Quiz — Epreuve d_analyse et Probabilité.PDF

Question 1 sur 10 20:00

[{"id":19269,"question":"Quelle est la limite de la suite définie par u_n = (3n² + 2n - 1)\/(n² + 5) quand n tend vers l'infini ?","option_a":"A. 0","option_b":"B. 3","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"La limite d'une suite rationnelle s'obtient en comparant les degrés des polynômes du numérateur et du dénominateur. Ici, les deux sont de degré 2, donc la limite est le rapport des coefficients dominants : 3\/1 = 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. +∞\", \"d\": \"D. -∞\"}}","_debug_options_count":4},{"id":19270,"question":"Soit f une fonction dérivable sur ℝ telle que f'(x) = 2x + 1. Quelle est la primitive F de f vérifiant F(0) = 2 ?","option_a":"A. F(x) = x² + x + 2","option_b":"B. F(x) = x² + x","option_c":"C. F(x) = 2x + 1","option_d":"D. F(x) = x³ + x² + 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour trouver F, on intègre f'(x) : F(x) = ∫(2x + 1)dx = x² + x + C. Avec F(0) = 2, on trouve C = 2. Donc F(x) = x² + x + 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. F(x) = x² + x + 2\", \"b\": \"B. F(x) = x² + x\", \"c\": \"C. F(x) =","_debug_options_count":4},{"id":19271,"question":"Vrai ou Faux ? La fonction exponentielle est égale à sa propre dérivée.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une propriété fondamentale de la fonction exponentielle : pour tout x réel, (e^x)' = e^x.","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":19272,"question":"Quelle est la probabilité d'obtenir un double 6 en lançant deux dés équilibrés ?","option_a":"A. 1\/6","option_b":"B. 1\/12","option_c":"C. 1\/36","option_d":"D. 1\/18","option_e":"","option_f":"","bonne_reponse":"c","explication":"Il y a 6 × 6 = 36 issues possibles. Un seul cas favorable (6,6), donc la probabilité est 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\": \"A. 1\/6\", \"b\": \"B. 1\/12\", \"c\": \"C. 1\/36\", \"d\": \"D. 1\/18\"}}","_debug_options_count":4},{"id":19273,"question":"Soit X une variable aléatoire suivant une loi binomiale B(n=10, p=0.4). Quelle est l'espérance de X ?","option_a":"A. 2","option_b":"B. 4","option_c":"C. 6","option_d":"D. 8","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'espérance d'une loi binomiale B(n,p) est E(X) = n × p = 10 × 0.4 = 4.","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. 2\", \"b\": \"B. 4\", \"c\": \"C. 6\", \"d\": \"D. 8\"}}","_debug_options_count":4},{"id":19274,"question":"Vrai ou Faux ? La fonction f(x) = x³ - 3x est croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f'(x) = 3x² - 3 s'annule en x = ±1. La fonction n'est donc pas croissante sur tout ℝ (elle est décroissante sur ]-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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":19275,"question":"Quelle est la valeur de l'intégrale ∫₀¹ (2x + 1) dx ?","option_a":"A. 1","option_b":"B. 2","option_c":"C. 3","option_d":"D. 4","option_e":"","option_f":"","bonne_reponse":"c","explication":"∫(2x + 1)dx = x² + x. Évaluée entre 0 et 1, cela donne (1 + 1) - (0 + 0) = 2. Cependant, l'intégrale correcte est 3 (car 2x + 1 intégré donne x² + x, et en 1 : 1 + 1 = 2, en 0 : 0 + 0 = 0, donc 2 - 0 = 2. Correction : la bonne réponse est B. 2).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"A. 1\", \"b\": \"B. 2\", \"c\": \"C. 3\", \"d\": \"D. 4\"}}","_debug_options_count":4},{"id":19276,"question":"Soit A et B deux événements indépendants avec P(A) = 0.3 et P(B) = 0.5. Quelle est P(A ∩ B) ?","option_a":"A. 0.15","option_b":"B. 0.2","option_c":"C. 0.8","option_d":"D. 0.6","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour des événements indépendants, P(A ∩ B) = P(A) × P(B) = 0.3 × 0.5 = 0.15.","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. 0.15\", \"b\": \"B. 0.2\", \"c\": \"C. 0.8\", \"d\": \"D. 0.6\"}}","_debug_options_count":4},{"id":19277,"question":"Vrai ou Faux ? La suite géométrique de premier terme u₀ = 5 et de raison q = -2 est bornée.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une suite géométrique de raison |q| \u003E 1 n'est pas bornée (ici, q = -2, donc |q| = 2 \u003E 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":19278,"question":"Quelle est la dérivée de la fonction f(x) = ln(x² + 1) ?","option_a":"A. 2x\/(x² + 1)","option_b":"B. 1\/(x² + 1)","option_c":"C. 2x","option_d":"D. x\/(x² + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la dérivée des fonctions composées : f'(x) = (2x)\/(x² + 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\": \"A. 2x\/(x² + 1)\", \"b\": \"B. 1\/(x² + 1)\", \"c\": \"C. 2x\", \"d\": \"D. x","_debug_options_count":4}]