Quiz — 204.._Magazine-25 FONCTIONS LOGARITHMES 2.pdf

Question 1 sur 10 20:00

[{"id":12208,"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":"Par définition, ln(e^x) = x pour tout x réel. Ici, ln(e^5) = 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\": \"5\", \"b\": \"e^5\", \"c\": \"1\", \"d\": \"0\"}}","_debug_options_count":4},{"id":12209,"question":"L'équation ln(x) = -2 admet-elle une solution ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai, car ln(x) prend toutes les valeurs réelles pour x \u003E 0. La solution est x = e^{-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":12210,"question":"Quelle est la dérivée de la fonction f(x) = ln(3x + 1) ?","option_a":"3\/(3x+1)","option_b":"1\/(3x+1)","option_c":"3x+1","option_d":"1\/(x+1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la règle de dérivation des fonctions composées, 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+1\", \"d\": \"1\/(x+1)\"}}","_debug_options_count":4},{"id":12211,"question":"Pour x \u003E 1, laquelle de ces fonctions croît le plus vite : ln(x) ou √x ?","option_a":"ln(x)","option_b":"√x","option_c":"Elles croissent à la même vitesse","option_d":"ln(x) croît plus vite pour x \u003E e^2","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour x \u003E 1, √x croît plus vite que ln(x). On peut le vérifier en comparant leurs dérivées.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"ln(x)\", \"b\": \"√x\", \"c\": \"Elles croissent à la même vitesse\", ","_debug_options_count":4},{"id":12212,"question":"L'inéquation ln(x) \u003E 1 est équivalente à :","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai, car ln(x) \u003E 1 ⇔ x \u003E e^1 = e.","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":12213,"question":"Quelle est la limite de ln(x)\/x quand 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) croît plus lentement que x, donc ln(x)\/x → 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\": \"+∞\", \"c\": \"1\", \"d\": \"-∞\"}}","_debug_options_count":4},{"id":12214,"question":"La fonction f(x) = ln(x^2) est définie pour :","option_a":"x \u003E 0","option_b":"x ≠ 0","option_c":"x ≥ 0","option_d":"x ≤ 0","option_e":"","option_f":"","bonne_reponse":"b","explication":"ln(x^2) est défini pour x ≠ 0, car x^2 \u003E 0 pour tout x ≠ 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\": \"x \u003E 0\", \"b\": \"x ≠ 0\", \"c\": \"x ≥ 0\", \"d\": \"x ≤ 0\"}}","_debug_options_count":4},{"id":12215,"question":"L'égalité ln(a) + ln(b) = ln(a+b) est-elle vraie ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. La bonne égalité est ln(a) + ln(b) = ln(ab).","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":12216,"question":"Quelle est la solution de l'équation ln(x-1) + ln(x+1) = ln(3) ?","option_a":"x = 2","option_b":"x = √2","option_c":"x = 1","option_d":"x = 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"En combinant les logarithmes : ln((x-1)(x+1)) = ln(3) ⇒ x^2 - 1 = 3 ⇒ x = 2 (car x \u003E 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\": \"x = 2\", \"b\": \"x = √2\", \"c\": \"x = 1\", \"d\": \"x = 0\"}}","_debug_options_count":4},{"id":12217,"question":"Pour x \u003E 0, ln(x) est toujours :","option_a":"positif","option_b":"négatif","option_c":"inférieur à x","option_d":"supérieur à x","option_e":"","option_f":"","bonne_reponse":"c","explication":"Pour 0 \u003C x \u003C 1, ln(x) \u003C 0 \u003C x. Pour x = 1, ln(1) = 0 = x. Pour x \u003E 1, ln(x) \u003C x (croissance comparée).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"positif\", \"b\": \"négatif\", \"c\": \"inférieur à x\", \"d\": \"supérie","_debug_options_count":4}]