Mathématiques MPSI : Testez vos connaissances en algèbre et analyse

Question 1 sur 10 10:00

[{"id":67112,"question":"Soit E un espace vectoriel de dimension 3. Quelle est la dimension maximale d'un sous-espace vectoriel de E ?","option_a":"0","option_b":"1","option_c":"2","option_d":"3","option_e":"","option_f":"","bonne_reponse":"D","explication":"La dimension maximale d'un sous-espace vectoriel d'un espace de dimension n est n. Ici, E est de dimension 3, donc la dimension maximale est 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":67113,"question":"La suite (u_n) définie par u_0 = 1 et u_{n+1} = 2u_n + 1 est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La suite est strictement croissante et non majorée, donc elle diverge vers +∞. Elle n'est pas convergente.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":67114,"question":"Quelle est la dérivée de la fonction f(x) = ln(x² + 1) ?","option_a":"f'(x) = 2x \/ (x² + 1)","option_b":"f'(x) = 2x","option_c":"f'(x) = 1 \/ (x² + 1)","option_d":"f'(x) = x \/ (x² + 1)","option_e":"","option_f":"","bonne_reponse":"A","explication":"En appliquant la règle de la chaîne, f'(x) = (2x) \/ (x² + 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":67115,"question":"Soit A une matrice carrée inversible. Quelle est la valeur de det(A⁻¹) ?","option_a":"det(A)","option_b":"1\/det(A)","option_c":"0","option_d":"det(A)²","option_e":"","option_f":"","bonne_reponse":"B","explication":"On sait que det(A⁻¹) = 1 \/ det(A) pour toute matrice inversible A.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":67116,"question":"La fonction f(x) = x³ - 3x + 2 admet-elle un extremum local en x = 1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"f'(x) = 3x² - 3. f'(1) = 0, mais f''(1) = 6 \u003E 0, donc x = 1 est un minimum local. L'affirmation est vraie.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":67117,"question":"Quelle est la solution générale de l'équation différentielle y' + 2y = 0 ?","option_a":"y(x) = Ce^{-2x}","option_b":"y(x) = Ce^{2x}","option_c":"y(x) = Cx + C","option_d":"y(x) = C \/ x","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'équation est linéaire du premier ordre. La solution générale est y(x) = Ce^{-2x}, où C est une constante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":67118,"question":"Soit u et v deux vecteurs d'un espace euclidien. Quelle est la valeur de ||u + v||² ?","option_a":"||u||² + ||v||²","option_b":"||u||² + ||v||² + 2\u003Cu,v\u003E","option_c":"||u||² + ||v||² - 2\u003Cu,v\u003E","option_d":"||u||² - ||v||²","option_e":"","option_f":"","bonne_reponse":"B","explication":"Par la formule de polarisation, ||u + v||² = ||u||² + ||v||² + 2\u003Cu,v\u003E.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":67119,"question":"La série de terme général 1\/n² converge-t-elle ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La série de Riemann de terme général 1\/n^α converge si et seulement si α \u003E 1. Ici, α = 2 \u003E 1, donc la série converge.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":67120,"question":"Quelle est la probabilité d'obtenir un double six en lançant deux dés équilibrés ?","option_a":"1\/6","option_b":"1\/12","option_c":"1\/36","option_d":"1\/18","option_e":"","option_f":"","bonne_reponse":"C","explication":"Il y a 6 × 6 = 36 issues possibles. Une seule issue correspond à un double six, donc la probabilité est 1\/36.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":67121,"question":"Soit f une fonction continue sur [a, b]. Quelle est la valeur moyenne de f sur cet intervalle ?","option_a":"(f(b) - f(a)) \/ (b - a)","option_b":"(b - a) \/ (f(b) - f(a))","option_c":"1\/(b-a) ∫[a à b] f(x) dx","option_d":"∫[a à b] f(x) dx","option_e":"","option_f":"","bonne_reponse":"C","explication":"La valeur moyenne d'une fonction continue f sur [a, b] est donnée par (1\/(b-a)) ∫[a à b] f(x) dx.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]