Quiz — Séance N°4 Enoncés+Correction possible ( Suites Réelles+Nombres complexes) Bac Sciences 2022-2023.pdf

Question 1 sur 10 20:00

[{"id":9733,"question":"Soit la suite (u_n) définie par u_0 = 2 et u_{n+1} = 3u_n - 1. Quelle est la nature de cette suite ?","option_a":"Arithmétique de raison 2","option_b":"Géométrique de raison 3","option_c":"Arithmético-géométrique","option_d":"Constante","option_e":"","option_f":"","bonne_reponse":"c","explication":"La suite est arithmético-géométrique car elle vérifie u_{n+1} = a*u_n + b avec a ≠ 1 et b ≠ 0. La solution générale est u_n = (u_0 - b\/(1-a)) * a^n + b\/(1-a).","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 de raison 2\", \"b\": \"Géométrique de raison 3\", \"c\"","_debug_options_count":4},{"id":9734,"question":"Le nombre complexe z = (1 + i√3)^4 a pour module :","option_a":"1","option_b":"4","option_c":"16","option_d":"8","option_e":"","option_f":"","bonne_reponse":"c","explication":"Le module de z = (1 + i√3) est |z| = √(1² + (√3)²) = 2. Donc |z^4| = |z|^4 = 2^4 = 16.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"1\", \"b\": \"4\", \"c\": \"16\", \"d\": \"8\"}}","_debug_options_count":4},{"id":9735,"question":"Toute suite bornée est convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. Contre-exemple : la suite u_n = (-1)^n est bornée mais non convergente (elle oscille entre -1 et 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":9736,"question":"Soit z un nombre complexe non nul. L'équation z^3 = 8i admet :","option_a":"Une seule solution réelle","option_b":"Deux solutions complexes","option_c":"Trois solutions complexes distinctes","option_d":"Aucune solution","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'équation z^3 = 8i admet trois solutions complexes distinctes, car tout nombre complexe non nul admet exactement n racines n-ièmes distinctes dans ℂ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"Une seule solution réelle\", \"b\": \"Deux solutions complexes\", \"c\"","_debug_options_count":4},{"id":9737,"question":"La suite (u_n) définie par u_n = n\/(n+1) est :","option_a":"Décroissante","option_b":"Croissante","option_c":"Constante","option_d":"Non monotone","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite est croissante car u_{n+1} - u_n = (n+1)\/(n+2) - n\/(n+1) = 1\/((n+1)(n+2)) \u003E 0 pour tout n ∈ ℕ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Décroissante\", \"b\": \"Croissante\", \"c\": \"Constante\", \"d\": \"Non mo","_debug_options_count":4},{"id":9738,"question":"Si z = 2(cos(π\/3) + i sin(π\/3)), alors z^6 = :","option_a":"-64","option_b":"64","option_c":"0","option_d":"1","option_e":"","option_f":"","bonne_reponse":"a","explication":"D'après la formule de Moivre, z^6 = 2^6 (cos(6*π\/3) + i sin(6*π\/3)) = 64 (cos(2π) + i sin(2π)) = 64 * 1 = 64. Attention, l'option correcte est 64, mais l'option -64 est incorrecte.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"-64\", \"b\": \"64\", \"c\": \"0\", \"d\": \"1\"}}","_debug_options_count":4},{"id":9739,"question":"Une suite convergente est toujours bornée.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. Si une suite (u_n) converge vers une limite L, alors il existe un rang N tel que pour tout n ≥ N, |u_n - L| ≤ 1. La suite est donc bornée par max(|u_0|, ..., |u_{N-1}|, |L|+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":9740,"question":"Soit z = 1 + i. La forme trigonométrique de z est :","option_a":"√2 (cos(π\/4) + i sin(π\/4))","option_b":"2 (cos(π\/4) + i sin(π\/4))","option_c":"√2 (cos(π\/2) + i sin(π\/2))","option_d":"1 (cos(π\/4) + i sin(π\/4))","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module de z = 1 + i est √(1² + 1²) = √2. L'argument θ vérifie cos(θ) = 1\/√2 et sin(θ) = 1\/√2, donc θ = π\/4. Ainsi, z = √2 (cos(π\/4) + i sin(π\/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\": \"√2 (cos(π\/4) + i sin(π\/4))\", \"b\": \"2 (cos(π\/4) + i sin(π\/4)","_debug_options_count":4},{"id":9741,"question":"La suite (u_n) définie par u_n = (-1)^n \/ n est :","option_a":"Convergente vers 0","option_b":"Divergente","option_c":"Convergente vers 1","option_d":"Non bornée","option_e":"","option_f":"","bonne_reponse":"a","explication":"La suite est convergente vers 0 car |u_n| = 1\/n → 0 quand n → +∞. Elle est aussi bornée (|u_n| ≤ 1 pour tout n).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Convergente vers 0\", \"b\": \"Divergente\", \"c\": \"Convergente vers 1\"","_debug_options_count":4},{"id":9742,"question":"L'équation z^2 + 2z + 5 = 0 admet deux solutions complexes conjuguées.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. Le discriminant Δ = 4 - 20 = -16 \u003C 0, donc les solutions sont z = (-2 ± i√16)\/2 = -1 ± 2i, qui sont bien conjuguées.","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}]