Quiz — P-EX06-18-OB.pdf

Question 1 sur 10 20:00

[{"id":45848,"question":"Quelle est la limite de la fonction f(x) = (2x² + 3x - 5)\/(x² + 1) lorsque x tend vers +∞ ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 2","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"La limite d'un quotient de polynômes de même degré est égale au rapport des coefficients dominants. Ici, 2x²\/x² = 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. +∞\"}}","_debug_options_count":4},{"id":45849,"question":"Soit une suite définie par u₀ = 1 et uₙ₊₁ = 2uₙ + 3. Cette suite est-elle arithmétique ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une suite arithmétique a une différence constante entre deux termes consécutifs. Ici, uₙ₊₁ - uₙ = uₙ + 3, qui n'est pas constant.","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":45850,"question":"Quelle est la dérivée de la fonction f(x) = ln(3x² + 2x) ?","option_a":"A. (6x + 2)\/(3x² + 2x)","option_b":"B. (3x² + 2x)\/(6x + 2)","option_c":"C. (6x + 2)\/x","option_d":"D. (3x² + 2x)\/x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(u) est u'\/u. Ici, u = 3x² + 2x, donc u' = 6x + 2.","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. (6x + 2)\/(3x² + 2x)\", \"b\": \"B. (3x² + 2x)\/(6x + 2)\", \"c\": \"C","_debug_options_count":4},{"id":45851,"question":"La fonction f(x) = x³ - 3x² + 4 est-elle croissante sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f'(x) = 3x² - 6x = 3x(x - 2) change de signe (négative entre 0 et 2), donc la fonction n'est pas croissante 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},{"id":45852,"question":"Quelle est la valeur de l'intégrale ∫₀¹ (2x + 1) dx ?","option_a":"A. 1","option_b":"B. 2","option_c":"C. 3","option_d":"D. 4","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'intégrale d'une fonction affine est une fonction quadratique. ∫(2x + 1)dx = x² + x. Évaluée entre 0 et 1, cela donne (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\": \"c\", \"options\": {\"a\": \"A. 1\", \"b\": \"B. 2\", \"c\": \"C. 3\", \"d\": \"D. 4\"}}","_debug_options_count":4},{"id":45853,"question":"Soit A = (1 2; 3 4) une matrice 2x2. Quel est le déterminant de A ?","option_a":"A. -2","option_b":"B. 0","option_c":"C. 2","option_d":"D. 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le déterminant d'une matrice 2x2 (a b; c d) est ad - bc. Ici, (1*4) - (2*3) = 4 - 6 = -2.","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. -2\", \"b\": \"B. 0\", \"c\": \"C. 2\", \"d\": \"D. 5\"}}","_debug_options_count":4},{"id":45854,"question":"Une variable aléatoire X suit une loi normale N(0,1). Quelle est la probabilité P(X ≤ 1) ?","option_a":"A. 0,68","option_b":"B. 0,84","option_c":"C. 0,95","option_d":"D. 0,99","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour une loi normale centrée réduite, P(X ≤ 1) ≈ 0,8413 (valeur lue dans la table 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. 0,68\", \"b\": \"B. 0,84\", \"c\": \"C. 0,95\", \"d\": \"D. 0,99\"}}","_debug_options_count":4},{"id":45855,"question":"La fonction f(x) = e^(x²) est-elle convexe sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée seconde f''(x) = (2 + 4x²)e^(x²) est toujours positive, donc la fonction est convexe sur ℝ.","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":45856,"question":"Quelle est la solution de l'équation différentielle y' + 2y = 0 avec y(0) = 3 ?","option_a":"A. y = 3e^(-2x)","option_b":"B. y = 3e^(2x)","option_c":"C. y = -3e^(-2x)","option_d":"D. y = 6e^(-x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation différentielle linéaire du premier ordre a pour solution générale y = Ce^(-2x). Avec y(0) = 3, on trouve C = 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. y = 3e^(-2x)\", \"b\": \"B. y = 3e^(2x)\", \"c\": \"C. y = -3e^(-2x)\",","_debug_options_count":4},{"id":45857,"question":"Soit une suite géométrique de premier terme u₀ = 5 et de raison q = 2. Quelle est la valeur de u₅ ?","option_a":"A. 10","option_b":"B. 160","option_c":"C. 320","option_d":"D. 640","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le terme général d'une suite géométrique est uₙ = u₀ * qⁿ. Donc u₅ = 5 * 2⁵ = 5 * 32 = 160.","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. 10\", \"b\": \"B. 160\", \"c\": \"C. 320\", \"d\": \"D. 640\"}}","_debug_options_count":4}]