Quiz — 63ef910912472_énoncé_Série 9 (Fonction logarithme).pdf

Question 1 sur 10 20:00

[{"id":67561,"question":"Quelle est la valeur de ln(1) ?","option_a":"0","option_b":"1","option_c":"e","option_d":"−1","option_e":"","option_f":"","bonne_reponse":"a","explication":"ln(1) = 0 car e^0 = 1. C'est une propriété fondamentale du logarithme népérien.","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\": \"e\", \"d\": \"−1\"}}","_debug_options_count":4},{"id":67562,"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":"Faux. La fonction ln(x) n'est définie que pour x \u003E 0, car elle est l'inverse de la fonction exponentielle définie sur ℝ.","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":67563,"question":"Quelle est la dérivée de f(x) = ln(3x + 1) ?","option_a":"3\/(3x + 1)","option_b":"1\/(3x + 1)","option_c":"3x\/(3x + 1)","option_d":"1\/(x + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(u(x)) est u'(x)\/u(x). Ici, u(x) = 3x + 1, donc u'(x) = 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\": \"a\", \"options\": {\"a\": \"3\/(3x + 1)\", \"b\": \"1\/(3x + 1)\", \"c\": \"3x\/(3x + 1)\", \"d\": \"1\/(x + ","_debug_options_count":4},{"id":67564,"question":"L'équation ln(x² − 4) = 0 admet deux solutions réelles.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. ln(x² − 4) = 0 ⇒ x² − 4 = 1 ⇒ x² = 5 ⇒ x = √5 ou x = −√5. Les deux solutions sont valides car x² − 4 \u003E 0 pour x = ±√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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":67565,"question":"Quelle est la limite de ln(x) lorsque x tend vers 0+ ?","option_a":"−∞","option_b":"0","option_c":"+∞","option_d":"1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction ln(x) tend vers −∞ lorsque x tend vers 0+, car ln(x) décroît sans borne inférieure sur ]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\": \"−∞\", \"b\": \"0\", \"c\": \"+∞\", \"d\": \"1\"}}","_debug_options_count":4},{"id":67566,"question":"La fonction ln(x) est-elle croissante sur son ensemble de définition ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. La dérivée de ln(x) est 1\/x, qui est positive pour tout x \u003E 0. La fonction est donc 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":67567,"question":"Résoudre l'inéquation ln(x) ≤ 1.","option_a":"x ≤ e","option_b":"0 \u003C x ≤ e","option_c":"x ≥ e","option_d":"x ∈ ]0, e]","option_e":"","option_f":"","bonne_reponse":"b","explication":"ln(x) ≤ 1 ⇒ x ≤ e^1 = e. Comme ln(x) est défini pour x \u003E 0, la solution est 0 \u003C x ≤ e.","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 ≤ e\", \"b\": \"0 \u003C x ≤ e\", \"c\": \"x ≥ e\", \"d\": \"x ∈ ]0, e]\"","_debug_options_count":4},{"id":67568,"question":"Quelle est la valeur de ln(e^5) ?","option_a":"5","option_b":"e^5","option_c":"1","option_d":"0","option_e":"","option_f":"","bonne_reponse":"a","explication":"ln(e^5) = 5 car ln(e^k) = k pour tout réel k, par définition du logarithme népérien.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"5\", \"b\": \"e^5\", \"c\": \"1\", \"d\": \"0\"}}","_debug_options_count":4},{"id":67569,"question":"La fonction f(x) = ln(x) + ln(1\/x) est-elle constante ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. ln(x) + ln(1\/x) = ln(x × 1\/x) = ln(1) = 0. La fonction est donc constante et égale à 0 pour tout x \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":67570,"question":"Quelle est la limite de (ln(x))\/x lorsque x tend vers +∞ ?","option_a":"0","option_b":"+∞","option_c":"1","option_d":"−∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par croissance comparée, (ln(x))\/x tend vers 0 lorsque x tend vers +∞, car ln(x) croît plus lentement que 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\": \"+∞\", \"c\": \"1\", \"d\": \"−∞\"}}","_debug_options_count":4}]