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Question 1 sur 10 10:00

[{"id":75252,"question":"Quelle est la limite de la fonction f(x) = (2x² + 3x - 5)\/(x² - 1) quand x tend vers l'infini ?","option_a":"A. 0","option_b":"B. 2","option_c":"C. -1","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, 2\/1 = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":75253,"question":"L'équation x³ - 6x² + 11x - 6 = 0 admet-elle trois solutions réelles distinctes ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"En factorisant, on obtient (x-1)(x-2)(x-3) = 0, donc trois solutions réelles distinctes : 1, 2 et 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":75254,"question":"Quelle est la dérivée de la fonction f(x) = ln(x² + 1) ?","option_a":"A. 2x\/(x² + 1)","option_b":"B. x\/(x² + 1)","option_c":"C. 2x","option_d":"D. 1\/(x² + 1)","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de ln(u) est u'\/u. Ici, u = x² + 1, donc u' = 2x. Ainsi, f'(x) = 2x\/(x² + 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":75255,"question":"La fonction f(x) = x² - 4x + 3 est-elle convexe sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Une fonction est convexe si sa dérivée seconde est positive. Ici, f''(x) = 2 \u003E 0 pour tout x ∈ ℝ, donc elle est convexe.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":75256,"question":"Quelle est la solution de l'inéquation 3x - 5 \u003E 2x + 7 ?","option_a":"A. x \u003E 2","option_b":"B. x \u003C 12","option_c":"C. x \u003E 12","option_d":"D. x \u003C 2","option_e":"","option_f":"","bonne_reponse":"C","explication":"En simplifiant, on obtient x \u003E 12. L'inéquation est vérifiée pour tous les x supérieurs à 12.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":75257,"question":"Le vecteur u(2, -3) et le vecteur v(-4, 6) sont-ils colinéaires ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Deux vecteurs sont colinéaires si l'un est un multiple de l'autre. Ici, v = -2u, donc ils sont colinéaires.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":75258,"question":"Quelle est l'intégrale de f(x) = 3x² - 2x + 1 entre 0 et 2 ?","option_a":"A. 4","option_b":"B. 6","option_c":"C. 8","option_d":"D. 10","option_e":"","option_f":"","bonne_reponse":"C","explication":"L'intégrale de 3x² - 2x + 1 est x³ - x² + x. Évaluée entre 0 et 2, on obtient (8 - 4 + 2) - (0 - 0 + 0) = 6.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":75259,"question":"La fonction f(x) = sin(x) est-elle périodique de période 2π ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La fonction sinus est périodique de période 2π, car sin(x + 2π) = sin(x) pour tout x ∈ ℝ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":75260,"question":"Quelle est la solution de l'équation e^(2x) = 5 ?","option_a":"A. x = ln(5)\/2","option_b":"B. x = 5\/2","option_c":"C. x = e^5","option_d":"D. x = ln(2)\/5","option_e":"","option_f":"","bonne_reponse":"A","explication":"En prenant le logarithme des deux côtés, on obtient 2x = ln(5), donc x = ln(5)\/2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":75261,"question":"Le nombre complexe z = 3 + 4i a-t-il un module égal à 5 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le module de z est √(3² + 4²) = √(9 + 16) = √25 = 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]