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[{"id":25498,"question":"Quelle est la limite de la fonction f(x) = (2x² + 3x - 5)\/(x² - 4) lorsque x tend vers +∞ ?","option_a":"A. 0","option_b":"B. 2","option_c":"C. +∞","option_d":"D. -2","option_e":"","option_f":"","bonne_reponse":"B","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},{"id":25499,"question":"Soit la suite définie par u₀ = 1 et uₙ₊₁ = 2uₙ + 1. 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, u₁ - u₀ = 3 et u₂ - u₁ = 7, donc elle n'est pas arithmétique.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":25500,"question":"Quelle est la dérivée de la fonction f(x) = ln(3x² + 2x) ?","option_a":"A. (6x + 2)\/(3x² + 2x)","option_b":"B. (3x + 1)\/(3x² + 2x)","option_c":"C. (6x + 2)\/(x² + 2x\/3)","option_d":"D. (6x + 2)\/(3x² + 2)","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de ln(u) est u'\/u. Ici, u = 3x² + 2x, donc u' = 6x + 2. La dérivée est donc (6x + 2)\/(3x² + 2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":25501,"question":"L'équation x² - 5x + 6 = 0 admet-elle deux solutions réelles distinctes ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le discriminant Δ = 25 - 24 = 1 \u003E 0, donc l'équation admet deux solutions réelles distinctes : x = 2 et x = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":25502,"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. Entre 0 et 1, cela donne (1 + 1) - (0 + 0) = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":25503,"question":"La fonction f(x) = x³ - 3x 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 - 3 s'annule en x = ±1. La fonction est décroissante sur [-1, 1], donc elle n'est pas croissante sur ℝ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":25504,"question":"Quelle est la valeur de la somme S = 1 + 2 + 4 + 8 + ... + 2ⁿ ?","option_a":"A. 2ⁿ⁺¹ - 1","option_b":"B. 2ⁿ - 1","option_c":"C. 2ⁿ⁺¹ + 1","option_d":"D. 2ⁿ⁺¹","option_e":"","option_f":"","bonne_reponse":"A","explication":"C'est une suite géométrique de raison 2. La somme est S = 2ⁿ⁺¹ - 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":25505,"question":"L'équation cos(x) = 0 admet-elle des solutions dans l'intervalle [0, π] ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"cos(π\/2) = 0, donc l'équation admet une solution dans [0, π].","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":25506,"question":"Quelle est la valeur de la limite lim (x→0) (sin(x)\/x) ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. +∞","option_d":"D. -1","option_e":"","option_f":"","bonne_reponse":"B","explication":"Cette limite est une limite fondamentale en mathématiques et vaut 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":25507,"question":"La fonction f(x) = eˣ 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) = eˣ \u003E 0 pour tout x, donc la fonction est convexe sur ℝ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]