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Question 1 sur 10 10:00

[{"id":27698,"question":"Quelle est la dérivée de la fonction f(x) = 3x² - 5x + 2 ?","option_a":"A) 6x - 5","option_b":"B) 3x - 5","option_c":"C) 6x² - 5","option_d":"D) 3x² - 5","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée d'une fonction polynôme est obtenue en appliquant la formule (ax^n)' = n*ax^(n-1). Ici, f'(x) = 6x - 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27699,"question":"L'équation x² + 4x + 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 = b² - 4ac = 16 - 20 = -4 \u003C 0, donc l'équation n'a pas de solution réelle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27700,"question":"Quel est le module du nombre complexe z = 3 + 4i ?","option_a":"A) 5","option_b":"B) 7","option_c":"C) √7","option_d":"D) 25","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le module d'un nombre complexe z = a + bi est donné par |z| = √(a² + b²). Ici, |z| = √(9 + 16) = 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27701,"question":"La fonction f(x) = e^x est toujours croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de f(x) = e^x est f'(x) = e^x, qui est toujours positive pour tout x ∈ ℝ. La fonction est donc strictement croissante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27702,"question":"Quelle est la limite de la fonction f(x) = (2x² + 3x - 1)\/(x² - 4) lorsque x tend vers +∞ ?","option_a":"A) 0","option_b":"B) 2","option_c":"C) +∞","option_d":"D) -∞","option_e":"","option_f":"","bonne_reponse":"B","explication":"Pour les limites à l'infini des fonctions rationnelles, on compare les degrés des polynômes. Ici, les degrés sont égaux (2), donc la limite est le rapport des coefficients dominants : 2\/1 = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27703,"question":"L'intégrale ∫(0 à 1) x² dx vaut :","option_a":"A) 1\/3","option_b":"B) 1\/2","option_c":"C) 1","option_d":"D) 2","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'intégrale de x^n est (x^(n+1))\/(n+1). Ici, ∫x² dx = x³\/3, donc ∫(0 à 1) x² dx = 1\/3 - 0 = 1\/3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27704,"question":"La suite définie par uₙ = (n+1)\/n est convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La suite uₙ = (n+1)\/n = 1 + 1\/n tend vers 1 lorsque n tend vers +∞. Elle est donc convergente.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27705,"question":"Quel est le résultat de l'opération (3 + 2i)(1 - i) ?","option_a":"A) 5 - i","option_b":"B) 5 + i","option_c":"C) 1 - 5i","option_d":"D) 3 - 2i","option_e":"","option_f":"","bonne_reponse":"A","explication":"En développant (3 + 2i)(1 - i) = 3*1 + 3*(-i) + 2i*1 + 2i*(-i) = 3 - 3i + 2i - 2i² = 3 - i + 2 = 5 - i.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27706,"question":"La fonction f(x) = ln(x) est définie pour tout x ∈ ℝ.","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},{"id":27707,"question":"Quelle est la solution de l'inéquation 2x - 5 \u003E 3x + 1 ?","option_a":"A) x \u003C -6","option_b":"B) x \u003E -6","option_c":"C) x \u003C 6","option_d":"D) x \u003E 6","option_e":"","option_f":"","bonne_reponse":"A","explication":"En résolvant 2x - 5 \u003E 3x + 1, on obtient -5 - 1 \u003E 3x - 2x, soit -6 \u003E x, ou x \u003C -6.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]