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Question 1 sur 10 10:00

[{"id":23078,"question":"Quelle est la limite de la fonction f(x) = (3x² - 2x + 1)\/(x² + 1) lorsque x tend vers l'infini ?","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 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":23079,"question":"L'équation différentielle y' + 2y = 0 a pour solution générale :","option_a":"A. y = Ce^(2x)","option_b":"B. y = Ce^(-2x)","option_c":"C. y = Ce^(x\/2)","option_d":"D. y = Ce^(-x\/2)","option_e":"","option_f":"","bonne_reponse":"B","explication":"L'équation différentielle linéaire du premier ordre y' + ay = 0 a pour solution y = Ce^(-ax). Ici, a = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":23080,"question":"Vrai ou Faux ? La dérivée de la fonction f(x) = ln(x² + 1) est f'(x) = 2x\/(x² + 1).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de ln(u) est u'\/u. Ici, u = x² + 1, donc u' = 2x. La dérivée est bien 2x\/(x² + 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":23081,"question":"Quel est le produit scalaire des vecteurs u(2, -1, 3) et v(1, 4, -2) ?","option_a":"A. -4","option_b":"B. 0","option_c":"C. 4","option_d":"D. 8","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le produit scalaire se calcule par u·v = 2*1 + (-1)*4 + 3*(-2) = 2 - 4 - 6 = -8. Aucune option ne correspond, vérifiez les données.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":23082,"question":"Vrai ou Faux ? Toute fonction continue sur un intervalle fermé est dérivable sur cet intervalle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Une fonction continue n'est pas nécessairement dérivable (exemple : f(x) = |x| en x = 0).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":23083,"question":"Quelle est la solution de l'inéquation 2x² - 5x + 2 \u003C 0 ?","option_a":"A. x ∈ ]-∞, 0,5[","option_b":"B. x ∈ ]1, 2[","option_c":"C. x ∈ ]-∞, 1[ ∪ ]2, +∞[","option_d":"D. x ∈ ]0,5, 2[","option_e":"","option_f":"","bonne_reponse":"B","explication":"Les racines sont x = 0,5 et x = 2. Le polynôme est négatif entre les racines (coefficient dominant positif).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":23084,"question":"Quelle est la probabilité d'obtenir un double 6 en lançant deux dés équilibrés ?","option_a":"A. 1\/6","option_b":"B. 1\/12","option_c":"C. 1\/36","option_d":"D. 1\/18","option_e":"","option_f":"","bonne_reponse":"C","explication":"Il y a 6*6 = 36 issues possibles. Une seule issue favorable (6,6), donc probabilité = 1\/36.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":23085,"question":"Vrai ou Faux ? La matrice A = [[1, 2], [3, 4]] est inversible.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le déterminant de A est 1*4 - 2*3 = -2 ≠ 0, donc A est inversible.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":23086,"question":"Quelle est la primitive de f(x) = 3x² - 4x + 5 qui s'annule en x = 1 ?","option_a":"A. x³ - 2x² + 5x - 4","option_b":"B. x³ - 2x² + 5x + 4","option_c":"C. x³ - 2x² + 5x - 5","option_d":"D. x³ - 2x² + 5x","option_e":"","option_f":"","bonne_reponse":"A","explication":"La primitive générale est F(x) = x³ - 2x² + 5x + C. F(1) = 1 - 2 + 5 + C = 0 ⇒ C = -4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":23087,"question":"Quelle est l'équation de la tangente à la courbe y = x² - 3x + 2 au point d'abscisse x = 2 ?","option_a":"A. y = x - 2","option_b":"B. y = x + 2","option_c":"C. y = 2x - 3","option_d":"D. y = x - 4","option_e":"","option_f":"","bonne_reponse":"A","explication":"La tangente a pour équation y = f'(2)(x-2) + f(2). f'(x) = 2x - 3 ⇒ f'(2) = 1. f(2) = 0. Donc y = 1*(x-2) + 0 = x - 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]