Quiz — 6321ad047836a_série 2-.pdf

Question 1 sur 10 20:00

[{"id":45453,"question":"Quelle est la limite de la fonction f(x) = (2x² + 3x - 5)\/(x² - 1) lorsque x tend vers l'infini ?","option_a":"A. 2","option_b":"B. 0","option_c":"C. 1","option_d":"D. -1","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une fonction rationnelle, la limite à l'infini est égale au rapport des coefficients dominants des polynômes du numérateur et du dénominateur. Ici, 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\": \"A. 2\", \"b\": \"B. 0\", \"c\": \"C. 1\", \"d\": \"D. -1\"}}","_debug_options_count":4},{"id":45454,"question":"La dérivée de la fonction f(x) = x³ - 2x² + 5x - 7 est f'(x) = 3x² - 4x + 5.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En appliquant les règles de dérivation, f'(x) = 3x² - 4x + 5 est correct. La dérivée est bien calculée.","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":45455,"question":"Quel est le produit scalaire de deux vecteurs u(2; -1; 3) et v(-1; 4; 2) ?","option_a":"A. 0","option_b":"B. 4","option_c":"C. -4","option_d":"D. 8","option_e":"","option_f":"","bonne_reponse":"c","explication":"Le produit scalaire se calcule par u·v = (2)(-1) + (-1)(4) + (3)(2) = -2 -4 +6 = 0. Le résultat est donc 0.","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. 4\", \"c\": \"C. -4\", \"d\": \"D. 8\"}}","_debug_options_count":4},{"id":45456,"question":"La fonction f(x) = x² - 4x + 3 est croissante sur l'intervalle [2; +∞[.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 2x - 4 est positive pour x \u003E 2, donc la fonction est bien croissante sur [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":45457,"question":"Quelle est la probabilité d'obtenir un nombre pair en lançant un dé équilibré à 6 faces ?","option_a":"A. 1\/6","option_b":"B. 1\/3","option_c":"C. 1\/2","option_d":"D. 2\/3","option_e":"","option_f":"","bonne_reponse":"c","explication":"Il y a 3 nombres pairs (2, 4, 6) sur 6 faces possibles. La probabilité est donc 3\/6 = 1\/2.","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. 1\/6\", \"b\": \"B. 1\/3\", \"c\": \"C. 1\/2\", \"d\": \"D. 2\/3\"}}","_debug_options_count":4},{"id":45458,"question":"La fonction f(x) = (x + 1)\/(x - 2) admet une asymptote verticale en x = -1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction admet une asymptote verticale en x = 2 (où le dénominateur s'annule), pas en x = -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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":45459,"question":"Quel est le coefficient directeur de la tangente à la courbe de f(x) = x³ + 2x² - x + 1 au point d'abscisse x = 1 ?","option_a":"A. 3","option_b":"B. 5","option_c":"C. 6","option_d":"D. 7","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le coefficient directeur est donné par f'(1). f'(x) = 3x² + 4x - 1, donc f'(1) = 3 + 4 - 1 = 6.","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. 3\", \"b\": \"B. 5\", \"c\": \"C. 6\", \"d\": \"D. 7\"}}","_debug_options_count":4},{"id":45460,"question":"La fonction f(x) = √(x² + 4) est définie pour tout x réel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction racine carrée est définie lorsque l'expression sous la racine est positive ou nulle. Ici, x² + 4 \u003E 0 pour tout x réel, donc la fonction est bien définie.","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":45461,"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 = -2e^(3x)","option_d":"D. y = 2e^(-3x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'équation différentielle y' + 2y = 0 a pour solution générale y = Ce^(-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\": \"b\", \"options\": {\"a\": \"A. y = 3e^(2x)\", \"b\": \"B. y = 3e^(-2x)\", \"c\": \"C. y = -2e^(3x)\", ","_debug_options_count":4},{"id":45462,"question":"La fonction f(x) = |x - 1| + |x + 2| est dérivable en x = -2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction valeur absolue n'est pas dérivable aux points où l'expression sous le module change de signe, ici en x = -2 et x = 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]