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Question 1 sur 10 10:00

[{"id":77792,"question":"Quelle est la limite de la fonction f(x) = (3x² + 2x - 1)\/(x² - 4) lorsque x tend vers l'infini ?","option_a":"A. 0","option_b":"B. 3","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"B","explication":"La limite d'un quotient de polynômes de même degré est égale au rapport des coefficients dominants. Ici, 3x²\/x² = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":77793,"question":"La fonction f(x) = x³ - 3x + 1 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 ℝ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":77794,"question":"Quelle est la valeur de l'intégrale ∫₀¹ (2x + 1) dx ?","option_a":"A. 1","option_b":"B. 2","option_c":"C. 3","option_d":"D. 4","option_e":"","option_f":"","bonne_reponse":"C","explication":"L'intégrale se calcule comme [x² + x]₀¹ = (1 + 1) - (0 + 0) = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":77795,"question":"Le nombre complexe z = 1 + i√3 a pour module :","option_a":"A. 1","option_b":"B. 2","option_c":"C. √2","option_d":"D. √3","option_e":"","option_f":"","bonne_reponse":"C","explication":"Le module de z = a + ib est √(a² + b²) = √(1 + 3) = √4 = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":77796,"question":"La suite définie par uₙ = (n² + 1)\/n converge vers 0 lorsque n tend vers l'infini.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La suite uₙ = (n² + 1)\/n = n + 1\/n tend vers +∞, donc elle ne converge pas vers 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":77797,"question":"Quelle est la dérivée de la fonction f(x) = ln(2x + 1) ?","option_a":"A. 2\/(2x + 1)","option_b":"B. 1\/(2x + 1)","option_c":"C. 2x\/(2x + 1)","option_d":"D. 1\/x","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de ln(u) est u'\/u. Ici, u = 2x + 1, donc u' = 2. Ainsi, f'(x) = 2\/(2x + 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":77798,"question":"L'équation x² + 2x + 5 = 0 admet deux solutions réelles distinctes.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Le discriminant Δ = 4 - 20 = -16 \u003C 0. L'équation n'a donc pas de solutions réelles.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":77799,"question":"Quelle est la valeur de l'intégrale ∫₀^(π\/2) sin(x) dx ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. π\/2","option_d":"D. 2","option_e":"","option_f":"","bonne_reponse":"B","explication":"L'intégrale de sin(x) est -cos(x). Ainsi, [-cos(x)]₀^(π\/2) = -cos(π\/2) + cos(0) = 0 + 1 = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":77800,"question":"Le nombre complexe z = 3e^(iπ\/3) a pour forme algébrique :","option_a":"A. 3 + i√3","option_b":"B. (3√3)\/2 + i(3\/2)","option_c":"C. 3\/2 + i(3√3)\/2","option_d":"D. 3 - i√3","option_e":"","option_f":"","bonne_reponse":"C","explication":"z = 3(cos(π\/3) + i sin(π\/3)) = 3(1\/2 + i√3\/2) = 3\/2 + i(3√3)\/2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":77801,"question":"La fonction f(x) = e^(-x²) est paire.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Une fonction f est paire si f(-x) = f(x). Ici, f(-x) = e^(-(-x)²) = e^(-x²) = f(x). La fonction est donc paire.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]