Quiz — item 64 fmm 2017 SMG-résidanat-NR-2016-2017 f.pdf

Question 1 sur 10 20:00

[{"id":8625,"question":"Quelle est la solution générale de l'équation différentielle y' + 2y = 0 ?","option_a":"A) y = Ce^(2x)","option_b":"B) y = Ce^(-2x)","option_c":"C) y = Ce^(x\/2)","option_d":"D) y = Ce^(-x\/2)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La solution générale d'une équation différentielle linéaire du premier ordre y' + ay = 0 est y = Ce^(-ax). Ici, a = 2, donc y = Ce^(-2x).","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) y = Ce^(2x)\", \"b\": \"B) y = Ce^(-2x)\", \"c\": \"C) y = Ce^(x\/2)\", ","_debug_options_count":4},{"id":8626,"question":"La fonction f(x) = x^3 - 3x^2 + 2 est convexe sur l'intervalle ]-∞, 1[.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La convexité se détermine par le signe de la dérivée seconde. f''(x) = 6x - 6. Sur ]-∞, 1[, f''(x) \u003C 0, donc f est concave, pas convexe.","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":8627,"question":"Quelle est la valeur de l'intégrale ∫(de 0 à π) sin(x) dx ?","option_a":"A) 0","option_b":"B) 1","option_c":"C) 2","option_d":"D) π","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'intégrale de sin(x) entre 0 et π est [-cos(x)] de 0 à π = -cos(π) - (-cos(0)) = 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\": \"A) 0\", \"b\": \"B) 1\", \"c\": \"C) 2\", \"d\": \"D) π\"}}","_debug_options_count":4},{"id":8628,"question":"Soit une 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) -1","option_d":"D) +∞","option_e":"","option_f":"","bonne_reponse":"d","explication":"Cette suite est arithmético-géométrique. Sa limite est solution de L = 2L + 1, soit L = -1. Cependant, comme le coefficient est \u003E1, 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\": \"d\", \"options\": {\"a\": \"A) 0\", \"b\": \"B) 1\", \"c\": \"C) -1\", \"d\": \"D) +∞\"}}","_debug_options_count":4},{"id":8629,"question":"La probabilité d'obtenir un double six en lançant deux dés est de 1\/36.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Il y a 6 faces sur chaque dé, donc 6x6 = 36 combinaisons possibles. Une seule donne un double six, donc la probabilité est bien 1\/36.","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":8630,"question":"Quelle est la dérivée de la fonction f(x) = ln(x^2 + 1) ?","option_a":"A) 2x\/(x^2 + 1)","option_b":"B) 1\/(x^2 + 1)","option_c":"C) 2x","option_d":"D) x\/(x^2 + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la dérivée de ln(u), f'(x) = u'(x)\/u(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\": \"A) 2x\/(x^2 + 1)\", \"b\": \"B) 1\/(x^2 + 1)\", \"c\": \"C) 2x\", \"d\": \"D) x","_debug_options_count":4},{"id":8631,"question":"Soit une variable aléatoire X 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) 0,9986","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour une loi normale centrée réduite, P(X ≤ 1) ≈ 0,8413 (valeur lue dans les tables de la loi normale).","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) 0,9986\"}","_debug_options_count":4},{"id":8632,"question":"La série ∑(de n=1 à ∞) 1\/n^2 converge.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une série de Riemann avec α = 2 \u003E 1, donc elle converge (vers π²\/6).","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":8633,"question":"Quelle est la solution de l'équation x^2 - 5x + 6 = 0 ?","option_a":"A) x = 2 ou x = 3","option_b":"B) x = -2 ou x = -3","option_c":"C) x = 1 ou x = 6","option_d":"D) x = 0 ou x = 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"En factorisant, (x-2)(x-3) = 0, donc les solutions sont x = 2 et x = 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\": \"A) x = 2 ou x = 3\", \"b\": \"B) x = -2 ou x = -3\", \"c\": \"C) x = 1 ou","_debug_options_count":4},{"id":8634,"question":"Soit f une fonction dérivable sur ℝ. Si f'(x) = 0 pour tout x, alors f est constante.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est le théorème des accroissements finis : si la dérivée est nulle partout, la fonction est 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}]