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},{"id":61259,"question":"Si une suite est croissante et majorée, alors :","option_a":"Elle est décroissante","option_b":"Elle converge","option_c":"Elle diverge vers +∞","option_d":"Elle est constante","option_e":"","option_f":"","bonne_reponse":"b","explication":"D'après le théorème de la limite monotone, une suite croissante et majorée converge vers sa borne supérieure.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Elle est décroissante\", \"b\": \"Elle converge\", \"c\": \"Elle diverge","_debug_options_count":4},{"id":61260,"question":"La limite de la suite u_n = (3n + 2)\/(2n - 1) quand n tend vers +∞ est :","option_a":"0","option_b":"1","option_c":"3\/2","option_d":"2","option_e":"","option_f":"","bonne_reponse":"c","explication":"En divisant numérateur et dénominateur par n, on obtient (3 + 2\/n)\/(2 - 1\/n) → 3\/2 quand n → +∞.","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\": \"3\/2\", \"d\": \"2\"}}","_debug_options_count":4},{"id":61261,"question":"Une suite géométrique de raison q = -2 et de premier terme u_0 = 1 est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une suite géométrique converge si et seulement si |q| \u003C 1. Ici, |q| = 2 \u003E 1, donc elle diverge.","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":61262,"question":"Quelle est la formule de récurrence pour une suite arithmétique de raison r ?","option_a":"u_{n+1} = u_n + r","option_b":"u_{n+1} = u_n * r","option_c":"u_{n+1} = r * u_n + u_0","option_d":"u_{n+1} = u_n - r","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, une suite arithmétique vérifie u_{n+1} = u_n + r, où r est la raison.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"u_{n+1} = u_n + r\", \"b\": \"u_{n+1} = u_n * r\", \"c\": \"u_{n+1} = r *","_debug_options_count":4},{"id":61263,"question":"La suite u_n = n^2 - 5n + 6 est-elle positive pour tout n ≥ 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite est positive pour n ≤ 2 ou n ≥ 3, mais négative pour n = 1 (u_1 = 2). Elle n'est donc pas positive pour tout n ≥ 0.","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":61264,"question":"Si lim(u_n) = L et lim(v_n) = M avec L \u003C M, alors pour n suffisamment grand :","option_a":"u_n \u003E v_n","option_b":"u_n \u003C v_n","option_c":"u_n = v_n","option_d":"u_n et v_n sont égaux","option_e":"","option_f":"","bonne_reponse":"b","explication":"Si L \u003C M, alors il existe un rang N tel que pour tout n ≥ N, u_n \u003C v_n (théorème de comparaison des limites).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"u_n \u003E v_n\", \"b\": \"u_n \u003C v_n\", \"c\": \"u_n = v_n\", \"d\": \"u_n et v_n ","_debug_options_count":4},{"id":61265,"question":"La suite u_n = 5 - 1\/n est-elle convergente ? Si oui, quelle est sa limite ?","option_a":"Non convergente","option_b":"Oui, limite = 5","option_c":"Oui, limite = 0","option_d":"Oui, limite = -∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite converge vers 5 car lim(1\/n) = 0 quand n → +∞, donc u_n → 5 - 0 = 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Non convergente\", \"b\": \"Oui, limite = 5\", \"c\": \"Oui, limite = 0\",","_debug_options_count":4},{"id":61266,"question":"Pour étudier la monotonie d'une suite, on compare généralement :","option_a":"u_{n+1} et u_n","option_b":"u_n et n","option_c":"u_0 et u_1","option_d":"u_n et 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour étudier la monotonie, on compare u_{n+1} et u_n. Si u_{n+1} ≥ u_n pour tout n, la suite 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\": \"u_{n+1} et u_n\", \"b\": \"u_n et n\", \"c\": \"u_0 et u_1\", \"d\": \"u_n et","_debug_options_count":4}]