Quiz — ex4(7)c.pdf

Question 1 sur 10 20:00

[{"id":22649,"question":"Quelle est la limite de la suite définie par u\u003Csub\u003En+1\u003C\/sub\u003E = (2u\u003Csub\u003En\u003C\/sub\u003E + 1)\/(u\u003Csub\u003En\u003C\/sub\u003E + 1) avec u\u003Csub\u003E0\u003C\/sub\u003E = 0 ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. -1","option_d":"D. La suite diverge","option_e":"","option_f":"","bonne_reponse":"c","explication":"La suite converge vers la solution de l'équation L = (2L + 1)\/(L + 1), soit L = 1. Cependant, en vérifiant la convergence, on trouve que la limite est -1 (solution stable).","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. -1\", \"d\": \"D. La suite diverge\"}}","_debug_options_count":4},{"id":22650,"question":"Soit f(x) = e\u003Csup\u003Ex\u003C\/sup\u003E - x. La fonction f est-elle toujours positive sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction f(x) = e\u003Csup\u003Ex\u003C\/sup\u003E - x est toujours positive car sa dérivée f'(x) = e\u003Csup\u003Ex\u003C\/sup\u003E - 1 s'annule en x=0 et f(0)=1 \u003E 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":22651,"question":"Quelle est la valeur de l'intégrale ∫\u003Csub\u003E0\u003C\/sub\u003E\u003Csup\u003E1\u003C\/sup\u003E x² e\u003Csup\u003Ex\u003C\/sup\u003E dx ?","option_a":"A. e - 2","option_b":"B. 2e - 5","option_c":"C. e - 1","option_d":"D. 1 - e","option_e":"","option_f":"","bonne_reponse":"b","explication":"En utilisant une intégration par parties, on obtient ∫x²e\u003Csup\u003Ex\u003C\/sup\u003Edx = x²e\u003Csup\u003Ex\u003C\/sup\u003E - 2∫xe\u003Csup\u003Ex\u003C\/sup\u003Edx = x²e\u003Csup\u003Ex\u003C\/sup\u003E - 2(xe\u003Csup\u003Ex\u003C\/sup\u003E - e\u003Csup\u003Ex\u003C\/sup\u003E) + C. Évaluée entre 0 et 1, cela donne 2e - 5.","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. e - 2\", \"b\": \"B. 2e - 5\", \"c\": \"C. e - 1\", \"d\": \"D. 1 - e\"}}","_debug_options_count":4},{"id":22652,"question":"Dans une classe de 30 élèves, 60% sont des filles. On choisit 3 élèves au hasard. Quelle est la probabilité que les 3 soient des filles ?","option_a":"A. 0.216","option_b":"B. 0.128","option_c":"C. 0.36","option_d":"D. 0.48","option_e":"","option_f":"","bonne_reponse":"a","explication":"La probabilité est (18\/30) × (17\/29) × (16\/28) ≈ 0.216. On utilise la loi hypergéométrique pour des tirages sans remise.","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. 0.216\", \"b\": \"B. 0.128\", \"c\": \"C. 0.36\", \"d\": \"D. 0.48\"}}","_debug_options_count":4},{"id":22653,"question":"L'équation différentielle y' + 2y = 0 a pour solution générale y(x) = Ce\u003Csup\u003E-2x\u003C\/sup\u003E. Cette affirmation est-elle correcte ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution générale de y' + 2y = 0 est bien y(x) = Ce\u003Csup\u003E-2x\u003C\/sup\u003E, car la dérivée de Ce\u003Csup\u003E-2x\u003C\/sup\u003E est -2Ce\u003Csup\u003E-2x\u003C\/sup\u003E, ce qui vérifie l'équation.","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":22654,"question":"Quel est le module du nombre complexe z = 3 + 4i ?","option_a":"A. 5","option_b":"B. 7","option_c":"C. 25","option_d":"D. 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module de z = a + bi est √(a² + b²). Ici, √(3² + 4²) = 5.","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. 5\", \"b\": \"B. 7\", \"c\": \"C. 25\", \"d\": \"D. 1\"}}","_debug_options_count":4},{"id":22655,"question":"Soit la fonction f définie par f(x) = ln(x) + x. Quel est le sens de variation de f sur ]0, +∞[ ?","option_a":"A. Strictement croissante","option_b":"B. Strictement décroissante","option_c":"C. Constante","option_d":"D. Non monotone","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 1\/x + 1 \u003E 0 pour tout x \u003E 0, donc f est strictement croissante.","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. Strictement croissante\", \"b\": \"B. Strictement décroissante\", ","_debug_options_count":4},{"id":22656,"question":"La probabilité d'obtenir un 6 en lançant un dé équilibré est de 1\/6. Après 5 lancers sans obtenir de 6, la probabilité d'en obtenir un au 6ème lancer est-elle toujours de 1\/6 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Les lancers sont indépendants. La probabilité reste 1\/6 à chaque lancer, quel que soit l'historique.","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":22657,"question":"Quelle est la dérivée de la fonction f(x) = (x² + 1)e\u003Csup\u003Ex\u003C\/sup\u003E ?","option_a":"A. (2x + 1)e\u003Csup\u003Ex\u003C\/sup\u003E","option_b":"B. (x² + 2x + 1)e\u003Csup\u003Ex\u003C\/sup\u003E","option_c":"C. (x² + 1 + 2x)e\u003Csup\u003Ex\u003C\/sup\u003E","option_d":"D. 2xe\u003Csup\u003Ex\u003C\/sup\u003E","option_e":"","option_f":"","bonne_reponse":"b","explication":"En utilisant la règle du produit, f'(x) = 2x e\u003Csup\u003Ex\u003C\/sup\u003E + (x² + 1)e\u003Csup\u003Ex\u003C\/sup\u003E = (x² + 2x + 1)e\u003Csup\u003Ex\u003C\/sup\u003E.","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. (2x + 1)e\u003Csup\u003Ex\u003C\/sup\u003E\", \"b\": \"B. (x² + 2x + 1)e\u003Csup\u003Ex\u003C\/sup\u003E\",","_debug_options_count":4},{"id":22658,"question":"Quel est le résultat de l'intégrale ∫\u003Csub\u003E-1\u003C\/sub\u003E\u003Csup\u003E1\u003C\/sup\u003E x³ dx ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 2","option_d":"D. -2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction x³ est impaire, donc son intégrale sur un intervalle symétrique autour de 0 est nulle.","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\", \"b\": \"B. 1\", \"c\": \"C. 2\", \"d\": \"D. -2\"}}","_debug_options_count":4}]