Quiz — BD_examen1011.pdf

Question 1 sur 10 20:00

[{"id":71377,"question":"Quelle est la limite de la fonction f(x) = (2x² + 3x - 5)\/(x² - 4) lorsque x tend vers +∞ ?","option_a":"A) 2","option_b":"B) 0","option_c":"C) +∞","option_d":"D) -2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La limite d'un polynôme en +∞ est déterminée par le terme de plus haut degré. Ici, les termes dominants sont 2x²\/x² = 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\": \"A) 2\", \"b\": \"B) 0\", \"c\": \"C) +∞\", \"d\": \"D) -2\"}}","_debug_options_count":4},{"id":71378,"question":"Soit une suite (uₙ) définie par u₀ = 1 et uₙ₊₁ = 2uₙ + 3. Cette suite est-elle arithmétique ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une suite arithmétique a une différence constante entre deux termes consécutifs. Ici, uₙ₊₁ - uₙ = uₙ + 3, qui n'est pas constant.","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":71379,"question":"Quel est le produit scalaire de deux vecteurs orthogonaux dans un plan ?","option_a":"A) 0","option_b":"B) 1","option_c":"C) -1","option_d":"D) Indéterminé","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, le produit scalaire de deux vecteurs orthogonaux est nul, car cos(90°) = 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) 1\", \"c\": \"C) -1\", \"d\": \"D) Indéterminé\"}}","_debug_options_count":4},{"id":71380,"question":"La fonction f(x) = x³ - 3x² + 2x est-elle 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 est positive sur [0, 2] (car son discriminant est négatif), 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":71381,"question":"Quelle est la dérivée de la fonction f(x) = ln(3x + 1) ?","option_a":"A) 3\/(3x + 1)","option_b":"B) 1\/(3x + 1)","option_c":"C) 3\/(x + 1)","option_d":"D) 1\/x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(u) est u'\/u. Ici, u = 3x + 1, donc u' = 3. Ainsi, f'(x) = 3\/(3x + 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\": \"A) 3\/(3x + 1)\", \"b\": \"B) 1\/(3x + 1)\", \"c\": \"C) 3\/(x + 1)\", \"d\": \"","_debug_options_count":4},{"id":71382,"question":"Un événement A a une probabilité de 0,4. Quelle est la probabilité de l'événement contraire ?","option_a":"A) 0,4","option_b":"B) 0,6","option_c":"C) 0,5","option_d":"D) 1","option_e":"","option_f":"","bonne_reponse":"b","explication":"La probabilité de l'événement contraire est 1 - P(A) = 1 - 0,4 = 0,6.","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) 0,4\", \"b\": \"B) 0,6\", \"c\": \"C) 0,5\", \"d\": \"D) 1\"}}","_debug_options_count":4},{"id":71383,"question":"Soit un triangle ABC rectangle en A. Quel est le produit scalaire des vecteurs AB et AC ?","option_a":"A) AB × AC","option_b":"B) 0","option_c":"C) AB² + AC²","option_d":"D) AB × AC × cos(90°)","option_e":"","option_f":"","bonne_reponse":"b","explication":"Dans un triangle rectangle en A, les vecteurs AB et AC sont orthogonaux, donc leur produit scalaire est nul.","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) AB × AC\", \"b\": \"B) 0\", \"c\": \"C) AB² + AC²\", \"d\": \"D) AB × ","_debug_options_count":4},{"id":71384,"question":"La fonction f(x) = e^(2x) est-elle solution de l'équation différentielle y' = 2y ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de f(x) = e^(2x) est f'(x) = 2e^(2x) = 2f(x), donc f est solution de y' = 2y.","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":71385,"question":"Quel est le module d'un nombre complexe z = 3 + 4i ?","option_a":"A) 5","option_b":"B) 7","option_c":"C) 25","option_d":"D) 12","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module de z = a + bi est √(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) 25\", \"d\": \"D) 12\"}}","_debug_options_count":4},{"id":71386,"question":"Soit une variable aléatoire X suivant une loi normale N(0, 1). Quelle est la probabilité P(X ≤ 1) ?","option_a":"A) 0,68","option_b":"B) 0,84","option_c":"C) 0,95","option_d":"D) 0,99","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour une loi normale centrée réduite, P(X ≤ 1) ≈ 0,8413 (valeur lue dans les tables de la loi normale).","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) 0,68\", \"b\": \"B) 0,84\", \"c\": \"C) 0,95\", \"d\": \"D) 0,99\"}}","_debug_options_count":4}]