Maths BAC : Testez vos connaissances en algèbre et analyse

Question 1 sur 10 10:00

[{"id":10713,"question":"Quelle est la solution de l'équation 2x² - 5x + 2 = 0 ?","option_a":"x = 1 ou x = 2","option_b":"x = 0,5 ou x = 2","option_c":"x = -1 ou x = 2","option_d":"x = 0,5 ou x = -2","option_e":"","option_f":"","bonne_reponse":"B","explication":"Résolvez l'équation en utilisant la formule du discriminant : Δ = b² - 4ac = 25 - 16 = 9. Les solutions sont x = (5 ± 3)\/4, soit x = 2 ou x = 0,5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10714,"question":"La fonction f(x) = x³ - 3x² + 4 est croissante sur l'intervalle [0 ; 2].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Calculez la dérivée f'(x) = 3x² - 6x. Sur [0 ; 2], f'(x) = 3x(x-2) ≤ 0, donc la fonction est décroissante sur cet intervalle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10715,"question":"Quelle est la limite de (3x² + 2x - 1)\/(x² - 4) quand x tend vers +∞ ?","option_a":"3","option_b":"0","option_c":"1","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"A","explication":"Divisez le numérateur et le dénominateur par x² : (3 + 2\/x - 1\/x²)\/(1 - 4\/x²). Quand x → +∞, les termes en 1\/x tendent vers 0, donc la limite est 3\/1 = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10716,"question":"L'équation ln(x) = 2 admet pour solution x = e².","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Par définition du logarithme, ln(x) = 2 équivaut à x = e². La solution est donc correcte.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10717,"question":"Quelle est la dérivée de la fonction f(x) = (2x + 1)\/(x - 3) ?","option_a":"f'(x) = 2\/(x-3)²","option_b":"f'(x) = (2x - 7)\/(x-3)²","option_c":"f'(x) = (2x + 7)\/(x-3)²","option_d":"f'(x) = -2\/(x-3)²","option_e":"","option_f":"","bonne_reponse":"B","explication":"Utilisez la formule de dérivation d'un quotient : (u\/v)' = (u'v - uv')\/v². Ici, u = 2x + 1, v = x - 3, donc u' = 2, v' = 1. f'(x) = (2(x-3) - (2x+1)(1))\/(x-3)² = (2x - 6 - 2x - 1)\/(x-3)² = -7\/(x-3)².","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10718,"question":"La fonction f(x) = √(x² + 1) est définie pour tout x réel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La racine carrée est définie pour tout x réel car x² + 1 ≥ 1 \u003E 0. La fonction est donc définie sur ℝ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10719,"question":"Quelle est la valeur de l'intégrale ∫(2x + 3) dx entre 0 et 1 ?","option_a":"5","option_b":"4","option_c":"6","option_d":"3","option_e":"","option_f":"","bonne_reponse":"A","explication":"Calculez l'intégrale : ∫(2x + 3) dx = x² + 3x. Évaluez entre 0 et 1 : (1 + 3) - (0 + 0) = 4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10720,"question":"La suite uₙ = n²\/(n+1) est bornée.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Pour n ≥ 1, uₙ = n²\/(n+1) = n - 1 + 1\/(n+1). Quand n → +∞, uₙ → +∞, donc la suite n'est pas bornée.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10721,"question":"Quelle est la solution de l'inéquation x² - 4x + 3 ≤ 0 ?","option_a":"x ∈ [1 ; 3]","option_b":"x ∈ ]-∞ ; 1] ∪ [3 ; +∞[","option_c":"x ∈ [0 ; 4]","option_d":"x ∈ ]-∞ ; +∞[","option_e":"","option_f":"","bonne_reponse":"A","explication":"Résolvez l'équation x² - 4x + 3 = 0 : solutions x = 1 et x = 3. Le trinôme est négatif entre les racines, donc x ∈ [1 ; 3].","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10722,"question":"La fonction f(x) = e^(2x) a pour dérivée 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}]