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It compiles key formulae spanning Pure Mathematics topics such as mensuration, algebraic equation solving, arithmetic and geometric series, binomial and Maclaurin series, trigonometric identities, differentiation rules, and integration methods. It also includes Further Pure Mathematics extensions, Mechanics motion and string-spring relationships, and Probability & Statistics summary statistics with discrete and continuous distributions, expectations, variances, and sampling concepts.",{"@graph":69,"@context":122},[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/list-of-formulae-and-statistical-tables-cambridge-international-as-a-level-mathematics-9709-and-further-mathematics-9231/642742/",{"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/list-of-formulae-and-statistical-tables-cambridge-international-as-a-level-mathematics-9709-and-further-mathematics-9231/642742.png","ImageObject",300,407,{"name":92,"@type":93},"River Wang","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-02","2026-10-23",true,{"@type":102,"interactionType":103,"userInteractionCount":29},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What subjects and syllabus scope does the formula list cover?","Question",{"text":112,"@type":113},"It covers Cambridge International AS & A Level Mathematics (9709) and Further Mathematics (9231), for use from 2020, with sections spanning Pure Mathematics, Further Pure Mathematics, Mechanics, Further Mechanics, and Probability & Statistics.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"Which Pure Mathematics topics are included?",{"text":117,"@type":113},"Mensuration, algebra (including quadratic equations and series), trigonometric identities, differentiation, and integration are included as formula summaries.",{"name":119,"@type":110,"acceptedAnswer":120},"What probability and statistics formulas are provided?",{"text":121,"@type":113},"The list provides summary statistics for ungrouped and grouped data, formulas for expectation and variance of discrete variables, distribution-specific results (binomial, geometric, Poisson), and expectation integrals for continuous variables.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},642742,1792876934,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":29,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"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":144,"read_time":36},1099514067438,"https://ap-avatar.wpscdn.com/avatar/100002539ee87300030?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780474512215547542","List MF19  \nList of formulae and statistical tables  \nCambridge International AS & A Level  \nMathematics (9709) and Further Mathematics (9231)  \nFor use from 2020 in all papers for the above syllabuses.  \nCST319  \n*2994182781*  \nPURE MATHEMATICS  \nMensuration  \nVolume of sphere = 43~~ ~~ πr3  \nSurface area of sphere = 4πr2  \nVolume of cone or pyramid = 13 × base area × height  \nArea of curved surface of cone = πr × slant height  \nArc length of circle = r θ (θ in radians)  \nArea of sector of circle = ~~ ~~12~~ ~~ r 2 θ (θ in radians)  \nAlgebra  \nFor the quadratic equation ax2 + bx + c = 0 :  \n−b ±  b2 − 4ac  \nx = ~~ ~~  \n2a  \nFor an arithmetic series:  \nun = a + (n −1)d , Sn = ~~ ~~12~~ ~~ n(a + l) = ~~ ~~12~~ ~~ n{2a + (n −1)d } For a geometric series:  \nun = arn−1 , Sn =  a(1 − rn )  (r ≠ 1) , S∞ =~~ ~~a~~ ~~ (  r  \u003C 1 )  \n1 − r 1 − r  \nBinomial series:  \n(a + b)n = an + 􀂧􀂩􀂨 n1􀂷􀂹􀂸 an−1b + 􀂧􀂨􀂩 n2􀂷􀂹􀂸 an−2b2 + 􀂧􀂩􀂨 n3􀂷􀂹􀂸 an−3b3 + K + bn , where n is a positive integer and 􀂧􀂨􀂩 nr􀂷􀂹􀂸 = ~~ ~~r!(nn−!~~ ~~r)!  \n(1 + x)n = 1 + nx + ~~ ~~n(n2−!~~ ~~1)~~ ~~ x2 + ~~ ~~n(n~~ ~~−~~ ~~13)!(n~~ ~~−~~ ~~2)~~ ~~ x3 + K , where n is rational and  x  \u003C 1  \n2  \nTrigonometry  \ntan θ ≡ ~~ ~~scθθ  \ncos2 θ + sin 2 θ ≡ 1 , 1 + tan 2 θ ≡ sec2 θ , cot 2 θ +1 ≡ cosec2 θ  \nsin(A ± B) ≡ sin AcosB ± cosA sin B  \ncos(A ± B) ≡ cosAcos B m sinA sin B  \ntan(A ± B) ≡  tan A ± tan B   \n1 m tan A tan B  \nsin 2A ≡ 2sinAcosA cos 2A ≡ cos2 A − sin2 A ≡ 2cos2 A −1 ≡ 1− 2sin 2 A  \ntan 2 A ≡ 2 tan A   \n1 − tan2 A Principal values:  \n− ~~ ~~12~~ ~~ π ⩽ sin −1 x ⩽ ~~ ~~12~~ ~~ π , 0 ⩽ cos−1 x ⩽ π , − ~~ ~~12~~ ~~ π \u003C tan −1 x \u003C ~~ ~~12~~ ~~ π  \nDifferentiation  \nf( x)  \nxn  \nln x  \nx  \nesin x cos x  \ntan x  \nsec x  \ncosec x  \ncot x  \n−1 tan x  \nuv  \nu  \nv  \nIf x = f(t ) and y = g(t) then dydx = ~~ ~~dydt ÷ ~~ ~~ddxt  \nf ′( x )  \nn−1 nx  \n 1  \nx  \nxe  \ncos x  \n−sin x  \n2 sec x  \nsec x tan x  \n− cosec x cot x  \n2  \n− cosec x  \n1  \n1 + x2  \ndu dv v dx + u dx du dv v dx − u dx  \n2 v  \n3  \nIntegration  \n(Arbitrary constants are omitted; a denotes a positive constant.)  \nf( x)  \nxn  \n1  \nx  \nex  \nsin x  \ncos x  \nsec2 x  \nx2  \n2  \n1  \n+  \n1  \na 2  \n2  \nx − a  \n1  \n2 2 a − x  \n∫ f( x ) dx  \nn+1 x  \nn +1  \n1  \na  \nln x  \nxe  \n− cos x sin x  \ntan x  \n1 􀂧 x 􀂷 tan − 􀂩􀂨 a 􀂹􀂸  \n1 ln x − a 2a x + a  \n1 ln a + x 2a a − x  \n∫u  \ndv  \ndx  \ndx = uv − ∫v  \ndu  \ndx  \ndx  \n∫  \nf ′(x) f(x)  \ndx = ln f (x)  \n(n ≠ −1)  \n(x > a)  \n( x \u003C a)  \nVectors  \nIf a = a1i + a2 j + a3k and b = b1i + b2 j + b3k then  \na.b = a1b1 + a2b2 + a3b3 = a b cos θ  \n4  \nFURTHER PURE MATHEMATICS  \nAlgebra  \nSummations:  \n∑r1 r = ~~ ~~12~~ ~~ n(n + 1) , ∑r1 r 2 = ~~ ~~16~~ ~~ n(n + 1)(2n + 1) , ∑r1 r 3 = ~~ ~~14~~ ~~ n2 (n + 1)2 Maclaurin’s series:  \nf(x) = f(0) + xf ′(0) + ~~ ~~x22! f ′′(0) + K + ~~ ~~xrr!~~ ~~ f (r ) (0) + K  \nex = exp(x) = 1 + x + ~~ ~~x22! + K + ~~ ~~xrr! + K (all x)  \n2 3 r  \nln(1 + x) = x − ~~ ~~x~~ ~~+ ~~ ~~x~~ ~~ − K + (−1)r+1 ~~ ~~x~~ ~~+ K (–1 \u003C x ⩽ 1)  \n2 3 r  \n3 5 2r+1  \nsin x = x − ~~ ~~x3! + ~~ ~~x5! − K + (−1)r ~~ ~~(x2r~~ ~~+~~ ~~1)! + K (all x)  \n2 4 2r  \ncos x = 1 − ~~ ~~x2! + ~~ ~~x4! − K + (−1)r ~~ ~~(x2r)! + K (all x)  \n3 5 2r+1  \ntan −1 x = x − ~~ ~~x~~ ~~ + ~~ ~~x~~ ~~ − K + (−1)r ~~ ~~x~~ ~~ + K (–1 ⩽ x ⩽ 1)  \n3 5 2r +1  \n3 5 2r+1  \nsinh x = x + ~~ ~~x3! + ~~ ~~x5! + K + ~~ ~~(x2r~~ ~~+~~ ~~1)! + K (all x)  \n2 4 2r  \ncosh x = 1 + ~~ ~~x2! + ~~ ~~x4! + K + ~~ ~~(x2r)! + K (all x)  \n3 5 2r +1  \ntanh −1 x = x + ~~ ~~x~~ ~~ + ~~ ~~x~~ ~~ + K + ~~ ~~x~~ ~~ + K (–1 \u003C x \u003C 1)  \n3 5 2r +1  \nTrigonometry  \nIf t = tan ~~ ~~12~~ ~~ x then:  \nsin x = ~~ ~~1tt~~ ~~2~~ ~~ and cos x = ~~ ~~11~~ ~~tt22  \nHyperbolic functions  \ncosh 2 x − sinh2 x ≡ 1 , sinh 2x ≡ 2sinh x cosh x , cosh 2x ≡ cosh 2 x + sinh 2 x  \nsinh −1 x = ln(x + x2 +1) cosh −1 x = ln(x + x2 −1) (x ⩾ 1) tanh −1 x = ~~ ~~12~~ ~~ ln 􀂨􀂧􀂩 ~~ ~~11~~ ~~xx~~ ~~ 􀂷􀂹􀂸 (| x | \u003C 1)  \n5  \nDifferentiation  \nf( x)  \nsin −1 x  \n−1 cos x  \nsinh xcosh x  \ntanh x sinh −1 xcosh −1 xtanh −1 x  \nf ′( x )  \n1  \n1  \n− ~~ ~~  \n1 − x2 cosh x sinh x  \nsech 2 x","cbCaidccPXwzMe14","https://ap.wps.com/l/cbCaidccPXwzMe14","pdf",426414,16,"English","# Mensuration\n# Algebra\n# Trigonometry\n# Differentiation\n# Integration\n# Further Pure Mathematics\n## Summations and series\n## Hyperbolic and inverse functions\n# Mechanics\n## Uniformly accelerated motion\n## Projectile motion and springs\n## Motion in a circle\n## Centres of mass\n# Further Mechanics\n# Probability & Statistics\n## Summary statistics and distributions","[{\"question\":\"What subjects and syllabus scope does the formula list cover?\",\"answer\":\"It covers Cambridge International AS \\u0026 A Level Mathematics (9709) and Further Mathematics (9231), for use from 2020, with sections spanning Pure Mathematics, Further Pure Mathematics, Mechanics, Further Mechanics, and Probability \\u0026 Statistics.\"},{\"question\":\"Which Pure Mathematics topics are included?\",\"answer\":\"Mensuration, algebra (including quadratic equations and series), trigonometric identities, differentiation, and integration are included as formula summaries.\"},{\"question\":\"What probability and statistics formulas are provided?\",\"answer\":\"The list provides summary statistics for ungrouped and grouped data, formulas for expectation and variance of discrete variables, distribution-specific results (binomial, geometric, Poisson), and expectation integrals for continuous variables.\"}]","List of formulae and statistical tables - Cambride International AS & A Level Mathematics (9709) and Further Mathematics (9231) | PDF",1792722769]