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Question 1 sur 10 10:00

[{"id":51601,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"6x² + 2","option_c":"3x + 2","option_d":"6x - 5","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·a·x^(n-1). Ici, f'(x) = 6x + 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":51602,"question":"La suite définie par uₙ = (-1)^n est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Une suite convergente doit avoir une limite finie. Or, uₙ alterne entre -1 et 1 selon la parité de n, donc elle n'a pas de limite.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":51603,"question":"Quel est le coefficient directeur de la tangente à la courbe de f(x) = x³ en x = 2 ?","option_a":"3","option_b":"6","option_c":"12","option_d":"24","option_e":"","option_f":"","bonne_reponse":"C","explication":"Le coefficient directeur de la tangente est égal à f'(2). Or f'(x) = 3x², donc f'(2) = 12.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":51604,"question":"Dans une expérience aléatoire, P(A) = 0.4 et P(B) = 0.5. Si A et B sont indépendants, quelle est P(A ∩ B) ?","option_a":"0.2","option_b":"0.9","option_c":"0.1","option_d":"0.25","option_e":"","option_f":"","bonne_reponse":"A","explication":"Pour deux événements indépendants, P(A ∩ B) = P(A) × P(B) = 0.4 × 0.5 = 0.2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":51605,"question":"La fonction f(x) = e^x est-elle 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 sur ℝ. Donc f est strictement croissante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":51606,"question":"Quelle est la limite de la suite uₙ = (n² + 1)\/(2n² - 3) quand n tend vers +∞ ?","option_a":"0","option_b":"1\/2","option_c":"1","option_d":"+∞","option_e":"","option_f":"","bonne_reponse":"C","explication":"En divisant numérateur et dénominateur par n², on obtient uₙ = (1 + 1\/n²)\/(2 - 3\/n²). Quand n→+∞, uₙ → 1\/2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":51607,"question":"Dans un repère orthonormé, quelle est l'équation du plan passant par A(1,2,3) et de vecteur normal n(2,-1,4) ?","option_a":"2x - y + 4z = 0","option_b":"2x - y + 4z = 12","option_c":"x + 2y + 3z = 14","option_d":"2x + y - 4z = 0","option_e":"","option_f":"","bonne_reponse":"B","explication":"L'équation d'un plan de vecteur normal n(a,b,c) passant par A(x₀,y₀,z₀) est a(x-x₀) + b(y-y₀) + c(z-z₀) = 0. Ici : 2(x-1) -1(y-2) +4(z-3)=0 → 2x - y + 4z = 12.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":51608,"question":"Si X suit une loi binomiale B(n,p), alors E(X) = np.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'espérance mathématique d'une loi binomiale B(n,p) est bien E(X) = n × p.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":51609,"question":"Quelle est la primitive de f(x) = 5x⁴ qui s'annule en x = 1 ?","option_a":"x⁵ + C","option_b":"x⁵ - 1","option_c":"5x⁵ + C","option_d":"x⁵ - 1 + C","option_e":"","option_f":"","bonne_reponse":"B","explication":"La primitive de 5x⁴ est x⁵ + C. Pour qu'elle s'annule en x=1, on pose 1⁵ + C = 0 → C = -1. Donc F(x) = x⁵ - 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":51610,"question":"Dans un triangle ABC, si AB = 5, AC = 7 et BC = 8, alors l'angle BÂC est-il aigu ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"D'après la réciproque du théorème de Pythagore : AB² + AC² = 25 + 49 = 74 \u003E BC² = 64. Donc l'angle BÂC est aigu.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]