Quiz — Série Analyse Homme.pdf

Question 1 sur 5 10:00

[{"id":577,"question":"Quelle est la limite de la fonction f(x) = (3x² + 2x - 1)\/(x² - 4) lorsque x tend vers +∞ ?","option_a":"3","option_b":"0","option_c":"1","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"En divisant le numérateur et le dénominateur par x², on obtient f(x) = (3 + 2\/x - 1\/x²)\/(1 - 4\/x²). Lorsque x → +∞, les termes en 1\/x tendent vers 0, donc la limite est 3\/1 = 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\", \"c\": \"1\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":578,"question":"Soit f(x) = x³ - 3x + 2. Quel est le nombre de points où la tangente à la courbe est parallèle à l'axe des abscisses ?","option_a":"0","option_b":"1","option_c":"2","option_d":"3","option_e":"","option_f":"","bonne_reponse":"c","explication":"La tangente est parallèle à l'axe des abscisses lorsque f'(x) = 0. f'(x) = 3x² - 3 = 0 ⇒ x² = 1 ⇒ x = 1 ou x = -1. Il y a donc 2 points.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"3\"}}","_debug_options_count":4},{"id":579,"question":"Quelle est la valeur de l'intégrale ∫(de 0 à 1) (2x + 1) dx ?","option_a":"1","option_b":"2","option_c":"3","option_d":"4","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'intégrale se calcule comme suit : [x² + x] de 0 à 1 = (1 + 1) - (0 + 0) = 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\": \"1\", \"b\": \"2\", \"c\": \"3\", \"d\": \"4\"}}","_debug_options_count":4},{"id":580,"question":"Soit l'équation différentielle y' + 2y = 0. Quelle est la solution générale ?","option_a":"y = Ce^(-2x)","option_b":"y = Ce^(2x)","option_c":"y = Cx + 2","option_d":"y = Cx - 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une équation différentielle linéaire du premier ordre. La solution générale est y = Ce^(-2x), où C est une constante.","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 = Cx + 2\", \"d\": \"y = C","_debug_options_count":4},{"id":581,"question":"Quelle est la dérivée de la fonction f(x) = ln(3x² + 1) ?","option_a":"f'(x) = 6x\/(3x² + 1)","option_b":"f'(x) = 3x\/(3x² + 1)","option_c":"f'(x) = 6x","option_d":"f'(x) = 3x","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la formule de dérivation des fonctions composées, f'(x) = (1\/(3x² + 1)) * (6x) = 6x\/(3x² + 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\": \"f'(x) = 6x\/(3x² + 1)\", \"b\": \"f'(x) = 3x\/(3x² + 1)\", \"c\": \"f'(x)","_debug_options_count":4}]