Quiz — TD4.pdf

Question 1 sur 10 20:00

[{"id":61287,"question":"Quelle est la limite de la suite définie par uₙ = (3n² + 2n - 1)\/(2n² + 5) quand n tend vers +∞ ?","option_a":"A. 0","option_b":"B. 3\/2","option_c":"C. +∞","option_d":"D. 1","option_e":"","option_f":"","bonne_reponse":"b","explication":"On divise numérateur et dénominateur par n² : uₙ = (3 + 2\/n - 1\/n²)\/(2 + 5\/n²). Quand n→+∞, les termes en 1\/n et 1\/n² tendent vers 0, donc la limite est 3\/2.","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\/2\", \"c\": \"C. +∞\", \"d\": \"D. 1\"}}","_debug_options_count":4},{"id":61288,"question":"La fonction f(x) = x³ - 3x² + 2x est strictement croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 3x² - 6x + 2 s'annule en x = 1 ± √(1\/3). La fonction n'est donc pas strictement croissante sur tout ℝ.","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":61289,"question":"Quelle est la probabilité d'obtenir exactement deux 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. (1\/6)² * (5\/6)³","option_c":"C. 5 * (1\/6)² * (5\/6)³","option_d":"D. C(5,2) * (1\/6)³ * (5\/6)²","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une loi binomiale de paramètres n=5 et p=1\/6. La probabilité est C(5,2) * (1\/6)² * (5\/6)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(5,2) * (1\/6)² * (5\/6)³\", \"b\": \"B. (1\/6)² * (5\/6)³\", \"c\":","_debug_options_count":4},{"id":61290,"question":"Soit f une fonction dérivable sur ℝ telle que f'(x) = 2x + 1. Alors f(x) = x² + x + C où C est une constante.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Toute primitive de f'(x) = 2x + 1 est de la forme f(x) = x² + x + C, 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":61291,"question":"Quel est le coefficient directeur de la tangente à la courbe de f(x) = ln(x) au point d'abscisse x = 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 f'(x) = 1\/x. En x=1, f'(1) = 1, donc le coefficient directeur est 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":61292,"question":"La suite définie par u₀ = 1 et uₙ₊₁ = 2uₙ + 3 est arithmético-géométrique.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une suite arithmético-géométrique est de la forme uₙ₊₁ = a*uₙ + b. Ici, a=2 et b=3, donc c'est bien le cas.","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":61293,"question":"Quelle est la valeur de ∫₀¹ (3x² + 2x + 1) dx ?","option_a":"A. 2","option_b":"B. 3","option_c":"C. 1","option_d":"D. 4","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale se calcule comme [x³ + x² + x]₀¹ = (1 + 1 + 1) - 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. 2\", \"b\": \"B. 3\", \"c\": \"C. 1\", \"d\": \"D. 4\"}}","_debug_options_count":4},{"id":61294,"question":"Soit A et B deux événements indépendants avec P(A) = 0.4 et P(B) = 0.5. Alors P(A ∩ B) = 0.2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour des événements indépendants, P(A ∩ B) = P(A) * P(B) = 0.4 * 0.5 = 0.2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":61295,"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 = Cx²","option_d":"D. y = C","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation caractéristique est r - 2 = 0, donc r=2. La solution générale est y = Ce^(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 = Ce^(2x)\", \"b\": \"B. y = Ce^(-2x)\", \"c\": \"C. y = Cx²\", \"d\":","_debug_options_count":4},{"id":61296,"question":"La fonction f(x) = e^(-x²) est paire.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une fonction paire vérifie f(-x) = f(x). Ici, f(-x) = e^(-(-x)²) = e^(-x²) = f(x), donc elle est paire.","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}]