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Question 1 sur 10 10:00

[{"id":3729,"question":"Quelle est la dérivée de la fonction f(x) = x² + 3x - 5 ?","option_a":"A. 2x + 3","option_b":"B. 2x + 3x","option_c":"C. x² + 3","option_d":"D. 2x - 5","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée d'une fonction polynôme se calcule terme par terme : f'(x) = 2x + 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":3730,"question":"La fonction f(x) = sin(x) est-elle périodique de période 2π ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La fonction sinus est périodique de période 2π, car sin(x + 2π) = sin(x) pour tout x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":3731,"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 + 2","option_d":"D. y = Cx²","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'équation différentielle linéaire du premier ordre y' + 2y = 0 a pour solution générale y = Ce^(-2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":3732,"question":"Le déterminant d'une matrice 2x2 est-il égal à ad - bc pour une matrice [[a, b], [c, d]] ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Pour une matrice 2x2 [[a, b], [c, d]], le déterminant est bien égal à ad - bc.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":3733,"question":"Quelle est la limite de la suite u_n = (n² + 1)\/n² lorsque n tend vers l'infini ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. +∞","option_d":"D. -1","option_e":"","option_f":"","bonne_reponse":"B","explication":"En divisant numérateur et dénominateur par n², on obtient u_n = 1 + 1\/n², donc lim(u_n) = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":3734,"question":"L'intégrale ∫(x² + 1) dx entre 0 et 1 vaut-elle 4\/3 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"L'intégrale ∫(x² + 1) dx entre 0 et 1 vaut [x³\/3 + x] de 0 à 1 = 1\/3 + 1 = 4\/3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":3735,"question":"Quelle est la nature de la série ∑(1\/n²) pour n ≥ 1 ?","option_a":"A. Convergente","option_b":"B. Divergente","option_c":"C. Oscillante","option_d":"D. Indéterminée","option_e":"","option_f":"","bonne_reponse":"A","explication":"La série ∑(1\/n²) est une série de Riemann convergente (série p avec p = 2 \u003E 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":3736,"question":"Le produit scalaire de deux vecteurs u et v est-il commutatif ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le produit scalaire est commutatif : u·v = v·u.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":3737,"question":"Quelle est la solution générale de l'équation différentielle y'' - 3y' + 2y = 0 ?","option_a":"A. y = Ce^x + De^(2x)","option_b":"B. y = Ce^(-x) + De^(-2x)","option_c":"C. y = Ce^(3x)","option_d":"D. y = Cx + D","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'équation caractéristique r² - 3r + 2 = 0 a pour racines r = 1 et r = 2, donc la solution générale est y = Ce^x + De^(2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":3738,"question":"La fonction f(x) = e^x est-elle toujours positive sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La fonction exponentielle e^x est strictement positive pour tout x réel.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]