x. x > 1 x 6 x. , 0 b ±a c Ø d 0, 4a. Exercise a x a b. e f b i c d + +

Transcript

1 MATHEMATICS Higher Level (Core) ANSWERS ANSWERS Eercise. 0 0 c d 0 0 e f 0 [] ] [ c ]0] d ] 0] e ][ f ] [ ] [ c + c d + + c + d e f + i - + ii 0 + i + ii + 0 {±} {±0} c Ø d {} e {} f {0} c d 0 0 ] [ ] [ c ][ + Eercise.. c d e f c d e f c d e f 0 c d + e f g 0 h i + c d e f g h 0 i 0 j - k ± l c d 0 e f 0 ± c Ø d 0 ± Eercise.. < c d e f 0 > c < < c > d ( + ) c d e f g h i c d e f g < > h i p < ( + ) ( ) < > > < < > > 0 > < < < > >

2 MATHEMATICS Higher Level (Core) ANSWERS < < c 0 < < < < c Eercise.. c d e c + c c d < < - < < < < / - - / f g h i j k l m n o p q -/ t - - -/ t / - / / + c d 0 c d f() f() f() Note signs! Eercise.. c d 0 e f c 0 0 d e f c. m m 0 c m. + c d 0 e f - + Eercise.. z 0 z c 0 z d z e f t t z t g z 0 h Eercise.. c 0 d e f g h i j 0 c d e f ± ± c ± d - e f c d e - f - ± - ± g ± h - i j no rel solutions k ± ± ± ± + - ± ± f() - ± - ± ±

3 MATHEMATICS Higher Level (Core) ANSWERS - ± ± l no rel solutions m n o p < p < p ± c p < or p > m m < c m > - ± m ± ] [ ] [ c ] [ k ± ] [ ] [ c ] [ 0 Eercise.. Grphs re shown using the ZOOM viewing window: c - ± c ( 0) ( 0) (0 )( 0) ( 0) (0 )(0. 0) ( 0) (0 ) d e f d e f ( 0) ( 0) (0 )(. 0) (. 0) (0 )( 0) ( 0) (0 ) g h i g h i (0. 0) (. 0) (0 )(. 0) ( 0) (0 )( 0) (0. 0) (0 ) j k l j k l ( 0) (0 )( 0) ( 0) (0 )(0. 0) (. 0) (0 ) Grphs re shown using the ZOOM viewing window: ( ) c i ± 0 ii (0 )

4 MATHEMATICS Higher Level (Core) ANSWERS ( ) c i ± 0 ii (0 ) k k < c k k k c k ± < k < c k < k > ( )( ) c ( ) d + 0 c ( + ) d + Eercise.. ] [ ] [ [] c ] 0] [ [ d ] [ e ].] [ [ f ]0..[ ] [ ] [ ][ c ] 0.] [ [ d [] e ] - [ f ] ] [ [ g [ ] h [.] i ] [ ]0. [ j ][ k ]0.[ l Ø m Ø n [.] o ] [ ] [ < k < 0 < k < c n 0. i ] [ ] [ ii [] i ] [ ] [ ii [] c i ][ ii ] ] [ [ d i ] [ ii ] ] [ [ e i ] [ ] [ ii [] f i ] + [ ii ] ] [ + [ ]0[ [0.] ( ) < k > > ( ) + ( + ) i ] [ ] + + [ ii ] ( ) ( + ) \{} iii ] i [- ] ii ll rel vlues {: < } {: > } {: < < } c i {: < 0.} ii {: < < 0} i ]0[ (k ); ]0[ (k ) ii Ø k >. Eercise.. ( ) ( ) ( ) ( ) c d ( ) e f ( ) g - h no rel solutions i - - j ( ) ( ) k no rel solutions ( ) ( ) c (0 ) ( ) d ( ) e Ø f ( ) g Ø h ± m < m > c < m < 0. ( ) ( ) 0. c i ( ) - ii ( ) c i A() B( ) ii sq. units Eercise c + + d + + e f - m ( ) ( + ) ( 0 )

5 MATHEMATICS Higher Level (Core) ANSWERS 0 Eercise...% c % digrm % c 0 Continuous Eercise.. ( )( ) ( )( ) + 0 ( + )( + ) + ( )( + ) + ( + )( + ) + ( )( ) Eercise. c d. e 0 c + Eercise. ( )( )( + ) ( ) c d ( )( + )( + ) e ( )( + ) f g ( ) ( + ) h ( ) i j ( ) + ( + ) ( ) + - ( + + ) ( )( )( + ) ( ) ( )( )( + + ) ( ) ( + )( )( + ) 0 ( + )( + )( ) ( + )( + ) ( )( + + ) ( ) ( )( + ) ( )( + )( ) c ( + ) m ( ) ( + ) m n k ; c ( α + αβ + β ) αβ ( α + β ) Eercise.. c d e f g h ± 0 i ± j ± + + d c d ( )( + )( + ) n k k n ; ( + )( )( + ) 0 ( + )( 0. )( ) 0 c ± c d + ( )( + )( + )

6 MATHEMATICS Higher Level (Core) ANSWERS c 0.0 d No other solutions m n ; Eercise.. ][ ] [ ] ] [] c [] [ [ d ]0 [ \{} e {} [0. [ f ] [ ][ g ] [ ] [\{} h ] ] {} ][ ] [ ] ] [] c [] [ [ d ] [ ] [ e [ [ f ] [ g ] [ h [ 0] [ + 0 [ i ] ] [ + + j ] [ ] [ k ] ] l ] - [ ] - [ Eercise. c 0. ] j k l m n o p q r. d e f / 0. / s t 0. c g h i d e f 000

7 MATHEMATICS Higher Level (Core) ANSWERS g h i j 0. i ii i ii c d ( + ) e 0 ( ) f + + ( + )( ) c ( + )( )( ) 0. ( + ) ( ) ( + ) ( + )( )( ) ( )( ) c c + {( 0 ) ( + )} { : > + } ( ) ( + ) ( )( ) or Eercise. Sum Product Sum Product c Sum Product d Sum Product e Sum Product f Sum Product g Sum Product h Sum Product - i Sum Product Consider the possiilit of zero denomintor! c d e f g h i The cuic cse: c d gives The fctorized version is: ( α )( β )( γ ) 0. The onl simple conclusion is tht the product of the roots is αβγ d. This is relted to the conjugte root theorem. The coefficients must e rel. Eercise.. + c + c + g + g + g c d e f + g h + 0 p q + + q + 0q - p + 0q - p + 0q - p + - 0q p + p r Eercise.. 0 c d 0 e p f 0p q g 0p c 0 d 0 e f % 0 c 0. d 0.% ā - c ā - d i 0 + j + k l + d + d + d m n o - p p + p p 00

8 MATHEMATICS Higher Level (Core) ANSWERS 0 0 n n ± n ± ± Eercise. dom { } rn { } dom { } rn { 0} c dom {0 } rn { } ] [ [0 [ c ] [ d ] ] e [ ] f ] [ g ] 0] h [0 ] i [0 [ j [ ] k ]0 [ l ] ] [ [ r [ [ d [0 [ r { : 0 }\{} d c r [0 [ \{} d [ [ \{0} d r [ 0[ d [ [ e r ] [ d ] ] [ [ f r [] d [0] one to mn mn to one c mn to one d one to one e mn to mn f one to one \{} ] [ c [] d ] ] [ [ e \{0} f g \{} h [ [ i [0 [ \{ } j ] ] [ [ k l \{ } ] [ ]0] c ] ] d [ [ e \{} f ] [ g [ [ h ] 0[ Eercise. Grphs with grphics clcultor output hve stndrd viewing window unless otherwise stted. i (+) + ii c 0 0 c ± c no solution 0 i ii i { } ii { } Window [] [] Rnge: [ ] c d e d e f Window [] [] [0 [ 0 {: > } {:.} 0 onl it is the onl one with identicl rules nd domins [ [ [0] c [ [ d [.[ ] [ i p ( ) + 0< < ii A ( ) 0< < i ii r ][ r ]0] 00

9 MATHEMATICS Higher Level (Core) ANSWERS Eercise.. c d () c d () c d Eercise.. c d e f 0. c d g h i 0 00

10 MATHEMATICS Higher Level (Core) ANSWERS c c d e f d e f g h i g h i c () [ [ [ [ [0 [ d e f ] 0] c d c d 00

11 MATHEMATICS Higher Level (Core) ANSWERS i Ø ii [] iii {±}. 0. { : } { : } hs diltion effect on f () (long the is). c d Eercise.. ( ) ( ) ( 0) ( 0) ( ) ( ) ( ) ( ) c d ( ) ]0 [ ( ) ]0 [ ( ) ]0 [ (.) ]0 [ [] [] c [0.] d [0.] e [0.0.] f [0.0] c d ] [ e ] [ ] e[ ] [ e f g h (.) ]0 [ (.) ]0 [ ( ) ]0 [ ( ) ]0 [. c i f g: ii f > g: < g i j k l ] + e [ [ [ c [ e + e ] ( ) ]0 [ ( /) ]0 [ ( /) ]0 [ ( 0/) ]0 [ 00 f

12 MATHEMATICS Higher Level (Core) ANSWERS 0 c d e f c c d () () ]0[ {} ] ] () c d [ [ [ [ c [0/] ] ] [0 [ c ]0] d e f [[ [ [ ] [ ]0 [ > ]0 [ < ]0 [ {} [0 [ ( ) EXERCISE.. c d ] [ ]0 [ ] [ 00

13 MATHEMATICS Higher Level (Core) ANSWERS e f g h c 0. ]0. [ ] [ ]0 [ 0 ]0 [ ]0 [ \{0} \{0} c ]0 [ 0 ]0 [ d e f ] [ / ]/ [ ] [ ] [ d e f ]0 [ ][ c ]0 [ ] [ ] [ ] [ ]0 [ c e e ]0 [ ]0 [ e ]e [ d e f ( ) ]0 [ \{} e \{0} d e f ] /e[ /e ]0 [ e ]0 [ ii 00 i

14 MATHEMATICS Higher Level (Core) ANSWERS c d /e / / : ā - < < + 0 < < ~. ] 0[ [e [ c d [0 [ ] ] [ [ EXERCISE.. c d / /. c e f g h / / c + ] [ ]e/ [ d e f \{e} e e - ] 0/[ 0 ] [ c d \{e} e 00

15 MATHEMATICS Higher Level (Core) ANSWERS iii f/g: \{} where ( f/g ) - + ( ) fog ( ) + gof ( ) ( + ) ] [ ] [. c i ii d e f / / ii fog ( ) + 0 gof ( ) + [ [ [ [ iii fog ( ) gof ( ) ( + ) [0 [ [ [ iv fog ( ) 0 gof ( ) 0 \{0} \{0} v fog ( ) 0 gof ( ) [0 [ [0 [ vi fog ( ) gof() does not eist. ] [ 0 vii fog ( ) 0 gof ( ) 0 ]0 [ ]0 [ viii fog ( ) gof ( ) [ [ [0 [ i fog ( ) + gof ( ) [ [ [0 [ fog() does not eist gof ( ) ( ) [0 [ i fog ( ) [0[ [0 [ gof ( ) EXERCISE.. i f+ g: [ 0 [ where ( f+ g) ( ) + [0 [ ii f+ g: ]0 [ where ( f+ g) ( ) + ln( ) [ [ iii f + g: [ ] [ ] where ( f+ g) ( ) + [ 0] i fg: [ 0 [ where ( fg )( ) / ii iii fg: [ ] [ ] where ( fg )( ) ( )( ) i f g: ] [ where ( f g) ( ) e ] [ ii f g: ] [ where ( f g) ( ) ( + ) + ]0. [ iii f g: ] [ where ( f g) ( ) + [] i ii fg: ]0 [ where ( fg )( ) f/g: \ { 0 } where ( f/g )( ) ln( ) - e e f/g: ] [ where ( f/g )( ) + ii fog ( ) + + gof ( ) + + [ [ [ 0. [ iii fog ( ) gof ( ) [ [ ]0 [ iv fog() does not eist gof ( ) - \{} + v fog() does not eist gof ( ) + ] [ vi [ [ ]0 [ fog ( ) + gof ( ) + c fof ( ) + g ( ) + fog ( ) + + \{ 0} ] ] [ [ gof() does not eist. fog ( ) 0 gof ( ) 0. ± c gog ( ) ].] [. [ + 00

16 MATHEMATICS Higher Level (Core) ANSWERS 0 + rnge ] [ ( ) hof ( ) ( ) + < r f d g nd r gof d h g ( ) ( + ) fog ( ) ]0 [ rnge ]0 [ gof ( ) ( ln ( e ( ) rnge ] [ ) + ) c fof ( ) e ( e ) rnge ]e [ hok does not eist. koh ( ) log ( ) > S \][; T T { : 0 }; S ] ] [ [ gof does not eist fog ( ) 0 Dom f ]0 [ rn f ]e [ Dom g ]0 [ rn g fog does not eist: r g d f ]0 [ gof eists s r f ]e [ d g ]0 [ c gof: ]0 [ where gof ( ) ( + ) + ln ( fog )( ) ; rnge [0 [ c ( ) fog ( ) 0 fof ( ) dom rn ]0 [ gof ( ) +ln 0 d f \ ( ) in f not g g rnge ] [ c rf \ c rf d fof ( ) f EXERCISE.. ( ) c ( + ) d e > 0 f ( ) g > 0 h ( 0 ) d gof: ] [ where gof ( ) fog*: ] [ where gof ( ) d fog [ ] fog c d gof / / [ ] fog rnge [ 0 ] ( + ) ( ) > c d e f g h + + ( 0 ) ( 0 ) (0) 00 f

17 MATHEMATICS Higher Level (Core) ANSWERS ± - < < c d / e f g h () () f ( ) f ( ) 0 c f ( ) d f ( ) + + e f ( ) / + f f ( ) 0 ( ) f ( ) + + > ( ) dom [ [ rn [ [ 0 f ( ) f ( ) c + f ( ) f ( ) log ( ) > f ( ) log ( + ) c f ( ) d ( log ) > 0 g ( ) + log 0 ( ) e h ( ) log \[0] f g ( ) log + > - + c inverse inverse < > inverse f ( ) ( ) [ [ + \{.} Inverse eists s f is one:one Cse : S ]0 [ Cse : S ] 0[ g ( ) + + g ( ) + d e f ( ) f ( ) ( + ) 0 { : f ( ) f ( )} ( ) inverse inverse inverse 0

18 MATHEMATICS Higher Level (Core) ANSWERS f ( + ) < ( ) f ( ) ln ( ) 0 < e e > e iii iv { 0 } c d f ( ) e + < /f + f ( ) ( ) > 0 < < 0 i tom ( ) e 0 ii mot ( ) e i ( tom ) ( ) ( ln( )) > ii ( mot ) ( ) ln > 0 c i & ii neither eist d Adjusting domins so tht the functions in prt c eist we hve: t om ( ) ( mot ) ( ) nd m ot ( ) ( tom ) ( ) e Yes s rules of composition OK. 0.0 (tom) 0. (mot) gof eists s r f d g. It is one:one so the inverse eists: f : f ( ) + e i ii f is one:one f ( ) ( ) ( ) < ( + ) > fog eists ut is not one:one c i B [ln [ ii iii EXERCISE. even even c neither d neither e even f odd g odd h even i odd Not if 0 is ecluded from the domin. ln f ( ) 0 ln R g ( fog ) : [0 [ where ( fog ) ( ) ln( + ) 0 f f

19 MATHEMATICS Higher Level (Core) ANSWERS EXERCISE. c d e + f + 0 g h ( ) ( + ) + ( ) + i ( + ) j k 0 l + i i ii ii iii iii ( + ) iv iv 0 c () () + c c First function in lck second function in lue c d e f.... g h i 0

20 MATHEMATICS Higher Level (Core) ANSWERS j 0 c d e f g h i j 0 0 Note: coordintes were sked for. We hve lelled most of these with single numers. c d ( ) 0. ( ) k l k m n o h g ( ) f ( ) + g ( ) f ( + ) c g ( ) f ( ) d g ( ) f ( ) + e g ( ) f ( ) + 0 i ii ( ) (0. 0) ( ) e f g h ( ) iii iv ( ) ( ) ( ) i j k l ( ) ( ) i ii 0 / ( ) m n o ( ) ( ) iii iv c i ii 0 0

21 MATHEMATICS Higher Level (Core) ANSWERS iii iv d i ii ( ) ( ) iii iv f ( + ) + f ( + ) + EXERCISE. c d ( ) i i ii ii iii iii iv iv c d 0

22 MATHEMATICS Higher Level (Core) ANSWERS f i ii iii iv f( ) (0 ) ( 0) 0 ( ) f( ) k ( ) i ii iii f ( ) h ( ) if ( + ) if 0 if < 0 if < ( ) iv v vi h ( ) if k ( ) if < ( ) if f ( ) if < if if < ( ) ( + ) if 0 if < 0 i ii ( ) (. ) (0 ) f( ) 0 iii iv f( ) (0 ) ( 0) 0 ( ) f( ) ( ) ( ) ( 0) f( ) ( ) f ( ) ( 0) ( ) (/ ) ( ) f ( 0) f ( ) f( ) + f ( ) f ( ) c f ( ) f d f ( ) f e f ( ) f f f ( ) f( ) + c d ( 0) > 0 ( ) ā - ( ) ā - < 0 0

23 MATHEMATICS Higher Level (Core) ANSWERS EXERCISE. i ii 0 0 d i ii () e i ii ( ) 0 0 i ii 0 0 c i (0 ) ii ( ) ( ) 0 () (0 (0 ) ) ( 0) ( ) () ( ) ( ) ( ) (0 ) ( ) 0 ( ) f i ii ( ) 0 ( ) c d. (0. ) (0. ) f( ) f( ) c f( + ) d f( ) e f( ) c d ( 0) ( 0) ( 0) ( 0) (0. ).. (0. ) (0. ) (0. ). ( ) 0

24 MATHEMATICS Higher Level (Core) ANSWERS e f g h i j k l - m n o p ( ) ( ) c d e f ( ) g h i j ( ) (0 ) ( ) i ii ( ) ( ) ( ) 0 (0. ) (0. ) 0 ā - ( ) ( ) ( ) ( 0) ( ) ( ) ( 0) ( ) ( ) ( ) ( ) ( ) q r ( ) (0 ) ( ) ( ) ( ) i ii c i ii c - - ( ) ( ) ( ) ( ) 0

25 MATHEMATICS Higher Level (Core) ANSWERS d e f ( ) ( ) ( ) EXERCISE. c ( ) 0. ( ) j k l 0. / () d e f / c EXERCISE. i ii d e f / ( ) ( 0.) i ii (0 ) (0 ) g h i 0. c i ii / (0 ) / / (0 ) / 0

26 MATHEMATICS Higher Level (Core) ANSWERS c d c i ii ( ) g i g ii i ii c i ii iii + iv v vi g g c 0. EXERCISE.. - c d e n + - f n + + g n + h ( n + ) i - c) d e f + n + z c d e n n + n + f n g + n n h n n i f f f f

27 MATHEMATICS Higher Level (Core) ANSWERS - m m kg c. rs d W + - c d e f + - c d e f - ( + h) ( + )( ) n c d e f g h mn p + - q pq - / 0.0.% c. m d. m e I 0 I t / n n c n d m n e n f EXERCISE.. c d e f. g h. i c d. e 0. f 0. g h i. EXERCISE.... c d. e. f. g. h i 0 0. c 0 d e f 0 c d e - ± EXERCISE.. i. ii. iii. i. ii. iii. c i. ii. iii. d i. ii. iii c 0. d0. 0 e. f 0 0. EXERCISE.. c 0. d c 0 0 c d e 0 f.. c. d Ø c 0.0 d c 0. 0 e k ln( ) EXERCISE. 000 c 000 d 0 ds + n + i ii iii. rs c 0. rs d c. kg d W e C i 00 C ii 00 C c.0 million rs d T gm c 0 ers d $ million $. mil c 0. ers d 0.0 c 0 d. ers 0 0 cm. cm c 00 ds d ds t 00 Q N 0 t V 0 W t t t 0 t

28 MATHEMATICS Higher Level (Core) ANSWERS i 0 ii c. erl 0 i $. ii $.0. ers c 0.t c T (.0.). C t ~ midnight. c 00 cm cm c. m d m e i.r ii.r iii.r f g t mg/min. min c i. ii. iii min d. mg e f No R A i $ ii $ iii $ c d i $ k ii $. mil iii $. mil f g $. mil & e 00 h Eercise. c d e f g 0 h 0 i j k 0. l log log c log 0 ( + ) d log 0 p e log ( ) f log ( ) t p R A t t c t d 0 z e 0 f c d e f g h i j k l.. c. d. e. f 0. g. h. i 0.0 Eercise. c d e f log log + log c log log + log c c log log c d log log + 0. log c e log log + log c f log log 0. log c c 0.0 z c + d e f c d e f no rel soln g h i j k l log w log c log ( + ) [ ] d ( )( + ) log e log 0 f c d e f ± c d c log -. log log log - + d 0. e log 0 log 0. f. g h. i 0. j no rel solution k log 0. l log 0 0. c d 00 0 e f ( log ) 000 c z c e c d Ø - e ln.0 ln 0.0 c ln. d e ln.0 f ln g 0. e log 0.0 log0 ± log - log -. log log log. 0. log ln 0. ( + ) - log 0. log 0 0

29 MATHEMATICS Higher Level (Core) ANSWERS h. 0. c 00. d 0. e.0.0 f 0. g 0. h i. j Eercise. 0 0 c 0. kg. c. [0[ i. ii. iii 0. rs c As c increses reliilit reduces. d 0 ct e 0.0 λ λ 0 0 k c.% d k Eercise.. i c t n n ii c t n n + iii c t n n + iv 0. c t n 0.n v c t n + n vi c t n n + e.0 i ± e ± 0.0 j Ø k e. l e 0 ln ln c ln ln d 0 e 0 f d e W. I n k L L 0 0 W. 0 0.h h h. W m. L ln ln 0 c d L 0 m 0.0 m λ log λ 0 L 0 L 0 st 0 i ii i ii 0 t n + ( n ) 0 Eercise.. 00 c 0 0 c c 00 c th week 0 0 Eercise d $ weeks 0 $ 0 i m ii 0 m m c Dist d e plers 00 m c 0 c n Eercise.. r u u n n r u - u n n c r d r u u n u u n n e r f r n n nn ( ) u u n u u n n n ( ) n 0 t

30 MATHEMATICS Higher Level (Core) ANSWERS ± ± ± th u n 0 c n times i $0 ii $0.. rs 000 u n - n 0..0 or $ $ Eercise.. c d e. f. 0 0 c d e ; ; c ; 0.00 d ;. e ;. f ;. c d 0 e - ; 0 $0.0. cm n 0. gms; 0 weeks V n V 0 0. n r. 0 0 $0.. 0 or out 00 illion tonnes. Eercise.. Term AP 0 GP. Sum to terms AP 0 GP weeks Ken $0 & Bo-Youn $) week week. [~00 depends on rounding errors] Eercise.. c 000 d fish. [NB: t <. If we use n then ns is 0 fish]; fish. Overfishing mens tht fewer fish re cught in the long run. or c 0 cm 0 Eercise.. 0. c ( t ) n + t ( t ) - n + t t - t n n - + t

31 MATHEMATICS Higher Level (Core) ANSWERS ± ( ) ( 0 0) c n m Eercise. $.0 $. $. $ $. $0. $00.0 $. $. $. 0 $. $. interest $.. Flt interest $000 $. $0. 0.0% /month (or.% p..) Eercise. cm cm c cm A B C ( + ) c d ( + ) e ( + ) f 0 0 ( + ) Eercise. i 00 T ii 0 T iii T iv 00 T i N E ii S iii S0 W iv N0 W.m.m ' m/s N W. km. m 0. m. m 0. km. m.0 km N. km E. km E 0. km S c. km E. km N d. km T m Eercise. ' '. cm. cm c ' d 0. cm e 0 ' ' ' c '.. m 0. m. c.. c. m d. m. m ( c) h h + tn c d ( + c) h + + c ( ) + h 0.0 m 0. m. m 0. cm ' c '. m ' c ' h tn c Eercise.. cm. cm c. cm d 0. cm e.0 cm f. cm g. cm h. cm i. cm j. cm k. cm 0

32 MATHEMATICS Higher Level (Core) ANSWERS l. cm m.0 cm n. cm o. cm m 00 cm. cm.sq units.0 sq units c. sq units. cm ' ( + tn θ ) tn θ Are of ΔACD 0. cm Are of ΔABC. cm Eercise.. cm cm c cm A B C Eercise.. c A B C c* B* C* d no tringles eist. Eercise.. 0. km. m. m 0 'T. m.0 m. m.000 m c. m. min hr. min c.0 km $ 0 0 m Eercise.. cm cm c cm A B C

33 MATHEMATICS Higher Level (Core) ANSWERS Eercise.. 0. km T '. cm. m. m. km W ' S Eercise...0 m. m 0. m 0 T. 0. cm cm left. km 0 m. cm m 0. cm. cm 0 m m c 0 d 0.0. m c. m m m c 0 m Eercise.. cm. m m 0. m.. min. ~0: m d sin φ sin( φ θ) d sin θ sin( φ θ) d sin φtn α i ii or c d - sin( φ θ) d sin θtn β sin( φ θ) sin φcos θ sin( φ θ) Eercise. - cm 0. c cm + cm d m. e f - cm. g.m +.cm h cm 00. i j cm. k cm + cm l - cm m n - cm. o cm c 0.0 m. c cm + - cm + - cm + - cm + - cm cm. cm. m 0. m c. c 0. c 0. c i.0 cm ii. cm c 0. cm.. cm 0. cm. cm cm cm c c tnα + cm 0. α 0. α. cm - cm - cm. + - cm + - m cm + - cm cm. + - cm cm cm + - cm 0

34 MATHEMATICS Higher Level (Core) ANSWERS Eercise c d 0 c c c c d - c c d e f g h i j - k l m n o p q 0 r s 0 t undefined 0 c 0 d e f g h i j k l m n o p q r s t - c d e f g h - c d e f g h - i j k l c d c - d + 0 c c k c k k c - - c c k k c k - k c k sin θ cot θ c d e cot θ f tn θ c d e f Eercise k k k c ( ) + ( ) 0 d e + + i ii i ii - c 0 d 0 i ii c ± + k k or i ii i ii + k ( k ) + k i ii k - k + - k i ii iii i ii iii ± + k k ( - ) + ( - ) - 0

35 MATHEMATICS Higher Level (Core) ANSWERS Eercise 0.. sin αcos φ+ cos αsin φ cos α cos β sin α sin β c sin cos cos sin d cos φ cos α+ sin φsin α e tn θ tn α f - tn φ tn ω + tn θtn α + tn φ tn ω sin ( α β ) cos ( α + β ) c sin ( + ) d cos( ) e tn ( α β) f tn g h i c c c d c d c d c d + + tn φ c 0 α ± α α α 0 R + tn α 0 R + tn α Eercise ( + ) sin + α+ β sin c d e f c d 0. c d e f g h i j tn - c d e f g h / c d / / 0. / 0. e f g h / / / c d.. / 0

36 MATHEMATICS Higher Level (Core) ANSWERS e f g h / / / c sec cot cosec c d e f g h i j k l m n Eercise 0. c d e f g.0 c h 0. c i 0.0 c j. c k 0. c l. c m undefined n. c o.0 c c - c d e f c d undefined e f g h - k - k + k 0 0. [ ] c d [ 0 ] [ ] 0. 00

37 MATHEMATICS Higher Level (Core) ANSWERS i ii c Cos Sin / tn - n + Eercise 0. c d - e f c - d e - f c d tn e - f c 0 0 d e - - f 0 g h c d ' e f g h. c.0 c i j k l ' ' m n o Ø ± c d e f g h - + tn - c 0 ± cos tn + tn( ) c / - - tn ± - ( ) - d 0 sin + 0 sin - - sin - + sin - sin - + sin - ii 0 ii 0 i { k + α( ) k k } ii { k + α ( k + ) α k } c 0 c ± ± ± 0 '0 ' ( '0 ') tn tn( ) Eercise 0. T sin t + c.. ( k + ) k cos cos + tn tn( ) { k } k k k - cos ( ) k + k L sin - t +. ( ) k k + k 0

38 MATHEMATICS Higher Level (Core) ANSWERS V c. m. m. sec. sec c mid-april to end of August 000 months c R d months m c m P S. t sin + sin ( t ) + P 0. sin ( t ) + D 0. t sin + sin - ( t. ) +. t F(t) G(t) c d.% d months d i ii [ 0 ] [ ] [ ] c. m Eercise. i ii iii iv 0 v vi i ii iii iv v vi c i i ii i iii + i iv v vi + i i c i d i e f + i i c + i d i e f ( + i ) c d e f + i i c i d ( + i ) e i f c c 0 i i i i ii i i i i ii i iii iv 0 or 0 or oth i i i c i ( + i ) ± ( + i ) i ( + ) ( + i ) ( uv ) ( + ) ( ) i i + i 0 + i + i i + i + i ( + i ) ( + i ) 0 t t

39 MATHEMATICS Higher Level (Core) ANSWERS cos ( θ+ α) + i sin( θ + α) cos ( θ α) + i sin( θ α) c r r ( cos ( θ+ α) + i sin( θ + α) ) d cos( θ ) + e + sin ( α ) + + i c ( + ) z cos ( θ ) + i sin( θ ) cos ( θ ) + i sin( θ ) α 0 α i 0 0 c α d 0 β 0 β 0 i β sin( θ ) + i cos( θ ) d cos ( θ ) i sin( θ ) Eercise. The points to plot re: () (0) () () ()()(0.0.)(). i i ii ( + i) iii i iv + i; Anticlockwise rottion of 0. i Reflection out the Re(z) is. ii Results will lws e rel numer so the point will lws lie on Re(z) is. iii Point will lws lie on the Im ( z ) is. + i c d e i f i g ( + i ) h ( i ) ; ; c ; d ; e ; f ; g ; 0 h ; rctn ; ; i or ii 0 or + ± ( i ) + i + i rctn ( ) Tringle propert; the sum of the lengths of two sides of tringle is lrger thn the third side. 0 c 0 + i ;. ; c ; 0 α n 0 0 β n ; θ ; θ c ; θ sec α α (for Principl rgument) otherwise α + k where k is n integer. sec α α + (for Principl rgument) otherwise α + k is n integer. + k θ cos θ c (for Principl rgument) otherwise + k k is n integer. i ii - + i sin θ i ii θ θ - Im(z) θ Eercise. cis cis c cis c d e f cis ' g h i i c d e f c 0 i i c ( ) + ( + )i c d e f ( + i ) c + i d i e f ( + i ) c d e ( + i ) f ( + i ) 0 i c d e f ( i ) i c i d e f - i ii iii i i ( + i ) c + + i O sin θ cis cis cis Re(z) ( ) cis cis + i i i + i ( + i ) ( + i ) i cis ( ' ) cis ( ' ) - 0cis ( ' ) ( + i ) -i ( + i ) i i ( + i ) 0 i i ( + i ) 0

40 MATHEMATICS Higher Level (Core) ANSWERS ; ( + i ) i ii c ( cis ( θ )) ( cos θ + cos α )( cos ( θ α) i sin( θ α) ) [or cos( α θ) ] cosec θ θ Eercise.. ( + i )( i ) ( + + i )( + i ) c ( + i )( i ) d ( z + + i )( z + i ) e z ( + i ) z ( + i ) f ( z + + i )( z + i ) g w i w + i i h ( w + i )( w i ) i w - i w + - i ± i c d e f ± i g ± i h ± i i ±i ± ± i ± ± i c ± ± i ( z i )( z + i ) ( z i ) z i c d ( z + + i )( z + i ) e z i f Eercise.. ( z + )( z + i) ( z i) ( z )( z + i) ( z i) c ( z )( z + i )( z i ) ( w + i )( w + + i )( w ) ( z )( z + i )( z i ) c ( z )( z + + i )( z + i ) d ( + )( )( + i) ( i) e ( w + )( w + i )( w i ) f ( z + )( z )( z + i )( z i ) ± i ± i c ± i d ± i e f - ( + i ) ± i - (( ) + ( + )i ) ± i ( z i )( z + i )( z )( z + ) - ± i ± i ( z )( z + i )( z i ) ± i ± i + - ± ( + ) z + + i ( + ) ± i ( )( z + i) ( )( z+ i )( z )( z + ) - ( ) ± i ± i ( z )( z + i) z i ± i 0 ± i ± i ( z )( z + i )( z i ) ( z )( z + i )( z i ) ± i ± i ± i ± i c ± ± i d ± ± i c d i ± ± i ( ± ) Eercise.. ± i + i i c i d e ( + + i ) f ± i ± i ± i ; ( z i )( z + i )( z + i )( z + + i ) ± + i i c ± cis cis cis cis cis c f ( ) ( z + )( z )( z + i )( z i ) ± ± i i + i i + i ( + i ) - ( + i ) cis cis cis cis cis cis cis ( + i ) ( + + i ) ± ± i - ( + i ) - cis cis cis c cis cis - i ± - i d cis cis cis e ( ± + i ) i ± cis (( + ) + ( )i ) - 0

41 MATHEMATICS Higher Level (Core) ANSWERS n n + n c d e f g h n + n + n + n c n n d e Eercise.. - n ( n + n + ) n ( n + ) c d n n ( n ) e f ± -i ± i Eercise.. - n ( n + n + ) nn ( + ) 0 ( n ) n n - + n + nn ( + )( n + ) n n + n n n + n n + n n + n n + nn ( + ) nn ( + ) n + - ( + ) nn - n n n n - + n n + n + n n Eercise. i 00 ii (0.) Smple size is lrge ut m e issed fctors such s the loction of the ctch. Popultion estimte is 000. i 00 ii 0 00 c 000 c numericl; d e ctegoricl d discrete; c e continuous Eercise. 0 0 Set A Mode. Men. Medin. Set B Mode Men. Medin. Set B is much more spred out thn set A nd lthough the two sets hve similr men the hve ver different mode nd medin. Eercise. Mode g; Men g; Medin g Mode.. g; Men. g; Medin.0 g Set A Mode.; Men.; Medin. Set B Mode ; Men.; Medin. $ $0 c Medin $00 $000 c Medin.. A:. hr B:. hr c Tpe B. crds 0 00 Eercise. Smple A Men. kg; Smple B Men.00 kg Smple A Smple std 0.0 kg; Smple B Smple std 0. kg c Smple A Popultion std 0.0 kg; Smple B Popultion std 0. kg.. Men.; Std. $. $.. $ $ c 0 i 0. ii. c

42 MATHEMATICS Higher Level (Core) ANSWERS Eercise. Med Q Q IQR Med. Q. Q. IQR. c Med. Q Q IQR d Med.0 Q 0. Q. IQR. e Med. Q 0 Q IQR 0 Med Q Q IQR Med Q Q IQR c Med Q Q. IQR 0. d Med 0 Q 0 Q 0 IQR 0 e Med 0 Q Q. IQR Eercise. Smple00 rndoml selected ptients popultion ll suffering from AIDS Smple000 working ged people in N.S.W popultion ll working ged people in N.S.W. c Smple John s I.B Higher Mths clss popultion ll seniors t Npp Vlle High School. Discrete: d; Continuous: c e f g. Frequenc score Cumultive frequenc score suggested nswers onl: 00; ; 0;... 00; 0; c 0; 0; Mke use of our grphics clcultor. grphics clcultor c. d. 0 grphics clcultor c 0. d.0. c. d.0 sec.. 0 Q~ Q~ ~ 0 c % d. rnge s..;.0.;... Eercise c 0 0!! c i! ii! 0 00 Eercise Eercise. 0 s n. s n. s n. s n. 0

43 MATHEMATICS Higher Level (Core) ANSWERS ; c n n C C c ; Eercise. c {HH HT TH TT} {HHHHHTHTHTHHTTTTTHTHTHTT} c c d c 0 0 c d {GGG GGB. GBG BGG BBB BBG BGB GBB} c 0 c c d {( H)( H)( H)(H)( H)( H)( T)( T)( T)( T)( T)(T)} - c Eercise. c c d.0 0. c c c 0.0 c c d e 0 c d 0. i 0. ii 0. c d Eercise c 0.0 d c 0.0 d 0.0 {TTTTTHTHTHTTHHHHHTHTHTHH} i ii iii iv 0

44 MATHEMATICS Higher Level (Core) ANSWERS c d c 0. T c c / / H F /0 /0 / c 0. d c 0. d c c c 0. d c 0. Eercise i ii 0. 0 R R _ / / 0 0 / / / / / / / / / H T H T H T Y B G / / / / Y B G Y B G R R _ R R _ i ii - N m c 0.0 d c c M 0. Eercise. - c - c d 0 c - - c N m N c ( ) N m - 0 m m+ ( N m) n 0

45 MATHEMATICS Higher Level (Core) ANSWERS Eercise i 0. ii 0. p0 ( ) p( ) p ( ) p ( ) c { 0 } 0 p() c d p() 0 H p() c p0 ( ) H T / i ii c i 0.0 ii p ( 0 ) p ( ) p ( ) i ii n 0 T p0 ( ) p ( ) p ( ) n P(N n) P(S s) T H T p( ) p ( ) H T H T H T 0 / 0 p ( ) p ( ) p ( ) p() 0 p() P(N n) s 0

46 MATHEMATICS Higher Level (Core) ANSWERS Eercise... i ii c i ii 0. i. ii. iii 0. i 0. ii. c i ii 0. σ 0. μ. np.. c. 0. i 0. ii c i 0 ii iii p ( 0 ) p ( ) - p ( ) - p ( ) i 0. ii 0. c W N E(W) 0. $.00 oth the sme 0 c c E(X) p Vr(X) p( p) i n( p) ii np( p) n 0 P(N n) W. 0 E ( X ) + - Vr( X ) ( + ) E(X) Vr(X) 0 EXERCISE c c 0.0 d c c 0. d c c 0.0 d c 0.00 d 0. e c 0. d c 0. d c c $0 d 0.0 i. ii iii.0 iv 0.0 v 0.00 i.0 ii iii. iv 0.0 v 0.0. i 0.0 ii. 0 iii 0. t lest 0 c d i ii. i ii $. $ $ c 0 p c. np np ( p ) - n 0 < p < ( p ) n np ( p ) n EXERCISE c c p 00

47 MATHEMATICS Higher Level (Core) ANSWERS P(X ) c 0 0 hpergeometric c P(X ) i 0. ii 0.00 c 0. 0 c 0 ds eforehnd (plce order on Jul) reminder ~ 0 P(X ) ~ 0% P(Accept) EXERCISE. e - 0! PX ( ) i 0. ii 0.0 iii 0.0 iv c c 0. 0.; 0. No c 0. i p ii pln p iii p + pln p c 0. EXERCISE c 0. d 0. e 0. f 0.00 g 0.0 h 0.0 i 0.00 j 0. k 0. l c 0.0 d 0.0 e 0.0 f 0.0 g 0. h 0.0 i 0.0 j 0.0 k 0.0 l 0. m 0.00 n 0. o 0.0 p 0.0 q 0. r 0. EXERCISE c 0.0 d 0. e 0.0 f c 0. d 0. e 0. f c c c c c c c c.0..0 % c 0. % 0. 0

48 MATHEMATICS Higher Level (Core) ANSWERS 0 % % % i 0.00 ii 0.0 iii $. $ i 0. ii 0.0 c $ μ. σ 0. $0.S μ. σ. 0 (.) i 0. ii 0. i 0. ii 0. c 0. Eercise. c d e f 0 0. c 0.0 d 0. e 0.0 f. g. h 0 m/s 0 m/s c + h + h m/s m/s + h.ºc/sec cm /cm i. cm /cm ii. cm /cm iii cm /cm..ºc/min t to t 0 m m/s c verge speed d m e m/s $0 $. $0.0 $0. $ 00. $0.0 per er Eercise. h t h t h c h t d h e f h t Eercise. h + + h c d h +h - + h c d + h ( + h) c ( + ) + h d + + h + h e ( + h + h ) f + ( )h + h g ( + h ) h i ( )( + h ) + h+ ; + h ; c + h + h ; d + h + h + h ; (t) i ms ii ms iii. ms d Find (limit) s h 0 e t t 0 s(t) i 0 cm ii. cm iii. iv. cm /d c 0 ( 0.h ) cm /d d i. cm /d ii. m /d t t h h t t t 0

49 MATHEMATICS Higher Level (Core) ANSWERS Eercise. c d. e f. m. ( h + h ) m c. m/s 0 c d e f 0 c + d e ( + ) f 0. / ms ( ) ms (t) i ms ms ii ms c t t ms d sec Eercise. c d e f g h 0 + i + j k l c d e f g h i 0 j - k l c d e f g l - Eercise.. m PQ + h ; lim m PQ h 0 P ( ) Q + h - ; ; + h m PQ - lim + h h 0 m PQ c d e f g h t h i j k ± c ( ) + 0 c ( ± ± ) ±- (0 0) : - < < 0 0 f ' ( + ) ( + ) + 0 Eercise.. c ' 0 > 0 ' : > - d e f ' ' g h i ' ' ' ' ' c d 0

50 MATHEMATICS Higher Level (Core) ANSWERS Eercise.. t - n c d t n n r + θ r r θ + e 0 L f 00 - g h i v l + + h n - + n c s / d + e f t r 0 r Eercise c d + + c - ( ) ( d + + ) e f ( ) + ( + ) ( ) ( sin + cos )e ln + c e ( ) d e sin + cos f tn + ( + )sec g ( cos sin ) h e ( cos + sin + sin ) i ( ln + + ln )e sin cos [ sin ( + ) + cos ] e c d e sin - cos sin ( + ) - ( e + ) ( f - + ) ln e g + h i ( + ) ( + ) ( sin cos ) e + cos + sin c ( + ) s + ( + ) d cos + e e sec + e f sin ( ) + e g cos sin cos( ) h 0 i j k l - t e - t t - + θ + - / m m + ln ( + ln ) + sin cos sin + - ln ( ln ) cos + sin cos sec θ cosθ c d e sin θ cos θ f e e cos( ) g sec h i ( log e ) - sin cos θ sin( sin θ ) cos j sin θ sec θ k cos csc ( ) l csc ( ) e e c e d e e f e e g e + h - i e ( )e + j cos ( θ )e sin θ k sin ( θ )e cos θ l m e n ( e + ) ( e + e )( e e ) o e p + )e + ( cos θ + e c + e d e f + g h sin cos i j k + cot l ln( + ) + sin c + sin cos - sin θ cos θ + θ d ( )e + e ( ln + ) sin( ln ) f ln ( ) sin( g ) cos ( )( ) 0 - h - ( ln ( 0 + ) ) ( sin ) [ ln( 0 + )] i ( cos sin )e j ln ( sin ) + cot k ( cos sin ) e l ( sin + cos ) sin( sin ) m + ( 0 ) n ( ) o - + p q r ( + ) + + ( + ) / s t u v w n n ln( n ) ( ) + sin θ + θ - ln( + ) ( + ) e e ( + ) sin θ cos + n + n n cos ( ln ) sin - ln + tn e cos θ + sin θln( sin θ) sinθcos θ ( + ) e + 0

51 MATHEMATICS Higher Level (Core) ANSWERS e - cos 0 cos sin c - sin 0 i sincos + cos - sin ii e ( cos ln cos sin ln cos sin tn ) i ( ln ) ii - i ii e cos( e ) cos e sin k m n + m+ n 0 { θ:n tn θ m tn θ n mθ m n } csc( ) sec ( ) tn ( ) c cot ( ) csc ( ) d sin( ) e csc f sec ( ) tn( ) sec ( ) tn ( ) sec c tn d cot csc e cos + sin f cot csc g csc ( ) cot ( ) csc( ) h cot sec ( ) csc tn( ) i - sec tn sin cos + sec e sec sec tn e sec ( e ) tn ( e ) c e sec ( ) + e sec ( ) tn( ) d csc ( log ) e csc( ) sec( ) f cot( ) csc ( ) log g cos cot( sin ) csc ( sin ) h cos( csc ) cot csc i 0 Eercise. c d e + - c d - f - g - h i ( + ) j - k l sin cos - ( ) + + if sin > 0 if sin < 0 e e f g h i j k l ( + )[ tn ( )] ( sin ( )) / m n o Tn sin c i Sin - d tn + tn e - log + sin ( ) ( + ) f - cos g e tn ( e ) e h 0 k k / - Sin f ' ( ) > nd ; () ] f ' ( ) - ; dom ( f ' ) > c f ' ( ) ( ) < < ; dom () f [ ] - Cos d f ' ( ) > 0 nd ; ] [ + e f ' ( ) ; < + e dom () f f f ' ( ) nd f ' ( ) - - < < - or - < < ; dom () f [ ] e + cos ( cos ) - e rcsin ( cos ( )) + + e tn + < dom f dom () f - Sin < 0 dom () f + - < <- [ ] [ 0

52 MATHEMATICS Higher Level (Core) ANSWERS g f ' ( ) > 0 nd ; ] [ h f ' ( ) < nd ; ] [ n 0 c d e f ( + ) 0 Eercise. ( ln ) ( ln ) c ( ln ) d ( ln ) e f g ( ln ) h ( ln ) + i ( ln ) ( ln ) + cos ( ) + ln ( ) sin( ) c ln ( ) e d ( ) ( + ) ( + ) ln( ) ln e - f sin + ln c d e n n + n ( ln ) + + < 0 dom () f + > dom () f ( rctn ) n + - > 0 ( + ) - ( ln0 ) ( ln ) ( + ) ( ln )( + ) f g log h log i ( ln ) log ( ln ) ( ) ( ln ) log ( ln( 0 )) log 0 ( + ) j - k l ( ln )( log ) ln ( 0 )( log 0 ( + )) ln 0 ln - ln + ln ln 0 ( ln ) cos( ) c d 0 + ( ln ) ln 0 ln0 + ( ln ) ( )( ) ( ln ) ( ln 0 )0 ( ln )( + ) e ( ln( )) log - ln ( )( log ) ( ) cos + ( ln ) 0 + ln ( ) ln c 0 ( ) ln 0 d ln e ( ) + f cos ln sin g cos ln h sin cos ln i ( ln + ) sin sin c d cos ln + ( ln ) ln ( ) ln Eercise. 0 c d e f g h i j k l cos sin m cos + sin sin n 0 o p q ( + ) e + e + e r cos ( sin s + ) 0 t c d e ( + ) ( ) / ( ) / ( + ) f g h + / i ( + ) ln cos ln sin - cos - cot c d ln sec ( ) ln ( 0 ) ln 0 ln ( tn ) e f g sin ln ( ( ) + ) tn ( ) ln ( ) ln h i - ( ) ln ( ) ln 0 ( + ) tn ( + ) ( ) sin θ - rctn ( ) - / ( ) / ln n ln + n ( ) ln n - n + ( + ) ( + ) Sin - ( ) + - ( + ) - ( ) sin cos e e ( + e ) ( e ) e ( ) ( ) / 0

53 MATHEMATICS Higher Level (Core) ANSWERS c f ' ( ) f (v) ( ) f ( ) 0 [0.0[ ].] Eercise.. sin c d e f g h i j k l () c d e Hperol Dom Rn [] c d smll ( ) e Dom Rn [kk] f ν pγ ( + ) f '' ( ) ( + ) f (iii) ( ) ( + ) f (iv) ( ) 0 ( + ) f (n) ( ) ( ) n n! ( + ) n n f '' ( ) n e ( ) n n n! ( + ) n + n k : n k : n ( n + ) + ( ) - n ( n ) ( ) ( ) k k sin ( + ) k ( + ) ( n ) ( ) ( ) k + k cos ( + ) k ± 0 - cos + sin nm ( ) m mn ( + ) n d d - + d d + n n e - + e ± 0-0 ( + )( tn ) undefined At (0.0 0.) grd.; t (0.0 0.) grd. Eercise.. f d cos d - sin g d cos d - cos tn cos + - cos h d + d i d + d ( + ) c d c 0 d Dom [0] Rn [] Eercise. 0 c d cos ( ) + sin( ) e ( + ) f ( + ) g e h l e cos ( ) + 0e sin( ) sin ( ) cos( ) cos( ) c sin( ) d tn ( ) + tn ( ) + 0 ( ln( )) ln( ) c d d d d d - d 0 c d - ( ln( )) d - d d d d d d d - + ( + ) e - + d - d d - d d - d d d e + t 0 ( ) ln ln( ) ( ) [ + ln( ) ] ( + ) d d d - d ( + ) - - ln( ) d - d

54 MATHEMATICS Higher Level (Core) ANSWERS Eercise c + d + e + f + g h + 0 c + d e f + g + h e e e c d e f g e h + e + e + c d e + f ( e ) e+ e e + g e h + A: B: Isosceles. sq. units 0 log e ( + ) ; ( + ) + + A: + B: + Tngents: tngent nd norml meet t (0. 0.) m n + 0 c + d e At ( ) ; At ( ) l : + l : + l : l : c l : + l : + Q ( ) z ( 0 ) e e + + ( ) e ( ) + e ( ) Eercise 0. c d ( ) ( ) ( 0) (0 ) m t ( ) min t c min t ( ) m ( ) d m t (0 ) min t ( -) e m t ( ) min t ( ) f min t + m t g min t ( ) 0 - h m t (0 ) min t ( 0) min t ( 0) i min t ( 0) m t j min t k min t ( ) m t ( ) l min t ( ) min t ( ) ( 0) c d (..) ( ). e f g h ( ) ( 0) (..0) ( ) (..0) ( 0) (0 ) 0

55 MATHEMATICS Higher Level (Core) ANSWERS i j - 0 () min t ( ) m t ( ) non-sttionr infl ( ) d ¼/ ¼/ i ( )e ii ( )e i ii c St. pt. ( e ) Inf. pt. ( e ) ( e ) ( e ) ( ) ( ) i (cos - sin)e ii cos.e i ii c Inf. e e d i e (sin + cos) ii e cos i ii c St. pts. -e Infl. pts. - e e d i e (cos sin) ii sin.e i ii 0 c St.pts. -e Inf. pts. (0 ) ( e ) ( e ) - e / / e 0 0 c d min vlue m vlue pt A: i Yes ii non-sttionr pt of inflect; pt B: i Yes ii Sttionr point (locl/ glol min); pt C: i Yes ii non-sttionr pt of inflect. pt A: i No ii. Locl/glol m; pt B: i No ii Locl/glol min; pt C: i Yes ii Sttionr point (locl m) c pt A: i Yes ii Sttionr point (locl/glol m); pt B: i Yes ii Sttionr point (locl min); pt C i Yes ii non-sttionr pt of inflect. d pt A: i Yes ii Sttionr pt (locl/glol m); pt B: i No ii Locl min; pt C: i Yes ii Sttionr point (locl m) e pt A: i No ii Cusp (locl min); pt B: i Yes ii Sttionr pt of inflect; pt C: i Yes ii Sttionr point (locl m) f pt A: i Yes ii Sttionr point (locl/glol m); pt B: i Yes ii Sttionr point (locl/glol min); pt C: i No ii Tngent prllel to is. i A ii B iii C i C ii B iii A c f ' ( ) f'' ( ) f ' ( ). f'' ( ) f ' ( ) f'' ( ) 0

56 MATHEMATICS Higher Level (Core) ANSWERS f ( ) m 0. n. f ( ) 0 ln ( ) i ii i ii c 0 Sttionr points: locl min t ( 0) nd locl m t ( e ). Inflection pts re: ( + ( + )e ( + ) ) nd ( ( )e ( ) ) + Asolute min t ~ -. locl m t ~ Inflection pts t ~ (0..) nd (. 0.) re left s questions for clssroom discussion c d c - d f ( ) 0... Use grphics clcultor to verif our sketch. Eecise 0. Locl min. t locl m t Locl m. t 0 locl min. t ± c Locl m. t 0. d Locl m. t e none f Locl m. t 0. locl min. t 0 g Locl m. t locl min. t h none m. 0 min. m. min. c m. 0. min. 0 d m. min. 0.. (.) (/) (//) (//) ¼ ( 0) -e - ( ) - 00

57 MATHEMATICS Higher Level (Core) ANSWERS Sttionr points occur where tn Glol min. t ( c ); c ( ) Locl min. t ( ); infl. pt. t + Locl min. t ( ); locl m. t ( ) c none ( ) ( e /.e ) 0 e / e Glol m. t ( e 0. 0.e ); infl. pt. t e / e. Verif our grphs with grphics clcultor. Glol min. t (0 0); locl m. t ( e ) Infl. pts. ( ( )e ( ) ) ( + ( + )e ( + ) ) Glol m. t ( 0 e ) infl. pt. t ± - e. c Locl m. t e Glol m. t ( e e ). Infl. pt. t e..e. ( ) Glol min. t - + ln c Glol min. t ( + ln); Infl. pt. t ( + ln) d none 0 Glol min. t f ' ( ) ( ) ( + ) (( + ) + ( ) ) i f ( ) - ; none + ii f ( ) ( ) ( + ); locl m. t - ; locl min. t ( 0) iii f ( ) ( ) ( + ) ; locl min. t (± 0) locl m. t (0 ). Locl min. t ( c c + c ) c Eercise 0. c d e 0 f c c 0. - d e f /

58 MATHEMATICS Higher Level (Core) ANSWERS Asmptotes: c d + 0 i (0 ) ( 0) ii ( ) i (0 ) ( 0) ii iii iv d \{} ( ) 0. f : \{-} where f ( ) ( ) c ( + ) c dom \{0} rn \{} 0. f( ) Asmptotes: 0 0 c 0 d 0 0 Asmptotes: 0 0 c 0 d 0 Rnge \{} dom \{0.0} rn \{0.} g( ) ( ) Eercise. i < 0 ii > iii 0 i < < ii < < < iii c i < < ii < iii d i 0 < < ii < < iii < 0 < e i ii < < iii f i < < < < ii < < < < iii Eercise.. ( deer per er to nerest integer) 00 cm. cm /d 0

59 MATHEMATICS Higher Level (Core) ANSWERS No $0. $ 0. per er c $. per er.0. c.0 0 < < 0 ppro. i 0 < < 000 ii 000 < < 0000 to nerest integer to nerest integer. d. e 0 < < ( + ) D' ( ) -. items/dollr ( ) i ii ( + ) 0 i 0 mm/s ii ~ 0. mm/s 0. sec. cm/s never c never e ms This question is est done using grphics clcultor: From the grph the prticles pss ech other three times c 0 s; s; s d i v ms ii ms A 0.e 0.t v B 0e t ( t ) e Yes on two occssions. m in positive direction i s ii never c 00 ms 0 A B Eercise. i v t > ii t > i v ( e t e t ) t 0 ( t ) ( t ) ii e t e t ( + ) t 0 c i v - 0 t < ii t t d i v ii ( t + ) ln 0 t + log 0 ( + ) t 0 ln0 e i v te t t 0 ii e t ( t ) t 0 f i v ( ln ) t + ( ln ) t t 0 ii v ms never t rest c i m from O in negtive direction ii ms d 0 m e s ms never c t or t d 0 ms t v t + ; t ~ sec c once d use grphics clcultor m in positive direction i m ii m c ms e oscilltion out origin with mplitude m nd period seconds 00 m in negtive direction times c i 0 ms ii ms d m m. units min. unit s c i cos( t ) ii ( ) 0 m ove i v.e 0.t ii 0.e 0.t c 0 m d 0. ( v + ) 0 0 < t < 0 or t > t > 0 c t or t - ( t + ) ( ln ) t t ( t ) 0 t < + t 0 t + + ( ln ) t t 0 window: [0 ] [0 ] i ms ii 0 ms c s d m window: [0 ] [ ] Eercise. r cm s cms cm s da - dt cm s ( side length) cms. cm h 0 cm h c 0. g cm h ~ 0. cms 0. cm min 0.0 ms 0.0 ms cm min kmh 0 rd s V h + h m min c 0. m min 0 m min 00 0 /

60 MATHEMATICS Higher Level (Core) ANSWERS 0 cm s 0. ms. ms 0 0.t [0 00] c i cm s ii.0 cm s d V 0 units c At infl. pts. when. cos ¼ ~. ms ~0.0 ms 0 + 0t t ~0. ms 0.0 cms 0. cm s i 0t ii 0t 0t c 0 kmh d. kmh 0. ms 0.0 ms. ms 0.0 m min. per sec rd per second % per second % per second per er % per second 0. rd per second Eercise.. m. mh $. per km 00 $00000 $. $0.0. m 0. m m m r dim of rect. i.e. prro.00 m.00 m θ m A (00 ) 0 < < 0 c < < 0 cm 00 mls 0 s c R.. c d 0 00 $000 0 & 0 ~. cm rdius 0 cm height 0 cm cm (0 00) 00 C (0.) 0 t 0 t

61 MATHEMATICS Higher Level (Core) ANSWERS h r r c r h r r : h : 0 ~ (0..). m where XP : PY : km r : h : cm : km from P r h r ltitude height of cone ~.0 m wide nd.00 m high when θ rcsin i.e. ppro..00 km from P. tnθ l k + kl + kl ( + k) c if k < c swimmer should row directl to Q. i r h + r ii c r : h : r + rh ( / + / ) / km long the ech c row directl to destintion R First integer greter thn α βeln - lnα β ln R S lnα - β ln 0 0 m isosceles tringle isoscles right-ngled tringle r + sq. units k + k sq. units k( k+ ) c r Eercise. c d e f g + c h + c + c c 0 + c d + c e + c f + c g + c h + c c d e f g h i c f + c + c + c + c / + + c + c + + c / + + c + c d e c / / + + / + c + c u u + c + c + c c d e f + c t + t + c t + + c + + c c + + z + z + c d e t t + c f z z c u ( + ) + c + + c c + c + + c + c + c t + t+ c ( ) + c / / + c - + c + + c + c u + u + u+ c βln α lnα - β ln S 0

62 MATHEMATICS Higher Level (Core) ANSWERS Eercise. + + c d e f g h $.0. cm + P ( ) N + + Vol ~ 0 cm 0 cm Eercise. c d e f g 0.e 0. + c h e + c i e + + c j e + c k e / + c l e + c t t 0 + ( + + ) f ( ) 0 e + c + 0 e + c log + c > 0 log + c > 0 c e + + e + c 0e 0. + c d log e ( + ) + c > e log f e + c > 0 log e g h + log e + c > 0 ln ( + ) + c c d cos ( ) + c e sin ( ) + c log e + c > 0 tn ( ) + c cos ( ) + c ( + ) e + c e + c + c > 0 c d e e f e + log g e + c > 0 h cos + log + c > 0 i j k l m n cos + o c c d e f + c g h i ( + ) j k + c ln ( + ) + c > ln ( + ) + c > l ln( ) + c m ln ( ) + c < n + c o i cos( ) ( ) e ( ) + c d 0 tn ( 0. ) + c e ln ( + ) + e +c f g + ln ( + ) ln ( + ) + c h ln ( + ).ms or.ms 0. cm e sin( ) + + c sin( ) + c cos + c e e + c ( ) + c ln ( + ) + c sin e + + c + e + < > - e ( + ) + c 0 + cos ( ) + c e + e + + c ( ) + c tn ( ) c cos + c sin c + ln + ( + ) + c f ( ) ( + ) f ( ) ln ( ) + c f ( ) sin ( + ) + d f ( ) e + c + / + c > 0 log e e cos( + ) + c sin ( ) + c ( ) + c ( + ) + c + + sin ln( ) ( + ) + c + ln + e e ln ( + ) + c ( ) + c ( + ) + c e < + c 0

63 MATHEMATICS Higher Level (Core) ANSWERS p + q g V' () t..% c ~. litres 0. m noon pm pm m V' () t B 000 c. d d ds Eercise. + c d c d 0 e f g 0 h i j 0 k l 0 e e ( e ) c 0 d e ( e ) e e + e f g e h e i / e ( ) ( e e ) ln ln c + ln d e f ln g h ln i ln A e ( sin cos ) + c c d e f 0 g 0 h i 0 j - c 0 d e f ln g t t ln - ( e e ) h i ( e + ) / ( e / ) 0 ln sin + cos ; 0 m n m + c n d m ( ) e n e e 0. ; 0e 0. 00e 0. + c i ccidents ii N t + 0te 0.t 00e 0.t + suscriers 0 ~ flies Eercise. sq.units sq.units c sq.units d sq.units e sq.units e sq.units ( e sq.units c sq.units e ) ( e+ e ) d ( e e ) sq.units ln sq.units ln sq.units c ln sq.units d 0. sq.units sq.units sq.units c sq.units d sq. units e sq.units sq. units sq.units. ln +. sq.units. sq.units. 0 sq. units 0. sq. units sq. unit c ( ) sq. units tn; ln sq.units sq. units sq. units sq.unit 0 sq. units ln + c sq. unit sq. units + 0 /

64 MATHEMATICS Higher Level (Core) ANSWERS sq. units sq. units i sq. units ii sq. units 0 sq. units i e + e sq. units ii sq. unit iii ln() sq. units.0 sq. units sq. units e sq. units e sq. units c e e e ~ 0.00 sq. units Eercise. t + t + 0 t 0 sin t + cos t t 0 c t e + t + t 0 t t t 0 00 c 00 m ( + t ) + t +. m - m - s;. m s m 0. m st () t 0. m c. m d. m v + k - t > 0 k c. m m Eercise.. k 0. k ln k t cos t t 0. c.% d c 0. d 0. 0 Eercise.. Both 0. Vrince ; SD 0. Mode ; Men 0.; Medin 0. Vr. 0.0; SD 0. All g Vr. 0.; SD 0. c [.0.] Mode Men Medin. SD 0. c Men 0. s c s d s e s All. Men. cm Vr... 0 k c. ds d. ds ( ) / i. ii / 0. ; 0 iii 0. Eercise. All vlues re in cuic units. ln 0 ln.0 Ft () e c Use grphics clcultor. t 0 t 0 c mode. ( e 0 e ) 0 - ln - ( ) ( sin ) / /

65 MATHEMATICS Higher Level (Core) ANSWERS 0 k 0 k - i ii + ( + ) Two possile solutions: solving ; solving 0 then c - Eercise.. c d ( + ) / + c ( + ) / + c ( + c + ) e f g h ( + e ) / + c i j k l m n ( e ) + c o p ( + ) + c q r s ( + sin ) / + c t u v cos w + sin + c ( ) + c ( + + ) + c + + c + c ( + ) ( + ) / + c e + + c + c ( + ) / c + c ( + tn ) c e + + c e + c c d e + sin + c ( / + ) / + c ( + cos ) / + c ( + / ) + c - ( ) + c e tn + c e ( + ) + c e ( + ) + c cos + c f e ( + ) + c g cos ( e ) + c h + i ln e j ln ( + e ) + c k ( + e ) / + c l cos ( + ) + c 0 cos + c c sin + d + c e log ( cos ) + c f log ( + tn ) + c g ( tn ( ) + ) h sin ( ln ) + c i ( + cos ) / c j k ln sin l + c m sec + c n o + c e Sin f + c Tn + c Sin c + c Sin + c d e Sin + c f g h i j k l + + e c ln ( + ) d Tn e n Tn ( ) o Tn p Sin q Sin r Tn s t u Tn Tn - v w + c ( e c ) ( ln( + e )) + c - + c ( + ) + c ( cos ) / + c + sin ( e ) + c e ( + ) + c [ ln( + e )] + c Tn Tn c Tn d Tn + c Cos Sin Tn + c + c + c Sin - + c -Tn + c Sin Sin + c tn + c + c + c Tn f cos g h i j k 0 l m / 0 e e ln -Sin sin + c e sin( e ) 0

66 MATHEMATICS Higher Level (Core) ANSWERS Eercise.. c + d ln e - f g h ( + ) + c e ( + ) + c ln ( z + z ) + c i e sin + c j ln [ e + ] + c k l c d + e ln ( ) + c f g ( ln ) h i + c ln ( + e ) + c ln ( ln ) + c 0 c d e f g h i c d e f g h i c d e ln f 0 c d e f + ln Tn ( + ) + c c d e sin f sin g h ( rccos ) + c i A B Tn k i ii c 0 ( + ) / + c 0 ( + ) / + c ( ) / + ( ) / + c sin + c ( ) / + ( ) / + c e tn c - ln ( + e ) / ( e / ) ln ln ( cos ) Sin c ln( + ) ln k ln -Tn ln 0 - ln ( ). c ( ) / + c Sin + c ( rcsin ) + c ( + ) + c ( t ) / + c ( + ) / ( + ) / + c ( ) / + ( ) / + c ( e e ) + c + c + c ( rcsin ) + c c d e f g Tn Eercise.. sin cos + c cos + sin + c c sin cos + c d e ( + ) + c e e + f g + c ln + c h ( cos + sin ) + c i cos sin j + c ln cos + tn + c k ( )( + ) / + c c Cos + c Tn ln ( + ) + c c Sin Cos + c c ( )Sin + + c c d e f - + sin ( ln ) + c c Sin ln ln + + c ( e + ) [ + ln( + ) ] cos( ln ) ( e ) ( )( + ) + c Sin ( ) ( + )( ) / + c Eercise.. e ( + ) + c cos + sin c + c e d ( cos sin ) + c e cos + sin + c cos( ln ) ( + )Tn + c + sin ( ln ) + c ln + c ( + )( + ) / + c log + + c + c 00

67 MATHEMATICS Higher Level (Core) ANSWERS e f ( cos sin ) + c g cos sin cos + sin + c h + i ln ( ) + ( ln( )) + c j ( ln ) c k l e + e cos + sin + c m n sin ln + c o ( + ln( + )) + c p ln q Tn c ( e e / ) d ln e + e f + e + c e cos Eercise. ln. m m c. m c i sq. units ii sq. units Cos / Sin cos + c c ( + ) + c ( ln ) + c vector sclr sclr Eercise. c d d cuic units d c {egu}; {df} {df}; {c}; {e} c {g}{cg} d {df} {e} e {df} {e} {cg} c d e f g -/ ( e + e + e ) ( e ) cuic units 0. Eercise. vector sclr sclr vector vector 0 AC AB c AD d BA e 0 Y N c Y d Y e N 0

68 MATHEMATICS Higher Level (Core) ANSWERS. N E N N long river 0 i 00 kph N ii. kph N W i 00 ii. Eercise. c c c c d 0 PS c AY d OC ( + ) ( + ) c c c+ c + c m n m Eercise. i + j k i + j + k c i j k d i j + k i + j + k c i j d 0 c d i i c i C A N W E S km 0 B C 0 km A 0 km 0 0 km B 0 0 m/s 0 0 m/s ii iv 0 ( cos 0 ) v 0 cos 0 ( + ) ( + ) ( + + c ) + i k + k i + j + k i j k i j k c i + j + k d 0i + j 0k 0 c d A B ( ) ( ) c ( ) 0 Depends on sis used. Here we used: Est s i North j nd verticll up k D 00i 00j + 0k A 00i 00j + 0k c 00i 00j Eercise. 0 c 0 d e f g h - ( i+ j) ( i + j ) c - ( i j ) d ( i + j k ) e - ( i + k ) f ( i j k ) g h Depends on the sis: i + j+ k or i j ( i j + k ) Eercise.. c c d f g h i j 0 0 c d 0 e f g 0 h 0. c Not possile d e Not possile f 0 c d Not possile 0. ( ) 0 ± ± ( i + j + k ) ( i j+ k ) k 0

69 MATHEMATICS Higher Level (Core) ANSWERS λ ( i 0j + k ) e.g. i+ j+ k c if c or c θ vˆ û i û ( i j ) ii vˆ - ( i + j ) 0 c. 0 ( i + j + k ) Use i s km estwrd vector nd j s km northwrd vector. WD i + j WS i+ j nd DS i j c ( i + j ) 0 d d ( i + j ) e i + j 0 Eercise.. i r i + j ii r i + j iii r i j line joins ( ) nd ( ) r i + j + λ ( i j ) r i + j + λ ( i + j ) c r j + λ ( i + j ) d r i j + λ ( i + j ) e r or + λ r i j + λ ( i + 0j ) 0 f r or + λ r i + j + λ ( i + j) r i + j + λ ( i + j ) r i + j + λ ( i j ) c r i j + λ ( i+ j) r i + j + λ ( i j ) r i j + t( i j ) c r i + j + λ ( i + j ) d r i + j + μ i + μ μ +.μ c d 0 + μ μ + 0.μ j 0. 0.t t c d e f r j + t ( i + j ) r i + t ( i+ j) c r i + t ( i+ j) i + j i j r i + j + t ( i + j ) ( ) ( ) ( ) d r i j + λ ( i + j ) e i M L ii + ii nd iii ( ) Eercise.. r i+ j+ k + t ( i j + k ) r i j k + t ( i + k ) r i + k + t ( i + j + k ) r i j + k + t ( i + j k ) c r i + j + k + t ( i + k ) c r k ( i + 0j ) + t t c t d 0 ( 0) Ø c Lines re coincident ll points re common. t + t z t 0 + t r z + 0.t - + z + t t z t z + z z + z t z + t z z z + t + t z + 0.t 0. M L 0

70 MATHEMATICS Higher Level (Core) ANSWERS + t t z z 0 r 0. t. +. Line psses through ( 0. ) nd is prllel to the vector i j + k.. c. ( 0. ) Does not intersect. L: M: Ø c. d i ( 00 ) ii 0 0 k 0 or z z z plne z i + j k (or n multiple thereof) Not prllel. Do not intersect. Lines re skew. Eercise.. c 0 d e 0 + t z 0 z + t z plne c Eercise.. i + k 0i j k c i j d e i + j + k f 0i j k 0 i + j k i 0 ii 0 i + j k 0. The must e prllel. Eercise.. c. 0 sq. units λk λ ( i j + k ) ( i j + k ) i + j k 0 0

71 MATHEMATICS Higher Level (Core) ANSWERS OA cos αi + sin αj cuic units k 0. OB βi Eercise.. r i+ k+ λ ( i + j+ k) + μ( i j + k ) r i + j + k + λ ( i j + k ) + μ( i j + k ) c r i+ j+ k + λ ( i + j k) + μ( i j+ k ) d r i j k + λ ( i + j k ) + μ i j + k + z z c d + 0z i r λ μ + + ii + z i r λ μ + + ii z i r λ μ + + ii r λ i + z ii + z 0 c d Coefficients re the negtive of those in prt. Eercise.. + z + z c d + + z 0 c nd d + z c + z. 0 c 0 d 0 c z + z + z cos + sin βj z + 0 r i + j + k + t ( i + j + k ) i + j+ k c z + μ i j + k + z 0 Eercise.. + z ; r ; 0 r z 0 c d Eercise.. ( ) Lines tht intersect re nd c; ( 0);. ( ) ( 0 ) Eercise.. i ( ) ii. i ( ) ii 0. c i ( ) ii. d i ( ) ii. (0 ) Plne is prllel to the z-is slicing the - plne on the line +. forms plne. z is in this plne prllel to the -z plne. ( ) Eercise z or - z ; or ; c plnes prllel + z + z + z 0-0z d z ; Eercise.. e.g. the fces of tringulr prism. z or - z or z 0 z z

72 MATHEMATICS Higher Level (Core) ANSWERS No solution Unique solution ( ) c Unique solution ( ) d Intersect on plne C D z E( 0) + None of these plnes is prllel ut the lines of intersection of pirs of plnes re skew. 0 k ; r. t. + or r i j + k c i iii not r t ( + c + c + + c) c 0 c For k Revision Eercises Set A i ] [ ii f ( ) ln( + ) c 0 i 0 ii c 0 F z B G(0 ) t s (.) O H. 0. t z t s z s c + z d e. 0 + λ z c ā c c f Asolute m. t ± - ; locl min. t (0 0); -intercept t (± 0) Locl min. t ± - ; smptotes t 0. ± ( ) ( ) nd ( 0) S [0 [ rnge [ [ c f : [ [ f () (ln) ( ) ( ) c d ( ) f() g() ( 0) ( ) ( ) ( 0) ( ) (0 ) ( ) ( 0) ( 0) ( ) 0 f

73 MATHEMATICS Higher Level (Core) ANSWERS i ii i h + h + h ii + h + h 0 i or ii i \{ } ii 0. iii 0 e i or ii ( e ) i 0 < < ii iii log 0. iv gf ( ( )) \{ ± } P ( ) i ii + c i fg ( ( )) ii []\{0} ln ln 0. + Cuic through ( 0) ( 0) ( + 0) with locl m. t ( ) nd locl min. t ( ) c i k < ii k ± iii < k < c ( 0) (0 ) ( 0) k 0 or ( )( + )( + ) c 0 < < 0 < < 0 c i P ( ) ( + )( )( ) ii < < i ii { > } 0 ± + c < < 0 or > < < c + e gf ( ( )) 0 ii p + p p p c < < or > ( ) e e 0. ( ) 0 c P ( ) ( + )( + + ) ( + ) d nd or nd ( ) ( ) c { < } { < < } < k < p q 0 c i { < < } { < < } ii { < } - n 0 ii {±} i ( ) ii 0 c ] ] c ] [ sq units d 0 c d c d c c d 0 c d c c c i & ii i ii $00 iii $ iv t f ( ) + - f ( ) g f g ( ) e + e 00 e 00 c d t 0

74 MATHEMATICS Higher Level (Core) ANSWERS i ]0 [ ii c ( log d e e ) 00 c 0 d 000 e t > f Bt () t 0 0 Bt At h ( ) 0 rnge ] ] Use grphics clcultor. f ( ) log e ( ) < c Use grphics clcultor. 0 0 λ λ z λ λ r g d f fog eists; r f d g gof doesn t eist. < or > λ λ z + λλ λ λ λ z - λ c S ][ f ( ) ( ) < r g d f f og does not eist; r f d g gof eists. c F ( ) t or t c + λ λ z λ λ i 0 ii 0e. c d i 0 ii. 00 Q(t) P(t) 0 0 cm cm c hrs d [0 ] e f Use grphics clcultor. g. hrs z ]0 ] c No ( 0). 0 c r f d g i.e. does not eist g ( ) + t h ( ) g( ). t - 0. g ( ) f Incresing t decresing rte g ~ 0 wsps h ii t 0 nd Revision Eercises Set B c d km c d km e 0 i A: $000; B: $00; C: $00 ii A: $000; B: $000; C: $00 % c i months ii C never reches its trget ( i) c r 0 cm or c i ii iii ( k + )( k )! - i ii + i or i cis ( θ ) c i ii 0 R α t 0 log e 0