Quiz — Solutions 2011.pdf

Question 1 sur 10 20:00

[{"id":12508,"question":"Quelle est la dérivée de la fonction f(x) = e^(2x+1) ?","option_a":"A. e^(2x+1)","option_b":"B. 2e^(2x+1)","option_c":"C. e^(2x+1) + 2","option_d":"D. 2e^(x+1)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée d’une fonction exponentielle e^u est u’·e^u. Ici, u = 2x+1, donc u’ = 2. Ainsi, f’(x) = 2e^(2x+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\": \"A. e^(2x+1)\", \"b\": \"B. 2e^(2x+1)\", \"c\": \"C. e^(2x+1) + 2\", \"d\": \"","_debug_options_count":4},{"id":12509,"question":"L’équation ln(x) = 2 admet-elle une solution réelle ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L’équation ln(x) = 2 admet une solution unique x = e^2, car la fonction ln est bijective 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":12510,"question":"Quelle est la limite de la suite u_n = (3n+1)\/(2n-5) quand n tend vers l’infini ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 3\/2","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"En divisant numérateur et dénominateur par n, on obtient u_n = (3 + 1\/n)\/(2 - 5\/n). Quand n → +∞, u_n → 3\/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. 0\", \"b\": \"B. 1\", \"c\": \"C. 3\/2\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":12511,"question":"La fonction f(x) = x^3 - 3x^2 + 2x est-elle croissante sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f’(x) = 3x^2 - 6x + 2 s’annule en x = 1 ± √(1\/3). La fonction n’est donc pas croissante sur tout 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":12512,"question":"Quelle est la solution de l’inéquation e^(x) \u003E 5 ?","option_a":"A. x \u003E ln(5)","option_b":"B. x \u003E 5","option_c":"C. x \u003C ln(5)","option_d":"D. x \u003C 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle est croissante. L’inéquation e^x \u003E 5 équivaut à x \u003E ln(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\": \"A. x \u003E ln(5)\", \"b\": \"B. x \u003E 5\", \"c\": \"C. x \u003C ln(5)\", \"d\": \"D. x \u003C","_debug_options_count":4},{"id":12513,"question":"La suite u_n = (-1)^n est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite alterne entre -1 et 1, donc elle n’a pas de limite. Elle est divergente.","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":12514,"question":"Quelle est la primitive de f(x) = 1\/(1+x^2) ?","option_a":"A. ln(1+x^2)","option_b":"B. arctan(x)","option_c":"C. x\/(1+x^2)","option_d":"D. 1\/(1+x^2)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La primitive de 1\/(1+x^2) est arctan(x) + C, car la dérivée de arctan(x) est 1\/(1+x^2).","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. ln(1+x^2)\", \"b\": \"B. arctan(x)\", \"c\": \"C. x\/(1+x^2)\", \"d\": \"D.","_debug_options_count":4},{"id":12515,"question":"L’équation x^2 - 4x + 5 = 0 admet-elle des solutions réelles ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le discriminant Δ = (-4)^2 - 4·1·5 = 16 - 20 = -4 \u003C 0. L’équation n’a donc pas de solutions réelles.","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":12516,"question":"Quelle est la valeur de l’intégrale ∫(0 à 1) x·e^x dx ?","option_a":"A. e","option_b":"B. 1","option_c":"C. e - 1","option_d":"D. 0","option_e":"","option_f":"","bonne_reponse":"c","explication":"L’intégrale se calcule par parties : u = x, v’ = e^x ⇒ u’ = 1, v = e^x. Ainsi, ∫x·e^x dx = x·e^x - ∫e^x dx = x·e^x - e^x + C. Évaluée entre 0 et 1, on obtient (1·e - e) - (0·1 - 1) = e - 1.","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. e\", \"b\": \"B. 1\", \"c\": \"C. e - 1\", \"d\": \"D. 0\"}}","_debug_options_count":4},{"id":12517,"question":"La fonction f(x) = x^2 - 4x + 3 est-elle convexe sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée seconde f''(x) = 2 \u003E 0, donc la fonction est convexe sur ℝ.","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}]