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1 PhysicsAdMthsTutor.com

2 PhysicsAdMthsTutor.com Jue Give tht y rsih ( ), > 0, () fid d y d, givig your swer s simplified frctio. () Leve lk () Hece, or otherwise, fid 4 d, 4 [ ( )] givig your swer i the form l 5, where d re itegers. (6) 0 *M545A008*

3 5. I 5 0 ( 5 ) d, 0 Leve lk () Fid epressio for d, 0 5. ( 5 ) () () Usig your swer to prt (), or otherwise, show tht I 5( ) I (5) (c) Fid I4 i the form kπ, where k is frctio. (4) *M545A08*

4 Questio 5 cotiued Leve lk *M545A08* Tur over

5 8. A curve, which is prt of ellipse, hs prmetric equtios π cos θ, y 5siθ, 0 θ. The curve is rotted through rdis out the -is. Leve lk () Show tht the re of the surfce geerted is give y the itegrl k ( 6c 9) dc, where c cos, 0 d where k d re costts to e foud. (6) () Usig the sustitutio c sih u, or otherwise, evlute the itegrl, showig ll of 4 your workig d givig the fil swer to sigifict figures. (5) 4 *M545A048*

6 Questio 8 cotiued Leve lk Q8 END (Totl mrks) TOTAL FOR PAPER: 75 MARKS 8 *M545A088*

7 . Use clculus to fid the ect vlue of d 4. (5) Leve lk 4 *N589RA048*

8 4. I ( ) cos d, > 0, 0 0 Leve lk () Show tht, for, () Hece evlute π 0 π ( ) (5) cos d. () 8 *N589RA088*

9 Questio 4 cotiued Leve lk *N589RA098* 9 Tur over

10 . The curve C hs equtio y, 0. The curve C is rotted through rdis out the -is. Usig clculus, fid the re of the surfce geerted, givig your swer to sigifict figures. (5) Leve lk *P544A08*

11 . Show tht 8 () d kπ, givig the vlue of the frctio k, (5) () 5 8 ( 0 4) d l( A ), givig the vlues of the itegers A d. (4) Leve lk 6 *P544A068*

12 4. () Prove tht, for, I e (l ) d, 0 Leve lk I e I (4) () Fid the ect vlue of I. (4) 8 *P544A088*

13 Questio 4 cotiued Leve lk *P544A098* 9 Tur over

14 . y Leve lk O Figure l The curve C, show i Figure, hs equtio y cosh, 0 l where is costt d > Usig clculus, show tht the legth of curve C is ( k ( d stte the vlue of the costt k. (6) 4 *P40A04*

15 4 4. I si d, 0 π 0 Leve lk () Prove tht, for, π I I ( ) (5) () Fid the ect vlue of I (4) (c) Show tht I 4 ( π 4π 48) 64 () 0 *P40A00*

16 Questio 4 cotiued Leve lk *P40A0* Tur over

17 PhysicsAdMthsTutor.com Jue 0 5. () Differetite rsih with respect to. () Hece, or otherwise, fid the ect vlue of rsih d 0 () Leve lk givig your swer i the form Al B C, where A, B d C re rel. (7) 4 *P40A04*

18 PhysicsAdMthsTutor.com Jue 0 Questio 5 cotiued Leve lk *P40A05* 5 Tur over

19 PhysicsAdMthsTutor.com Jue 0 7. f( ) 5cosh4sih, R () Show tht f() ( e 9e ) Hece () Leve lk () solve f( ) 5 (c) show tht l l 5cosh 4sih d π 8 (4) (5) *P40A0*

20 PhysicsAdMthsTutor.com Jue 0 Questio 7 cotiued Leve lk *P40A0* Tur over

21 5. 5 I ( ) d, 0 Leve lk () Prove tht, for, ( ) I I 5 (5) () Usig the reductio formul give i prt (), fid the ect vlue of I (5) 6 *P4956A06*

22 Questio 5 cotiued Leve lk *P4956A07* 7 Tur over

23 PhysicsAdMthsTutor.com Jue 0 (R) 7. y Leve lk O Figure The curves show i Figure hve equtios y 6 cosh d y 9 sih () Usig the defiitios of sih d cosh i terms of e, fid ect vlues for the -coordites of the two poits where the curves itersect. (6) The fiite regio etwee the two curves is show shded i Figure. () Usig clculus, fid the re of the shded regio, givig your swer i the form l c, where, d c re itegers. (6) 4 *P4956A04*

24 PhysicsAdMthsTutor.com Jue 0 (R) Questio 7 cotiued Leve lk *P4956A05* 5 Tur over

25 8. y C Leve lk O 8 Figure The curve C, show i Figure, hs equtio y, 8 () Show tht the legth s of curve C is give y the equtio s 8 d () () Usig the sustitutio sih u, or otherwise, fid ect vlue for s. Give your swer i the form l( c) where, d c re itegers. (9) 8 *P4956A08*

26 Questio 8 cotiued Leve lk Q8 (Totl mrks) TOTAL FOR PAPER: 75 MARKS END *P4956A0*

27 . () Fid d ( 4 9) () Use your swer to prt () to fid the ect vlue of () Leve lk d ( 4 9) givig your swer i the form k l( 5), where d re itegers d k is costt. () 4 *P44A048*

28 . The curve with prmetric equtios Leve lk cosh, y 4 sih, 0 is rotted through rdis out the -is. Show tht the re of the surfce geerted is (cosh ), where d is costt to e foud. (7) 8 *P44A088*

29 6. Give tht 4 I ( 6 ) d, 0, 0 Leve lk () prove tht, for, ( ) I 6( ) I (6) () Hece, showig ech step of your workig, fid the ect vlue of I 5 (5) 6 *P44A068*

30 Questio 6 cotiued Leve lk *P44A078* 7 Tur over

31 0 Edecel AS/A level Mthemtics Formule List: Further Pure Mthemtics FP Issue Septemer 009 Further Pure Mthemtics FP Cdidtes sittig FP my lso require those formule listed uder Further Pure Mthemtics FP, d Core Mthemtics C C4. Vectors The resolved prt of i the directio of is. The poit dividig AB i the rtio μ λ : is μ λ λ μ Vector product: ˆ si k j i θ ) ( ) ( ) ( c. c. c. c c c If A is the poit with positio vector k j i d the directio vector is give y k j i, the the stright lie through A with directio vector hs crtesi equtio ) ( λ z y The ple through A with orml vector k j i hs crtesi equtio. d d z y 0 where The ple through o-collier poits A, B d C hs vector equtio c c r μ λ μ λ μ λ ) ( ) ( ) ( The ple through the poit with positio vector d prllel to d c hs equtio c r t s The perpediculr distce of ),, ( γ β α from 0 d z y is d γ β α.

32 Hyperolic fuctios cosh sih sih sih cosh cosh cosh sih rcosh l{ } ( ) rsih l{ } rth l ( < ) Coics Ellipse Prol Hyperol Rectgulr Hyperol Stdrd Form y y y 4 y c Prmetric Form ( cosθ, siθ ) ( t, t) ( sec θ, t θ ) (± cosh θ, sih θ ) ct, c t Eccetricity e < ( e ) e e > e ( ) e Foci ( ± e, 0) (, 0) ( ± e, 0) (± c, ± c) Directrices ± e ± y ± c e Asymptotes oe oe y ± 0, y 0 Edecel AS/A level Mthemtics Formule List: Further Pure Mthemtics FP Issue Septemer 009

33 Differetitio f() f () rcsi rccos rct sih cosh cosh sih th sech rsih rcosh rth Itegrtio ( costt; > 0 where relevt) f() f( ) d sih cosh cosh sih th l cosh rcsi rct ( < ) rcosh, l{ } rsih, l l l { } rth ( ( > ) < ) Edecel AS/A level Mthemtics Formule List: Further Pure Mthemtics FP Issue Septemer 009

34 Edecel AS/A level Mthemtics Formule List: Further Pure Mthemtics FP Issue Septemer 009 Arc legth y s d d d (crtesi coordites) t t y t s d d d d d (prmetric form) Surfce re of revolutio S y ds π y y d d d π t t y t y d d d d d π

35 Further Pure Mthemtics FP Cdidtes sittig FP my lso require those formule listed uder Core Mthemtics C d C. Summtios r r r r 6 4 ( )( ) ( ) Numericl solutio of equtios The Newto-Rphso itertio for solvig f( ) 0 : f( ) f ( ) Coics Prol Rectgulr Hyperol Stdrd Form y 4 y c Prmetric Form (t, t) ct, c t Foci (, 0) Not required Directrices Not required Mtri trsformtios Aticlockwise rottio through θ out O: cosθ siθ siθ cosθ Reflectio i the lie cos θ si θ y (tθ ) : si θ cos θ I FP, θ will e multiple of Edecel AS/A level Mthemtics Formule List: Further Pure Mthemtics FP Issue Septemer 009

36 Core Mthemtics C4 Cdidtes sittig C4 my lso require those formule listed uder Core Mthemtics C, C d C. Itegrtio ( costt) f() f( ) d sec k t k k t l sec cot l si cosec l cosec cot, l t( ) sec l sec t, l t( 4 π ) dv du u d uv v d d d Edecel AS/A level Mthemtics Formule List: Core Mthemtics C4 Issue Septemer 009 7

37 Core Mthemtics C Cdidtes sittig C my lso require those formule listed uder Core Mthemtics C d C. Logrithms d epoetils e l Trigoometric idetities si ( A ± B) si Acos B ± cos Asi B cos( A ± B) cos Acos B si Asi B t A ± t B t ( A ± B) ( A ± B ( k ) t A t B A B A B si A si B si cos A B A B si A si B cos si A B A B cos A cos B cos cos A B A B cos A cos B si si π ) Differetitio f() t k sec cot cosec f( ) g( ) f () k sec k sec t cosec cosec cot f ( )g( ) f( )g ( ) (g( )) 6 Edecel AS/A level Mthemtics Formule List: Core Mthemtics C Issue Septemer 009

38 Edecel AS/A level Mthemtics Formule List: Core Mthemtics C Issue Septemer Core Mthemtics C Cdidtes sittig C my lso require those formule listed uder Core Mthemtics C. Cosie rule c c cos A Biomil series ) ( r r r ( ) where )!!(! C r r r r < r r r, ( ) ( ) ( ) ( ) ( ) Logrithms d epoetils log log log Geometric series u r S r r ) ( S r for r < Numericl itegrtio The trpezium rule: y d h{(y 0 y ) (y y... y )}, where h

39 Core Mthemtics C Mesurtio Surfce re of sphere 4π r Are of curved surfce of coe π r slt height Arithmetic series u ( )d S ( l) [ ( )d] 4 Edecel AS/A level Mthemtics Formule List: Core Mthemtics C Issue Septemer 009

PhysicsAndMathsTutor.com

PhysicsAndMathsTutor.com PhysicsAdMthsTuto.com PhysicsAdMthsTuto.com Jue 009 7. () Sketch the gph of y, whee >, showig the coodites of the poits whee the gph meets the es. () Leve lk () Solve, >. (c) Fid the set of vlues of fo