Quiz — 69e0d83b4d531_corrigé série 19.pdf

Question 1 sur 10 20:00

[{"id":27980,"question":"Quelle est la solution de l'équation 2x² - 5x + 2 = 0 ?","option_a":"x = 1 ou x = 2","option_b":"x = 0.5 ou x = 2","option_c":"x = -1 ou x = -2","option_d":"x = 0.5 ou x = -2","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'équation 2x² - 5x + 2 = 0 se résout en utilisant le discriminant Δ = b² - 4ac = 25 - 16 = 9. Les solutions sont x = (5 ± 3)\/4, soit x = 2 ou x = 0.5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"x = 1 ou x = 2\", \"b\": \"x = 0.5 ou x = 2\", \"c\": \"x = -1 ou x = -2\"","_debug_options_count":4},{"id":27981,"question":"La fonction f(x) = x³ - 3x² + 4 est décroissante sur l'intervalle [-1, 1].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 3x² - 6x = 3x(x - 2). Sur [-1, 1], f'(x) ≤ 0 (car x(x-2) ≥ 0), donc la fonction est décroissante sur cet intervalle. L'affirmation est donc vraie.","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":27982,"question":"Quelle est la limite de la fonction f(x) = (2x² + 3x - 1)\/(x² - 4) lorsque x tend vers l'infini ?","option_a":"2","option_b":"0","option_c":"1","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour x tendant vers l'infini, on divise le numérateur et le dénominateur par x². On obtient f(x) ≈ (2 + 3\/x - 1\/x²)\/(1 - 4\/x²) → 2\/1 = 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\": \"2\", \"b\": \"0\", \"c\": \"1\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":27983,"question":"Dans un triangle rectangle, si l'hypoténuse mesure 5 cm et un côté mesure 3 cm, quelle est la longueur du troisième côté ?","option_a":"4 cm","option_b":"2 cm","option_c":"√34 cm","option_d":"√16 cm","option_e":"","option_f":"","bonne_reponse":"a","explication":"D'après le théorème de Pythagore, le troisième côté mesure √(5² - 3²) = √(25 - 9) = √16 = 4 cm.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"4 cm\", \"b\": \"2 cm\", \"c\": \"√34 cm\", \"d\": \"√16 cm\"}}","_debug_options_count":4},{"id":27984,"question":"La probabilité d'obtenir un nombre pair en lançant un dé équilibré est de 1\/2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Un dé équilibré a 6 faces (1, 2, 3, 4, 5, 6). Les nombres pairs sont 2, 4 et 6, soit 3 faces. La probabilité est donc 3\/6 = 1\/2. L'affirmation est vraie.","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":27985,"question":"Quelle est la dérivée de la fonction f(x) = ln(3x + 1) ?","option_a":"f'(x) = 1\/(3x + 1)","option_b":"f'(x) = 3\/(3x + 1)","option_c":"f'(x) = 3x\/(3x + 1)","option_d":"f'(x) = 1\/(x + 1)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de ln(u) est u'\/u. Ici, u = 3x + 1, donc u' = 3. Ainsi, f'(x) = 3\/(3x + 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\": \"f'(x) = 1\/(3x + 1)\", \"b\": \"f'(x) = 3\/(3x + 1)\", \"c\": \"f'(x) = 3x\/","_debug_options_count":4},{"id":27986,"question":"Un sac contient 4 boules rouges et 6 boules bleues. On tire deux boules successivement sans remise. Quelle est la probabilité d'obtenir deux boules rouges ?","option_a":"4\/25","option_b":"2\/15","option_c":"1\/5","option_d":"3\/10","option_e":"","option_f":"","bonne_reponse":"b","explication":"La probabilité de tirer une boule rouge en premier est 4\/10. Sans remise, la probabilité de tirer une deuxième boule rouge est 3\/9. La probabilité totale est (4\/10)*(3\/9) = 12\/90 = 2\/15.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"4\/25\", \"b\": \"2\/15\", \"c\": \"1\/5\", \"d\": \"3\/10\"}}","_debug_options_count":4},{"id":27987,"question":"La fonction f(x) = e^(2x) est toujours croissante.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de f(x) = e^(2x) est f'(x) = 2e^(2x). Comme e^(2x) \u003E 0 pour tout x, f'(x) \u003E 0. La fonction est donc toujours croissante. L'affirmation est vraie.","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":27988,"question":"Quelle est l'intégrale de f(x) = 3x² + 2x - 1 entre 0 et 2 ?","option_a":"10","option_b":"14\/3","option_c":"8","option_d":"12","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale de f(x) est F(x) = x³ + x² - x. Entre 0 et 2, F(2) - F(0) = (8 + 4 - 2) - 0 = 10.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"10\", \"b\": \"14\/3\", \"c\": \"8\", \"d\": \"12\"}}","_debug_options_count":4},{"id":27989,"question":"Dans une classe de 30 élèves, 18 sont des filles. Quelle est la probabilité qu'un élève choisi au hasard soit une fille ?","option_a":"3\/5","option_b":"2\/3","option_c":"3\/10","option_d":"1\/2","option_e":"","option_f":"","bonne_reponse":"a","explication":"Il y a 18 filles sur 30 élèves. La probabilité est donc 18\/30 = 3\/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\": \"3\/5\", \"b\": \"2\/3\", \"c\": \"3\/10\", \"d\": \"1\/2\"}}","_debug_options_count":4}]