Quiz — correction ct 2 4M 204 2050001.pdf

Question 1 sur 10 20:00

[{"id":12758,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"f’(x) = 6x + 2","option_b":"f’(x) = 3x + 2","option_c":"f’(x) = 6x² + 2","option_d":"f’(x) = 3x² + 2x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d’une fonction polynôme se calcule terme par terme : (3x²)’ = 6x, (2x)’ = 2 et (-5)’ = 0. Ainsi, f’(x) = 6x + 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\": \"f’(x) = 6x + 2\", \"b\": \"f’(x) = 3x + 2\", \"c\": \"f’(x) = 6x² ","_debug_options_count":4},{"id":12759,"question":"Soit une suite géométrique de premier terme u₀ = 2 et de raison q = 3. Quelle est la valeur de u₃ ?","option_a":"u₃ = 18","option_b":"u₃ = 54","option_c":"u₃ = 6","option_d":"u₃ = 162","option_e":"","option_f":"","bonne_reponse":"b","explication":"La formule d’une suite géométrique est uₙ = u₀ × qⁿ. Ainsi, u₃ = 2 × 3³ = 2 × 27 = 54.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"u₃ = 18\", \"b\": \"u₃ = 54\", \"c\": \"u₃ = 6\", \"d\": \"u₃ = 162\"}","_debug_options_count":4},{"id":12760,"question":"La fonction f(x) = x³ - 3x² + 2x est croissante sur l’intervalle [0 ; 2].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f’(x) = 3x² - 6x + 2. En étudiant son signe sur [0 ; 2], on constate qu’elle est positive, donc f est croissante sur cet intervalle.","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":12761,"question":"Quelle est la limite de la suite uₙ = (2n + 1)\/(n - 3) lorsque n tend vers l’infini ?","option_a":"Limite = 2","option_b":"Limite = 0","option_c":"Limite = 1","option_d":"Limite = -2","option_e":"","option_f":"","bonne_reponse":"a","explication":"En divisant numérateur et dénominateur par n, on obtient uₙ = (2 + 1\/n)\/(1 - 3\/n). Lorsque n tend vers l’infini, 1\/n et 3\/n tendent vers 0, donc la limite est 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\": \"Limite = 2\", \"b\": \"Limite = 0\", \"c\": \"Limite = 1\", \"d\": \"Limite =","_debug_options_count":4},{"id":12762,"question":"Soit f une fonction dérivable sur ℝ. Si f’(x) \u003E 0 pour tout x ∈ ℝ, alors f est strictement croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si la dérivée est strictement positive sur un intervalle, alors la fonction est strictement croissante sur cet intervalle.","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":12763,"question":"Quelle est la solution générale de l’équation différentielle y’ + 2y = 0 ?","option_a":"y(x) = Ce^{-2x}","option_b":"y(x) = Ce^{2x}","option_c":"y(x) = Cx + C","option_d":"y(x) = Csin(2x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"Cette équation est de la forme y’ + ay = 0. Sa solution générale est y(x) = Ce^{-ax}, donc ici y(x) = Ce^{-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\": \"y(x) = Ce^{-2x}\", \"b\": \"y(x) = Ce^{2x}\", \"c\": \"y(x) = Cx + C\", \"d","_debug_options_count":4},{"id":12764,"question":"Soit une suite arithmétique de premier terme u₀ = 5 et de raison r = -2. Quelle est la valeur de u₅ ?","option_a":"u₅ = -5","option_b":"u₅ = 5","option_c":"u₅ = -3","option_d":"u₅ = 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La formule d’une suite arithmétique est uₙ = u₀ + nr. Ainsi, u₅ = 5 + 5×(-2) = 5 - 10 = -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\": \"u₅ = -5\", \"b\": \"u₅ = 5\", \"c\": \"u₅ = -3\", \"d\": \"u₅ = 1\"}}","_debug_options_count":4},{"id":12765,"question":"La dérivée de la fonction f(x) = ln(x) est f’(x) = 1\/x.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de la fonction logarithme népérien ln(x) est bien 1\/x 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\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":12766,"question":"Quelle est la valeur de la limite lim (x→+∞) (x² + 3x)\/(2x² - 5) ?","option_a":"Limite = 1\/2","option_b":"Limite = 0","option_c":"Limite = +∞","option_d":"Limite = 3\/2","option_e":"","option_f":"","bonne_reponse":"a","explication":"En divisant numérateur et dénominateur par x², on obtient (1 + 3\/x)\/(2 - 5\/x²). Lorsque x tend vers +∞, les termes 3\/x et 5\/x² tendent vers 0, donc la limite est 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\": \"Limite = 1\/2\", \"b\": \"Limite = 0\", \"c\": \"Limite = +∞\", \"d\": \"Lim","_debug_options_count":4},{"id":12767,"question":"Soit f une fonction deux fois dérivable sur ℝ. Si f''(x) \u003E 0 pour tout x ∈ ℝ, alors f est convexe sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si la dérivée seconde est strictement positive sur un intervalle, alors la fonction est strictement convexe sur cet intervalle.","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}]