Suggested Solution for Pure Mathematics 2011 By Y.K. Ng (last update: 8/4/2011) Paper I. (b) (c)
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1 per I. Le α 7 d β 7. The α d β re he roos o he equio, such h α α, β β, --- <*> α β d αβ. For, α β For, α β α β αβ 66 The seme is rue or,. ssume Cosider, α β d α β y deiiio α α α α β or some posiive ieer. α β α β β α β β β y <*> The seme is lso rue or. By iducio, he seme is rue or ll posiive ieers.. c u, d d d e
2 . Le B C. The B C u, d, we hve ii N N, B, C / / / / 96 N N N N / / / / 96 i 96 / 96 / / / s N 5 96 / / / / e
3 . Mehod I cos θ cosθ θ Mehod II cos θ cos θ si θ si θ cos θ cos θ si θ cos θ si θ cos cos θ cos θ θ cos θ cos θ cos cos θ Re cis θ Re[ cisθ cos cos θ cos θ si θ θ cos θ cos cos θ cos θ ] θ cos θ 6 cos θ cos θ cos θ cos θ cos θ. θ 5. θ θ cos si si θ cos θ si θ, where < θ < cos θ si θ cos θ oly sice si θ > i u ii α, cos θ Q si θ cos α Q si α si θcos α cos θ si α cos θ cos α si θ si α cos θ si α si θ cos α cos θ α si θ α si θ α cos θ α where < θ < si α cos α si α cos α cos θ si α si θ cos α cos θ cos α si θ si α θ ± or Z θ α which represes relecio i he lie y. 6 ± θ 9 cos θ or Z cos, cos, cos e
4 6. i ii oe: > or,,, > u or,,, such h > d [... ] By ii,. s >. 7. i coeicie deermi D λ λ λ λ λ λ λ λ λ Homoeous sysem S hs o-rivil soluios D λ λ λ Whe λ, µ S: y z y z. y z Solvi y o he ove equios, z, y z Soluio se {,, : R } No-homoeous sysem S hs uique soluio D λ λ λ λ λ λ λ, d λ < or < λ < or < λ < or λ > λ D µ µ [ λ µ λ ] µ λ λ D λ µ µ λ λµ y µ e
5 D λ µ µ λµ z λ µ I cse o uique soluio, Cse iii λ : S ecomes y z y z µ y z µ Firs wo equios ive D D D y y D Dz z D µ [ λ µ λ ] λ λ λ µ λ λµ λ λ λ µ λµ λ λ λ µ z, y z For cosisece, µ z z z µ µ µ µ µ Cse i λ : S ecomes Cse ii λ : S ecomes y z µ y z z µ which hs o soluio s µ y z y z µ y z µ Firs wo equios ive µ z, y z For cosisece, µ z z z µ µ µ µ µ µ d λ µ d λ The ive sysem equls o S y pui µ d λ Soluios re For cosisece, z, y z Cse iii o ii y z z z z 9z z z or 9, y, z or, y, z e 5
6 e 6. i Suppose he coverse h is siulr. The, Bu < which leds o cordicio. ii iii,, d, d u, such h > By, B where, 5 Now, B
7 9. i C: α β > or ll Z. roo: is rivil. ssume α β > or some ieer * β α β > Q α, β > * α β α α β β α β α β β α > α β Q α, β > By iducio, α β > or ll Z. α α α α β β α α α β α β Q α β > β β α β β β β β α β Q α β > ii Now, β β... β α... α α { α } is decresi d ouded elow y β s { β } is icresi d ouded ove y α α, β B oh eis. β α β, β α β B B, B B iii α β α α β β α β Deie α β α β... α β Te i o oh sides, α β α β α α d y B α β α > ives α β or ll posiive ieers. The, α > β > s > y > y By, For, α, β α β y y y y y α oh eis. y α β y β α β B oly sice B β β β >, y oh eis. e 7
8 . Deie h p p. C: h or ll posiive ieers. roo: Whe, h p p ive ssume h or some posiive ieers. Cosider, h p p p p h y ssumpio The seme is lso rue or By iducio, h or ll posiive ieers. c i u,, u,, u,, ii Le Q or some polyomil Q. Q Q Q Q Q or ieers Q Q or ll R y Q C where C Q is cos. Furhermore, h p p is polyomil wih iie deree, d h hs iiiy my disic roos. h p p i u, Q Q or ieer ii C: or ll ieers. roo: u,, ssume or some ieers. By i, The seme is lso rue or By iducio, or ll ieers. Thereore, or ll posiive ieers. or ll R y e
9 . Im z z Im cisθ cis θ si θ si θ si θ cos θ iii For where is posiive ieer. si θ or θ or z cis, cis,,, cos θ ± where Z cis or cis i or i i z i, z i, ω cis cis ω ii Le where is posiive ieer. r ω S S ω ω oe ω ω r ω ω z z cis cis cis iv S S r r r r ω ω ω ω ω For where is posiive ieer. S r r ω ω ω ω r r S ω ω ω ω ω ω m 9 S S S 669 S 67 S 67 S m ω ω Cosider m m m ω ω 9 m m Which hs o soluio or ieer m. v { S S, S } {,, ω }, S S S For is eve, For is odd, ω ω ω ω, should e muliple o. 6N where N is posiive ieer ω ω ω, hs o soluio i sice ω ω 6N where N is posiive ieer --- Ed o Soluios o per I --- m m e 9
10 per II. si 5 cos e y 6 e e 5 e s si 5 cos 5 cos 5si is o diereile c For >, he ucio hs o sympoe. For <, ovious here is o vericl sympoe. Le m y m c e he o-vericl sympoe. e c [ m] he oly sympoe is y 6. 6 e 6 e 6 O c i ii y 5 5 O 5 5 e
11 . l l 5 ' 5 5 ' Mehod I Mehod II 5 [ 5 ' ] [ ] Show y iducio. 5 5 C C u, or. Furher, d 5 [ 5] ' ' 6 5 The, 5 C d 9 d d9 9 [ d 9 C 9 ] [ ] e
12 5. 5 si θ, d cos θdθ 6. u 6sec θ, y θ 7 d d y 6 sec lies o H. θ θ sec θ θ 5 d dy i y, d dy d 6 sec θ θ si θ s < θ < cos θ cos θd θ cos θ dθ θ si θ C θ si θ cos θ C si θ L: y θ 6sec θ u y, sec θ ; pu, y 5 θ -iercep sec θ, y-iercep 5 θ ii 5 re sec θ 5 θ 5 sec θ θ s < θ < Volume si si C C sec θ θ si si θ, si θ cos θ θ si θ si θ si θ θ 6 7 y d 6 sec θ, θ, 7 7 d si e
13 7. ', i R ii > iii < < c mi p.,, l 9 d " oi o ileio, d,, l d, l e l ives or re 6 l d 6 l l d [ l ] 6 l d d d y, l, l O, l 9 d [ l ] d e
14 . i d si cos cos cos si si d si d ii si d si d si d d d d sec d u d u pu u d c cos cos u u l e l e d d u l cos d e cos [l e l e ] d cos cos d l e d l e d cos d d cos l e si cos cos l e d cos d cos cos si cos d cos si d d y c, sei cos cos si such h y, sei cos si such h d u du d cos si d si e
15 e 5 si cos d si si si cos d 9. i d d df d d d d d d [ ] d [ ] d d C: or d lies ewee d. roo: I For < <, d Q oh re icresi II For < <, d Q oh re icresi For >, d d df For, d d df For <, d d df Les o F F ii F F, d d d d d d
16 iii, he seme is rivil. ssume d d d d d or some posiive ieer.. Noe h u s s s s s > d u s s > u u i ' ξ s u where u < ξ < ' u Q ' is decresi d d u u u ' ξ where < ξ < u y ssumpio d oi h oh ierls re posiive s > d y ii, such h ' ' d By iducio, he seme is rue or ll posiive ieers. d 5 5 h u d u h pu 5u 5 h 5u du 5u 5 h du y d oi h h is icresi coiuous ucio d h > 5 h d pu 5u ii u ' u s u s u ' u Q ' is decresi u y i [ u] s[ u ] Q s,, > s s u u s Cse h : LS h p q h p q h p q h ph q RS Cse Wihou loss o eerliy, we ssume h < : Se l or ll R such h is wice diereile d " or ll R By ii, p h q ph q p l h q l l ph q l h h p p q q l ph q ph q y e is icresi e 6
17 c, LS λ λ s λ. ssume he seme is rue or some posiive ieer. Cosider, λ λ λ λ λ... where λ λ λ λ λ λ λ λ λ... λ λ λ λ λ.... : L: i dy dy, y, d d y oe s s s S s,s sy s ss s, sy s L : sy s : y λ λ λ λ λ λ λ λ λ... λ λ λ y ssumpio d oice h λ λ... λ λ λ λ λ... λ λ λ λ λ λ λ λ λ λ y d oice h λ λ λ λ λ λ... λ λ λ The seme is lso rue or By iducio, he seme is rue or ll posiive ieers. y s y s hs roos α d β. α β α β sum o roos s s αβ 6 α β 6 produc o roos s ii Noe: αβ >, α, β Le : m e he slope o he lie L. y dy, y d dy m, m d α α dy d B β β 6 s s m m θ m m m m m m m m α β αβ αβ αβ α β αβ αβ e 7
18 s s s s s s s > θ s s s s y.m. G.M. iii L : L : rees vlue o θ is αy α αα α βy β α β β αβ α β αβ s α α β β α α β α β y α β s α α β β C s, s Le C, y s The,. Eii s, we hve y 6 y s --- Ed o Soluios o per II --- e
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