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Question 1 sur 10 10:00

[{"id":13643,"question":"Quelle est la limite de la fonction f(x) = (3x² + 2x - 1)\/(x² - 4) lorsque x tend vers l'infini ?","option_a":"A. 3","option_b":"B. 0","option_c":"C. +∞","option_d":"D. -1","option_e":"","option_f":"","bonne_reponse":"A","explication":"La limite d'un quotient de polynômes lorsque x tend vers l'infini est donnée par le rapport des coefficients des termes de plus haut degré. Ici, 3x²\/x² = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":13644,"question":"La dérivée de la fonction f(x) = e^(2x) est f'(x) = 2e^(2x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de e^(u(x)) est u'(x)e^(u(x)). Ici, u(x) = 2x, donc u'(x) = 2. Ainsi, f'(x) = 2e^(2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":13645,"question":"Quel est le produit scalaire de deux vecteurs u et v orthogonaux ?","option_a":"A. 0","option_b":"B. ||u|| * ||v||","option_c":"C. 1","option_d":"D. -1","option_e":"","option_f":"","bonne_reponse":"A","explication":"Deux vecteurs orthogonaux ont un produit scalaire égal à 0, car cos(90°) = 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":13646,"question":"La fonction f(x) = x³ - 3x + 1 admet-elle un extremum local en x = 1 ?","option_a":"A. Oui, un maximum","option_b":"B. Oui, un minimum","option_c":"C. Non","option_d":"D. Oui, un point d'inflexion","option_e":"","option_f":"","bonne_reponse":"B","explication":"La dérivée f'(x) = 3x² - 3 s'annule en x = 1. La dérivée seconde f''(x) = 6x est positive en x = 1, donc c'est un minimum local.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":13647,"question":"L'équation différentielle y' + 2y = 0 a pour solution générale y = Ce^(-2x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'équation différentielle linéaire du premier ordre y' + ay = 0 a pour solution générale y = Ce^(-ax). Ici, a = 2, donc y = Ce^(-2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":13648,"question":"Quelle est la valeur de l'intégrale ∫(0 à 1) x² dx ?","option_a":"A. 1\/3","option_b":"B. 1","option_c":"C. 0","option_d":"D. 2","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'intégrale de x² entre 0 et 1 est [x³\/3] de 0 à 1 = 1\/3 - 0 = 1\/3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":13649,"question":"Dans un triangle ABC, si AB = 5, AC = 7 et BC = 8, quel est l'angle en A arrondi au degré près ?","option_a":"A. 30°","option_b":"B. 45°","option_c":"C. 60°","option_d":"D. 75°","option_e":"","option_f":"","bonne_reponse":"D","explication":"En utilisant la loi des cosinus : cos(A) = (AB² + AC² - BC²)\/(2*AB*AC) = (25 + 49 - 64)\/70 = 10\/70 ≈ 0,142. A ≈ arccos(0,142) ≈ 81,8°, arrondi à 75° près.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":13650,"question":"La suite définie par uₙ = (n² + 1)\/n converge-t-elle ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La suite uₙ = (n² + 1)\/n = n + 1\/n tend vers +∞ lorsque n tend vers l'infini, donc elle ne converge pas.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":13651,"question":"Quel est le coefficient directeur de la tangente à la courbe y = ln(x) au point d'abscisse x = 1 ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. e","option_d":"D. -1","option_e":"","option_f":"","bonne_reponse":"B","explication":"La dérivée de ln(x) est 1\/x. En x = 1, la pente de la tangente est 1\/1 = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":13652,"question":"Si P(A) = 0,3 et P(B) = 0,5, avec A et B indépendants, quelle est P(A ∪ B) ?","option_a":"A. 0,65","option_b":"B. 0,8","option_c":"C. 0,15","option_d":"D. 0,2","option_e":"","option_f":"","bonne_reponse":"B","explication":"Pour deux événements indépendants, P(A ∪ B) = P(A) + P(B) - P(A)*P(B) = 0,3 + 0,5 - 0,15 = 0,65.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]