Quiz — EXAMEN 2018.pdf

Question 1 sur 10 20:00

[{"id":20709,"question":"Quelle est la limite de la fonction f(x) = (2x² + 3x - 5)\/(x² - 4) lorsque x tend vers +∞ ?","option_a":"A. 0","option_b":"B. 2","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par x², on obtient f(x) = (2 + 3\/x - 5\/x²)\/(1 - 4\/x²). Lorsque x tend vers +∞, les termes en 1\/x et 1\/x² 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\": \"b\", \"options\": {\"a\": \"A. 0\", \"b\": \"B. 2\", \"c\": \"C. +∞\", \"d\": \"D. -∞\"}}","_debug_options_count":4},{"id":20710,"question":"Soit une suite géométrique de raison q = 1\/2 et de premier terme u₀ = 8. Quelle est la limite de cette suite lorsque n tend vers +∞ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une suite géométrique de raison |q| \u003C 1 converge vers 0. Ici, q = 1\/2, donc la suite tend vers 0 lorsque n tend vers +∞.","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":20711,"question":"Quel est le module du nombre complexe z = 3 + 4i ?","option_a":"A. 5","option_b":"B. 7","option_c":"C. 3","option_d":"D. 4","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module d'un nombre complexe z = a + bi est donné par √(a² + b²). Ici, |z| = √(3² + 4²) = √(9 + 16) = √25 = 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. 5\", \"b\": \"B. 7\", \"c\": \"C. 3\", \"d\": \"D. 4\"}}","_debug_options_count":4},{"id":20712,"question":"Soit une variable aléatoire X suivant une loi normale N(μ, σ²). Si μ = 10 et σ = 2, quelle est la probabilité P(X ≤ 12) ?","option_a":"A. 0.1587","option_b":"B. 0.8413","option_c":"C. 0.6915","option_d":"D. 0.3085","option_e":"","option_f":"","bonne_reponse":"b","explication":"On standardise X en Z = (X - μ)\/σ. Pour X = 12, Z = (12 - 10)\/2 = 1. La probabilité P(Z ≤ 1) ≈ 0.8413 d'après la table de la loi normale centrée réduite.","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. 0.1587\", \"b\": \"B. 0.8413\", \"c\": \"C. 0.6915\", \"d\": \"D. 0.3085\"}","_debug_options_count":4},{"id":20713,"question":"La fonction f(x) = x³ - 3x + 1 est strictement croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de f est f'(x) = 3x² - 3. f'(x) = 0 pour x = ±1. f'(x) \u003C 0 sur ]-1, 1[, donc f n'est pas strictement croissante sur 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":20714,"question":"Quelle est l'équation de la tangente à la courbe y = x² + 2x + 1 au point d'abscisse x = 1 ?","option_a":"A. y = 4x - 1","option_b":"B. y = 4x + 1","option_c":"C. y = 2x + 3","option_d":"D. y = 3x - 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de f(x) = x² + 2x + 1 est f'(x) = 2x + 2. En x = 1, f'(1) = 4 et f(1) = 4. L'équation de la tangente est y = f'(1)(x - 1) + f(1) = 4(x - 1) + 4 = 4x.","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 = 4x - 1\", \"b\": \"B. y = 4x + 1\", \"c\": \"C. y = 2x + 3\", \"d\": ","_debug_options_count":4},{"id":20715,"question":"Soit un cube de côté 3 cm. Quel est le volume de la sphère inscrite dans ce cube ?","option_a":"A. (9\/2)π cm³","option_b":"B. (27\/6)π cm³","option_c":"C. (9\/6)π cm³","option_d":"D. (3\/2)π cm³","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le rayon de la sphère inscrite est égal à la moitié du côté du cube, soit r = 3\/2 cm. Le volume de la sphère est V = (4\/3)πr³ = (4\/3)π(27\/8) = (9\/2)π cm³.","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. (9\/2)π cm³\", \"b\": \"B. (27\/6)π cm³\", \"c\": \"C. (9\/6)π cm³\"","_debug_options_count":4},{"id":20716,"question":"La fonction f(x) = ln(x) est définie pour tout x réel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction ln(x) est définie uniquement pour x \u003E 0. Elle n'est pas définie pour 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":20717,"question":"Soit une suite arithmétique de premier terme u₀ = 5 et de raison r = 3. Quel est le terme u₅ ?","option_a":"A. 14","option_b":"B. 20","option_c":"C. 17","option_d":"D. 23","option_e":"","option_f":"","bonne_reponse":"c","explication":"Le terme général d'une suite arithmétique est uₙ = u₀ + n·r. Pour n = 5, u₅ = 5 + 5·3 = 5 + 15 = 20.","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. 14\", \"b\": \"B. 20\", \"c\": \"C. 17\", \"d\": \"D. 23\"}}","_debug_options_count":4},{"id":20718,"question":"Quel est le résultat de l'intégrale ∫(2x + 1) dx entre 0 et 2 ?","option_a":"A. 4","option_b":"B. 6","option_c":"C. 8","option_d":"D. 10","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégrale de 2x + 1 est x² + x. Évaluée entre 0 et 2, on obtient (2² + 2) - (0² + 0) = 4 + 2 = 6.","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. 4\", \"b\": \"B. 6\", \"c\": \"C. 8\", \"d\": \"D. 10\"}}","_debug_options_count":4}]