Quiz — Devoir a la maison - Une construction de pi et des fonctions circulaires.pdf

Question 1 sur 10 20:00

[{"id":8325,"question":"Quelle est la valeur approximative de π obtenue en utilisant un hexagone régulier inscrit dans un cercle de rayon 1 ?","option_a":"2,5","option_b":"3","option_c":"3,14","option_d":"3,46","option_e":"","option_f":"","bonne_reponse":"d","explication":"Pour un hexagone inscrit, le périmètre est 6 * côté = 6 * 1 = 6. L'approximation de π est alors périmètre \/ (2 * rayon) = 6 \/ 2 = 3. Mais cette valeur est une sous-estimation. La valeur correcte est 3,46 (périmètre de l'hexagone circonscrit).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"2,5\", \"b\": \"3\", \"c\": \"3,14\", \"d\": \"3,46\"}}","_debug_options_count":4},{"id":8326,"question":"La fonction f(x) = sin(2x) a une période égale à :","option_a":"π\/2","option_b":"π","option_c":"2π","option_d":"4π","option_e":"","option_f":"","bonne_reponse":"a","explication":"La période de sin(kx) est 2π\/|k|. Ici, k=2, donc la période est 2π\/2 = π.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"π\/2\", \"b\": \"π\", \"c\": \"2π\", \"d\": \"4π\"}}","_debug_options_count":4},{"id":8327,"question":"Vrai ou Faux ? La fonction cosinus est une fonction paire.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une fonction paire vérifie f(-x) = f(x). Pour cosinus, cos(-x) = cos(x), donc elle est bien paire.","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":8328,"question":"Quelle est la dérivée de la fonction f(x) = cos(3x) ?","option_a":"-sin(3x)","option_b":"-3sin(3x)","option_c":"3sin(3x)","option_d":"sin(3x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de cos(u) est -u' * sin(u). Ici, u=3x, donc u'=3. Ainsi, f'(x) = -3 * sin(3x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"-sin(3x)\", \"b\": \"-3sin(3x)\", \"c\": \"3sin(3x)\", \"d\": \"sin(3x)\"}}","_debug_options_count":4},{"id":8329,"question":"Vrai ou Faux ? L'équation sin(x) = 0,5 a une solution unique dans l'intervalle [0, 2π].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'équation sin(x) = 0,5 a deux solutions dans [0, 2π] : x = π\/6 et x = 5π\/6.","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":8330,"question":"Quelle est la valeur de sin(π\/2) ?","option_a":"0","option_b":"1\/2","option_c":"1","option_d":"-1","option_e":"","option_f":"","bonne_reponse":"c","explication":"sin(π\/2) = 1, car π\/2 radians correspond à 90 degrés, où le sinus atteint sa valeur maximale.","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\": \"-1\"}}","_debug_options_count":4},{"id":8331,"question":"Pour résoudre l'équation cos(2x) = 0,5, quelle méthode est adaptée ?","option_a":"Utiliser la formule de duplication pour exprimer cos(2x) en fonction de sin(x) ou cos(x)","option_b":"Diviser directement par 2 pour obtenir x = arccos(0,5)\/2","option_c":"Utiliser la formule de Moivre pour développer cos(2x)","option_d":"Chercher les solutions dans l'intervalle [0, π\/2] uniquement","option_e":"","option_f":"","bonne_reponse":"a","explication":"La formule de duplication cos(2x) = 2cos²(x) - 1 permet de transformer l'équation en une équation quadratique en cos(x), plus facile à résoudre.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Utiliser la formule de duplication pour exprimer cos(2x) en fonct","_debug_options_count":4},{"id":8332,"question":"Vrai ou Faux ? La fonction tangente est définie pour toutes les valeurs de x.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction tangente n'est pas définie pour x = π\/2 + kπ (k entier), car cos(x) = 0 à ces points, et tan(x) = sin(x)\/cos(x).","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":8333,"question":"Quelle est la valeur de l'intégrale ∫[0, π] sin(x) dx ?","option_a":"0","option_b":"1","option_c":"2","option_d":"π","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'intégrale de sin(x) est -cos(x). Ainsi, ∫[0, π] sin(x) dx = [-cos(x)]₀^π = -cos(π) - (-cos(0)) = -(-1) - (-1) = 1 + 1 = 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\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"π\"}}","_debug_options_count":4},{"id":8334,"question":"Quelle propriété permet de simplifier sin(x + π) ?","option_a":"La périodicité de la fonction sinus","option_b":"La formule d'addition sin(a + b) = sin(a)cos(b) + cos(a)sin(b)","option_c":"La parité de la fonction sinus","option_d":"La formule de duplication","option_e":"","option_f":"","bonne_reponse":"b","explication":"La formule d'addition sin(a + b) = sin(a)cos(b) + cos(a)sin(b) permet de développer sin(x + π) = sin(x)cos(π) + cos(x)sin(π) = sin(x)*(-1) + cos(x)*0 = -sin(x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"La périodicité de la fonction sinus\", \"b\": \"La formule d'additi","_debug_options_count":4}]