Quiz Maths Terminale : Devoir de Synthèse n°1 — 1er Trimestre

Question 1 sur 10 10:00

[{"id":5545,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"6x² + 2","option_c":"3x + 2","option_d":"3x² + 2x","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée d'une fonction polynôme se calcule terme par terme : (3x²)' = 6x, (2x)' = 2, et (-5)' = 0. Donc f'(x) = 6x + 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":5546,"question":"La limite de la suite uₙ = (2n + 1)\/(n + 3) quand n tend vers +∞ est égale à 2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"En divisant numérateur et dénominateur par n, on obtient uₙ = (2 + 1\/n)\/(1 + 3\/n). Quand n → +∞, 1\/n → 0, donc uₙ → 2\/1 = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":5547,"question":"Quelle est la formule de l'intégrale de f(x) = x² entre 0 et 1 ?","option_a":"1\/3","option_b":"1\/2","option_c":"1","option_d":"2","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'intégrale de x² est (x³)\/3. Entre 0 et 1, cela donne (1³)\/3 - (0³)\/3 = 1\/3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":5548,"question":"Si A et B sont deux événements indépendants, alors P(A ∩ B) = P(A) × P(B).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Par définition, deux événements sont indépendants si et seulement si P(A ∩ B) = P(A) × P(B).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":5549,"question":"Résoudre l'équation différentielle y' + 2y = 0.","option_a":"y = Ce^(-2x)","option_b":"y = Ce^(2x)","option_c":"y = C + e^(-2x)","option_d":"y = Cx^(-2)","option_e":"","option_f":"","bonne_reponse":"A","explication":"C'est une équation différentielle linéaire du 1er ordre. La solution générale est y = Ce^(-2x), où C est une constante réelle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":5550,"question":"La suite définie par u₀ = 1 et uₙ₊₁ = 2uₙ + 1 est croissante.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Calculons les premiers termes : u₁ = 3, u₂ = 7, u₃ = 15. La suite est strictement croissante car uₙ₊₁ - uₙ = uₙ + 1 \u003E 0 pour tout n.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":5551,"question":"Quelle est la primitive de f(x) = 4x³ qui s'annule en x = 1 ?","option_a":"x⁴ - 1","option_b":"x⁴","option_c":"4x⁴","option_d":"x⁴ + 1","option_e":"","option_f":"","bonne_reponse":"A","explication":"La primitive de 4x³ est x⁴ + C. Pour qu'elle s'annule en x = 1, on pose 1⁴ + C = 0 ⇒ C = -1. Donc la primitive est x⁴ - 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":5552,"question":"Si P(A) = 0.4 et P(B) = 0.5, alors P(A ∪ B) ≤ 0.9.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"D'après la formule des probabilités totales, P(A ∪ B) = P(A) + P(B) - P(A ∩ B). Comme P(A ∩ B) ≥ 0, alors P(A ∪ B) ≤ 0.4 + 0.5 = 0.9.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":5553,"question":"Quelle est la limite de la fonction f(x) = (x² - 1)\/(x - 1) quand x tend vers 1 ?","option_a":"0","option_b":"1","option_c":"2","option_d":"La limite n'existe pas","option_e":"","option_f":"","bonne_reponse":"B","explication":"En factorisant le numérateur, on obtient f(x) = (x - 1)(x + 1)\/(x - 1) = x + 1 pour x ≠ 1. Donc lim(x→1) f(x) = 1 + 1 = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":5554,"question":"La fonction exponentielle est toujours positive sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Par définition, e^x \u003E 0 pour tout x réel. La fonction exponentielle est strictement positive sur ℝ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]