Quiz — Centrale_2009_MP_M2_Enonce.pdf

Question 1 sur 10 20:00

[{"id":153897,"question":"Soit une matrice carrée A d'ordre 3. Si det(A) = 0, alors :","option_a":"A est inversible","option_b":"A est diagonalisable","option_c":"A n'est pas inversible","option_d":"A est symétrique","option_e":"","option_f":"","bonne_reponse":"c","explication":"Le déterminant d'une matrice est nul si et seulement si la matrice n'est pas inversible.","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 est inversible\", \"b\": \"A est diagonalisable\", \"c\": \"A n'est pas","_debug_options_count":4},{"id":153898,"question":"La suite (u_n) définie par u_0 = 1 et u_{n+1} = 2u_n + 1 est :","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Cette suite est bien définie par la relation de récurrence donnée, mais elle n'est pas 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":153899,"question":"Quelle est la limite de la suite (v_n) définie par v_n = (1 + 1\/n)^n ?","option_a":"0","option_b":"1","option_c":"e","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"La suite (v_n) converge vers le nombre d'Euler e ≈ 2.718.","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\": \"e\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":153900,"question":"Soit X une variable aléatoire suivant une loi normale N(0,1). Quelle est la probabilité P(X ≤ 0) ?","option_a":"0","option_b":"0.5","option_c":"1","option_d":"0.25","option_e":"","option_f":"","bonne_reponse":"b","explication":"La loi normale centrée réduite est symétrique autour de 0, donc P(X ≤ 0) = 0.5.","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\": \"0.5\", \"c\": \"1\", \"d\": \"0.25\"}}","_debug_options_count":4},{"id":153901,"question":"La fonction f(x) = x^3 - 3x est convexe sur l'intervalle :","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée seconde f''(x) = 6x est négative sur ]-∞, 0[, donc la fonction n'est pas convexe 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":153902,"question":"Quelle est la valeur de l'intégrale ∫(de 0 à π) sin(x) dx ?","option_a":"0","option_b":"1","option_c":"2","option_d":"π","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'intégrale de sin(x) entre 0 et π vaut [-cos(x)] de 0 à π = -(-1) - (-1) = 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\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"π\"}}","_debug_options_count":4},{"id":153903,"question":"Soit une suite (w_n) définie par w_n = n\/(n+1). Cette suite est :","option_a":"Décroissante","option_b":"Croissante","option_c":"Constante","option_d":"Non monotone","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite w_n = n\/(n+1) est croissante car w_{n+1} - w_n \u003E 0 pour tout n.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Décroissante\", \"b\": \"Croissante\", \"c\": \"Constante\", \"d\": \"Non mo","_debug_options_count":4},{"id":153904,"question":"Soit A une matrice symétrique. Alors :","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, une matrice symétrique vérifie A = A^T.","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":153905,"question":"Quelle est la dérivée de la fonction f(x) = ln(x^2 + 1) ?","option_a":"2x\/(x^2 + 1)","option_b":"2x","option_c":"1\/(x^2 + 1)","option_d":"x\/(x^2 + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la règle de dérivation des fonctions composées, f'(x) = (2x)\/(x^2 + 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\": \"2x\/(x^2 + 1)\", \"b\": \"2x\", \"c\": \"1\/(x^2 + 1)\", \"d\": \"x\/(x^2 + 1)\"}","_debug_options_count":4},{"id":153906,"question":"Soit X une variable aléatoire discrète prenant les valeurs 1, 2 et 3 avec les probabilités respectives 0.2, 0.5 et 0.3. Quelle est l'espérance E(X) ?","option_a":"1.8","option_b":"2.0","option_c":"2.1","option_d":"2.3","option_e":"","option_f":"","bonne_reponse":"c","explication":"E(X) = 1×0.2 + 2×0.5 + 3×0.3 = 0.2 + 1.0 + 0.9 = 2.1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"1.8\", \"b\": \"2.0\", \"c\": \"2.1\", \"d\": \"2.3\"}}","_debug_options_count":4}]