Quiz — modakirafinal.pdf

Question 1 sur 10 20:00

[{"id":58596,"question":"Quelle est la dérivée de la fonction f(x) = e^(3x) ?","option_a":"3e^(3x)","option_b":"e^(3x)","option_c":"3xe^(3x-1)","option_d":"e^(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction de la forme e^(u(x)) est u'(x)e^(u(x)). Ici, u(x) = 3x, donc u'(x) = 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\": \"3e^(3x)\", \"b\": \"e^(3x)\", \"c\": \"3xe^(3x-1)\", \"d\": \"e^(x)\"}}","_debug_options_count":4},{"id":58597,"question":"Une suite géométrique de raison q = -2 et de premier terme u0 = 3 est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une suite géométrique converge si et seulement si |q| \u003C 1. Ici, |q| = 2 \u003E 1, donc la suite diverge.","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":58598,"question":"Soit X une variable aléatoire suivant une loi normale N(0,1). Que vaut P(X ≤ 1) ?","option_a":"0.6826","option_b":"0.8413","option_c":"0.9772","option_d":"0.3413","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour une loi normale centrée réduite, P(X ≤ 1) ≈ 0.8413 (valeur lue dans la table 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\": \"0.6826\", \"b\": \"0.8413\", \"c\": \"0.9772\", \"d\": \"0.3413\"}}","_debug_options_count":4},{"id":58599,"question":"L'équation de la tangente à la courbe de f(x) = ln(x) au point d'abscisse x = 1 est :","option_a":"y = x - 1","option_b":"y = x + 1","option_c":"y = 1","option_d":"y = 2x - 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La tangente a pour équation y = f'(a)(x - a) + f(a). Ici, f'(1) = 1 et f(1) = 0, donc y = 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\": \"y = x - 1\", \"b\": \"y = x + 1\", \"c\": \"y = 1\", \"d\": \"y = 2x - 1\"}}","_debug_options_count":4},{"id":58600,"question":"La probabilité P(A ∩ B) est toujours inférieure ou égale à P(A) × P(B).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est l'inégalité de Bonferroni : P(A ∩ B) ≥ P(A) + P(B) - 1, mais P(A ∩ B) ≤ min(P(A), P(B)) ≤ P(A) × P(B) si A et B sont indépendants.","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":58601,"question":"Quelle est la limite de la suite u_n = (n² + 1)\/(2n² - 3) quand n tend vers l'infini ?","option_a":"0","option_b":"1\/2","option_c":"1","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"En divisant numérateur et dénominateur par n², on obtient lim (1 + 1\/n²)\/(2 - 3\/n²) = 1\/2. Cependant, la limite correcte est 1\/2, donc la bonne réponse est 1\/2 (erreur dans les options).","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\": \"1\/2\", \"c\": \"1\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":58602,"question":"Soit f une fonction dérivable sur R. Si f'(x) \u003E 0 pour tout x ∈ R, alors f est strictement croissante sur R.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si f est dérivable et f'(x) \u003E 0 pour tout x, alors f est strictement croissante (théorème fondamental de l'analyse).","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":58603,"question":"Le produit scalaire de deux vecteurs orthogonaux est égal à :","option_a":"0","option_b":"1","option_c":"||u|| × ||v||","option_d":"-1","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, deux vecteurs sont orthogonaux si et seulement si leur produit scalaire est nul.","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\", \"b\": \"1\", \"c\": \"||u|| × ||v||\", \"d\": \"-1\"}}","_debug_options_count":4},{"id":58604,"question":"Soit X une variable aléatoire suivant une loi binomiale B(n,p). Que vaut E(X) ?","option_a":"np","option_b":"n(1-p)","option_c":"p(1-p)","option_d":"n²p","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'espérance d'une loi binomiale B(n,p) est E(X) = n × p.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"np\", \"b\": \"n(1-p)\", \"c\": \"p(1-p)\", \"d\": \"n²p\"}}","_debug_options_count":4},{"id":58605,"question":"L'équation différentielle y' + 2y = 0 admet pour solution générale :","option_a":"y = Ce^(2x)","option_b":"y = Ce^(-2x)","option_c":"y = C + e^(-2x)","option_d":"y = 2Ce^x","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'équation différentielle linéaire du premier ordre y' + 2y = 0 a pour solution y = Ce^(-2x), où C est une constante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"y = Ce^(2x)\", \"b\": \"y = Ce^(-2x)\", \"c\": \"y = C + e^(-2x)\", \"d\": \"","_debug_options_count":4}]