Quiz : Fractions, Polynômes et Équations Rationnelles — Terminale Math

Question 1 sur 10 10:00

[{"id":17098,"question":"Quelle est la forme irréductible de la fraction \\frac{x^2 - 4}{x^2 - 2x} ?","option_a":"A. \\frac{x + 2}{x}","option_b":"B. \\frac{x - 2}{x}","option_c":"C. \\frac{x + 2}{x - 2}","option_d":"D. \\frac{x - 2}{x + 2}","option_e":"","option_f":"","bonne_reponse":"B","explication":"On factorise le numérateur et le dénominateur : x² - 4 = (x - 2)(x + 2) et x² - 2x = x(x - 2). Après simplification, on obtient \\frac{x + 2}{x}.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17099,"question":"L'équation \\frac{2x + 1}{x - 3} = 0 a pour solution x = -0.5.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Une fraction est nulle si et seulement si son numérateur est nul. Ici, 2x + 1 = 0 donne x = -0.5, mais il faut vérifier que le dénominateur x - 3 ≠ 0, ce qui est le cas. La solution est donc correcte.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17100,"question":"Quel est le reste de la division euclidienne de P(x) = 2x³ - 5x² + 3x - 1 par D(x) = x - 2 ?","option_a":"A. 5","option_b":"B. 7","option_c":"C. -1","option_d":"D. 3","option_e":"","option_f":"","bonne_reponse":"A","explication":"On utilise le théorème du reste : P(2) = 2(8) - 5(4) + 3(2) - 1 = 16 - 20 + 6 - 1 = 1. Le reste est donc 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17101,"question":"La fonction f(x) = \\frac{x^2 - 1}{x^2 + 1} est définie pour tout x réel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le dénominateur x² + 1 est toujours strictement positif pour tout x réel, donc la fonction est définie sur ℝ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17102,"question":"Quelle est la solution de l'inéquation \\frac{x + 1}{x - 2} ≥ 0 ?","option_a":"A. x ∈ ]-∞ ; -1] ∪ ]2 ; +∞[","option_b":"B. x ∈ [-1 ; 2[","option_c":"C. x ∈ ]-∞ ; -1] ∪ [2 ; +∞[","option_d":"D. x ∈ ]-1 ; 2[","option_e":"","option_f":"","bonne_reponse":"A","explication":"On étudie le signe du numérateur et du dénominateur. La solution est x ∈ ]-∞ ; -1] ∪ ]2 ; +∞[ (en excluant x = 2 où la fraction n'est pas définie).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17103,"question":"Le polynôme P(x) = x³ - 3x² + 4x - 2 admet x = 1 comme racine.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"P(1) = 1 - 3 + 4 - 2 = 0, donc x = 1 est bien une racine du polynôme.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17104,"question":"Quelle est la limite de f(x) = \\frac{2x^2 + 3x}{x^2 - 1} quand x tend vers +∞ ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 2","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"C","explication":"On divise numérateur et dénominateur par x² : f(x) = \\frac{2 + 3\/x}{1 - 1\/x²}. Quand x → +∞, f(x) → 2\/1 = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17105,"question":"L'équation \\frac{1}{x} + \\frac{1}{x - 1} = 2 admet deux solutions réelles.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"En mettant au même dénominateur, on obtient \\frac{2x - 1}{x(x - 1)} = 2. L'équation 2x - 1 = 2x(x - 1) donne 2x² - 3x + 1 = 0, dont le discriminant est 1 \u003E 0. Il y a donc deux solutions réelles.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17106,"question":"Quelle est la dérivée de f(x) = \\frac{x^2 + 1}{x} ?","option_a":"A. f'(x) = 1 - \\frac{1}{x^2}","option_b":"B. f'(x) = 1 + \\frac{1}{x^2}","option_c":"C. f'(x) = x - \\frac{1}{x^2}","option_d":"D. f'(x) = 2x - \\frac{1}{x^2}","option_e":"","option_f":"","bonne_reponse":"A","explication":"On utilise la formule de dérivation d'un quotient : f'(x) = \\frac{2x \u0007b\u0007bcdot\u0007b\u0007d x - (x² + 1) \u0007b\u0007cdot\u0007b\u0007d 1}{x²} = \\frac{2x² - x² - 1}{x²} = \\frac{x² - 1}{x²} = 1 - \\frac{1}{x²}.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17107,"question":"La fonction f(x) = \\frac{x^2 - 4x + 3}{x - 1} est-elle une fonction rationnelle ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Une fonction rationnelle est le quotient de deux polynômes. Ici, f(x) = \\frac{(x - 1)(x - 3)}{x - 1} = x - 3 (pour x ≠ 1), ce qui est un polynôme. Cependant, avant simplification, c'est bien une fonction rationnelle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]