Quiz — 178.._Magazine-24 logarithmes népériens 1.pdf

Question 1 sur 10 20:00

[{"id":37440,"question":"Quelle est la valeur de ln(e^3) ?","option_a":"3","option_b":"e^3","option_c":"0","option_d":"1","option_e":"","option_f":"","bonne_reponse":"a","explication":"ln(e^3) = 3 car ln et exp sont des fonctions inverses.","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\", \"b\": \"e^3\", \"c\": \"0\", \"d\": \"1\"}}","_debug_options_count":4},{"id":37441,"question":"La dérivée de ln(2x) est 1\/x.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de ln(2x) est 2\/(2x) = 1\/x, donc 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":37442,"question":"Résoudre l'équation ln(x) + ln(2) = ln(8).","option_a":"x = 4","option_b":"x = 6","option_c":"x = 8","option_d":"x = 16","option_e":"","option_f":"","bonne_reponse":"a","explication":"ln(x) + ln(2) = ln(2x) = ln(8) ⇒ 2x = 8 ⇒ x = 4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"x = 4\", \"b\": \"x = 6\", \"c\": \"x = 8\", \"d\": \"x = 16\"}}","_debug_options_count":4},{"id":37443,"question":"La fonction ln(x) est définie pour tout x réel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"ln(x) n'est défini que pour x \u003E 0.","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":37444,"question":"Quelle est la dérivée de f(x) = ln(x^2 + 1) ?","option_a":"2x\/(x^2 + 1)","option_b":"x\/(x^2 + 1)","option_c":"1\/(x^2 + 1)","option_d":"2x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(u(x)) est u'(x)\/u(x). Ici, u(x) = x^2 + 1 ⇒ u'(x) = 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\": \"2x\/(x^2 + 1)\", \"b\": \"x\/(x^2 + 1)\", \"c\": \"1\/(x^2 + 1)\", \"d\": \"2x\"}","_debug_options_count":4},{"id":37445,"question":"ln(1) = 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"ln(1) = 0 car e^0 = 1.","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":37446,"question":"Résoudre l'inéquation ln(x) \u003E 2.","option_a":"x \u003E e^2","option_b":"x \u003E 2","option_c":"x \u003E 0","option_d":"x \u003E 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"ln(x) \u003E 2 ⇒ x \u003E e^2 car la fonction ln est 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\": \"x \u003E e^2\", \"b\": \"x \u003E 2\", \"c\": \"x \u003E 0\", \"d\": \"x \u003E 1\"}}","_debug_options_count":4},{"id":37447,"question":"La fonction ln(x) est toujours croissante.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(x) est 1\/x \u003E 0 pour x \u003E 0, donc la fonction est 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":37448,"question":"Quelle est la limite de ln(x)\/x quand x tend vers +∞ ?","option_a":"0","option_b":"1","option_c":"+∞","option_d":"-∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par croissance comparée, ln(x)\/x tend vers 0 quand x → +∞.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"+∞\", \"d\": \"-∞\"}}","_debug_options_count":4},{"id":37449,"question":"La fonction f(x) = ln(x) + x est-elle convexe ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"f''(x) = -1\/x^2 + 0 = -1\/x^2 \u003C 0 pour x \u003E 0, donc f est concave (non convexe).","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}]