Quiz — EXO Fonctions circulaires et hyperboliques.pdf

Question 1 sur 10 20:00

[{"id":46148,"question":"Quelle est la dérivée de la fonction f(x) = sin(2x) ?","option_a":"2cos(2x)","option_b":"cos(2x)","option_c":"2sin(2x)","option_d":"-2cos(2x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de sin(u) est u'cos(u). Ici, u = 2x, donc u' = 2. Ainsi, f'(x) = 2cos(2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"2cos(2x)\", \"b\": \"cos(2x)\", \"c\": \"2sin(2x)\", \"d\": \"-2cos(2x)\"}}","_debug_options_count":4},{"id":46149,"question":"La fonction cosh(x) est-elle toujours positive ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, cosh(x) = (e^x + e^{-x})\/2. Comme e^x \u003E 0 pour tout x réel, cosh(x) \u003E 0 pour tout 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},{"id":46150,"question":"Quelle identité trigonométrique est correcte ?","option_a":"sin²(x) + cos²(x) = 1","option_b":"sin²(x) - cos²(x) = 1","option_c":"1 + tan²(x) = -1","option_d":"cos²(x) - sin²(x) = 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'identité fondamentale en trigonométrie est sin²(x) + cos²(x) = 1, valable pour tout x réel.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"sin²(x) + cos²(x) = 1\", \"b\": \"sin²(x) - cos²(x) = 1\", \"c\": \"1","_debug_options_count":4},{"id":46151,"question":"Quelle est la valeur de sinh(0) ?","option_a":"0","option_b":"1","option_c":"-1","option_d":"2","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, sinh(x) = (e^x - e^{-x})\/2. Pour x = 0, sinh(0) = (1 - 1)\/2 = 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\": \"0\", \"b\": \"1\", \"c\": \"-1\", \"d\": \"2\"}}","_debug_options_count":4},{"id":46152,"question":"La fonction tan(x) est-elle 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 tan(x) = sin(x)\/cos(x) n'est pas définie lorsque cos(x) = 0, c'est-à-dire pour x = π\/2 + kπ (k ∈ ℤ).","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":46153,"question":"Quelle est la primitive de f(x) = cos(x) ?","option_a":"sin(x) + C","option_b":"cos(x) + C","option_c":"-sin(x) + C","option_d":"tan(x) + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de cos(x) est sin(x) + C, car la dérivée de sin(x) est cos(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\": \"sin(x) + C\", \"b\": \"cos(x) + C\", \"c\": \"-sin(x) + C\", \"d\": \"tan(x) ","_debug_options_count":4},{"id":46154,"question":"Quelle relation lie cosh²(x) et sinh²(x) ?","option_a":"cosh²(x) - sinh²(x) = 1","option_b":"cosh²(x) + sinh²(x) = 1","option_c":"sinh²(x) - cosh²(x) = 1","option_d":"cosh(x) = sinh(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'identité fondamentale pour les fonctions hyperboliques est cosh²(x) - sinh²(x) = 1, analogue à l'identité trigonométrique sin²(x) + cos²(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\": \"cosh²(x) - sinh²(x) = 1\", \"b\": \"cosh²(x) + sinh²(x) = 1\", \"c\"","_debug_options_count":4},{"id":46155,"question":"La fonction sin(x) est-elle périodique ? Si oui, quelle est sa période ?","option_a":"Vrai, période = π","option_b":"Vrai, période = 2π","option_c":"Faux","option_d":"Vrai, période = π\/2","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction sin(x) est périodique de période 2π, car sin(x + 2π) = sin(x) pour tout x réel.","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, période = π\", \"b\": \"Vrai, période = 2π\", \"c\": \"Faux\", \"","_debug_options_count":4},{"id":46156,"question":"Quelle est la valeur de tan(π\/4) ?","option_a":"0","option_b":"1","option_c":"√2\/2","option_d":"√3\/2","option_e":"","option_f":"","bonne_reponse":"b","explication":"tan(π\/4) = sin(π\/4)\/cos(π\/4) = (√2\/2)\/(√2\/2) = 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\": \"0\", \"b\": \"1\", \"c\": \"√2\/2\", \"d\": \"√3\/2\"}}","_debug_options_count":4},{"id":46157,"question":"La fonction f(x) = tanh(x) est-elle bornée ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction tanh(x) = sinh(x)\/cosh(x) est bornée entre -1 et 1 pour tout x réel, car |sinh(x)| ≤ |cosh(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}]