ANSWERS. mathematical. StUDieS StaNDaRD LeVeL. Peter Blythe Jim Fensom Jane Forrest Paula Waldman de Tokman

Transcription

1 O X F O R D I D I p l O m a p R O g R a m m NSWERS mathmatial StDiS StaNDaRD LVL O RS E O M P N I O N Ptr lth Jim Fnsom Jan Forrst Paula Walman Tokman

2 Numr an algra nswrs Skills hk a = ( ) = (. ) (. ) =. = = ( ) (. ). =. = ( )( + ) = ( (. ))(. + ) =. a 7= = + 7 = = 7 ( ) = = = + = ( ) = = = = = or = a % of =..% of = a > > > = =. rmmr that th asolut valu of a numr is alwas gratr than or qual to zro ut nvr ngativ. a = = 7 = = = Eris a i a+ = + = = ii ( a+ ) = ( + ) = iii a = = = iv ( a ) = ( ) = ( ) = i Ys ii Ys iii No iv Ys Eris a + = = = ( or =. ) It is not an intgr. a = = or = oth ar intgrs. a i a = a + + = = ii a = ( ) = = i It is an intgr. ii It is not an intgr. Eris ( or 7. ) Look for th imal pansion of ah of th frations =.... Thrfor th imal pansion of this fration rurs. =.. Thrfor th imal pansion of this fration is finit. Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

3 7 =.... Thrfor th imal pansion of this fration rurs. = Thrfor th imal pansion of this fration rurs. =.. Thrfor th imal pansion of this fration is finit. a a =.. a =..... a =..... a a = a = a = =.. =. =..... = 7 = = a It oul.;.;.;.; t It oul. ;. ;. ;. ; t It oul. 7;. ;. Eris D ithr work out th arithmti man of ths numrs as shown in th ook or look for thir imal pansion. Th numrs ar an Thrfor an. Numrs in twn ma for ampl.;.;. a ( ) whn = an = ( ) = (or.) is a rational numr a Th numrs ar an Thrfor. an.. Numrs in twn ma for ampl.;.;.. i Th numrs ar an ii Thrfor.... an Numrs in twn ma for ampl.;.;. infinit Eris E a It is a right angl triangl. h = +. h =. h =. m h is rational. a r = = m = π = π m is irrational. Eris F a i. <. ii i multipl. <. < mak th sujt of th inqualit + i q =. is solution as <.. t = ii is solution as <. ii q =. is solution as.. t = is not solution as th inqualit is not tru. a i + > > > ii iii > > i iii ii 7 sustitut ah of ths numrs in th inqualitis p Inqualit + > + > π Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

4 Eris G i. = to th narst unit ii iii. = to th narst unit. = to th narst unit iv. = to th narst unit i. = orrt to th narst ii = orrt to th narst iii. = orrt to th narst iv = 7 orrt to th narst i = orrt to th narst. ii = orrt to th narst. iii = orrt to th narst. iv = orrt to th narst. i 7 = orrt to th narst. ii = orrt to th narst. iii = orrt to th narst. iv = orrt to th narst. n whr < n whr < n whr. <. Eris H i.7 =.7 orrt to.p. ii iii. =. orrt to.p.. =. orrt to.p. iv. =. orrt to.p. i.7 =. orrt to.p. ii iii. =. orrt to.p.. =. orrt to.p. iv.7 =. orrt to.p. i. =. orrt to th narst thousanth. ii. =. orrt to th narst thousanth. iii.7 =.7 orrt to th narst thousanth. iv. =. orrt to th narst thousanth.... i ii iii. (.p.) =.7..7 (.p.).7 (.p.) iv orrt to th narst v orrt to th narst ( p + q) p + q =. i. (.p.) ii. (.p.) iii orrt to th narst unit iv orrt to th narst n whr. <. 7 7 n whr. <. Eris I i has signifiant figurs as all non-zro igits ar signifiant an zros twn nonzro igits ar signifiant. ii iii has signifiant figur as trailing zros in a whol numr ar not signifiant.. has signifiant figur as all non-zro igits ar signifiant an zros to th lft of th first non-zro igit ar not signifiant. iv has signifiant figurs as trailing zros in a whol numr ar not signifiant. v has signifiant figurs as all non-zro igits ar signifiant an zros twn non-zro igits ar signifiant. i = ( s.f.) ii iii.7 =.7 ( s.f.). = ( s.f.) iv. =. ( s.f.) i = ( s.f.) ii iii. =. ( s.f.).7 =.( s.f.) iv. = ( s.f.) i 7 = 7 ( s.f.) ii iii. =. ( s.f.) = ( s.f.) iv. =.( s.f.) a =. orrt to signifiant figurs orrt to signifiant figurs. orrt to imal pla. orrt to th narst hunrth π =. a orrt to th narst unit. orrt to.p.. orrt to s.f.. orrt to.p. Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

5 7 a = ( s.f.) a = ( s.f.).7 =.7 ( s.f.) = i.7 ii.7 iii.7 Eris J a = πr a. = πr. p = r r =. p r =. m ( s.f.) = πr. = m ( s.f.) + =. ( s.f.) sustitut th valus of p an q in th formula. ( p+ q) = + ( ) =. ( s.f.) =. m ( s.f.) Eris K a. 7 = = = = = population nsit = = pips. total population lan ara 7 7 population nsit = 77 population nsit population nsit popl pr km Numr of rams = Numr of rams Numr of rams vrag sp = vrag sp =. istan travll tim takn vrag sp vrag sp km h Numr of visitors pr ar = Numr of visitors pr ar Numr of visitors pr ar 7 stimat th ara of th squar using rasonal numrs. ra of squar =.. ra of squar = ra of squar = m Thrfor Ptr s alulation is not orrt. is far iggr than. Eris L a sustitut th valus of a an of in th givn formula. a + =. +.7 =. Prntag rror = v v Prntag rror = E ve %.. % Prntag rror = 7.% ( s.f.) a tual final gra = tual final gra =.7 ( s.f.) Th thr gras roun ar, 7 an. pproimat final gra = pproimat final gra = Prntag rror =. %. Prntag rror =.% ( s.f.) a Eat ara =.. Eat ara =. m Lngth =. m With =. m pproimat ara =. m a r.. Prntag rror =. Prntag rror =.7% ( s.f.) r r = p m r =. m (.p.) = πr = p p =. m ( s.f.) % Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

6 pproimat valu for primtr = m pt valu for primtr =. m Prntag rror =.. Prntag rror = % ( s.f.) Eris M. ; % a numr is writtn in stanar form if it is writtn as a k whr a < an k is an intgr. a =. or. ( s.f.). =. =.. =.. =. First, writ ah numr in stanar form... 7 =. =. Now writ in orr. ;. 7 =. ; =. ;... =.. =... ;. =. ;. =.. Eris N a =.. =.7 7 or.7 7 ( s.f.). =. =.. =. =. a th arithmti man twn a an is simpl a+. rithmti man =. +. rithmti man = rithmti man =. narst million is th narst multipl of = orrt to th narst million or a t =. t =. t q =.. 7 = 7 a = =. ( ) = =. > aus oth hav th sam ponnt for whn writtn in stanar form an. > thrfor th statmnt is tru. i sustitut th valu of in th givn prssion. ii Eris O = = Writ our answr in stanar form =. a km h or km/h kg m or kg/m m s or m/s a i agrams ii ntison iii millimtr iv imtr a km = m = m.7 m =.7 am =.7 am m = m =. m a g = =. kg 7 kg = 7 ag = 7 ag g = hg =. hg a. s =. = ms s = as = as. hs =. s = a 7 mg = 7 = 7. kg = kg orrt to th narst kg. m = km =. km = km orrt to th narst km. g =. mg =. mg Eris P a. m =. m = m. m =. am =. am mm = m = m. m =. mm = mm. km =. hm = hm f m = km =. km a m = m = m. am =. m = m mm = m =. m m = mm =. mm m = am =. am Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

7 f.7 hm =.7 am = 7. am = 7. am ( s.f.) th ara of a squar with si lngth l is l. a ra = l l ra = ra = m m = m =. m th volum of a u with si lngth (or g) l is l. a Volum = l Volum =. Volum =. m or. m ( s.f.). m =. m = m or m ( s.f.) onvrt all th masurmnts to th sam unit.. am =. m ; mm = m ; m =. m 7 m =.7 m. m Thrfor th list from smallst is 7 m ;. m ; m mm ;. am onvrt all th masurmnts to th sam unit.. m ; m =. m. am = m mm =. m m =. m Thrfor th list from smallst is m ;. am ; m ; mm ;. m Eris Q a hang all to sons = h = min = s = s h = min = s = 7 s min = s = s Thrfor h m = s + 7 s + s = s s = (narst ) a hang all to sons = h = min = s = 7 s min = s = s Thrfor min = 7 s + s = 7 s 7 s =.7 s or.7 s ( s.f.) a l = ml = ml. ml =. hl =. hl l = m al = l = l a l = m = m = m. l =. l =. m or. m ( s.f.) hl = l = m = m = m a. m =. l = orrt to th narst unit.. m =. m = m = l =. hl = hl orrt to th narst unit. m = m =. m =. l =. l = l orrt to th narst unit. istan travll vrag sp = tim takn a vrag sp = m min = m tim takn tim takn = istan travll tim takn m m min tim takn = 7 min 7 min = 7 min = s 7 volum of a u = l a Volum =. =.7 m.7 m =.7 m = 7 m a 7 m = 7 l an 7 l < l thrfor l of watr annot pour in this ontainr. Onl 7 l an pour. of m = 7 m 7 m = 7 m = =. ta ups thrfor Mrs an srv up to ta ups. a vrag sp = istan travll km h = tim takn tim takn km tim takn = km km h tim takn =. h or. h ( s.f.) vrag sp = istan travll tim takn vrag sp = km h vrag sp =. km h or 7 km h Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

8 Tim travlling =. h + h +. h =. h rrival tim = +. =. h or : PM Eris R a t = t k 7. t = 7. t = 7. =.. =. orrt to on tnth of gr = + t t = ( ) t = =... =.7 orrt to on tnth of gr a t F = + t F = =.. F = 7 F orrt to th narst gr. t F = + 7 t F = =.. F = F orrt to th narst gr. a t = 7. =. Thrfor K =. or. ( s.f.) hn mans us th pring answr to solv this part qustion. K =. lso t F =. + t F =. + =. Thrfor K =. F or. F ( s.f.) a mak t K th sujt of th formula. t = t K 7. t K = t + 7. mak t th sujt of th formula t t F = t + = ( t ) F Rviw ris Papr stl qustions π. a =.. =.... =. =.. ;. ; ; ;. a kg =. kg i.7 kg = 7 kg ii prntag rror formula Prntag rror = v v E % v Prntag rror = v Prntag rror = E v E ve 7 % % Prntag rror =.7% ( s.f.) a 7 m s = m s m s mans mtrs pr son thrfor th answr from a givs ou th istan travl in son. travls s m m = km = km travls s km travls s km = km km =. km Thrfor th avrag vloit is. km h a Eat wight of on ook = g = kg =. kg. kg =. kg ( s.f.) pt valu =. kg Estimat valu =. kg Prntag rror = v Prntag rror = v E ve... % Prntag rror =.% ( s.f.) % = g a m = m =. m. m =. l of. l = 7. l i. jars 7. Thrfor San pours jars. ii.7 =.7.7 =. Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr 7

9 7 a = = + (. ). + whn =. =.. =. ( s.f.). =. a = i. km =. m = m = = = m ii Primtr = Primtr = m a t = t. 7 F K t F =. 7 t F =. or. ( s.f.) t = t K K. 7 = t K. 7 t K = ( +. 7 ) t F =.7... = orrt to th narst unit a + > + > l = = < p < = >.. =.... > =. >. =. < Thrfor ;. ; a ra = mm 7 mm ra = 7 mm 7 mm = 7 m =.7 m wighs m 7 g on pag wighs. 7 m.7 7 =.777 g =. g ( s.f.). = g g = kg =. kg Rviw ris Papr stl qustions a P rimtr of th fil = + Primtr of th fil = 7 m 7 m = 7 km = 7. km ost of fning th fil = ost of fning th fil =. (.p.) V = =. Prntag rror = v v Prntag rror = E ve % %... Prntag rror =.% ( s.f.) ra of th fil = ra of th fil = m ra of th fil = km =. km a Raius of smiirls = = m Lngth of irumfrn = πr Lngth of irumfrn = π = π Primtr = + π Primtr =.7... m = 7 m orrt to th narst mtr. Numr of laps that Elgr runs total istan run Elgr = primtr of running trak Numr of laps that Elgr runs = Numr of laps that Elgr runs =.7 Thrfor Elgr runs omplt laps aroun th trak. onvrt th istan to km.7... m =.7... km =.7... km avrag sp = istan travll tim takn km h =.7... km tim takn tim takn =.7... km km h tim takn =. h ( s.f.) avrag sp = km h m min m m min = tim takn m m min km m h min tim takn = tim takn =. min ( s.f.) Prntag rror = v v Prntag rror = E ve.. % Prntag rror =.% ( s.f.) % Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

10 a Diamtr =. m Raius =. =. m Volum of on hoolat = πr Volum of on hoolat = π (.) Volum of on hoolat =... m =. m (.p.) first onvrt th masurmnts to m. Raius of linrial o =. mm =. m Volum of linrial o = πr h Volum of linrial o = π (.) Volum of linrial o = m = 7. m (.p.) Numr of hoolats in th o = =. hoolats Volum oupi th hoolats =... =.7... m Volum not oupi th hoolats = volum of o volum oupi hoolats Volum not oupi th hoolats = =. m ( s.f.). m =. mm = mm f. mm Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

11 Dsriptiv statistis nswrs Eris a Disrt ontinuous Disrt Disrt ontinuous f Disrt g ontinuous h ontinuous i ontinuous j Disrt k ontinuous l Disrt a ias Ranom ias Ranom ias Eris Numr of goals Frqun 7 7 Numr of has Frqun 7 7 g Frqun 7 Numr of risps Frqun Numr Frqun 7 7 m =, n = Eris nswrs will pn on with of lass intrvals hosn. Eampl: a Numr Frqun < < 7 < < < < < < < < Numr Frqun < 7 < < 7 < < 7 < 7 7 < < < Numr Frqun < < 7 < 7 7 < < < < < 7 7 < < Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

12 Eris D a lass Lowr ounar ppr ounar Tim (t sons) Lowr ounar ppr ounar. t <.... t <.... t <... Eris E f a Lowr ounaris ar,,,, ppr ounaris ar,,,, 7 f g a Lowr ounar of th thir lass is. an th uppr ounar is. f.... a Lowr ounaris ar.,.,.,., 7.,.,. ppr ounaris ar.,.,., 7.,.,.,. Frqun 7 Lngth (m) a Numr Frqun < < < 7 < < < < 7 7 < < Frqun Numr of tims travl train a Numr of frqun ws < < < < < < 7 < 7 7 < Frqun Numr of ws 7 a Th lowr ounar of th fourth group is. an th uppr ounar is. f Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

13 Numr of visitors Tim (7 + + ) a = ( + ) = So, + = () = 7 = 77 a Zo s total = = Shun s total = 7 = + = () = = 7 + = () = = Eris F a rrang in orr: 7 Mo = Mian = th ntr = 7 7 Man = =.7 ( sf) 7 7 rrang in orr: Mo = Mian =.th ntr = Man = 7 =. ( sf) a rrang in orr: Mian =.th ntr = Mo =. Man = = 7.7( sf) 7 =. rrang in orr: Mian =.th ntr = =. Man = =. a rrang in orr: Mian =.th ntr = =. Man = =. If th mo is thn s = aus w n mor s than othr numrs. If th man is. thn ( t ++) =. + t =. = So, t = Eris G a Moal sor = (it has th highst frqun) Mian = =th ntr = Man = 7 =. 7 a Numr of hilrn = = Highst frqun =, thrfor moal numr of visits = Man = =. a n = ( ) = 7 Man = 7 =.7 7 aus it has th highst frqun. a Man = 7 = =. ( + ) = % aus it has th highst frqun Eris H a t < s GD. S haptr for hlp. a 7 s < s GD. S haptr for hlp. a < Eris I s GD. S haptr for hlp. a N = th total numr of tims = + a = a = = ( ) = = + = Qustions : ll th answrs an ra from th graphs. Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

14 Eris J ll th answrs an ra from th graphs. Eris K ll th answrs an ra from th graphs. Eris L a i rang = = IQR (from GD) = = i rang = = IQR (from GD) = = i rang = = 7 IQR (from GD) =. =. Eris M s GD. S haptr in th ook for hlp. Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

15 Statistial appliations nswrs Skills hk a + h = h = m + = = = = m or 7.7 m ( s.f.) a i sing th mipoint formula. Lt M th mipoint twn an. 7 M =, M = (, ) ii Lt th istan twn an. = ( ) + 7 = ( ) ( ) or. ( s.f.) sing th mipoint formula an st two quations in p an q. q p (.,) = +, + q =. an p Thrfor q = an p = Eris = sing th graint formula. a m = 7 m = m = 7 m = m = 7 ( ) m = m = m = ( 7) a i (, ); (, ) ii m = m = i (, ); (, ) ii m = m = ( ) i (, ); (, ) ii m = m = i (, ); (, ) ii m = ( ) m = i (, ); (, ) ii m = m = ( ) ( ) f i (, ); (, ) m = m = Eris a plot th givn point an thn using that th graint is m = stp fin mor points ling on th lin. stp Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

16 a i m = 7 ii m = graint of = p graint of = graint of = graint of Thrfor p = p = i m = ii m = graint of = t graint of = Thrfor t = t = i m = m = ii graint of = q graint of = Thrfor q q = i m = ( ) ii m = graint of = s graint of = Thrfor s ( ) = s = i m = ( ) m = ii graint of = graint of = r r = = ( ) ( ) r ( ) a ( ) a m = m = a + Thrfor a + quating answr to a to th graint = a + a + = a = a m =. m = t m = t using our answrs to a an. t =. t =. Eris a m = m =. L L L Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

17 if th -oorinats ar th sam thn th lin is paralll to th -ais an if th -oorinats ar th sam th lin is paralll to th -ais. a paralll to th -ais. paralll to th -ais. nithr. a n horizontal lin is paralll to th...-ais. n vrtial lin is paralll to th...-ais. n horizontal lin has graint qual to... zro... If th lin is paralll to th -ais thn -oorinat of an point on that lin will alwas th sam. a =. oth (; ) an(, a) li on th sam lin paralll to th -ais thrfor th hav th sam -oorinat. If th lin is paralll to th -ais thn -oorinat of an point on that lin will alwas th sam. m = oth (m, ) an ( ; ) li on th sam lin paralll to th -ais thrfor th hav th sam -oorinat. Eris D ngativ riproals ar numrs that multipli togthr giv a an ar ngativ riproals. an ar ngativ riproals. an ar ngativ riproals. prpniular lins hav graints that ar ngativ riproals an ar graints of prpniular lins. an ar graints of prpniular lins. a Lt m th graint of a prpniular lin to. m = m = Lt m th graint of a prpniular lin to. m = m = or. Lt m th graint of a prpniular lin to. m = m = Lt m th graint of a prpniular lin to. m = m = Lt m th graint of a prpniular lin to. m = m = ( ) a m = m = 7 Lt m th graint of a prpniular lin to. m 7 = m = 7 m = m = Lt m th graint of a prpniular lin to. m = m = a i ii iii i ii iii Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

18 i ii iii a m = a m = a = m + = + = + = = a i m = ii iii Th point of intrstion with th -ais has th form (, ) = + = Thrfor th point is (, ) Th point of intrstion with th -ais has th form (, ) = + = + m = s answrs to a an to st an quation whr th unknown is a. a = a = 7 a m = m = m = m = = = m t t = t t = t = Eris E a = m + = + = + = = + = m + = + = + = = + = Thrfor th point is i m =, ii = + = Thrfor th point is (, ) iii = + = + = Thrfor th point is, i m = ii = + = Thrfor th point is (, ) iii = + = + = Thrfor th point is (, ) i m = ii = = = Thrfor th point is (, ) iii = = Thrfor th point is, Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

19 a Epan th numrator an writ th whol prssion as a sum. ( ) = = = or =. m =. th -intrpt is, = Th point of intrstion with th -ais has th form (, ) =. =. = Thrfor th point is (, ) a sing th graint formula m = ( ) m = = + = + = = + a m = m = = m+ = + sustitut th graint into th quation = + sustitut P or Q in th quation to fin = = + m = m = or. = m+ =. + =. + = =. + a = m + = + = = + t th -ais th point has =. = + = + 7 a m m m = m m m = thrfor m = = + = m + = m m = = = m + = m m = thrfor m = = WORKED SOLTIONS Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

20 two points from th graph (, ) an (, ) sing th formula m = ( ) m = Or using th graph m = = + = ( ) + = = f two points from th graph (, ) an (, ) m = m = Or using th graph m = = + = = Eris F a Lt (, ) a point on this lin. Sustituting in th graint formula (, ) an (, ). m = = ( ) = + = + = or an multipl of this quation with a,,,. Lt (, ) a point on this lin. Sustituting in th graint formula (, ) an (, ) m = = ( ) = ( ) = + = or an multipl of this quation with a,,,. ( ) m = m = Lt (, ) a point on this lin. Sustituting in th graint formula (, ) an (, ) = ( ) ( ) ( ) = + + = + + 7= or an multipl of this quation with a,,,. (, ) an (,. ) m = m = = ( ) + = + = or an multipl of this quation with a,,,. Mak th sujt of th formula. a Mak th sujt of th formula. + = = + + = = + = = + = = = or =. + = = + = + = + Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

21 a Mak th sujt of th formula. + = + = + = = + or =. + t th -intrpt th -oorinat is. = + = + = Th -intrpt is. = a Point (, ) = = = Whn =, = thrfor th point lis on this lin. Point (, ) = = = Whn = th valu of is not thrfor th point os not li on this lin. Point (, ) = = = Whn =, = thrfor th point lis on this lin. Point D(, ) = = = Whn =, = thrfor th point D lis on this lin. Point E(,) = = = Whn =, th valu of is not thrfor th point E os not li on this lin. Point F(, ) = = = Whn =, = thrfor th point F lis on this lin. = 7= a a = = or a =. t = 7 t = a Point (, ) + = + = = Thrfor point lis on this lin. Point (, ) + = + = = Thrfor point lis on this lin. Point (, ) + = + = = whih is not tru thrfor point os not li on this lin. Point D(, ) + = + = = whih is not tru thrfor point D os not li on this lin. Point E, + = + = = Thrfor point E lis on this lin. + = a + = a = + = + t = t = Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr 7

22 Thr svral was to solv this qustion. On of thm is to hoos on lin an s whih of th onitions sri in th son olumn vrifis. : + = W writ th quation in th form = m+ + = = + = + = + Th graint is an th -intrpt is thrfor it maths with H. : = Th graint is thrfor it is not F an th -intrpt is thrfor it is not E. It is G. : + + = Th -intrpt is. mans that th lin passs through th point (., ). Sustitut (., ) in th givn quation.. thrfor (.,) os not li on this lin. Thrfor it is not E an so it is F. D: = + It is E. Th -intrpt is an whn is. th valu of is. sing th point (, ) = + = = + Points ar ollinar if th li on th sam lin. Putting th oorinats of in th quation of th lin givs = + = Thrfor lis on this lin an, an ar ollinar. Eris G i Vrtial lins hav quations of th form = k = ii Horizontal lins hav quations of th form = k = a us our GD. In th graph mo input oth quations an fi n th intrstion point. mak th sujt of th formula. 7 a + = = + Th graint of L is. Th -intrpt of L is. sustitut into th quation. = + =. t = + t = paralll lins hav qual graints so th graint of L is. if it passs through (, ) thn th -intrpt is. f a m = m = Th intrstion point is (, ). Mtho : writ own oth quations in th form = m+ an thn us th GD as shown in a. + = = an + = Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

23 Th intrstion point is (, ). Mtho : solv th simultanous quations in th quations mo. + = + = + = Th intrstion point is (, ) Th intrstion point is (, ) Mtho : solv th simultanous quations in th GD quations mo. + + = + = + = + = Th intrstion point is ( 7, ). GD. =. + an = Th intrstion point is (, ) Mtho : writ own oth quations in th form = m + an thn us th GD + + = an = + = = + = Th intrstion point is (, ) f Th lins ar = an = Th intrstion point is (, ) Writ th quations in th form = m+ an ompar th graints. L : + + = = an L : + = = + Graint of L = graint of L = Thrfor oth lins ar paralll. a = ( ) = = + an + = = + = + oth graints ar qual an th hav iffrnt -intrpt thrfor ths lins o not mt at an point. Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

24 + = + = ( ) = + an = + Th ar th sam lin (sam graint an sam -intrpt) thrfor th mt at an infinit numr of points. = an = = Th hav iffrnt graints ( an ) an iffrnt -intrpts ( an ) thrfor th mt at onl on point. + = = + an + + = = = + Th ar th sam lin (sam graint an sam -intrpt) thrfor th mt at an infinit numr of points. a Point lis on oth lins. = + (, ) lis on L = + = = Graint of L = = + QR PR PR PQ PQ QR EF ED ED DF DF EF a os ; sin ; tan os ; sin ; tan os ; sin ; tan fin first th missing si. a hp hp i ii iii sin sin os os tan tan opp hp aj hp opp aj opp opp = opp i sin sin opp hp (, ) lis on L = + = = + Eris H ii iii os aj hp os tan tan opp aj Triangl Hpotnus Si opposit Si ajant to X a XZ YZ XY Y Z a R P RQ PR PQ a Q Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

25 a + aj = aj = aj i sin ii iii opp hp sin os os tan tan aj hp opp aj sin tan sin f Eris I tan h h =. ( s.f.) os tan os 7 Sum of th intrior angls of a triangl is. Qˆ Qˆ tan a tan QR PR = QR QR =.7 m ( s.f.) T m Ŝ + + = a Ŝ = os = T T =. m ( s.f.) Z V m S W os. os. =. (.p) tan m m =. (.p) Ẑ + + = a Ẑ = 7 tan VZ VZ =. m ( s.f) L sin. sin. m =. (.p) sin t sin t t =.7 (.p) tan s tan s s =. (.p) Eris J a m R P Q N Mˆ Mˆ 7 os LM LM os M LM =. m ( s.f) a D is right-angl triangl. tan =. m ( s.f.) Primtr of th rtangl D = D + Primtr of th rtangl D = Primtr of th rtangl D = 7. m ( s.f.) Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

26 ra of th rtangl D = D ra of th rtangl D =.... ra of th rtangl D =. m ( s.f.) tan = h 7 a h = 7. m ( s.f.) 7 7 m sin = =. m ( s.f.) 7 os = =. m ( s.f.) Eris K 7 a sin (. ) mans th angl with a sin of. tan os us our GD. a sin (. ) =. tan os a sin. mans th angl with a tangnt of mans th angl with a osin of =. =. sin.. os os. tan tan. a tan = = tan 7. 7 =. =.... =. os R = R = os R =. =.... =. os M =. M = os M =.. =.... =. sin Z = f a Z = sin Z =. Y =.... Y =.. tan J = 7 J = tan J = I = I =.. os F = F = os F =.. E =.... E =. m D tan = m = tan =. Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

27 a E m a 7 a G m os F os F F. J H m m sin H = H = sin H =. I D is a right-angl triangl. tan D D tan D. ( s.f.) m F tan = m = tan m =. m tan = = tan = 7. ( s.f.) a m m sin = = sin =. ( s.f.) Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

28 is th rqust angl. W first fin O whih is half of.. tan O = O =. tan O =.... = O =.... =. tan = a = tan =. 7 m Eris L a 7 m 7 m Th hight of triangl ists (an is prpniular to ). raw th shortr iagonal an rmmr that th two iagonals of th rhomus ist ah othr at right-angls. 7 m. m Lt half of th shortr iagonal.. tan =. = tan = =. m ( s.f.) 7 m 7 m 7 m rop a prpniular to D from. Lt half of. os = 7 =.... m = =.... =. m Primtr of = Primtr of = Primtr of =. 7 Primtr of =. m = m orrt to th narst m. m D 7 m m O. m a DE = DE = m os D = D = os D =7. ( s.f.) a Drop a prpniular from Q to SR. Lt T th point of whr th prpniular an SR mt. ppl Pthagoras in QTR. QT QT QT SP = QT = m ra of PQRS = ( + 7) ra of PQRS = m ossrq = SRQ = os SRQ =. ( s.f.) sin or tangnt an also us. Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

29 a m m m m tan = = tan =. ( s.f.) m m h tan = h m h = m ( s.f.) 7 a sin sin =. D Put all th masurmnts in th sam unit. m. km. km = m tan = = tan =. ( s.f.) i D. m ii (, ) i = ii = D Lt th rquir angl. tan = = tan =. ( s.f.) Eris M m m tan = = tan = 7. 7 a m Lt th hight of th tr. Th = +. tan = = m = +. =. m..7 m km. 7 tan. = =. 7 tan. =. m = s oth an angls ar altrnat intrior angls. Th vrtial istan is th lngth of th si opposit angl (foun in a). Lt th rquir istan. tan = =. km = m (narst mtr) Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

30 Eris N Sustitut into th sin rul formula. a Y X sin 7 = 7 sin = 7sin 7 sin =. 7 km ( s.f.) P m r Q Z m sin = sin = = sin = sin = sin sin =. m ( s.f.) P R r sin = sin Q km Q = Q = r sin r = sin = sin sin r =. 7m ( s.f.) sin = sin 7. sin 7.sin sin. km ( s.f.) sin = sin sin =. m m 7. km Q QR sin QR = R sin = sin sin QR =7. km ( s.f.) Sustitut into th sin rul formula. a m 7 m sin 7 = sin sin = = sin P sin 7 sin 7 =7. ( s.f.) m m sin = sin R sin R = R = sin sin Q R sin R =. ( s.f.) Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

31 Y km using triangl sin sin m ( s.f.) X km Z sin = siny siny = Y = sin sin sin Y = 7. ( s.f.) 7 m Eris O sing osin rul formula a X km Y. 7 km 7 7 os =. km ( s.f.) Z m P m 7 Q m p 7 sin = sin sin = = sin 7 sin 7sin =. ( s.f.) 7 Q m P. m R 7. sin7 = sin R sin7 sin R =. sin7 R = sin. R =. ( s.f.) a X X using triangl X sin = sin X X X sin = sin = sin sin =. m ( s.f.) R p os7 p.7... p.7 m ( s.f.). m.7 m.7..7.os m ( s.f.) In ths ass w ar looking for angls. a m m m os = + os =. 77 P = os (. 77 ) =. ( s.f.) 7. m. m Q. m R os os = = os (. 7...) =. ( s.f.) Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr 7

32 Y km X osa a km km Z osa... a os... a. ( s.f.) m m 7 a osz = + os Z =.... Z = os (....) X. ( s.f.) m J m WORKED SOLTIONS S SJ = + os SJ = = + os = = m ( s.f.) R.7 m. m P SJ =. m ( s.f.) Etn th lin J an raw a prpniular from S to J. m m S 7 m Q PQ = os PQ =.... PQ =7. m ( s.f.) m a Y X m m os X = + os X =. X = os (. ) X =. ( s.f.) km Y X km km Z Z O = O J os7 = = 7. m ( s.f.) os O = + os O =.... O = os.... O = ( s.f.) ( ) a In triangl PQR, PR = os PR =.... PR =. m ( s.f.) ou an appl ithr sin rul or osin rul..... sin =. sin PRQ. sin sin PRQ =.... sin PRQ = PRQ = sin (. 7...) PRQ =. ( s.f.) Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

33 QPR=. QPR=. RPM=. RPM=. sin RPM sin. = 7. h PR 7. + h.... a WORKED SOLTIONS sin... = m (narst m ) = sin =. =. or quivalnt.... = 7. + h Eris P h =. m ( s.f.) s th ara of a triangl formula. a = 7 sin =. km ( s.f.) =. 7. sin = m ( s.f.) a is an isosls triangl = sin =. m a = = = sin =. m Fin first th siz of on angl. a os X = + os X =. X = os (. ) X =.... = sin.... =. km ( s.f.) sin = siny sin siny = sin Y =.... Y = sin (....) Y =.... Z = Z =. ( s.f.) =. = = m 7 a D is a right-angl triangl. D = + D = m or 7. m ( s.f.) in triangl D D m m 7º º = D sin sin7 D = from parts a an. D = =. 7.. sin =. m ( s.f.) sin7 sin D =. 7 m ( s.f.) ra of D = ra of D + ra of D ra of D = ra of D = 7. m ( s.f.) Rviw ris Papr stl qustions a (, ) an (, ) m = m = paralll lins hav th sam graint. = + L passs through (, ) = + or quivalnt forms. a us th graint formula (, ) an (, ) m = m = Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

34 prpniular lins hav graints that ar opposit an riproal. m m = m m = m = m = = + L passs through O (, ) = + = a i lin mts th -ais at th point whr = = + = + = (or.) Point is, ii lin mts th -ais at th point whr = = + = + = Point is (, ) s th two points foun in a an raw th lin. tan. tan α = α = tan α =. ( s.f.) If a point lis on a lin thn its oorinats vrif th quation of th lin. a = + (a, ) lis on L = a + a = = + (., ) lis on L =. + = a us th GD. + = = + an = + Th point is (, ) m tan tan = 7 m ( s.f.) a Sum of th intrior angls of a triangl is. + = = = sin sin sin = sin =. m ( s.f.) = sin =. m ( s.f.) 7 a = = m sing th osin rul, os = os ( ) =. ( s.f.) sin. =. m a O = O = os O = m 7. os O =. O = os (.) O = 7. = sin =. m ( s.f.) Sha ara =.... Sha ara =. m ( s.f.) Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

35 Rviw ris Papr stl qustions a WORKED SOLTIONS.... =. m ( s.f.) Th lngth of th lar is still th sam. P m. m M i D ii D(,) (, ) an (, ) m = m = or D an ar prpniular lins. m m = m = m = us th graint formula = ( ) = ( ) + = f i (, ) an D(, ) ( ) ( ) = + = ii (, ) an (, ) ( ) ( ) = + = g tan D = D = tan D =. ( s.f.) a Lt th lngth of th lar. os = = os = m Lt th hight of th pol. tan = =. m ( s.f.) Nw istan Lt th nw istan +... = =. ( s.f.)... tan... β = tan β =. ( s.f.) a in triangl D, D = + D = m in triangl D, tan D = D = tan D =.7 (.p.) angl D = D =.7 = 7. ( s.f.) In triangl D. = + os 7. = m ( s.f.) i Primtr = Primtr = m ( s.f.) f ii vloit = istan tim. = tim tim =. tim = sons tim = minuts = minuts (narst minut) split th quarilatral in two triangls ra D = ra D + ra D ra D = sin 7. + ra D = m = km =. km. Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

36 Mathmatial mols nswrs Skills hk a =. + whn =, =. h = t whn t =, h = = t t + whn t =, = - a + = =.,. t t = t t = t =.7,. = = =, = a (7, ) (, ) m = = = (, ) (, ) m = = = Eris a Mik Lu Liia Diana a 7 Mrs. rquiza Mr. Gnzr This is a funtion sin ah stunt is in onl on mathmatis lass. Mrs. rquiza Mr. Gnzr Mik Lu Liia Diana This is not a funtion sin ah tahrs tahs two of th stunt. This is a funtion sin ah lmnt of is rlat to on an onl on lmnt of. 7 This is not a funtion sin on lmnt of () is not rlat to an lmnt of. 7 This is a funtion sin ah lmnt of is rlat to on an onl on lmnt of. a i ii iii iv v This is not a funtion sin on lmnt of () is not rlat to an lmnt of. This is not a funtion sin on lmnt of () is not rlat to an lmnt of. This is not a funtion sin on lmnt of () is not rlat to an lmnt of. This is a funtion sin ah lmnt of is rlat to on an onl on lmnt of. This is a not a funtion sin on lmnt of () is not rlat to an lmnt of. a = = = a Funtion Funtion = not a funtion sin ngativ lmnts in th first st ar not rlat to an lmnt in th son st Funtion Eris a i. = 7 ii omain is th st of all ral numrs iii s, sin = is th imag of = Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

37 i ii omain is th st of all ral numrs iii no, sin thr is no solution to th quation = + i ii iii i omain is th st of all ral numrs pt = no, sin thr is no solution to th quation = - ii omain is th st of all non-ngativ ral numrs iii s, = is th imag of = a Fals, thr is no solution to th quation = tru, = for all valus of tru, = + for all valus of tru, = whn = ± tru, = - = f fals, th imag of = is = Eris a = i (, ) = + ii (, ) (, ) no (, ) = = a i {: } ii {: } 7 iii (, ) iv (, ) i {: } ii {: } iii (, ), (, ) iv (, ) i {: } ii {: } iii (, ), (, ) iv (, ) i { : } ii { : } iii no points iv (, ) a i F ii F iii T i F ii T iii F i F ii T iii F i F ii F iii T a = + = Eris D f () = ( ) ( +) a f () = () () = f = 7 = 7 f ( ) = ( )() = = + f ( ) = ( )() = (, ) lis on th graph of f (t) = t t a t (.) = (.). =. () = = () = () = = () = () (n) = n a () = = = () = = 7 (a) = a = a = a i v () = ii v () = m + = m = t = v (t) < for t > Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

38 f () =. ( ) a =.( ) (, ) (,.). ( ) = = = h () = a i h() = ii h( ) = = = = = Eris E a i l = ii w = V = ( )( ) i V () is th volum of th o whn th squars ut from ah ornr hav si lngth m. ii V () = () () () = m iii V (.) = (.) (.). =. m iv No, < 7. sin th with of th ar is onl m a with = = ( ) i () is th ara of th rtangl whn th lngth is m ii () = () = m No, if = th with woul. a = + n () = + () = SD i + n < ii + () =, no iii n <, n <. as. () =. + I() =. + a P() = I() () =. + (. + ) = + P() = + () = 7 D, a loss of 7 D i P() = + () = D ii I() =.() + () = ssuming all ooks hav sam pri, on ook osts I() = D or Eris F a kg = pouns Pouns (l) =. Kilograms (kg) graint =. p () =. p (7) = p () = 7 =. =. f k(7) =. k() =. = S$. a = S$. SGD ($) =. GP ( ) graint =. s () =. s () = s () = 7 p () = = $. a. GP = SD SD ($) =. GP ( ) p () = 7. graint =. u () =. u() = u () =. p () =. f p () = p (77) = k() =. Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

39 Eris G a a Tmpratur ( ) 7 Tim (minuts) º T () = + Lngth of spring (mm) Wight (g) mm g. mm L () =. + T () = +.7º m m g L () =. + Eris H a Flour = s + f Fat = s + f s + f = s + f = s = f = 7 spong aks, 7 fruit aks t + = t + = 7 t = = tals, hairs v + = 7v + = 7 v =. = 7. 7 vans, ars p + t = p + t = p = t = passngr plans, transport plans 7 + = = + = = = = volum, volum Eris I = + (, ) = + (, ) (, ) = + 7 ( 7, ) (, ) Eris J (, ) = ( ), (, ) (, 7) ( ) 7, (, ) = (, ) = (, ) = (, 7) = (, ) = Eris K = ( ) a = (, ) (, ) (, ) = ( + ) a = (, ) (, ) (, ) = = ( ) a = (, ) (, ) (, ) = = ( ) a = (, ) (, ), Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

40 = = ( ) a = (, ) (, ) (, ) = = ( ) a = (, ) (, ), 7 = + = ( + ) a = (, ) (, ) (, ) = + = ( + ) a = (, ) (, ), = ( + ) ( ) a = (, ) (, ) (, ) = ( ) ( + ) a = (, ) (, ) (, ) = ( ) ( ) a = (, ) (, ) (, ) = ( + ) ( ) a = (, ) (, ) (, ) Eris L = + a = no points (, ) = + = ( + ) ( ) a = (, ), (, ) (, ) = + + a = (.7, ), (., ) (, ) = + a = (., ), (., ) (, ) = a = (., ), (., ) (, ) = + 7 a = (., ), (., ) 7 =. + a = no points =. + a = (., ), ( 7., ) Eris M f () = + = ( + ) ( ) a i (, ) ii = iii (, ) iv (, ), (, ) v,,, 7 = + (, ) f () = = ( + ) ( + 7) a i (, 7) ii = iii (, ) iv ( 7, ), (, ) v = (, ) f () = 7 = ( 7) ( + ) a i (, 7) ii = iii (, ) iv (, ), (7, ) v (, ) = 7 f () = = ( ) ( + ) a i (, ) ii = iii, v iv (, ), (, ) = (, ) Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

41 f () = = ( ) ( + ) a i (, ) ii = iii, v iv (, ), (, ) = (, ) f () = + = ( + ) ( ) a i (, ) ii = iii, v iv,, (, ) v Eris N = (, ) (, ) (, ) = (, ) (, ) = + = (, ) 7 f () = + = ( ) ( + ) a i (, ) ii = iii, v = + (, ) iv,, (, ) f () = = ( ) ( + ) = (, ) (, ) (, ) =. (., ) = (, ) (, ) (, ) a i (, ) ii = iii, iv,, (, ) Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

42 (, ) =. (, ) (, ) Eris O f () = + = g () a (, ), (, ) + = + = ( + ) ( ) = = or (Sam as part a) h() = + = + = ( + ) ( ) = = or (, 7), (, ) f () = + a + + = = + = + = ( + ) = = or (, ) (, ) a f () = + g() = + = = ( )( + ) = = or (, ), (, ) f () = + h() = + + = + = ( )( + ) = = or (, 7), (, ) f : rang = { :. } g : rang = { : } = or f () = h() = + = ( )( + ) = = or f (, 7), (, ) a (.,.), (.,.) f () < g(). < <. Eris P f () = a + + = a = = a (, ) = a a = a = a a = a = a = = f () = + g() = a + + = a = = a (, ) = a a = a a = a = = g() = + f () = a + + a = = a (, ) = a + + a + = a + = a a = a =, = f () = + g() = a + + a = = a (, ) = a + + a + = a a = a + = g() = + f () = a + + a = = a (, ) = a + + a + = a = a = = f () = + + g() = a + + a = = a (, ) = a + + a + = a = a = = g() = + + Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr 7

43 f () = a + + a = = a (, ) = a + a = a = a = = f () = + g() = a + a = = a (, ) = a a = a = a = = g() = f () = a + a = = a, a = a = a a = = f () = + g() = a + + a = = (, ) = a + a = g() = + Eris Q a l + w = 7 l + w = l = w = lw = w( w) = w( w) For maimum ara, w =., l =. lngth =. m, with =. m l + w + (w ) = l + w = l =. w = w (. w) For maimum ara, w =., l =. lngth =. m, with =. m H (t) = 7t t a 7 m. m 7 s Eris R For all qustions: intrpt is (, ), horizontal asmptot is = Eris S f () = a (, ) = = f () = a (, ) = f () = = f () = a (, ) = = ( ) a (, ) = p (u) =.u + u a. rials rials units = ( ) Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

44 f () = () + a (, 7) = = () + f () = () a (, ) = = () 7 f () = () + a (, ) = = () + f () = () a (, ) = = () f () =.() + a (,.) = =.() + f () = (.) + a (, ) = = (.) + f () =. + a (, ) = =. + f () = (.) a (, ) = = (.) Eris T f () = () + a (, ) = = () + f () = + a (, ) = = + f () = () + a (, ) = = () + f () = () a (, ) = = () Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

45 f () =.() + a (,.) = =.() + f () =. + a (, ) = f () = () a (, ) = = () Eris f() = +. =. + g() = +. 7 f () = (.) a (, ) = = (.) a (,.) =. v(t) = t a uros = =. v (t) = (.) t v () =. v() =.7 ars M(t) = (.) t f () =. + a (, ) = a M(t) M(t) = (.) t t =. + M = M ()=. g M () = 7.7 M (7) = ars (t) = (.) t f () = (.) + a (, 7) = = (.) + a (t) (t) = (.) t t as for st Jun. ()= m () = 7.7 () =. t = Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

46 k + = an = k +, so = k = T (t) = + () t a T(t) 7 T(t) = + () t t T ()=.7 T (.) =.7 T (.) =.. minuts s t inrass T gts losr to (T = is an asmptot). 7 D (t) = (.) t a SD D () =. SD D () =. D (7) =. 7 ars f () = a (, ) (,.) = a. = a a = =. = + (, a) (, ) a a = + a = = + = = a a =.7 = f() f() = () + rang = { : > } (or f () > ) Eris V a f () = a.77 hours. hours, 7. hours ( ) + = ( ) = = f () Eris W. = + 7 = 7 = = f f () = = = =.7 minuts = 7 =. minuts =, = f () >.7 = + = = =, = = = ±.7 rang is all ral numrs pt { :, } Eris X minimum valu = 7. (whn =.7) 7. ms = + + = =. s,.7 s a v = () = = = = + = + = + For minimum ara, =., =. =. lngth =. m, rath =. m, hight =. m Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

47 a v = h h l l = h + = + l = l = l h h + + = + l = h h = = + f For minimum ara, =.7, h =.7 =. si lngth =.7 m, hight =. m Lt with =, lngth =, hight = h = + h h = viwing ara = h + h + h = h = ( ) maimum viwing ara =.7 = 7 m Eris Y f () = +, a { :, }.. f ()..... f ()... = = = + f () = +, a { :, } f ().. f () 7.. = = Eris Z Skth graphs Rang :.. = + < or. Eris a a, = ( ) = (.,.), (.,.), (.,.7) g() = f() = + =, = Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

48 + = = or. { :, } a (.,.), (.,.) a 7 = g() = f() = =. or. = = solutions = + (., 7.), (.,.), (.,.) = + = (.,.), (.7,.), (.,.) = + = a =. or. = = Eris a tim in hours, watr onsumption in litrs 7 7,, (loal maimum at ) f 7, (loal minimum at ) a tim in minuts, tmpratur in a minut no f approimatl t N s ( ) = = a m. s an. s. s. s 7 s m,. s f all rturns to groun lvl a i. m ii. m iii an < t < a twi f an g no, th tmpratur at th start of th following a is whras it was at th start of this a. 7 a = =. = f ().. = 7 tns to zro Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

49 a m = = = + = + (m) () (m ) (sf ) f =. = + Rviw ris Papr stl qustions a 7 = nr + s a = r + s = r + s r = r = SGD = + s s = SGD a + = ( + ) (, ) (, ) = + (.,.) =. h(t) = t t t a h() = m m from t = to t =, s a f () = m (,.). = f () = (, n) n = f () = = m m =. = a = ( )( + ) i t, = = (, ) 7 a ii ii t, = = (, ) i iii iv a i (.,.) ii (.,.) f () < g (). < <. = a with =. = (. ) a For maimum ara, =. m = = =, = =. Papr stl qustions n = (.) t a, a n n = (.) t t i n = (.). = ii t =. hours = hours mins = Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

50 =, = g() = = ±. { :, } f () = (.) + a a =. = 7. 7 f() = (.) + f () > =. = a (.,.), (., 7.) f =.,. f() = = 7 P = = () + = (.7,.), (.,.), (.,.) + a 7 P f (t) = + 77(.) t a f () = + 77 = = f () =. P = + a f() = g() = (.,.) (, ) a i uros ii iii or 7 iv uros (, 7) (., ) (., ) = 7 (, 7), (., ), (., ) Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

51 a = 7 (., ) (., ) (, 7) = ±. =,,,,, + = ( ) = = or (, ), (, ) = = a = = k = f() = (, ) (, ) g() = + f () < g () < < Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

52 Statistial appliations nswrs Skills hk a man =. stanar viation =. Th small stanar viation implis that th ata ar los to th man man = stanar viation =. Th man is th mil ata valu () sin th frqunis ar smmtrial aout this valu. Th stanar viation is vr small sin most of th ata valus qual th man an th rst ar los to it a = + Eris a a f() = n v n v n σ n n + v n + v n + v 7 Hight (m)... = f() n v n σ n v n n + v n + v n + v 7 Tim (s).%. = a a f() n v n σ n v n n + v n + v n + v Tim (min) %. =. or f() n v n σ n v n n + v n + v n + v Volum (ml). 7. = Eris a f().7 7 Tim (min).7 = 7. or as a i P(IQ < ) =. ii P(IQ > ) =. iii P( < IQ < ) =. P(IQ > ) =.,. = a...7 = 7 a f() Monthl inom (uro).. =.7 or a 7.%.. =. or. 7 a.%. =. or Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

53 man =.7 m stanar viation =. m a. Eris p =. h = k =. w = a.7 to. kg. =. 7.7% w =. a a =, =, =. =.... = (apt to ) 7 a.. t = a f() Sp (km h ).% p =. = 7. = 7 a f() 7 7 Mass (g)... =.7 or 7 p = a.% a wighing. kg (. is narr th man than.).. =. or w =. Eris D a strong positiv linar morat ngativ linar morat positiv linar f g h a wak positiv linar non prft ngativ linar non-linar wak ngativ linar morat positiv linar orrlation morat ngativ linar orrlation Eris E i a, strong positiv orrlation man of =, man of =.7 ii a, strong ngativ orrlation man of =., man of =. a, Wight(kg) Hight (m) morat positiv orrlation man hight =.7 m man wight = kg kg Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

54 a, ITGS(%) 7 7 hmistr (%) wak positiv orrlation hmistr man =. ITGS man =. % a, a ri(hours) a hl (hours) morat ngativ orrlation man numr of hours hl =. man numr of hours spnt ring =.. hours a, ost($) Siz (inhs) morat positiv orrlation man srn siz =. inhs man ost = $ $ Eris F r =., strong positiv orrlation a r =. strong positiv orrlation r =., strong ngativ orrlation r =., strong positiv orrlation r =., strong positiv orrlation r =., vr wak positiv orrlation 7 r =., strong positiv orrlation r =., morat positiv orrlation Eris G a r =., strong positiv orrlation =.7 + =.7 () + =, ( s.f) a r =.7 =. +. =.() +. =.,. m a =. s =. = s =.7 r =. strong ngativ orrlation =.7 + =.7 (7) + = 7., 7 sons a r =.7 =. +. =. () +. =.,. a =.7 +. =.7 (7) +. =.77,. D a =.7 +. =.7 () +. =., situps 7 a =.. =. (). =.7,. a =.7 +. =.7() +. =.,. hours. Eris H a H : Gnr of ook is inpnnt of ag H : Gnr of ook is pnnt on ag 7 =. ( ) ( ) = χ al =.. >., thrfor w rjt th null hpothsis. Thr is nough vin to onlu that gnr of ook is pnnt on ag. ( p-valu =.7 <.) a H : Hair olor an olor ar inpnnt H : Hair olor an olor ar pnnt =.7 ( ) ( ) = χ al =.. > 7.77, thrfor w rjt th null hpothsis. Thr is nough vin to onlu that hair olour an olor ar pnnt. ( p-valu =. <.) Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

55 a H : Favorit flavor is inpnnt of ra. H : Favorit flavor is pnnt on ra. = ( ) ( ) = χ al =.7.7 <., thrfor w o not rjt th null hpothsis. Thr is nough vin to onlu that favourit flavor is inpnnt of ra. ( p-valu =. >.) a H : Film gnr is inpnnt of gnr H : Film gnr is pnnt on gnr =. ( ) ( ) = χ al =.. >., thrfor w rjt th null hpothsis. Thr is nough vin to onlu that film gnr is pnnt on gnr. ( p-valu =.7 <.) a H : Gra is inpnnt of th numr of hours H : Gra is pnnt on th numr of hours =. ( ) ( ) = χ al =.. >., thrfor w rjt th null hpothsis. Thr is nough vin to onlu that gra is pnnt on numr of hours spnts plaing omputr gams. ( p-valu =. <.) a H : Emplomnt gra is inpnnt of gnr H : Emplomnt gra is pnnt on gnr Dirtors Managmnt Tahrs Mal. 7.. Fmal. 7.. ( )( ) = χ al = >., thrfor w rjt th null hpothsis. Thr is nough vin to onlu that mplomnt gra is pnnt on gra. ( p-valu =. <.) 7 a H : mount of sushi sol is inpnnt th a of th wk H : mount of sushi sol is pnnt on th a of th wk =. 7 ( ) ( ) = χ al =.. <., thrfor w o not rjt th null hpothsis. Thr is nough vin to onlu that th amount of sushi sol is inpnnt of th a of th wk. ( p-valu. >.) a H : pupp s wight is inpnnt of its parnt s wight. H : pupp s wight is pnnt on its parnt s wight =. ( )( ) = χ al =..7 >.77, thrfor w rjt th null hpothsis. Thr is nough vin to onlu that a pupp s wight is pnnt on its parnt s wight. a H : Musi prfrn is inpnnt of ag H : Musi prfrn is pnnt on ag 7 =. ( ) ( ) = χ al =.. >., thrfor w rjt th null hpothsis. Thr is nough vin to onlu that musi prfrn is pnnt on ag. ( p-valu =. <.) a H : g at whih a a is pott train is inpnnt of gnr. H : g at whih a a is pott train is pnnt on gnr. =. ( ) ( ) = χ al =.. >., thrfor w rjt th null hpothsis. Thr is nough vin to onlu that th ag at whih a a in pott train is pnnt on gnr. ( p =. <.). Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

56 a H : Gra is inpnnt of gnr H : Gra in pnnt on gnr, or 7 or or Mal... Fmal... ( ) ( ) = χ al =.7.7 <., thrfor w o not rjt th null hpothsis. Thr is nough vin to onlu that gra is inpnnt of gnr. ( p =.7 >.) Rviw ris Papr stl qustions a f() Volum (ml).. =., ans a.%. =., popl a.% p =. a strong positiv orrlation a no orrlation morat ngativ orrlation strong positiv orrlation = = a r =., strong positiv orrlation =. 77. =. (7) 77. =.77,. m 7 a r =., strong positiv orrlation =. +. =. () +. =. sons H : Flavor of i rams is inpnnt of ag H : Flavor of i rams is pnnt on ag Ept valus < < Vanilla... Strawrr... hoolat... grs of from = ( ) ( ) = p-valu =. >., χ al =. W o not rjt th null hpothsis. Thr is nough vin to onlu that flavor of i ram is inpnnt of ag. (ritial valu =., (χ al =. <.) a H : Th numr of pins knok own is inpnnt of whih han is us. ( )( ) = = p-valu =. >. (signifian valu). Thrfor w o not rjt th null hpothsis. Thr is nough vin to onlu that th numr of pins knok own is inpnnt of whih han is us. a H : Th outom is inpnnt of th tim spnt prparing for a tst. ( )( ) = p-valu =. >., thrfor w o not rjt th null hpothsis. Thr is nough vin to onlu that th outom is inpnnt of th tim spnt prparing for a tst. Papr stl qustions a f()..7 7 Hight (m). =., mn k = a f(). 7 Wight (g). = 7., 7 swts Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

57 a, wight( kg) 7 =. = 7. 7 Hight ( m) i =.7. kg a r =. strong positiv orrlation =. +. =.() +. =. = hours (narst hr) a r =. =. (.p.) strong positiv orrlation =.. a r =.7 =. =. (.p.) strong positiv orrlation =. +. =.() +. =.7,. uros 7 a =.. =. (7). =.7, rss siz r =.7 morat positiv orrlation H : hoi of gam is inpnnt of gnr H : hoi of gam pns on gnr Ept valus: aminton Tal tnnis Darts Mal... Fmal... grs of from = ( ) ( ) = χ al =.7 p-valu =.77 >. W o not rjt th null hpothsis. Thr is nough vin to onlu that hoi of gam is inpnnt of gnr. (ritial valu =., χ al =.7 <.) a p =. q =. r =. i H : Th tra-urriular ativit is inpnnt of gnr ii ( )( ) = χ al =.. >., thrfor w rjt th null hpothsis. Thr is nough vin to onlu that tra-urriular ativit is pnnt on gnr. a i = ii = = = H : position in uppr managmnt is inpnnt of gnr H : position in uppr managmnt is pnnt on gnr i χ al =. ii ( )( ) = iii. >., thrfor w rjt th null hpothsis. Thr is nough vin to onlu that position in uppr managmnt is pnnt on gnr. a H : Th hoi of aniat is pnnt on whr th votr livs. 7 = 7 i χ al =. ii ( )( ) = i Th hoi of aniat is pnnt on gnr. ii. >., thrfor w rjt th null hpothsis. a =. i H : Gra is inpnnt of gnr ii ( )( ) = iii χ al =.. <., thrfor w o not rjt th null hpothsis. Thr is nough vin to onlu that gra is inpnnt of gnr Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

58 Introuing iffrntial alulus nswrs Skills hk a f () = 7, f ( ) = f () =, f ( ) = g () =, g g () =, g () = f () =., f ( ) =. a r = r V h =± r V r r f r a = = = = = =± = a a + = ( ) = = ( ) = + Eris a f g h i j k a f g h i 7 j k a + + a Eris a = t t = t t = t + = t = t t = t t t = t + t = t + t = + 7t t = 7 t t f = t t = t t g h = t t + t = t t + t = t t = t t a f (r) = (r ) = f (r) = r f (r) = (r + ) = r + f (r) = (r ) = (r ) = r f (r) = ( r) = ( r) = r f (r) = (r + ) = r + f f (r) = (7 r) = (7 r) = r 7 Eris = f ( ) = = 7 f ( ) = = = 7 f ( ) = g ( ) = g ( ) = + = + g ( ) = + ( ) Eris D = whn =, = () = = whn =, = () = Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr

59 = whn =, = ( ) ( ) = = + = + whn =, = ( ) + = = whn =, = () = 7 = whn =, = ( ) = = whn =, = () = = t (, ), = ( ) = 7 s = t t s t t t =, = () = s = t + t s = + t = at t = t t t =, s = + () = 7 t v = 7 t v t W p V r r V r r V r r =.7 h = h. t h =, =.7p. Whn p =, W p r = r r =. Whn r =, V r h =. Whn r =, = r = r +. Whn r =, V r r. Whn r =, = πr + r =. Whn r =, r r = Eris E a = + V r V r. Whn r =, r t P, = + = 7 = = t P, = () + () = + = =.7 = 7 = = = = + = = π at r = = = = = a a 7 = + t Q, t Q, = = t R, = + = = = = = + + = = = = = lso, = + () () = + = So, R = (, ) = t R, = (a) = a = a = lso, = () () = So R is (, ) + = + = whn graint is. = =, = point is (, ) = whn graint is = = = = lso, = ( ) ( ) = = So th point is (, ) = + whn graint is, whn = = + = = = = ± = () + () + = whn = = ( ) + ( ) + = So points ar (, ) an (, ) Ofor nivrsit Prss : this ma rprou for lass us soll for th purhasr s institut Work solutions: haptr