Quiz — cnc-mp-2019-maths-1-epreuve.pdf

Question 1 sur 10 20:00

[{"id":6919,"question":"Quelle est la dérivée de la fonction f(x) = x³ - 3x² + 2x - 1 ?","option_a":"A. 3x² - 6x + 2","option_b":"B. 3x² - 6x + 1","option_c":"C. 3x² - 3x + 2","option_d":"D. 3x² - 6x - 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'un polynôme s'obtient en appliquant la règle de dérivation terme à terme : f'(x) = 3x² - 6x + 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. 3x² - 6x + 2\", \"b\": \"B. 3x² - 6x + 1\", \"c\": \"C. 3x² - 3x + ","_debug_options_count":4},{"id":6920,"question":"La fonction f(x) = x² est 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) = 2 \u003E 0, donc la fonction est 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":6921,"question":"Quelle est la solution de l'équation différentielle y' + 2y = 0 ?","option_a":"A. y = Ce^(2x)","option_b":"B. y = Ce^(-2x)","option_c":"C. y = Ce^(x\/2)","option_d":"D. y = Ce^(-x\/2)","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'équation différentielle linéaire du premier ordre y' + 2y = 0 a pour solution générale 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\": \"b\", \"options\": {\"a\": \"A. y = Ce^(2x)\", \"b\": \"B. y = Ce^(-2x)\", \"c\": \"C. y = Ce^(x\/2)\", ","_debug_options_count":4},{"id":6922,"question":"Soit la suite définie par u₀ = 1 et uₙ₊₁ = 2uₙ + 1. Quelle est la valeur de u₃ ?","option_a":"A. 7","option_b":"B. 15","option_c":"C. 31","option_d":"D. 63","option_e":"","option_f":"","bonne_reponse":"c","explication":"Calculons les termes successifs : u₁ = 2*1 + 1 = 3, u₂ = 2*3 + 1 = 7, u₃ = 2*7 + 1 = 15. La réponse correcte est donc 15.","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. 7\", \"b\": \"B. 15\", \"c\": \"C. 31\", \"d\": \"D. 63\"}}","_debug_options_count":4},{"id":6923,"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 logarithme népérien ln(x) est définie uniquement pour x \u003E 0. Elle n'est donc pas définie pour x ≤ 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":6924,"question":"Quelle est l'intégrale de f(x) = 3x² - 2x + 1 entre 0 et 1 ?","option_a":"A. 1","option_b":"B. 2","option_c":"C. 3","option_d":"D. 4","option_e":"","option_f":"","bonne_reponse":"b","explication":"Calculons l'intégrale : ∫(3x² - 2x + 1)dx = [x³ - x² + x] de 0 à 1 = (1 - 1 + 1) - (0 - 0 + 0) = 1. La réponse correcte est donc 1.","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},{"id":6925,"question":"Soit une variable aléatoire X suivant une loi normale N(0,1). Quelle est la probabilité P(X ≤ 1) ?","option_a":"A. 0.6826","option_b":"B. 0.8413","option_c":"C. 0.9772","option_d":"D. 0.9987","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour une loi normale centrée réduite N(0,1), P(X ≤ 1) ≈ 0.8413 (valeur lue dans les tables de la loi normale).","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. 0.6826\", \"b\": \"B. 0.8413\", \"c\": \"C. 0.9772\", \"d\": \"D. 0.9987\"}","_debug_options_count":4},{"id":6926,"question":"La suite (uₙ) définie par uₙ = (n+1)\/n est décroissante.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Étudions le rapport uₙ₊₁\/uₙ = ((n+2)\/(n+1)) \/ ((n+1)\/n) = n(n+2)\/(n+1)². Pour n ≥ 1, ce rapport est supérieur à 1, donc la suite est croissante.","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":6927,"question":"Quelle est la solution de l'inéquation x² - 5x + 6 \u003C 0 ?","option_a":"A. x ∈ ]2; 3[","option_b":"B. x ∈ ]-∞; 2[ ∪ ]3; +∞[","option_c":"C. x ∈ ]-∞; +∞[","option_d":"D. x ∈ ∅","option_e":"","option_f":"","bonne_reponse":"a","explication":"Résolvons l'inéquation : x² - 5x + 6 = (x-2)(x-3) \u003C 0. Le produit est négatif entre les racines, donc 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\": \"A. x ∈ ]2; 3[\", \"b\": \"B. x ∈ ]-∞; 2[ ∪ ]3; +∞[\", \"c\": \"","_debug_options_count":4},{"id":6928,"question":"Soit f une fonction dérivable sur ℝ. Si f'(x) = 0 pour tout x, alors f est constante.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"D'après le théorème des accroissements finis, si la dérivée d'une fonction est nulle sur un intervalle, alors cette fonction est constante 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}]