Algèbre 2 : Maîtrisez polynômes, matrices et équations !

Question 1 sur 10 10:00

[{"id":54791,"question":"Quelle est la solution de l'équation x² - 5x + 6 = 0 ?","option_a":"x = 2 ou x = 3","option_b":"x = -2 ou x = -3","option_c":"x = 1 ou x = 6","option_d":"x = 0 ou x = 5","option_e":"","option_f":"","bonne_reponse":"A","explication":"Factorisation : (x-2)(x-3) = 0 ⇒ solutions x=2 et x=3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54792,"question":"Le déterminant d'une matrice 2x2 [[a, b], [c, d]] est égal à ad - bc.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La formule du déterminant pour une matrice 2x2 est bien ad - bc.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54793,"question":"Soit P(x) = 2x³ - 3x² + x - 5. Quelle est la valeur de P(1) ?","option_a":"-5","option_b":"-3","option_c":"1","option_d":"5","option_e":"","option_f":"","bonne_reponse":"A","explication":"P(1) = 2(1)³ - 3(1)² + 1 - 5 = 2 - 3 + 1 - 5 = -5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54794,"question":"Une matrice carrée est inversible si et seulement si son déterminant est non nul.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"C'est une propriété fondamentale de l'algèbre linéaire.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54795,"question":"Quelle est la dérivée de f(x) = 3x⁴ - 2x² + 5 ?","option_a":"f'(x) = 12x³ - 4x","option_b":"f'(x) = 3x³ - 2x","option_c":"f'(x) = 12x³ - 4x + 5","option_d":"f'(x) = 3x⁴ - 4x","option_e":"","option_f":"","bonne_reponse":"A","explication":"Dérivée terme à terme : 3*4x³ - 2*2x + 0 = 12x³ - 4x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54796,"question":"Le produit de deux matrices A et B est commutatif, c'est-à-dire AB = BA.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Le produit matriciel n'est généralement pas commutatif.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54797,"question":"Résoudre l'inéquation 2x² - 8x + 6 ≤ 0.","option_a":"x ∈ [1, 3]","option_b":"x ∈ ]-∞, 1] ∪ [3, +∞[","option_c":"x ∈ [0, 4]","option_d":"x ∈ [-1, 5]","option_e":"","option_f":"","bonne_reponse":"A","explication":"Résolution : 2(x-1)(x-3) ≤ 0 ⇒ solution entre les racines.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54798,"question":"Soit A = [[1, 2], [3, 4]]. Quelle est la trace de A ?","option_a":"5","option_b":"7","option_c":"-3","option_d":"10","option_e":"","option_f":"","bonne_reponse":"A","explication":"La trace est la somme des éléments diagonaux : 1 + 4 = 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54799,"question":"Toute matrice carrée peut être diagonalisée.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Seules les matrices ayant n vecteurs propres linéairement indépendants le peuvent.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54800,"question":"Quelle est la solution de l'équation matricielle AX = B, où A est inversible ?","option_a":"X = A⁻¹B","option_b":"X = BA⁻¹","option_c":"X = AB","option_d":"X = B - A","option_e":"","option_f":"","bonne_reponse":"A","explication":"Multiplier les deux côtés par A⁻¹ donne X = A⁻¹B.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]