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Problems require finding exact values of expressions involving sin and cos, simplifying nested logarithms, applying given relations to select the correct multiple-choice option, and evaluating a composite expression through identities. Later questions determine a minimum value of an expression and compute the area enclosed by two curves using intersection points and integral area setup.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/math-contract-answers-solution-to-the-sample-test-for-socie/144186/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/math-contract-answers-solution-to-the-sample-test-for-socie/144186.png","ImageObject",300,407,{"name":92,"@type":93},"Theodora","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-21","2026-08-25",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",16,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What is required in Question 1?","Question",{"text":113,"@type":114},"Find the value of sin and cos related expression, then choose the correct option.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"What does Question 2 ask you to compute?",{"text":118,"@type":114},"Compute a product/combination of logarithms and select the correct answer choice.",{"name":120,"@type":111,"acceptedAnswer":121},"How are Questions 5 and 6 solved conceptually?",{"text":122,"@type":114},"Question 5 determines the minimum value of a given expression, while Question 6 computes the enclosed area between two curves using their intersection points and an appropriate integral setup.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},144186,1787698637,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":14,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":130,"read_time":24},687197207919,"https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552","Solution to the Sample Test for SOCIE  \n􀄭  \n1. When sin 􀆝 cos􀆝 􀅀 􀅕 􀄮  \n① 􀄿 􀅦 ② 􀄿 􀅦􀄯 􀄯  \n􀆥  \nfor 􀄶 􀅎 􀆝 􀅎  , find the value  \n􀅕 􀅕  \n③ 􀄿 􀅦 ④ 􀄿 􀅦  \n􀄯 􀄯  \nof sin 􀆝 􀄿 cos 􀆝 .  \n􀄭  \n⑤ 􀄿 􀅦􀄯  \n(Sol) Since 􀄽 sin 􀆝 􀄿 cos 􀆝􀄾􀄮 􀅀 􀄭 􀄿 􀄮 sin 􀆝 cos􀆝 􀅀  , it follows that sin 􀆝 􀄿 cos 􀆝 􀅀 ±  .  \n􀆥 􀅕 Since 􀄶 􀅎 􀆝 􀅎  , we have sin 􀆝 􀄿 cos􀆝 􀅎 􀄶 . Hence , sin 􀆝 􀄿 cos 􀆝 􀅀 􀄿  .  \nThe answer is ③.  \n􀄮 􀄯 􀄰 􀄮􀄱  \n2. Compute log􀄭􀄶  􀅁 log􀄭􀄶  􀅁 log􀄭􀄶  􀅁 ⋯ 􀅁 log􀄭􀄶  .  \n① log􀄭􀄶􀄮􀄱 ② log􀄭􀄶 􀄯􀄱 ③ log􀄭􀄶􀄰􀄱 ④ log􀄭􀄶􀄱􀄱 ⑤ log􀄭􀄶 􀄲􀄱  \n(Sol) log 􀄭􀄶  􀅁 log 􀄭􀄶  􀅁 log 􀄭􀄶  􀅁 ⋯ 􀅁 log 􀄭􀄶  􀅀 log 􀄭􀄶 􀄽    ⋯  􀄾 􀅀 log 􀄭􀄶 􀄮􀄱 . The answer is ①.  \n3. For 􀃹 􀅀 􀄽􀄮􀄭 􀄭􀄭 􀄾 and 􀃺 􀅀 􀄽􀄭􀄶 􀄮􀄭􀄾 , we write 􀃹􀃺􀃹 􀄿 􀄭 􀅀 􀄽 􀇞􀇠 􀇟􀇡􀄾 . Find 􀇞􀅁 􀇟 􀅁 􀇠 􀅁 􀇡 .  \n① 􀄮 ② 􀄰 ③ 􀄲 ④ 􀄴 ⑤ 􀄭􀄶  \n(Sol) For 􀃹 􀅀 􀄽 􀄮􀄭 􀄭􀄭 􀄾 , 􀃹 􀄿 􀄭 􀅀 􀄽 􀄿􀄭 􀄭 􀄿􀄮 􀄭 􀄾 . Thus 􀃹􀃺􀃹 􀄿 􀄭 􀅀 􀄽 􀄮􀄭 􀄭􀄭 􀄾 􀄽􀄭􀄶 􀄮􀄭 􀄾􀄽 􀄿􀄭 􀄭 􀄿􀄮 􀄭 􀄾 􀅀 􀄽 􀄿 􀄯􀄮 􀄴􀄱 􀄾 , and 􀇞 􀅁 􀇟 􀅁 􀇠 􀅁 􀇡 􀅀 􀄴 .  \nThe answer is ④.  \n4. Evaluate l􀆝􀄶 􀄽 o􀄯􀆝s􀄾􀄮􀄾􀆝􀄮 , where 􀇱􀇤􀃹 􀅀 ions􀃹􀃹 .  \n􀄭 􀄮 􀄭 􀄰 􀄱  \n① 􀅦 ② 􀅦 ③ 􀅦 ④ 􀅦 ⑤ 􀅦  \n􀄵 􀄵 􀄯 􀄵 􀄵  \n(Sol) We note that  \n􀄭 􀄿 cos􀄮􀆝 cos􀄮 􀄯􀆝 􀄽􀄭 􀄿 cos􀄮􀆝􀄾􀄽􀄭 􀅁 cos􀄮􀆝􀄾 cos􀄮 􀄯􀆝 􀄭 􀄿 cos􀄮 􀄮􀆝 cos􀄮 􀄯􀆝􀅦 􀅀 􀄽􀄭 􀄿 cos􀄮􀆝 􀄾 ⋅ 􀅦 􀅀 􀅦 􀅦 􀅀 􀅦 􀅦􀄽􀇱􀇤􀄽􀄯􀆝􀄾􀄾 􀄮 sin 􀄮 􀄯􀆝 􀄽􀄭 􀅁 cos􀄮􀆝 􀄾 sin 􀄮 􀄯􀆝 􀄽􀄭 􀅁 cos􀄮􀆝 􀄾 sin 􀄮 􀄯􀆝  \nsin 􀄮 􀄮􀆝 cos􀄮 􀄯􀆝 􀄽 sin 􀄮􀆝􀄾 􀄮 􀄰 􀄽􀄯􀆝􀄾 􀄮 cos􀄮 􀄯􀆝  \n􀅀 cos􀄮􀆝􀄾 sin 􀄮 􀄯􀆝 􀅀 􀄾 􀄮 ⋅  ⋅ sin 􀄯􀆝􀄾􀄮 ⋅  􀅋  \n􀄭 􀄿 cos􀄮􀆝 􀄽 sin 􀄮􀆝􀄾 􀄮 􀄰 􀄽􀄯􀆝􀄾 􀄮 cos􀄮 􀄯􀆝 􀄰 􀄭 􀄮  \nwhich yields l􀆝􀄶 􀄮 􀄯􀆝 􀅀 l􀆝􀄶 􀄾 􀄮  sin 􀄯􀆝􀄾􀄮  􀅀 􀄭 ․  ․ 􀄭 ․  􀅀  . The answer is ②.  \n5. Find the minimum value of 􀇣􀄽􀇵􀄾 􀅀 􀄯􀇵􀄰 􀄿 􀄮􀇵􀄯 􀅁 􀄲􀇵􀄮 􀄿 􀄲􀇵 􀅁 􀄮 .  \n①  ②  ③  ④  ⑤ 􀄭􀄲 􀄭􀄲 􀄭􀄲 􀄭􀄲 􀄭􀄲  \n(Sol) Since 􀇣 ′ 􀄽􀇵􀄾 􀅀 􀄭􀄮􀇵􀄯 􀄿 􀄲􀇵􀄮 􀅁 􀄭􀄮􀇵 􀄿 􀄲 􀅀 􀄲􀄽􀄮􀇵 􀄿 􀄭􀄾􀄽􀇵􀄮 􀅁 􀄭􀄾 , the minimum value is  \n􀇣 􀄽  􀄾 􀅀  .  \nThe answer is ④.  \n6. Find the area of the region enclosed by 􀇶 􀅀 􀇵 􀄰 􀄿 􀄯􀇵􀄯 􀅁 􀇵 􀄮 􀄿 􀄭 and 􀇶 􀅀 􀇵 􀄰 􀄿 􀄯􀇵􀄯 􀅁 􀇵 􀅁 􀄭 .  \n􀄭 􀄯 􀄱 􀄳 􀄵  \n① 􀅦 ② 􀅦 ③ 􀅦 ④ 􀅦 ⑤ 􀅦  \n􀄮 􀄮 􀄮 􀄮 􀄮  \n(Sol) Since 􀄽􀇵􀄰 􀄿 􀄯􀇵􀄯 􀅁 􀇵􀅁 􀄭􀄾 􀄿 􀄽􀇵􀄰 􀄿 􀄯􀇵􀄯 􀅁 􀇵 􀄮 􀄿 􀄭􀄾 􀅀 􀄿 􀄽􀇵 􀄮 􀄿 􀇵 􀄿 􀄮􀄾 􀅀 􀄿 􀄽􀇵 􀅁 􀄭􀄾􀄽􀇵 􀄿 􀄮􀄾 􀅋 two graphs meet at 􀇵 􀅀 􀄿 􀄭 and 􀇵 􀅀 􀄮 . Since 􀇵 􀄰 􀄿 􀄯􀇵􀄯 􀅁 􀇵 􀅁 􀄭 ≥ 􀇵 􀄰 􀄿 􀄯􀇵􀄯 􀅁 􀇵 􀄮 􀄿 􀄭 on the interval 􀅂 􀄿 􀄭 􀅋 􀄮 􀅃 , the area is given by  \n􀅔 􀄮 􀄽 􀄽􀇵􀄰 􀄿 􀄯􀇵 􀄯 􀅁 􀇵 􀅁 􀄭􀄾 􀄿 􀄽􀇵􀄰 􀄿 􀄯􀇵 􀄯 􀅁 􀇵 􀄮 􀄿 􀄭􀄾􀄾􀇡􀇵 􀅀 􀅔 􀄮 􀄿 􀄽􀇵 􀄮 􀄿 􀇵 􀄿 􀄮􀄾 􀇡􀇵  \n􀄿 􀄭 􀄿 􀄭  \n􀅀  􀄿  􀇵 􀄯 􀅁  􀇵 􀄮 􀅁 􀄮􀇵  The answer is ⑤.  \n􀄮  \n􀄿 􀄭  \n􀅀 􀄽 􀄿  􀅁 􀄮 􀅁 􀄰 􀄾􀄿 􀄽  􀅁  􀄿 􀄮 􀄾 􀅀  .","cbCain5FvaBeBBBV","https://ap.wps.com/l/cbCain5FvaBeBBBV","pdf",102332,"English","# Question 1\n# Question 2\n# Question 3\n# Question 4\n# Question 5\n# Question 6","[{\"question\":\"What is required in Question 1?\",\"answer\":\"Find the value of sin and cos related expression, then choose the correct option.\"},{\"question\":\"What does Question 2 ask you to compute?\",\"answer\":\"Compute a product/combination of logarithms and select the correct answer choice.\"},{\"question\":\"How are Questions 5 and 6 solved conceptually?\",\"answer\":\"Question 5 determines the minimum value of a given expression, while Question 6 computes the enclosed area between two curves using their intersection points and an appropriate integral setup.\"}]","math contract answers - Solution to the Sample Test for SOCIE | PDF"]