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Question 1 sur 10 10:00

[{"id":42206,"question":"Quelle est la dérivée de la fonction f(x) = x² + 3x - 5 ?","option_a":"A. 2x + 3","option_b":"B. x² + 3","option_c":"C. 2x + 5","option_d":"D. x + 3","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de x² est 2x, celle de 3x est 3, et celle de -5 est 0. Donc f'(x) = 2x + 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42207,"question":"La fonction exponentielle est 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, car e^x \u003E 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42208,"question":"Quelle est la limite de (sin x)\/x lorsque x tend vers 0 ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"B","explication":"C'est un résultat fondamental en analyse : lim(x→0) (sin x)\/x = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42209,"question":"Dans une loi normale centrée réduite, quelle est la probabilité P(-1 ≤ Z ≤ 1) ?","option_a":"A. 0.3413","option_b":"B. 0.6826","option_c":"C. 0.9544","option_d":"D. 0.9974","option_e":"","option_f":"","bonne_reponse":"B","explication":"Environ 68,26% des valeurs d'une loi normale centrée réduite se situent entre -1 et 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42210,"question":"L'équation différentielle y' = 2y admet 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 de y' = 2y est bien y = Ce^(2x), où C est une constante réelle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42211,"question":"Quelle est la valeur de l'intégrale ∫(0 à 1) e^(-x) dx ?","option_a":"A. 1 - 1\/e","option_b":"B. e - 1","option_c":"C. 1\/e","option_d":"D. 1 + 1\/e","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'intégrale de e^(-x) est -e^(-x). Évaluée entre 0 et 1, on obtient -e^(-1) - (-e^0) = 1 - 1\/e.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42212,"question":"Dans un triangle rectangle, le carré de l'hypoténuse est égal à la somme des carrés des deux autres côtés.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"C'est l'énoncé du théorème de Pythagore, valable pour tout triangle rectangle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42213,"question":"Quelle est la dérivée seconde de f(x) = x³ - 2x² + x ?","option_a":"A. 6x - 4","option_b":"B. 3x² - 4x + 1","option_c":"C. 6x² - 4x","option_d":"D. 6x - 2","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée première est f'(x) = 3x² - 4x + 1. La dérivée seconde est f''(x) = 6x - 4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42214,"question":"La fonction f(x) = ln(x) est définie pour x \u003E 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le logarithme népérien ln(x) est défini uniquement pour x \u003E 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42215,"question":"Quelle est la solution de l'équation différentielle y'' + y = 0 ?","option_a":"A. y = Ce^x + De^(-x)","option_b":"B. y = C cos x + D sin x","option_c":"C. y = Cx + D","option_d":"D. y = e^(ix)","option_e":"","option_f":"","bonne_reponse":"B","explication":"L'équation caractéristique est r² + 1 = 0, de solutions r = ±i. La solution générale est donc y = C cos x + D sin x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]