Quiz — math_c.pdf

Question 1 sur 10 20:00

[{"id":64876,"question":"Quelle est la limite de la fonction f(x) = (3x² + 2x - 1)\/(x² - 4) lorsque x tend vers +∞ ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 3","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"La limite d'un polynôme de même degré au numérateur et au dénominateur est égale au rapport des coefficients dominants. 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\": \"c\", \"options\": {\"a\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. 3\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":64877,"question":"La fonction f(x) = x³ - 3x² + 2 est croissante sur l'intervalle [-1, 1].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f'(x) = 3x² - 6x s'annule en x=0 et x=2. Sur [-1,1], f'(x) ≤ 0, donc la fonction est décroissante sur cet intervalle.","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":64878,"question":"Quel est le produit scalaire de deux vecteurs u(2, -1) et v(-3, 4) ?","option_a":"A. -10","option_b":"B. 5","option_c":"C. -5","option_d":"D. 10","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le produit scalaire se calcule par u·v = (2)(-3) + (-1)(4) = -6 -4 = -10.","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. -10\", \"b\": \"B. 5\", \"c\": \"C. -5\", \"d\": \"D. 10\"}}","_debug_options_count":4},{"id":64879,"question":"La probabilité d'obtenir un nombre pair en lançant un dé équilibré est-elle égale à 1\/2 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Un dé équilibré a 3 nombres pairs (2,4,6) sur 6 faces, 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":64880,"question":"Quelle est la solution générale de l'équation différentielle y' + 2y = 0 ?","option_a":"A. y = Ce^{2x}","option_b":"B. y = Ce^{-2x}","option_c":"C. y = C + e^{2x}","option_d":"D. y = C - e^{-2x}","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'équation différentielle linéaire du premier ordre y' + ay = 0 a pour solution générale y = Ce^{-ax}. Ici, a=2, donc y = Ce^{-2x}.","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. y = Ce^{2x}\", \"b\": \"B. y = Ce^{-2x}\", \"c\": \"C. y = C + e^{2x}\"","_debug_options_count":4},{"id":64881,"question":"Le déterminant d'une matrice 2x2 [[a, b], [c, d]] est égal à ad - bc.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La formule du déterminant pour une matrice 2x2 est bien ad - bc.","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":64882,"question":"Quelle est la dérivée de la fonction f(x) = ln(3x² + 1) ?","option_a":"A. (6x)\/(3x² + 1)","option_b":"B. (3x)\/(3x² + 1)","option_c":"C. 6x","option_d":"D. (6x²)\/(3x² + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la règle de dérivation des fonctions composées, f'(x) = (1\/(3x² + 1)) * 6x = (6x)\/(3x² + 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. (6x)\/(3x² + 1)\", \"b\": \"B. (3x)\/(3x² + 1)\", \"c\": \"C. 6x\", \"d\"","_debug_options_count":4},{"id":64883,"question":"La suite définie par u_{n+1} = 2u_n + 3 avec u_0 = 1 est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Cette suite est arithmético-géométrique. Sa limite, si elle existe, vérifie L = 2L + 3 ⇒ L = -3. Mais u_n = 2^n - 3, donc la suite diverge vers +∞.","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":64884,"question":"Quel est le module du nombre complexe z = 3 + 4i ?","option_a":"A. 5","option_b":"B. 7","option_c":"C. √7","option_d":"D. 25","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module d'un nombre complexe z = a + bi est |z| = √(a² + b²) = √(9 + 16) = 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. √7\", \"d\": \"D. 25\"}}","_debug_options_count":4},{"id":64885,"question":"La fonction f(x) = e^{x²} est-elle paire ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une fonction paire vérifie f(-x) = f(x). Ici, f(-x) = e^{(-x)²} = e^{x²} = f(x), donc la fonction est paire.","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}]