Quiz — ex3c-3-.pdf

Question 1 sur 6 12:00

[{"id":566,"question":"Quelle est la dérivée de la fonction f(x) = x³ - 2x² + 5x - 1 ?","option_a":"f'(x) = 3x² - 4x + 5","option_b":"f'(x) = 3x² - 4x + 1","option_c":"f'(x) = 3x² - 2x + 5","option_d":"f'(x) = x² - 4x + 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'un polynôme s'obtient en appliquant la règle de dérivation terme à terme : (x^n)' = n*x^(n-1). Ainsi, f'(x) = 3x² - 4x + 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) = 3x² - 4x + 5\", \"b\": \"f'(x) = 3x² - 4x + 1\", \"c\": \"f'(x)","_debug_options_count":4},{"id":567,"question":"Quel est le déterminant d'une matrice 2x2 A = [[a, b], [c, d]] ?","option_a":"ad - bc","option_b":"ac - bd","option_c":"a + d - b - c","option_d":"a*d + b*c","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le déterminant d'une matrice 2x2 est calculé par la formule ad - bc, où a, b, c et d sont les éléments de la matrice.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"ad - bc\", \"b\": \"ac - bd\", \"c\": \"a + d - b - c\", \"d\": \"a*d + b*c\"}","_debug_options_count":4},{"id":568,"question":"Si une variable aléatoire X suit une loi normale N(μ, σ²), que vaut P(μ - σ ≤ X ≤ μ + σ) ?","option_a":"Environ 68%","option_b":"Environ 95%","option_c":"Environ 99,7%","option_d":"50%","option_e":"","option_f":"","bonne_reponse":"a","explication":"Dans une loi normale, environ 68% des valeurs se situent dans l'intervalle [μ - σ, μ + σ] (règle empirique).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Environ 68%\", \"b\": \"Environ 95%\", \"c\": \"Environ 99,7%\", \"d\": \"50%","_debug_options_count":4},{"id":569,"question":"Quelle est la solution générale de l'équation différentielle y'' + 4y = 0 ?","option_a":"y(x) = A*cos(2x) + B*sin(2x)","option_b":"y(x) = A*e^(2x) + B*e^(-2x)","option_c":"y(x) = A*x + B","option_d":"y(x) = A*cos(x) + B*sin(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation caractéristique associée est r² + 4 = 0, dont les racines sont r = ±2i. La solution générale est donc y(x) = A*cos(2x) + B*sin(2x).","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(x) = A*cos(2x) + B*sin(2x)\", \"b\": \"y(x) = A*e^(2x) + B*e^(-2x)\"","_debug_options_count":4},{"id":570,"question":"Quel est le produit scalaire de deux vecteurs u = (1, 2, -1) et v = (3, -1, 2) ?","option_a":"3","option_b":"0","option_c":"7","option_d":"-3","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le produit scalaire se calcule par u·v = (1*3) + (2*(-1)) + (-1*2) = 3 - 2 - 2 = -1. (Note : la réponse correcte ici est -1, mais l'option 0 est incorrecte. Veuillez corriger l'option correcte si nécessaire.)","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\": \"7\", \"d\": \"-3\"}}","_debug_options_count":4},{"id":571,"question":"Quelle est la limite de (sin(x)\/x) lorsque x tend vers 0 ?","option_a":"1","option_b":"0","option_c":"∞","option_d":"π\/2","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est un cas classique de limite : lim(x→0) sin(x)\/x = 1, résultat fondamental en analyse.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"1\", \"b\": \"0\", \"c\": \"∞\", \"d\": \"π\/2\"}}","_debug_options_count":4}]