Quiz — 2012c.pdf

Question 1 sur 10 20:00

[{"id":36460,"question":"Quelle est la dérivée de la fonction f(x) = x² + 3x - 5 ?","option_a":"A) 2x + 3","option_b":"B) x² + 3","option_c":"C) 2x - 5","option_d":"D) 3x² + 3","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme s'obtient en appliquant la règle : (x^n)' = n*x^(n-1). Ici, f'(x) = 2x + 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) 2x + 3\", \"b\": \"B) x² + 3\", \"c\": \"C) 2x - 5\", \"d\": \"D) 3x² + ","_debug_options_count":4},{"id":36461,"question":"La limite de (sin(x)\/x) quand x tend vers 0 est égale à 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une limite fondamentale en analyse : lim(x→0) sin(x)\/x = 1, utilisée pour les dérivées des fonctions trigonométriques.","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":36462,"question":"Quelle est l'espérance d'une variable aléatoire suivant une loi binomiale B(n=10, p=0.3) ?","option_a":"A) 3","option_b":"B) 0.3","option_c":"C) 7","option_d":"D) 10","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une loi binomiale B(n,p), l'espérance est E(X) = n*p. Ici, E(X) = 10*0.3 = 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) 3\", \"b\": \"B) 0.3\", \"c\": \"C) 7\", \"d\": \"D) 10\"}}","_debug_options_count":4},{"id":36463,"question":"Soit une suite définie par u₀ = 2 et uₙ₊₁ = 3uₙ - 1. Que vaut u₂ ?","option_a":"A) 5","option_b":"B) 14","option_c":"C) 11","option_d":"D) 8","option_e":"","option_f":"","bonne_reponse":"c","explication":"Calculons : u₁ = 3*2 - 1 = 5, puis u₂ = 3*5 - 1 = 14. La réponse correcte est donc 14.","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) 5\", \"b\": \"B) 14\", \"c\": \"C) 11\", \"d\": \"D) 8\"}}","_debug_options_count":4},{"id":36464,"question":"Le produit scalaire de deux vecteurs orthogonaux est toujours nul.","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":36465,"question":"Quelle est la solution de l'équation différentielle y' = 2y avec y(0) = 3 ?","option_a":"A) y = 3e^(2x)","option_b":"B) y = 2e^(3x)","option_c":"C) y = 3e^(-2x)","option_d":"D) y = 2x + 3","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution générale de y' = ky est y = Ce^(kx). Avec y(0)=3, on trouve C=3. La solution est 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 = 2e^(3x)\", \"c\": \"C) y = 3e^(-2x)\", \"","_debug_options_count":4},{"id":36466,"question":"Soit f une fonction continue sur [a,b]. Le théorème des valeurs intermédiaires garantit que f prend toutes les valeurs comprises 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 dans [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},{"id":36467,"question":"Quel est le coefficient directeur de la tangente à la courbe de f(x) = ln(x) au point d'abscisse 1 ?","option_a":"A) 0","option_b":"B) 1","option_c":"C) e","option_d":"D) -1","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de ln(x) est 1\/x. Au point x=1, le coefficient directeur est f'(1) = 1\/1 = 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) e\", \"d\": \"D) -1\"}}","_debug_options_count":4},{"id":36468,"question":"Soit une série statistique de moyenne 5 et d'écart-type 2. Quelle est la variance de cette série ?","option_a":"A) 2","option_b":"B) 4","option_c":"C) 5","option_d":"D) 10","option_e":"","option_f":"","bonne_reponse":"b","explication":"La variance est le carré de l'écart-type. Ici, variance = 2² = 4.","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) 4\", \"c\": \"C) 5\", \"d\": \"D) 10\"}}","_debug_options_count":4},{"id":36469,"question":"La fonction f(x) = x³ - 3x² + 2x est croissante sur l'intervalle [0, 2].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Étudions le signe de f'(x) = 3x² - 6x + 2. Sur [0,2], f'(x) est positif (ex: f'(1) = -1 \u003C 0, donc la fonction n'est pas croissante sur tout 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]