Quiz — 6a0723ebad8a4_Main Bac exam 1 (1).pdf

Question 1 sur 10 20:00

[{"id":31490,"question":"Quelle est la limite de la fonction f(x) = (3x² + 2x - 1)\/(x² - 4) lorsque x tend vers l'infini ?","option_a":"A) 3","option_b":"B) 0","option_c":"C) +∞","option_d":"D) -∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"La limite d'un quotient de polynômes de même degré est égale au rapport des coefficients des termes de plus haut degré. Ici, 3x²\/x² = 3.","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) 3\", \"b\": \"B) 0\", \"c\": \"C) +∞\", \"d\": \"D) -∞\"}}","_debug_options_count":4},{"id":31491,"question":"La fonction f(x) = x³ - 3x² + 4 est convexe sur l'intervalle ]-∞, 1[.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée seconde f''(x) = 6x - 6 est négative sur ]-∞, 1[, donc la fonction est concave 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":31492,"question":"Quel est le produit scalaire des vecteurs u(2, -1) et v(3, 4) dans un repère orthonormé ?","option_a":"A) 2","option_b":"B) 5","option_c":"C) 10","option_d":"D) -2","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le produit scalaire se calcule par u·v = (2×3) + (-1×4) = 6 - 4 = 2. L'option B est incorrecte.","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) 2\", \"b\": \"B) 5\", \"c\": \"C) 10\", \"d\": \"D) -2\"}}","_debug_options_count":4},{"id":31493,"question":"La suite définie par uₙ = (2n + 1)\/n converge vers :","option_a":"A) 0","option_b":"B) 1","option_c":"C) 2","option_d":"D) +∞","option_e":"","option_f":"","bonne_reponse":"d","explication":"En divisant numérateur et dénominateur par n, on obtient uₙ = (2 + 1\/n)\/1, qui tend vers 2 lorsque n tend vers l'infini.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"A) 0\", \"b\": \"B) 1\", \"c\": \"C) 2\", \"d\": \"D) +∞\"}}","_debug_options_count":4},{"id":31494,"question":"L'équation différentielle y' + 2y = 0 a pour solution générale :","option_a":"A) y = Ce^(-2x)","option_b":"B) y = Ce^(2x)","option_c":"C) y = Cx^2","option_d":"D) y = C","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une équation différentielle linéaire du premier ordre. La solution générale est y = Ce^(-2x), où C est une constante.","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) y = Ce^(-2x)\", \"b\": \"B) y = Ce^(2x)\", \"c\": \"C) y = Cx^2\", \"d\":","_debug_options_count":4},{"id":31495,"question":"La probabilité d'obtenir un nombre pair en lançant un dé équilibré est de 1\/2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Il y a 3 nombres pairs (2, 4, 6) sur 6 faces possibles, donc la probabilité est 3\/6 = 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":31496,"question":"Quel est le coefficient directeur de la tangente à la courbe de f(x) = ln(x) au point d'abscisse 1 ?","option_a":"A) 0","option_b":"B) 1","option_c":"C) e","option_d":"D) -1","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de ln(x) est f'(x) = 1\/x. En x = 1, f'(1) = 1, donc le coefficient directeur est 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) 0\", \"b\": \"B) 1\", \"c\": \"C) e\", \"d\": \"D) -1\"}}","_debug_options_count":4},{"id":31497,"question":"La fonction f(x) = x² - 4x + 5 admet un minimum égal à :","option_a":"A) 0","option_b":"B) 1","option_c":"C) 5","option_d":"D) -3","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le minimum d'une fonction quadratique f(x) = ax² + bx + c est atteint en x = -b\/(2a). Ici, x = 2, et f(2) = 4 - 8 + 5 = 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) 0\", \"b\": \"B) 1\", \"c\": \"C) 5\", \"d\": \"D) -3\"}}","_debug_options_count":4},{"id":31498,"question":"L'intervalle de fluctuation asymptotique d'une proportion p à 95% est donné par :","option_a":"A) [p - 1.96√(p(1-p)\/n), p + 1.96√(p(1-p)\/n)]","option_b":"B) [p - 2√(p(1-p)\/n), p + 2√(p(1-p)\/n)]","option_c":"C) [p - 1.96\/n, p + 1.96\/n]","option_d":"D) [p - √(p(1-p)), p + √(p(1-p))]","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intervalle de fluctuation asymptotique à 95% pour une proportion est [p - 1.96√(p(1-p)\/n), p + 1.96√(p(1-p)\/n)].","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) [p - 1.96√(p(1-p)\/n), p + 1.96√(p(1-p)\/n)]\", \"b\": \"B) [p -","_debug_options_count":4},{"id":31499,"question":"La suite géométrique de premier terme 3 et de raison 2 est définie par :","option_a":"A) uₙ = 3×2ⁿ","option_b":"B) uₙ = 3×2^(n-1)","option_c":"C) uₙ = 2×3ⁿ","option_d":"D) uₙ = 3 + 2n","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une suite géométrique de premier terme u₀ = 3 et de raison 2 est définie par uₙ = 3×2ⁿ. Cependant, si u₁ = 3, alors uₙ = 3×2^(n-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) uₙ = 3×2ⁿ\", \"b\": \"B) uₙ = 3×2^(n-1)\", \"c\": \"C) uₙ = ","_debug_options_count":4}]