Maths 3ème : Testez vos connaissances en dérivées et géométrie !

Question 1 sur 10 10:00

[{"id":78922,"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 + 5","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée d'une fonction polynôme se calcule terme par 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":78923,"question":"L'équation x² - 4x + 4 = 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 - 16 = 0. Une seule solution réelle double x = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":78924,"question":"Quelle est l'équation de la tangente à la courbe y = x³ au point d'abscisse 1 ?","option_a":"A. y = 3x - 2","option_b":"B. y = x + 1","option_c":"C. y = 3x + 1","option_d":"D. y = x² + 1","option_e":"","option_f":"","bonne_reponse":"A","explication":"La tangente en x=1 a pour équation y = f'(1)(x-1) + f(1). f'(x)=3x² ⇒ f'(1)=3, f(1)=1. Donc y = 3(x-1) + 1 = 3x - 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":78925,"question":"Dans un repère orthonormé, la distance entre les points A(2;3) et B(-1;5) est :","option_a":"A. √13","option_b":"B. 5","option_c":"C. √17","option_d":"D. √5","option_e":"","option_f":"","bonne_reponse":"C","explication":"La distance AB = √[(xB-xA)² + (yB-yA)²] = √[(-1-2)² + (5-3)²] = √[9 + 4] = √13.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":78926,"question":"La fonction f(x) = 1\/x est dérivable en x = 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La fonction n'est pas définie en x=0, donc elle n'est pas dérivable en ce point.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":78927,"question":"Quelle est la solution générale de l'équation différentielle y' + 2y = 0 ?","option_a":"A. y = Ce^{-2x}","option_b":"B. y = Ce^{2x}","option_c":"C. y = Cx + 2","option_d":"D. y = C\/x","option_e":"","option_f":"","bonne_reponse":"A","explication":"C'est une équation différentielle linéaire du premier ordre. La solution générale est y = Ce^{-2x}, où C est une constante réelle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":78928,"question":"Le point M(1;2) appartient-il à la droite d'équation 2x - y + 3 = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"En remplaçant x=1 et y=2 dans l'équation : 2(1) - 2 + 3 = 3 ≠ 0. Donc M n'appartient pas à la droite.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":78929,"question":"Quelle est la dérivée de la fonction f(x) = e^{2x} ?","option_a":"A. 2e^{2x}","option_b":"B. e^{2x}","option_c":"C. 2x e^{2x}","option_d":"D. e^{x}","option_e":"","option_f":"","bonne_reponse":"A","explication":"En utilisant la formule de dérivation des fonctions exponentielles : (e^{u})' = u' e^{u}. Ici u=2x ⇒ u'=2. Donc f'(x)=2e^{2x}.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":78930,"question":"L'intégrale de 0 à 1 de la fonction f(x) = x² est égale à :","option_a":"A. 1\/3","option_b":"B. 1","option_c":"C. 1\/2","option_d":"D. 2","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'intégrale ∫₀¹ x² dx = [x³\/3]₀¹ = 1\/3 - 0 = 1\/3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":78931,"question":"La fonction f(x) = ln(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 logarithme népérien est définie uniquement pour x \u003E 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]