Quiz — 63317b5761187_Exercice 5.pdf

Question 1 sur 10 20:00

[{"id":12498,"question":"Quelle est la limite de la suite définie par uₙ = (n² + 3n) \/ (2n² - 5) lorsque n tend vers l'infini ?","option_a":"A. 0","option_b":"B. 1\/2","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"La limite se calcule en divisant le numérateur et le dénominateur par n², ce qui donne (1 + 3\/n) \/ (2 - 5\/n²) → 1\/2 lorsque n → ∞.","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. 1\/2\", \"c\": \"C. +∞\", \"d\": \"D. -∞\"}}","_debug_options_count":4},{"id":12499,"question":"La fonction f(x) = x³ - 3x² + 2x est décroissante sur l'intervalle [-1, 1].","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 [-1, 1] (par exemple, f'(0) = 2 \u003E 0), donc la fonction 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":12500,"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 = Cx + e^(-2x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'équation est de la forme y' + ay = 0, sa solution générale est 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":12501,"question":"Soit A et B deux événements indépendants tels que P(A) = 0,4 et P(B) = 0,5. Quelle est P(A ∩ B) ?","option_a":"A. 0,2","option_b":"B. 0,4","option_c":"C. 0,5","option_d":"D. 0,9","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour des événements indépendants, P(A ∩ B) = P(A) × P(B) = 0,4 × 0,5 = 0,2.","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. 0,2\", \"b\": \"B. 0,4\", \"c\": \"C. 0,5\", \"d\": \"D. 0,9\"}}","_debug_options_count":4},{"id":12502,"question":"Le nombre complexe z = (1 + i)² est égal à :","option_a":"A. 2i","option_b":"B. 1 + 2i","option_c":"C. 2","option_d":"D. 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"En développant (1 + i)² = 1 + 2i + i² = 1 + 2i - 1 = 2i.","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. 2i\", \"b\": \"B. 1 + 2i\", \"c\": \"C. 2\", \"d\": \"D. 0\"}}","_debug_options_count":4},{"id":12503,"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) n'est définie que pour x \u003E 0. Elle n'est pas définie pour les réels négatifs ou nuls.","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":12504,"question":"Quelle est la dérivée de la fonction f(x) = e^(3x) ?","option_a":"A. e^(3x)","option_b":"B. 3e^(3x)","option_c":"C. 3x e^(3x-1)","option_d":"D. e^(3x) \/ 3","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de e^(u(x)) est u'(x) × e^(u(x)). Ici, u(x) = 3x, donc u'(x) = 3, d'où f'(x) = 3e^(3x).","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. e^(3x)\", \"b\": \"B. 3e^(3x)\", \"c\": \"C. 3x e^(3x-1)\", \"d\": \"D. e^","_debug_options_count":4},{"id":12505,"question":"Soit une variable aléatoire X suivant une loi binomiale B(n=10, p=0,3). Quelle est l'espérance de X ?","option_a":"A. 3","option_b":"B. 7","option_c":"C. 10","option_d":"D. 0,3","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'espérance d'une loi binomiale B(n, p) est E(X) = n × p = 10 × 0,3 = 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. 7\", \"c\": \"C. 10\", \"d\": \"D. 0,3\"}}","_debug_options_count":4},{"id":12506,"question":"La fonction f(x) = x^4 - 4x³ + 6x² est convexe sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée seconde f''(x) = 12x² - 24x + 12 = 12(x² - 2x + 1) = 12(x-1)² ≥ 0 pour tout x. La fonction est donc convexe sur ℝ.","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":12507,"question":"Quelle est la valeur de l'intégrale ∫₀¹ (3x² + 2x) 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 se calcule en trouvant une primitive : [x³ + x²]₀¹ = (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}]