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1 PhysicsAMthsTuto.com

2 . M Leve lk 6 () Show tht 7 is eigevlue of the mti M fi the othe two eigevlues of M. (5) () Fi eigevecto coespoig to the eigevlue 7. *M545A068* (4)

3 Questio cotiue Leve lk *M545A078* 7 Tu ove

4 6. 0 M 0, whee is costt. k 0 Leve lk 6 Give tht is eigevecto of M, 6 () fi the eigevlue of M coespoig to () show tht k, (c) show tht M hs ectly two eigevlues. 6, 6 () () (4) A tsfomtio T : is epesete y M. The tsfomtio mps the lie l, with ctesi equtios the lie l. y z 4, oto () Tkig k, fi ctesi equtios of l. (5) 6 *N589RA068*

5 Questio 6 cotiue Leve lk *N589RA078* 7 Tu ove

6 7. The mti M is give y k M 0, k Leve lk () Show tht et M k. () Fi M, i tems of k. () (5) The stight lie l is mppe oto the stight lie l y the tsfomtio epesete y the mti 0. The equtio of l is ( ) 0, whee 4i j 7k 4i j k. (c) Fi vecto equtio fo the lie l. (5) 0 *P544A008*

7 Questio 7 cotiue Leve lk *P544A08* Tu ove

8 8. The mti M is give y 0 M Leve lk () Show tht 4 is eigevlue of M, fi the othe two eigevlues. () Fo the eigevlue 4, fi coespoig eigevecto. (5) () The stight lie l is mppe oto the stight lie l y the tsfomtio epesete y the mti M. The equtio of l is ( ) 0, whee i j k i j k. (c) Fi vecto equtio fo the lie l. (5) 6 *P40A06*

9 Questio 8 cotiue Leve lk Q8 (Totl mks) TOTAL FOR PAPER: 75 MARKS END *P40A0*

10 4. hs vecto equtio Leve lk s t, 0 whee s t e el pmetes. y the tsfomtio epesete y the mti T, whee 0 T 0 0 i the fom. p (9) *P4956A0*

11 Questio 4 cotiue Leve lk *P4956A0* Tu ove

12 6. It is give tht 0 is eigevecto of the mti A, whee 4 A 0 8 Leve lk e costts. () Fi the eigevlue of A coespoig to the eigevecto () Fi the vlues of.. 0 () () (c) Fi the othe eigevlues of A. (5) 0 *P4956A00*

13 Questio 6 cotiue Leve lk *P4956A0* Tu ove

14 5. The mti M is give y Leve lk M c, whee, c e costts. 0 () Give tht j k i k e two of the eigevectos of M, fi (i) the vlues of, c, (ii) the eigevlues which coespo to the two give eigevectos. (8) () The mti P is give y P 0, whee is costt, 0 Fi (i) the etemit of P i tems of, (ii) the mti P i tems of. (5) *P44A08*

15 Questio 5 cotiue Leve lk *P44A08* Tu ove

16 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.

17 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

18 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

19 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

20 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 Eecel AS/A level Mthemtics Fomule List: Futhe Pue Mthemtics FP Issue Septeme 009

21 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

22 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

23 Eecel AS/A level Mthemtics Fomule List: Coe Mthemtics C Issue Septeme 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

24 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