Testez vos connaissances : CAPES Mathématiques 2010 — Algèbre, Analyse & Probabilités

Question 1 sur 10 10:00

[{"id":43136,"question":"Soit une suite définie par u_{n+1} = 2u_n - 1 et u_0 = 3. Que vaut u_2 ?","option_a":"5","option_b":"7","option_c":"9","option_d":"11","option_e":"","option_f":"","bonne_reponse":"B","explication":"Calculons u_1 = 2*3 - 1 = 5, puis u_2 = 2*5 - 1 = 9. La bonne réponse est donc 9.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":43137,"question":"La fonction f(x) = x^3 - 3x est-elle strictement croissante sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La dérivée f'(x) = 3x^2 - 3 s'annule en x = ±1. La fonction n'est donc pas strictement croissante sur ℝ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":43138,"question":"Quelle est la solution générale de l'équation différentielle y'' + 4y = 0 ?","option_a":"y(x) = A cos(2x) + B sin(2x)","option_b":"y(x) = A e^{2x} + B e^{-2x}","option_c":"y(x) = A x + B","option_d":"y(x) = A cos(x) + B sin(x)","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'équation caractéristique est r^2 + 4 = 0, soit r = ±2i. La solution générale est donc y(x) = A cos(2x) + B sin(2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":43139,"question":"Soit X une variable aléatoire suivant une loi binomiale B(n=10, p=0.3). Que vaut E(X) ?","option_a":"3","option_b":"7","option_c":"10","option_d":"30","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'espérance d'une loi binomiale B(n,p) est E(X) = n*p = 10*0.3 = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":43140,"question":"La série ∑_{n=1}^∞ 1\/n diverge-t-elle ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La série harmonique ∑_{n=1}^∞ 1\/n est une série de Riemann divergente (p=1 ≤ 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":43141,"question":"Quelle est la valeur de l'intégrale ∫_{0}^{π} sin(x) dx ?","option_a":"0","option_b":"1","option_c":"2","option_d":"π","option_e":"","option_f":"","bonne_reponse":"C","explication":"∫ sin(x) dx = -cos(x) + C. Ainsi, ∫_{0}^{π} sin(x) dx = [-cos(π)] - [-cos(0)] = 1 - (-1) = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":43142,"question":"Soit A une matrice carrée d'ordre 2. Si det(A) = 0, alors A est inversible.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Une matrice est inversible si et seulement si son déterminant est non nul. Si det(A) = 0, A n'est pas inversible.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":43143,"question":"Quelle est la limite de la suite u_n = (n^2 + 1)\/(n^2 - 1) quand n tend vers l'infini ?","option_a":"0","option_b":"1","option_c":"2","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"B","explication":"En divisant numérateur et dénominateur par n^2, on obtient u_n = (1 + 1\/n^2)\/(1 - 1\/n^2). La limite quand n→∞ est donc 1\/1 = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":43144,"question":"Soit f une fonction continue sur [a,b]. Le théorème des valeurs intermédiaires garantit que pour tout y entre f(a) et f(b), il existe c ∈ [a,b] tel que f(c) = y.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"C'est exactement l'énoncé du théorème des valeurs intermédiaires (TVI).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":43145,"question":"Quelle est la dérivée de la fonction f(x) = ln(x^2 + 1) ?","option_a":"2x\/(x^2 + 1)","option_b":"1\/(x^2 + 1)","option_c":"(2x + 1)\/(x^2 + 1)","option_d":"2\/(x^2 + 1)","option_e":"","option_f":"","bonne_reponse":"A","explication":"En utilisant la dérivée de ln(u) = u'\/u, on obtient f'(x) = (2x)\/(x^2 + 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]