Maths SL Answers. Exercise Exercise Exercise i b 4 c t n = 4n 2

Transcription

1 Mths SL Answers Eerise... i t n = n 8 9, ii t n = n + iii t n = n + 6 iv 0. t n = 0.n v t n = + n vi t n = n + 9 st 0, i ii i ii 8 0 t n = + ( n ) 0 Eerise th week = 9, = 7 Eerise.., 0.,,., 7, 9.,. = d = $7, 7 weeks 0 $ 70 i 8 m ii 0 m 8 m Dist = n n = n( n ) d 8 e 6 plers, 00 m n 07

2 Mths SL Answers Eerise.. r =, u = 8 u n =, n r r =, u =, u 7 n = n =, u = -, u 6 n = n d r =, u = 6, u n = ( ) n e f r r, u = =, u n = n, u, u n = = = n ± ± ± 96 th u n = n = times 888, 6 i $096 ii $ rs u n - = n , ,,0 or 0,, $6 6 $99 8 Eerise.. f d e d - e ; 9 7; 7 8; d 8; e ;.6 f ;. 7-8 d 60 e ; $ m 8 V n = V n gms; 0 weeks ,

3 Mths SL Answers r =, $ or out 00 illion tonnes. Eerise..6 Term 9 AP = 80, GP = 6. Sum to terms AP = 60, GP = , 8 weeks Ken $0 & Bo-Youn $) 6 week 8 week [~00, depends on rounding errors] Eerise d fish. [NB: t <. If we use n = then ns is 6660 fish]; fish. 7 Overfishing mens tht fewer fish re ught in the long run. 8,, or 6,, m ( t) n + t ( t ) - n + t + t - + t Eerise..8, t n = 6n , 07

4 Mths SL Answers ± (,, ), (, 0, 0), 7,, 8 n 6 m Eerise..9 $77.08 $77.6 $78.9 $ 6 $ $ $ $ , $8. 9 $ $98.6, $967., interest $67.. Flt interest = $6000 $., $790., 0.60% /month (or 7.% p..) ½, The sequene ½, ½, ½,.. is rithmeti. Proof m = 9, n = 07

5 Mths SL Answers Eerise n + d e 8 f n + + g n + h ( n + ) i d e 6 f 9 n + z 7n n + d 9 e 6n + f n g + n n h n + n + i 7 - m m d e f - ( + h) d e 7 8 n - f ( + ) ( ) 7 d mn e p - + q f pq - g 7 8 h 7/ 8 8 / n n n d 7 m n 8 e 6 n n + Eerise.. d e 6 f. g h. i 6 d. e 0. f 0. g h i. 8 Eerise.. d e f g 0 h 0 i j k 0. l 07

6 Mths SL Answers log00000 = log00.00 = log0 ( + ) = d log0p = 7 e log ( ) = f log ( ) = 9 = = = t d 0 = z e 0 = f = 6 d 9 e f g h 9 i j k l d 7.89 e.687 f g.98 h.78 i Eerise.. d e f log e = log + log log = log + log log = log d log = log + 0.log log = log + log f log = log 0.log = z = = + - d = + e = f = ( + ) 7 d e f no rel soln g,7 h - i j 0 + k 6 l 6 6 log w log log 7 ( + ) [ ] d ( )( + ) log e log f 0 log - 7 d 9 e f 9 8,, ±, 9 log - =.8 log - log8 = 0.90 log0 d,± log - =.9 log d log log = 0. e log 0 log = log 0.7 f. g - log = log

7 Mths SL Answers h 7.7 i 0.9 j no rel solution k log = 0. log l log. = 0.7 log 0 0.,, d 0,0 0 e f (, log ) 00,0, = z = = e - e d Ø ln =.0 ln0 =.06 ln7 =.99 d ln = 0.69 e ln =.0986 f ln = g e = 0.08 h e =.60 i ± e 9 ± = j Ø k e =.89 l e 9 = , ln ln ln, ln d 0 e 0, ln f ln ,0.69 d,0.9 e.9,0.660 f 0,.898 g 0., h i. j 07

8 Mths SL Answers Eerise g + g + g d e f g h i j k l + 9 d + 7d + 7d m n o - p p + 8p p p q q + 0q - 0q p q p q p p + p r Eerise d 80 e 6p f 0p q g 680p d 0 e 868 f % d 0.% = ± n = n =

9 Mths SL Answers =, n = 8 = ±, = ± 6. = n = 6 = ¹/₃ n = = n = d = 6 n = 07

10 Eerise.. dom = {,, }, rn = {, 9, 9} dom = {,,,, 7, 9}, rn = {,,, 6, 8, 0} dom = {0, }, rn = {, } ], [ [0, [ ]9, [ d ], ] e [, ] f ], [ g ], 0] h [0, ] i [0, [ j [, ] k ]0, [ l ], ] [, [ r = [, [, d = [0, [ r = { : 0}\{}, d = r = [0, [ \{}, d = [, [ \{0} d r = [, 0[, d = [, [ e r = ], [ d =], ] [, [ f r = [,], d = [0,8] \{ } ], 9[ [,] d ], ] [, [ e \{0} f g \{ } h [, [ i [0, [ \{ } j ], ] [, [ k l \{ } ], [ ]0,] ], ] d [, [ Mths SL Answers e \{} f ],[ g [, [ h ], 0[ Eerise.., i (+) + ii 0, 0 0, 0 +, + + ± no solution = 0, Window [,], [,] Rnge: [, ] i ii i {, } ii {, } 6,, d, e 8, d, e, f 9 Window [,], [,] [0, [ 0 {: > } {:.} 0 07

11 Mths SL Answers onl it is the onl one with identil rules nd domins [, [ [,0] [, [ d [.,[ ], [ i p( ) = 8 + 6, 0 < < ii A( ) = 6, 0 < < i 6 r = ]8,6[ ii 8 r = ]0,8] 8 Eerise.. even even neither d neither e even f odd g odd h even i odd Not if 0 is eluded from the domin. 6 f( ) = 0, R Eerise.. i f + g: [ 0, [ where ( f + g) ( ) = + [0, [ ii f + g: ]0, [ where ( f + g) ( ) = + ln( ) [, [ iii f + g: [, ] [, ] where ( f + g) ( ) = 9 +, [, 0] i fg: [ 0, [ where ( fg) ( ) = = / ii fg: ]0, [ where ( fg) ( ) = ln( ) iii fg: [, ] [, ] where ( fg) ( ) = ( 9 )( ) 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 f/g: ], [ where ( f/g) ( ) = + iii f/g: \{ } where( f/g) ( ) =

12 Mths SL Answers fog( ) = +, gof( ) = ( + ) ], [, ], [ ii fog( ) = +, 0, gof( ) = + [, [, [, [ iii fog( ) =, gof( ) ( + ) = [0, [, [, [ iv fog( ) =, 0, gof( ) =, 0 \{0}, \{0} v vi vii fog( ) =, 0, gof( ) = [0, [, [0, [ fog( ) =, 0, gof() does not eist. ], [ fog( ) =, 0, gof( ) =, 0 ]0, [, ]0, [ viii fog( ) =, gof( ) = [, [, [0, [ i fog( ) = +, gof( ) = [, [, [0, [ fog() does not eist, gof( ) = ( ), [0, [ i fog( ) =, gof( ) + = - +, [0,[, [0, [ ii fog( ) = + +, gof( ) = + + [, [, [ 0.7, [ iii fog( ) =, gof( ) = [, [, ]0, [ iv fog() does not eist, gof( ) v fog() does not eist, gof( ) vi fog( ) =, 0, gof = -, \{ } + = + ], [ ( ) 0. = [, [, ]0, [ fog( ) = +, gof( ) = +, fof( ) = +, g( ) = +, 6 fog( ) = + +, \{ 0}, ], ] [, [ gof() does not eist. gog( ) = + +, 0, ],.] [., [ g ( )= + 07

13 Mths SL Answers 0 g ( )= + = ± =, rnge =], [ (, ) hof( ) = + ( ) +,, < r f d g nd r gof d h g( ) = + ( ), fog( ) =, ]0, [ rnge = ]0, [ gof( ) = ( ln( e ) + ), (= ) rnge = ], [ fof( ) = e ( e ), rnge = ]e, [ 6 hok does not eist. koh( ) = log( ), >, 7 S = \],[; T = 8 T { : 6, = 0} = ; S = ], ] [, [ gof does not eist = fog( ) = fog( ) = gof( ) 9 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, [ gof: ]0, [, where gof( ) = ( + ) + ln 0 ( fog) ( ), = ; rnge =[0, [ +ln (, ) fof( ) = dom = rn = ]0, [ f g d gof: ], [ fog*: ], [, where gof( ) =, where gof( ) = (, ) in f, not g rnge = ], [ d f = \ rf \, =, rf d f, fof( ) = 07

14 Mths SL Answers d fog = [, ], fog = d gof / / = [, ], fog =, rnge = [ 0, ] Eerise.. ( ),, ( + ), d ( ), e, > 0 f ( ), g, > 0 h ( + ), > d 9 ( 0, ) e f g h (0,) +, +, ( 0, ) ( 0, ) ± -, < < d / e f g h (, ) (8,) 6 f ( ) = log( ), > f ( ) = log( + ), > f ( ) = ( log ), > 0 d g ( ) = + log 0( ), < e h ( ) = log f g ( ) +, = log - +, \[,0] > 07

15 Mths SL Answers 7 inverse inverse inverse d e f (, ) inverse (, ) inverse inverse 8 f f ( ) ( ) =, = 0, f ( ) =, d f ( ) = + +, e f ( ) = / +, f f ( ) = 0 ( ), 9 f ( ) = + +, > (, ) dom = [, [, rn = [, [ 0 f ( ) = f ( ) = + f ( ) = f ( ) = (, ) [, [ + \{.} Inverse eists s f is one:one Cse : S = ]0, [ Cse : S = ], 0[ g ( ) = + + g ( ) = + f ( ) = ( + ), 0 { : f( ) = f ( ) } = 07

16 Mths SL Answers 6 f ( + ), < ( ) =, f ( ) = ln( ), 0 < e e, > e f ( ) = + e, < 0, 0 d f ( ) = ( ), >, 0 < < 7 + f /f + f :,f ( ) = + e 8 gof eists s r f d g. It is one:one so the inverse eists: 9 i ii f is one:one f( ) = ( ) < ( + ) > 6 (, ) 6 iii iv{, 0, } 0 i tom( ) = e, 0 ii mot( ) = e, i ( tom) ( ) = ( ln( ) ), > ii ( mot) ( ) = ln, > 0 i & ii neither eist d Adjusting domins so tht the funtions in prt eist, we hve: t om ( ) ( mot) ( ) = nd m ot ( ) = ( tom) ( ) e Yes s rules of omposition OK (tom) 0. (mot) f g fog eists ut is not one:one 07

17 Mths SL Answers i B = [ln, [ ii ( fog) : [0, [ where, ( fog) ( ) = ln( + ) iii ln ln 07

18 Mths SL Answers Eerise... (0,) (,0). (,0) (,0) (0, ), 9. 07

19 Mths SL Answers. (,0) (,0) (0, ). (0,) 6. (,0) 07

20 Mths SL Answers 7. The -is interept of this grph is t 0 0. The vlue of this epression remins topi of nimted disussion mongst mthemtiins. The minimum point is t pproimtel ( ). 8. (,0) 9. 0, 07

21 Mths SL Answers 0. Eerise.... C = +0.0n. Cost (C) (0,). 800 Crds ordered (n) ( ). If the origin is hosen t the top left of the urve nd using, s vriles: = = 0 A m wide ship will need gp of metres t wter level. The point on the grph is (, 6.) so the mimum drught is 6. metres - ut this leves no sfet mrgin.. Asmptotes: =, =. Lines of smmetr = +, =. Rottionl smmetr out (,) of order i 6 ii ds kg.6 ers d Weight (W) (0,) Time (n) 07

22 Mths SL Answers 6. Inverse is lue. 7. = %. m d. m e Intensit Depth 9. Month vlue $0.00 $0.8 $.67 $. $. $.9 $.0 $.89 $6.7 $7.60 $8.6 Month vlue $9. $60.9 $6.0 $6.9 $6.80 $6.67 $6. $6. $66. $67. Month vlue $68.0 $68.99 $69.89 $70.79 Vlue($) The simple interest option ields $7 nd is etter. Time 07

23 Mths SL Answers Eerise... i (,) ii (,) iii (,) iv (,) v vi (,) (,) vii (,) viiii (,) 07

24 Mths SL Answers. d e. = = = d = 07

25 Mths SL Answers Eerise.. i ii i 0 0 ii 0 0 i ii (0, ) (, 0) (0, ) d i (,6) ii (, 6) e i ii = f( ) = f( ) = f( + ) d = f( ) e = f( ) 07

26 Mths SL Answers Eerise.. d 9 (, ) (, 0) (, 0) (, 0) (, 0) e f g h = 6 (, ) i j k l m 0. (, ) (, ) (0, ) (, ) d e f (0, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, 0) (, ) g h (, ) ( 0., ) (, ) (0., ) 07

27 Mths SL Answers Eerise... iv iii vii vi ii (0,) i v (0,) (, ) (, ) viii ( ) ( ) ( ) = f = f + = f = f ( )+ ( ) = f + = f ( )+ = f ( +)+ ( ) = f (0,) iv vii (0,) ii ( ) ( ) ( ) = f = f + = f = f ( )+ i iii v ( ) = f + = f ( )+ = f ( +)+ ( ) = f vi viii. d e 07

28 Mths SL Answers. = = = d Eerise... d e f d. 07 =

29 Mths SL Answers e f d e f.. = ( ) = 8 +. = + 6. ½<< 07

30 Mths SL Answers 7. Profitle from 0. to 9.7 ers. 8. Yes - h(.) = 0. (. metres) so the projetile psses over the dek. 07

31 Mths SL Answers Eerise... i ii iii iv v vi. =,. = +,. =, =. = ± 6. P = 70 V 07

32 Mths SL Answers Eerise.. =, = =, = =, = d =, = e =, = 0 f =, = = - - = 0. = = 0. d e f = = = =. =, = = 0. = = = = 6 i (0, ), (, 0) ii =, = iii iv d = \{ } = = f : \{-}, wheref ( ) = ( ) ( + ) = = 7 = 8, = = 8 Rnge = \{8} 8 dom = \{0}, rn = \{} dom = \{ 0.,0}, rn = \{0.} = f( ) = g( ) 0. 07

33 Mths SL Answers 9 Asmptotes: =, = 0 0 Asmptotes:, 0 Asmptotes:, i (0, ), (, 0) ii, =, = 0 =, = 0 d =, = 0 = = =, = 0 =, = 0 d =, = 0 = + = 0 = +, = 0 =, = 0 d = +, = 0 = = (, ) = + = + 0. (, ) = (, ) 07

34 Mths SL Answers 07

35 Mths SL Answers Eerise.6. d (, ) ]0, [ (, ) ]0, [ (, ) ]0, [ (,.) ]0, [ e f g h (,.) ]0, [ (,.8) ]0, [ (, ) ]0, [ (, ) ]0, [ i j k l (, ) (, /) ]0, [ ]0, [ (, 8/) (, 0/7) ]0, [ ]0, [. 0. hs diltion effet on f () = (long the is). d (, ) (, ) (, 0) (, 0) (, ) (, ) (, ) (, ) [,6] [,7] [0.,6] d [0.,] e [0.,0.] f [0.,0] 6 d e ], [ ],[ ],e[ ], [ 7. i f = g: = ii f > g: < f g 8 ], + e [ [, [ [ e, + e ] 07

36 Mths SL Answers 9 0 d (,) (,) ]0,[ {} ],] (,) ],] [0,/] d [, [ [, [ [0, [ (, ) d e f 07

37 Mths SL Answers 6 ]0,] d e f [, [ > ]0, [ = {} < ]0, [ [0, [ [,[ ], [ ]0, [ Eerise.6. d ]0, [ ], [ ], [ e f g h 7 0. ]0., [ ],[ ]0, [ 0 ]0, [ ]0, [ ]0, [ 0 ], [ d e f ],[ / ] /, [ ], [ e e ]0, [ ]0, [ e ]e, [ d e f ],/e[ /e ]0, [ e ]0, [ 07

38 Mths SL Answers ]0, [ \{0} \{0} d e f ]0, [ ],[ ], [ ]0, [ ], [ ], [ ]0, [ (, ) d e f e ]0, [ \{ } \{0} 6 i ii d /e 7 0 < < ~. 8 d [0, [ ],] [, [ 9 07

39 Mths SL Answers 0 ],0/[ + ], [ e - ]e/, [ 0 d e f \{e} e ],[ \{e} e / / { : < <+ } ],0[ [e, [ 07

40 Mths SL Answers Eerise.7.. i ii iii iv 0 v vi vii viii i 0 0. i ± 7 ii ± iv ± v vi ± iii no rel roots vii no rel roots viii i, ± 7. i 0 ii iii! (ever rel numer) iv v 0 vi vii viii, i no rel roots.. i.66 ii.0 iii iv 0.60 v.69 vi.0 vii no rel roots viii 0.9 i < or >. 7. Proof Proof 0.. ± k ±. ±. 9,.. (there is firl lrge negtive solution) 07

41 Mths SL Answers Eerise.8.. h( )= , m.. 0 moles per litre in terms of H-ion on Aout times times.. or ppro.6 kph hrs 7. i ~7.m ii m 8. Mn orret nswers :. 7 07

42 Mths SL Answers 07

43 Mths SL Answers 07

44 Mths SL Answers 07

45 Mths SL Supplementr Questions Eerise.. Simplif the following. e 6 8 f - n g - n + 6 h - n + 6 i 6 Simplif the following. e ( ) f 7 n + 6 n + Simplif the following. e n - n + n f n + n ( n ) g n + ( n + ) ( n ) h n + n ( n + ) ( n ) i ( )( + )( ) ( ) Simplif the following, leving our nswer in positive power form. ( ) e ( ) ( ) f ( ) 6 Simplif the following. e ( ) - f ( ) + ( + ) ( ) Simplif the following. n + n n z z d - m + n m n - n m m n n e p q p - q f - + g n + ( n ) ( n + ) - h 8 Simplif the following. n + 8n ( - ) ( ) n + n 6 - n n d 7 m + 7 m 7 n 7 n + e n + + n n + + n 07

46 Mths SL Supplementr Questions Eerise... Solve the following equtions. g { = } h { + = 8} i { 7 = } 07

47 Mths SL Supplementr Questions Eerise.. Use the definition of logrithm to determine the following. g log h log 0 i log j log 9 k log l log Chnge the following logrithmi epression into its equivlent eponentil form. f log ( ) = Solve for in eh of the following. g log 6 = h log 8 = i log = j log ( ) = k log 8 = + l log ( ) = Solve for in eh of the following, giving our nswer to d.p. g log e ( + ) = h log e ( ) = i log e = 07

48 Mths SL Supplementr Questions 07

49 Mths SL Supplementr Questions 07

50 Mths SL Supplementr Questions 07

51 Mths SL Supplementr Questions Eerise.. Without using lultor, evlute the following. e log 0 log f log 0 log Write down n epression for log in terms of log nd log for the following. e = f = - Epress eh of the following s n eqution tht does not involve logrithm. d log + = e log = log f log ( + ) = log Solve the following equtions. d log 0( + ) log 0 = log 0 + log 0 e log 0( + ) log 0 = f log ( + 8) log ( ) = g log 0( + ) = + log 0 ( ) h log ( + = log ( ) i log ( + ) + log 6 = 6 j log ( ) + log ( ) = k log log ( ) = log l log 0( + ) log 0 = log 0 6 Simplif the following log + log ( + ) d log log ( ) + log + ( ) e 0 log + log 6 log 0 f log log log 7 Solve the following d e f log + log ( 8) = log + log = log + log = 7 07

52 Mths SL Supplementr Questions 8 Solve for. log = ( log ) d log = ( log ) e Investigte the solution to log n n = ( log n ) n. 9 Solve the following, giving n et nswer nd n nswer to d.p. e + = 0 f 0.8 = 0. g 0 = h.7 0. = 9 i 0. = 0 j + 0. = k - = l - + = - 0 Solve for. log 0 ( + 6 ) = d ( log 0 ) log = 0 e log ( + 0 ) = f log + ( + ) = Solve the following simultneous equtions. = log log = Epress eh of the following s n eqution tht does not involve logrithm. ln = Solve the following for. log ( + ) + log e = 0 e d log ( + ) log e = 0 e Solve the following for. + e = d 00e = 0 e e = f 70e + = 60 g ln = h ln ( ) = i ln( ) = 9 j ln ln( + ) = k ln + = l ln( ) = 9 07

53 Mths SL Supplementr Questions Solve the following for. e e + 6 = 0 d e e + = 0 e e 6e + = 0 f e 9e 0 = 0 6 Solve eh of the following. = = = 0 d = 0 e = 0 f = 0 g log + log = log( 9 ) h log log = log( ) i log + log 8 = 9 j log + log = 07

54 Mths SL Answers Eerise.. Find the rnge for eh of the following. i =, 0 j =, k = -, > 0 + l {(, ): =, } Determine the implied domin for eh of the following reltions. h = +, > 0 i =, > 0 j = k = Find the rnge of the following reltions. f =, > 0 g =, > 0 h =, < 0 07

55 Mths SL Supplementr Questions Eerise... The funtion f is defined s f:], [,where f( ) =. Sketh the grph of: i f ii = +, ], [ i Find: { :f( ) = } ii { :f( ) = + } 6 Whih of the following reltions re lso funtions? d e f 7 Use oth visul nd lgeri tests to show tht the following reltions re lso funtions: +, ]0, [ +, [ 0, 9[ {(, ): = +, } d {(, ):= +, } 8 Use n lgeri method to deide whih of the following reltions re lso funtions: e f:, \{ 0} {(, ): = 9, 9} (, ): f( ) = { = 9, 9} d +, 0 f( ) =, f f: -, \{ } + 9 Sketh the grph of f: - +, nd use it to: show tht f is funtion determine its rnge. 0 A funtion is defined f: Determine the rnge of f. Find the vlue of suh tht f( ) =. Consider the funtions h( ) Show tht [ h( ) ] = h( ) , 8 8 nd 0. = ( + ) nd k( ) = ( ). If [ h( ) ] [ k( ) ] =, find the onstnt. Whih of the following funtions re identil? Eplin. f( ) = nd h( ) =. f( ) = nd h( ) =. f( ) = nd h( ) = d f( ) = nd h( ) = ( ). 07

56 Mths SL Supplementr Questions Find the lrgest possile suset X of, so tht the following reltions re one-to-one inresing funtions: f : X, where f( ) = f : X, where f( ) = 9 f : X, where f( ) = 9 d f : X, where f( ) = -, 0, An isoseles tringle ABC hs two side lengths mesuring m nd vrile ltitude. Let the ltitude e denoted m. Find, in terms of, reltion for: i ii its perimeter, p( ) m nd speif its implied domin. its re, A( ) m nd speif its implied domin. Sketh the grph of: i ii p( ) nd determine its rnge. A( ) nd determine its rnge. 07

57 Mths SL Answers Eerise.. All of the following funtions re mppings of unless otherwise stted. viii i i iii v Determine the omposite funtions ( fog) ( ) nd ( gof) ( ), if the eist. For the omposite funtions in prt tht do eist, find their rnge. f( ) =, g( ) = f( ) =, g( ) = + f( ) = -,, g( ) = + ii f( ) =,, g( ) = f( ) = + +, g( ) = f( ) =, g( ) = iv f( ) = -,, g( ) = + f( ) =, >, g( ) = + vi f( ) =, g( ) = Find ( hof) ( ), given tht h( ) +, = nd f:,., < Sketh the grph of ( hof) ( ) nd use it to find its rnge. Given three funtions, f, g nd h, when would hogof eist? If f: +,, g:, nd h:,, find ( hogof) ( ). Given the funtions f( ) = e nd g( ) ( fog) ( gof) ( fof) In eh se find the rnge of the omposite funtion. = ( ln + ) find, where the eist: 6 Given tht h( ) = log ( ), > nd k() =, ], [, find, where the eist : 0 ( hok) ( koh). 7 Given the funtions f( ) = 9, S nd g( ) =, T, find the lrgest positive susets of so tht: gof eists fog eists. 8 For eh of the following funtions: determine if fog eists nd sketh the grph of fog when it eists. determine if gof eists nd sketh the grph of gof when it eists. i = = f( ) g( ) ii = g( ) = f( ) Given the funtions f : S where f( ) = e + nd g : S where g( ) Stte the domin nd rnge of oth f nd g. = ln where S = ]0, [. 07

58 Mths SL Answers Giving resons, show tht gof eists ut fog does not eist. Full define gof, sketh its grph nd stte its rnge. 0 The funtions f nd g re given f( ) = if nd g( ) = +. if 0 < < Show tht fog is defined. Find ( fog) ( ) nd determine its rnge., Let f : where f( ) 0 < + + =. -, > Sketh the grph of f. Define the omposition fof, justifing its eistene. Sketh the grph of fof, giving its rnge. Consider the funtions f : ], [ where f() = nd g : \{0} where g( ) =. Sketh the grphs of f nd g on the sme set of es. Prove tht gof eists nd find its rule. Prove tht fog nnot eist. d If new funtion g * : S where g * ( ) = g( ) is now defined, find the lrgest positive suset of so tht fog * does eist. Find fog *, sketh its grph nd determine its rnge. Given tht f( ) =, show tht fof eists nd find its rule. Sketh the grphs of f( ) = nd g( ) =, where > 0. Show tht fog eists, find its rule nd stte its domin. Let S e the lrgest suset of so tht gof eists. i Find S. ii Full define gof, sketh its grph nd find its rnge. 07

59 Mths SL Answers Eerise.. Sketh the inverse of the following funtions. e f (0, ) (, ) g h (, 8) 8 Consider the funtions f nd g: = = g( ) f( ) Does gof eist? Justif our nswer. Does ( gof) eist? Justif our nswer. If it does eist, sketh the grph of ( gof). 9 On the sme set of es, sketh the grph of f( ) = nd its inverse, f ( ). The funtion g is given g( ) i Sketh the grph of g. +, < =,., > ii Full define its inverse, g, stting wh it eists. iii Sketh the grph of g. iv Find { : g( ) = g ( ) }. 0 Consider the funtions t( ) = e nd m( ) =. Find, where the eist, the omposite funtions: i ( tom) ( ) ii. ( mot) ( ) With justifition, find nd sketh the grphs of: i ( tom) ( ) ii ( mot) ( ) Find: i t om ( ) ii m ot ( ) d Wht onlusion(s) n ou mke from our results of prts nd? e Will our results of prt d work for n two funtions f nd g? Eplin. 07

60 Mths SL Answers Find { : + = 0}. If f( ) = -, sketh the grph of = f( ) nd find { : f( ) = f ( ) }. Consider the funtions f( ) =, A nd g( ) = e, B. Sketh the grphs of: i f if A = ii g if B =. With A nd B s given in prt, give resons wh ( fog) will not eist. i Find the lrgest set B whih inludes positive vlues, so tht ( fog) eists. ii iii Full define ( fog). On the sme set of es, sketh the grphs of ( fog) ( ) nd ( fog) ( ). 07

61 Mths SL Supplementr Questions Eerise... Desrie the trnsformtion(s) under the following mppings. + ( ) d ( ) e ( ) f Consider the funtion g( ) Find n epression for: if =. 6 if < i iv f( ) = g( + ) ii h( ) = g( ) iii h( ) = g( ) k( ) = g( ) v k( ) = g( ) vi f( ) = On seprte sets of es, sketh the grphs of eh of the funtions in prt. g ( + ) 7. Given the reltion f( ) = = f ( ) = f - ( ) if < if if ( ) if <, sketh the grphs of: 8. Given the funtion f( ) =, sketh the grphs of: = f( ), > 0 = f( ), > 0 = f( + ), > 0 d = f ( ), 0 9. Given the funtion f( ) =, sketh the grphs of: = f( ),, >0 = f( ),, > 0 07

62 Mths SL Supplementr Questions 07

63 Mths SL Supplementr Questions 07

64 Mths SL Supplementr Questions 07

65 Mths SL Supplementr Questions Eerise.. 6. Consider the funtion f( ) i = -. + Find the oordintes of the interepts with the es. ii Determine the equtions of the smptotes of f. iii Hene, sketh the grph of f. iv Determine the domin nd rnge of f. Find f, the inverse funtion of f. Dedue the grph of ( f( ) ). 7. Epress 8 in the form A + B, where A nd B re integers. Hene, stte the equtions of the vertil nd horizontl smptotes of the funtion f( ) Sketh the grph of f( ) = 8 nd use it to determine its rnge. = On different sets of es, sketh the grphs of f( ) = + nd g( ) = -, stting their domins nd rnges. f( ) 9. Sketh the grphs of the following funtions, lerl lelling ll smptotes. f( ) = +, 0 g( ) = +, 0 g( ) = +, 0 d f( ) =, 0 0. Sketh the grphs of the following funtions, lerl lelling ll smptotes. h( ) = +, 0 f( ) = +, 0 g( ) =, 0 d f( ) = +, 0. Sketh the grphs of the following funtions, lerl lelling ll smptotes. f( ) = + +, 0 f( ) = + +, 0 g( ) = +, 0 d f( ) + = -, 0. For the funtion f( ) = + : i determine ll il interepts nd the oordintes of its sttionr points. ii write down the eqution of ll the smptotes. Sketh the grph of = f( ) lerl lelling ll the informtion from prt. 07

66 Mths SL Supplementr Questions. Sketh the grphs of: f( ). Sketh the grphs of the following funtions., =. g( ) ( + ) ( ) =, 0. f( ) = + + = = Sketh the grph of f( ) + =, lerl identifing ll smptotes nd turning points. 07

67 Mths SL Supplementr Questions Eerise On the sme set of es, sketh the grphs of f( ) = nd g( ) =. Find: i {(, ) : f( ) = g( ) } ii { : f( ) > g( ) }. 8. Find the rnge of the following funtions. f: ]0, [, where f() = e ( + ) +. g() = e +, ],0]. e +, [,] 9. Sketh the grph of f( ) =, lerl lelling ll interepts with the es nd the eqution of the smptote. Solve for, where =. 0. Sketh the grphs of the following funtions: f( ) = g( ) = h( ) =. Sketh the grphs of the following funtions. f( ) = g( ) = + h( ) =. Sketh the grphs of the following funtions nd find their rnge. f( ) =, < f( ) = e, > 0, +, 0 f( ) = -, +, < d g( ) =, < <,. Sketh the grphs of the following, nd hene stte the rnge in eh se. f:, = + f:, = + f:, = d f:, =. Sketh the grph of the funtions. e g( ) = ), > 0 h( ) =, 0 < < f( ) =, > d f( ) =, 0 < < g( ) =, > f h( ) = +, >. On the sme set of es, sketh f( ) = nd g( ) = where >. Hene, sketh the grph of the funtion h( ) = +, where >. On the sme set of es, sketh f( ) = nd g( ) = +, where >. Hene, dedue the grph of h( ) = ( ) +, where >. 07

68 Mths SL Supplementr Questions 6. Sketh the grph of the following funtions nd determine their rnge. e f( ) =, > f( ) =, 0 < < g( ) = ( ), > d h( ) =, > f( ) = -, > f g( ) =, > 07

69 Mths SL Supplementr Questions 07

70 Mths SL Supplementr Questions 07

71 Mths SL Supplementr Questions Eerise Given the funtion = f( ), sketh the grphs of: = f( ) = f( ) e f( ) = log 0 ( ) d f( ) = ln e p f( ) = ln( e ) f f( ) = log ( ) 7. On the sme set of es, sketh the grphs of f( ) = ln nd g( ) = ln( e). Find : ln > ln( e) + { }. 8. Sketh the grphs of the following funtions nd find their rnges. log f( ) 0, log = f( ) ( ), =, <, < f( ) ln, e =, e < e d g( ) = +, > log +, 0 < 0 9. Sketh the grphs of the following funtions. f( ) = log f( ) = log ( ) f( ) = log + 0. Sketh the grph of the following funtions, lerl stting domins nd lelling smptotes. e f( ) = log ( ), > f( ) = ln( e), > e g( ) = log0 ( 0 ), < < 0 d g( ) = ln e, > g( ) = ln e, > f h( ) = log, 0 < <. Sketh the grph of f( ) = log ( ), 0 < < lerl lelling its smptote, nd interept(s) with the es. Hene, find : f( ) >.. Sketh the grph of: f( ) ln =, > 0 g( ) Given tht f( ) e for ll rel > 0, stte the rnge of g( ). =, > 0 ln 07

72 Mths SL Supplementr Questions 07

73 Mths SL Supplementr Questions 07

74 Mths SL Supplementr Questions 07

75 Mths SL Answers Eerise.. r =, u = 8 u n =, n r r =, u =, u 7 n = n =, u = -, u 6 n = n d r =, u = 6, u n = ( ) n e f r r, u = =, u n = n, u, u n = = = n ± ± ± 96 th u n = n = times 888, 6 i $096 ii $ rs u n - = n , ,,0 or 0,, $6 6 $99 8 Eerise.. f d e d - e ; 9 7; 7 8; d 8; e ;.6 f ;. 7-8 d 60 e ; $ m 8 V n = V n gms; 0 weeks ,

76 Mths SL Answers r =, $ or out 00 illion tonnes. Eerise..6 Term 9 AP = 80, GP = 6. Sum to terms AP = 60, GP = , 8 weeks Ken $0 & Bo-Youn $) 6 week 8 week [~00, depends on rounding errors] Eerise d fish. [NB: t <. If we use n = then ns is 6660 fish]; fish. 7 Overfishing mens tht fewer fish re ught in the long run. 8,, or 6,, m ( t) n + t ( t ) - n + t + t - + t Eerise..8, t n = 6n , 07

77 Mths SL Answers ± (,, ), (, 0, 0), 7,, 8 n 6 m Eerise..9 $77.08 $77.6 $78.9 $ 6 $ $ $ $ , $8. 9 $ $98.6, $967., interest $67.. Flt interest = $6000 $., $790., 0.60% /month (or 7.% p..) ½, The sequene ½, ½, ½,.. is rithmeti. Proof m = 9, n = 07

78 Mths SL Answers Eerise m,.+ m 9 m, + 8 m m, 88 + m d 6 7 m, m e 96 6 m,.8+ m f 6 m,.+ 9 m g 8.8 m, m h 9 00 m, m i 9 7 m,. + m j 88 m, + 8 m k m, + m l 98 m, 8+ m m 96 7 m,.6+ 8 m n m, m o 0 m,. + 0 m 0.6, m.6 79 m 6. m m 06.8 m i 7.09 m ii 88.7 m 70.9 m 6.7 m 9. m 0. m 8.8 m 9 m m 6 07

79 Mths SL Answers = - tn α. =. α 0.9 α m 07

80 Mths SL Answers Eerise d 0 7 d d e f - g - h i - j - k l m - n - o p q 0 r s 0 t undefined 0 0 d e - f - g h i j - k - l m - n - d e - f g 6 - d e f g - h - i - j - k - 07

81 Mths SL Answers 7, -, - -, - d -, d + 0 k k k k k k - k 7 k k - k - k 8 sin θ ot θ d e ot θ f tn θ 9,,, d, e, - f 6 6 7,

82 Mths SL Answers Eerise.. + = k, k k + =, ( ) + ( ) =, 0 d ( - ) ( ) + - = e = 6 i ii i - ii 7-7,,, 7,, ,,,, d 6 6, i ii + i 6 ii iii 9 8 i ii iii ± + k, k 6 or 7 + k, k 6 i ii i 7 ii + k ( k) + k 6 - i + ii 7 0, ± 8 0,,, 07

83 Mths SL Answers Eerise.. sin αosφ + osαsinφ os αosβ sinαsinβ sinos ossin d os φosα + sinφsinα tnθ tnα e + tnθtnα f - tnφ tnω + tnφtnω sin( α β) os( α + β) sin( + ) d os ( ) e tn ( α β) f tn g tn φ h sin + α + β i sin d d d ( + ) d 0,,,,,, 6 6 0,,, α, ± α, α, α = tn - R = +, tnα = 0 6 R = +, tnα = 8 07

84 Mths SL Answers Eerise.. d e f d 0., 6, d e, f, g 6 h, i d e f g h / / / j 8, / d.. e f g h / / / 6 d / e f g h / / / 07

85 Mths SL Answers 7 d e f g h i j k l m n 8 se ose ot 07

86 Mths SL Answers Eerise.., 7, - 6 6, d,, -, 7 -, -, e, f 7,,,,,, 7,, d e, 6, 7 6, f,,, 7 6 6, 7, d tn e,, 6, - 6 f 90, 0 80,0 90,70 d 6, e,, -, 7 - f 0,, g,,, h, 7, -, ,00,, d,6 e,,, f, g, h.9,.0 i, j,,, k,, 7, l 68,8 6 6 m, n 7,,, o Ø 6, ± 7,,, d e ± f, 7 8, 9 8, - g 8 8, h, 07

87 Mths SL Answers 7 6, 6,, 6,7 6,7,0,,,7 d 6, 6,, 6,7 6, 6,9 6 e n ± sin ± n ±, ( ),n =, 8 7 tn,,, + tn,,,,, 7 9,, 7, -, -, 7 -, 9 -, -, 0,,,,,,6 0,, ± os, 7, tn ( ), + tn ( ) 7,,, 6 6 d tn tn, ( ), + tn, tn ( ) sin + 6 0,, sin, 6,, 6-7 6, sin - sin -, + sin - sin -, ii 0,, ii 0 6,, 6, 7 i { = k + α( ) k, k } ii { k + α ( k + ) α, k } = ( k + ) { = k}, k k - = + k 0 =, k 9 0,,, -, os, os, 7 os ±, ±, ± 07

88 Mths SL Answers 90,99 8,0 (99 8,0 ) (, ) = k +, = k (, ) k =, = k +, k Eerise.. t,,, 9 T = sin t,.,, 7 L = sin t,, 0, 7 V = sin - + 7,,, P = sin ( t ) +.6, 7,, 6 S =.6sin ( t ) t 6 0.6,., 0, P = 0.6sin ,.6,.7, D = 0.8sin- ( t.7) , m, 7. m.8 se,. se 0 70, 80. mid-april to end of August 000 months R 9 d months d months t s 6m 0s d [8,] d 8m e s f D ( t )= 8sin (for emple) g m 07

89 Mths SL Answers D ( t )= 0sin 6 ( + 0.) + (for emple) 7m 6m d 0.86s,, m m t F(t) G(t) d 8.% 6 d 9 i 7,,9, ii [ 0, 7] [, 9] [, ].9 m t 07

90 Mths SL Answers Eerise.6. m m m A B C Eerise.6. A B C * B* C* d no tringles eist. 07

91 Mths SL Answers Eerise km.7 m 76. m 0 7 T.9 m.0 m. m m.7 m 7.9 min hr.96 min.08 km 8 $ m Eerise.6. m m m A B C Eerise km T 7 07

92 Mths SL Answers. m 7. m. m 6. km W8 7 S Eerise m 76.8 m 8.8 m d 70.9 m e 8.0 m f 7. m g 9. m h. m i 6. m j 8. m k 98.8 m l 87. m m 9.0 m n 8.7 m o.6 m 69 m m 7. m 6.77sq units.70 sq units 6. sq units 6.6 m 7 7 ( + tnθ) 8 tnθ 9 Are of ACD = 0.78 m, Are of ABC = 6.8 m Eerise m.8 m 0. m,, T m 7 6 m 8, 6, left. km 0 6 m. m 86 m 0. m.7 m 90 m m 60 d m.99 m 8 m 86 m 0 m 07

93 Mths SL Supplementr Questions Eerise... Find the res nd perimeters of the following setors. h i j Rdius 8.6 m 6. m 76 m Angle k m 0 l m 60 m.8 m 0 n.8 m 70 o. m 8.. The digrm shows running trk. The perimeter of the inside line is 00 metres nd the length of eh stright setion is 00 metres. Find the rdius of eh of the semiirulr prts of the inner trk. If the width of the lne shown is metre, find the perimeter of the outer oundr of the lne. A seond lne is dded on the outside of the trk. The strting positions of runners who hve to run (ntilokwise) in the two lnes re shown. Find the vlue of ngle A (to the nerest degree) if oth runners re to run 00 metres. 9. Find the ngle sutended t the entre of rdius length m whih forms setor of re 80 sq. m. 0. Find the ngle sutended n r of irle of rdius length 0 m whih forms setor of re 7 sq. m. 07

94 Mths SL Supplementr Questions. A hord of length m is drwn in irle of rdius 0 m. Find the ngle it sutends t the entre. Find: i the minor r length ii the mjor r length. Find the re of the minor setor.. Two irles of rdii 6 m nd 8 m hve their entres 0 m prt. Find the re ommon to oth irles... Two pulles of rdii 6 m nd 0 m hve their entres 0 m prt. Find the length of the piee of string tht will e required to pss tightl round the irles if the string does not ross over.. Two pulles of rdii 7 m nd m hve their entres m prt. Find the length of the piee of string tht will e required to pss tightl round the irles if: the string nnot ross over. the string rosses over itself.. A setor of irle hs rdius of m nd n ngle of 6. The setor is folded in suh w tht it forms one, so tht the two stright edges of the setor do not overlp. Find the se rdius of the one. Find the vertil height of the one. Find the semi vertil ngle of the one. 6. A tut elt psses over two diss of rdii m nd m s shown in the digrm. If the totl length of the elt is 88 m, show tht = (. α) tnα i On the sme set of es, sketh the grphs of: = - tnα α ii =. α. Hene find { α : = (. α) tnα}, giving our nswer to two d.p. 7. The digrm shows dis of rdius 0 m with prts of it pinted. The smller irle (hving the sme entre s the dis) hs rdius of 0 m. Wht re of the dis hs not een pinted in lue? 07

95 Mths SL Supplementr Questions Eerise.. 7. Find the oordintes of the point P on the following unit irles. d P P 60 0 P 0 P 8. Find the et vlue of: - sin os sin os sin os tn tn 6 d + tn tn 6 os os + sin sin 9. Show tht the following reltionships re true. sinθ = sinθosθ, where θ = osθ = os θ, where θ = 6 tnθ. tnθ = - tn, where θ θ = d sin( θ φ) = sinθ osφ sinφosθ, where θ = nd φ = 0. Given tht sinθ = nd 0 < θ <, find: sin ( + θ) sin ( θ) os + θ. Given tht osθ = nd 0 < θ <, find: os ( θ) se θ sin θ. Given tht tnθ = k nd 0 < θ <, find: tn ( + θ) tn + θ tn( θ). Given tht sinθ = nd 0 < θ <, find: 07

96 Mths SL Supplementr Questions os θ se θ os( + θ). Given tht osθ = nd < θ <, find: sin θ tn θ os( + θ). Given tht tnθ = nd < θ <, find: sin θ tn + θ seθ 6. Given tht osθ = k nd < θ <, find: os( θ) sinθ otθ 7. Given tht sinθ = k < θ < nd, find: osθ tnθ ose + θ 8. Simplif the following. sin ( θ) os + θ sin( + θ) + θ sin θ os - sin θ sin θ osθ d tn ( + θ) otθ e os ( θ)oseθ f - seθ oseθ 9. If 0 θ, find ll vlues of suh tht: sin = - os = tn = d os = - e tn = - f sin = 07

97 Mths SL Supplementr Questions Eerise.. 6. Prove sin ( + not ) + os ( + ntn ) = sin ( n + ot ) + os ( n + tn ). 7. If kseφ = m tnφ, prove tht seφtnφ mk = m k. 8. If = kse φ + mtn φ nd = lse φ + ntn φ, prove tht k k + m l =. l + n 9. Given tht tnθ = -, 0 < θ <, find: sin θ osθ 0. If sin + os =, find the vlues of: i sin + os ii sin + os Hene, dedue the vlue of sin k + os k, where k is positive integer.. If tnφ =, < φ <, find, in terms of, sin φ + osφ sin φ osφ sin φ os φ. Find: the mimum vlue of the minimum vlue of i os θ + ii sin iii os θ + θ + sinθ. Given tht sinφ = nd osφ =, find. Hene, find ll vlues of φ tht stisf the reltionship desried in prt.. Find: the mimum vlue of the minimum vlue of i sin θ ii osθ. Given tht sinθosθ = k, find: ( sinθ + osθ), sinθ + osθ > 0. sin θ + os θ, sinθ + osθ > 0 07

98 Mths SL Supplementr Questions 6. Given tht sinφ = -, 0 < φ <, find tn φ. + Given tht sinφ =, < φ <, find : i osφ ii otφ 7. Find: the vlue(s) of os, where ot = ( ose tn), 0 < <. the vlues of sin, where os = +, 0. os 8. Given tht sin = sin os, find ll vlues of, suh tht sin = tn, 0. 07

99 Mths SL Supplementr Questions Etr Emples Emple..8 If sinθ = nd osφ =, where 0 θ nd φ, find: sin ( θ + φ) os ( θ + φ) tn( θ φ) We strt drwing two right-ngled tringles stisfing the given onditions: θ osθ = tnθ = φ sin( θ + φ) = sinθosφ + sinφosθ However, we nnot simpl sustitute the ove rtios into this epression s we now need to onsider the sign of the rtios. As 0 θ then osθ = nd s φ then sinφ =. sinφ tnφ = = Therefore, sin( θ + φ) = + = os( θ + φ) = osθosφ sinθ sinφ 6 6 As 0 θ then osθ = nd s φ then sinφ =. Therefore, os( θ + φ) = = 6 Emple..9 If sinθ =, where 7 θ, find: sin θ os θ tnθ We strt drwing the relevnt right-ngled tringle: sinθ = sinθosθ = θ = 9 osθ sin = θ = 7 = 9 07

100 Mths SL Supplementr Questions tnθ sinθ 9 = = - = osθ 9 Emple..0 Prove tht: sinαtnα + osα = otβ = otβ tnβ L.H.S = sinαtnα + osα = sin sinα αosα - + ( sin α) osα = = sin α + sin α = R.H.S R.H.S = otβ tnβ = = = = = - osβ - sinβ sinβ osβ os β sin β sinβosβ osβ sinβ osβ sinβ otβ = L.H.S Notie tht, when proving identities, when ll else fils, then epress everthing in terms of sine nd osine. This will lws led to the desired result even though sometimes the working seems like it will onl grow nd grow eventull, it does simplif. Be persistent. To prove given identit, n one of the following pprohes n e used:. Strt with the L.H.S nd then show tht L.H.S = R.H.S. Strt with the R.H.S nd then show tht R.H.S = L.H.S. Show tht L.H.S = p, show tht R.H.S = p L.H.S = R.H.S. Strt with L.H.S = R.H.S L.H.S R.H.S = 0. When using pprohes nd, hoose whihever side hs more to work with. 07

101 Mths SL Supplementr Questions Emple.. Find ll vlues of, suh tht sin = os, where 0. sin = os sinos = os O 90 = 0 = 6 O 0 = 6 sin os os = 0 os( sin ) = 0 os = 0 or sin = Now, os = 0, 0 =, nd sin =, 0 =, 6 6 Emple.. Simplif θ sin +. Epress os θ sinθ in the form Ros ( θ + α), where R nd α re rel numers. Hene find the mimum vlue of os θ sinθ. sin θ + sinθ os + osθsin θ = = sin - + osθ - = sinθ + osθ In this instne, s the sttement needs to e true for ll vlues of θ, we will determine the vlues of R nd α setting Ros ( θ + α) osθ sinθ. Now, Ros( θ + α) = R[ osθosα sinθ sinα] = Rosθosα Rsinθ sinα Therefore, we hve tht Rosθosα Rsinθsinα osθ sinθ Rosθ osα = osθ Rosα = () Rsinθsinα = sinθ Rsinα = () 07

102 Mths SL Supplementr Questions Dividing () () we hve - Rsinα Rosα = tn α = α = Sustituting into () we hve Ros = R - = R =. Therefore, osθ sinθ os θ + Then, s the mimum vlue of the osine is, the mimum of os θ + is. Eerise.. 0. Prove tht: + sin + os + sin os = ot os = 8os 8os + sin φ = + osφ osφ d 8 8 sin = tn + tn. For the right-ngled tringle shown, prove tht: B A C sinα = - osα = - sinα = d osα = - +. Find the et vlue tn. 8. Given tht α + β + γ =, prove tht sinα + sinβ + sinγ = sinαsinβsinγ.. Solve the following for 0. sin = sin sin = os tn = tn 07

103 Mths SL Supplementr Questions. Given tht sinθ + osθ Rsin( θ + α), epress R nd α in terms of nd. Find the mimum vlue of + sinθ + osθ. 6. Given tht osθ + sinθ Ros( θ α), epress R nd α in terms of nd. Find the minimum vlue of + osθ + sinθ. 7. Prove tht tn + = se + tn. 8. Show tht if t = tn then t + t =. Hene find the et vlue of tn. 07

104 Mths SL Supplementr Questions Eerise.. 7. Sketh grphs of the following funtions for -vlues in the intervl [,]. = sin( ) = os = tn d = sin e = sin( ) f = os - g = tn + h = os + i = sin + j = tn( + ) + k - ( ) = sin l = sin m = os( ) n = sin[ ( + ) ] 8. i Sketh one le of the grph of the funtion f( ) = sin. ii iii For wht vlues of is the funtion = - not defined? f( ) Hene, sketh one le of the grph of the funtion g( ) = ose. i ii iii i ii iii Sketh one le of the grph of the funtion f( ) = os. For wht vlues of is the funtion = - not defined? f( ) Hene, sketh one le of the grph of the funtion g( ) = se. Sketh one le of the grph of the funtion f( ) = tn. For wht vlues of is the funtion = - not defined? f( ) Hene, sketh one le of grph of the funtion g( ) = ot. 07

105 Mths SL Supplementr Questions Eerise.. 6. Solve the following equtions for the intervls indited, giving et nswers: e os = os, f se =, 0 g sin os =, 0 h sin = os, 0 7. Find: tn + tn =, 0. tn + tn = tn +, If 0, find: sin = 0 tn = 0 os ( ) = 9. If 0, find: se + se = 8 se = tn + ot ot = 0 d 6ose = 8 + ot 0. Epress sin + os in the form Rsin ( + α). Solve sin + os =, 0.. Epress sin os in the form Rsin ( + α). Solve sin os =, 0.. Find if + sin + sin =, 0.. Sketh the grph of f( ) = sin, 0. Hene, find: i sin > { 0 < < }. ii { sin < } { 0 < < }. 07

106 Mths SL Supplementr Questions Eerise... A hill hs its ross-setion modelled the funtion, h : [ 0, ], h( ) = + os( k), where h( ) mesures the height of the hill reltive to the horizontl distne m from O. Determine the vlues of h m i k ii iii O.0.0 m How fr, horizontll from O, would n nt liming this hill from O e, when it first rehes height of metre? How muh further, horizontll, will the nt hve trvelled when it rehes the sme height of metre one over the hill nd on its w down?. A nurser hs een infested two inset pests: the Fruitfl nd the Gretfl. These insets pper t out the sme time tht prtiulr plnt strts to flower. The numer of Fruitfl (in thousnds), t weeks fter flowering hs strted is modelled the funtion F( t) = 6 + sin( t), 0 t Wheres the numer of Gretfl (in thousnds), t weeks fter flowering hs strted is modelled the funtion G( t) = 0.t +, 0 t Cop nd omplete the following tle of vlues, giving our nswers orret to the nerest hundred. t F(t) G(t) i d On the sme set of es drw the grphs of: F( t) = 6 + sin( t), 0 t. ii G( t) = 0.t +, 0 t. On how mn osions will there e equl numers of eh inset? For wht perentge of the time will there e more Gretflies thn Fruitflies? 6. The depth, d( t) metres, of wter t the entrne to hrour t t hours fter midnight on prtiulr d is given Sketh the grph of d( t) for 0 t. d( t) = + sin t 6, 0 t For wht vlues of t will: i d( t) = 0., 0 t ii d( t) 0., 0 t. Bots requiring minimum depth of metres re onl permitted to enter the hrour when the depth of wter t the entrne of the hrour is t lest metres for ontinuous period of one hour. Find the lrgest vlue of, orret to two deiml ple, whih stisfies this ondition. 07

107 Mths SL Supplementr Questions Eerise The frmework for n eperimentl design for kite is shown. Mteril for the kite osts $ per squre m. How muh will it ost for the mteril if it is to over the frmework of the kite. 0 C 8 m A 0 m B D 9. A o wlking long stright rod noties the top of tower t ering of 8 T. After wlking further. km he noties tht the top of the tower is t ering of 9 T. How fr from the rod is the tower? 07

108 Mths SL Supplementr Questions 07

109 Mths SL Supplementr Questions 07

110 Mths SL Supplementr Questions 07

111 Mths SL Supplementr Questions Eerise A sndpit in the shpe of pentgon ABCDE is to e uilt in suh w tht eh of its sides is of equl length, ut its D ngles re not ll equl. The pentgon is smmetril out DX, where X is the midpoint of AB. The ngle AXE nd BXC re oth nd eh side is m long. E C Find XEA. Find the length of EX. A X B How muh snd is required if the sndpit is 0 m deep? Give our nswer to three deiml ples. 8. A tringulr region hs een set side for housing development whih is to e divided into two setions. Two djent street frontges, AB nd AC mesure 00 m nd 0 m respetivel, with the 00 m frontge running in n esterl diretion, while the 0 m frontge runs in north-est diretion. A pln for this development is shown longside. Give ll nswers to the nerest metre. Find the re overed the housing development. During the development stges, n environmentl group speified tht eisting trees were not to e removed from the third frontge. This mde it diffiult for the surveors to mesure the length of the third frontge. 0 m C Clulte the length of the third frontge, BC. The estte is to e divided into regions, iseting the ngle t A with stepping wll running from A to the frontge BC. A 00 m B How long is this stepping wll? 07

112 Mths SL Supplementr Questions 07

113 Mths SL Supplementr Questions 07

114 Mths SL Supplementr Questions 07

115 Mths SL Answers Eerise.. d d {,,e,g,u}; {d,f} {d,f}; {,}; {,e} {,g},{,g} d {d,f}, {,e} e {d,f}, {,e}, {,,g} d e f g AC AB AD d BA e 0 6 Y N Y d Y e N 7 i i i A C 0 m/s C 0 km 60 N km 0 km 80 W E B B S 0 km A 0 m/s N, E N 9 79 N long river ii iv 0 ( os0 ) v 0 os0 0 i 00 kph N ii.6 kph, N 7 7 W i 00 ii

116 Mths SL Answers Eerise.. i + 8j k i + j + k i + 7j 7k d 6i k i j + k 8i + j + k 8i j + k d i + 6j + k d 6, (, ) 6 8i j 8k 9i 7j 6k 7i + j + k d 0i + j 0k d A =, B = 7 9 (, ) (, ) ( 6, ) 0 Depends on sis used. Here we used: Est s i, North j nd vertill up k D = 600i 800j + 60k, A = 00i 00j + 60k 800i 00j 07

117 Mths SL Answers Eerise d 9 f g h 6 i d 0 e 7 f 7 g 80 h Not possile d e Not possile f 0 6 d Not possile = 6, 7 = 7 0 ± ( i + j + k) λ( 6i 0j + k) e.g. i + j + k 7 if or = θ vˆ 6 û - i û = ( i j) ii vˆ = - ( i + j) ( i + j + k) Use i s km estwrd vetor nd j s km northwrd vetor. WD = i + 8j, WS i + j = nd DS 9i 7j = ( i + 8j) 80 d d ( i + 8j) e i + 6j 80 07

118 Mths SL Answers Eerise.. 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) r = j + λ( 7i + 8j) d r = i 6j + λ( i + j) e r = + λ 0 or r = i j + λ( i + 0j) f r = + λ or r = i + j + λ( i + j) r = i + j + λ( i + j) r = i + j + λ( i j) r = i j + λ( i + j) r = 9i + j + λ( i j) r = 6i 6j + t( i j) r = i + j + λ( i + 8j) d r i j µ i = + + j = 8 + µ = 0 + µ = 7 µ = µ = +.µ = + 0.µ d = 0. 0.t = t 6 = = = - 8 d 0. = 0. e = 7 7 r = j + t( i + j) r = i + t( i + j) r = 6i + t( i + j) 8 6i + j 6 i 8 j 9 r = i + 7j + t( i + j) (, ), (, ), (9, ) d r = i j + λ( i + j) e i M L ii M = L + =,, 07

119 Mths SL Answers ii nd iii ( 8, ) 6 r = k ( 9i + 0j) 7 7 9, Ø Lines re oinident, ll points re ommon. Eerise.. 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 + 7k + t( i + 9j k) r = i + j + k + t( 7i + 7k) = = z - + = z +, = = = z = 7t = + t z = 6 t r 7 = + t 6 7 = = z 6,, 0 6 = + t = + t z = + 0.t = +.t = t z = t = t = t z = + t d = + t = + t z = + 0.t 7 = z + = - =, = z = = z = z, = 0 (,, 0) =, = 07

120 Mths SL Answers = + t = + t = t z = z 0 z= plne + = = z = 0 z = + t z = plne r 0. t. = +. Line psses through (, 0., ) nd is prllel to the vetor i j + k (, 0., ) Does not interset. L: z = = M: z, = - = Ø 8.9 d i ( 0,, 0) ii 0,, 0 8 = = 9 z 9 k = or i + 6j 7k (or n multiple thereof) Not prllel. Do not interset. Lines re skew. 07

121 Mths SL Answers Eerise = = z = z, = (,, 0) =, = = + t = + t = t z = z 0 z= plne + = = z = 0 z = + t z = plne r 0. t. = +. Line psses through (, 0., ) nd is prllel to the vetor i j + k (, 0., ) Does not interset. 7 L: z = = M: z, = - = Ø 8.9 d i ( 0,, 0) ii 0,, 0 0 k = 6 = = 9 7 or z i + 6j 7k (or n multiple thereof) Not prllel. Do not interset. Lines re skew. 6 t= 7 (,,), no 8 ~ 07

122 Mths SL Supplementr Questions Eerise... Ptrik wlks for 00 m to point P due est of his in t point O, then 00 m due north where he rehes vertil liff, point Q. Ptrik then lims the 80 m liff to point R. Drw vetor digrm showing the vetors OP, PQ nd QR. Find: i OQ ii OR 07

123 Mths SL Supplementr Questions 07

124 Mths SL Supplementr Questions 07

125 Mths SL Supplementr Questions 07

126 Mths SL Supplementr Questions Emple Find the ngle etween the lines: r = + λ,r = λ It is neessr to find the ngle etween the two vetors tht represent the diretions of the lines: These re:,. Using the dot produt method : = ( ) + = 0 = + = = + = osθ= 0 θ = Eerise... The line L is defined the prmetri equtions = k nd = + k. Find the oordintes of three points on L. Find the vlue of k tht orresponds to the point (, 8). Show tht the point (, ) does not lie on the line L. d Find the vetor form of the line L. e A seond line, M, is defined prmetrill = + 0λ nd = 6λ. Desrie the reltionship etween M nd L for the se tht: i = 8 nd = ii = nd =. Find the Crtesin eqution of the line tht psses through the point A(, ) nd suh tht it is perpendiulr to the vetor i + j.. Find the diretion osines for eh of the following lines: r = + µ r = + λ 9. 07

127 Mths SL Supplementr Questions. Show tht the line + + = 0 hs diretionl vetor nd norml vetor. B mking use of diretionl vetors, whih of the following lines re prllel to L : + = 0? i = 0 ii = 0 iii + 6 = 0. Find the point of intersetion of the lines r = + λ 8 nd =. 6. Find vetor eqution of the line pssing through the origin tht lso psses through the point of intersetion of the lines: u = + λ nd v = + µ. 7. Consider the line with vetor eqution r = ( i j) + λ( i + j). Find the points of intersetion of this line with the line: u = ( i + j) + µ ( i j) v = ( i + j) + t( 6i 8j) w = ( i + 9j) + s( i + j) 07

128 Eerise.. 8. Show tht the lines Mths SL Supplementr Questions = = z nd = + = + z re prllel. 9. Find the Crtesin eqution of the lines joining the points (,, ) to (,, ) (,, ) to (,, ) 0. Find the oordintes of the point where the line r + t = intersets the - plne. The line = + = z psses through the point (,, ). Find the vlues of nd.. Find the Crtesin eqution of the line hving the vetor form: r = + t r 0 t 0 = +. 0 In eh se, provide digrm showing the lines.. Find the vetor eqution of the line represented the Crtesin form Clerl desrie this line. = = z.. Find the ute ngle etween the following lines. r 0 s = + nd r = + t. r + s 0 = nd r = s + z = = nd = = z 07

129 Mths SL Supplementr Questions. Find the point of intersetion of the lines: z 9 = 0 = nd =, 9 = z z = = nd + = - = z -. Find the Crtesin form of the lines with prmetri eqution given : L : = λ, = λ +, z = λ nd M : = µ, = + µ, z = µ Find the point of intersetion of these two lines. Find the ute ngle etween these two lines. Find the oordintes of the point where: i L uts the - plne. ii M uts the -z plne. 6. Show tht the lines z + - = = nd z = = re oinident. 7. Show tht the lines 7 z = = nd z 7 = = re skew. 8. Find the eqution of the line pssing through the origin nd the point of intersetion of the lines with equtions, z = = nd 6 = 0 = z. 9. The lines = = + z nd z k = =, k \{0} meet t right ngles. Find k. 0. Consider the lines L : = 0, = z + nd M : z 0 = = -. Find, orret to the nerest degree, the ngle etween the lines L nd M.. Find the vlue(s) of k, suh tht the lines k z = = nd k + - z = = re perpendiulr.. Find diretion vetor of the line tht is perpendiulr to oth z + - = = nd + z = = Are the lines + - Wht do ou onlude? z = = nd z = = prllel? Find the point of intersetion of these lines. 07

130 Mths SL Answers Eerise.. i 00 ii (0.) Smple size is lrge ut m e issed ftors suh s the lotion of the th. Popultion estimte is 000. i 00 ii , numeril;, d, e tegoril, d disrete;,, e ontinuous Eerise.. % 9.% Sore 7% 80 Continuous Rinfll mm 07

131 Mths SL Answers Eerise.... The vlue 7.6 hs prol een mis-reorded n should hve een.76. It should e disrded. Bering in mind the errors evident in the dt, the result should e reported s.7 gm/ s the men is.7.. There is no orret nswer. Most dontions re $ to $ with the medin $. 07