Quiz — Mathematiques Exercices incontournables (MP) ( PDFDrive.com ).pdf

Question 1 sur 10 20:00

[{"id":22429,"question":"Soit la suite définie par u_{n+1} = 2u_n + 1 avec u_0 = 0. Quelle est la limite de u_n quand n tend vers l'infini ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. +∞","option_d":"D. -1","option_e":"","option_f":"","bonne_reponse":"c","explication":"La suite est arithmético-géométrique. En résolvant l'équation l = 2l + 1, on trouve l = -1. Comme u_0 = 0 \u003E -1, la suite diverge vers +∞.","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. +∞\", \"d\": \"D. -1\"}}","_debug_options_count":4},{"id":22430,"question":"La fonction f(x) = x^2 * sin(1\/x) est-elle dérivable en 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction est dérivable en 0 car la limite du taux d'accroissement existe et vaut 0. On peut prolonger f par f(0) = 0 et f'(0) = 0.","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":22431,"question":"Quel est le rayon de convergence de la série entière Σ (n^2 * x^n) \/ 3^n ?","option_a":"A. 1\/3","option_b":"B. 3","option_c":"C. ∞","option_d":"D. 0","option_e":"","option_f":"","bonne_reponse":"b","explication":"En utilisant le critère de d'Alembert, on trouve que le rayon de convergence est R = 3, car lim |a_{n+1}\/a_n| = 1\/3.","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. 1\/3\", \"b\": \"B. 3\", \"c\": \"C. ∞\", \"d\": \"D. 0\"}}","_debug_options_count":4},{"id":22432,"question":"Soit A une matrice carrée d'ordre 2. Si A^2 = 0, alors A est-elle nécessairement la matrice nulle ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Non, par exemple A = [[0, 1], [0, 0]] vérifie A^2 = 0 mais A ≠ 0. Une matrice nilpotente d'ordre 2 peut satisfaire cette propriété.","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":22433,"question":"Quelle est la valeur de l'intégrale impropre ∫_{0}^{∞} e^{-x} dx ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. ∞","option_d":"D. -1","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégrale converge et vaut 1, car ∫_{0}^{∞} e^{-x} dx = [-e^{-x}]_{0}^{∞} = 0 - (-1) = 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. ∞\", \"d\": \"D. -1\"}}","_debug_options_count":4},{"id":22434,"question":"Soit X une variable aléatoire suivant une loi normale N(0,1). Quelle est la probabilité P(X ≤ 1) ?","option_a":"A. 0.6826","option_b":"B. 0.8413","option_c":"C. 0.9772","option_d":"D. 0.3413","option_e":"","option_f":"","bonne_reponse":"b","explication":"P(X ≤ 1) ≈ 0.8413, car dans une loi normale centrée réduite, environ 68% des valeurs sont dans [-1,1], et la probabilité pour X ≤ 1 est la moitié de 1 + 0.6826.","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. 0.6826\", \"b\": \"B. 0.8413\", \"c\": \"C. 0.9772\", \"d\": \"D. 0.3413\"}","_debug_options_count":4},{"id":22435,"question":"L'équation différentielle y'' + 4y = 0 admet-elle des solutions périodiques ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, les solutions sont de la forme y(x) = A*cos(2x) + B*sin(2x), qui sont périodiques de période π.","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":22436,"question":"Soit f une fonction continue sur [a,b]. Quel théorème garantit l'existence d'un c ∈ [a,b] tel que f(c) = (f(a)+f(b))\/2 ?","option_a":"A. Théorème des valeurs intermédiaires","option_b":"B. Théorème de Rolle","option_c":"C. Théorème de Cauchy","option_d":"D. Théorème de Bolzano-Weierstrass","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le théorème des valeurs intermédiaires garantit l'existence d'un tel c, car f prend toutes les valeurs entre f(a) et f(b).","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. Théorème des valeurs intermédiaires\", \"b\": \"B. Théorème d","_debug_options_count":4},{"id":22437,"question":"Quelle est la dimension de l'espace vectoriel des matrices 3x3 symétriques réelles ?","option_a":"A. 3","option_b":"B. 6","option_c":"C. 9","option_d":"D. 12","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une matrice 3x3 symétrique a 6 coefficients indépendants (3 sur la diagonale et 3 au-dessus), donc la dimension est 6.","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. 3\", \"b\": \"B. 6\", \"c\": \"C. 9\", \"d\": \"D. 12\"}}","_debug_options_count":4},{"id":22438,"question":"Soit (X_n) une suite de variables aléatoires indépendantes de même loi. Sous quelle condition la loi des grands nombres s'applique-t-elle ?","option_a":"A. E[X_1] existe et est finie","option_b":"B. Var(X_1) existe et est finie","option_c":"C. X_1 suit une loi normale","option_d":"D. X_1 est bornée","option_e":"","option_f":"","bonne_reponse":"a","explication":"La loi des grands nombres (faible ou forte) s'applique dès que l'espérance E[X_1] existe et est finie, sans nécessiter de condition sur la variance.","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. E[X_1] existe et est finie\", \"b\": \"B. Var(X_1) existe et est f","_debug_options_count":4}]