Quiz — Centrale_2009_MP_M1_Corrige.pdf

Question 1 sur 10 20:00

[{"id":54197,"question":"Quelle est la solution générale de l'équation différentielle y'' + 4y = 0 ?","option_a":"A. y = C1 cos(2x) + C2 sin(2x)","option_b":"B. y = C1 e^(2x) + C2 e^(-2x)","option_c":"C. y = C1 x + C2","option_d":"D. y = C1 e^(x) + C2 e^(-x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation caractéristique associée est r² + 4 = 0, dont les racines sont r = ±2i. La solution générale est donc y = C1 cos(2x) + C2 sin(2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. y = C1 cos(2x) + C2 sin(2x)\", \"b\": \"B. y = C1 e^(2x) + C2 e^(-","_debug_options_count":4},{"id":54198,"question":"Soit f(x) = x² e^(-x). La dérivée de f est-elle toujours positive ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f'(x) = e^(-x)(2x - x²) change de signe. Elle est positive pour x ∈ ]0, 2[ et négative ailleurs, donc f n'est pas toujours croissante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":54199,"question":"Quelle est la limite de la suite u_n = (n² + 1)\/(n² + n + 1) quand n tend vers l'infini ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. +∞","option_d":"D. -1","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par n², on obtient lim(u_n) = lim(1 + 1\/n²)\/(1 + 1\/n + 1\/n²) = 1\/1 = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. +∞\", \"d\": \"D. -1\"}}","_debug_options_count":4},{"id":54200,"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,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":54201,"question":"Quelle est la solution de l'équation différentielle y' + 2y = e^(-x) ?","option_a":"A. y = C e^(-2x) + e^(-x)","option_b":"B. y = C e^(-x) + x e^(-x)","option_c":"C. y = C e^(-2x) + x e^(-x)","option_d":"D. y = C e^(-x)","option_e":"","option_f":"","bonne_reponse":"c","explication":"La solution générale est y = C e^(-2x) + x e^(-x), où C est une constante. Le terme x e^(-x) est une solution particulière.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"A. y = C e^(-2x) + e^(-x)\", \"b\": \"B. y = C e^(-x) + x e^(-x)\", \"c","_debug_options_count":4},{"id":54202,"question":"Soit f(x) = ln(x² + 1). Quelle est la dérivée de f ?","option_a":"A. f'(x) = 2x\/(x² + 1)","option_b":"B. f'(x) = 1\/(x² + 1)","option_c":"C. f'(x) = 2x","option_d":"D. f'(x) = x\/(x² + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la formule de dérivation des fonctions composées, f'(x) = (2x)\/(x² + 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. f'(x) = 2x\/(x² + 1)\", \"b\": \"B. f'(x) = 1\/(x² + 1)\", \"c\": \"C.","_debug_options_count":4},{"id":54203,"question":"Soit (u_n) une suite définie par u_0 = 1 et u_{n+1} = (u_n + 2)\/3. Cette suite est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La suite est convergente car elle est contractante (|u_{n+1} - u_n| ≤ (2\/3)|u_n - u_{n-1}|) et bornée. Sa limite est L = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":54204,"question":"Quelle est la valeur propre d'une matrice triangulaire supérieure ?","option_a":"A. Les valeurs propres sont les éléments diagonaux","option_b":"B. Les valeurs propres sont toujours nulles","option_c":"C. Les valeurs propres sont les éléments hors diagonale","option_d":"D. Les valeurs propres sont complexes","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une matrice triangulaire, les valeurs propres sont exactement les éléments de la diagonale principale.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. Les valeurs propres sont les éléments diagonaux\", \"b\": \"B. L","_debug_options_count":4},{"id":54205,"question":"Soit f(x) = x³ - 3x² + 2. Combien de racines réelles cette fonction possède-t-elle ?","option_a":"A. 1","option_b":"B. 2","option_c":"C. 3","option_d":"D. 0","option_e":"","option_f":"","bonne_reponse":"b","explication":"En étudiant les variations de f (f'(x) = 3x² - 6x), on trouve deux extrema locaux. La fonction coupe l'axe des abscisses en deux points distincts.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A. 1\", \"b\": \"B. 2\", \"c\": \"C. 3\", \"d\": \"D. 0\"}}","_debug_options_count":4},{"id":54206,"question":"Soit A une matrice symétrique. Ses valeurs propres sont-elles toujours réelles ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, une matrice symétrique à coefficients réels a toujours des valeurs propres réelles (théorème spectral).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]