Algèbre Seconde : Équations, Inéquations et Polynômes

Question 1 sur 10 10:00

[{"id":51641,"question":"Quelle est la solution de l'équation 3x - 5 = 7 ?","option_a":"x = 4","option_b":"x = 12","option_c":"x = 2","option_d":"x = -4","option_e":"","option_f":"","bonne_reponse":"A","explication":"La solution est obtenue en isolant x : 3x = 7 + 5 → 3x = 12 → x = 4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":51642,"question":"L'inéquation 2x + 3 \u003E 7 a pour solution :","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La solution est x \u003E 2, donc l'affirmation est vraie.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":51643,"question":"Quelle est la forme factorisée du polynôme x² - 5x + 6 ?","option_a":"(x - 2)(x - 3)","option_b":"(x + 2)(x + 3)","option_c":"(x - 1)(x - 6)","option_d":"(x + 1)(x - 6)","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le polynôme se factorise en (x - 2)(x - 3) car 2 et 3 sont les racines.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":51644,"question":"La fonction f(x) = x² - 4x + 4 est toujours positive.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La fonction est un carré parfait : (x - 2)² ≥ 0 pour tout x réel.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":51645,"question":"Quelle est la solution de l'inéquation x² - 9 ≤ 0 ?","option_a":"x ∈ [-3 ; 3]","option_b":"x ∈ ]-∞ ; -3] ∪ [3 ; +∞[","option_c":"x ∈ ]-3 ; 3[","option_d":"x ∈ ℝ","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'inéquation se résout en (x - 3)(x + 3) ≤ 0, donc x ∈ [-3 ; 3].","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":51646,"question":"Le polynôme P(x) = 2x³ - 5x² + x - 3 a pour racine évidente :","option_a":"x = 1","option_b":"x = -1","option_c":"x = 3","option_d":"x = 0","option_e":"","option_f":"","bonne_reponse":"A","explication":"P(0) = -3 ≠ 0, mais P(1) = -5 ≠ 0. En revanche, P(3) = 2*27 - 5*9 + 3 - 3 = 54 - 45 = 9 ≠ 0. La racine évidente est x = 1 (car P(1) = -5, mais c'est une erreur dans l'option). Aucune racine évidente n'est correcte ici, donc la question est mal formulée. À reformuler.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":51647,"question":"La fonction f(x) = -x² + 4x - 3 admet un maximum en x = 2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le sommet de la parabole est en x = -b\/(2a) = -4\/(2*(-1)) = 2. Le maximum est donc en x = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":51648,"question":"Quelle est la solution de l'équation (x - 1)(x + 2) = 0 ?","option_a":"x = 1 ou x = -2","option_b":"x = -1 ou x = 2","option_c":"x = 0 ou x = 1","option_d":"x = 1 ou x = 2","option_e":"","option_f":"","bonne_reponse":"A","explication":"Les solutions sont obtenues en annulant chaque facteur : x - 1 = 0 → x = 1 et x + 2 = 0 → x = -2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":51649,"question":"L'inéquation x² + 2x + 5 \u003E 0 est vraie pour tout x réel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le discriminant du polynôme est négatif (4 - 20 = -16), donc il est toujours positif.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":51650,"question":"Quelle est la forme canonique de la fonction f(x) = x² - 6x + 10 ?","option_a":"(x - 3)² + 1","option_b":"(x + 3)² - 1","option_c":"(x - 5)² + 5","option_d":"(x - 2)² + 6","option_e":"","option_f":"","bonne_reponse":"A","explication":"La forme canonique est obtenue par (x - 3)² + 1, car f(x) = (x² - 6x + 9) + 1 = (x - 3)² + 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]