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Question 1 sur 10 10:00

[{"id":42576,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"A) 6x + 2","option_b":"B) 3x + 2","option_c":"C) 6x² + 2","option_d":"D) 6x + 2x","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée d'une fonction polynôme se calcule terme à terme : (3x²)' = 6x, (2x)' = 2, et (-5)' = 0. Donc f'(x) = 6x + 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42577,"question":"L'équation x² - 4x + 4 = 0 admet-elle 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 - 16 = 0. Une équation du second degré avec Δ = 0 admet une solution réelle double (ici, x = 2).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42578,"question":"Quelle est la limite de la fonction f(x) = (2x + 1)\/(x - 3) lorsque x tend vers l'infini ?","option_a":"A) 0","option_b":"B) 2","option_c":"C) 1","option_d":"D) -3","option_e":"","option_f":"","bonne_reponse":"C","explication":"Pour les fonctions rationnelles, la limite en l'infini est égale au rapport des coefficients dominants : 2x\/x = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42579,"question":"Le vecteur u(2; -1) et le vecteur v(-4; 2) 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 = -2 × u, donc ils sont colinéaires.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42580,"question":"Quelle est la primitive de la fonction f(x) = 5x⁴ - 3x² + 1 ?","option_a":"A) x⁵ - x³ + x + C","option_b":"B) 20x³ - 6x + C","option_c":"C) x⁵ - x³ + C","option_d":"D) 5x⁵ - 3x³ + x + C","option_e":"","option_f":"","bonne_reponse":"A","explication":"La primitive d'un polynôme s'obtient en augmentant chaque exposant de 1 et en divisant par ce nouvel exposant : ∫5x⁴ dx = x⁵, ∫-3x² dx = -x³, ∫1 dx = x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42581,"question":"La fonction f(x) = x³ - 3x + 2 est-elle croissante sur l'intervalle [-2; 2] ?","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. Sur [-2; -1], f'(x) \u003E 0 (croissante) ; sur [-1; 1], f'(x) \u003C 0 (décroissante) ; sur [1; 2], f'(x) \u003E 0 (croissante). Donc la fonction n'est pas croissante sur tout l'intervalle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42582,"question":"Quelle est la valeur de l'intégrale ∫₀¹ (3x² + 2x) dx ?","option_a":"A) 1","option_b":"B) 2","option_c":"C) 3","option_d":"D) 4","option_e":"","option_f":"","bonne_reponse":"B","explication":"L'intégrale se calcule comme suit : [x³ + x²]₀¹ = (1 + 1) - (0 + 0) = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42583,"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 d'un nombre complexe z = a + bi est |z| = √(a² + b²). Ici, |z| = √(9 + 16) = √25 = 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42584,"question":"Quelle est la solution de l'inéquation 2x - 5 ≤ 3x + 1 ?","option_a":"A) x ≥ -6","option_b":"B) x ≤ -6","option_c":"C) x ≥ 6","option_d":"D) x ≤ 6","option_e":"","option_f":"","bonne_reponse":"B","explication":"En résolvant : 2x - 5 ≤ 3x + 1 → -5 - 1 ≤ 3x - 2x → -6 ≤ x → x ≥ -6.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42585,"question":"La fonction f(x) = e^(2x) est-elle une solution de l'équation différentielle y' = 2y ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de f(x) = e^(2x) est f'(x) = 2e^(2x) = 2f(x). Donc f satisfait bien l'équation différentielle y' = 2y.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]