Quiz — td2bis.pdf

Question 1 sur 10 20:00

[{"id":102703,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"3x² + 2x","option_c":"6x - 5","option_d":"3x + 2","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 + 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\": \"6x + 2\", \"b\": \"3x² + 2x\", \"c\": \"6x - 5\", \"d\": \"3x + 2\"}}","_debug_options_count":4},{"id":102704,"question":"La suite (u_n) définie par u_n = 2n + 1 est-elle arithmétique ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une suite est arithmétique si la différence entre deux termes consécutifs est constante. Ici, u_{n+1} - u_n = 2, donc la suite est arithmétique.","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":102705,"question":"Quelle est la limite de la fonction f(x) = (2x² - 3x + 1)\/(x² + 1) quand x tend vers l'infini ?","option_a":"2","option_b":"1","option_c":"0","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour les fonctions rationnelles, la limite en l'infini est égale au rapport des coefficients dominants. Ici, lim (2x²\/x²) = 2, mais après simplification, la limite est 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\": \"2\", \"b\": \"1\", \"c\": \"0\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":102706,"question":"Soit X une variable aléatoire suivant une loi binomiale B(10, 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":"L'espérance d'une loi binomiale B(n, p) est E(X) = n*p. Ici, E(X) = 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":102707,"question":"La fonction f(x) = x³ - 3x² + 4 est-elle croissante sur l'intervalle [0, 2] ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour étudier la croissance, calculons la dérivée f'(x) = 3x² - 6x. Sur [0, 2], f'(x) ≤ 0, donc la fonction est décroissante sur cet intervalle.","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":102708,"question":"Quelle est la solution de l'équation 2^(x+1) = 16 ?","option_a":"x = 3","option_b":"x = 2","option_c":"x = 4","option_d":"x = 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"16 s'écrit 2^4, donc 2^(x+1) = 2^4 ⇒ x+1 = 4 ⇒ x = 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\": \"x = 3\", \"b\": \"x = 2\", \"c\": \"x = 4\", \"d\": \"x = 1\"}}","_debug_options_count":4},{"id":102709,"question":"Soit (u_n) une suite géométrique de raison q = -2 et de premier terme u_0 = 5. Quelle est la valeur de u_3 ?","option_a":"-40","option_b":"40","option_c":"20","option_d":"-20","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une suite géométrique, u_n = u_0 * q^n. Ici, u_3 = 5 * (-2)^3 = 5 * (-8) = -40.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"-40\", \"b\": \"40\", \"c\": \"20\", \"d\": \"-20\"}}","_debug_options_count":4},{"id":102710,"question":"La fonction f(x) = ln(x) est-elle 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) est définie uniquement pour x \u003E 0. Elle n'est pas définie pour les réels négatifs ou nuls.","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":102711,"question":"Quelle est la probabilité d'obtenir exactement 2 fois le chiffre 6 en lançant 5 fois un dé équilibré ?","option_a":"5\/324","option_b":"10\/243","option_c":"5\/243","option_d":"1\/36","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est une loi binomiale B(5, 1\/6). P(X=2) = C(5,2) * (1\/6)^2 * (5\/6)^3 = 10 * (1\/36) * (125\/216) = 1250\/7776 ≈ 10\/243.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"5\/324\", \"b\": \"10\/243\", \"c\": \"5\/243\", \"d\": \"1\/36\"}}","_debug_options_count":4},{"id":102712,"question":"Soit f une fonction dérivable sur R. Si f'(x) = 0 pour tout x, alors f est une fonction constante.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"D'après le théorème des accroissements finis, si la dérivée est nulle sur un intervalle, alors la fonction est constante sur cet intervalle.","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}]