f 5 x 3 d x x 12 x 16 x a x < 1 b x < 2 a c x e f b i c d + +

Transcription

1 ANSWERS Chper 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} c Ø d {,} e {,} f {0,} c d 0 ],[ ],[ c ],[ + Eercise.. - c d e f - c d e f - c d e f c d e f g h - + i , c, d e f g h 0 i, 0 j k l Eercise.. c d e f - 0 c - < < c - d - + c d e f g h i - c d e f g h - i p < Eercise.. c d e / / - / 0

2 c + c c d + / f g h i j c d k l m n o p q -/ - / - -/ c d f() f() f() Noe signs! f() c. m, m 0, c m,. + c d e f - + Eercise.. z 0 z c 0 d z e f z 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 soluions k l no rel soluions m n o < p < p ± c p < or p > m m < c m > 0 0 m ] [ ] [ c ] [ k ] [ ] [ c ] [ Eercise.. Grphs re shown using he ZOOM viewing window: z - c Eercise..,, c, d 0, e, f, - - c 0, 0 d - - e f

3 d e f 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, ) (, ) c i - 0 ii (0, ) Grphs re shown using he ZOOM viewing window: c (, ) c i - 0 ii (0, ) (, 0), (, 0), (0, ) (, 0), (, 0), (0, ) (0., 0), (, 0), (0, ) d e f (, 0), (, 0), (0, ) (., 0), (., 0), (0, ) (, 0), (, 0), (0, ) k k c k k - k c - k - k k c k k - c d c d Eercise.. ],[[ [,] c ],0][ d ],[ e ],.][ f ]0.,.[ + 0-0

4 MATHEMATICS Sndrd Level ANSWERS ],[[ ],[ c ],0.][ d [,] e ] - [ f ],][ g [ ] h [.,] i ],[[ 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.] {: < }{: > } {: < < } Eercise.. (, ) (, ) (, ) (, ) c d - 0 e f g h no rel soluions + + i - j (, ), (, ) + + k no rel soluions (, ), (, ) - c (0, ), (, ) d e Ø f (, ) g Ø h - m. m c m 0 - c - m i (, ), - ii (, ), c i A(,), B(, ) ii sq. unis - Chper Eercise.., 0., nd or nd, 0 m kmh, ;, ds $0 ech. kmh hours hrs 0 hrs Chir-one: 0; Chir-wo: km. km 0. hrs, 0. hrs Eercise.. i 00 ii 0 < < 0 (Noe: if 0 or 0, A 0 nd so here is no enclosure) i A ii 0 m 0 m or 0 m 0 m iii 0 iv m 0 m ii 0 < < i 0 ii iii m m m m c A d m m 0 C $

5 R p i $0000 ii or (s nswer mus e ineger vlues) iii $ i A 0 ii 0 < < 0 c i - m ii ; dimensions 0 m m. kmh 00% 0. (firs ime) hen gin.0 c i.% ii. weeks 0 < < A c i ii. iii + i 00 m ii 0 m i 0. sec nd. sec ii. s c s d 0 m 0. s (on he w up) nd. s (on he w down) 0 s c 00 m d.0 s e 0 m $00 No. (loss of $0000) c 00 ($) i $0000 ii $0000 C A 0 [0, 0] [0, ] [0, 0] [0, ] d As ime increses, ogen level will e 00%. Cos $ R Gin Loss 0 usge ('00 MJ) 0 B(, ), C(, ) unis in meres C c $.0 d $..0 $C Loss Brek-even poins c Fied cos (e.g. slr, elecrici,...) d See grph in pr e $ (o neres dollr) f i P ii $000 iii 00 g i 0 0 or ii h See grph in pr (km)

6 MATHEMATICS Sndrd Level ANSWERS ($) liner c i ii M 0,, i.e. m R Gin C 000 Loss Brek-even poins Loss i P ii d The compn will rek even rdios nd rdios. Provided he compn sells eween nd rdios he will mke profi. e 0 Eercise.. ii ii c ii Second difference & c + 0 c prold P + + P k + + k k i 000, ii k 0 k S (, 0000) i prolic + c i p 0.q + ii p 0.0q +. d Opimium scenrio: demnd suppl. This occurs when p.0, q C p.m. The model is no vlid ouside d rnge, herefore erpolion will no necessril work Equion of ph:. Grees heigh:. m. ii h c i. m ii. m [BE] : +., [BO] :., 0 c second difference is consn 0 c d $00 per cr e i $0 ii $0

7 i & ii hve consn grdien iii resuls impl qudric form ii p , C + 000, R c P , m. profi P Chper Eercise.. + c + c + g + g + g c d e f g h + 0 p - q q 0q 0q - 0q p - 0q + + p + - p + p + - p r Eercise.. 0 c d 0 e p f 0p q g 0p c 0 d 0 e f % c 0. d 0.% 0 0 n n i 0 + j + k l + d + d + d m n o - 0 n p p + p p Chper 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 o mn mn o one c mn o one d one o one e mn o mn f one o one \{} ], [ c [,] d ] [ e \{0} f g \{} h [, [ i [0, [ \{ } j ] [ k l \ ],[ ]0,] c ] ] d [,[ e \{} f ],[ g [,[ h ], 0[ Eercise. Grphs wih grphics clculor oupu hve sndrd viewing window unless oherwise sed., i (+) + ii c 0-0, c , c no soluion 0, + + i ii Window [,], [,] Rnge: [, ]

8 i ii {, }, c, d, e, d, e, f Window [,], [,] [0, [ 0 {: > } {:.} 0 onl i is he onl one wih idenicl rules nd domins [,[ [,0] c [,[ d [.,[ ],[ i p + 0 ii A i ii r ],[ Eercise.. r ]0,] (, ) 0 c d c d c d c d (, )

9 Eercise.. c g h i (, ) d e f c 0. g h i c 0 [, [ [, [ d e f ],0] [0, [ c d d e f c

10 d e f : : g h i Eercise.. c d (, ) ]0,[ (, ) ]0,[ (, ) ]0,[ e f g h (,.) ]0,[ c d (,.) ]0,[ (,.) ]0,[ (, ) ]0,[ i j k l (, ) ]0,[ i Ø ii [,] iii {±} (, ) ]0,[ (, /) ]0,[ (, /) ]0,[ (, 0/) ]0,[. 0. hs dilion effec on f () (long he is).

11 c d c (, ) (, ) (, 0) (, 0) (, ) (, ) (, ) (, ) [,] [,] c [0.,] d [0.,] e [0.,0.] f [0.,0] c d ],[ e ],[ ],e[. c i f g: ii f > g: < g ],[ c d (,) (,) ]0,[{} ],] (,) [0,/] c d ],] [,[ [,[ [0,[ ], + e [ [, [ c [ e, + e ] c (, ) 0 c d e f f

12 c Eercise.. c ]0,] d e f [,[ [,[ ],[ ]0,[ c d ],[ ],[ ]0,[ e f g h > ]0,[ {} < ]0,[ [0,[ ]0,[ ]0,[ 0 d e f ],[ / ]/,[ c e e ]0,[ ]0,[ d e f ],/e[ /e ]0,[ ],[ ],[ ]e,[ e ]0,[ c ]0,[ \{0} \{0} e 0. ]0.,[ ],[ ]0,[ 0 ]0,[ d e f ]0,[ ],[ ],[ ],[

13 c ]0,[ (, ) ]0,[ ],[ d e f \{} ]0,[ e \{0} c d [0,[ ],] c [,[ ii 0 c + ],[ e ]e/,[ ],0/[ 0 - c d e d e f \{e} ],[ /e 0 < < ~. / \{e} e / : ā - + ],0[[e,[ i

14 Eercise.. c d d e f / e f g h / / / 0... Eercise.. i f+ g: 0 where f+ g + [0,[ / ii f+ g: 0 where f+ g + ln [,[ / c d. c i ii iii f+ g: where f + g +, 0 i ii fg: 0 where fg 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 f/g: \ 0 where f/g - e e ii fg: 0 where fg f/g: ] where f/g / ln + iii f/g: \{} where f/g - + i fog +, gof + ii ], [, ], [ i fog + 0, gof + ii [, [, [, [ c i, + ii [0, [, [, [ d i fog 0, gof 0 ii \{0}, \{0} e i fog 0, gof ii [0, [, [0, [ fog gof f i fog 0, gof() does no eis. ii ], [ g i fog 0, gof 0 ii ]0, [, ]0, [ 0

15 h i fog, gof ii [, [, [0, [ i i fog +, gof ii [, [, [0, [ j i fog() does no eis, gof ii [0, [ k i fog, ii [0,[, [0, [ gof l i fog + +, gof + + ii [, [, 0. [ S \],[; T T : 0 ; S ],] [,[ gof does no eis fog 0 fog 0 gof m i fog, gof ii [, [, ]0, [ n i fog() does no eis, gof - ii \{} + o i fog() does no eis, gof + ii ], [ p i fog 0, gof 0. ii [, [, ]0, [ fog + gof + c fof + g + fog + + \ 0, ],] [,[ gof() does no eis. 0 + c gog + + 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 no eis. koh log, Dom f ]0,[, rn f ]e,[, Dom g ]0,[, rn g fog does no eis: r g d f ]0,[ gof eiss s r f ]e [ d g ]0,[ c gof: ]0,[, where gof + + ln fog ; rnge [0, [ c 0 d f \ (, ) (, ) in f, no g g rnge ],[ c rf \ c rf d fof f d fof dom rn ]0,[ +ln gof: ],[, where gof fog*: ],[, where gof d fog fog c d, gof / / fog rnge 0 f

16 Eercise.. c + d e 0 f g 0 h c d e f g h c d / (0, ) f log f log + c f 0 d g + log log + e h log \[,0] f g f f c f, d f + + e f / + f f 0 0 log - + c inverse inverse inverse d e f f + + (, ) inverse 0 (, ) inverse inverse e f g h (,) (,) (, ) dom [, [, rn [, [ 0 f f c + f

17 f (, ) [, [ + \{.} Inverse eiss s f is one:one Cse : S ]0, [ Cse : S ], 0[ g f + 0 g - + : f f + /f gof eiss s r f d g. I is one:one so he inverse eiss: + f :, f + e i ii f is one:one f (, ) + iii iv {, 0, } f + f c d f e ln 0 e e e f 0 i om e 0 ii mo e i om ln ii mo ln 0 c i & ii neiher eis d Adjusing domins so h he funcions in pr c eis, we hve: om mo nd m o om e Yes s rules of composiion OK. f

18 f MATHEMATICS Sndrd Level ANSWERS fog eiss u is no one:one g (om) (mo) c i B [ln, [ ii fog : [0,[ where, fog ln + iii iii iii iv iv c (,) ln Chper Eercise. ln c d e + f + 0 g h 0 i + j k 0 l + i i ii ii + + c c (,).

19 Firs funcion in lck, second funcion in mroon c d e f. Noe: coordines were sked for. We hve lelled mos of hese wih single numers. c d (, ) e f g h (, ) 0. (, ).. g h i j (, ) i j k l (, ) (, ) / (, ) m n o (, ) (, ) 0 c d e f 0 g h i j k l k m n h o g f + g f + c g f

20 d g f + e g f + 0 i ii iii (0., 0) Eercise. c d iv (, ) (, ) 0 (, ) i ii (, ) c d iii iv c i ii (, ) (, ) 0 iii iv d i ii iii iv (, ) i i ii ii iii iii f + + f + +

21 iv iv i ii 0 f f + f f c f f d f f e f f f f f + (, ) (, ) (0, ) 0 (,) (0, ) (, ) (,) (, ) (, ) (, ) f iii iv f (0, ) (, 0) i ii (, ) f c d. (0., ) (0., ) (0., ). + if 0 if 0. (0., ) c f h if if (0., ) (0., ) h. if if f 0 iii iv f (0, ) (, 0) 0 (, ) f k (, ) d e f if k if (, ) if if f (., ) (, ) + if 0 if 0 (0, )

22 0 (, ) (, ) f (, 0) (, ) f (, 0) (, ) (/, ) c d (, 0) Eercise. i ii 0 0 > 0 - (, ) ā - (, ) - ā - < 0 (, ) 0 f (, 0) i ii 0 0 c i ii (, 0) (0, ) d i ii (,) e i ii (, ) 0 (, ) (, ) (, ) (0, ) 0 (, )

23 f i ii (, ) (, ) (, ) f f c f+ d f e f c d 0 0 e f g h i j k l m n o p 0. q r 0 0 (, ) (, ) (, ) c d e f g h i j i ii (, ) (, ) (0, ) (0, ) (, ) i ii (, ) (, ) (, ) (, ) (, ) - 0 (0., ) (0., ) 0 ā - (, ) (, ) (, 0) (, ) (, ) (, 0) (, ) (, ) (, ) (, ) (, )

24 c i ii d e f c (, ) d e f (, ) (, ) (, ) Eercise. c d e f c (, ) (, ) (, ) (, ) / 0. (, ) g h i j k l Eercise. Miscellneous quesions i ii / (,) / (, ) (, 0.) 0

25 i ii (0, ) (0, ) c 0. c i ii / (0, ) / / (0, ) / c i ii c d i ii (, ) g i g ii + c i ii iii g g iv v vi f f f f

26 Chper Eercise.. - c d e f n + + g n + h n + i c) d e f z c d e f g n n + n + n h n + n + i m m + c d e f + c d e f - + h + n c d e f g h / n c d e f Eercise.. n mn p + - q pq - n + - c d e f. g h. i n n m n n n + c d. e 0. f 0. g h i. Eercise.... c d. e. f. g. h i 0 0., c 0, d, e, f 0,,,,,, c,, d,, e,,, Eercise.. i. ii. iii. i. ii. iii. c i. ii. iii. d i. ii. iii c 0. d0., 0 e. f 0, 0. Eercise.. c 0. d c 0 + n n / + - 0,, c d, e 0, f.. c. d Ø c 0.0 d c 0. 0 e k ln Eercise. 000 c 000 d 0 ds kg c. ers d W 0.0.% c. m d. m e I I 0 i ii iii. ers c 0. ers d c. kg d W e W C i 00 C ii 00 C c.0 million ers d T N 0

27 0.0.0 gm c 0 ers d 00 Q mg/min. min c i. ii. iii min d. mg e f No A $ million $. million c 0. ers d V R i $ ii $ iii $ c d i $000 ii $. million iii $. million f g $. million & e R 0.0 c 0 d. ers 0 0 cm. cm c 00 ds d ds i 0 ii c. erl 0 i $. ii $.0. ers c c T (.0,.). C ~ midnigh. c 00 cm cm c. m d m e i. ers ii. ers iii. ers f g A h Eercise. c d e f g 0 h 0 i j k 0. l log log c log 0 + d e log f log c 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 0 + k - l log w log c d log e f log - - c d e f,, c, d, log + p log 0 p + - log + 0

28 MATHEMATICS Sndrd Level ANSWERS log -. log c log log - h. i 0. j no rel soluion k 0. l 0 0., c, d 0, 0 0 e f 00, 0 c, c c d Ø ln.0 ln 0.0 c ln. d e ln.0 f g h. 0. c 0, 0. d, 0. e., 0.0 f 0,. g 0., h i. j..0 c 0., 0. d. e.0 f. Eercise. 0 0 c 0. kg. c - log 0.0 log0 log. log d 0. e 0 log 0. f. g log log z - e e - log log log log ln - 0. e 0.0 m L.. L L 0 0 c d 0. log 0 log log ln 0. i e 0.0 j Ø k e. l e ln ln c ln ln d 0 e 0 ln f ln 0 d e W. W. 0 0.h h h. W L 0 e.0 [0,[ i. ii. iii 0. ers c As c increses, reliili reduces. d 0 c e I n k m m L 0 L k c.% d k log - 0 Chper Eercise.. i c n n ii c n n + iii c n n + iv 0. c n 0.n v c n + n vi c n n, 0 s 0 i ii i ii n n 0 Eercise.. 00 c 0 0 c c 00 c h week 0 0 Eercise.. Miscellneous quesions, 0.,,.,,.,. d $, weeks 0 $ 0 i m ii 0 m m c Dis n n nn d e plers, 00 m c +

29 0 c n Eercise.. r u u n n r u c r d r u u n u e r f r - h u n 0 c n imes i $0 ii $0.. ers 000 u n - - n 0..,, 0 or 0,,. 0 0 $ $ Eercise.. c d e. f. 0 0 c d e ; ; c ; 0.00 d ;. e ;. f ;. - c d 0 e ; 0 $0.0. cm 0. gms; 0 weeks. 0., 0. u n - u n n V n V 0 0. n 0 n n.0 - u - u n u - u n n n n r,. 0 0 $0.. 0 or ou 00 illion onnes. Eercise.. Term AP 0, GP. Sum o erms AP 0, GP 0., weeks Ken $0 nd Bo-Youn $) week week. (~00, depends on rounding errors) Eercise.. c 000 d fish. (Noe:. If we use n hen ns is 0 fish); fish. Overfishing mens h fewer fish re cugh in he long run.,, or,, - - c 0 cm 0 Eercise.. Miscellneous quesions, c 00 + n + - n

30 0 n n, ± (,, ), (, 0, 0),,, c n m Eercise. $.0 $. $. $ $. $0. $00.0 $., $. $. 0 $., $., ineres $.. Fl ineres $000 $., $0., 0.0% /monh (or.% p..) Chper Eercise. cm cm c cm A B C c... 0 d e... 0 f.. 0 g... 0 h i... 0 j... 0 k... 0 l... 0 m n... 0 o... 0 p q r... 0 s u... 0 v w c d + e f + 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 c d + c h 0 h n + + c + h m 0. m. m 0. cm ' c '. m ' c ' Eercise.. cm. cm c. cm d 0. cm e.0 cm h n c

31 f. cm g. cm h. cm i. cm j. cm k. cm l. cm m.0 cm n. cm o. cm m 00 cm. cm. sq unis.0 sq unis c. sq unis. cm ' + n - n Are of ACD 0. cm, Are of ABC. cm Eercise.. cm cm c cm A B C c d... e... f..0. g.0.. h. 0.. i j... k. 0.. l... m n... 0 o... p... q... 0 r s Eercise.. c A B C c* B* C* c d e f g h i j k l m n o p q r s d no ringles eis. 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 c... 0 d...0 e f g... h i..0. j... 0 k... l... m... n o p... q 0... r... 0 s u... 0

32 v.. 0. w Eercise.. 0. km T '. cm. m. m. km W ' S Eercise.. Miscellneous quesions.0 m. m 0. m,, 0 T. 0. cm cm,, lef. 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 sin i ii or - c d Eercise. d sin sin - cm 0. + cm d sin n sin - cm,, m. c cm, + cm d, d sin n sin + cm + m sin cos sin cm. + cm e, f, g.m, +.cm h, cm. i, j, k cm, + cm l, m, - cm. n, o, 0. c, 0.0 m. c cm. cm. m 0. m c. c 0. c 0. c i.0 cm ii. cm c 0. cm cm.. cm 0. cm. cm cm cm c c 0... cm Chper 0 - n + - cm + - cm. 0. Eercise c d 0 c - c c - c d c - cm. + cm - cm cm cm + cm - cm cm cm + -cm 0 - cm. + cm

33 c d e f g h i j k l m n o p q 0 r s 0 undefined 0 c 0 d e f g h i j k l m n o p q r s c d e f g h c d e f g h i j k l c d 0 c d c c k c k c c c k k c k sin co c d e co f n k k c k k k k 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 Eercise 0.. sin cos + cos sin cos cos sin sin c sin cos cos sin d cos cos + sin sin e - n n f n n + n n + n n sin cos + c sin + d cos e n f n g h i + - i ii iii i ii iii k k n sin + + sin

34 - - c - - c c d c d c d c - + d R + n 0 R + n Eercise 0. c d e f c d c d e f g h i j c n - c d e f g h / c d e f g h / / / c d e f g h / / c d /.. / / / / 0

35 e f g h k k + k i j k l m n c d 0 i ii c Cos / / c sec 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 cosec c d e f c d undefined e f g h - co Sin n - n + Eercise 0. - c d e - f c d e - - f c - d n e - - f 0, 0 0, 0 c 0, 0 d, e f 0,, g - h

36 MATHEMATICS Sndrd Level ANSWERS 0, 00 - c d, ' e - f - g - h. c,.0 c i j - k l ', ' m n - o Ø c d e f g h - - c 0,,,,,,, cos n + n c d n - - ii 0 ii n n n i k + k ii k + k + k n sin sin n sin + sin sin + sin k c cos - - 0, ',0 ' ( ',0 ') Eercise 0. cos -,,, T sin - + c.,.,, L sin +.,, 0, V cos- k k+ k k + k,,, P.,,, S. 0.,., 0, P 0. 0.,.,., D , 000 c sin - + sin - + sin - +. m,. m. sec,. sec 0 0, 0. c mid-april o end of Augus sin monhs c R d monhs sin. +. d monhs k + k k c k k k,, m c m

37 F() G() c d.% d c d i,,, ii 0 c. m Chper Eercise. vecor sclr sclr vecor vecor vecor sclr sclr Eercise. c d {,,e,g,u}; {d,f} {d,f}; {,c}; {,e} c {,g},{c,g} d {d,f}, {,e} e {d,f}, {,e}, {,c,g} c d e f g AC AB c AD d BA e 0 Y N c Y d Y e N ii iv 0 cos 0 v 0 cos 0. 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 i i C c i A N W E S km B + C 0 km A 0 km km B 0 0 m/s 0 0 m/s

38 + c c c c 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 m - n, (, ) c c i j k i j k c i j k d 0 0 c d 0 + i k + k i + j + k + + 0i + j 0k A, B (, ) (, ) c (, ) 0 Depends on sis used. Here we used: Es s i, Norh j nd vericll 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 he sis: i + j+ k or i j + k i j+ k i j + k Eercise.. c c d f g h i j 0 0 c d 0 e f g 0 h 0. c No possile d e No possile f 0 c d No possile i + j + k i 0j + k e.g. i j c if c or.,., 0. i + j + k Use i s km eswrd vecor nd j s km norhwrd vecor. d WD i + j, WS i + j nd DS i j c d e i + j - c vˆ û + + k i û - i j ii vˆ i + j 0 c. - i + j 0 - i + j 0

39 Eercise.. i r i + j ii r i + j iii r i j line joins (, ) nd (, ) r i + j + i j r i + j + i + j c r j + i + j d r i j + i + j e r or + r i j + i + 0j 0 f r or + r i + j + i + j r i + j + i + j r i + j + i j c r i j + i+ j r i + j + i j r i j + i j c r i + j + i + j d r i + j + i + +. c d c d e f r j + i+ j r i + i+ j c r i + i+ j i + j -i -j r i + j + i + j (, ), (, ), (, ) d r i j + i + j e i M L ii + ii nd iii (, ) r k i + 0j Eercise.. No. mins fer A j Ø c Lines re coinciden, ll poins re common M L i r A ii No c i ii.m. + r B + + Eercise.. r i + j + k + i j + k r i j k + i + k r i + k + i + j + k r i j + k + i + j k c r i + j + k + i + k, c z - + z + z + c d,, 0 0 r Line psses hrough (, 0., ) nd is prllel o he vecor i j + k.. c. 0. Does no inersec. L: M: Ø c. + z d i 00 ii 0 0 r z z 0 z z z + +. z z + z z plne - + z + z z z z + z + + z z + z plne

40 Se A Mode. Men. Medin. Se B Mode Men. Medin. Se B is much more spred ou hn se A Med Q 0 Q IQR Sles Eercise. i 00 ii (0.) Smple size is lrge u m e ised fcors such s he locion of he cch. Populion esime is 000. i 00 ii 0 00 c 000, c numericl,, d, e cegoricl, d discree,, c, e coninuous Eercise. i + j k (or n muliple hereof) No prllel. Do no inersec. Lines re skew. Chper 0 or nd lhough he wo ses hve similr men, he hve ver differen mode nd medin. Eercise. Mode g, Men g, Medin g Mode.. g, Men. g, Medin.0 g Se A Mode., Men., Medin.; Se B Mode, Men., Medin. $ $0 c Medin $ 00 $ 000 c Medin.. Eercise. Smple A Men. kg; Smple B Men.00 kg Smple A Smple sd 0.0 kg; Smple B Smple sd 0. kg c Smple A Populion sd 0.0 kg; Smple B Populion sd 0. kg.. Men., Sd. 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 $. $. c $ d Q $.0, Q $ IQR $0.0 e Medin nd IQR... c d Q, Q, IQR $ $ ce 0 Med $0 0 Q $0 Q $0 0 IQR $0 k MATHEMATICS Sndrd Level ANSWERS

41 Eercise. Miscellneous quesions Smple00 rndoml seleced piens, populion ll suffering from AIDS Smple000 working ged people in N.S.W, populion ll working ged people in N.S.W. c Smple John s I.B. Higher Mhs clss, populion ll seniors Npp Vlle High School. Discree:,, d; Coninuous: c, e, f, g. Frequenc Cumulive frequenc non-liner relionship!) f i Decresing ii Appro. liner iii Mild Posiive ssociion, liner, srengh: ver srong Posiive ssociion, liner, srengh: ver srong D displs srong posiive ssociion. Increse in led conen cn e riued o increse in rffic flow. suggesed nswers onl: 00; ; 0;... 00; 0; c 0; 0; Mke use of our grphics clculor. grphics clculor c. d. 0 grphics clculor c 0. d.0. c. d.0 sec.,. 0 Q~, Q~ ~ 0 c % d. rnge, s..;.0.;.., Chper score s n. s n. s n. s n. score Eercise. i Incresing, posiive ii Appro. liner iii Mild (o wek) i No ssociion iiiii 0 i Incresing, posiive ii Liner iii Ver srong d i Incresing, posiive ii Squre roo iii Mild (srengh no pproprie s i is non-liner relionship!) e i Decresing, negive ii Eponenil iii Mild (sengh no pproprie s i is Eercise. r 0. Worksfe polic hs hd desired effec, i.e. numer of ccidens hs decresed. D displs srong negive ssociion.

42 r 0.0 (sumed liner) iii r 0. No, no liner! i ii iii No. The relionship is no liner. i % ii % i % ii % c i ii. +. iii r 0. ± 0. % 0 r 0. There is srong evidence o sugges h suden who does well in Mhs will lso do well in Biolog. c Vlues of r re he sme. Eercise. i ii. +. d i ii iii i ii r 0. c.% d. +. e i ii 0..

43 r 0. c i ii Liner rend eiss, r 0. Bsed on he scer digrm, here is definie liner relionship. Therefore, owner is jusified. c i r 0. ii C. +.w d i 0., i.e. ii., i.e. iii From ii, serving people per hour is unrelisic. Scer digrm shows liner relionship. Therefore sisic is pproprie, r 0.. c i. +. ii d i. ii.0 r 0. 0.% c.0 +. d (,.), (., ), (,.), (0.0, ). The equion is used o predic from -vlues, no from -vlues. We would need o find he regression of on. 0 i r 0. ii c M 0.T +. c i ii Remins he sme. 0. d i. ii. iii is fir w ou from he se of vlues used o oin he regression line Scer digrm shows liner relionship. Therefore sisic is pproprie, r 0. c i ii c i ii d i. ii 0. d (0). (inerpolion); (0) 0. (erpolion)

44 Eercise. Miscellneous quesions i ii r c 0. Becuse of he srong posiive liner ssociion, nd he high r vlue, we cn s h he ller he suden he greer heir weigh. 0.0 B 0. Srong posiive relionship. 0. i P.0M. ii % c i M 0.E +. ii 00 iii Erpoled. Coninued liner rend highl likel. Therefore confiden d Find regression equion of E on M, hen use M 0 ino his new equion. Posiive Liner c Ver srong 0.,. 0. c c. 0 i ii 0.0 r., h is,.% c.. d e,.; Ependiure is $ i.;.0 ii.0;. 0.; r 0. c r d Regression equion is \ when.,. c d When 00,.. The crcss weighs. ls. Chper Eercise c 0 0!! c i! ii! 0 00 Eercise Eercise. Miscellneous quesions

45 ; c n n C C c , ; Chper Eercise. c - - {HH, HT, TH, TT} {HHH,HHT,HTH,THH,TTT,TTH,THT,HTT} c c d - c 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 c c c c - c d e - 0 c - d - 0. i 0. ii 0. c d Eercise c 0.0 d c 0.0 d c d - - {TTT,TTH,THT,HTT,HHH,HHT,HTH,THH} i ii iii iv - / / - R R _ - / / / / R R _ R R _ - - -

46 c 0. c c c 0. d c 0. d c c c 0. d c 0. Eercise i ii 0. i ii - N m H T F /0 /0 /0 - N m N 0 0 / / / / / / / / / / / c 0.0 d c c c 0. H T H T H T Y B G Y B G Y B G N m - m - m + N m n Eercise. - c - c d 0 c c c Chper Eercise i 0. ii

47 - p 0 p - p p c {,,,,,,,, 0,, } c d 0 p() p() T - i ii c i 0.0 ii T H T p 0 - p p i ii - H T H T H T - - / H p() c p 0 H T / - p 0 p p - p - p p p p Eercise... i ii c i ii 0. i. ii. iii 0. i 0. ii. c i ii 0.. np. c. 0. i 0. ii c i 0 ii iii. 0 p 0 p p p i 0. ii 0. c W N, E(W) 0. $.00 oh he sme 0 c c p() n 0 P(N n) - n P(N n) - s P(S s) E(X) p, Vr(X) p( p) i n( p) ii np( p) n 0 P(N n) - - W

48 E(X), Vr(X) 0 Eercise 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. les 0 c d i ii. i ii. 0..,. 0. $. $ $ c 0 EX Vr X 0 p 0. p c. Chper Eercise 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. Eercise 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 % % % i 0.00 ii 0.0 iii $. $ i 0. ii 0.0 c $ μ., 0. $0.S μ.,. 0 (.) i 0. ii 0. i 0. ii 0. c 0.

49 Chper 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. C/sec cm /cm i. cm /cm ii. cm /cm iii cm /cm.. C/min o 0 m m/s c verge speed d m e m/s $0, $., $0.0, $0., $ 00. $0.0 per er Eercise. h Eercise. h + + h c d h +h c d + h ( + h) c ( + ) + h d + + h + h - h c h d h e f h - + h h e ( + h + h ) f + ( )h + h g - + h h i - + h + h+ ; + h ; c + h + h ; d + h + h + h ; h h () i ms ii ms iii. ms i 0 cm ii. cm iii. cm 0 s() iv. cm /d c 0 0.h cm /d d i. cm /d ii. cm /d Eercise. c 0 d e 0 f g h e i j c 0. d 0 e 0 undefined c d e undefined i 0 ii 0 i ii undefined c 0. d e c undefined c d e 0 c d 0 e c. d e 0 0 i ii iii iv 0 d Find (limi) s h 0 e + 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

50 MATHEMATICS Sndrd Level ANSWERS () i ms ii ms c ms d sec Chper Eercise. c d e f g h 0 + i + j k l c d e f g h 0 i j k l c d e f g h i j k l - + Eercise.. m PQ h ; P Q + h - ; ; + h m PQ c d e f g h c (, ) + 0 c -,(0, 0) : 0 : lim m PQ h lim m + h PQ h ,, 0 Eercise.. c ' f ' d e f ' ' g h i ' ' ' ' ' ' c d

51 Eercise.. n - c d n - n r r r + e 0 L f 00 g h i v l + + h n - + n c s / d + e f - r 0 - r Eercise c d + + c d e f sin + cos e ln + c e d cos sin e sin + cos f n + + sec g cos sin h e cos + sin + sin i ln + + ln e sin cos sin + + cos e c - - d cos sin e ln sin + e + ln f - + ln e g + h i ln + + sin cos + ln e + cos + sin c d s + e sec + e f sin + e g h 0 + i j k l cos sin cos cos + sin cos sec cos c d - e e sin cos f e e cos g sec h i sin log e cos sinsin cos j sin sec k cos csc l csc - - sin / m m + cos + e + sin - cos sin c e d e f g h i + j cos k sin e cos l m n e + o e + p + e + cos + e c + e d e f g h sin cos i j k + co l ln + + sin c + sin cos sin cos + d e + e ln + sin ln f ln sin g cos 0 - h ln 0 + sin ln0 + i cos sin e j ln sin + co k cos sin e l sin + cos sin sin m n o + p + 0 q r / s u v w - n n ln n e e + e e sin + - ln + cos sin c e + e - + cos 0 - e e - + i sincos + cos - sin ii e cos ln cos sin ln cos sin n e e + e - + cos + sin 0 n + - n n e -e e + e sin e e ln ln + n -e cos + sin lnsin - sincos e +

52 i ii i cos e ii cos e sin 0 :n n m n n m m n csc sec n c co csc d sin e csc f sec n c d e cos + sin f co csc g csc co csc h co sec csc n i sec n sin cos + sec e sec sec n e sec e n e c e sec + e sec n d - csc log e csc sec f - co g cos cosin csc sin h coscsc co csc i 0 Eercise. 0 + c d e f g h k ln m n - + m+ n i j + k l m sin cos sin cos + sin sin 0 sin cos n o p q r cos sin s + [0,.0[ ].,] - e sec n sec n co csc e + e + e - n ln + n ln f ln n - n n f '' n n n csc log e - Chper 0 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. unis, 0 log e + ; + A: +, B: +,, Tngens: ngen nd norml mee (0., 0.) m, n Eercise 0. Q + + z 0 e e e c d (, ) (, ) (, 0) (0, ) m (, ) min c min (, ) m (, ) - d m (0, ), min (, -) e m (, ), min (, ) (, 0) + e f min + 0 -, m g min (, ) 0 + (, )

53 h m (0, ), min (, 0), min (, 0) i min (, 0) m - j min - k min (, ), m (, ) l min (, ), min (, ) c d (.,.) (, ) -. e f g h (, ) (, 0) (, 0) (0, ) (.,.0) (.,.0) - c Inf. e - e d / i e (sin + cos) ii e cos i - ii - - c S. ps. - e, Infl. ps., - e e d i e (cos sin) ii sin.e i ii 0,, - c S.ps. e, Inf. ps. (0, ), (, e ), (, e ) - e d / e i j min (, ), m (, ), non-sionr infl (, ) i (cos - sin)e ii cos.e i ii (,) - - i ( )e ii ( )e i ii c S. p. (, e ) Inf. p. (, e ) (, e ) 0 0 c d ¼/ ¼/. (, e ) min vlue m vlue p A: i Yes ii non-sionr p of inflec; p B: i Yes ii Sionr poin (locl/ glol min); p C: i Yes ii non-sionr p of inflec. p A: i No ii. Locl/glol m; p B: i No ii Locl/glol min; p C: i Yes ii Sionr poin (locl m) c p A: i Yes ii Sionr poin (locl/glol m); p B: i Yes ii Sionr poin (locl min); p C i Yes ii non-sionr p of inflec. d p A: i Yes ii Sionr p (locl/glol m); p B: i No ii Locl min; p C: i Yes ii Sionr poin (locl m)

54 MATHEMATICS Sndrd Level ANSWERS e p A: i No ii Cusp (locl min); p B: i Yes ii Sionr p of inflec; p C: i Yes ii Sionr poin (locl m) f p A: i Yes ii Sionr poin (locl/glol m); p B: i Yes ii Sionr poin (locl/glol min); p C: i No ii Tngen prllel o is. i A ii B iii C i C ii B iii A c f ' f '' f ' f '' f ' f '' Sionr poins: locl min (, 0) nd locl m e. Inflecion ps re: + + e + nd e + Asolue min ~ -., locl m ~ Inflecion ps ~ (0.,.) nd (., 0.) re lef s quesions for clssroom discussion.,, c, d (, 0) (, ) (, ) c d f e f f ln ,.,. Use grphics clculor o verif our skech. Eercise 0. Locl min., locl m Locl m. 0, locl min. ± c Locl m. 0. d Locl m. e none f Locl m. 0., locl min., 0 g Locl m., locl min. h none m. 0, min. m., min. - c m. 0., min. 0 d m., min. 0.. (,.) 0.0 m 0., n. 0 i ii i ii 0 (,/) (/,/) (/,/) - ¼,, c

55 Sionr poins occur where n e /.e 0 e / e Locl min. (, ); infl. p. + Locl min. (, ); locl m. (, ) c none - - (, ) Verif our grphs wih grphics clculor. Glol min. (0, 0); locl m. e Infl. ps. e + + e + Glol m. 0 e, infl. p. e. c Locl m. e Glol m. e e. Infl. p. e..e. Glol min. + ln c Glol min. (, + ln); Infl. p. (, + ln) d none 0 f ' i f - ; none + ii f + ; locl m. ; locl min. (, 0) iii f + ; locl min. (±, 0), locl m. (0, ). Glol min. (, c ); c (, ) Glol m. e 0. 0.e ; infl. p. e e. Eercise 0.,, c, d, e, 0 f, c, c / 0. d e f /

56 / MATHEMATICS Sndrd Level ANSWERS i (0, ), (, 0) ii, iii iv d \{} f : \{-} where f - c +, Chper 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, o neres ineger) 00 cm. cm /d No $0. $ 0. per er c $. per er.0. c.0 0 < < 0 ppro. i 0 < < 000 ii 000 < < 0000 o neres ineger, o neres ineger. d. e 0 < < 00 D' -. iems/dollr i ii i 0 mm/s ii ~ 0. mm/s 0. sec. cm/s never c never e ms Rnge \{} Eercise. i v - ii - i v e e 0 ii e e + 0 c i v - 0 ii - 0 d i v - + log + 0 ii ln 0 e i v e 0 ii e 0 f i v ln + ln 0 ii v ln + ln ms never res c i m from O in negive direcion ii ms d 0 m e s ms never c or d 0 ms v + ; ~ sec c once d use grphics clculor m in posiive direcion i m ii m c ms e oscillion ou origin wih mpliude m nd period seconds 00 m, in negive direcion imes c i 0 ms ii ms d m m. unis, min. uni s c i cos ii 0 m ove i v.e 0. ii 0.e 0. c 0 m d 0. v < < 0 or > > 0 c or This quesion is es done using grphics clculor: From he grph he pricles pss ech oher hree imes c 0 s; s; s d i v ms ii ms A 0.e 0. v B 0e e Yes, on wo occssions. m in posiive direcion i s ii never c 00 ms 0 0 window: [0, ] [0, ] ln A B

57 Eercise.. m. mh $. per km 00 $00000 $. $0.0. m 0. m m m r, dim of rec. i.e. prro.00 m.00 m m A (00 ), 0 < < 0 c, 0 00 mls 0 s c R.,. c d 0 00, $ & unis c A infl. ps. when. - - cos ¼ cm (0, 00) 00 C (0,,.) 0 0 ~. cm 0 rdius - cm, heigh 0 - cm - cm 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 liude heigh of cone ~.0 m wide nd.00 m high - when rcsin, i.e. ppro..00 km from P. n - l k + kl + kl + k c if k < c, swimmer should row direcl o Q. i r h + r ii c r : h : r + rh / + / / km long he ech c row direcl o desinion

58 MATHEMATICS Sndrd Level ANSWERS Chper Eercise. c d e f g h + c + c c 0 + c d e f g h c d e f / + + c g + + h + c i c d e c + f c d e + c f + + c / + / + c c d e + c f Eercise. + + c d e c + c + c f g h $.0. cm + c + c + c + c + + c c / / + + / + c + c u u - z + + z + z+ c z c u + + c + + c + + c c + c + c + c + c + c + c + c + + c + c + c / / + c c + c + c + + c + c / + + c u + u + u+ c +, P Vol ~ 0 cm 0 cm Eercise. c d e f g 0.e 0. + c h e i j k e / + c l e + c log e + c 0 log e + c 0 c d log e + + c e log + c 0 f log e + c 0 g + log + c 0 h cos + c sin + c c d cos + c + c sin + c d e e f + + c 0 g h cos + log e + c 0 i j e + c k l m sin + c + N f 0 - e + c e - e + c + + cos + c e log e e e + c 0e 0. + c + c e + + c e + c e sin ln + + c e log + c 0 e + c e + c n + c cos + c + cos + c e + e + + c e + + c e + c n + / + c 0 log e e cos + + c

59 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 - + c - + c ln + + c d 0 n 0. + c e ln + + e +c f g + ln + ln + + c h ln + 0.ms or.ms. cm e / sin 0. g + e + - e c 0 + c c cos + c sin c + ln c f + f ln + c f sin + + d f p, - + q c c + + sin ln c + ln + e e ln + + c -e sin cos + c + c + + c e + c V'..% c ~. lires 0. m noon pm pm m V' B 000 c. d d ds Eercise.. c d + / + c + / + c e f g h + e / + c i j k l m n e + c o p - q r + + c + / c s + sin / + c u v cos w + sin + c A + c + e + + c e + c c e n + c d e + + c e e f e + + c g cos e + c h + i ln e + c c + + c + c / + c e + + c + c - + c + n c + sin + c / + / + c e c + cos / + c + / + c + c + + c cos + c + + c j ln + e + c k + e / + c l - - ln + e + c cos + + c 0 cos + c c sin + + c d cos / + c /

60 e log cos + c f g log + n + c + n + c h sin ln + c i + cos / c j k l + c m sec + c n o ln sin e c ln + d sin e sine cos / e f g h i j 0 k l m n - 0 Eercise.. + / + c c + d e f + g ln z + z + c h i e sin + c j ln e + + c k e n + c ln + c c d e ln + e + c f 0 c d e f - c d e f c d c d e + ln c d e f g Tn Eercise. - - c d e e ln + / + c c e + c - ln Sin - ln ln + c ln - ln cos sin e + c e + + c ln + e + c sin + c - + c ln -. c n + c c d 0 e f g h i j ln Sin - 0 ln + c e e - ln + + c / + c 0 - k l e e e c 0 d e e e e + e f g e h i ln ln c + ln d e f ln g h ln i ln c d e f 0 g 0 h i 0 c 0 d e f ln ln - - sin + cos ; 0 m n m+ c n d m e n e e 0. ; 0e 0. 00e 0. + c i ccidens ii N + 0e 0. 00e 0. + suscriers 0 ~ flies Eercise. sq.unis sq.unis c sq.unis d sq.unis e sq.unis e sq.unis sq.unis c e+ e sq.unis d e e sq.unis ln sq.unis ln sq.unis c ln sq.unis d 0. sq.unis sq.unis sq.unis c sq.unis d sq. unis e sq.unis sq. unis sq.unis. ln +. sq.unis. sq.unis. - 0 sq. unis e / e - - e e e e ln e e +

61 0. sq. unis sq. uni c sq. unis n; ln sq.unis sq. unis sq. unis sq.uni 0 sq. unis ln + c sq. uni - sq. unis sq. unis sq. unis i - sq. unis ii - sq. unis 0 - sq. unis i e + e sq. unis ii sq. uni iii ln() sq. unis.0 sq. unis - sq. unis e sq. unis e sq. unis c e e e ~ 0.00 sq. unis Eercise sin + cos 0 c e c 00 - m m m s;. m s m 0. m s 0. m c. m d. m 0 k cos - v + k - 0 k c. m m Eercise. All vlues re in cuic unis. ln 0 k e 0 e ln sin 0 - k Two possile soluions: solving + 0 0,.; solving 00 0, hen Revision Se A i ], [ ii f ln + c 0 i 0 ii c 0 f / f

62 (, ), (, ) nd (, 0) S [0, [, rnge [, [ c f : [, [, f () (ln) i ii i h + h + h ii + h + h i or ii i ii 0. iii 0 0 i or ii i 0 < < ii iii log e 0. iv gf, i ii c i, ii [,]\{0} 0. (, ) (, ) (, 0) (, ) c d ln - ln (, ) \ e e (, ) (0, ) (, 0) f() g() \ P - + fg (, 0) (, ) (, ) (, ) (, 0) gf e 0. + e c (0, ) (, ) (, 0) (, 0) k 0 or + + c 0 < < 0 < < 0 c,, ± 0 ii p p, + + p ii {±} i ii c p n 0 ],] c ],[ sq unis d 0 c, d c, d - c c 0 d 0 c, d c, d c c c (, ) 0

63 0 i & ii i ii $00 iii $ iv 0, 0 0 f + f g f g e + e 00,00 0 cm cm c hrs d [0, ] e f Use grphics clculor. g. hrs ]0, ] c No ( 0) 0 c r f d g, i.e. does no eis - 0 g + h g g i ]0, [ ii - c log d e e e, 00 c 0 d 000 e > f 000 B 0 0 B h 0, rnge ], ] Use grphics clculor. f log e c Use grphics clculor. 0 r g d f fog eiss; r f d g gof doesn eis. < or > f r g d f f og does no eis; r f d g gof eiss. c F or c + z i 0 ii 0e. c d i 0 ii. 00 Q() P() 00 A f Incresing decresing re g ~ 0 wsps h ii 0 nd Revision Se B c d km c d km e 0 0 log e

64 MATHEMATICS Sndrd Level ANSWERS i A: $000; B: $00; C: $00 ii A: $000; B: $000; C: $00 % c i monhs ii C never reches is rge r 0 cm or M. vlue is for or, where k is n ineger; min. vlue is for, where k is n ineger u n n n c, i cosec ii ~ 0 erms c 0 < < d {,,,,... } e f $. 0 cm i 0. m ii 0. m ~. m c cm - - $ k f cos - c i BP 0 m, PQ m 0 m + k p u u m c $ k - p + p n rnge [,.] c i ii - - u n n 0 i W. P. ii W 0 0. P 0.0 iii W. P. Amp, period 0 weeks c d $.0 e during h & h weeks $000, $00, $0 $0. c. ers d ~$ n ii crds, 0,, c, d crds e n n + ~. m i.0 m ii.0 m c. pm d Use grphics clculor. e. h - - c m. C D + d Use grphics clculor. e m o midnigh m 0 ii N , 00, 0,,. c hrs d cm cm c ii r iii d i ii e Geomeric Revision Se C i + j + k c P() W() cos - - cm cm + z + i+ j+ k A n z. - n n 0

65 , (,, ) i j k 00 c i j k r B min i 0 ii uni i s + p ii s + p iii s + p iv i ii iii i + j c i no ii lines re skew 0 c 0 r A 0 + (unis in meres) c The do no collide. r c ~. km + LP + c c i ii uni s + p - i + j + k or - i + j + k OA i j + k, OB i k ; 0 Yes i (,, ); ii (,, ); lines do no mee z + - or c PX vlues re: 0, 0.0),, 0.0),, 0.0) d EX 0.0 vr X c 0.0 d. < Y < 0. e.0 0 PX 0, i.e. geomeric i 0.00 ii 0.0 iii PX vlues re: -,,,, c EX, vrx d c c PX vlues re:,0.,,0.,,0.,,0. c i ii iii PX vlues re: 0 -,,, ii iii i 0.0 ii 0. i 0. ii 0. i 0. ii EX 0. vr X - 0 i ii iii iv v PX vlues re: 0 ; c EX vr X c 0. d 0. -

66 MATHEMATICS Sndrd Level ANSWERS PX vlues re: 0,, c d i i iii iv v c d. c no independen i 0. ii iii c $0. i ii frequenc EX. vr X 0. p q+ 00q 00 0 cum. freq 0 i. ii. c i min, m, med,, 0 ii med, mode iii i 0. ii Q c Q >. use grphics clculor i A., B. ii A: s n.0, B: s n.0 c i Clss A: min, m, med, Q, Q Clss B: min, m, med, Q, Q Q. Q. ii Clss A: med ; mulimodl,,,, Clss B: med ; mulimodl,,,, iii Clss A: IQR, Clss B: IQR d Resuls from oh clsses re ver close, however, Clss B does slighl eer s i hs lrger medin s well s he lrger mimum vlue. Revision Se D cos - c sq. unis -. C. C per minue c. min m - or + sin cos sin i 0 ii c 0 d - + h - + h h Locl min ; smpoes, 0. sin + cos - + / Clss A Clss B log e sin ms Asolue mimum ; locl min 0, 0; -inercep ±, 0 e + e + c m

67 (, ) A ln ln sq. unis c i A (0,): + e A e : e e ii (,0) (0,0) (0, ) i ii cm Are A -h, Volume / V 0.h m/min (, ), (, ), (, 0) use grphics clculor c - log sq. unis e 0 (m, n) iii e sq. unis d ii cuic unis - e e + i 0 < < 0. ii 0. iii < 0 or > 0. c ln d e i + ii ln 000 h - rdius cm, heigh. cm r - e - + e h + h + h + h + h c log e p' e 0.0 ~. million c i decresing ii 0. million/er d 0 ers ime, i.e. 00;. million cm A B C 0 use grphics clculor c log sq. unis + cos + sin + cos + + h + h h 0 0 A e d i ii - e iii d k e / + e c i e ii 0e cuic unis log e log e i sec ii 0 sec c sec e f i ln sq. unis ii + ln sq. unis e.0 i cos ii 0.0 log e cuic unis,, c - i sin cos ii + A 0, B 0. 0 sq. unis cuic unis sin + c V r kv 0 h c d 0 r r + r P krh + kr P V / 0 - r + kr c cuic unis log e + ; log e c, e; e d e sq. unis V / / / /

68 MATHEMATICS Sndrd Level ANSWERS ; f +, 0 cuic unis i ii [0, [ 0. c {0, } d i ii { > g() e i sq. unis ii cuic unis sin cos e + sin + cos sin + cos e e e + + c e + e curve sq. unis f() norml - 0 sq. unis cuic unis [0, ] use grphics clculor c 0. d c Minimum, / ; Mimum / v 0 seconds c meres e cos + sin + ln c use grphics clculor A cos 0 d. sq. unis