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Question 1 sur 10 10:00

[{"id":17588,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"3x² + 2","option_c":"6x - 5","option_d":"3x + 2","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée d'une fonction polynôme s'obtient en appliquant la règle de dérivation : (ax^n)' = n*ax^(n-1). Ici, f'(x) = 6x + 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17589,"question":"L'équation x² - 4x + 3 = 0 admet deux solutions réelles distinctes.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le discriminant Δ = b² - 4ac = 16 - 12 = 4 \u003E 0, donc il y a bien deux solutions réelles distinctes.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17590,"question":"Quelle est la limite de la fonction f(x) = (2x² + 3x - 1)\/(x² - 1) lorsque x tend vers +∞ ?","option_a":"2","option_b":"1","option_c":"0","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"A","explication":"Pour x → +∞, on divise numérateur et dénominateur par x² : f(x) ≈ (2 + 3\/x - 1\/x²)\/(1 - 1\/x²) → 2\/1 = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17591,"question":"La fonction f(x) = x³ - 3x + 2 est strictement croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La dérivée f'(x) = 3x² - 3 s'annule en x = ±1. La fonction n'est donc pas strictement croissante sur tout ℝ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17592,"question":"Quelle est la solution de l'inéquation 2x - 5 ≤ 3x + 1 ?","option_a":"x ≥ -6","option_b":"x ≤ -6","option_c":"x ≥ 6","option_d":"x ≤ 6","option_e":"","option_f":"","bonne_reponse":"B","explication":"En isolant x : 2x - 3x ≤ 1 + 5 → -x ≤ 6 → x ≥ -6 (attention au sens de l'inégalité lors de la multiplication par un nombre négatif).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17593,"question":"La fonction f(x) = e^x 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 pour tout x ∈ ℝ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17594,"question":"Quelle est la primitive de la fonction f(x) = 4x³ - 2x + 1 ?","option_a":"x⁴ - x² + x + C","option_b":"x⁴ - x² + C","option_c":"4x⁴ - x² + x + C","option_d":"x⁴ - 2x² + x + C","option_e":"","option_f":"","bonne_reponse":"A","explication":"La primitive s'obtient en intégrant terme à terme : ∫4x³ dx = x⁴, ∫-2x dx = -x², ∫1 dx = x. La constante C est ajoutée.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17595,"question":"L'ensemble des solutions de l'équation |2x - 3| = 5 est S = {-1, 4}.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'équation |2x - 3| = 5 donne deux cas : 2x - 3 = 5 → x = 4, ou 2x - 3 = -5 → x = -1. L'ensemble S est correct.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17596,"question":"Quelle est la valeur de la limite lim (x→0) (sin(x))\/x ?","option_a":"0","option_b":"1","option_c":"∞","option_d":"Non définie","option_e":"","option_f":"","bonne_reponse":"B","explication":"C'est une limite fondamentale en analyse : lim (x→0) (sin(x))\/x = 1, résultat essentiel pour les dérivées trigonométriques.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17597,"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) est définie uniquement pour x \u003E 0. Elle n'est pas définie pour x ≤ 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]