Quiz interactif : Maîtrisez les séries de Terminale Mathématiques

Question 1 sur 10 10:00

[{"id":17738,"question":"Quelle est la limite de la fonction f(x) = (2x² + 3x - 5)\/(x² - 4) lorsque x tend vers l'infini ?","option_a":"A. 0","option_b":"B. 2","option_c":"C. ∞","option_d":"D. -2","option_e":"","option_f":"","bonne_reponse":"B","explication":"La limite d'un quotient de polynômes de même degré est égale au rapport des coefficients dominants. Ici, 2x²\/x² = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17739,"question":"La fonction f(x) = x³ - 3x² + 2x est croissante sur l'intervalle [-1, 1].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée f'(x) = 3x² - 6x + 2 s'annule en x = 1 ± √(1\/3). Sur [-1, 1], f'(x) \u003E 0, donc la fonction est bien croissante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17740,"question":"Quelle est la valeur de l'intégrale ∫(0 à 1) (e^x + 1) dx ?","option_a":"A. e","option_b":"B. e + 1","option_c":"C. e - 1","option_d":"D. 2","option_e":"","option_f":"","bonne_reponse":"C","explication":"∫(e^x + 1) dx = e^x + x. Évaluée entre 0 et 1, cela donne (e + 1) - (1 + 0) = e.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17741,"question":"Dans un triangle ABC, si cos(A) = 3\/5 et sin(B) = 4\/5, alors sin(C) est égal à :","option_a":"A. 7\/25","option_b":"B. 24\/25","option_c":"C. 12\/13","option_d":"D. 5\/13","option_e":"","option_f":"","bonne_reponse":"B","explication":"Utilisez la formule sin(C) = sin(π - A - B) = sin(A + B). Avec sin(A) = 4\/5 et cos(B) = 3\/5, sin(A + B) = sin(A)cos(B) + cos(A)sin(B) = (4\/5)(3\/5) + (3\/5)(4\/5) = 24\/25.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17742,"question":"La suite définie par u₀ = 2 et uₙ₊₁ = (uₙ + 3)\/(uₙ + 1) est convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La suite converge vers la solution de L = (L + 3)\/(L + 1), soit L² - 2L - 3 = 0. Les solutions sont L = 3 ou L = -1. Comme u₀ = 2 \u003E 0, la suite converge vers 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17743,"question":"Quelle est la dérivée de la fonction f(x) = ln(x² + 1) ?","option_a":"A. 2x\/(x² + 1)","option_b":"B. 1\/(x² + 1)","option_c":"C. 2x","option_d":"D. x\/(x² + 1)","option_e":"","option_f":"","bonne_reponse":"A","explication":"En utilisant la formule de dérivation des fonctions composées, f'(x) = (2x)\/(x² + 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17744,"question":"Un dé équilibré est lancé deux fois. Quelle est la probabilité d'obtenir au moins un 6 ?","option_a":"A. 1\/6","option_b":"B. 11\/36","option_c":"C. 1\/3","option_d":"D. 5\/12","option_e":"","option_f":"","bonne_reponse":"B","explication":"La probabilité d'obtenir au moins un 6 est égale à 1 - (5\/6)² = 11\/36.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17745,"question":"La fonction f(x) = x² - 4x + 5 admet un minimum en x = 2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée f'(x) = 2x - 4 s'annule en x = 2. Comme f''(x) = 2 \u003E 0, il s'agit bien d'un minimum.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17746,"question":"Quelle est la solution de l'équation différentielle y' + 2y = 0 avec y(0) = 1 ?","option_a":"A. y = e^(-2x)","option_b":"B. y = e^(2x)","option_c":"C. y = 1 - 2x","option_d":"D. y = 2e^(-x)","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'équation différentielle a pour solution générale y = Ce^(-2x). Avec y(0) = 1, on trouve C = 1, donc y = e^(-2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17747,"question":"Dans un repère orthonormé, la distance entre les points A(1, 2) et B(4, 6) est :","option_a":"A. 5","option_b":"B. √7","option_c":"C. 7","option_d":"D. √13","option_e":"","option_f":"","bonne_reponse":"A","explication":"La distance AB = √[(4-1)² + (6-2)²] = √(9 + 16) = √25 = 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]