Quiz — EpreuveII_2001.pdf

Question 1 sur 10 20:00

[{"id":19879,"question":"Quelle est la dérivée de la fonction f(x) = 3x² - 5x + 2 ?","option_a":"f'(x) = 6x - 5","option_b":"f'(x) = 3x - 5","option_c":"f'(x) = 6x² - 5x","option_d":"f'(x) = 5 - 6x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme s'obtient en appliquant la formule (ax^n)' = n*ax^(n-1). Ici, f'(x) = 6x - 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\": \"f'(x) = 6x - 5\", \"b\": \"f'(x) = 3x - 5\", \"c\": \"f'(x) = 6x² - 5x\",","_debug_options_count":4},{"id":19880,"question":"Une suite géométrique de raison q = 2 et de premier terme u₀ = 3 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 sa raison vérifie |q| \u003C 1. Ici, q = 2, donc la suite est divergente.","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":19881,"question":"Soit X une variable aléatoire suivant une loi binomiale B(n=10, p=0.3). Quelle est la probabilité P(X=3) ?","option_a":"0.2668","option_b":"0.2334","option_c":"0.1559","option_d":"0.3000","option_e":"","option_f":"","bonne_reponse":"b","explication":"P(X=k) = C(n,k) * p^k * (1-p)^(n-k). Ici, P(X=3) = C(10,3) * (0.3)^3 * (0.7)^7 ≈ 0.2334.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0.2668\", \"b\": \"0.2334\", \"c\": \"0.1559\", \"d\": \"0.3000\"}}","_debug_options_count":4},{"id":19882,"question":"La fonction f(x) = x³ - 3x² + 2 admet-elle un extremum local en x = 1 ?","option_a":"Oui, c'est un maximum local","option_b":"Oui, c'est un minimum local","option_c":"Non, car f'(1) = 0 mais f''(1) = 0","option_d":"Non, car f'(1) ≠ 0","option_e":"","option_f":"","bonne_reponse":"c","explication":"f'(x) = 3x² - 6x. f'(1) = -3 ≠ 0, donc x=1 n'est pas un point critique. L'extremum n'existe pas en ce point.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"Oui, c'est un maximum local\", \"b\": \"Oui, c'est un minimum local\",","_debug_options_count":4},{"id":19883,"question":"Si A et B sont deux événements indépendants, alors P(A ∩ B) = P(A) * P(B).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est la définition de l'indépendance de deux événements. La formule est correcte.","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":19884,"question":"Quelle est la limite de la suite uₙ = (2n + 1)\/(n + 3) quand n tend vers l'infini ?","option_a":"0","option_b":"1","option_c":"2","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par n, on obtient uₙ = (2 + 1\/n)\/(1 + 3\/n). Quand n→∞, uₙ → 2\/1 = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":19885,"question":"Soit f une fonction dérivable sur ℝ. Si f'(x) \u003E 0 pour tout x ∈ ℝ, alors f est strictement croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est un théorème fondamental du cours : si la dérivée est strictement positive, la fonction est strictement 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":19886,"question":"Une suite arithmétique de premier terme u₀ = 5 et de raison r = -2 est-elle décroissante ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une suite arithmétique est décroissante si sa raison r est négative. Ici, r = -2 \u003C 0, donc la suite est décroissante.","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":19887,"question":"Quelle est la dérivée de la fonction g(x) = e^(2x) ?","option_a":"g'(x) = e^(2x)","option_b":"g'(x) = 2e^(2x)","option_c":"g'(x) = 2x e^(2x)","option_d":"g'(x) = e^(x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de e^(u(x)) est u'(x) * e^(u(x)). Ici, u(x) = 2x, donc u'(x) = 2. Ainsi, g'(x) = 2e^(2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"g'(x) = e^(2x)\", \"b\": \"g'(x) = 2e^(2x)\", \"c\": \"g'(x) = 2x e^(2x)\"","_debug_options_count":4},{"id":19888,"question":"Si P(A) = 0.4 et P(B) = 0.5, alors P(A ∪ B) = 0.9 si A et B sont incompatibles.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour deux événements incompatibles, P(A ∪ B) = P(A) + P(B) = 0.4 + 0.5 = 0.9. La formule est correcte.","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}]