Quiz — Lycée Pilote Sousse - DS 3 2012 (+Corrigé).pdf

Question 1 sur 10 20:00

[{"id":83745,"question":"Quelle est la dérivée de la fonction f(x) = x² * e^x ?","option_a":"A) (2x + x²) * e^x","option_b":"B) (x + 2x²) * e^x","option_c":"C) (2x + x²) * e^(-x)","option_d":"D) (x + x²) * e^x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'un produit u*v est u'v + uv'. Ici, u = x² (u' = 2x) et v = e^x (v' = e^x). Donc f'(x) = 2x*e^x + x²*e^x = (2x + x²)*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) (2x + x²) * e^x\", \"b\": \"B) (x + 2x²) * e^x\", \"c\": \"C) (2x + ","_debug_options_count":4},{"id":83746,"question":"La fonction f(x) = ln(x) est définie pour tout x réel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction ln(x) est définie uniquement pour x \u003E 0. Elle n'est pas définie pour x ≤ 0.","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":83747,"question":"Quelle est la limite de la suite u_n = (3n² + 2n - 1)\/(n² + 5) quand n tend vers l'infini ?","option_a":"A) 0","option_b":"B) 3","option_c":"C) +∞","option_d":"D) -∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant le numérateur et le dénominateur par n², on obtient u_n = (3 + 2\/n - 1\/n²)\/(1 + 5\/n²). Quand n → ∞, 2\/n, 1\/n² et 5\/n² tendent vers 0, donc la limite est 3\/1 = 3.","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) +∞\", \"d\": \"D) -∞\"}}","_debug_options_count":4},{"id":83748,"question":"Dans une loi binomiale B(n, p), l'espérance mathématique est égale à :","option_a":"A) n * p","option_b":"B) n * (1 - p)","option_c":"C) p * (1 - p)","option_d":"D) n² * p","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'espérance d'une loi binomiale B(n, p) est E(X) = n * p, où n est le nombre d'épreuves et p la probabilité de succès.","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) n * p\", \"b\": \"B) n * (1 - p)\", \"c\": \"C) p * (1 - p)\", \"d\": \"D)","_debug_options_count":4},{"id":83749,"question":"Le produit scalaire de deux vecteurs orthogonaux est toujours égal à 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, deux vecteurs sont orthogonaux si et seulement si leur produit scalaire est nul.","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":83750,"question":"Quelle est la solution de l'équation différentielle y' + 2y = 0 avec y(0) = 3 ?","option_a":"A) y = 3e^(-2x)","option_b":"B) y = 3e^(2x)","option_c":"C) y = -3e^(-2x)","option_d":"D) y = 3e^(-x\/2)","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation y' + 2y = 0 a pour solution générale y = C*e^(-2x). Avec y(0) = 3, on trouve C = 3, donc y = 3e^(-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 = 3e^(-2x)\", \"b\": \"B) y = 3e^(2x)\", \"c\": \"C) y = -3e^(-2x)\",","_debug_options_count":4},{"id":83751,"question":"Dans un arbre pondéré, la somme des probabilités issues d'un même nœud est toujours égale à 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une propriété fondamentale des arbres pondérés : la somme des probabilités des branches issues d'un même nœud est égale 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":83752,"question":"Quelle est la dérivée de la fonction f(x) = cos(3x) ?","option_a":"A) -3sin(3x)","option_b":"B) 3sin(3x)","option_c":"C) -sin(3x)","option_d":"D) sin(3x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de cos(u) est -u'*sin(u). Ici, u = 3x, donc u' = 3. Ainsi, f'(x) = -3*sin(3x).","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) -3sin(3x)\", \"b\": \"B) 3sin(3x)\", \"c\": \"C) -sin(3x)\", \"d\": \"D) s","_debug_options_count":4},{"id":83753,"question":"La fonction f(x) = x^3 - 3x + 2 admet un extremum local en x = 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"f'(x) = 3x² - 3. f'(1) = 0 et f''(x) = 6x, donc f''(1) = 6 \u003E 0. La fonction admet donc un minimum local en 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":83754,"question":"Quelle est la probabilité d'obtenir exactement 2 fois le chiffre 6 en lançant un dé équilibré 5 fois ?","option_a":"A) C(5,2) * (1\/6)^2 * (5\/6)^3","option_b":"B) C(5,2) * (1\/6)^3 * (5\/6)^2","option_c":"C) (1\/6)^2 * (5\/6)^3","option_d":"D) 5 * (1\/6)^2","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une loi binomiale B(5, 1\/6). La probabilité d'obtenir exactement 2 succès est C(5,2) * (1\/6)^2 * (5\/6)^(5-2).","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(5,2) * (1\/6)^2 * (5\/6)^3\", \"b\": \"B) C(5,2) * (1\/6)^3 * (5\/6)","_debug_options_count":4}]