Quiz — mp-2021-2022-dm07.pdf

Question 1 sur 10 20:00

[{"id":54557,"question":"Quelle est la dérivée de la fonction f(x) = x² + 3x - 5 ?","option_a":"2x + 3","option_b":"2x² + 3","option_c":"x² + 3","option_d":"2x + 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme se calcule terme par terme : (x²)' = 2x, (3x)' = 3 et (-5)' = 0. Donc f'(x) = 2x + 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\": \"2x + 3\", \"b\": \"2x² + 3\", \"c\": \"x² + 3\", \"d\": \"2x + 5\"}}","_debug_options_count":4},{"id":54558,"question":"L'équation x² + 4x + 5 = 0 admet-elle des solutions réelles ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le discriminant Δ = b² - 4ac = 16 - 20 = -4 \u003C 0. Comme Δ \u003C 0, l'équation n'a pas de solutions réelles.","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":54559,"question":"Quelle est la limite de la fonction f(x) = (3x² + 2x - 1)\/(x² + 1) quand x tend vers +∞ ?","option_a":"0","option_b":"3","option_c":"1","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"Pour les limites en +∞ de fonctions rationnelles, on compare les termes de plus haut degré : (3x²)\/(x²) = 3. Donc lim(x→+∞) f(x) = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"0\", \"b\": \"3\", \"c\": \"1\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":54560,"question":"Le nombre complexe z = 3 - 4i a pour module :","option_a":"5","option_b":"7","option_c":"1","option_d":"√7","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module d'un nombre complexe z = a + bi est donné par |z| = √(a² + b²). Ici, |z| = √(3² + (-4)²) = √(9 + 16) = √25 = 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\": \"7\", \"c\": \"1\", \"d\": \"√7\"}}","_debug_options_count":4},{"id":54561,"question":"La fonction f(x) = 1\/x est continue sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction f(x) = 1\/x n'est pas définie en x = 0, donc elle n'est pas continue sur tout ℝ. Elle est continue sur ℝ* (ℝ privé de 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":54562,"question":"Quelle est la primitive de la fonction f(x) = 4x³ - 2x + 1 ?","option_a":"x⁴ - x² + x + C","option_b":"x⁴ - x² + C","option_c":"x⁴ - 2x² + x + C","option_d":"4x⁴ - 2x² + x + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une primitive de f(x) = 4x³ - 2x + 1 est F(x) = x⁴ - x² + x + C, car (x⁴)' = 4x³, (-x²)' = -2x et (x)' = 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\": \"x⁴ - x² + x + C\", \"b\": \"x⁴ - x² + C\", \"c\": \"x⁴ - 2x² + x","_debug_options_count":4},{"id":54563,"question":"L'intégrale ∫₀¹ (2x + 1) dx vaut :","option_a":"1","option_b":"2","option_c":"3","option_d":"4","option_e":"","option_f":"","bonne_reponse":"c","explication":"Calculons l'intégrale : ∫(2x + 1) dx = x² + x + C. Évaluée entre 0 et 1 : (1² + 1) - (0² + 0) = 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\": \"1\", \"b\": \"2\", \"c\": \"3\", \"d\": \"4\"}}","_debug_options_count":4},{"id":54564,"question":"Le produit scalaire de deux vecteurs orthogonaux est toujours nul.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, deux vecteurs sont orthogonaux si et seulement si leur produit scalaire est égal à 0.","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":54565,"question":"Quelle est la solution de l'inéquation x² - 5x + 6 ≤ 0 ?","option_a":"x ∈ [2; 3]","option_b":"x ∈ ]-∞; 2] ∪ [3; +∞[","option_c":"x ∈ ℝ","option_d":"x ∈ ∅","option_e":"","option_f":"","bonne_reponse":"a","explication":"Résolvons l'inéquation : x² - 5x + 6 = (x-2)(x-3) ≤ 0. Le tableau de signes montre que la solution est x ∈ [2; 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\": \"x ∈ [2; 3]\", \"b\": \"x ∈ ]-∞; 2] ∪ [3; +∞[\", \"c\": \"x ∈ ","_debug_options_count":4},{"id":54566,"question":"La fonction exponentielle est-elle toujours positive 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 est strictement positive pour tout x ∈ ℝ, car eˣ \u003E 0 quel que soit 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}]