Quiz — Fiche-série-entière.pdf

Question 1 sur 10 20:00

[{"id":78251,"question":"Quelle est la limite de la suite définie par uₙ = (3n² + 2n - 1)\/(2n² - n + 4) lorsque n tend vers l'infini ?","option_a":"A. 0","option_b":"B. 3\/2","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"La limite d'une suite rationnelle est égale au rapport des coefficients dominants des polynômes au numérateur et au dénominateur. Ici, 3\/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. 3\/2\", \"c\": \"C. +∞\", \"d\": \"D. -∞\"}}","_debug_options_count":4},{"id":78252,"question":"Soit f une fonction dérivable sur ℝ telle que f'(x) = 2x - 3. Si f(0) = 5, alors f(2) est égal à :","option_a":"A. 3","option_b":"B. 5","option_c":"C. 7","option_d":"D. 9","option_e":"","option_f":"","bonne_reponse":"a","explication":"On intègre f'(x) pour obtenir f(x) = x² - 3x + C. Avec f(0) = 5, on trouve C = 5. Donc f(2) = 4 - 6 + 5 = 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. 3\", \"b\": \"B. 5\", \"c\": \"C. 7\", \"d\": \"D. 9\"}}","_debug_options_count":4},{"id":78253,"question":"Vrai ou Faux ? Toute fonction continue sur un intervalle fermé et borné est 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 sur un intervalle fermé et borné n'est pas nécessairement dérivable. Exemple : la fonction valeur absolue 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":78254,"question":"Quelle est la dérivée de la fonction f(x) = ln(3x² + 2x) ?","option_a":"A. (6x + 2)\/(3x² + 2x)","option_b":"B. (3x² + 2x)\/(6x + 2)","option_c":"C. 6x + 2","option_d":"D. 1\/(3x² + 2x)","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. (3x² + 2x)\/(6x + 2)\", \"c\": \"C","_debug_options_count":4},{"id":78255,"question":"Soit (uₙ) une suite géométrique de raison q = -2 et de premier terme u₀ = 3. Quelle est la valeur de u₃ ?","option_a":"A. -24","option_b":"B. 24","option_c":"C. -12","option_d":"D. 12","option_e":"","option_f":"","bonne_reponse":"a","explication":"La formule d'une suite géométrique est uₙ = u₀ × qⁿ. Donc u₃ = 3 × (-2)³ = 3 × (-8) = -24.","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. -24\", \"b\": \"B. 24\", \"c\": \"C. -12\", \"d\": \"D. 12\"}}","_debug_options_count":4},{"id":78256,"question":"Vrai ou Faux ? Si une suite est croissante et majorée, alors elle converge.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est le théorème de convergence monotone : une suite croissante et majorée converge vers sa borne supérieure.","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":78257,"question":"Quelle est la solution de l'équation différentielle y' + 2y = 0 avec la condition initiale y(0) = 1 ?","option_a":"A. y = e^(-2x)","option_b":"B. y = e^(2x)","option_c":"C. y = -2e^(-2x)","option_d":"D. y = 2e^(-x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation différentielle est de la forme y' + ay = 0, dont la solution générale est y = Ce^(-ax). Ici, a = 2, donc y = Ce^(-2x). Avec y(0) = 1, on trouve C = 1. Ainsi, y = e^(-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. y = e^(-2x)\", \"b\": \"B. y = e^(2x)\", \"c\": \"C. y = -2e^(-2x)\", \"","_debug_options_count":4},{"id":78258,"question":"Soit f une fonction telle que f(x) = x³ - 3x² + 2x. Quel est le coefficient directeur de la tangente à la courbe de f au point d'abscisse x = 1 ?","option_a":"A. -1","option_b":"B. 0","option_c":"C. 1","option_d":"D. 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le coefficient directeur de la tangente est égal à f'(1). f'(x) = 3x² - 6x + 2, donc f'(1) = 3 - 6 + 2 = -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. -1\", \"b\": \"B. 0\", \"c\": \"C. 1\", \"d\": \"D. 2\"}}","_debug_options_count":4},{"id":78259,"question":"Vrai ou Faux ? La fonction exponentielle est la seule fonction dérivable sur ℝ telle que f' = f.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est un résultat fondamental : l'équation différentielle f' = f admet une unique solution à une constante multiplicative près, et la fonction exponentielle est la solution particulière avec f(0) = 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":78260,"question":"Quelle est la valeur de l'intégrale ∫₀¹ (2x + 1) dx ?","option_a":"A. 1","option_b":"B. 2","option_c":"C. 3","option_d":"D. 4","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégrale d'une fonction linéaire est calculée en trouvant une primitive. ∫(2x + 1) dx = x² + x + C. Donc ∫₀¹ (2x + 1) dx = (1 + 1) - (0 + 0) = 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. 1\", \"b\": \"B. 2\", \"c\": \"C. 3\", \"d\": \"D. 4\"}}","_debug_options_count":4}]