Quiz — Centrale-supélec 2007 (doc réponse).pdf

Question 1 sur 10 20:00

[{"id":99070,"question":"Quelle est la dérivée de la fonction f(x) = ln(x² + 1) ?","option_a":"f'(x) = 2x \/ (x² + 1)","option_b":"f'(x) = 1 \/ (x² + 1)","option_c":"f'(x) = 2x \/ (x² - 1)","option_d":"f'(x) = 1 \/ (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\": \"f'(x) = 2x \/ (x² + 1)\", \"b\": \"f'(x) = 1 \/ (x² + 1)\", \"c\": \"f'(x","_debug_options_count":4},{"id":99071,"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, la matrice 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":99072,"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 e^(x) + B e^(-x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation caractéristique est r² + 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,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"y(x) = A cos(2x) + B sin(2x)\", \"b\": \"y(x) = A e^(2x) + B e^(-2x)\"","_debug_options_count":4},{"id":99073,"question":"Soit X une variable aléatoire suivant une loi normale centrée réduite. Quelle est la probabilité P(-1 ≤ X ≤ 1) ?","option_a":"≈ 0.6827","option_b":"≈ 0.9545","option_c":"≈ 0.9973","option_d":"≈ 0.3413","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une loi normale centrée réduite, environ 68,27 % des valeurs se situent entre -1 et 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\": \"≈ 0.6827\", \"b\": \"≈ 0.9545\", \"c\": \"≈ 0.9973\", \"d\": \"≈ 0.34","_debug_options_count":4},{"id":99074,"question":"Soit f une fonction continue sur [a, b]. Si f(a) \u003C 0 et f(b) \u003E 0, alors il existe c ∈ ]a, b[ tel que f(c) = 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est le théorème des valeurs intermédiaires : une fonction continue sur un intervalle qui prend des valeurs de signes opposés s'annule nécessairement.","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":99075,"question":"Quelle est la valeur de l'intégrale ∫(de 0 à π) sin(x) dx ?","option_a":"0","option_b":"1","option_c":"2","option_d":"π","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'intégrale de sin(x) de 0 à π est [-cos(x)] de 0 à π = -cos(π) - (-cos(0)) = 1 - (-1) = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"π\"}}","_debug_options_count":4},{"id":99076,"question":"Soit (u_n) une suite définie par u₀ = 1 et u_{n+1} = 2u_n + 1. Quelle est l'expression de u_n en fonction de n ?","option_a":"u_n = 2^n - 1","option_b":"u_n = 2^n + 1","option_c":"u_n = 3^n - 1","option_d":"u_n = 3^n + 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La suite est arithmético-géométrique. En résolvant, on trouve u_n = 2^n - 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\": \"u_n = 2^n - 1\", \"b\": \"u_n = 2^n + 1\", \"c\": \"u_n = 3^n - 1\", \"d\": ","_debug_options_count":4},{"id":99077,"question":"Soit A et B deux matrices carrées de même taille. L'égalité (A + B)² = A² + 2AB + B² est toujours vraie.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Cette égalité n'est vraie que si A et B commutent (AB = BA). Sinon, il faut développer (A + B)² = A² + AB + BA + B².","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":99078,"question":"Quelle est la limite de (1 + 1\/n)^n lorsque n tend vers l'infini ?","option_a":"0","option_b":"1","option_c":"e","option_d":"+∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"Cette limite est la définition du nombre e, base du logarithme naturel.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"e\", \"d\": \"+∞\"}}","_debug_options_count":4},{"id":99079,"question":"Soit f une fonction dérivable sur ℝ. Si f'(x) = 0 pour tout x ∈ ℝ, alors f est une fonction constante.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si la dérivée d'une fonction est nulle sur un intervalle, alors cette fonction est constante sur cet intervalle (théorème de Rolle).","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}]