Quiz — 6297851fae52f_Corrigé1-Maths-Prototype d’examen Bac N 8.pdf

Question 1 sur 10 20:00

[{"id":31110,"question":"Quelle est la limite de la suite définie par u_{n+1} = 2u_n + 3 et u_0 = 1 ?","option_a":"A. 0","option_b":"B. -3","option_c":"C. 3","option_d":"D. La suite diverge","option_e":"","option_f":"","bonne_reponse":"c","explication":"La suite est arithmético-géométrique. En résolvant l'équation L = 2L + 3, on trouve L = -3. Comme |2| \u003E 1, la suite diverge.","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. -3\", \"c\": \"C. 3\", \"d\": \"D. La suite diverge\"}}","_debug_options_count":4},{"id":31111,"question":"Soit f une fonction dérivable sur ℝ telle que f'(x) = 3x² - 2x + 1. La fonction f est-elle convexe sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une fonction est convexe si sa dérivée seconde est positive. Ici, f''(x) = 6x - 2, qui n'est pas toujours positive (ex. : pour x \u003C 1\/3). Donc f n'est pas 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":31112,"question":"Quelle est la solution générale de l'équation différentielle y' + 2y = 4 ?","option_a":"A. y = Ce^{-2x} + 2","option_b":"B. y = Ce^{2x} + 2","option_c":"C. y = Ce^{-2x} - 2","option_d":"D. y = Ce^{x} + 4","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution générale d'une équation différentielle linéaire du premier ordre est y = Ce^{-ax} + b\/a, où b\/a est une solution particulière. Ici, a = 2 et b = 4, donc y = Ce^{-2x} + 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. y = Ce^{-2x} + 2\", \"b\": \"B. y = Ce^{2x} + 2\", \"c\": \"C. y = Ce^","_debug_options_count":4},{"id":31113,"question":"Dans un plan muni d'un repère orthonormé, l'équation x + 2y - 3 = 0 représente :","option_a":"A. Une droite parallèle à l'axe des abscisses","option_b":"B. Une droite parallèle à l'axe des ordonnées","option_c":"C. Une droite passant par le point (1, 1)","option_d":"D. Une droite passant par le point (3, 0)","option_e":"","option_f":"","bonne_reponse":"d","explication":"L'équation x + 2y - 3 = 0 peut s'écrire y = -0.5x + 1.5. Elle passe par le point (3, 0) car 3 + 2*0 - 3 = 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"A. Une droite parallèle à l'axe des abscisses\", \"b\": \"B. Une dr","_debug_options_count":4},{"id":31114,"question":"Soit X une variable aléatoire suivant une loi binomiale B(n=10, p=0.3). Quelle est la probabilité P(X ≤ 2) ?","option_a":"A. 0.3828","option_b":"B. 0.1493","option_c":"C. 0.6496","option_d":"D. 0.2668","option_e":"","option_f":"","bonne_reponse":"b","explication":"P(X ≤ 2) = P(X=0) + P(X=1) + P(X=2) = C(10,0)(0.3)^0(0.7)^10 + C(10,1)(0.3)^1(0.7)^9 + C(10,2)(0.3)^2(0.7)^8 ≈ 0.1493.","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. 0.3828\", \"b\": \"B. 0.1493\", \"c\": \"C. 0.6496\", \"d\": \"D. 0.2668\"}","_debug_options_count":4},{"id":31115,"question":"La fonction f définie par f(x) = x^3 - 3x + 1 admet un extremum local en x = 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"f'(x) = 3x² - 3. f'(1) = 0 et f''(x) = 6x. Comme f''(1) = 6 \u003E 0, x = 1 est un minimum local. Donc l'affirmation est vraie.","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":31116,"question":"Quelle est la valeur de l'intégrale ∫_{0}^{1} (3x² + 2x) dx ?","option_a":"A. 2","option_b":"B. 1","option_c":"C. 3","option_d":"D. 4","option_e":"","option_f":"","bonne_reponse":"a","explication":"∫ (3x² + 2x) dx = x³ + x² + C. Évaluée entre 0 et 1, on obtient (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\": \"a\", \"options\": {\"a\": \"A. 2\", \"b\": \"B. 1\", \"c\": \"C. 3\", \"d\": \"D. 4\"}}","_debug_options_count":4},{"id":31117,"question":"Soit (u_n) une suite géométrique de raison q = -2 et de premier terme u_0 = 5. Quelle est la valeur de u_3 ?","option_a":"A. -40","option_b":"B. 40","option_c":"C. -20","option_d":"D. 20","option_e":"","option_f":"","bonne_reponse":"a","explication":"u_n = u_0 * q^n. Donc u_3 = 5 * (-2)^3 = 5 * (-8) = -40.","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. -40\", \"b\": \"B. 40\", \"c\": \"C. -20\", \"d\": \"D. 20\"}}","_debug_options_count":4},{"id":31118,"question":"Dans un espace vectoriel, deux vecteurs non nuls sont colinéaires si et seulement si :","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Deux vecteurs non nuls sont colinéaires si l'un est multiple de l'autre, c'est-à-dire s'il existe un scalaire λ tel que u = λv. Donc l'affirmation est vraie.","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":31119,"question":"Quelle est la dérivée de la fonction f(x) = ln(x² + 1) ?","option_a":"A. 2x \/ (x² + 1)","option_b":"B. 1 \/ (x² + 1)","option_c":"C. 2x","option_d":"D. x \/ (x² + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la dérivée d'une fonction composée, f'(x) = (1 \/ (x² + 1)) * 2x = 2x \/ (x² + 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. 2x \/ (x² + 1)\", \"b\": \"B. 1 \/ (x² + 1)\", \"c\": \"C. 2x\", \"d\": \"","_debug_options_count":4}]