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Question 1 sur 10 10:00

[{"id":10703,"question":"Quelle est la dérivée de la fonction f(x) = x² + 3x - 5 ?","option_a":"f'(x) = 2x + 3","option_b":"f'(x) = x² + 3","option_c":"f'(x) = 2x - 5","option_d":"f'(x) = 3x + 2","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée d'une fonction polynomiale s'obtient en appliquant la règle de dérivation terme à terme : (x²)' = 2x, (3x)' = 3 et (-5)' = 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10704,"question":"La limite de (sin x)\/x quand x tend vers 0 est égale à 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"C'est un résultat fondamental en analyse : lim(x→0) (sin x)\/x = 1, souvent démontré par le théorème des gendarmes ou le développement limité.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10705,"question":"Quelle est la solution de l'équation différentielle y' = 2y ?","option_a":"y = Ce^(2x)","option_b":"y = 2x + C","option_c":"y = Ce^(-2x)","option_d":"y = x² + C","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'équation y' = ky a pour solution générale y = Ce^(kx). Ici, k = 2, donc y = Ce^(2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10706,"question":"Le discriminant d'une équation du second degré ax² + bx + c = 0 est donné par :","option_a":"Δ = b² - 4ac","option_b":"Δ = b - 4ac","option_c":"Δ = a² - 4bc","option_d":"Δ = 4b² - ac","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le discriminant est une formule clé pour déterminer le nombre de solutions réelles d'une équation du second degré : Δ = b² - 4ac.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10707,"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 exp(x) = e^x est strictement positive pour tout x réel, car e^x \u003E 0 quel que soit x ∈ ℝ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10708,"question":"Quelle est l'intégrale de f(x) = 3x² entre 0 et 2 ?","option_a":"8","option_b":"6","option_c":"4","option_d":"12","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'intégrale de 3x² est x³. En évaluant entre 0 et 2 : 2³ - 0³ = 8.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10709,"question":"La suite définie par uₙ = (-1)ⁿ est convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La suite uₙ = (-1)ⁿ alterne entre -1 et 1, elle n'a donc pas de limite finie : elle est divergente.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10710,"question":"Quel est le module d'un nombre complexe z = 3 + 4i ?","option_a":"5","option_b":"7","option_c":"25","option_d":"12","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le module de z = a + bi est |z| = √(a² + b²). Ici, |z| = √(3² + 4²) = 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10711,"question":"La fonction f(x) = ln(x) est définie pour tout x réel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La fonction logarithme népérien ln(x) n'est définie que pour x \u003E 0. Elle n'est pas définie pour x ≤ 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10712,"question":"Quelle est la primitive de f(x) = 5x⁴ ?","option_a":"F(x) = x⁵ + C","option_b":"F(x) = 5x³ + C","option_c":"F(x) = x⁴ + C","option_d":"F(x) = 20x³ + C","option_e":"","option_f":"","bonne_reponse":"A","explication":"La primitive de xⁿ est x^(n+1)\/(n+1). Ici, n = 4, donc F(x) = 5x⁵\/5 + C = x⁵ + C.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]