Quiz — Lycée pilote Ariana DS 1 2014 corr.pdf

Question 1 sur 10 20:00

[{"id":4996,"question":"Quelle est la solution de l’équation x² - 5x + 6 = 0 ?","option_a":"A : x = 2 ou x = 3","option_b":"B : x = -2 ou x = -3","option_c":"C : x = 1 ou x = 6","option_d":"D : x = -1 ou x = 6","option_e":"","option_f":"","bonne_reponse":"a","explication":"L’équation x² - 5x + 6 = 0 se factorise en (x-2)(x-3) = 0, donc les solutions sont x = 2 et x = 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 : x = 2 ou x = 3\", \"b\": \"B : x = -2 ou x = -3\", \"c\": \"C : x = 1","_debug_options_count":4},{"id":4997,"question":"La fonction f(x) = x³ - 3x² + 2x est croissante sur l’intervalle ]-∞, 0[.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f’(x) = 3x² - 6x + 2 est positive sur ]-∞, 0[ seulement si x \u003C 0, mais pas pour tout x dans cet intervalle. Par exemple, f’(-1) = 11 \u003E 0, mais f’(0) = 2 \u003E 0. La fonction est croissante sur ]-∞, 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":4998,"question":"Quelle est la limite de (3x² + 2x - 1)\/(x² - 4) lorsque x tend vers +∞ ?","option_a":"A : 0","option_b":"B : 3","option_c":"C : +∞","option_d":"D : -∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par x², on obtient (3 + 2\/x - 1\/x²)\/(1 - 4\/x²), donc la limite est 3\/1 = 3.","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 : 3\", \"c\": \"C : +∞\", \"d\": \"D : -∞\"}}","_debug_options_count":4},{"id":4999,"question":"Soit une suite définie par u₀ = 1 et uₙ₊₁ = 2uₙ + 1. Quelle est la valeur de u₂ ?","option_a":"A : 3","option_b":"B : 5","option_c":"C : 7","option_d":"D : 9","option_e":"","option_f":"","bonne_reponse":"c","explication":"u₁ = 2*1 + 1 = 3, u₂ = 2*3 + 1 = 7.","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 : 3\", \"b\": \"B : 5\", \"c\": \"C : 7\", \"d\": \"D : 9\"}}","_debug_options_count":4},{"id":5000,"question":"La dérivée de la fonction f(x) = ln(x² + 1) est f’(x) = 2x\/(x² + 1).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la formule de dérivation des fonctions composées, f’(x) = (2x)\/(x² + 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":5001,"question":"Quelle est la primitive de f(x) = 4x³ - 2x + 1 ?","option_a":"A : x⁴ - x² + x + C","option_b":"B : x⁴ - x² + C","option_c":"C : 12x² - 2 + C","option_d":"D : x⁴ - 2x² + x + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de 4x³ est x⁴, celle de -2x est -x², et celle de 1 est x. Donc F(x) = x⁴ - x² + x + C.","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 : x⁴ - x² + x + C\", \"b\": \"B : x⁴ - x² + C\", \"c\": \"C : 12x","_debug_options_count":4},{"id":5002,"question":"Soit une variable aléatoire X suivant une loi binomiale B(n=5, p=0.3). Quelle est P(X=2) ?","option_a":"A : 0.3087","option_b":"B : 0.1323","option_c":"C : 0.3601","option_d":"D : 0.2304","option_e":"","option_f":"","bonne_reponse":"a","explication":"P(X=2) = C(5,2) * (0.3)² * (0.7)³ = 10 * 0.09 * 0.343 = 0.3087.","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 : 0.3087\", \"b\": \"B : 0.1323\", \"c\": \"C : 0.3601\", \"d\": \"D : 0.23","_debug_options_count":4},{"id":5003,"question":"La fonction f(x) = e^x est 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) = e^x est toujours positive, 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},{"id":5004,"question":"Quelle est la solution de l’inéquation ln(x) \u003E 1 ?","option_a":"A : x \u003E 0","option_b":"B : x \u003E e","option_c":"C : x \u003E 1","option_d":"D : x \u003E 1\/e","option_e":"","option_f":"","bonne_reponse":"b","explication":"ln(x) \u003E 1 équivaut à x \u003E e (car ln(e) = 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 : x \u003E 0\", \"b\": \"B : x \u003E e\", \"c\": \"C : x \u003E 1\", \"d\": \"D : x \u003E 1\/e","_debug_options_count":4},{"id":5005,"question":"Soit une suite géométrique de premier terme u₀ = 2 et de raison q = -1\/2. Quelle est la limite de uₙ lorsque n tend vers +∞ ?","option_a":"A : 0","option_b":"B : +∞","option_c":"C : -∞","option_d":"D : n’existe pas","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une suite géométrique de raison |q| \u003C 1 converge vers 0. Ici, |q| = 1\/2 \u003C 1, donc la limite est 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\": \"A : 0\", \"b\": \"B : +∞\", \"c\": \"C : -∞\", \"d\": \"D : n’existe pa","_debug_options_count":4}]