For use only in [the name of your school] 2014 FP2 Note. FP2 Notes (Edexcel)

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1 For use oly i [the ame of your school] 04 FP Note FP Notes (Edexcel) Copyright wwwpgmathscouk - For AS, A otes ad IGCSE / GCSE worksheets

2 For use oly i [the ame of your school] 04 FP Note BLANK PAGE Copyright wwwpgmathscouk - For AS, A otes ad IGCSE / GCSE worksheets

3 For use oly i [the ame of your school] 04 FP Note Copyright wwwpgmathscouk - For AS, A otes ad IGCSE / GCSE worksheets

4 For use oly i [the ame of your school] 04 FP Note Copyright wwwpgmathscouk - For AS, A otes ad IGCSE / GCSE worksheets 4

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8 For use oly i [the ame of your school] 04 FP Note BLANK PAGE Copyright wwwpgmathscouk - For AS, A otes ad IGCSE / GCSE worksheets 8

9 For use oly i [the ame of your school] 04 FP Note Iequalities The maipulatio ad solutio of algebraic iequalities ad iequatios the solutio of x iequalities such as > x a x b Iequalities of the form x ( x ) so x < 7 + > have bee met ad rearraged to give x+ > x 6 ad Quadratic iequalities of the form x 7x+ < 0 were factorised i C to give ( x )( x 4) < 0 ad, by cosiderig the graph of y x 7x+, it ca be see that < x < 4 Now cosider more complicated iequalities Example Solve > x Method Simply look to the graphs of y x ad y y x y From this it ca be see that > x whe < x < 4 Copyright wwwpgmathscouk - For AS, A otes ad IGCSE / GCSE worksheets 9

10 For use oly i [the ame of your school] 04 FP Note Method 4 x Alteratively cosider f( x) ad solve f( x ) > 0 Establish the values of x which x x make the umerator ad the deomiator equal to zero, ie 4 ad Set up a table as follows: Sig of f( x ) x < < x < 4 x > Hece the solutio set is < x < 4 Example Solve x x 5 x 8 Method Simply look to the graph ad establish the solutio set from this x y y x 8 x x 5 y x 8 y x 5 So the itersectio poits of is x 4 ad 5< x < 8 x y ad x 8 y x 5 are x ad x 4 ad so the solutio set Copyright wwwpgmathscouk - For AS, A otes ad IGCSE / GCSE worksheets 0

11 For use oly i [the ame of your school] 04 FP Note Method Alteratively cosider x f( x) x 5 x 8 x 8 x( x 5) x 5 x 8 x + x 6 8 ( x 5)( x 8) ( x 6x 8) + ( x 5)( x 8) ( x )( x 4) ( x 5)( x 8) ( x)( x 4) ( x 5)( x 8) ad solve f( x ) > 0 Establish the values of x which make the umerator ad the deomiator equal to zero, ie, 4, 5 ad 8 Set up a table as follows: Sig of f( x ) x < < x < 4 4< x < 5 5< x < 8 x > NB : If the iitial iequality ivolves or the the iequalities ivolvig the critical values from the top of the fractio will be or If the iitial iequality ivolves < or > the the iequalities ivolvig the critical values from the top of the fractio will be < or > The iequalities ivolvig the critical values from the bottom of the fractio will always be < or > x So sice the iequality is the iequalities ivolvig ad 4 will be or whilst x 5 x 8 the iequalities ivolvig 5 ad 8 will be < or > The solutio set is x 4 ad 5< x < 8 Copyright wwwpgmathscouk - For AS, A otes ad IGCSE / GCSE worksheets

12 For use oly i [the ame of your school] 04 FP Note icludig those ivolvig the modulus sig The solutio of iequalities such as x > x+ It is best to draw the graph ad to see where the two curves or lies itersect x + Example Solve x Graphical Method y x + y x The equatios of the differet parts of the curve y are show below x This part is y x This part is y x This part is y x Copyright wwwpgmathscouk - For AS, A otes ad IGCSE / GCSE worksheets

13 For use oly i [the ame of your school] 04 FP Note x + ± 7 Solve x ad so x x 0 This gives x but from the graph it is clear that the larger of the two solutios is required so the solutio is + 7 NB The other solutio has othig to with the aswer to the questio y x y x + x + ± Solve x ad so x + x 0 This gives but from the graph it is clear that the larger of the two solutios is required so the solutio is NB The other solutio has othig to with the aswer to the questio y y x x + So the solutio set is x ad + 7 x Copyright wwwpgmathscouk - For AS, A otes ad IGCSE / GCSE worksheets

14 For use oly i [the ame of your school] 04 FP Note No-Graphical Method x + Solve x x + It is worth recogisig at the outset that sice ay possible solutio to x must satisfy x + 0 ad so x Aythig outside of this rage ca be igored x + x + With this coditio i mid, simply solve x ad x x + Solvig x gives x x+ ad so x x 0 Formula gives ± Usig C method this gives x ad x Sice 7 7 <, the iequality x ca be igored ad so coditio from this part of the solutio + 7 x is the oly x + Solvig x gives x x+ ad so x + x 0 Formula gives + Usig C method this gives x ± Sice <, the iequality x ca be igored ad so coditio from this part of the solutio + x is the oly So, overall, + x or + 7 x Copyright wwwpgmathscouk - For AS, A otes ad IGCSE / GCSE worksheets 4

15 For use oly i [the ame of your school] 04 FP Note Series Summatio of simple fiite series usig the method of differeces Studets should be able to sum series such as by usig partial fractios such as r r r + r r + r r + Cosider the series 00 r r r + This ca be writte as more simply, as but there does ot seem to be a obvious way for calculatig this or, This is where the method of differeces ca be used Sice r r+ r r+ it follows that r r+ r r r r So r r+ r r r r Settig it out as follows shows how this ca be simplified: r r r r+ r r May of the middle terms cacel So 00 r 00 r r 0 0 ( + ) The above method is a example of the method of differeces Copyright wwwpgmathscouk - For AS, A otes ad IGCSE / GCSE worksheets 5

16 For use oly i [the ame of your school] 04 FP Note Example (a) Calculate ad hece fid r r + r r r+ (b) Use this to fid 0 r r r 0 + r+ r r r+ r r+ r r+ r r+ So r r r r+ r r Notice there that there are two positive terms at the start do ot cacel ad the two egative terms at the ed do ot cacel So r r r Ad so + + r r r Copyright wwwpgmathscouk - For AS, A otes ad IGCSE / GCSE worksheets 6

17 For use oly i [the ame of your school] 04 FP Note Example Cosider r + r ad hece fid r + r r + r r + r r r + So r + by usig the fact that r r r r + r r r + r + r May of the middle terms cacel ( ) ( ) + leavig r+ ( + ) + ( + ) r Example (a) Calculate r r r r ( + ) ( + )( + ) (b) Hece fid r r + ( r + )( r ) (a) r r r r r r r ( + ) ( + )( + ) ( + ) ( + ) r+ r r ( r+ ) r( r+ ) r( r+ ) ( + )( r+ ) r Copyright wwwpgmathscouk - For AS, A otes ad IGCSE / GCSE worksheets 7

18 For use oly i [the ame of your school] 04 FP Note (b) From the above it follows that r r r r+ r+ r r+ r+ r+ 6 6 This leaves ( + )( + ) ( + )( + ) r r r r + ( + )( + ) ( + ) ( + )( + ) + ( ) ( + ) ( + ) ( + )( + ) Hece ( + ) r r r+ r The expressio for the sum, S, of the first terms shows whether the sum is coverget or diverget as teds to ifiity eg If eg If S the S coverges to as teds to ifiity + S the S diverges as teds to ifiity ( + )( + ) Copyright wwwpgmathscouk - For AS, A otes ad IGCSE / GCSE worksheets 8

19 For use oly i [the ame of your school] 04 FP Note iθ Euler's relatio e cosθ i siθ siθ e e i iθ iθ Complex Numbers + ad iθ iθ + Studets should be familiar with cosθ ( e e ) Cosider the differetial equatio d y d y 0 θ + From FP the equatio m + 0 gives m ± i Hece the geeral solutio is e i θ iθ y A + Be where A ad B are costats However lettig y cosθ + isiθ gives d y d y 0 θ + d y dθ cosθ i siθ ad so satisfies the equatio It therefore follows that values of A ad B ca be foud such that cos si e i θ iθ θ + i θ A + Be Pluggig i θ 0 gives A+ B Differetiatig both sides gives si cos e i θ iθ θ + i θ Ai Bie Pluggig i θ 0 gives i Ai Bi ad so A B Hece A ad B 0 iθ From this i So cos i si e i θ θ + θ This is kow as Euler s relatio e cos θ + si θ cosθ isiθ i iθ iθ iθ iθ Hece it follows cosθ ( e + e ) ad that siθ ( e e ) Example Express + i i the form re iθ where θ is i radias ( π < θ π ) r θ Pythagoras gives that must be positive) Trigoometry gives that π 6 So + i e i r + 4 so π θ 6 r (r Copyright wwwpgmathscouk - For AS, A otes ad IGCSE / GCSE worksheets 9

20 For use oly i [the ame of your school] 04 FP Note It is worth beig familiar with these three triagles: π π i e i π 4 π i e i 6 + i e i π 6 π From these it ca be see that + i e i e i e πi πi πi If z re iθ ad z se iφ the: z r ad arg ( z ) θ s arg z φ z ad (a) iθ zz re se rse ( θ+ φ) i iφ + i e i e i e πi 4 πi 4 πi 4 + i e i e i e 5 πi 6 5 πi 6 πi 6 Hece zz rs z z (b) z z Hece i re i se r s e θ φ ad arg ( zz ) θ + φ arg ( z) + arg ( z ) ( θ φ) i z r z z s z z ad arg θ φ arg ( z) arg ( z) z NB Argumets follow the same rule as logarithms: That is, just as log ab log a + log b, so ( zz ) ( z) + ( z ) arg arg arg ad just as log a log a log b, b arg z arg z arg z z so So i summary (i) zz z z (ii) z z z z (iii) arg ( zz ) arg ( z) + arg ( z ) (iv) arg arg ( z ) arg ( z ) z z Copyright wwwpgmathscouk - For AS, A otes ad IGCSE / GCSE worksheets 0