Mathématiques Bac : Testez vos connaissances en algèbre, analyse et géométrie

Question 1 sur 10 10:00

[{"id":71872,"question":"Quelle est la solution de l’équation 2x² - 5x + 2 = 0 ?","option_a":"x = 1 ou x = 2","option_b":"x = 0,5 ou x = 2","option_c":"x = -1 ou x = -2","option_d":"x = 1 ou x = -2","option_e":"","option_f":"","bonne_reponse":"B","explication":"Résolvons l’équation : Δ = b² - 4ac = 25 - 16 = 9. Les solutions sont x = (5 ± √9)\/4, soit x = 2 ou x = 0,5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":71873,"question":"La dérivée de la fonction f(x) = x³ - 3x² + 2x est f’(x) = 3x² - 6x + 2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée correcte est f’(x) = 3x² - 6x + 2. L’affirmation est donc vraie.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":71874,"question":"Quelle est l’image de la fonction f(x) = 2x + 1 pour x = -3 ?","option_a":"f(-3) = -5","option_b":"f(-3) = -7","option_c":"f(-3) = 5","option_d":"f(-3) = 7","option_e":"","option_f":"","bonne_reponse":"B","explication":"En remplaçant x par -3 dans f(x) = 2x + 1, on obtient f(-3) = 2*(-3) + 1 = -6 + 1 = -5. La bonne réponse est donc f(-3) = -5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":71875,"question":"L’inéquation x² - 4x + 3 \u003E 0 a pour solution :","option_a":"x ∈ ]1 ; 3[","option_b":"x ∈ ]-∞ ; 1[ ∪ ]3 ; +∞[","option_c":"x ∈ ]-∞ ; 1[","option_d":"x ∈ ]3 ; +∞[","option_e":"","option_f":"","bonne_reponse":"B","explication":"Résolvons l’inéquation : les racines de x² - 4x + 3 = 0 sont x = 1 et x = 3. Le polynôme est positif à l’extérieur des racines, donc x ∈ ]-∞ ; 1[ ∪ ]3 ; +∞[.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":71876,"question":"La fonction f(x) = 1\/x est définie pour tout x réel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La fonction f(x) = 1\/x n’est pas définie pour x = 0. L’affirmation est donc fausse.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":71877,"question":"Quelle est la limite de la fonction f(x) = (x² - 1)\/(x - 1) lorsque x tend vers 1 ?","option_a":"0","option_b":"1","option_c":"2","option_d":"La limite n’existe pas","option_e":"","option_f":"","bonne_reponse":"B","explication":"Factorisons le numérateur : (x² - 1) = (x - 1)(x + 1). La fonction devient f(x) = x + 1 pour x ≠ 1. La limite lorsque x tend vers 1 est donc 1 + 1 = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":71878,"question":"Le vecteur u(2 ; -3) et le vecteur v(-4 ; 6) sont colinéaires.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Les vecteurs u et v sont colinéaires car v = -2u (coordonnées proportionnelles). L’affirmation est donc vraie.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":71879,"question":"Quelle est la dérivée de la fonction f(x) = sin(2x) ?","option_a":"f’(x) = 2cos(2x)","option_b":"f’(x) = cos(2x)","option_c":"f’(x) = 2sin(x)","option_d":"f’(x) = -2cos(2x)","option_e":"","option_f":"","bonne_reponse":"A","explication":"En utilisant la formule de dérivation des fonctions composées, f’(x) = 2cos(2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":71880,"question":"L’équation de la tangente à la courbe y = x² au point d’abscisse x = 1 est :","option_a":"y = 2x - 1","option_b":"y = x + 1","option_c":"y = 2x + 1","option_d":"y = x - 1","option_e":"","option_f":"","bonne_reponse":"A","explication":"La tangente a pour équation y = f’(1)(x - 1) + f(1). Avec f’(x) = 2x, f’(1) = 2 et f(1) = 1. L’équation est donc y = 2(x - 1) + 1 = 2x - 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":71881,"question":"La fonction f(x) = e^x est toujours positive pour tout x réel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La fonction exponentielle e^x est strictement positive pour tout x réel. L’affirmation est donc vraie.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]