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. Leve lk A O c C B Figue The poits A, B C hve positio vectos, c espectively, eltive to fie oigi O, s show i Figue. It is give tht i j, i j k c i j k. Clculte () c, ().( c), (c) the e of tigle OBC, () () () () the volume of the tetheo OABC. () 4 *M545A048*
Questio cotiue Leve lk Q (Totl 8 mks) *M545A058* 5 Tu ove
Leve lk 7. The lies l l hve equtios α 0 4. 4 0 If the lies l l itesect, fi () the vlue of, (4) () equtio fo the ple cotiig the lies l l, givig you swe i the fom y cz 0, whee,, c e costts. (4) Fo othe vlues of, the lies l l o ot itesect e skew lies. Give tht, (c) fi the shotest istce etwee the lies l l. () 0 *M545A008*
Questio 7 cotiue Leve lk *M545A08* Tu ove
7. The ple hs vecto equtio Leve lk i k ( 4i j) (6i j k) () Fi equtio of i the fom. p, whee is vecto pepeicul to is costt. (5) The poit hs cooites (6,, 5). The lie psses though is pepeicul to. The lie itesects t the poit. () Show tht the cooites of e (,, ). (4) The poit lies o hs cooites (, 0, ). (c) Fi the pepeicul istce fom to the lie. Give you swe to sigifict figues. (5) 0 *N589RA008*
Questio 7 cotiue Leve lk *N589RA08* Tu ove
6. The ple P hs equtio 0 λ μ Leve lk () Fi vecto pepeicul to the ple P. () The lie l psses though the poit A (,, ) meets P t (,, ). The cute gle etwee the ple P the lie l is. () Fi to the eest egee. (4) (c) Fi the pepeicul istce fom A to the ple P. (4) 6 *P544A068*
Questio 6 cotiue Leve lk *P544A078* 7 Tu ove
. The positio vectos of the poits A, B C eltive to oigi O e ij k, 7i k 4i 4j espectively. Leve lk Fi () AC BC, () the e of tigle ABC, (c) equtio of the ple ABC i the fom. p (4) () () 6 *P40A06*
Questio cotiue Leve lk *P40A07* 7 Tu ove
. Two skew lies l l hve equtios Leve lk l : (i j k) (4i j k) l : (i 7j k) ( 4i 6j k) espectively, whee e el pmetes. () Fi vecto i the iectio of the commo pepeicul to l l () () Fi the shotest istce etwee these two lies. (5) 4 *P4956A04*
Questio cotiue Leve lk *P4956A05* 5 Tu ove
8. The ple hs vecto equtio Leve lk.(i 4j k) 5 () Fi the pepeicul istce fom the poit (6,, ) to the ple () The ple hs vecto equtio (i j 5k) (i j k), whee e scl pmetes. () Fi the cute gle etwee givig you swe to the eest egee. (5) (c) Fi equtio of the lie of itesectio of the two ples i the fom, whee e costt vectos. (6) 4 *P44A048*
Questio 8 cotiue Leve lk Q8 (Totl 4 mks) TOTAL FOR PAPER: 75 MARKS END 8 *P44A088*
0 Eecel AS/A level Mthemtics Fomule List: Futhe Pue Mthemtics FP Issue Septeme 009 Futhe Pue Mthemtics FP Cites sittig FP my 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 y k j i, the the stight lie though A with iectio vecto hs ctesi equtio ) ( λ z y The ple though A with oml vecto k j i hs ctesi equtio. z y 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 y is γ β α.
Hypeolic fuctios cosh sih sih sih cosh cosh cosh sih cosh l{ } ( ) sih l{ } th l ( < ) Coics Ellipse Pol Hypeol Rectgul Hypeol St Fom y y y 4 y c Pmetic Fom ( cosθ, siθ ) ( t, t) ( sec θ, t θ ) (± cosh θ, sih θ ) ct, c t Ecceticity e < ( e ) e e > e ( ) e Foci ( ± e, 0) (, 0) ( ± e, 0) (± c, ± c) Diectices ± e ± y ± c e Asymptotes oe oe y ± 0, y 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 y s (ctesi cooites) t t y t s (pmetic fom) Sufce e of evolutio S y s π y y π t t y t y π
Futhe Pue Mthemtics FP Cites sittig FP my 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 Hypeol St Fom y 4 y 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 y (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 my 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 my 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 my 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: y h{(y 0 y ) (y y... y )}, 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