Quiz — Lycée Pilote Sfax - DC 3 2010.pdf

Question 1 sur 10 20:00

[{"id":23324,"question":"Quelle est la dérivée de la fonction f(x) = x² + 3x - 5 ?","option_a":"f'(x) = 2x + 3","option_b":"f'(x) = x² + 3","option_c":"f'(x) = 2x - 5","option_d":"f'(x) = 2x + 3x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme se calcule terme par terme : la dérivée de x² est 2x, celle de 3x est 3, et celle de -5 est 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\": \"f'(x) = 2x + 3\", \"b\": \"f'(x) = x² + 3\", \"c\": \"f'(x) = 2x - 5\", \"","_debug_options_count":4},{"id":23325,"question":"Une suite géométrique de raison q = 2 et de premier terme u₀ = 3 a pour terme général uₙ = 3 × 2ⁿ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, le terme général d'une suite géométrique est uₙ = u₀ × qⁿ. Ici, u₀ = 3 et q = 2, donc uₙ = 3 × 2ⁿ est correct.","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":23326,"question":"Soit une variable aléatoire X suivant une loi normale N(0,1). Quelle est la probabilité 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 N(0,1), 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\": \"≈ 0.34","_debug_options_count":4},{"id":23327,"question":"L'équation différentielle y' + 2y = 0 a pour solution générale y = Ce^(-2x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation y' + 2y = 0 est une équation différentielle linéaire du premier ordre. Sa solution générale est bien 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":23328,"question":"Quelle est la limite de la suite uₙ = (n² + 1)\/(2n² - 3) quand n tend vers l'infini ?","option_a":"0","option_b":"1\/2","option_c":"1","option_d":"L'infini","option_e":"","option_f":"","bonne_reponse":"c","explication":"Pour trouver la limite d'une suite rationnelle, on divise le numérateur et le dénominateur par le terme dominant (ici n²) : uₙ = (1 + 1\/n²)\/(2 - 3\/n²) → 1\/2 quand n → ∞.","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\": \"L'infini\"}}","_debug_options_count":4},{"id":23329,"question":"Dans un repère orthonormé, l'équation d'une sphère de centre O(0,0,0) et de rayon R est x² + y² + z² = R².","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, l'équation d'une sphère de centre O(0,0,0) et de rayon R est bien x² + y² + z² = R².","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":23330,"question":"Soit f une fonction définie par f(x) = ln(x² + 1). Quelle est sa dérivée f'(x) ?","option_a":"f'(x) = 2x\/(x² + 1)","option_b":"f'(x) = 1\/(x² + 1)","option_c":"f'(x) = 2x","option_d":"f'(x) = x\/(x² + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la formule de dérivation d'une fonction composée (ln(u))' = u'\/u, avec u = x² + 1, on obtient f'(x) = (2x)\/(x² + 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\": \"f'(x) = 2x\/(x² + 1)\", \"b\": \"f'(x) = 1\/(x² + 1)\", \"c\": \"f'(x) = ","_debug_options_count":4},{"id":23331,"question":"La probabilité d'obtenir un double 6 en lançant deux dés équilibrés est de 1\/36.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Il y a 6 × 6 = 36 issues possibles en lançant deux dés. Un seul cas favorable (6,6), donc la probabilité est 1\/36.","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":23332,"question":"Quelle est la solution générale de l'équation différentielle y'' + y = 0 ?","option_a":"y = Ce^x","option_b":"y = C₁cos(x) + C₂sin(x)","option_c":"y = Cx + C","option_d":"y = Ce^(-x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'équation y'' + y = 0 est une équation différentielle linéaire du second ordre à coefficients constants. Sa solution générale est y = C₁cos(x) + C₂sin(x), où C₁ et C₂ sont des constantes 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\": \"y = Ce^x\", \"b\": \"y = C₁cos(x) + C₂sin(x)\", \"c\": \"y = Cx + C\",","_debug_options_count":4},{"id":23333,"question":"Soit une fonction f dérivable sur ℝ telle que 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 sur ℝ et que f'(x) \u003E 0 pour tout x ∈ ℝ, alors f est strictement croissante sur ℝ (théorème de la dérivée).","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}]