Quiz — 13

Question 1 sur 10 20:00

[{"id":25919,"question":"Quelle est la dérivée de la fonction f(x) = x³ - 2x² + 5x - 1 ?","option_a":"3x² - 4x + 5","option_b":"x² - 4x + 5","option_c":"3x² - 4x - 1","option_d":"x³ - 2x + 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme est obtenue en appliquant la règle de dérivation terme à terme : f'(x) = 3x² - 4x + 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\": \"3x² - 4x + 5\", \"b\": \"x² - 4x + 5\", \"c\": \"3x² - 4x - 1\", \"d\": \"","_debug_options_count":4},{"id":25920,"question":"Soit une matrice A = [[2, 1], [3, 4]]. Quel est son déterminant ?","option_a":"5","option_b":"10","option_c":"8","option_d":"1","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le déterminant d'une matrice 2x2 [[a, b], [c, d]] est ad - bc. Ici, (2×4) - (1×3) = 8 - 3 = 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\": \"5\", \"b\": \"10\", \"c\": \"8\", \"d\": \"1\"}}","_debug_options_count":4},{"id":25921,"question":"Vrai ou Faux ? La fonction exponentielle est toujours croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle f(x) = eˣ a une dérivée f'(x) = eˣ \u003E 0 pour tout x ∈ ℝ, donc elle est strictement croissante.","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":25922,"question":"Quelle est la solution de l'équation différentielle y' + 2y = 0 ?","option_a":"y = Ce⁻²ˣ","option_b":"y = Cx²","option_c":"y = C\/x","option_d":"y = Ce²ˣ","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une équation différentielle linéaire du premier ordre. La solution générale est y = Ce⁻²ˣ, où C est une constante réelle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"y = Ce⁻²ˣ\", \"b\": \"y = Cx²\", \"c\": \"y = C\/x\", \"d\": \"y = Ce²ˣ","_debug_options_count":4},{"id":25923,"question":"Dans une expérience aléatoire, la probabilité d'un événement A est P(A) = 0,3. Quelle est la probabilité de l'événement contraire A̅ ?","option_a":"0,7","option_b":"0,3","option_c":"0,5","option_d":"0,2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La probabilité de l'événement contraire est P(A̅) = 1 - P(A) = 1 - 0,3 = 0,7.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"0,7\", \"b\": \"0,3\", \"c\": \"0,5\", \"d\": \"0,2\"}}","_debug_options_count":4},{"id":25924,"question":"Vrai ou Faux ? Le produit de deux matrices inversibles est toujours inversible.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si A et B sont inversibles, alors (AB)⁻¹ = B⁻¹A⁻¹, donc AB est inversible.","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":25925,"question":"Quelle est la limite de la suite uₙ = (n² + 3n) \/ (2n² - 5) quand n tend vers +∞ ?","option_a":"1\/2","option_b":"3\/2","option_c":"0","option_d":"+∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"En divisant numérateur et dénominateur par n², on obtient lim uₙ = lim (1 + 3\/n) \/ (2 - 5\/n²) = 1\/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\": \"1\/2\", \"b\": \"3\/2\", \"c\": \"0\", \"d\": \"+∞\"}}","_debug_options_count":4},{"id":25926,"question":"Soit un vecteur u(2, -3, 5) dans l'espace. Quelle est sa norme ?","option_a":"√38","option_b":"√14","option_c":"√43","option_d":"√30","option_e":"","option_f":"","bonne_reponse":"a","explication":"La norme d'un vecteur (x, y, z) est √(x² + y² + z²) = √(4 + 9 + 25) = √38.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"√38\", \"b\": \"√14\", \"c\": \"√43\", \"d\": \"√30\"}}","_debug_options_count":4},{"id":25927,"question":"Vrai ou Faux ? Une fonction paire vérifie f(-x) = f(x) pour tout x de son ensemble de définition.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, une fonction paire est symétrique par rapport à l'axe des ordonnées, donc f(-x) = f(x).","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":25928,"question":"Quelle est la valeur de l'intégrale ∫₀¹ (3x² + 2x) dx ?","option_a":"2","option_b":"3","option_c":"1","option_d":"4","option_e":"","option_f":"","bonne_reponse":"b","explication":"∫ (3x² + 2x) dx = x³ + x² + C. En évaluant 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\": \"b\", \"options\": {\"a\": \"2\", \"b\": \"3\", \"c\": \"1\", \"d\": \"4\"}}","_debug_options_count":4}]