[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-207801-en":3,"doc-seo-207801-105":29,"detail-sidebar-cat-0-en-105":88},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":20,"language":21,"language_code":22,"site_id":23,"html_lang":22,"table_of_contents":24,"faqs":25,"seo_title":26,"seo_description":14,"update_tm":27,"read_time":28},207801,962088121634,"supergirl","https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc",4,"Exam","Revision Sheet - Unit 1 WJEC","A comprehensive Unit 1 WJEC revision sheet covering key A Level and AS Level mathematics techniques. It consolidates proof methods, rules of indices, surds, rationalising denominators, the discriminant and completing the square. It also summarises solving equations and inequalities, remainder and factor theorems, curve and graph transformations, and coordinate geometry including circle theorems and circle intersections. Further topics include the binomial theorem, trigonometry, exponentials and logarithms, differentiation, integration, stationary points, and vectors, ending with an examination overview and a completion checklist.","Unit 1 WJEC  \nRevision Sheet  \nProof  \n(a) Proof by deduction.(b) Proof by exhaustion.(c) Disproof by counterexample.  \nRules of Indices  \n􀝊 􀯔 × 􀝊 􀯕 = 􀝊 􀯔+􀯕 .  \n􀝊 􀯔 ÷ 􀝊 􀯕 = 􀝊 􀯔−􀯕 .  \n(􀝊􀯔 )􀯕 = 􀝊 􀯔×􀯕 .  \n􀝊 0 = 1.  \n􀝊  − 􀯔 = 􀯡1􀳌~~ ~~ .  \n1  \n􀝊 􀳌 = 􀳌√􀝊 .  \n( 􀳍√􀝊)􀯔 = 􀝊 ~~ ~~􀳌􀳍 = 􀳍√􀝊 􀯔  .  \nSurds  \n√􀜽 × √􀜾 = √􀜽􀜾 .  \n(√􀜽  )2 = 􀜽 .  \nRationalise the denominator:  \n􀯔+􀯕 √􀯖  (􀯔+􀯕 √􀯖)(􀯗−􀯘√􀯙)  \n~~  ~~ = ~~     ~~  \n􀯗+􀯘√􀯙 (􀯗+􀯘√􀯙)(􀯗−􀯘√􀯙) .  \nE.g. 2+4√2 = (2+4√2)(3−5√2) 3+5√2 (3+5√2)(3−5√2)    \n6−10√2+12√2−20×2  −34+2√2  \n= ~~     ~~ =  \n9−15√2+15√2−25×2 −41 .  \nThe Discriminant  \nTwo distinct real roots:  \n􀜾 2 − 4􀜽􀜿 > 0.  \nTwo real roots:  \n􀜾 2 − 4􀜽􀜿 ≥ 0.  \nOne real root (repeated):  \n􀜾 2 − 4􀜽􀜿 = 0. Two complex roots / No real roots:  \n􀜾 2 − 4􀜽􀜿 \u003C 0.  \nCompleting the Square  \n􀝔 2 + 2􀜽􀝔 = (􀝔 + 􀜽)2 − 􀜽 2 .􀜽􀝔2 + 􀜾􀝔 + 􀜿 = 􀜽 (􀝔 2 + 􀯕􀯔 􀝔 + 􀯖􀯔) =  \n􀜽 ((􀝔 + 􀯕2􀯔)2 − (􀯕2􀯔)2 + 􀯖􀯔).  \nSolving Equations and Inequalities  \n􀝔 =  \n−􀯕±√􀯕2 −4􀯔􀯖  \n2􀯔 .  \nSimultaneous equations.  \n􀝔 2 − 6􀝔 + 8 ≥ 0  \n(􀝔 − 2)(􀝔 − 4) ≥ 0 Either 􀝔 ≤ 2 or 􀝔 ≥ 4.  \n􀝕  \nThe Remainder Theorem  \nWhen the polynomial 􀝂(􀝔) is divided by the polynomial 􀝔 – 􀜽 , where 􀜽 is a constant, then the remainder at the end of the calculation is 􀝂(􀜽).  \nThe Factor Theorem  \n(a) The polynomial (􀝔 – 􀜽) is a factor of the polynomial 􀝂(􀝔) if 􀝂 (􀜽) = 0.  \n(b) If 􀝂(􀜽) = 0, then the polynomial (􀝔 – 􀜽) is a factor of the polynomial 􀝂(􀝔).  \nGraphs  \nPolynomial graphs, e.g.  \n􀝕 = 􀝔 3 − 4􀝔2 − 17􀝔 + 60. Reciprocal graphs, e.g. 􀝕 = 􀯫22. Asymptotes.  \nTransformations:  \n􀝕 = 􀝂 (􀝔 − 􀜽)  \nMove the graph 􀜽 units right.  \n􀝕 = 􀝂 (􀝔) + 􀜽  \nMove the graph 􀜽 units up.  \n􀝕 = 􀜽􀝂 (􀝔)  \nStretch along the 􀝕–axis.  \n􀝕 = 􀝂 (􀜽􀝔)  \nCompress along the 􀝔–axis.  \n􀝔  \nCo-ordinate Geometry  \nLet A = (􀝔1, 􀝕1) , B = (􀝔2, 􀝕2). Length of AB =  \n√(􀝔2 − 􀝔1)2 + (􀝕2 − 􀝕1)2. Gradient of AB: 􀝉 = 􀯬􀯫~~2~~2−􀯬􀯫~~1~~1 . Equation of AB:  \n􀝕 − 􀝕1 = 􀝉 (􀝔 − 􀝔1) .  \nMid-point of AB = (􀯫~~ 1 ~~+2􀯫~~2~~ , 􀯬~~1 ~~+2􀯬~~2~~) .  \nLines 􀜮1 and 􀜮 2 with gradients 􀝉1 and 􀝉2 :  \n􀜮 1 and 􀜮 2 are parallel: 􀝉1 = 􀝉2.  \n􀜮 1 and 􀜮 2 are perpendicular:  \n􀝉1 = − 1􀯠2 , 􀝉2 = − 1􀯠1,􀝉1􀝉2 = −1.  \nIf a tangent has gradient 􀝉 then the normal has gradient − 1􀯠 .  \nEquation of a Circle  \nEquation of a circle 􀜥 with centre (􀜽, 􀜾) and radius 􀝎:  \n(􀝔 − 􀜽)2 + (􀝕 − 􀜾)2 = 􀝎 2 or 􀝔 2 + 􀝕 2 + 2􀝃􀝔 + 2􀝂􀝕 + 􀜿 = 0, where 􀝎 = √􀝃 2 + 􀝂2 − 􀜿  ;  \n(􀜽, 􀜾) = (−􀝃 , −􀝂).  \nTangent and radius meet at 90°, so the gradient of the tangent is the negative reciprocal of the gradient of the radius.  \nIntersection of two circles 􀜥1 and 􀜥2 with centres 􀜣 , 􀜤; radii 􀝎1 , 􀝎2 : Two intersections:  \nLength 􀜣􀜤 \u003C 􀝎1 + 􀝎2. One intersection (the circles tough externally):  \nLength 􀜣􀜤 = 􀝎1 + 􀝎2.  \nOne intersection (the circles  \ntouch internally): Length 􀜣􀜤 = |􀝎1 − 􀝎2 |. No intersections: Length 􀜣􀜤 > 􀝎1 + 􀝎2.  \nLength 􀜣􀜤 \u003C |􀝎1 − 􀝎2 |.  \nCircle Theorems  \n(a) The angle in a semicircle is aright angle.  \n(b) The perpendicular from the centre to a chord bisects the chord.  \n(c) The radius of a circle at a given point on its circumference is perpendicular to the tangent to the circle at that point.  \nThe Binomial Theorem  \nPascal’s Triangle:  \n1  \n1 1 1 2 1  \n1 3 3 1 1 4 6 4 1  \n1 5 10 10 5 1 etc...  \n(􀜽 + 􀜾􀝔)􀯡  \n= 􀜽􀯡 + 􀝊􀜽􀯡−1 (􀜾􀝔)  \n􀝊 (􀝊 − 1)  \n+ 􀜽􀯡−2 (􀜾􀝔)2  \n2 + ⋯ + (􀜾􀝔)􀯡 .  \nTrigonometry  \nO A OS C T  \nH H A.  \nExact values for 30°, 45°, 60°. The graphs of sin, cos, tan.  \nSine rule: si􀯔n􀮺 = si􀯕n~~ ~~􀮻 = si􀯖n~~ ~~􀮼  \nor  \nsin 􀮺   sin 􀮻   sin 􀮼  \n= =  \n􀯔 􀯕 􀯖 .  \nCosine rule:  \n􀜽 2 = 􀜾 2 + 􀜿 2 − 2􀜾􀜿 cos 􀜣  \nor cos 􀜣 = 􀯕 2 +􀯖2 −􀯔2 2􀯕􀯖 .  \n a   \nc b  \nArea of a triangle = 12 􀜽􀜾 sin 􀜥 .  \nsin2 􀟠 + cos 2 􀟠 = 1. sin2 􀟠 = 1 − cos 2 􀟠 . cos 2 􀟠 = 1 − sin2 􀟠 . tan 􀟠 = sincos􀰏􀰏~~ ~~ .  \n􀝔  \nExponentials and Logarithms  \nIf 􀜾 􀯬 = 􀝔 then 􀝕 = log 􀯕 (􀝔).  \nSolving equations with logarithms.  \nThe graphs of 􀝕 = 􀜽􀯫 , 􀝕 = 􀝁 􀯫 and 􀝕 = ln (􀝔) .  \nExponential models: 􀝕 = 􀜣􀝁􀯞􀯧(growth) or 􀝕 = 􀜣􀝁 −􀯞􀯧 (decay) .􀯗􀯗􀯫 (􀝁 􀯞􀯫 ) = 􀝇􀝁 􀯞􀯫  \n.  \nLimitations of models.  \nRules:  \nlog 􀯔 (􀝔􀝕) = log 􀯔 (􀝔) + log􀯔 (􀝕). log 􀯔 (􀯫􀯬) = log 􀯔 (􀝔) − log􀯔 (􀝕). log 􀯔 (􀝔􀯡 ","cbCaipzwyMMnS8eT","https://ap.wps.com/l/cbCaipzwyMMnS8eT","pdf",1790684,1,"English","en",105,"# Proof\n# Rules of Indices - Surds - Rationalising Denominators\n# The Discriminant - Completing the Square\n# Solving Equations and Inequalities\n# The Remainder Theorem - The Factor Theorem\n# Graphs - Transformations\n# Co-ordinate Geometry\n# Equation of a Circle - Circle Theorems\n# The Binomial Theorem - Trigonometry\n# Exponentials and Logarithms\n# Differentiation - Stationary Points\n# Integration\n# Vectors\n# The Examination - Checklist","[{\"question\":\"What proof methods are included in the Unit 1 WJEC revision sheet?\",\"answer\":\"It lists proof by deduction and proof by exhaustion, and also notes disproof by counterexample.\"},{\"question\":\"How does the sheet define and use the discriminant?\",\"answer\":\"It classifies roots using the condition involving b^2 - 4ac: two distinct real roots if it is greater than 0, two real roots if it is at least 0, one repeated real root if equal to 0, and no real roots if less than 0.\"},{\"question\":\"What are the main topics covered in calculus and vectors?\",\"answer\":\"Calculus topics include differentiation from first principles, quick differentiation rules, stationary points using the second-derivative test, and integration formulas. Vectors cover position vectors, magnitudes, line division in ratios, and conditions for parallel vectors.\"}]","Revision Sheet - Unit 1 WJEC | PDF",1788601176,3,{"code":4,"msg":30,"data":31},"ok",{"site_id":23,"language":22,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":83,"head_meta":85,"extra_data":87,"updated_unix":27},"revision-sheet-unit-1-wjec","",{"@graph":35,"@context":82},[36,51,65],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,49],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":28},"https://docshare.wps.com/document/exam/",{"item":50,"name":13,"@type":42,"position":11},"https://docshare.wps.com/document/revision-sheet-unit-1-wjec/207801/",{"url":50,"name":13,"@type":52,"author":53,"headline":13,"publisher":55,"fileFormat":58,"inLanguage":22,"description":14,"dateModified":59,"datePublished":59,"encodingFormat":58,"isAccessibleForFree":60,"interactionStatistic":61},"DigitalDocument",{"name":9,"@type":54},"Person",{"url":40,"name":56,"@type":57},"DocShare","Organization","application/pdf","2026-09-05",true,{"@type":62,"interactionType":63,"userInteractionCount":4},"InteractionCounter",{"@type":64},"ViewAction",{"@type":66,"mainEntity":67},"FAQPage",[68,74,78],{"name":69,"@type":70,"acceptedAnswer":71},"What proof methods are included in the Unit 1 WJEC revision sheet?","Question",{"text":72,"@type":73},"It lists proof by deduction and proof by exhaustion, and also notes disproof by counterexample.","Answer",{"name":75,"@type":70,"acceptedAnswer":76},"How does the sheet define and use the discriminant?",{"text":77,"@type":73},"It classifies roots using the condition involving b^2 - 4ac: two distinct real roots if it is greater than 0, two real roots if it is at least 0, one repeated real root if equal to 0, and no real roots if less than 0.",{"name":79,"@type":70,"acceptedAnswer":80},"What are the main topics covered in calculus and vectors?",{"text":81,"@type":73},"Calculus topics include differentiation from first principles, quick differentiation rules, stationary points using the second-derivative test, and integration formulas. 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