Quiz — ex app 2.PDF

Question 1 sur 10 20:00

[{"id":54437,"question":"Quelle est la limite de la fonction f(x) = (2x² + 3x - 5)\/(x² - 1) lorsque x tend vers +∞ ?","option_a":"A) 2","option_b":"B) 0","option_c":"C) -∞","option_d":"D) +∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"La limite d'un polynôme rationnel lorsque x tend vers l'infini est égale au rapport des coefficients des termes de plus haut degré. Ici, 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) +∞\"}}","_debug_options_count":4},{"id":54438,"question":"Soit une suite définie par u₀ = 1 et uₙ₊₁ = 3uₙ + 2. 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 = 2uₙ + 2, qui n'est pas constante.","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":54439,"question":"Quelle est la dérivée de la fonction f(x) = ln(3x² + 2x) ?","option_a":"A) (6x + 2)\/(3x² + 2x)","option_b":"B) (6x + 2)\/ln(3x² + 2x)","option_c":"C) 1\/(3x² + 2x)","option_d":"D) (3x² + 2x)\/(6x + 2)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(u) est u'\/u. Ici, u = 3x² + 2x, donc u' = 6x + 2. Ainsi, f'(x) = (6x + 2)\/(3x² + 2x).","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 + 2)\/(3x² + 2x)\", \"b\": \"B) (6x + 2)\/ln(3x² + 2x)\", \"c\": ","_debug_options_count":4},{"id":54440,"question":"Le produit de deux matrices carrées A et B de même taille est-il toujours commutatif ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"En général, AB ≠ BA. Le produit matriciel n'est pas commutatif, sauf dans des cas particuliers (ex : matrices diagonales).","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":54441,"question":"Quelle est l'intégrale de f(x) = 5x⁴ entre 0 et 1 ?","option_a":"A) 1","option_b":"B) 5","option_c":"C) 0","option_d":"D) 1\/5","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégrale de xⁿ est xⁿ⁺¹\/(n+1). Ici, ∫5x⁴ dx = 5x⁵\/5 = x⁵. Entre 0 et 1, cela donne 1⁵ - 0⁵ = 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) 1\", \"b\": \"B) 5\", \"c\": \"C) 0\", \"d\": \"D) 1\/5\"}}","_debug_options_count":4},{"id":54442,"question":"Soit z = 2 + 3i un nombre complexe. Quel est son conjugué ?","option_a":"A) 2 - 3i","option_b":"B) -2 + 3i","option_c":"C) 3 + 2i","option_d":"D) 2 + 3i","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le conjugué d'un nombre complexe z = a + bi est z̄ = a - bi.","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 - 3i\", \"b\": \"B) -2 + 3i\", \"c\": \"C) 3 + 2i\", \"d\": \"D) 2 + 3i\"","_debug_options_count":4},{"id":54443,"question":"Une fonction continue sur un intervalle fermé [a, b] est-elle toujours dérivable sur cet intervalle ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une fonction continue n'est pas nécessairement dérivable. Exemple : f(x) = |x| est continue en 0 mais non dérivable en 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":54444,"question":"Quelle est la solution de l'équation différentielle y' + 2y = 0 ?","option_a":"A) y = Ce^{2x}","option_b":"B) y = Ce^{-2x}","option_c":"C) y = Cx","option_d":"D) y = C","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'équation y' + 2y = 0 est une équation différentielle linéaire du premier ordre. Sa solution générale est 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 = Cx\", \"d\": \"","_debug_options_count":4},{"id":54445,"question":"Le déterminant d'une matrice triangulaire est-il égal au produit des éléments de sa diagonale ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, pour une matrice triangulaire (supérieure ou inférieure), le déterminant est bien le produit des éléments de la diagonale.","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":54446,"question":"Quelle est la probabilité d'obtenir un double 6 en lançant deux dés équilibrés ?","option_a":"A) 1\/6","option_b":"B) 1\/12","option_c":"C) 1\/36","option_d":"D) 1\/18","option_e":"","option_f":"","bonne_reponse":"c","explication":"Il y a 6 × 6 = 36 issues possibles. Une seule issue correspond à un double 6 (6,6). La probabilité est donc 1\/36.","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) 1\/6\", \"b\": \"B) 1\/12\", \"c\": \"C) 1\/36\", \"d\": \"D) 1\/18\"}}","_debug_options_count":4}]