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Question 1 sur 10 10:00

[{"id":56751,"question":"Quelle est la limite de la fonction f(x) = (3x² + 2x - 5)\/(x² - 4) quand x tend vers l'infini ?","option_a":"A. 0","option_b":"B. 3","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"B","explication":"La limite d'un rapport 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":56752,"question":"La fonction f(x) = x³ - 3x + 2 est strictement croissante sur l'intervalle [-1, 1].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée f'(x) = 3x² - 3 est négative sur [-1, 1] (car x² ≤ 1), donc la fonction est décroissante sur cet intervalle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":56753,"question":"Quel est le discriminant de l'équation x² - 5x + 6 = 0 ?","option_a":"A. 1","option_b":"B. -1","option_c":"C. 13","option_d":"D. 25","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le discriminant Δ = b² - 4ac = (-5)² - 4*1*6 = 25 - 24 = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":56754,"question":"La probabilité de tirer un as dans un jeu de 52 cartes est de 1\/13.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Il y a 4 as dans un jeu de 52 cartes, donc P(as) = 4\/52 = 1\/13.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":56755,"question":"Quelle est la dérivée de la fonction f(x) = ln(2x + 1) ?","option_a":"A. 2\/(2x + 1)","option_b":"B. 1\/(2x + 1)","option_c":"C. 2x\/(2x + 1)","option_d":"D. 1\/x","option_e":"","option_f":"","bonne_reponse":"A","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},{"id":56756,"question":"L'équation de la tangente à la courbe y = x² + 3x - 2 au point d'abscisse x = 1 est y = 5x - 3.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La tangente en x = 1 a pour équation y = f'(1)(x - 1) + f(1). f(1) = 2 et f'(x) = 2x + 3, donc f'(1) = 5. Ainsi, y = 5(x - 1) + 2 = 5x - 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":56757,"question":"Quelle est la solution de l'inéquation x² - 4x + 3 \u003C 0 ?","option_a":"A. x ∈ ]1, 3[","option_b":"B. x ∈ ]-∞, 1[ ∪ ]3, +∞[","option_c":"C. x ∈ {1, 3}","option_d":"D. x ∈ ]-∞, +∞[","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'inéquation x² - 4x + 3 \u003C 0 est vérifiée entre les racines de l'équation x² - 4x + 3 = 0, qui sont x = 1 et x = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":56758,"question":"La fonction f(x) = e^(2x) est une solution de l'équation différentielle y' = 2y.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de e^(2x) est 2e^(2x) = 2f(x), donc f(x) est bien solution de y' = 2y.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":56759,"question":"Quel est le centre de symétrie de la fonction f(x) = (x + 1)\/(x - 2) ?","option_a":"A. (0, 0)","option_b":"B. (2, 1)","option_c":"C. (-1, 0)","option_d":"D. (1, -1)","option_e":"","option_f":"","bonne_reponse":"D","explication":"Le centre de symétrie d'une fonction homographique f(x) = (ax + b)\/(cx + d) est le point d'intersection des asymptotes, ici (-d\/c, a\/c) = (2, 1). Cependant, en vérifiant, le centre est bien (-1, 0) pour cette fonction.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":56760,"question":"La suite définie par uₙ = 2n + 1 est arithmétique de raison 2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La différence entre deux termes consécutifs est uₙ₊₁ - uₙ = 2(n+1) + 1 - (2n + 1) = 2, donc la suite est arithmétique de raison 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]