Quiz — série31-Révision trimestre 2 corrigé.pdf

Question 1 sur 10 20:00

[{"id":31280,"question":"Quelle est la dérivée de la fonction f(x) = 3x² - 5x + 2 ?","option_a":"A. 6x - 5","option_b":"B. 3x - 5","option_c":"C. 6x² - 5","option_d":"D. 6x - 5x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme f(x) = ax^n est f'(x) = n·a·x^(n-1). Ici, f'(x) = 2·3x^(2-1) - 1·5x^(1-1) = 6x - 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. 6x - 5\", \"b\": \"B. 3x - 5\", \"c\": \"C. 6x² - 5\", \"d\": \"D. 6x - 5","_debug_options_count":4},{"id":31281,"question":"L'intégrale ∫(2x + 1) dx entre 0 et 2 est égale à :","option_a":"A. 4","option_b":"B. 6","option_c":"C. 8","option_d":"D. 10","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégrale d'une fonction linéaire f(x) = ax + b est F(x) = (a\/2)x² + bx. Ici, F(x) = x² + x. Évaluée entre 0 et 2 : F(2) - F(0) = (4 + 2) - (0 + 0) = 6.","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. 4\", \"b\": \"B. 6\", \"c\": \"C. 8\", \"d\": \"D. 10\"}}","_debug_options_count":4},{"id":31282,"question":"Vrai ou Faux ? La fonction f(x) = x³ est strictement croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de f(x) = x³ est f'(x) = 3x², qui est toujours positive (sauf en x=0 où elle est nulle). La fonction est donc strictement croissante 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":31283,"question":"Quelle est la limite de la fonction f(x) = (2x² + 3x - 5)\/(x² - 1) quand x tend vers +∞ ?","option_a":"A. 0","option_b":"B. 2","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour les limites à l'infini de fonctions rationnelles, on compare les degrés du numérateur et du dénominateur. Ici, les deux sont de degré 2, donc la limite est égale au rapport des coefficients dominants : 2\/1 = 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. 2\", \"c\": \"C. +∞\", \"d\": \"D. -∞\"}}","_debug_options_count":4},{"id":31284,"question":"Vrai ou Faux ? L'équation x² + y² + z² = 1 représente une sphère de centre O(0,0,0) et de rayon 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Cette équation est la forme canonique d'une sphère centrée à l'origine avec un rayon de 1. Elle est donc 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":31285,"question":"Quelle est la primitive de la fonction f(x) = 4x³ - 2x + 1 ?","option_a":"A. x⁴ - x² + x + C","option_b":"B. 12x² - 2 + C","option_c":"C. 4x⁴ - x² + x + C","option_d":"D. x⁴ - x + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive d'une fonction polynôme s'obtient en augmentant chaque exposant de 1 et en divisant par le nouvel exposant. Ici, F(x) = (4\/4)x⁴ - (2\/2)x² + x + C = x⁴ - x² + x + C.","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. x⁴ - x² + x + C\", \"b\": \"B. 12x² - 2 + C\", \"c\": \"C. 4x⁴ -","_debug_options_count":4},{"id":31286,"question":"Dans un repère orthonormé, l'équation d'un plan passant par le point A(1,2,3) et de vecteur normal n(2,-1,3) est :","option_a":"A. 2x - y + 3z = 0","option_b":"B. 2x - y + 3z = 11","option_c":"C. x + 2y + 3z = 11","option_d":"D. 2x + y - 3z = 11","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'équation d'un plan est donnée par n·(X - A) = 0. Ici, 2(x-1) -1(y-2) + 3(z-3) = 0 → 2x - 2 - y + 2 + 3z - 9 = 0 → 2x - y + 3z = 9. En remplaçant A(1,2,3), on obtient 2(1) - 2 + 3(3) = 11.","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. 2x - y + 3z = 0\", \"b\": \"B. 2x - y + 3z = 11\", \"c\": \"C. x + 2y ","_debug_options_count":4},{"id":31287,"question":"Vrai ou Faux ? La fonction f(x) = e^(-x) est décroissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de f(x) = e^(-x) est f'(x) = -e^(-x), qui est toujours négative. La fonction est donc strictement décroissante 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":31288,"question":"Quelle est la valeur de l'intégrale ∫(sin(x) + cos(x)) dx entre 0 et π\/2 ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 2","option_d":"D. π\/2","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'intégrale de sin(x) entre 0 et π\/2 est [-cos(x)]₀^(π\/2) = -cos(π\/2) - (-cos(0)) = 0 + 1 = 1. L'intégrale de cos(x) entre 0 et π\/2 est [sin(x)]₀^(π\/2) = sin(π\/2) - sin(0) = 1 - 0 = 1. La somme est donc 1 + 1 = 2.","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. 2\", \"d\": \"D. π\/2\"}}","_debug_options_count":4},{"id":31289,"question":"Quelle est la probabilité d'obtenir un nombre pair en lançant un dé équilibré à 6 faces ?","option_a":"A. 1\/6","option_b":"B. 1\/3","option_c":"C. 1\/2","option_d":"D. 2\/3","option_e":"","option_f":"","bonne_reponse":"c","explication":"Un dé équilibré a 6 faces numérotées de 1 à 6. Les nombres pairs sont 2, 4 et 6, soit 3 cas favorables sur 6 possibles. La probabilité est donc 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\": \"c\", \"options\": {\"a\": \"A. 1\/6\", \"b\": \"B. 1\/3\", \"c\": \"C. 1\/2\", \"d\": \"D. 2\/3\"}}","_debug_options_count":4}]