Quiz — 276041995_3123050364573793_3931285017647909232_n.pdf

Question 1 sur 10 20:00

[{"id":47518,"question":"Quelle est la limite de la fonction f(x) = (3x² + 2x - 1)\/(x² - 4) lorsque x tend vers +∞ ?","option_a":"A. 3","option_b":"B. 0","option_c":"C. -∞","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"La limite d'un quotient de polynômes lorsque x tend vers l'infini est égale au rapport des coefficients dominants. 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\": \"a\", \"options\": {\"a\": \"A. 3\", \"b\": \"B. 0\", \"c\": \"C. -∞\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":47519,"question":"Soit une suite définie par uₙ₊₁ = 2uₙ - 1 avec u₀ = 3. Cette suite est-elle croissante ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Calculons u₁ = 2*3 - 1 = 5 \u003E u₀. La suite est donc croissante.","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":47520,"question":"Quelle est la dérivée de la fonction f(x) = ln(2x + 1) ?","option_a":"A. 1\/(2x + 1)","option_b":"B. 2\/(2x + 1)","option_c":"C. 1\/(x + 1)","option_d":"D. 2\/(x + 1)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de ln(u) est u'\/u. Ici, u = 2x + 1, donc u' = 2. Ainsi, f'(x) = 2\/(2x + 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. 1\/(2x + 1)\", \"b\": \"B. 2\/(2x + 1)\", \"c\": \"C. 1\/(x + 1)\", \"d\": \"","_debug_options_count":4},{"id":47521,"question":"La fonction f(x) = x³ - 3x² + 4 admet-elle un extremum local en x = 2 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"f'(x) = 3x² - 6x. f'(2) = 0, mais f''(x) = 6x - 6. f''(2) = 6 \u003E 0, donc x = 2 est un minimum local.","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":47522,"question":"Quelle est l'intégrale de f(x) = 3x² + 2x entre 0 et 1 ?","option_a":"A. 1","option_b":"B. 2","option_c":"C. 3","option_d":"D. 4","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégrale de 3x² + 2x est x³ + x². Évaluée entre 0 et 1, on obtient (1 + 1) - (0 + 0) = 2.","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\", \"b\": \"B. 2\", \"c\": \"C. 3\", \"d\": \"D. 4\"}}","_debug_options_count":4},{"id":47523,"question":"Soit Z = 3 + 4i. Quel est le module de Z ?","option_a":"A. 5","option_b":"B. 7","option_c":"C. √7","option_d":"D. 25","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module de Z = a + bi est √(a² + b²). Ici, √(3² + 4²) = √(9 + 16) = √25 = 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\": \"A. 5\", \"b\": \"B. 7\", \"c\": \"C. √7\", \"d\": \"D. 25\"}}","_debug_options_count":4},{"id":47524,"question":"La fonction exponentielle est-elle toujours positive sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour tout x réel, eˣ \u003E 0. La fonction exponentielle est donc toujours positive.","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":47525,"question":"Quelle est la solution de l'équation différentielle y' + 2y = 0 ?","option_a":"A. y = Ce²ˣ","option_b":"B. y = Ce⁻²ˣ","option_c":"C. y = C + e²ˣ","option_d":"D. y = Cx + e⁻²ˣ","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'équation y' + 2y = 0 a pour solution générale y = Ce⁻²ˣ, où C est une constante réelle.","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 = Ce²ˣ\", \"b\": \"B. y = Ce⁻²ˣ\", \"c\": \"C. y = C + e²ˣ\",","_debug_options_count":4},{"id":47526,"question":"Soit X une variable aléatoire suivant une loi binomiale B(10, 0.3). Quelle est l'espérance de X ?","option_a":"A. 3","option_b":"B. 0.3","option_c":"C. 10","option_d":"D. 7","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'espérance d'une loi binomiale B(n, p) est n*p. Ici, 10*0.3 = 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. 3\", \"b\": \"B. 0.3\", \"c\": \"C. 10\", \"d\": \"D. 7\"}}","_debug_options_count":4},{"id":47527,"question":"La fonction f(x) = x⁴ - 4x³ + 6x² est-elle convexe sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"f''(x) = 12x² - 24x + 12 = 12(x² - 2x + 1) = 12(x-1)² ≥ 0. La fonction est donc convexe sur ℝ.","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}]