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Question 1 sur 10 10:00

[{"id":11628,"question":"Quelle est la dérivée de la fonction f(x) = x³ + 2x² - 5x + 1 ?","option_a":"3x² + 4x - 5","option_b":"3x² + 4x - 5x","option_c":"x² + 2x - 5","option_d":"3x² + 4x - 1","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de x³ est 3x², celle de 2x² est 4x, celle de -5x est -5, et la constante 1 a une dérivée nulle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":11629,"question":"L'intégrale de 0 à 1 de la fonction f(x) = x² est égale à 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"L'intégrale de x² de 0 à 1 est [x³\/3] de 0 à 1 = 1\/3 - 0 = 1\/3 ≠ 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":11630,"question":"Quelle est la limite de la suite uₙ = (2n + 1)\/(n + 3) quand n tend vers l'infini ?","option_a":"0","option_b":"1","option_c":"2","option_d":"3","option_e":"","option_f":"","bonne_reponse":"C","explication":"En divisant numérateur et dénominateur par n, on obtient (2 + 1\/n)\/(1 + 3\/n) → 2\/1 = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":11631,"question":"L'équation différentielle y' + 2y = 0 a pour solution générale y = Ce^(-2x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La solution générale est bien y = Ce^(-2x), car la dérivée de Ce^(-2x) est -2Ce^(-2x), ce qui vérifie l'équation.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":11632,"question":"Quelle est la primitive de la fonction f(x) = 3x² - 4x + 1 ?","option_a":"x³ - 2x² + x + C","option_b":"x³ - 2x² + C","option_c":"3x³ - 4x² + x + C","option_d":"x³ - 4x² + x + C","option_e":"","option_f":"","bonne_reponse":"A","explication":"La primitive de 3x² est x³, celle de -4x est -2x², celle de 1 est x, et C est la constante d'intégration.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":11633,"question":"La fonction f(x) = e^(2x) est strictement croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Sa dérivée f'(x) = 2e^(2x) est toujours positive, donc la fonction est strictement croissante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":11634,"question":"Quelle est la valeur de l'intégrale ∫(de 0 à π) sin(x) dx ?","option_a":"0","option_b":"1","option_c":"2","option_d":"-1","option_e":"","option_f":"","bonne_reponse":"B","explication":"L'intégrale de sin(x) est -cos(x). Évaluée de 0 à π, on obtient -cos(π) - (-cos(0)) = 1 - (-1) = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":11635,"question":"La suite uₙ = n²\/(n + 1) est bornée.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La suite tend vers l'infini quand n → ∞, donc elle n'est pas bornée.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":11636,"question":"Quelle est la solution de l'équation différentielle y'' + y = 0 avec y(0) = 1 et y'(0) = 0 ?","option_a":"y = cos(x)","option_b":"y = sin(x)","option_c":"y = e^x","option_d":"y = e^(-x)","option_e":"","option_f":"","bonne_reponse":"A","explication":"La solution générale est y = A cos(x) + B sin(x). Avec y(0) = 1 et y'(0) = 0, on obtient A = 1 et B = 0, donc y = cos(x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":11637,"question":"La fonction f(x) = ln(x) est définie sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La fonction ln(x) est définie uniquement pour x \u003E 0, donc pas sur tout ℝ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]