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Question 1 sur 10 10:00

[{"id":53641,"question":"Quelle est la limite de la fonction f(x) = (3x² + 2x - 1)\/(x² + 1) lorsque x tend vers +∞ ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 3","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, 3x²\/x² = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":53642,"question":"La fonction f(x) = x³ - 3x² + 2 est convexe sur l'intervalle [-1, 3].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La convexité dépend du signe de la dérivée seconde : f''(x) = 6x - 6. f''(x) \u003C 0 pour x \u003C 1, donc la fonction n'est pas convexe sur [-1, 3].","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":53643,"question":"Soit une suite définie par u₀ = 2 et uₙ₊₁ = 0.5uₙ + 1. Que vaut u₂ ?","option_a":"A. 1.5","option_b":"B. 2","option_c":"C. 2.5","option_d":"D. 3","option_e":"","option_f":"","bonne_reponse":"A","explication":"u₁ = 0.5*2 + 1 = 2, u₂ = 0.5*2 + 1 = 2. La suite est constante à partir de u₁.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":53644,"question":"L'intégrale ∫₀¹ (2x + 1) dx vaut :","option_a":"A. 1","option_b":"B. 2","option_c":"C. 2.5","option_d":"D. 3","option_e":"","option_f":"","bonne_reponse":"C","explication":"L'intégrale d'une fonction affine se calcule avec la primitive : [x² + x]₀¹ = (1 + 1) - (0 + 0) = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":53645,"question":"La fonction f(x) = e^(-x²) est 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).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":53646,"question":"Quel est le discriminant de l'équation x² - 4x + 3 = 0 ?","option_a":"A. 0","option_b":"B. 4","option_c":"C. 16","option_d":"D. -4","option_e":"","option_f":"","bonne_reponse":"C","explication":"Le discriminant Δ = b² - 4ac = (-4)² - 4*1*3 = 16 - 12 = 4. Ici, Δ = 4 (et non 16).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":53647,"question":"Soit X une variable aléatoire suivant une loi normale N(0,1). P(X ≤ 1.96) est environ égal à :","option_a":"A. 0.5","option_b":"B. 0.95","option_c":"C. 0.975","option_d":"D. 0.99","option_e":"","option_f":"","bonne_reponse":"C","explication":"Pour une loi normale centrée réduite, P(X ≤ 1.96) ≈ 0.975 (valeur issue de la table de la loi normale).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":53648,"question":"La suite (uₙ) définie par uₙ = (-1)ⁿ est bornée.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Une suite est bornée si elle est majorée et minorée. Ici, uₙ ∈ {-1, 1} pour tout n, donc elle est bornée.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":53649,"question":"Quelle est la dérivée de f(x) = ln(x² + 1) ?","option_a":"A. 2x\/(x² + 1)","option_b":"B. 1\/(x² + 1)","option_c":"C. 2x","option_d":"D. x\/(x² + 1)","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de ln(u) est u'\/u. Ici, u = x² + 1, donc u' = 2x. Ainsi, f'(x) = 2x\/(x² + 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":53650,"question":"L'équation e^x = 5 admet une solution unique dans ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La fonction exponentielle est strictement croissante et bijective de ℝ vers ℝ⁺*, donc l'équation e^x = 5 admet une unique solution.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]