Quiz — M-EA-INT-JMF-03.pdf

Question 1 sur 10 20:00

[{"id":58305,"question":"Quelle est la solution générale de l'équation différentielle y' + 2y = 0 ?","option_a":"A. y = Ce^{-2x}","option_b":"B. y = Ce^{2x}","option_c":"C. y = Cx^2","option_d":"D. y = C\/x","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation différentielle y' + 2y = 0 est une équation linéaire du premier ordre. Sa solution générale est y = Ce^{-2x}, où C est une constante réelle.","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 = Ce^{-2x}\", \"b\": \"B. y = Ce^{2x}\", \"c\": \"C. y = Cx^2\", \"d\":","_debug_options_count":4},{"id":58306,"question":"La fonction f(x) = x^3 - 3x + 1 est convexe sur l'intervalle [0, +∞[.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction f est convexe si sa dérivée seconde est positive. Or f''(x) = 6x, qui est positive sur [0, +∞[. Donc l'affirmation est fausse : elle est convexe sur cet 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":58307,"question":"Soit une matrice A de taille 3x3 avec det(A) = 0. Que peut-on dire de ses colonnes ?","option_a":"A. Elles sont linéairement indépendantes","option_b":"B. Elles forment une base de R^3","option_c":"C. Elles sont linéairement dépendantes","option_d":"D. Elles sont orthogonales","option_e":"","option_f":"","bonne_reponse":"c","explication":"Si le déterminant d'une matrice est nul, ses colonnes (ou lignes) sont linéairement dépendantes. Elles ne peuvent donc pas former une base de R^3.","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. Elles sont linéairement indépendantes\", \"b\": \"B. Elles forme","_debug_options_count":4},{"id":58308,"question":"Quelle est la valeur de l'intégrale ∫_{0}^{π} sin(x) dx ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 2","option_d":"D. π","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'intégrale de sin(x) entre 0 et π est égale à [-cos(x)]_{0}^{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\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. 2\", \"d\": \"D. π\"}}","_debug_options_count":4},{"id":58309,"question":"Une variable aléatoire X suit une loi normale N(μ, σ²). Que vaut P(X ≤ μ) ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une loi normale centrée sur μ, P(X ≤ μ) = 0,5 car la distribution est symétrique autour de la moyenne.","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":58310,"question":"Soit la suite définie par u_{n+1} = 2u_n + 1 avec u_0 = 0. Quelle est la limite de la suite (u_n) ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. +∞","option_d":"D. -1","option_e":"","option_f":"","bonne_reponse":"c","explication":"La suite est strictement croissante et non majorée. En résolvant l'équation l = 2l + 1, on trouve l = -1, mais comme u_0 = 0 et la suite est croissante, elle tend vers +∞.","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. 0\", \"b\": \"B. 1\", \"c\": \"C. +∞\", \"d\": \"D. -1\"}}","_debug_options_count":4},{"id":58311,"question":"Quelle est la valeur de la somme S = Σ_{k=0}^{n} C(n,k) ?","option_a":"A. 2^n","option_b":"B. n^2","option_c":"C. n!","option_d":"D. 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"La somme des coefficients binomiaux pour un n donné est égale à 2^n, d'après la formule du binôme de Newton.","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. 2^n\", \"b\": \"B. n^2\", \"c\": \"C. n!\", \"d\": \"D. 0\"}}","_debug_options_count":4},{"id":58312,"question":"Soit f une fonction dérivable sur R. Si f'(x) \u003E 0 pour tout x ∈ R, alors f est strictement croissante sur R.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si la dérivée est strictement positive sur un intervalle, la fonction est strictement croissante sur cet intervalle. L'affirmation est donc vraie.","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":58313,"question":"Quelle est la solution de l'équation différentielle y'' + 4y' + 4y = 0 ?","option_a":"A. y = e^{-2x}(Ax + B)","option_b":"B. y = e^{2x}(Ax + B)","option_c":"C. y = Ax^2 + Bx + C","option_d":"D. y = A cos(2x) + B sin(2x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation caractéristique est r² + 4r + 4 = 0, qui a une racine double r = -2. La solution générale est donc y = e^{-2x}(Ax + B).","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^{-2x}(Ax + B)\", \"b\": \"B. y = e^{2x}(Ax + B)\", \"c\": \"C. y","_debug_options_count":4},{"id":58314,"question":"Soit X une variable aléatoire discrète prenant les valeurs 1, 2 et 3 avec P(X=1)=0.2, P(X=2)=0.5 et P(X=3)=0.3. Que vaut E(X) ?","option_a":"A. 2","option_b":"B. 2.1","option_c":"C. 1.9","option_d":"D. 2.2","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'espérance E(X) = 1×0.2 + 2×0.5 + 3×0.3 = 0.2 + 1 + 0.9 = 2.1.","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. 2\", \"b\": \"B. 2.1\", \"c\": \"C. 1.9\", \"d\": \"D. 2.2\"}}","_debug_options_count":4}]