Quiz Mathématiques — 4ème Année Secondaire : Algèbre, Analyse & Probabilités

Question 1 sur 10 10:00

[{"id":47321,"question":"Quelle est la solution de l'équation 2x² - 5x + 2 = 0 ?","option_a":"x = 1 ou x = 2","option_b":"x = 2 ou x = 1\/2","option_c":"x = -1 ou x = -2","option_d":"x = 1\/2 ou x = 2","option_e":"","option_f":"","bonne_reponse":"B","explication":"Résolvons l'équation 2x² - 5x + 2 = 0. Le discriminant Δ = (-5)² - 4*2*2 = 25 - 16 = 9. Les solutions sont x = (5 ± √9)\/4, soit x = 2 ou x = 1\/2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":47322,"question":"La fonction f(x) = x³ - 3x² + 2x est croissante sur l'intervalle [0, 1].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée f'(x) = 3x² - 6x + 2. Sur [0, 1], f'(x) = 3x² - 6x + 2. Calculons f'(0) = 2 \u003E 0 et f'(1) = -1 \u003C 0. La fonction n'est pas croissante sur tout l'intervalle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":47323,"question":"Quelle est la limite de la fonction f(x) = (3x² + 2x - 1)\/(x² - 4) lorsque x tend vers l'infini ?","option_a":"3","option_b":"2","option_c":"1","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"A","explication":"Pour x → ∞, on divise numérateur et dénominateur par x² : f(x) = (3 + 2\/x - 1\/x²)\/(1 - 4\/x²). Les termes en 1\/x et 1\/x² tendent vers 0, donc la limite est 3\/1 = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":47324,"question":"Une variable aléatoire X suit une loi normale N(μ, σ²). Si μ = 5 et σ = 2, quelle est la probabilité P(X ≤ 7) ?","option_a":"0,8413","option_b":"0,6915","option_c":"0,5","option_d":"0,3085","option_e":"","option_f":"","bonne_reponse":"A","explication":"On standardise : Z = (X - μ)\/σ = (7 - 5)\/2 = 1. P(X ≤ 7) = P(Z ≤ 1) ≈ 0,8413 (valeur lue dans la table de la loi normale).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":47325,"question":"La suite définie par uₙ = (n² + 1)\/n est convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"uₙ = n + 1\/n. Lorsque n → ∞, uₙ → ∞. La suite est donc divergente.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":47326,"question":"Quel est le développement limité de ln(1 + x) à l'ordre 2 autour de 0 ?","option_a":"x - x²\/2 + o(x²)","option_b":"x + x²\/2 + o(x²)","option_c":"x - x² + o(x²)","option_d":"x + x² + o(x²)","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le développement limité de ln(1 + x) à l'ordre 2 est ln(1 + x) = x - x²\/2 + o(x²).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":47327,"question":"Dans une classe de 30 élèves, 12 sont des filles. Quelle est la probabilité de choisir au hasard 2 filles ?","option_a":"12\/30 * 11\/29","option_b":"12\/30 * 12\/29","option_c":"12\/30 + 11\/29","option_d":"12\/30 * 30\/29","option_e":"","option_f":"","bonne_reponse":"A","explication":"La probabilité est P = (12\/30) * (11\/29) = 132\/870 ≈ 0,1517.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":47328,"question":"La dérivée de la fonction f(x) = e^(2x) est f'(x) = 2e^(2x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de e^(u(x)) est u'(x)e^(u(x)). Ici, u(x) = 2x, donc u'(x) = 2. Ainsi, f'(x) = 2e^(2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":47329,"question":"Quelle est la somme des 5 premiers termes de la suite géométrique de premier terme 3 et de raison 2 ?","option_a":"93","option_b":"48","option_c":"96","option_d":"31","option_e":"","option_f":"","bonne_reponse":"A","explication":"La somme S₅ = 3*(2⁵ - 1)\/(2 - 1) = 3*(32 - 1) = 93.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":47330,"question":"L'intégrale ∫₀¹ (3x² + 2x) dx est égale à :","option_a":"2","option_b":"3","option_c":"4","option_d":"1","option_e":"","option_f":"","bonne_reponse":"B","explication":"Calculons l'intégrale : ∫(3x² + 2x) dx = x³ + x². Évaluée de 0 à 1 : (1 + 1) - (0 + 0) = 2. L'intégrale vaut donc 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]