Quiz — Corrigé finale - prototype d_examen 2 1.pdf

Question 1 sur 10 20:00

[{"id":11963,"question":"Quelle est la dérivée de la fonction f(x) = x³ - 2x² + 5x - 1 ?","option_a":"3x² - 4x + 5","option_b":"3x² - 4x + 1","option_c":"x² - 2x + 5","option_d":"3x² - 2x + 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'un polynôme s'obtient en appliquant la formule (ax^n)' = n*ax^(n-1). Ici, f'(x) = 3x² - 4x + 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\": \"3x² - 4x + 5\", \"b\": \"3x² - 4x + 1\", \"c\": \"x² - 2x + 5\", \"d\": \"","_debug_options_count":4},{"id":11964,"question":"La suite définie par uₙ = (n+1)\/n est convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La suite uₙ = (n+1)\/n tend vers 1 lorsque n tend vers l'infini, donc elle est convergente.","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":11965,"question":"Quelle est la solution de l'équation différentielle y' + 2y = 0 ?","option_a":"y = Ce^(-2x)","option_b":"y = Ce^(2x)","option_c":"y = -2Ce^x","option_d":"y = Ce^(-x\/2)","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation différentielle linéaire du premier ordre y' + 2y = 0 a pour solution générale y = Ce^(-2x), où C est une constante réelle.","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 = -2Ce^x\", \"d\": \"y = C","_debug_options_count":4},{"id":11966,"question":"Dans une expérience aléatoire, A et B sont deux événements indépendants. Si P(A) = 0.4 et P(B) = 0.5, alors P(A ∩ B) = ?","option_a":"0.2","option_b":"0.9","option_c":"0.1","option_d":"0.45","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour deux événements indépendants, P(A ∩ B) = P(A) × P(B) = 0.4 × 0.5 = 0.2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"0.2\", \"b\": \"0.9\", \"c\": \"0.1\", \"d\": \"0.45\"}}","_debug_options_count":4},{"id":11967,"question":"La fonction f(x) = ln(x) est définie pour tout x réel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction logarithme népérien ln(x) n'est définie que pour x \u003E 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":11968,"question":"Quelle est la limite de la suite uₙ = (2n² + 3n - 1)\/(n² + 5) lorsque n tend vers l'infini ?","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 n², on obtient lim uₙ = lim (2 + 3\/n - 1\/n²)\/(1 + 5\/n²) = 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":11969,"question":"Soit f une fonction continue sur [a, b]. Si f(a) = -2 et f(b) = 3, alors il existe c ∈ [a, b] tel que f(c) = 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est le théorème des valeurs intermédiaires : comme f est continue et change de signe, elle s'annule nécessairement sur [a, b].","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":11970,"question":"Quelle est l'aire sous la courbe de f(x) = x² entre 0 et 2 ?","option_a":"4\/3","option_b":"8\/3","option_c":"2","option_d":"1","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'aire est donnée par l'intégrale ∫₀² x² dx = [x³\/3]₀² = 8\/3 - 0 = 8\/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\": \"4\/3\", \"b\": \"8\/3\", \"c\": \"2\", \"d\": \"1\"}}","_debug_options_count":4},{"id":11971,"question":"La fonction f(x) = e^(-x²) est paire.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une fonction f est paire si f(-x) = f(x). Ici, f(-x) = e^(-(-x)²) = e^(-x²) = f(x), donc f est paire.","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":11972,"question":"Quelle est la probabilité d'obtenir un double 6 en lançant deux dés équilibrés ?","option_a":"1\/6","option_b":"1\/12","option_c":"1\/36","option_d":"1\/18","option_e":"","option_f":"","bonne_reponse":"c","explication":"Il y a 36 issues possibles (6 × 6). Le double 6 ne correspond qu'à une seule issue, donc P = 1\/36.","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\/6\", \"b\": \"1\/12\", \"c\": \"1\/36\", \"d\": \"1\/18\"}}","_debug_options_count":4}]