Quiz — Centrale_2005_TSI_M1_Enonce.pdf

Question 1 sur 10 20:00

[{"id":121166,"question":"Quelle est la dérivée de la fonction f(x) = ln(x² + 1) ?","option_a":"A) 2x \/ (x² + 1)","option_b":"B) 2x \/ (x² + 1)²","option_c":"C) 1 \/ (x² + 1)","option_d":"D) x \/ (x² + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(u) est u'\/u. Ici, u = x² + 1, donc u' = 2x. Ainsi, 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) 2x \/ (x² + 1)\", \"b\": \"B) 2x \/ (x² + 1)²\", \"c\": \"C) 1 \/ (x²","_debug_options_count":4},{"id":121167,"question":"Soit une matrice A de taille 2x2. 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":121168,"question":"Quelle est la solution générale de l'équation différentielle y'' + 4y = 0 ?","option_a":"A) y = C1e^(2x) + C2e^(-2x)","option_b":"B) y = C1cos(2x) + C2sin(2x)","option_c":"C) y = C1e^(4x) + C2e^(-4x)","option_d":"D) y = C1x + C2","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'équation caractéristique est r² + 4 = 0, soit r = ±2i. La solution générale est donc y = C1cos(2x) + C2sin(2x).","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) y = C1e^(2x) + C2e^(-2x)\", \"b\": \"B) y = C1cos(2x) + C2sin(2x)\"","_debug_options_count":4},{"id":121169,"question":"Soit une suite définie par u_{n+1} = 2u_n + 3 avec u_0 = 1. Quelle est la limite de la suite si elle converge ?","option_a":"A) 0","option_b":"B) 3","option_c":"C) -3","option_d":"D) La suite diverge","option_e":"","option_f":"","bonne_reponse":"b","explication":"Si la suite converge vers L, alors L = 2L + 3 ⇒ L = -3. Mais la suite diverge car u_n = 2^n - 3, qui tend vers l'infini.","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) 3\", \"c\": \"C) -3\", \"d\": \"D) La suite diverge\"}}","_debug_options_count":4},{"id":121170,"question":"La matrice identité I_n est-elle diagonalisable ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La matrice identité est déjà diagonale, donc elle est diagonalisable (ses valeurs propres sont toutes égales à 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":121171,"question":"Quelle est la nature de la conique d'équation x² + y² - 2x + 4y + 5 = 0 ?","option_a":"A) Ellipse","option_b":"B) Hyperbole","option_c":"C) Cercle","option_d":"D) Parabole","option_e":"","option_f":"","bonne_reponse":"c","explication":"En complétant les carrés, on obtient (x-1)² + (y+2)² = 0, qui représente un point (cas dégénéré du cercle).","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) Ellipse\", \"b\": \"B) Hyperbole\", \"c\": \"C) Cercle\", \"d\": \"D) Para","_debug_options_count":4},{"id":121172,"question":"Soit f une fonction continue sur [a, b]. Quelle est la valeur moyenne de f sur cet intervalle ?","option_a":"A) (f(b) - f(a)) \/ (b - a)","option_b":"B) ∫[a à b] f(x) dx \/ (b - a)","option_c":"C) f((a+b)\/2)","option_d":"D) max(f(x)) - min(f(x))","option_e":"","option_f":"","bonne_reponse":"b","explication":"La valeur moyenne d'une fonction continue sur [a, b] est donnée par l'intégrale de f sur [a, b] divisée par la longueur de l'intervalle.","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) (f(b) - f(a)) \/ (b - a)\", \"b\": \"B) ∫[a à b] f(x) dx \/ (b - ","_debug_options_count":4},{"id":121173,"question":"Une matrice symétrique est-elle toujours diagonalisable ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Toute matrice symétrique réelle est diagonalisable dans une base orthonormée (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},{"id":121174,"question":"Quelle est la solution de l'équation différentielle y' = y avec y(0) = 1 ?","option_a":"A) y = e^x","option_b":"B) y = x + 1","option_c":"C) y = 1 \/ (1 - x)","option_d":"D) y = ln(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation y' = y a pour solution générale y = Ce^x. Avec y(0) = 1, on obtient C = 1, soit y = e^x.","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 = e^x\", \"b\": \"B) y = x + 1\", \"c\": \"C) y = 1 \/ (1 - x)\", \"d\":","_debug_options_count":4},{"id":121175,"question":"Soit A une matrice 3x3 avec det(A) = 5. Quelle est la valeur de det(2A) ?","option_a":"A) 5","option_b":"B) 10","option_c":"C) 40","option_d":"D) 80","option_e":"","option_f":"","bonne_reponse":"d","explication":"Pour une matrice A de taille n, det(kA) = k^n det(A). Ici, n = 3, donc det(2A) = 2^3 * 5 = 40.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"A) 5\", \"b\": \"B) 10\", \"c\": \"C) 40\", \"d\": \"D) 80\"}}","_debug_options_count":4}]