Maths Terminale : Testez vos connaissances en analyse, dérivées et probabilités

Question 1 sur 10 10:00

[{"id":47511,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"A) 6x + 2","option_b":"B) 3x + 2","option_c":"C) 6x² + 2x","option_d":"D) 3x² + 2","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de f(x) = 3x² + 2x - 5 est f'(x) = 6x + 2, car la dérivée de x² est 2x et celle de x est 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":47512,"question":"La fonction f(x) = x³ - 3x est décroissante sur l'intervalle [-1, 1].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La dérivée f'(x) = 3x² - 3 est négative sur [-1, 1] (car 3x² ≤ 3), donc la fonction est décroissante sur cet intervalle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":47513,"question":"Quelle est la limite de la suite uₙ = (2n + 1)\/(n + 3) quand n tend vers l'infini ?","option_a":"A) 0","option_b":"B) 2","option_c":"C) 1","option_d":"D) +∞","option_e":"","option_f":"","bonne_reponse":"C","explication":"En divisant numérateur et dénominateur par n, on obtient uₙ = (2 + 1\/n)\/(1 + 3\/n), donc la limite est 2\/1 = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":47514,"question":"Soit X une variable aléatoire suivant une loi normale N(0,1). Quelle est P(X ≤ 1) ?","option_a":"A) 0.6826","option_b":"B) 0.8413","option_c":"C) 0.9544","option_d":"D) 0.9974","option_e":"","option_f":"","bonne_reponse":"B","explication":"Pour une loi normale centrée réduite, P(X ≤ 1) ≈ 0.8413 (valeur lue dans la table de la loi normale).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":47515,"question":"La suite définie par u₀ = 1 et uₙ₊₁ = 2uₙ + 1 est convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Cette suite est croissante et non majorée (uₙ = 2ⁿ - 1), donc elle diverge vers +∞.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":47516,"question":"Quelle est la primitive de f(x) = 4x³ - 2x + 1 ?","option_a":"A) x⁴ - x² + x + C","option_b":"B) 12x² - 2 + C","option_c":"C) x⁴ - x² + C","option_d":"D) 4x⁴ - 2x² + x + C","option_e":"","option_f":"","bonne_reponse":"A","explication":"La primitive de 4x³ est x⁴, celle de -2x est -x², et celle de 1 est x. On ajoute la constante C.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":47517,"question":"Soit A et B deux événements indépendants avec P(A) = 0.4 et P(B) = 0.5. Quelle est P(A ∩ B) ?","option_a":"A) 0.2","option_b":"B) 0.9","option_c":"C) 0.1","option_d":"D) 0.5","option_e":"","option_f":"","bonne_reponse":"A","explication":"Pour des événements indépendants, P(A ∩ B) = P(A) × P(B) = 0.4 × 0.5 = 0.2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":47518,"question":"La fonction f(x) = e^x est convexe sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée seconde de f(x) = e^x est f''(x) = e^x \u003E 0 pour tout x, donc la fonction est convexe.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":47519,"question":"Quelle est la solution de l'équation différentielle y' + 2y = 0 ?","option_a":"A) y = Ce^{-2x}","option_b":"B) y = Ce^{2x}","option_c":"C) y = Cx","option_d":"D) y = C","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'équation différentielle y' + 2y = 0 a pour solution générale y = Ce^{-2x}, où C est une constante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":47520,"question":"Soit X une variable aléatoire discrète avec P(X=1) = 0.3, P(X=2) = 0.5 et P(X=3) = 0.2. Quelle est E(X) ?","option_a":"A) 1.9","option_b":"B) 2.0","option_c":"C) 2.1","option_d":"D) 2.2","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'espérance E(X) = 1×0.3 + 2×0.5 + 3×0.2 = 0.3 + 1.0 + 0.6 = 1.9.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]