Quiz — 63317fdca7ae1_Exercice 22.pdf

Question 1 sur 10 20:00

[{"id":39660,"question":"Quelle est la limite de la fonction f(x) = (3x² + 2x - 1)\/(x² - 4) lorsque x tend vers +∞ ?","option_a":"A. 0","option_b":"B. 3","option_c":"C. -1","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"La limite d'un quotient de polynômes lorsque x tend vers l'infini est égale au rapport des coefficients des termes de plus haut degré. Ici, 3x²\/x² = 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. 0\", \"b\": \"B. 3\", \"c\": \"C. -1\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":39661,"question":"Une suite croissante est toujours convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une suite croissante n'est pas forcément convergente. Par exemple, la suite u_n = n est croissante mais diverge vers +∞.","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":39662,"question":"Quel est le discriminant de l'équation différentielle y'' + 4y' + 4y = 0 ?","option_a":"A. 0","option_b":"B. 4","option_c":"C. 16","option_d":"D. -16","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le discriminant de l'équation caractéristique r² + 4r + 4 = 0 est Δ = b² - 4ac = 16 - 16 = 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\": \"A. 0\", \"b\": \"B. 4\", \"c\": \"C. 16\", \"d\": \"D. -16\"}}","_debug_options_count":4},{"id":39663,"question":"Soit X une variable aléatoire suivant une loi normale N(μ, σ²). Quelle est la probabilité P(X ≤ μ) ?","option_a":"A. 0","option_b":"B. 0.5","option_c":"C. 1","option_d":"D. Impossible à déterminer","option_e":"","option_f":"","bonne_reponse":"b","explication":"Dans une loi normale centrée réduite, la probabilité que X soit inférieure ou égale à la moyenne μ est de 0.5, car la loi est symétrique.","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. 0.5\", \"c\": \"C. 1\", \"d\": \"D. Impossible à déterm","_debug_options_count":4},{"id":39664,"question":"La fonction f(x) = x³ - 3x est convexe sur l'intervalle ]-∞, 0[.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction f est convexe si sa dérivée seconde est positive. Ici, f''(x) = 6x, qui est négative sur ]-∞, 0[, donc la fonction est concave sur cet intervalle.","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":39665,"question":"Quel est le développement limité de ln(1+x) à l'ordre 2 au voisinage de 0 ?","option_a":"A. x - x²\/2","option_b":"B. x + x²\/2","option_c":"C. x - x²","option_d":"D. x + x²","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le développement limité de ln(1+x) à l'ordre 2 est ln(1+x) ≈ x - x²\/2 + o(x²).","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 - x²\/2\", \"b\": \"B. x + x²\/2\", \"c\": \"C. x - x²\", \"d\": \"D. x","_debug_options_count":4},{"id":39666,"question":"Soit une suite définie par u_{n+1} = 2u_n + 1 avec u_0 = 0. Quelle est la limite de cette suite ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. -1","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"Cette suite est arithmético-géométrique. Sa limite L vérifie L = 2L + 1, donc L = -1. Cependant, avec u_0 = 0, la suite converge vers -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. -1\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":39667,"question":"La fonction f(x) = e^{-x²} est-elle paire ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une fonction f est paire si f(-x) = f(x). Ici, f(-x) = e^{-(-x)²} = e^{-x²} = f(x), donc la fonction est paire.","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":39668,"question":"Quel est le coefficient directeur de la tangente à la courbe de f(x) = x² - 3x + 2 au point d'abscisse x = 1 ?","option_a":"A. -1","option_b":"B. 1","option_c":"C. 2","option_d":"D. -2","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le coefficient directeur de la tangente est donné par f'(x). Ici, f'(x) = 2x - 3, donc f'(1) = -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. -1\", \"b\": \"B. 1\", \"c\": \"C. 2\", \"d\": \"D. -2\"}}","_debug_options_count":4},{"id":39669,"question":"Soit une variable aléatoire X suivant une loi binomiale B(n, p). Quelle est son espérance ?","option_a":"A. np","option_b":"B. n(1-p)","option_c":"C. p(1-p)","option_d":"D. n²p","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'espérance d'une loi binomiale B(n, p) est E(X) = np.","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. np\", \"b\": \"B. n(1-p)\", \"c\": \"C. p(1-p)\", \"d\": \"D. n²p\"}}","_debug_options_count":4}]