[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-572891-105":3,"detail-sidebar-cat-1-en-105":81,"doc-detail-572891-en":126},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","c4-cheat-sheet","C4 Cheat Sheet","","Study guide for C4 topics in algebra and calculus, organized as chapters with “usual types of questions,” practical tips, and common failure points. Covers splitting fractions into partial fractions (including squared factors), working with parametric equations and converting them to Cartesian form, expanding binomial expressions and products up to required powers, and differentiation including exponential growth/decay, implicit differentiation, and forming differential equations from rate descriptions. Includes worked examples and cautions about frequent mistakes like missing terms, sign errors, and incorrect algebraic handling.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/template/","Template",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/template/presentations/","Presentations",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/template/c4-cheat-sheet/572891/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/c4-cheat-sheet/572891.png","ImageObject",442,249,{"name":42,"@type":43},"Valentina","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-18","2026-10-13",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",5,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What should be done when a denominator contains squared factors in partial fraction decomposition?","Question",{"text":63,"@type":64},"Use two fractions for that squared factor in the partial fraction sum, not just one. Also remember the extra term that comes from the square.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"How can parametric equations involving trig functions be converted into a single Cartesian equation?",{"text":68,"@type":64},"Typically make sin x and cos x (or the relevant parameter forms) the subject first, then use the identity sin^2 x + cos^2 x = 1. Often squaring a relation helps create sin^2 and/or cos^2 terms.",{"name":70,"@type":61,"acceptedAnswer":71},"What are common pitfalls in binomial expansions?",{"text":72,"@type":64},"Avoid missing brackets when cubing or squaring, forgeting to apply the outer power correctly when factoring constants, and errors in binomial coefficients such as missing factorials in denominators. Be especially careful with signs and with minus signs inside exponents.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},572891,1792033378,{"code":4,"msg":82,"data":83},"success",[84,88,93,98,103,108,113,118,122],{"id":85,"doc_module":22,"doc_module_name":25,"category_name":29,"show_sort_weight":86,"slug":87},11,90,"presentations",{"id":89,"doc_module":22,"doc_module_name":25,"category_name":90,"show_sort_weight":91,"slug":92},12,"Resumes",80,"resumes",{"id":94,"doc_module":22,"doc_module_name":25,"category_name":95,"show_sort_weight":96,"slug":97},14,"Invoices",70,"invoices",{"id":99,"doc_module":22,"doc_module_name":25,"category_name":100,"show_sort_weight":101,"slug":102},15,"Posters",60,"posters",{"id":104,"doc_module":22,"doc_module_name":25,"category_name":105,"show_sort_weight":106,"slug":107},16,"Social Media",50,"social-media",{"id":109,"doc_module":22,"doc_module_name":25,"category_name":110,"show_sort_weight":111,"slug":112},17,"Forms",40,"forms",{"id":114,"doc_module":22,"doc_module_name":25,"category_name":115,"show_sort_weight":116,"slug":117},18,"Letters",30,"letters",{"id":119,"doc_module":22,"doc_module_name":25,"category_name":120,"show_sort_weight":55,"slug":121},21,"Paper Templates","papers-templates",{"id":123,"doc_module":22,"doc_module_name":25,"category_name":124,"show_sort_weight":4,"slug":125},158,"General","general-158",{"code":4,"msg":82,"data":127},{"doc_id":79,"user_id":128,"nickname":42,"user_avatar":129,"doc_module":22,"category_id":85,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":135,"language":136,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":12,"update_tm":140,"read_time":30},13056703020460,"https://ap-avatar.wpscdn.com/avatar/be000253dac470eee5d?_k=1778207105932848923","C4 Cheat Sheet  \n\n| Chapter | Usual types of questions | Tips | What can go ugly |  |  |  |  |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n| 1 – Partial\u003Cbr>Fractions | 􀁸 Be able to split a fraction whose denominator is a product of linear expressions, e.g. 􀯫2(􀯫􀯫31)\u003Cbr>􀁸 Be able to split a fraction where one (or more) of the factors in the denominator are squared, e.g. 􀯫~~ ~~22􀯫(􀯫31)\u003Cbr>􀁸 Deal with top-heavy fractions where the highest power in the denominator is greater or equal to the highest power in the denominator, e.g 􀯫􀯫(2􀯫21) | 􀁸 The textbook provides two methods for dealing with top heavy fractions. The algebraic long division method is miles easier!\u003Cbr>e.g. 􀯫􀯫(2􀯫21) = 􀯫􀯫222􀯫 . Using long division we get a quotient of 1 and a remainder of −􀝔 + 2, thus:\u003Cbr>􀝔 2 + 2 −􀝔 + 2\u003Cbr>~~ ~~ = 1 + ~~ ~~\u003Cbr>􀝔 2 + 􀝔 􀝔 (􀝔 + 1)\u003Cbr>Then split the 􀯫−(􀯫􀯫21) into partial fractions as normal.\u003Cbr>􀁸 Don’t forget that when you have a squared factor in the denominator, you need two fractions in your partial fraction sum:\u003Cbr> 2  ≡ 􀜣 +  􀜤  +  􀜥 \u003Cbr>􀝔 2 (􀝔 + 1) 􀝔 􀝔 2 􀝔 + 1\u003Cbr>􀁸 When you have three unknowns it’s generally easiest to use substitution to get two of them (e.g. the 􀜣 and the 􀜤) then compare the coefficients of 􀝔 2 to get the 􀜥 . For the above example:\u003Cbr>2 ≡ 􀜣􀝔(􀝔 + 1) + 􀜤 (􀝔 + 1) + 􀜥􀝔 2 We can see immediately, without needing to write out the expansion, that 0 = 􀜣 + 􀜥 , by comparing 􀝔 2 terms. | 􀁸 Forgetting the extra term when the denominator’s factors are squared.\u003Cbr>􀁸 Being sloppy at algebraic long division!\u003Cbr>􀁸 Be careful with substitution of negative values.\u003Cbr>􀁸 You may have to spot that you need to factorise the denominator first before expressing as partial fractions.\u003Cbr>􀁸 Not realising the fraction is top heavy and therefore trying to incorrectly do: |  |  |  |  |\n|  |  |  | 2􀝔2\u003Cbr>􀝔 (􀝔 + 1) | = | 􀜣􀝔 | + | 􀜤\u003Cbr>􀝔 + 1 |\n| 2 – Parametric Equations | 􀁸 Know that 􀯗􀯗􀯬􀯫 = ((􀳏~~􀳏~~􀳏􀳏􀳤~~􀳟~~􀳣􀳟))\u003Cbr>(This makes sense as we have just divided numerator and denominator by 􀝀􀝐)\u003Cbr>􀁸 Be able to integrate parametric equations.\u003Cbr>􀁸 Be able to convert parametric equations into a single Cartesian one. | 􀁸 Note: You will NOT be asked to sketch parametric equations.\u003Cbr>􀁸 To convert parametric equations involving trig functions to Cartesian ones, the strategy is usually to make sin 􀝔 and cos 􀝔 the subject before using the identity sin2 􀝔 + cos 2 􀝔 ≡\u003Cbr>1. Often squaring one of the parametric equations helps so that we have sin2 􀝔 and/or cos 2 􀝔:\u003Cbr>􀝔 = √3 sin 2􀝐 􀝕 = 4 cos 2 􀝐\u003Cbr>􀝔 = 2√3 sin 􀝐 cos 􀝐\u003Cbr>􀝔 2 = 12 sin2 􀝐 cos 2 􀝐\u003Cbr>􀝔 2 = 12 (1 − cos2 􀝐) cos 2 􀝐\u003Cbr>􀝔 2 = 12 ( 1 − (􀝕4)2) 􀝕4 | 􀁸 Hitting a dead end converting parametric equations to Cartesian. See tips on left.\u003Cbr>􀁸 Forgetting to multiply by 􀯗􀯗􀯫􀯧 when integrating parametric equations. Remember that the 􀝀􀝔 in ∫ 􀝕 􀝀􀝔 can be replaced with 􀯗􀯗􀯫􀯧 􀝀􀝐, which is easy to remember, as the 􀝀􀝐’s cancel if we think of 􀝀􀝔 and 􀝀􀝐 just as quantities. |  |  |  |  |\n\n\n| 3 – Binomial Expansion | 􀁸 Expanding out an expression of the form (1 + 􀝇􀝔)􀯡 , where 􀝊 is negative or fractional.\u003Cbr>􀁸 Expanding out an expression of the form (􀜽 + 􀝇􀝔)􀯡 , where 􀜽 needs to be factorised out first.\u003Cbr>􀁸 Finding the product of two Binomial expansions, e.g.\u003Cbr>\u003Cbr>√1 + 􀝔  1 − 1\u003Cbr>~~  ~~ → (1 + 􀝔)2 (1 − 􀝔) 2 √1 − 􀝔 | 􀁸 (1 + 􀝇􀝔)􀯡 = 1 + 􀝊(􀝇􀝔) + 􀯡~~ ~~(􀯡2−!~~ ~~1) (􀝇􀝔)2\u003Cbr>􀝊 (􀝊 − 1)(􀝊 − 2)\u003Cbr>+ ~~ ~~ (􀝇􀝔)3 + ⋯\u003Cbr>3!\u003Cbr>􀁸 Your expression may be a binomial expansion in disguise, e.g.√1~~ ~~~~ ~~2􀝔 = (1 − 2􀝔) −~~ ~~1~~2~~ = 1 + ( − 12) (−2􀝔) + −~~ ~~1~~2 ~~×2!~~ ~~−~~ ~~3~~2~~ (−2􀝔)2 + ⋯\u003Cbr>􀁸 When the first term is not 1, you have to factorise this number out, raised to the power outside the brackets. e.g.\u003Cbr>1\u003Cbr>(4 + 5􀝔)1~~2~~ = 41~~2~~ ( 1 + 54 􀝔 )2 = 2 [ 1 + 12 (54􀝔 ) + ⋯]\u003Cbr>Ensure the outer brackets are maintained till the very end, when you expand them out.\u003Cbr>􀁸 When finding the product of two expansions, then if you needed up to the 􀝔 2 term, then you only need to find up to the 􀝔 2 term in each of the two expansions. Onl","cbCairZrEc3bu1VJ","https://ap.wps.com/l/cbCairZrEc3bu1VJ","pdf",511918,8,"English","# Chapter 1 - Partial Fractions\n## Tips and common failure points\n# Chapter 2 - Parametric Equations\n## Converting to Cartesian and integration notes\n# Chapter 3 - Binomial Expansion\n## Expansion rules and typical errors\n# Chapter 4 - Differentiation\n## Exponential models, implicit differentiation, and differential equations","[{\"question\":\"What should be done when a denominator contains squared factors in partial fraction decomposition?\",\"answer\":\"Use two fractions for that squared factor in the partial fraction sum, not just one. Also remember the extra term that comes from the square.\"},{\"question\":\"How can parametric equations involving trig functions be converted into a single Cartesian equation?\",\"answer\":\"Typically make sin x and cos x (or the relevant parameter forms) the subject first, then use the identity sin^2 x + cos^2 x = 1. Often squaring a relation helps create sin^2 and/or cos^2 terms.\"},{\"question\":\"What are common pitfalls in binomial expansions?\",\"answer\":\"Avoid missing brackets when cubing or squaring, forgeting to apply the outer power correctly when factoring constants, and errors in binomial coefficients such as missing factorials in denominators. Be especially careful with signs and with minus signs inside exponents.\"}]","C4 Cheat Sheet | PDF",1791929912]