Quiz — devoir-de-synthèse-n°3--2010-2011(mr-rakrouki-mourad).pdf

Question 1 sur 10 20:00

[{"id":3661,"question":"Quelle est la dérivée de la fonction f(x) = 3x² - 5x + 2 ?","option_a":"A. 6x - 5","option_b":"B. 3x - 5","option_c":"C. 6x² - 5","option_d":"D. 6x - 5x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme s'obtient en appliquant la règle de dérivation terme à terme : (3x²)' = 6x, (-5x)' = -5 et (2)' = 0.","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. 6x - 5\", \"b\": \"B. 3x - 5\", \"c\": \"C. 6x² - 5\", \"d\": \"D. 6x - 5","_debug_options_count":4},{"id":3662,"question":"La fonction f(x) = (2x + 1)\/(x - 3) admet une asymptote verticale en x = 3.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une asymptote verticale existe lorsque le dénominateur s'annule sans que le numérateur ne s'annule au même point. Ici, x = 3 annule le dénominateur (x - 3) mais pas le numérateur (2*3 + 1 = 7 ≠ 0).","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":3663,"question":"Quelle est la solution de l'équation différentielle y' + 2y = 0 ?","option_a":"A. y = Ce^(-2x)","option_b":"B. y = Ce^(2x)","option_c":"C. y = C + 2x","option_d":"D. y = Cx^2","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une équation différentielle linéaire du premier ordre. La 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 = C + 2x\", \"d","_debug_options_count":4},{"id":3664,"question":"Si X suit une loi binomiale B(n=10, p=0,3), quelle est la probabilité P(X=3) ?","option_a":"A. C(10,3) * 0,3^3 * 0,7^7","option_b":"B. 0,3^3","option_c":"C. 10 * 0,3 * 0,7^9","option_d":"D. C(10,3) * 0,3^7 * 0,7^3","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une loi binomiale, P(X=k) = C(n,k) * p^k * (1-p)^(n-k). Ici, n=10, k=3, p=0,3.","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. C(10,3) * 0,3^3 * 0,7^7\", \"b\": \"B. 0,3^3\", \"c\": \"C. 10 * 0,3 *","_debug_options_count":4},{"id":3665,"question":"La fonction f(x) = x^3 - 3x est strictement croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f'(x) = 3x² - 3 s'annule en x = ±1. La fonction est décroissante sur ]-1, 1[ et croissante ailleurs.","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":3666,"question":"Quelle est l'équation de la tangente à la courbe y = ln(x) au point d'abscisse x = 1 ?","option_a":"A. y = x - 1","option_b":"B. y = x + 1","option_c":"C. y = 1\/x","option_d":"D. y = ln(1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La tangente en x=1 a pour équation y = f(1) + f'(1)(x-1). Ici, f(1)=0, f'(1)=1, donc y = x - 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. y = x - 1\", \"b\": \"B. y = x + 1\", \"c\": \"C. y = 1\/x\", \"d\": \"D. y","_debug_options_count":4},{"id":3667,"question":"Soit X une variable aléatoire suivant une loi normale N(μ=5, σ=2). Quelle est P(X ≤ 7) ?","option_a":"A. 0,8413","option_b":"B. 0,1587","option_c":"C. 0,6915","option_d":"D. 0,3085","option_e":"","option_f":"","bonne_reponse":"a","explication":"On standardise : Z = (7-5)\/2 = 1. P(X ≤ 7) = P(Z ≤ 1) ≈ 0,8413 (valeur lue dans la table de la loi normale centrée réduite).","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. 0,8413\", \"b\": \"B. 0,1587\", \"c\": \"C. 0,6915\", \"d\": \"D. 0,3085\"}","_debug_options_count":4},{"id":3668,"question":"La suite définie par u_{n+1} = 2u_n + 1 avec u_0 = 0 est convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Cette suite est arithmético-géométrique. Sa limite, si elle existe, vérifie L = 2L + 1 ⇒ L = -1. Cependant, la suite diverge vers +∞ car le terme constant domine.","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":3669,"question":"Quelle est la limite de (sin(x))\/x quand x tend vers 0 ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est un résultat fondamental en analyse : lim_{x→0} (sin(x))\/x = 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. -∞\"}}","_debug_options_count":4},{"id":3670,"question":"Soit f une fonction continue sur [a,b]. Le théorème des valeurs intermédiaires garantit que f prend toutes les valeurs entre f(a) et f(b).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le théorème des valeurs intermédiaires stipule que pour toute valeur k entre f(a) et f(b), il existe c ∈ [a,b] tel que f(c) = k.","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}]