Maths Terminale : Testez vos connaissances en analyse et algèbre

Question 1 sur 10 10:00

[{"id":53781,"question":"Soit la fonction f définie par f(x) = x²e^(-x). Quelle est la limite de f(x) quand x tend vers +∞ ?","option_a":"A. 0","option_b":"B. +∞","option_c":"C. -∞","option_d":"D. 1","option_e":"","option_f":"","bonne_reponse":"A","explication":"Par croissance comparée, x² croît moins vite que e^(-x) ne décroît. Donc lim(x→+∞) f(x) = 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":53782,"question":"L'équation différentielle y' + 2y = 0 admet pour solution générale y(x) = Ce^(-2x) où C est une constante réelle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"C'est une équation différentielle linéaire du 1er ordre à coefficients constants. La solution générale est bien y(x) = Ce^(-2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":53783,"question":"Quelle est la dérivée de la fonction g(x) = ln(3x² + 1) ?","option_a":"A. 6x\/(3x²+1)","option_b":"B. 3x\/(3x²+1)","option_c":"C. 6x²\/(3x²+1)","option_d":"D. 1\/(3x²+1)","option_e":"","option_f":"","bonne_reponse":"A","explication":"En utilisant la dérivée d'une fonction composée : g'(x) = (6x)\/(3x²+1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":53784,"question":"Soit P(x) = x³ - 4x² + 5x - 2. Le polynôme P admet une racine double en x=1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"P(1) = 0 et P'(1) = 3(1)² - 8(1) + 5 = 0. Donc 1 est bien une racine double de P.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":53785,"question":"Calculez l'intégrale ∫(de 0 à 1) (2x + 1) dx.","option_a":"A. 1","option_b":"B. 2","option_c":"C. 3","option_d":"D. 4","option_e":"","option_f":"","bonne_reponse":"B","explication":"L'intégrale vaut [x² + x] de 0 à 1 = (1+1) - (0+0) = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":53786,"question":"La suite définie par u₀ = 1 et uₙ₊₁ = (uₙ + 2)\/2 est convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La suite est croissante et majorée par 2, donc convergente (vers 2).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":53787,"question":"Soit A = [[1, 2], [3, 4]]. Le déterminant de A est égal à :","option_a":"A. -2","option_b":"B. 2","option_c":"C. -5","option_d":"D. 5","option_e":"","option_f":"","bonne_reponse":"A","explication":"det(A) = (1×4) - (2×3) = 4 - 6 = -2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":53788,"question":"Toute fonction continue sur un intervalle fermé [a,b] est dérivable sur cet intervalle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Contre-exemple : la fonction valeur absolue est continue sur [-1,1] mais non dérivable en 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":53789,"question":"Résolvez l'inéquation x² - 5x + 6 ≤ 0.","option_a":"A. x ∈ [2,3]","option_b":"B. x ∈ ]-∞,2] ∪ [3,+∞[","option_c":"C. x ∈ [1,6]","option_d":"D. x ∈ ∅","option_e":"","option_f":"","bonne_reponse":"A","explication":"Les racines sont 2 et 3. Le polynôme est négatif entre les racines : x ∈ [2,3].","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":53790,"question":"Soit f une fonction deux fois dérivable sur ℝ telle que f''(x) \u003E 0 pour tout x. Alors f est convexe sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Par définition, si f''(x) \u003E 0 alors f est convexe sur ℝ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]