Quiz — Corrige-Janvier.pdf

Question 1 sur 10 20:00

[{"id":10033,"question":"Soit la suite (u_n) définie par u_0 = 2 et u_{n+1} = 3u_n - 1. Que peut-on dire de cette suite ?","option_a":"Elle est arithmétique de raison 3","option_b":"Elle est géométrique de raison 3","option_c":"Elle est arithmético-géométrique","option_d":"Elle est constante","option_e":"","option_f":"","bonne_reponse":"c","explication":"La suite est arithmético-géométrique car elle s'écrit sous la forme u_{n+1} = a*u_n + b avec a ≠ 1. Ici, a = 3 et b = -1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"Elle est arithmétique de raison 3\", \"b\": \"Elle est géométrique","_debug_options_count":4},{"id":10034,"question":"La fonction f(x) = x^3 - 3x + 1 est dérivable sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Toute fonction polynôme est dérivable sur ℝ. Sa dérivée est f'(x) = 3x^2 - 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":10035,"question":"Quelle est la limite de la suite (v_n) définie par v_n = (2n^2 + 3n)\/(n^2 + 1) quand n tend vers +∞ ?","option_a":"2","option_b":"0","option_c":"1","option_d":"+∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"En divisant numérateur et dénominateur par n^2, on obtient lim v_n = 2\/1 = 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\": \"2\", \"b\": \"0\", \"c\": \"1\", \"d\": \"+∞\"}}","_debug_options_count":4},{"id":10036,"question":"Soit f une fonction continue sur [a, b] telle que f(a) = -2 et f(b) = 3. Le théorème des valeurs intermédiaires garantit qu'il existe c ∈ [a, b] tel que :","option_a":"f(c) = 0","option_b":"f(c) = 1","option_c":"f(c) = 2","option_d":"f(c) = 4","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le TVI garantit l'existence d'un c tel que f(c) = k pour tout k entre f(a) et f(b), ici k = 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"f(c) = 0\", \"b\": \"f(c) = 1\", \"c\": \"f(c) = 2\", \"d\": \"f(c) = 4\"}}","_debug_options_count":4},{"id":10037,"question":"L'équation différentielle y' + 2y = 0 a pour solution générale :","option_a":"y = Ce^{-2x}","option_b":"y = Ce^{2x}","option_c":"y = C + e^{-2x}","option_d":"y = Cx + e^{-2x}","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une équation différentielle linéaire du 1er ordre. La solution générale est y = Ce^{-2x} où C est une constante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"y = Ce^{-2x}\", \"b\": \"y = Ce^{2x}\", \"c\": \"y = C + e^{-2x}\", \"d\": \"","_debug_options_count":4},{"id":10038,"question":"Le nombre complexe z = 3 + 4i a pour module :","option_a":"5","option_b":"7","option_c":"1","option_d":"√7","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module de z = a + bi est |z| = √(a² + b²) = √(9 + 16) = 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"5\", \"b\": \"7\", \"c\": \"1\", \"d\": \"√7\"}}","_debug_options_count":4},{"id":10039,"question":"Soit X une variable aléatoire suivant une loi binomiale B(n=10, p=0.3). Quelle est l'espérance de X ?","option_a":"3","option_b":"0.3","option_c":"10","option_d":"7","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une loi binomiale B(n,p), l'espérance est E(X) = n*p = 10*0.3 = 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\": \"3\", \"b\": \"0.3\", \"c\": \"10\", \"d\": \"7\"}}","_debug_options_count":4},{"id":10040,"question":"La fonction f(x) = |x| est dérivable en x = 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction valeur absolue n'est pas dérivable en 0 car elle présente un point anguleux (dérivée à gauche ≠ dérivée à droite).","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":10041,"question":"Quelle est la solution générale de l'équation différentielle y'' - 4y' + 4y = 0 ?","option_a":"y = (C1 + C2x)e^{2x}","option_b":"y = C1e^{2x} + C2e^{-2x}","option_c":"y = C1e^{x} + C2e^{-x}","option_d":"y = C1cos(2x) + C2sin(2x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation caractéristique est r² - 4r + 4 = 0, de racine double r = 2. La solution générale est y = (C1 + C2x)e^{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\": \"y = (C1 + C2x)e^{2x}\", \"b\": \"y = C1e^{2x} + C2e^{-2x}\", \"c\": \"y =","_debug_options_count":4},{"id":10042,"question":"Soit z = 2e^{iπ\/3}. Que vaut z^3 ?","option_a":"8","option_b":"8i","option_c":"-8","option_d":"8e^{iπ}","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par la formule de Moivre, z^3 = 2^3 * e^{iπ} = 8*(-1) = -8.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"8\", \"b\": \"8i\", \"c\": \"-8\", \"d\": \"8e^{iπ}\"}}","_debug_options_count":4}]