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The document provides step-by-step reasoning to evaluate trigonometric values, compute logarithmic combinations, simplify complex expressions, determine geometric/number-based results, find the shortest distance between a curve and a point, and compute the volume of a solid of revolution around a specified axis. Each problem concludes with the selected option.",{"@graph":69,"@context":114},[70,84,105],{"@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/maths-scholarship-answers-2022-solution-to-socie-sample-test/144190/",{"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/maths-scholarship-answers-2022-solution-to-socie-sample-test/144190.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":56},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108],{"name":109,"@type":110,"acceptedAnswer":111},"What types of calculus problems are included?","Question",{"text":112,"@type":113},"There are problems finding the shortest distance between a curve and a point, and finding the volume of a solid formed by rotating a region about an axis.","Answer","https://schema.org",{"og:url":83,"og:type":116,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":118,"canonical":83},"index,follow",{"doc_id":120,"site_id":62},144190,1787698657,{"code":4,"msg":5,"data":123},{"doc_id":120,"user_id":124,"nickname":92,"user_avatar":125,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":126,"file_id":127,"file_url":128,"file_type":129,"file_size":130,"view_count":56,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":14,"language":131,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":132,"faqs":133,"seo_title":134,"seo_description":67,"update_tm":121,"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 (Scholarship)  \n􀄭  \n1. When sin 􀆝 cos􀆝 􀅀  , find the value of cos 􀄰􀆝 .  \n①  ②  ③  ④  ⑤   \n􀄵 􀄵 􀄯 􀄵 􀄵  \n(Sol) Since sin 􀄮􀆝 􀅀 􀄮 sin 􀆝 cos􀆝 􀅀  , it follows that cos􀄰􀆝 􀅀 􀄭 􀄿 􀄮 sin 􀄮 􀄮􀆝 􀅀  . The answer is ①.  \n2. Compute 􀇫􀅠 􀄭 log􀄭􀄶 􀄽 􀄭 􀄿  􀄾 .  \n① 􀄿 􀄰 ② 􀄿 􀄮 ③ 􀄶 ④ 􀄮 ⑤ 􀄰  \n(Sol) 􀇫􀅠 􀄭 log􀄭􀄶 􀄽 􀄭 􀄿  􀄾 􀅀 􀇫􀅠 􀄭 log􀄭􀄶 􀄽  􀄾 􀅀 log􀄭􀄶 􀄽  ․  ․ ⋯ ․  􀄾 􀅀 log􀄭􀄶  􀅀 􀄿 􀄮 . The answer is ②.  \n3 . When 􀃹 􀅀 􀄽 􀄭􀄶 􀄮􀄭 􀄾 and 􀄽􀃹 􀄮􀄶􀄮􀄮􀄾 􀄿 􀄭 􀅀 􀄽 􀇞􀇠 􀇟􀇡􀄾 , find 􀇞􀅁 􀇟 􀅁 􀇠 􀅁 􀇡 .  \n① 􀄿 􀄰􀄶􀄰􀄴 ② 􀄿 􀄰􀄶􀄰􀄲 ③ 􀄿 􀄰􀄶􀄰􀄰 ④ 􀄿 􀄰􀄶􀄰􀄮 ⑤ 􀄿 􀄰􀄶􀄰􀄶  \n(Sol) For 􀃹 􀅀 􀄽 􀄭􀄶 􀄮􀄭􀄾 , 􀃹 􀄿 􀄭 􀅀 􀄽 􀄭􀄶 􀄿􀄭 􀄮 􀄾 . Simple computation show that 􀃹 􀄮 􀅀 􀄽􀄭􀄶 􀄰􀄭􀄾 and 􀃹 􀇫 􀅀 􀄽 􀄭􀄶 􀄮􀄭􀇫 􀄾 . In particular , 􀃹 􀄮􀄶􀄮􀄮 􀅀 􀄽 􀄭􀄶 􀄰 􀄶􀄰􀄭􀄰 􀄾 , and hence 􀄽􀃹 􀄮􀄶􀄮􀄮􀄾 􀄿 􀄭 􀅀 􀄽 􀄭􀄶 􀄿 􀄰􀄶􀄭􀄰􀄰􀄾 , which shows 􀇞􀅁 􀇟 􀅁 􀇠 􀅁 􀇡 􀅀 􀄿 􀄰􀄶􀄰􀄮 .  \nThe answer is ④.  \n4. 􀆭  \n① 􀄿 􀇦 ②􀆭 􀄮  \n􀆭 􀄭􀄶􀄮 􀅀 􀄽􀆭 􀄯 􀄾 􀄯􀄰  \nThe answer  \n􀇵 􀄮 􀄿  \n􀄿 􀄭 ③ 􀄶  \n􀄿 􀆭 􀅁 􀄭 􀅀 􀄶  \n􀅀 􀄽􀄿 􀄭􀄾 􀄯􀄰 􀅀 􀄭 is ②.  \n􀄭􀄶􀄭  \n􀇵 􀅁 􀄭 􀅀 􀄶 􀄋 􀅀 􀇫􀄭 􀆭 􀇫④ 􀄭 ⑤ 􀇦  \n􀆭 􀄯 􀅀 􀆭 􀄮 􀄿 􀆭 􀅀 􀄿 􀄭  \n􀆭 􀄿 􀆭 􀄭􀄶􀄮 􀆭 􀄿 􀄭  \n􀄋 􀅀  􀅀  􀅀 􀄿 􀄭  \n5. Find the shortest distance between the curve 􀇶 􀅀 􀅕 􀅋 􀄽 􀇵 ≥ 􀄶 􀄾 and the point 􀄽 􀄭 􀅋 􀄶􀄾 .  \n􀄭 􀅕 􀅕 􀄯  \n①  ②  ③  ④ 􀄭 ⑤   \n(Sol) Since any point on the curve 􀇶 􀅀 􀅕 is expressed by 􀄽 􀇵 􀅋 􀅕 􀄾 , the distance 􀆴 􀄽􀇵􀄾 from 􀄽 􀇵 􀅋 􀅕 􀄾 to 􀄽 􀄭 􀅋 􀄶􀄾 is given by  \n􀆴􀄽􀇵􀄾 􀅀 􀅕􀄾 􀄮 􀅁 􀄽􀅕􀄿 􀄶􀄾􀄮 􀅀 􀅕􀄮 􀄿 􀇵 􀅁 􀄭 . Since 􀆴 ′ 􀄽􀇵􀄾 􀅀  􀆴􀄽􀇵􀄾 has  \n􀄮􀅕 􀇵 􀄮 􀄿 􀇵 􀅁 􀄭 ,  \n􀅕 􀄭  \nthe minimum value  at 􀇵 􀅀  .  \nThe answer is ③.  \n6. Find the volume of the solid obtained by rotating the region enclosed by  \n􀇶 􀅀 􀄿 􀇵 􀄮 􀅁 􀄰 and 􀇶 􀅀 􀇵 􀅁 􀄮 about the 􀇵-axis .  \n􀄲􀄮 􀄳􀄰 􀄴􀄲 􀄵􀄴 􀄭􀄶􀄴  \n①  􀆥 ②  􀆥 ③  􀆥 ④  􀆥 ⑤  􀆥  \n(Sol) From 􀄿 􀇵 􀄮 􀅁 􀄰 􀅀 􀇵 􀅁 􀄮 , it follows that 􀇵 􀄮 􀅁 􀇵 􀄿 􀄮 􀅀 􀄽􀇵 􀅁 􀄮􀄾􀄽􀇵 􀄿 􀄭􀄾 􀅀 􀄶 , which shows that two graphs meet at 􀇵 􀅀 􀄿 􀄮 and 􀇵 􀅀 􀄭 . Since 􀇵 􀅁 􀄮 ≤ 􀄿 􀇵 􀄮 􀅁 􀄰 on the interval 􀅂 􀄿 􀄮 􀅋 􀄭 􀅃 , the volume 􀄎 is given by  \n􀄎 􀅀 􀆥 􀅔 􀄭 􀄽􀄽􀄿 􀇵 􀄮 􀅁 􀄰􀄾 􀄮 􀄿 􀄽􀇵 􀅁 􀄮􀄾 􀄮 􀄾 􀇡􀇵 􀅀 􀆥 􀅔 􀄭 􀄽 􀇵􀄰 􀄿 􀄵􀇵 􀄮 􀄿 􀄰􀇵 􀅁 􀄭􀄮􀄾 􀇡􀇵  \n􀄿 􀄮 􀄿 􀄮  \n􀅀 􀆥   􀇵 􀄱 􀄿 􀄯􀇵 􀄯 􀄿 􀄮􀇵 􀄮 􀅁 􀄭􀄮􀇵   􀄮 􀅀 􀆥 􀄽 􀄽  􀄿 􀄯 􀄿 􀄮 􀅁 􀄭􀄮􀄾 􀄿 􀄽  􀅁 􀄮􀄰 􀄿 􀄴 􀄿 􀄮􀄰􀄾 􀄾  \n􀄭􀄶􀄴  \n􀅀 􀅦􀄱  \n􀆥 .  \nThe answer is ⑤.","cbCais0N4exCaJoh","https://ap.wps.com/l/cbCais0N4exCaJoh","pdf",102075,"English","# SOCIE (Scholarship) Sample Test - Solutions\n## Trigonometry and logarithms\n## Algebraic and calculus problems","[{\"question\":\"What types of calculus problems are included?\",\"answer\":\"There are problems finding the shortest distance between a curve and a point, and finding the volume of a solid formed by rotating a region about an axis.\"}]","maths scholarship answers 2022 - Solution to SOCIE sample test | PDF"]