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6. The hpeol H hs equtio, whee e costts. The lie L hs equtio m c, whee m c e costts. Leve lk () Give tht L H meet, show tht the -cooites of the poits of itesectio e the oots of the equtio ( m ) mc( c ) 0 () Hece, give tht L is tget to H, () show tht m c. The hpeol H hs equtio. 5 6 (c) Fi the equtios of the tgets to H which pss though the poit (, 4). () (7) 6 *M545A068*
Questio 6 cotiue Leve lk *M545A078* 7 Tu ove
. The lie 8 is iecti of the ellipse with equtio, > 0, >0, the poit (, 0) is the coespoig focus. Leve lk Fi the vlue of the vlue of. (5) *N589RA08*
8. The hpeol hs equtio. 6 4 Leve lk The lie l is the tget to t the poit (4sec, t). () Use clculus to show tht equtio of l is sit 4cost (5) The lie l psses though the oigi is pepeicul to l. The lies l l itesect t the poit. () Show tht, s vies, equtio of the locus of is ( ) 6 4 (8) 4 *N589RA048*
Questio 8 cotiue Leve lk Q8 (Totl mks) END TOTAL FOR PAPER: 75 MARKS 8 *N589RA088*
8. The hpeol H hs equtio () Use clculus to show tht the equtio of the tget to H t the poit ( cosh θ, sih θ ) m e witte i the fom Leve lk coshθ sihθ (4) The lie l is the tget to H t the poit ( cosh θ, sih θ ), θ. Give tht l meets the -is t the poit P, () fi, i tems of, the cooites of P. () The lie l is the tget to H t the poit (, 0). Give tht l l meet t the poit Q, (c) fi, i tems of,, the cooites of Q. () () Show tht, s vies, the locus of the mi-poit of PQ hs equtio (4 ) (6) 4 *P544A048*
Questio 8 cotiue Leve lk Q8 END (Totl 4 mks) TOTAL FOR PAPER: 75 MARKS 8 *P544A088*
. The hpeol H hs equtio Leve lk 6 9 Fi () the cooites of the foci of H, () the equtios of the iectices of H. () () *P40A0*
6. The ellipse E hs equtio Leve lk The lie l is tget to E t the poit P ( cos θ, si θ). () Usig clculus, show tht equtio fo l is cosθ siθ (4) The cicle C hs equtio The lie l is tget to C t the poit Q ( cos θ, si θ). () Fi equtio fo the lie l. () Give tht l l meet t the poit R, (c) fi, i tems of,, the cooites of R. () () Fi the locus of R, s vies. () 8 *P40A08*
Questio 6 cotiue Leve lk *P40A09* 9 Tu ove
. The hpeol H hs foci t (5, 0) ( 5, 0) iectices with equtios Leve lk 9 9. 5 5 Fi ctesi equtio fo H. (7) *P4956A0*
. The poit P lies o the ellipse E with equtio Leve lk 6 9 N is the foot of the pepeicul fom poit P to the lie 8 M is the mipoit of PN. () Sketch the gph of the ellipse E, showig lso the lie 8 possile positio fo the lie PN. () () Fi equtio of the locus of M s P moves ou the ellipse. (4) (c) Show tht this locus is cicle stte its cete ius. () 8 *P4956A08*
Questio cotiue Leve lk *P4956A09* 9 Tu ove
. A hpeol H hs equtio Leve lk, whee is positive costt. 5 The foci of H e t the poits with cooites (, 0) (, 0). Fi () the vlue of the costt, () () the equtios of the iectices of H. () *P44A08*
7. The ellipse E hs equtio, > > 0 Leve lk The lie l is oml to E t poit P ( cos θ, si θ), 0 < θ< () Usig clculus, show tht equtio fo l is π si cos ( )sicos (5) The lie l meets the -is t A the -is t B. () Show tht the e of the tigle OAB, whee O is the oigi, m e witte s ksi, givig the vlue of the costt k i tems of. (4) (c) Fi, i tems of, the ect cooites of the poit P, fo which the e of the tigle OAB is mimum. () 0 *P44A008*
Questio 7 cotiue Leve lk *P44A08* Tu ove
0 Eecel AS/A level Mthemtics Fomule List: Futhe Pue Mthemtics FP Issue Septeme 009 Futhe Pue Mthemtics FP Cites sittig FP m lso equie those fomule liste ue Futhe Pue Mthemtics FP, Coe Mthemtics C C4. Vectos The esolve pt of i the iectio of is. The poit iviig AB i the tio μ λ : is μ λ λ μ Vecto pouct: ˆ si k j i θ ) ( ) ( ) ( c. c. c. c c c If A is the poit with positio vecto k j i the iectio vecto is give k j i, the the stight lie though A with iectio vecto hs ctesi equtio ) ( λ z The ple though A with oml vecto k j i hs ctesi equtio. z 0 whee The ple though o-collie poits A, B C hs vecto equtio c c μ λ μ λ μ λ ) ( ) ( ) ( The ple though the poit with positio vecto pllel to c hs equtio c t s The pepeicul istce of ),, ( γ β α fom 0 z is γ β α.
Hpeolic fuctios cosh sih sih sih cosh cosh cosh sih cosh l{ } ( ) sih l{ } th l ( < ) Coics Ellipse Pol Hpeol Rectgul Hpeol St Fom 4 c Pmetic Fom ( cosθ, siθ ) ( t, t) ( sec θ, t θ ) (± cosh θ, sih θ ) ct, c t Ecceticit e < ( e ) e e > e ( ) e Foci ( ± e, 0) (, 0) ( ± e, 0) (± c, ± c) Diectices ± e ± ± c e Asmptotes oe oe ± 0, 0 Eecel AS/A level Mthemtics Fomule List: Futhe Pue Mthemtics FP Issue Septeme 009
Diffeetitio f() f () csi ccos ct sih cosh cosh sih th sech sih cosh th Itegtio ( costt; > 0 whee elevt) f() f( ) sih cosh cosh sih th l cosh csi ct ( < ) cosh, l{ } sih, l l l { } th ( ( > ) < ) Eecel AS/A level Mthemtics Fomule List: Futhe Pue Mthemtics FP Issue Septeme 009
Eecel AS/A level Mthemtics Fomule List: Futhe Pue Mthemtics FP Issue Septeme 009 Ac legth s (ctesi cooites) t t t s (pmetic fom) Sufce e of evolutio S s π π t t t π
Futhe Pue Mthemtics FP Cites sittig FP m lso equie those fomule liste ue Coe Mthemtics C C. Summtios 6 4 ( )( ) ( ) Numeicl solutio of equtios The Newto-Rphso itetio fo solvig f( ) 0 : f( ) f ( ) Coics Pol Rectgul Hpeol St Fom 4 c Pmetic Fom (t, t) ct, c t Foci (, 0) Not equie Diectices Not equie Mti tsfomtios Aticlockwise ottio though θ out O: cosθ siθ siθ cosθ Reflectio i the lie cos θ si θ (tθ ) : si θ cos θ I FP, θ will e multiple of 45. 8 Eecel AS/A level Mthemtics Fomule List: Futhe Pue Mthemtics FP Issue Septeme 009
Coe Mthemtics C4 Cites sittig C4 m lso equie those fomule liste ue Coe Mthemtics C, C C. Itegtio ( costt) f() f( ) 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 ) v u u uv v Eecel AS/A level Mthemtics Fomule List: Coe Mthemtics C4 Issue Septeme 009 7
Coe Mthemtics C Cites sittig C m lso equie those fomule liste ue Coe Mthemtics C C. Logithms epoetils e l Tigoometic ietities 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 ) Diffeetitio f() t k sec cot cosec f( ) g( ) f () k sec k sec t cosec cosec cot f ( )g( ) f( )g ( ) (g( )) 6 Eecel AS/A level Mthemtics Fomule List: Coe Mthemtics C Issue Septeme 009
Eecel AS/A level Mthemtics Fomule List: Coe Mthemtics C Issue Septeme 009 5 Coe Mthemtics C Cites sittig C m lso equie those fomule liste ue Coe Mthemtics C. Cosie ule c c cos A Biomil seies ) ( ( ) whee )!!(! C <, ( ) ( ) ( ) ( ) ( ) Logithms epoetils log log log Geometic seies u S ) ( S fo < Numeicl itegtio The tpezium ule: h{( 0 ) (... )}, whee h
Coe Mthemtics C Mesutio Sufce e of sphee 4π Ae of cuve sufce of coe π slt height Aithmetic seies u ( ) S ( l) [ ( )] 4 Eecel AS/A level Mthemtics Fomule List: Coe Mthemtics C Issue Septeme 009