Quiz — Ex8-7-c.pdf

Question 1 sur 10 20:00

[{"id":33530,"question":"Quelle est la limite de la fonction f(x) = (3x² + 2x - 1)\/(x² + 1) lorsque x tend vers +∞ ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 3","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"La limite d'un quotient de polynômes de même degré est égale au rapport des coefficients dominants. Ici, 3x²\/x² = 3.","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. 0\", \"b\": \"B. 1\", \"c\": \"C. 3\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":33531,"question":"Soit la suite définie par u₀ = 2 et uₙ₊₁ = 3uₙ - 1. Cette suite est-elle arithmétique ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une suite arithmétique a une différence constante entre deux termes consécutifs. Ici, u₁ - u₀ = 5 et u₂ - u₁ = 14, donc non arithmétique.","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":33532,"question":"Quel est le sens de variation de la fonction f(x) = x³ - 3x² + 4 sur ℝ ?","option_a":"A. Toujours croissante","option_b":"B. Croissante puis décroissante","option_c":"C. Décroissante puis croissante","option_d":"D. Toujours décroissante","option_e":"","option_f":"","bonne_reponse":"c","explication":"La dérivée f'(x) = 3x² - 6x s'annule en x=0 et x=2. Le signe de f' montre que f est décroissante sur ]-∞;0[, croissante sur ]0;2[, puis décroissante sur ]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. Toujours croissante\", \"b\": \"B. Croissante puis décroissante\",","_debug_options_count":4},{"id":33533,"question":"Soit X une variable aléatoire suivant une loi binomiale B(10; 0,3). Quelle est la probabilité P(X = 3) ?","option_a":"A. 0,2668","option_b":"B. 0,2334","option_c":"C. 0,1211","option_d":"D. 0,0474","option_e":"","option_f":"","bonne_reponse":"a","explication":"P(X=3) = C(10;3) × (0,3)³ × (0,7)⁷ ≈ 0,2668 (calcul avec la formule de la loi binomiale).","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. 0,2668\", \"b\": \"B. 0,2334\", \"c\": \"C. 0,1211\", \"d\": \"D. 0,0474\"}","_debug_options_count":4},{"id":33534,"question":"La fonction f(x) = ln(x) est-elle définie sur ℝ⁻* ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction ln(x) n'est définie que pour x \u003E 0. Elle n'est donc pas définie sur ℝ⁻* (nombres réels strictement négatifs).","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":33535,"question":"Quel est le coefficient directeur de la tangente à la courbe de f(x) = eˣ en x = 0 ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. e","option_d":"D. -1","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le coefficient directeur de la tangente en un point est égal à la valeur de la dérivée en ce point. f'(x) = eˣ, donc f'(0) = 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\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. e\", \"d\": \"D. -1\"}}","_debug_options_count":4},{"id":33536,"question":"Soit une suite géométrique de premier terme u₀ = 5 et de raison q = 2. Quel est le terme u₅ ?","option_a":"A. 10","option_b":"B. 160","option_c":"C. 80","option_d":"D. 320","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour une suite géométrique, uₙ = u₀ × qⁿ. Donc u₅ = 5 × 2⁵ = 5 × 32 = 160.","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. 10\", \"b\": \"B. 160\", \"c\": \"C. 80\", \"d\": \"D. 320\"}}","_debug_options_count":4},{"id":33537,"question":"La fonction f(x) = x² + 2x - 3 admet-elle un minimum sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction est une parabole de coefficient dominant positif, donc elle admet un minimum en x = -b\/(2a) = -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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":33538,"question":"Quel est le résultat de l'intégrale ∫₀¹ (2x + 1) dx ?","option_a":"A. 1","option_b":"B. 2","option_c":"C. 1,5","option_d":"D. 0,5","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'intégrale d'une fonction linéaire se calcule par la primitive : ∫(2x+1)dx = x² + x. Évaluée entre 0 et 1, cela donne (1 + 1) - (0 + 0) = 2. Attention, l'option C est 1,5, mais le calcul correct donne 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. 1,5\", \"d\": \"D. 0,5\"}}","_debug_options_count":4},{"id":33539,"question":"Soit z = 3 + 4i un nombre complexe. Quel est son module ?","option_a":"A. 5","option_b":"B. 7","option_c":"C. √7","option_d":"D. 25","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module d'un nombre complexe z = a + bi est |z| = √(a² + b²). Ici, |z| = √(3² + 4²) = √(9+16) = √25 = 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\": \"A. 5\", \"b\": \"B. 7\", \"c\": \"C. √7\", \"d\": \"D. 25\"}}","_debug_options_count":4}]