( ) ( ) Chapter 1 Exercise 1A. x 3. 1 a x. + d. 1 1 e. 2 a. x x 2. 2 a. + 3 x. 3 2x. x 1. 3 a. 4 a. Exercise 1C. x + x + 3. Exercise 1B.

Transcription

1 answrs Chaptr Ers A a g a a a 6 h ( + ) Ers B a g h j a a ( + ) Ers C a

2 a Ers D a a a Ers E a a a a + + Ers F + + ( ) a v v 6 6. Th ar th sam aus 6 6!!! an 6 6! so,!! nspton, th nomnators ar th sam.

3 answrs a mor sltons a C + C + C ts 6 ts 0 ts 7 ts Ers G a 0, 0, 9, 9 n n r n r n n r n r a n n n 7 n n n g n h n 7 a n n 8 n a n n n n a n n + n! n! +! n!! n! + ( ) nn nn n!! nn ( ) nn + ( n )!! nn ( ) + nn ( ) ( n )! nn ( ) [ + ( n ) ]! ( n + ) n( n )! ( n + ) n( n ) ( n )! ( n )! ( n + )! n +! ( n )! - Stunt s own answrs, ut shoul ollow smlar stps to Q part a gvn aov. Ers H a a + a + 6a + a + a a + a m + 6m 6m + 8m 6a + 96a + 6a + 6a g h + 0 g + 60g + 8g a m m + + m m 6 z + 6z + z z z z a 0 r r r 6 70 r r 8 8 r r r r r

4 answrs 7 r ( ) 60 r r r 8 8 r r r r a a a Ers I a ; ; ; ; ; ; g ; 7986 a ; ; Chaptr rvw a ( + ) ( + ) g + a n 7 n n Stunt s own answr, ut shoul ollow smlar stps to Ers G Q. a a + a + 6a + a a a r r r r r ; 0 6 r r r r ( ) ; 0 0 9

5 ANSWERS Chaptr Ers A a sn os sn + os sn 6( + ) g 0 h 9 ( + ) sn a ( 7) (8 7) os ( ) sn ( + )sn ( + ) g h os sn ( ) sn os or sn 6 os or sn tansn sn or tan os os a os (os )sn ( + ) ( + + ) sn (os ) sn os (sn ) os 6sn os 6 os (sn ) sn (sn ) os g ( + sn ) ( + os ) h ( + ) sn ( + + ) + + 6os( + ) sn ( + ) ( + )sn( + ) os ( + ) os(os )sn sn (os ) os( + ) sn ( + ) 7 os Ers B a + sn g 6( + ) h a ln tan + sn os

6 ANSWERS Ers C a 0 os + sn or ( os + sn ) 0 ( ) ( ) os(os sn ) (os 6sn ( )) 7( + ) 6 ( + 7) ( ) ( ) (7 ) g ( + ln ) h os sn ( + ) (os + sn ) a ( ln ) os ( sn + os ) ( ln + ( ) ) sn ( os ) 6 ( os sn ) (7 + ) ( + ) g sn + os h (sn ( ) + os ( )) + ln Whn 0, Whn, grant Proo Ers D a os os sn ( os + sn ) os a a ln 6 ( ) ( ) ( ) os sn ( + ) ( ) ( + ) ( + )( ) ( )( ) ( ) ( + ) ( + )( ) ( ) ln7 (ln 7) ln ( ) 6 6 Whn, 9 ln 7 sn ln Ers E a ( + sn )

7 ANSWERS 680 ( ) ( ) or ( ) () ( + )( ) O () a Mnmum SP (, 9( ln 7)) a Mnmum SP (ln 0, 0( ln 0)) + 6, mnmum SP whn 0, rsng PI whn sn os, mamum SP whn, mnmum SP whn snθ sn θ, mamum SP whn 0 67, mnmum SP whn 0 77 alraton atr sons 8 ms - (lraton) 6 alraton atr hours 60 mph 7 a s(0) 0 mtrs, v(0) 9 ms - a(0) 6 ms - o s nstantanousl at rst atr sons Ers F a s tan s os os ot os ( ) os + ot + + s ( lntan ) a ln s os (tan ) tan( ) 6 os ( ) ot ( ) ot ( ln os + ) ln 8 tan s a ot os (ot( + 7)) os ( + 7) tan ln s tan ' ( + )(s tan os ) (s + ot ) ( + ) s ( + os ) Ers G a 9 + ( ) 6

8 ANSWERS os g tan + + or ( ( + )tan + ) + h ( + ) ( tan ) ( + )tan whn, grant 7 whn t, rat o hang 0 whn, quaton o tangnt: Ers H a g + + tan s h at P (, ), quaton o tangnt: + at (, ), quaton o tangnt: + 0 at (, ), quaton o tangnt: 8 at T (, 7), quaton o tangnt: + Ers I a + ( + ) ( + ) ( ) + ( + ) + + a, Proo Mamum SP (, ) Ers J a Implt Eplt Implt Eplt Implt Eplt ln6 ln6 ln ln ln ln Implt ( ln ) Eplt ln ( ln ) Implt (sn + lnos ) sn Eplt (sn + lnos ) Proo + +

9 answrs whn, quaton o tangnt: 9 9 SP whn 9 0 Ers K t a t + ost + snt ost snt t + t + t t SPs (6, 6); (, 6) whn t, grant Proo whn t 0, quaton o tangnt: a SPs (0, ); (0, ) t, mn SP (0, ); ( t + ) ma SP (0, ) 7 a SP (, ) tt ( + ) ( t )( t + ), 8t ( t + ) mn SP 8 SPs (0 88, 78) an (0 960, 6 708) Ers L whn, quaton o th tangnt: 69 whn t, rat o hang whn θ, statonar pont mnmum TP whn, quaton o th tangnt: 9 whn t, grant o th tangnt 6 a t mn SP whn t t + 7 whn t 9, alraton A( 8 9 ) ms 8 whn, grant o th tangnt 6 9 whn t 0 sons, alraton, A 0 ms 0 a Gratst F whn, ul n km/ltr Last F whn, ul n.9 km/ltr v 00 km/h; v 60 km/h whn t s, rat o hang 808 Vs whn t 0 s, rat o hang 9 Vs Chaptr rvw a ot + (os sn ) (sn os ) ( + ) ( + ) os ( ot ) g os 6 ln 6 h (In + ) ( + ) ln sn sn os os os ln j s ()+ ( tan() + )

10 ANSWERS k s + ln tan + + ln l ot (ot tan ( ) tan os tan ( ) s ot ) 0 ( ) 9 6 8, mn SP whn 00,, ma SP whn 7 (, 6) a atr t 6 sons, v ms alraton 0 ms atr sons an 6 6 at (, ) ( ( 6 ) at (, ) 7 Proo 8 at (, ) an 9 Proo 0 otθ os θ )

11 ANSWERS Chaptr Ers A 7 a ( ) sn + + g os h ( 8t + ) + 6 ( ) j 6 ( ) k os( θ ) l sn( ) a sn ( ) 7 ( + ) a + ( + ) a 9 6 a Ers B a + sn sn + sn q q + os os q a 0 sn ) Ers C a ln ln + + ln 7 g + + h + ln j + k ln + C a.8 (.p.). (.p.) (.p.) ln (.p.) Ers D a tan + tan + 6 8tan +

12 answrs tan + tan + 9 tan + g tan + h os + tan + a (.p.) (.p.) Ers E a sn + sn + sn + tan tan + a 0 67 (.p.) (.p.) 0 0 (.p.) Ers F a ln ln 0 6 ln ln ln ln ln ln( ) + + ln + a 89 (.p.) 7ln + Ers G a sn ( + ) + os + ln + + sn ln os + sn + sn + + sn tan ot + Ers H a ln + + ln os ln + + ( ) +

13 ANSWERS + a ( ) + 8 ( + ) + a ( ) + ( 8) + 6 ( + ) + a ( ln ) + ln(ln ) + sn( ln ) + a ( ln ) + 6 ( ) ( + ) ( + ) + ( ) + 6 ( + 7) os Ers I (.p.) In ln Ers J a sn sn + os os os + a sn sn sn a + ln os + os s s + lnos + Ers K sn os tan + lnos + ( )tan + lnos sn + + Ers L (sn ( ) + os ) + 6 ( sn( ) + os ) + 6 ( ) os sn + (( ) 6 )sn ( 8 )os 8os + 6 ( + ) + 8 Ers M ( + ) + 7 sn sn os os ( sn os ) + ( os + sn) +

14 answrs + sn os sn + ( ossn snos) + 6 snos snos + Ers N A ln unts A 6( ln ) unts A 6 8 unts A unts A 0 unts 6 V unts 7 V m > m, so spaton mts th rqurmnts. 8 V 0 unts 9 a O ( + ) 0 V ltrs 0 V rato ull V unts unts a st () t sn t +. unts s 7 8 s ( 0) n th postv rton Chaptr rvw ln + + ln + + ln ln ( + 6 tan( ) ln os 0 ) 6 tan( + ) sn 0 6 tan (8) 9 09 ln ( + )( + ) ln ln ln tan + 9 a V 6 m Proo V 67 unts

15 ANSWERS Chaptr Compl numrs Ers A a a g + h 7 7 a g h a ± 0, ± ± ± g ± ± h ± 79 ± a,,,, 6, 7, 8, 9 n n n, 8,,, a n, 7,,, a 6 a a Ers B a g 7 h + 7 a a a a Stunt s own rt proo 8 a All ompl numrs wth R(z) 0 z + ; w a +, +, +,, + Ers C h lm a R 6 O 6 7 g 6

16 answrs a z plott at (, ) an z plott at (, ) z plott at (, ) an z plott at (, ) z s a rlton n th -as o z a z, q 0 97 z, q 0 6 z, q 6 z, q 0 z, q 0 78 z, q g z, q 6 h z, q a z 6, q 0 z, q z, q a os + sn + os + sn os sn os + os 6 + sn + sn 6 os sn + 6 a os + sn os + sn os sn os + sn os + sn os + sn g os + sn h 0 os sn z ± Ers D 7 a 6 7 os + sn 8 os + sn os sn + os sn + os + sn os + sn g os + sn h 6 os sn + os + sn j os 9 + sn

17 ANSWERS a g h a, lm z z z z z z z O 6 R z z z z (os + sn ) z z + os sn z z + os sn Th poston vtor o z, z an z hav n rotat n an antlokws rton Stunt s own nvstgaton Stunt s own nvstgaton a os + sn os + sn os + sn os + sn 6 6 n n os + sn 6 6 Ers E a os + sn os + sn 0 os + sn 000 os + sn a 6 a a a os + sn os + sn os + sn 6 6 os + sn os + sn os + sn a Stunt s own rt proo Stunt s own rt proo 7 a os q sn q + sn q os q os q + sn q os q os q sn q sn q sn q os q 8 a os q osq sn q + os q snq sn q os q + sn q os q os q os q sn q sn q os q sn q sn q sn q sn q sn q 9 os q 6os q 0os q + os q 0 os 7q 6os 7 q os q + 6os q 7os q a Smlar to Q8,9,0, Stunt s own rt proo tanθ tanθ tan θ 6tan θ + tan θ

18 ANSWERS Ers F a os sn ; os 7 9 sn 7 + ; 9 os sn Solutons v a rl o raus nto qual stors raans apart. os + sn ; os + sn ; 7 7 os sn + Solutons v a rl o raus nto qual stors raans apart. os + sn ; os + sn ; os + sn ; 8 8 os sn + ; 8 8 os sn + Solutons v a rl o raus nto qual stors raans apart. os + sn ; os + sn ; os sn + Solutons v a rl o raus nto qual stors raans apart. os + sn ; os sn Solutons v a rl o raus nto qual stors raans apart. os + sn ; 0 0 os 0 os 6 + sn ; 0 + sn ; 6 os sn ; os sn Solutons v a rl o raus nto qual stors raans apart. g os + sn ; os 6 + sn ; 6 os sn + ; os sn Solutons v a rl o raus nto qual stors raans apart. h 6 os 9 + sn ; os sn ; 6 os sn Solutons v a rl o raus 6 nto qual stors raans apart. a ; + ;,,,

19 ANSWERS ; os + sn ; os + sn ; os sn + ; os sn + a ; + ; ; + ; + ; ; ; a + ; + ; ; os + sn ; os + sn ; os + sn ; 0 0 os 7 sn + 7 ; 0 0 os sn ; ; os + sn ; 8 8 os 8 + sn ; os sn ; os sn os + sn ; os + sn ; os + sn ; os sn + ; os sn + + ; ; Ers G a ; + ; + ; + ; + ; + ; a ; + ; ; + ; ; + ; ; + ; ; ; + ; a ; ; ; + ; 6; + ; ; + ; ; + ; ; + ; a solutons ; + ; solutons ; + ; solutons ; ; + ; + ; ; + Ers H a Crl C(0, 0), raus unts Crl C(0, 0), raus unts Crl C(, 0), raus unts Crl C(0, ), raus unts Crl C(, ), raus unts

20 ANSWERS Crl C(, ), raus unts g Crl C,, raus unts h Crl C,, raus unts a Straght ln wth quaton Straght ln wth quaton Straght ln wth quaton Straght ln wth quaton a Crl C(0, 0), raus unts O Crl (, 0), raus unts O Crl (0, ), raus unts O Crl (, 0), raus unts O 6 7 a Straght ln wth quaton Straght ln wth quaton Straght ln wth quaton Straght ln wth quaton Chaptr rvw a , Whn a, Whn a, 0 z : Argan agram showng (, ) z : Argan agram showng (, ) os sn a sn 0sn + sn 7 os sn + os sn os sn + 8 ; +, 9 ±, ±

21 ANSWERS Chaptr Ers A a + tan + A whr A ln A 9 whr A ± 8 g ln + h A ( ), whr A a ln + or ln ( + ) ln t + ln os t( t) V P t + t + a A kt whn t hours, 6 atra a h kt + h 0 whn t 0 mnuts a n A kt k 96 (to.p.) t 8 wks (appro. wks 6 as) 6 a n A kt + 8 T 9 C (to s..) t 88 hours (appro. hour mnuts) 7 t 6 7 hours (appro. 6 hours mnuts) 8 a ln9 A, k t 9 as (to narst 0 a) Ers B + ( + ) ( + ) ( sn + ) ( os ) 6 ( ln + ) 7 ( + ) or + 8 ( os ) 9 sn 0 a Proo (Susttut u os ) ( + ) os a G k t ( ) k 0 (.p.) G(0) 09 m m, so th lam s just. at t, G 6 m Ers C a A + B 9 (A + B) A + B 7 (A sn + B os ) (A + B) (A os + B sn ) a sn + os ) ( ) Ers D a A + B (A + B) + + Asn Bos + (A + B) A 6 + B + os

22 ANSWERS ( ) ( ) 6 A + 6 B sn os + os sn a ( ) + os + sn + os ( sn + os ) + sn os Chaptr rvw a ± A( ) ln 6 + or ln 6 + tan ( + ) V t 9 ( + ) A ( +) whr A a m t km gnral soluton, m A kt t. months (to narst 0. months) a T 00 C T. mnuts (to narst 0. mnut) a ( + ) + kt A kt A 0 9t 00, 0 9 t whr t 0 00 t.6 as whn 0. 6 a (sn os + ) + ln ( os ) + os + 7 a (A sn +B os ) A + B 7 (A + B) 6 + (os 6 7 sn 6) sn os

23 answrs Chaptr 6 Ers 6A a,, g, 0, h, a 0, 79 6, 0 6 No vrtal asmptots a No vrtal asmptots, No vrtal asmptots g No vrtal asmptots h s a rpat root o th nomnator, so although + s a ommon ator o th numrator an nomnator, a ator o + rmans n th nomnator atr anllaton. a No vrtal asmptots Ers 6B a a 0 No horzontal asmptots a 0 a 9.8 ms (to.p.) ms v() 9. ms, whh s ar lss than 99% o ms. Th mol s not partularl aurat. Ers 6C a + a a Vrtal an horzontal Horzontal Vrtal an horzontal Vrtal an olqu Vrtal an olqu No asmptots a, 0, 0,,,, Ers 6D a loal mn (0, ), 9 8 loal ma loal ma,, loal mn, 0 9 loal ma (, 7), loal mn (, ) loal mn,, loal ma,, loal mn, 9 loal mn ( 0 79, 89) a pont o horzontal nlton (0, ) pont o horzontal nlton, loal ma (, 0 ), loal mn (0, 0)

24 answrs loal mn (, 0 ), pont o horzontal nlton (0, ) loal mn (0, 0), loal ma (, 0 ) loal mama at + k, 7 loal mnma at + k, or k,,, 0,,, Ers 6E a (0, ) (, 0) (, 7), (, ) No ponts o nlton (0, 0) a (, 0), (, 0), (0, 0), (, 0), (, 0) 7,,,,,,,,,,, 7,,,, 0 6,, 0, 6 6, 0, 7, 6 0, 0, 7, 6 0,, 6 0, 0,, 0, 0 () ( + ) whh nvr quals 0, so no ponts o nlton. Non horzontal pont o nlton at, ( 6,.6), (0, ), ( 6,.8) Ers 6F a ma valu, mn valu 9 ma valu 7, mn valu ma valu 80, mn valu 0 ma valu 6, mn valu 6 ma valu, mn valu 0 ma valu 8, mn valu 0 a ma valu, mn valu 0. ma valu 7 67, mn valu 7 ma valu 9, mn valu 0 7 ma valu 0, mn valu 89 ma valu 0, mn valu 0 ma valu 7, mn valu 0 a ma valu 6, mn valu 0 ma valu, mn valu ma valu, mn valu ma valu, mn valu 0 () tns to an as tns to 0 rom aov an low, rsptvl, so () has no mamum or mnmum valus on th gvn ntrval. Ers 6G a ( ) ( ) () g( ) g() h( ) os (( )) os ( ) os () h() r( t) 6( t) 6 ( t) + ( t) 6t 6 t + t r(t) ( ( t) ( t)) s( t) 6( t) ( t + t ) ( 6t) ( t t) ( 6t) ( t t ) s(t) 6t ( q ) 8( q )sn(( q)) 8q( sn q ) 8q sn q ) (q ) a ( ) q( ) ( ) + ( ) 7 7 ( + 7 ) q() h( q ) tan ( q) + sn ( q) tan q sn q (tan q + sn q ) h(q)

25 answrs s( t) ( t) (( t) ( t) ) t ( t t ) t (t t) s(t) ( ) ( ( ) ) ( ) ( + ) ( ) ( ) ( ( )) ( ) v( t) ( t) os ( t) t os t v(t) a o sum o two o untons nthr (0) 0 so not o, an ( ) 0 () so not vn vn sum o two vn untons (th raton s vn sn t s a prout o two o untons) vn sum o two vn untons (oth sn an os ar vn sn th ar prouts o two o untons an two vn untons, rsptvl) nthr (0) 0 so not o, an ( ) + () so not vn o sum o two o untons h( ) ( ) os ( ) ( ) os + ( os ) h() Th onstant unton 0 s th onl ral valu unton whh s oth vn an o. 6 a Lt () an g() vn untons. Thn ( + g)( ) ( ) + g( ) () + g() ( + g)(), so + g s vn. Lt () an g() o untons. Thn ( + g)( ) ( ) + g( ) () + ( g()) ( + g)(), so + g s o Lt () an g() vn untons. Thn ( g)( ) ( )g( ) ()g() ( g)(), so g s vn. Lt () an g() o untons. Thn ( g)( ) ( )g( ) ( ()) ( g()) ()g() ( g)(), so g s vn. Lt () an vn unton an g() an o unton. Thn ( g)( ) ( )g( ) ()( g()) ()g() ( g)(), so g s o. 7 a Ers 6H a, () s not n,, () not n at thr pont, th lmt as tns to rom low s, ut () () s ontnuous, th lmt as tns to rom low s, ut () 0, th lmt as tns to 0 rom aov os not st g () s ontnuous h, th lmt as tns to 0 rom low s, ut () Ers 6I a ( 0.77,.8) (0, ) (0.77, 0.6)

26 answrs (0, 0) (, 0) (0, 0.) (, 0) ( 0, ) (, ) (, ) (0, 0) (, 0) (0, 0) (, ) g (0, 0.) (, 0) (, 0) (.6, 0) (.6, 0)

27 answrs h Graph s sht lt unts thn own unts. (0, ) (., 0) (.6, 0) ( 0.76, 0) unton s unn whn 0 (, ) Ers 6J a Graph s sht up unts. Graph s omprss horzontall a ator o. (0, ) (0, 0) Graph s sht rght unts. ( + ) ( ( )). Th graph s sht rght unt, thn rlt n th as, thn rlt n th as an sal vrtall a ator o. (, 0) (0, ) (0, ) (, 0)

28 answrs Graph s sal vrtall a ator o, thn sht own unt. Not that ( ) ( 8 ), so th graph s sht rght 8 unts thn sal horzontall a ator o.. 0. ( 0.707, 0) (0.707, 0) (0, ) a Graph s sal vrtall a ator o, thn sht own unts. 6 7 Th graph s sal horzontall a ator o, thn sal vrtall a ator o, thn sht up unt. a Graph s rlt n th as (so no t sn os() s an vn unton) thn sht up unt. (0, ) (, 0) Graph s sht lt unts, thn rlt n th as. (, 0) (0, )

29 answrs (, ) (.7, 0) (0, ) (, ) (0, 0.68) g (0, ) h (, 0)

30 answrs j (0, 0) (, 0) 6 a Ers 6K a (, 0) (, 0) (, ) (, 0) ( 0., 0) (., 0) (, 0) ( 0., 0) (0., 0)

31 answrs (0, ) (, ) (, 0) (, ) a (0,.86) (, 0) (0, ) (, 0) (, 0) (.86, 0) (0, ) (0, ) (0, ) (.6, 0) (, 0) (0, ) (0,.6) (0, 0.69) (0.69, 0) (, 0)

32 answrs a (, ) a (0, ) (, 0)

33 answrs (0, ) (, ) (, 0.) (, 0.) (, 0) (, 0) (, ) (, ) (, 0) a (0, ) (, 0) (0, ) ( 0.68, 0) (, 0) (.68, 0)

34 answrs (0, ) (, 0 ) (0, 9) (, 0) (, 0) ( 0.667, 0.8) (, 0) (0, 0) g (, ) (, 0) (, 0) (0, 0)

35 answrs h 7 a (, ) (0, ) (, ), 0, 0 6 a,,, (, 0) (0, ), (, 0) (, 0) (0, ),,,,, 8 (, ) (, ) (0, ) (, ) (, ) (0, 0) (, 0) 9 a,,, (0, 0) (, 0) (, 0) (0, )

36 answrs For ah valu o, thr ar two possl valus o (.. () s not an njtv unton). Chaptr rvw a,, 0 a 0 (0, ) (, 0) a + (, 0) n + t + 0 (vrtal), + 7 (olqu) loal mn (, 6 7), pont o horzontal nlton (0, 0) 6 a loal ma (, 667), loal mn (, 9), pont o nlton 7 a ma valu, mn valu. ma valu 6, mn valu ma valu, mn valu 8 a Evn sum o two vn untons Nthr g( ) os( ) +, ut g() an g() (0, ) + (, 0) Evn prout o two o untons 9 a,

37 answrs 0 a a (0, ) (, 0) (0, 0) (0, 7) Graph s rlt n th as an sal vrtall a ator o. (, 0) (, ) (0, 0) (6, ) + (0, 0) (, 0) Graph s sht lt unts. (, 0)

38 answrs Graph s rlt n th as, thn sal vrtall a ator o, thn sht own 6 unts. Pro s oul, graph s rlt n th as, thn sal vrtall a ator o. a 7 (0, 6) (, 0) (0, 0.6) a (0, ) Graph s sht lt 6 unts. ( 8, 0) Pro s halv, thn graph s rlt n th as, thn sht up unts.. (0.68, 0)

39 answrs a (0, ) (0, 0) (, 0) (0, ) (0, ) (, 0) (, 0) (0, 6) g (0,.889) (, 0) (, 0) (, 0) (.9, 0) (.9, 0) (, 0)

40 answrs Chaptr 7 Ers 7a a a 8, Not an arthmt squn a, Not an arthmt squn a, 0 7 a, 6 6 g a, h Not an arthmt squn a n + n n + 90n + 0 n n + 0 a 9 a a a, a 0, 7 a, a, a 7, a a, th trm 9 a 8 9th trm 0,,, or,, 9 6 pn Ers 7B a g h 80 8 a 7 a 0 7 a a 7, a 8, a, n + n 0 8 u n 9 n a S n 0n n n or n a 60 S n n( + n) ars 6 a ars ponts 8 a 0 straws 0 trangls Ers 7C a a, r Not a gomtr squn (t s an arthmt squn) a 6, r a, r a, r a, r g Not a gomtr squn a a 800

41 ANSWERS a a a, r a 6, r a, r a 8, r st trm 9 th trm 0 Atr 8 ars 9 ars Da o tranng Ers 7D a (s) g 09 7 h 60 (s) a a 000, m grans ( s..) 0 6 mtrs Ers 7E a S os not st 6 79 S os not st 0 S os not st a a 0 r, S r, u a, r , 6,, 8, or, 0, 0 0, 6 a Ers 7F a a

42 answrs a < < < < a a ; s s tan s tan s tan, < Ers 7G a g a g h a Ers 7H a a ln ln

43 answrs Ers 7I a ( s.. ) g h a r + n r n r r n r 0 r r 6 r r + r 8 r r g r ( r + ) h r r r 0 r a Ers 7J a n ( n + ) n n( n + 7) n n n( 9 + n) a n( n + ) ; 06 n + n ; 9 n n; 977 n n ( ); p + p a n + n + n n + n os (n + ) os 8n + n + n 9 n n + n r + n + n n r r ( n + ) n + n n n +, 6 Smlar stps to Q Ers 7K a n( n + )( n + ) 6 n n n ( + + ) n n 6n ( + + ) n n + n + + n n n n n 9n 6 a n ( n + ) n ( n + ) + n n ( n + ) ( n + ) n n n ( + 7 n )( + + ) n n + 0 n + 7 n + 60

44 ANSWERS a a n( n + )( n + ) ; 80 6 n( n + )( n + )( n + ) ; 0 Chaptr rvw a u n n th trm a a ; 6 a a ; r a a r so S sts sn r < ; 6 6 a a a n( n + )( n + ) n( n + )( n + )( n ) 6

45 answrs Chaptr 8 Matrs Ers 8A a a,,,, a 6 7 a a g a Proo (stunt s own answrs) 8 9 a, 0 s, t Ers 8B a (7) (7) ( + + z) g j k a snθ 8 9 h l 0 6 AB BA

46 answrs a 9 A(BC) (AB)C ( AB) BA a,, a p r q s p r q s 6 a Proo (stunt s own answr) A A I 7 a A A 6I A 6A I 6 8 AB a BA 0 7 AB BA , 8 0 or or Ers 8C a 0 0 a t(ab) tatb 77 t(ba) t(ab) 77 t A t A a 8 a t(ab) tatb 0 t(ba) t(ab) 0 t A t A Ers 8D a a sngular matr

47 answrs Proo (stunt s own answrs) a,,, a t t + 6 t t t 6t 6 t t ± t 8 t 6 a, Proo (stunt s own answrs) A 0A 6I 7 a ± 6 A A, A 8 a m, n A A + I 9 A A A n Ers 8E a t p 0 p p 0 p n p 0 n p 0 t t, t + t 0 7 t A k 6 sngular matr a AB 8I,, z ,, z 0 0 Ers 8F A (, ), B (9, ), C (6, ), D (, ) 0 0 a

48 answrs a Proo (stunt s own answr) k a an 0 (0, ) Ers 8G a z z z 0 z,, z,, z + z z + z z a a 6 s, t s, t s Th sstm s 0 ll-onton as a small hang n onts gv a vr g hang n soluton. 8 B an C 7 Chaptr rvw a g h M a Proo (stunt s own answrs) P 0P 9I a p an q a 6 a + a a

49 answrs 6 7 a (, ) λ 6λ 8 z, λ 7( λ ), 7( λ ) whn l th sstm s nonsstnt an thr ar no solutons. 9 z,, 0 whn t th sstm s runant an th gnral 7z + z soluton s z z,, + 0 Soluton to s 97, Soluton to s 00, 9 Ths shows th sstm s ll-onton as a small hang n ont las to a larg hang n soluton.

50 ANSWERS Chaptr 9 Ers 9A a : : : 8 : : : 6 a,, 8, 8, 8 6, 6, 7 6 Proo (stunt s own answrs) Ers 9B a (a ) a a squar unts squar unts u ± 6( j k) 0 Ers 9C a V 6 a ( ) ( a) (a ) 7 Ers 9D a + z r + t r + t 7 a + z 6 No Ys Ys r + t 0 a z 7 + z z z 0 7 Ers 9E (9,, 0) 8 6 or 0 98 raans a POI (,, ); angl or 0 9 raans o not ntrst POI (, 6, ); angl 70 or 97 raans a (,, 7) (,, ) (, 6, 8); angl 0 89 or 0 9 raans Ers 9F +z a r + j + k + s( j + k) + t( 7j + k) s + t, s 7t, z + s+t, 8, z It ls on th plan wth s an t Proo (stunt s own answrs) a z 9 + z

51 answrs z a 6 6 z 6 Ers 9G a (, 0, ) 7 (, 0, ) 7 7 (,, ) 0 9 (,, 0) 06 (, 9, ) 78 (, 8, ) No ntrston so paralll t an an valu so th ln ls on th plan n n plans ar paralll a + z z z (,, ) 7 a z t, + 7t, + t Plans an ntrst along 8t, + t, z 7t. Plans an ntrst along 8t, + t, z 7t Plans an ar paralll. Chaptr rvw z 6 + z + 7z z 6 7 a t +, t +, z t + + z (, 8, ) 8 No solutons or a an nnt solutons or a, th ln woul l on th plan. 9 a + z z 6 0 a, POI (,, ) (, 0, )

52 ANSWERS Chaptr 0 Ers 0A a quotnt, rmanr q, r 0 q, r q 7, r 7 q, r q, r 0 Ers 0B a a a Ers 0C a a a a a Th rsults shoul th sam or ah mtho (atr lung an runant zros rom th start o th nal nar numr). Both mthos gv 00. No. Th mtho works whn th ntal as s th squar o th as ou ar hangng to. Lt a numr prss n as, an wrt (a n a n- a a 0 ), whr th a ar th gts o. So ah a s an ntgr twn 0 an, nlusv. Rplang ah gt a ts nar onvrson gvs a nar prsson q k r k q k- r k- q r q 0 r 0, whr q an r ar th quotnt an rmanr, rsptvl, whn a s v. W n to show that (q n rnq n- r n- q r q 0 r 0 ) (whr s th nar prsson or ). Sn a, Eulan vson a q + r or all, whr 0 q, r. Thror, usng as 0, 0 n (q n + r n ) + + (q + r ) + (q 0 + r 0 ) n (q n + r n ) + + (q + r ) + (q 0 + r 0 ) n+ q n + n r n q + r + q 0 + r 0 (q n r n q r q 0 r 0 ) as rqur. Th mtho osn t work, ut wll work w us thr nar gts whn onvrtng ah otal gt. Ths works sn 8. Ers 0D a g 90 0 h a 6 6D 6 E F6 6 99B 6

53 ANSWERS g 7 6 h D66 6 DC 6 a 9 6 FF Ers 0E a 8 a 8 a No (g(, 69) 8). No (g(, 69) ). Ys. s th largst ntgr whh vs vr numr n th gvn st. a a Th g s n all ass. m vs a an m vs a +, so w ma wrt a rm an a + sm or som ntgrs r an s. Thn (a + ) a sm rm (s r)m, an sn s r s an ntgr ths mpls that m vs. Us proo nuton. Whn k, g(u, u ) g(, ). Assum tru or k r, so g(u r, u r + ). Lt g(u r +, u r + ). W must show that. Clarl vs u r +, an sn u r + u r + + u r, vs u r + + u r. B part, must also v u r. Sn vs oth u r an u r +, must v g(u r, u r + ) whh quals th nuton hpothss. Thus. So th statmnt hols or k r t hols or k r +, an sn t hols or k, nuton t hols or all k. Conjtur: g(u k, u k + ) or all k. Us proo nuton. Whn k, g(u, u ) g(, ). Assum tru or k r, so g(u r, u r + ). Lt g(u r +, u r + ). W must show that. Clarl vs u, an sn u u + u, r + r + r + r + vs u r + + u r +. B part, must also v u r +. Sn vs oth u r + an u r +, must v g(u r +, u ) ( part ). Thus. r + So th statmnt hols or k r t hols or k r +, an sn t hols or k, nuton t hols or all k. Ers 0F a Sn m vs a, a rm or som ntgr r. Thror ka k(rm) (kr)m, an sn kr s an ntgr ths shows that m vs ka. Sn m vs oth a an, w ma wrt a rm an sm or ntgrs r an s. Thn a + rm + sm (r + s)m, an sn r + s s an ntgr ths shows that m vs a +. Sn m vs oth a an, w ma wrt a rm an sm or ntgrs r an s. Thn a rm sm (r s)m, an sn r s s an ntgr ths shows that m vs a. Ers 0G a p, q p, q p, q p, q p, q 6 p, q 6 a

54 ANSWERS a Sn g(a, ), must v oth a an. Thus w ma wrt a m an n or ntgrs m an n. Hn a + m + n (m + n). Sn m + n s an ntgr, ths shows that vs. Sn g(a, ), th tn Eulan algorthm thr st ntgrs m an n suh that am + n. Suppos that k or som ntgr k. Thn k k(am + n) (km)a + (kn), an takng km an kn gvs th rqur ntgr solutons. A lnar Dophantn quaton a + has ntgr solutons an an onl s a multpl o g(a, ). a 7, 6, 8, 8 Ers 0H a g 9, lm 9 g 8, lm 9008 g 88, lm 060 g, lm g 0, lm 000 g, lm Usng th gvn roots w hav () ( + )( )( )( 6). I a s an ntgr suh that (a), thn sn a +, a, a an a 6 ar stnt ntgrs ths mpls that atorss as a prout o our stnt ntgrs. Howvr, th Funamntal Thorm o Arthmt, th onl atorsatons o nto stnt ntgrs ar an ( ). Ths ontraton shows that thr s no ntgr a suh that (a). a Th quaton a + 7 has ntgr solutons an an onl a s oprm to 7. Sn 7 s prm, a s oprm to 7 an onl a s not a multpl o 7. Thr ar ourtn multpls o 7 twn an 00. Thror thr ar valus o a or whh th quaton has ntgr solutons. Argung as n part a, ut lung multpls o an (sn 6 ), thr ar valus o or whh th quaton has ntgr solutons. Chaptr rvw a F 6 a a g(8, 0) a,,,,,,, 8, 7, Whn k w hav u, whh s o. Assum tru or k r, so u r s o. Thn u r + u r + u r. u r s o an u r s vn, an th sum o an o ntgr wth an vn ntgr s o. Thror u r + s o. So th statmnt hols or k r t hols or k r +, an sn t hols or k nuton t hols or all k. Whn k, g(u, u ) g(, ). Assum tru or k r, so g(u r, u r + ). Lt g(u r +, u r + ). W must show that. Clarl vs u r +, an sn u u + u, vs r + r + r u r + + u r. Hn must also v u r. B part, u r + an u r + ar o, so must o (sn vs oth). Thror vs u r t

55 ANSWERS must v u r. Sn vs oth u r an u r +, must v g(u r, u r + ) whh quals th nuton hpothss. Thus. So th statmnt hols or k r t hols or k r +, an sn t hols or k nuton t hols or all k. g(u k, u k + ) or k 0. Whn k, g(u, u ) g(, ). Assum tru or k r, so g(u r, u r + ). Lt g(u r +, u r + ). W must show that. Clarl vs u r +, an sn u r + u r + + u r +, vs u r + + u. Hn must also v u. B r + part, g(u, u r ), so +. r + r + must v, an hn. So th statmnt hols or k r t hols or k r +, an sn t hols or k nuton t hols or all k. No. As a ountr-ampl, u, u 6, an g(, ) g(0, 796) a a 7,, g, lm 67 a 7, 7, g, lm 808 a,, g 8, lm 0 a,, g, lm 7 a, 7, g 88, lm 960 a 7, 9, g, lm 07

56 ANSWERS Chaptr Ers A a Estntal Unvrsal Unvrsal Estntal Unvrsal a a : a > a, a >, 0. < A M ( ) : t(a) q : q Ers B a > > a 8 a 6 n n () () g φ osφ h φ 0 tanφ 0 a P Q P Q P Q No mplatons. a P Q P Q P Q P Q P Q a No mplatons Q P P Q P Q Q P 6 a I a s an ntgr, thn a For a matr A, A s nvrtl thn ta 0. vs vs. vs onl vs. I B s a squar matr, thn B sts. () 0 sunt or (). g a 0 mpls a 0 or all. h Thr sts a ratonal numr whh vs onl thr sts an ntgr whh vs. 7 a Dav has travll to Europ Dav has travll to Ital. Th oo ng rut s nssar or th oo to a anana. Laura plas amnton onl Laura plas a sport. Ham havng a son s sunt or Ham to a athr. Bran havng a jo mpls Bran works or th loal ounl. Ers C a < z S : z r, rt A M ( ) : A 0 g, : h p : q, pq 0 a or an a an a 0 p : pq 0 or r q.6 or (. an 0) ( s ratonal or ) an ( s ratonal or ) Ers D a 0 0 whh s not postv 0 0 < 0 ( ) < Thr ar no ral numrs suh that 0.

57 ANSWERS a n ( ) 6 ( ) Th unton () + s an ampl wth no ral roots. Lt g(). Thn g() ( ) g( ), ut. Th matr A 0 0 s non-zro ut 0 not nvrtl. Ers E Whn n w hav so th rsult hols. Assum tru or n k. Whn n k + w hav: (k ) + ((k + ) ) k + ((k + ) ) ( n hp) k + k + (k + ) so th rsult hols or n k +. Sn t hols or n, nuton th rsult hols or all n. Whn n w hav 7 >, so th rsult hols. Assum tru or n k. Whn n k + w hav: k + k > k ( n hp) k > (k + ) (sn k > k + or all k ), so th rsult hols or n k +. Sn t hols or n, nuton th rsult hols or all ntgrs gratr than. Whn n w hav + 7 0, so th rsult hols. Assum tru or n k. Whn n k + w hav: (k + ) + 7(k + ) (k + k + 0k + 0k + k + ) + 7k + 7 (k + 7k) + k + 0k + 0k + k + 0 Th rakt trm s a multpl o th nutv hpothss, an all th othr trms n th sum ar larl multpls o. Thror th whol rght-han s s a multpl o, an th rsult hols or n k +. Sn t hols or n, nuton th rsult hols or all n. Whn n w hav , so th rsult hols. Assum tru or n k. Whn n k + w hav: (k +) + + (k+) k+ + k ( k+ + k ) k + k 9( k+ + k ) k (9 ) 9( k+ + k ) k 7 Sn k+ + k s a multpl o 7 th nutv hpothss, th whol rght-han s s a multpl o 7, an th rsult hols or n k +. Sn t hols or n, nuton th rsult hols or all n. Whn n w hav 0 so 0 th rsult hols. Assum tru or n k. Whn n k + w hav: k k k k + 0 k ( n hp) so th rsult hols or n k +. Sn t hols or n, nuton th rsult hols or all n.

58 ANSWERS a 6 Whn n w hav 0 a, so th rsult hols. 0 Assum tru or n k. Whn n k + w hav: k+ a a a k k a a 0 ( n hp) 0 k k + a 0 a a k k a( ) + ( + ) k+ k a 0 k+ k a + 0 k + so th rsult hols or n k +. Sn t hols or n, nuton th rsult hols or all n. 7 Whn n w hav (os q + sn q) os( q) + sn( q), so th rsult hols. Assum tru or n k. Whn n k + w hav: (os q + sn q) k+ (os q + sn q)(os q + sn q) k (os q + sn q)(os kq + sn kq) ( n hp) os q os kq sn q sn kq + (os q sn kq + sn q os kq) os((k + )q) + sn((k + )q) (usng multpl angl ntts or os an sn), so th rsult hols or n k +. Sn t hols or n, nuton th rsult hols or all n. 8 a Whn n w hav ( ) so th rsult hols. Assum tru or n k. Whn n k + w hav: ( ) k+ ( ) k k ( n hp) +k (k+) so th rsult hols or n k +. Sn t hols or n, nuton th rsult hols or all n N. Whn n w hav ln ln ln so th rsult hols. Assum tru or n k. Whn n k + w hav: (k + )ln kln + ln ln k + ln ( n hp) ln k ln k+ so th rsult hols or n k +. Sn t hols or n, nuton th rsult hols or all n. 9 Whn n w hav a a , so th rsult hols. Assum tru or n k. Whn n k + w hav: a k + a k + ( k ) + ( n hp) k + + k + so th rsult hols or n k +. Sn t hols or n, nuton th rsult hols or all n. 0 Whn n w hav u0 u, so ( u0 + ) ( + ) ( + ) th rsult hols. Assum tru or n k. Whn n k + w hav: uk uk + ( u + ) k ( k + ) + ( k + )

59 ANSWERS ( + k + ) (( k + ) + ) so th rsult hols or n k +. Sn t hols or n, nuton th rsult hols or all n. a, m+,, n Conjtur: kk n ( k + ) ( n + ) or all n. Whn n th rsult hols. Assum tru or n m. Whn n m + w hav: kk ( + ) k m kk + ( + ) ( m + )( m + ) k m + + m ( m + )( m + ) ( n hp) mm ( + ) + ( m + )( m + ) m + m + ( m + )( m + ) ( m + )( m + ) ( m + )( m + ) m + m + m + ( m + ) + so th rsult hols or n m +. Sn t hols or n, nuton th rsult hols or all n. a 0,,,, n Conjtur: k ( n! ) or k! k n! all n. Whn n th rsult hols. Assum tru or n m. Whn n m + w hav: m + k k m k k! ( m + ) + k! ( m + )! k m! m + ( n hp) m! ( m + )! ( m! )( m + ) m + ( m + )! ( m + )! ( m + )! - ( m + ) + m ( m + )! ( m + )! ( m + )! so th rsult hols or n m +. Sn t hols or n, nuton th rsult hols or all n. a Whn n th lt-han s s ( ), whl th rght-han s s ( + ), so th rsult hols. Assum tru or n k. Whn n k + w hav: k + k + r k r(r ) r( r ) + ( k + )(( k + ) ) r k ( k + ) + ( k + )( k + ) ( n hp) k + k + k + ( k + k + )( k + ) ( k + )(( k + ) + ) so th rsult hols or n k +. Sn t hols or n, nuton th rsult hols or all n N. Whn n th lt-han s s + + 8, whl th rght-han s s ( + ) 8, so th rsult hols. Assum tru or n k. Whn n k + w hav: (r + r + r) r k (r + r + r) + ( k + ) + ( k + ) + ( k + ) r

60 ANSWERS kk ( + ) + ( k + ) + ( k + ) + ( k + ) ( n hp) ( k + )( k( k + ) + ( k + ) + ( k + ) + ) ( k + )( k + 6k + k + 8) ( k + )( k + ) ( k + )(( k + ) + ) k + so th rsult hols or n k +. Sn t hols or n, nuton th rsult hols or all n. Whn n th lt-han s s ( ), whl th rghthan s s ( ) ( + ) so th rsult hols. Assum tru or n k. Whn n k + w hav: r r r k + ( ) r + ( ) ( k + ) r k r k ( ) kk ( + ) k + + ( ) ( k + ) ( n hp) k k+ ( ) kk ( + ) ( ) ( k + ) + k + ( kk ( + ) + ( k + )) k + (( k + )(( k + ) k) k + (( k + )( k + ) k + (( k + )(( k + ) + ) so th rsult hols or n k +. Sn t hols or n, nuton th rsult hols or all n. Whn n oth ss qual, so th rsult hols. Assum tru or n k. Whn n k + w hav: k + k + ( k + ) k + ( k + ) ( n hp) kk ( + ) ( ) + ( k + ) k ( k + ) ( k + ) + ( k + )( k + ( k + )) ( k + )( k + ) ( + + ( k )( k ) ) k + so th rsult hols or n k +. Sn t hols or n, nuton th rsult hols or all n. Ers F a 0 < 9 < 9 < 9 < 9 8 < 9 Thror < 9 or { 0,,,, }. ( ) ( ) 6 0 ( ) ( ) Thror < 9 or {,, 0,, }. whh s o 7 whh s o whh s o 7 whh s o Thror s an o ntgr n th st {,,,,, 6, 7} thn s an o ntgr.

61 ANSWERS Thror z s a ratonal numr wth {0,, } an {,, }, thn z. a Lt a r + an s or som ntgrs r an s. Thn a + (r + ) + s (r + s) +, so a + s o. Lt a r + an s + or som ntgrs r an s. Thn a + (r + ) + (s + ) r + s + (r + s + ), so a + s vn. Lt a r an s or som ntgrs r an s. Thn a (r)(s) (rs), so a s vn. Lt a r an s + or som ntgrs r an s. Thn a (r)(s + ) (rs + r), so a s vn. Lt a r + an s + or som ntgrs r an s. Thn a (r + )(s + ) (r)(s) + r + s + (rs + r + s) +, so a s o. a Tru a + > a + (a + ) a > (a + ) a (a a) + > (a a) + > Tru a < a (a ) a < (a ) a (a a) < (a a) < > Tru Not that sn a s a natural numr, a > 0. a > a a a > 0 a( ) > 0 > 0 > Fals. A ountr-ampl s a,,. Lt n an o natural numr, an wrt n k + or som ntgr k. Thn n (k + ) k + k + (k + k) +. Sn k + k s a natural numr, ths shows that n s o. Lt m, n. Thn (m + n ) m + mn + n > m + n sn mn s postv. 6 m (k + ) 9k + 6k + 9k + 6k + + (k + k + ) +, so m s qual to an ntgr whh s a multpl o plus, hn s not vsl. 7 Lt m an onsr th prout m(m + )(m + ). Evr son natural numr s vn, so at last on o m, m + an m + must vn, so vsl, so th prout must also vsl. Morovr, vr thr natural numr s vsl, so on o m, m + an m + s vsl, an so th prout must also vsl. I a natural numr s vsl oth an, t s vsl 6.

62 ANSWERS Hn th prout o an thr onsutv natural numrs s vsl 6. 8 Lt r an s onsutv o ntgrs, wth r < s. Thn w ma wrt r k an s k + or som ntgr k. W hav s r (k + ) (k ) k + k + (k k + ) 8k an so s r s a multpl o 8, as rqur. Also, r s (s r ) 8k whh s also a multpl o 8. 9 I, thn larl () (). Now assum that () (). Dvng oth ss o th quaton w gt () (), an sn s an njtv unton ths mpls that. Hn s also an njtv unton. 0 a Tru. Wrt n k or som ntgr k. Thn n (k) 6k (k ), so n s vsl. Fals. s a multpl o, ut s not a multpl o. a Tru. Suppos that A a an suppos wthout loss o gnralt that a 0 (th proo s ssntall hoosng, or to non-zro). Thn ka k a ka k, an k k sn k 0 w must hav ka 0, so ka s a non-zro matr. 0 Fals. Takng A an 0 0 B provs a ountr-ampl. 0 Fals. Takng A an 0 0 B 0 0 provs a 0 ountr-ampl. 0 Fals. I A 0 0 thn t(a) Fals. Takng A 0 0 gvs a ountr-ampl. Lt S n n. Consr S n. Ang S n to tsl ut wth th orr o th son sum rvrs, w hav: S n (n ) + (n ) + n + n + (n ) + (n ) Wrttn ths wa, th sum onssts o n olumns, wth th sum o th two ntrs n ah olumn qual to n +. Hn S n nn ( + ) n(n + ), an so Sn, as rqur. Ers G Suppos that m s not o, so m s vn. Thn w ma wrt m k or som ntgr k. W hav m (k) k (k ) whh s vn, so m s not o. Suppos that s not vn, so s o. Thn w ma wrt k + or som ntgr k. W hav + (k + ) (k + ) + k + k + k + k + k + + (k + k + ) + whh shows that + s o, so not vn.

63 ANSWERS In ah as, assum that th statmnt at last on o an s rratonal s als,.. assum that oth an ar m ratonal numrs, an wrt n an r whr m, n, r, s wth n, r, s 0. s a m r n s m r n s ms nr ns ns ( ms nr) ns Both ms nr an ns, an sn n, s 0, ns 0, so s a ratonal numr, so s not rratonal. m r n s ( ) mr ns Both mr an ns, an sn n, s 0, ns 0, so s a ratonal numr, so s not rratonal. m n ( r s ) ms nr Both ms an nr, an sn n, r 0, nr 0, so s a ratonal numr, so s not rratonal. Suppos that oth an ar lss than or qual to 8. Thn 8 8 6, so s not gratr than 6. Suppos that oth an l outs ( 0, 0 ). Thn > 0 an > 0, so > Hn annot l n th ntrval ( 0, 0 ). 6 Suppos that s not a transnntal numr. Thn s algra, so s th root o som polnomal, sa a n n + a n n + + a + a 0. Thror 0 a n () n + a n () n + + a () + a 0 ( n a n ) n + ( n a n ) n + + (a ) + a 0 whh shows that s a root o th polnomal ( n a n ) n + ( n a n ) n + + (a ) + a 0. Sn all th onts o ths polnomal ar ntgrs, ths shows that s an algra numr, so s not transnntal. Ers H Assum or ontraton that m s not vsl. Thn w ma wrt m k + or m k + or som ntgr k. W show n Eampl.8 that m k + thn m s not vsl, whl n Ers F, Q w show that m k + thn m s not vsl. Thror m s not vsl, thn m s not vsl. Howvr, ths ontrats th at that m s vsl. Thus our ntal assumpton was als, an m s vsl. Suppos or ontraton that + s not rratonal,.. s a ratonal numr. Thn w ma wrt + m or som ntgrs m an n, wth n n 0. W hav ( + ) an also ( + ) ( m n ) m n Thror + 6 m n 6 m n 6 m n n m n 6 n

64 ANSWERS whh mpls that 6 s ratonal, ontratng th at that 6 s rratonal. Thror our ntal assumpton was als, an + s rratonal. Suppos or ontraton that s ratonal, an wrt m n whr m an n ar ntgrs wth no ommon ators. W hav ( m n ) m n an so m n. Thror m s vsl, an w show n Q that ths mpls m must also vsl. Hn w ma wrt m k or som ntgr k, an w hav n m (k) 9k, whh mpls that n k. Thus n s vsl, an so n s also vsl. Thror oth m an n ar vsl. Howvr, w assum that m an n ha no ommon ators ths s a ontraton. Our ntal assumpton must thror als, an s rratonal. Assum or ontraton that log () s ratonal, an wrt log () m or som n ntgrs m an n, wth n 0. Thn log () m n m n n m Howvr, th lt-han s s o, whl th rght-han s s vn, whh s a ontraton. Thror log () s rratonal. Suppos or ontraton that +. Thn ( + ). W hav ( + ) , an so whh mpls 0 8 an thn 0. In turn ths mpls that 0 6, a ontraton. Hn our ntal assumpton was als, an + <. 6 Suppos or ontraton that s not transnntal, so s algra. Thn s th root o som polnomal, sa a n n + a n n + + a + a 0. Thror 0 a n () n + a n () n + + a () + a 0 ( n a n ) n + ( n a n ) n + + (a ) + a 0 whh shows that s a root o th polnomal ( n a n ) n + ( n a n ) n + + (a ) + a 0. Sn all th onts o ths polnomal ar ntgrs, ths shows that s an algra numr, ontratng th at that s transnntal. Hn our ntal assumpton was als, an s transnntal. 7 Suppos or ontraton thr ar a nt numr o prms o th orm n, an lt p, p,, p k a lst o ths prms. Dn p p p k (so n partular s an ntgr o th orm s ). Suppos rst that s vsl a prm o th orm n, an wthout loss o gnralt assum ths prm s p. Thn w ma wrt p or som ntgr, an w hav p p p p k whh rarrangs to gv p p p p p ( p p p ). Ths k k mpls that s vsl p, ontratng th at that s not vsl an prm. Th othr posslt s that s onl vsl prms o th orm n +. Howvr, whn panng a prout o ths orm w s that (n + )(n + ) (n s + ) r + whr r s som ntgr (ths an prov ormall usng nuton, or ampl). Ths ontrats ng an ntgr o th orm s. Sn w hav arrv at a ontraton n ah possl as, our ntal assumpton must als an thr ar an nnt numr o prms o th orm n.

65 ANSWERS 8 Suppos or ontraton that th prson s an nhatant o th slan, so s thr a Knght or a Knav. Frst assum that th prson s a Knght, so alwas tlls th truth. Thn th statmnt I am a lar must tru, so th Knght s a lar. Ths ontrats th at that Knghts alwas tll th truth. Now assum th prson s a Knav, so alwas ls. Thn th statmnt I am a lar must als, so th Knav s not a lar. Ths ontrats th at that Knavs ar lars. Sn oth posslts la to ontratons, th orgnal assumpton must als, an th prson s not an nhatant o th slan. α α α k 9 a Lt m p p... p k th prm omposton o m. α α α k Thn m p p... p k, an w s that a prm p appars n th prm omposton o m t must also appar n th prm omposton o m, so must v m. Whn usng proo ontrapostv w assum m was not vsl, rsptvl, an hk or ah as that m was not vsl, rsptvl. For p thr was on as to onsr, or p thr wr two ass to onsr. For an artrar prm p w annot us ths mtho, as th numr o stuatons to onsr s artrarl larg. Suppos or ontraton that p s m ratonal, an wrt p n whr m an n ar ntgrs wth no ommon ators. W hav m p n ( ) m n an so m pn. Thror m s vsl p, an part a m s also vsl p. Hn w ma wrt m pk or som ntgr k, an w hav pn m (pk) p k, whh mpls that n pk. Thus n s vsl p, an so n s also vsl p. Thror oth m an n ar vsl p. Howvr, w assum that m an n ha no ommon ators ths s a ontraton. Our ntal assumpton must thror als, an p s rratonal. I a s a squar numr whh vs an ntgr m, t n not th as that a vs m. For ampl, vs ut os not v. Chaptr rvw qustons a, () > 0 a : 9a 0 vs 6 vs 0 r 0 r a :, 8 an 0 an ( z, z ) a Th oman A ontans atl k lmnts, an sn s njtv th mag o ah lmnt s stnt. Hn th rang o ontans atl k lmnts. Sn B ontans atl k lmnts, t must th as that th rang o s qual to B. Hn B s surjtv. I s surjtv, thn s njtv. W an prov th ontrapostv statmnt. Suppos that s not njtv. Thn thr st, A suh that ut () (). Ths mpls that th rang o ontans wr than k lmnts, so annot

66 answrs qual th whol o B. Thus s not surjtv. I : A B s a unton, wth A an B ontanng atl k lmnts whr k s a natural numr, thn ng njtv s quvalnt to ng surjtv. W us proo nuton. Whn n w hav 6 >, so th rsult hols. Assum tru or n k. Whn n k + w hav k + k > k ( n hp) k > (k + ) sn k > k + or k. Hn th rsult hols or n k +, an sn t hols or n, nuton th rsult hols or all n. Whn n w hav ( ) ( ), so th rsult hols. Assum tru or n k. Whn n k + w hav k + ( ) ( ) k + (k + ) k(k ) + (k + ) ( n hp) k(k ) 6( k + ) + (k + k + ) ( k + )(k + ) ( k + )(( k + ) ) Hn th rsult hols or n k +, an sn t hols or n, nuton th rsult hols or all natural numrs n. 6 Us proo nuton. Whn n w hav () () so th rsult hols. Assum tru or n k. Whn n k + w hav k + () ( k ()) ( k ) ( n hp) ( k ) k k + Hn th rsult hols or n k +, an sn t hols or n, nuton th rsult hols or all natural numrs n. 7 a Sn vs w ma wrt k or som ntgr k. Thn k k, an so vs. Sn s vn w ma wrt k or som ntgr k. Thn + (k) + (k) k + 6k (k + k) whh shows that + s o. a > a a a > a a > < 0 8 a Fals. Takng A an 0 0 B 0 0 provs a 0 ountr-ampl. Tru. Usng proo ontrapostv, A 0 thn ka k k 0 k 0 k 0 k 0 0 0, so ka a Assum that oth an ar ratonal, so w ma wrt m n an

67 ANSWERS r whr m, n, r, s an n, s s 0. W hav + m r + n s ( ( ) ) m n + ms ns + r s nr ns ( ms + nr) ns Both ms + nr an ns, an sn n, s 0, ns 0, so + s a ratonal numr, so s not rratonal Assum oth an ar lss than or qual to (ut gratr than 0). Thn, so s not gratr than. 0 Assum or ontraton that 6 s a ratonal numr, an wrt 6 m n whr m an n ar ntgrs wth no ommon ators. Thn m 6 n whh mpls that m 6n. Thror 6 vs m, an n partular oth an v m. B rsults stalsh prvousl n ths haptr (s Eampl.9 an Ers H, Q), ths mpls that oth an v m, an so 6 must also v m. Wrtng m 6k or som ntgr k, w now hav 6n m (6k) 6k, an so n 6k an 6 vs n. Thror 6 must also v n, ontratng th at that m an n hav no ommon ators. Our orgnal assumpton was thror als, an 6 s rratonal. For ontraton assum that w an wrt log () m whr m an n ar n ntgrs wth n 0. Thn log () m n n m m n whh s a ontraton as n s o ut m s vn. Thror log () s rratonal.