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Question 1 sur 10 10:00

[{"id":8393,"question":"Quelle est la dérivée de la fonction f(x) = x² + 3x - 5 ?","option_a":"2x + 3","option_b":"2x² + 3","option_c":"x + 3","option_d":"2x + 5","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de x² est 2x, celle de 3x est 3, et celle de -5 est 0. Donc f'(x) = 2x + 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":8394,"question":"Une suite géométrique de raison q = -2 et de premier terme u₀ = 3 est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Une suite géométrique est convergente si et seulement si |q| \u003C 1. Ici, |q| = 2 \u003E 1, donc la suite diverge.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":8395,"question":"Quelle est la solution générale de l'équation différentielle y' + 2y = 0 ?","option_a":"y = Ce^(-2x)","option_b":"y = Ce^(2x)","option_c":"y = Cx + C","option_d":"y = C\/x","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'équation est de la forme y' + ay = 0, sa solution générale est y = Ce^(-ax). Ici, a = 2, donc y = Ce^(-2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":8396,"question":"Dans un triangle ABC, si sin(A) = 0,5, alors A mesure nécessairement 30°.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"sin(A) = 0,5 implique A = 30° ou A = 150° (car sin(150°) = 0,5). Donc A ne mesure pas nécessairement 30°.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":8397,"question":"Quelle est la valeur de la limite lim(x→+∞) (3x² + 2x - 1)\/(x² + 5) ?","option_a":"0","option_b":"3","option_c":"1\/5","option_d":"5","option_e":"","option_f":"","bonne_reponse":"B","explication":"En divisant numérateur et dénominateur par x², on obtient (3 + 2\/x - 1\/x²)\/(1 + 5\/x²). Quand x→+∞, les termes en 1\/x tendent vers 0, donc la limite est 3\/1 = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":8398,"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":"B","explication":"Le déterminant d'une matrice 2x2 [[a, b], [c, d]] est ad - bc, et non ad + bc.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":8399,"question":"Quelle est l'intégrale indéfinie de f(x) = 4x³ - 2x + 1 ?","option_a":"x⁴ - x² + x + C","option_b":"x⁴ - x² + C","option_c":"4x⁴ - 2x² + x + C","option_d":"x⁴ - x² + 1 + C","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'intégrale de 4x³ est x⁴, celle de -2x est -x², celle de 1 est x, et on ajoute la constante C.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":8400,"question":"Une variable aléatoire X suit une loi normale N(μ, σ²). Si μ = 5 et σ = 2, alors P(X ≤ 7) est-elle supérieure à 0,5 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Pour une loi normale, P(X ≤ μ) = 0,5. Ici, μ = 5, et 7 \u003E 5, donc P(X ≤ 7) \u003E 0,5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":8401,"question":"Quelle est la forme algébrique du nombre complexe z = 3(cos(π\/3) + i sin(π\/3)) ?","option_a":"3\/2 + i(3√3)\/2","option_b":"3 + i3","option_c":"-3\/2 + i(3√3)\/2","option_d":"3\/2 - i(3√3)\/2","option_e":"","option_f":"","bonne_reponse":"A","explication":"En utilisant cos(π\/3) = 1\/2 et sin(π\/3) = √3\/2, on obtient z = 3*(1\/2 + i√3\/2) = 3\/2 + i(3√3)\/2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":8402,"question":"La fonction f(x) = x³ - 3x² + 4 est-elle convexe sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La convexité dépend du signe de la dérivée seconde. f''(x) = 6x - 6. f''(x) = 0 en x = 1, et f''(x) \u003C 0 pour x \u003C 1, donc f n'est pas convexe sur ℝ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]