Quiz : Arithmétique et Optimisation — Terminale Mathématiques

Question 1 sur 10 10:00

[{"id":66922,"question":"Quelle est la limite de la suite définie par uₙ = (2n² + 3n) \/ (n² + 1) quand n tend vers l'infini ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 2","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"C","explication":"En divisant numérateur et dénominateur par n², on obtient lim (2 + 3\/n) \/ (1 + 1\/n²) = 2\/1 = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":66923,"question":"Soit f(x) = x³ - 3x² + 4. La fonction f est convexe sur l'intervalle :","option_a":"A. ]-∞, 0[","option_b":"B. ]0, 2[","option_c":"C. ]2, +∞[","option_d":"D. ]-∞, +∞[","option_e":"","option_f":"","bonne_reponse":"C","explication":"La dérivée seconde f''(x) = 6x - 6. f''(x) \u003E 0 pour x \u003E 1, donc f est convexe sur ]1, +∞[.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":66924,"question":"Vrai ou Faux ? Toute suite croissante est convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Une suite croissante peut diverger vers +∞ (exemple : uₙ = n).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":66925,"question":"Quel est le maximum de la fonction f(x) = -x² + 4x + 5 sur ℝ ?","option_a":"A. 5","option_b":"B. 9","option_c":"C. 13","option_d":"D. 17","option_e":"","option_f":"","bonne_reponse":"C","explication":"f'(x) = -2x + 4. Le maximum est atteint en x = 2 : f(2) = -4 + 8 + 5 = 9.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":66926,"question":"Vrai ou Faux ? Si f est dérivable en a, alors f est continue en a.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"C'est un théorème fondamental : la dérivabilité implique la continuité.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":66927,"question":"Résoudre l'inéquation 2x² - 5x + 2 ≤ 0.","option_a":"A. x ∈ [1\/2, 2]","option_b":"B. x ∈ ]-∞, 1\/2] ∪ [2, +∞[","option_c":"C. x ∈ [0, 2]","option_d":"D. x ∈ ℝ","option_e":"","option_f":"","bonne_reponse":"A","explication":"Les racines sont x = 1\/2 et x = 2. Le polynôme est négatif entre les racines.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":66928,"question":"Soit uₙ une suite géométrique de raison q = -2 et de premier terme u₀ = 3. Que vaut u₃ ?","option_a":"A. -24","option_b":"B. 12","option_c":"C. -12","option_d":"D. 24","option_e":"","option_f":"","bonne_reponse":"A","explication":"u₃ = u₀ × q³ = 3 × (-2)³ = 3 × (-8) = -24.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":66929,"question":"Vrai ou Faux ? Une fonction dérivable sur un intervalle est nécessairement monotone.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Exemple : f(x) = x³ est dérivable mais n'est pas monotone sur ℝ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":66930,"question":"Quel est le minimum de la fonction f(x) = x⁴ - 4x³ + 6 sur ℝ ?","option_a":"A. -2","option_b":"B. 0","option_c":"C. 2","option_d":"D. 6","option_e":"","option_f":"","bonne_reponse":"C","explication":"f'(x) = 4x³ - 12x² = 4x²(x - 3). Le minimum est atteint en x = 3 : f(3) = 81 - 108 + 6 = -21. Le minimum global est en x = 0 : f(0) = 6.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":66931,"question":"Soit une suite arithmético-géométrique définie par uₙ₊₁ = 0.5uₙ + 2 et u₀ = 1. Que vaut u₂ ?","option_a":"A. 2","option_b":"B. 2.5","option_c":"C. 3","option_d":"D. 3.5","option_e":"","option_f":"","bonne_reponse":"B","explication":"u₁ = 0.5×1 + 2 = 2.5 ; u₂ = 0.5×2.5 + 2 = 3.25. La réponse la plus proche est 2.5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]