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Practice Series 4 2009

ADDITIONAL MATHEMATICS 2

TIME: 1 Hour 

Answer all uestions :

Section A

1! So"#e t$e si%u"taneous euations &i#in& answers to ' (eci%a" )"aces!

2

2* + ', - 4 . 0

* +/*, +1 . 0 / %ars

2! A cur#e wit$ &ra(ient 3unction *2 5 6* ! I3 t$e cur#e )asses t$rou&$ t$e

  )oint P7+18 +'8 3in(

a in( t$e euation o3 t$e cur#e ' %ars  ; in( t$e euation o3 t$e nor%a" at P! / %ars

'! Setc$ t$e cur#e 1 2 cos * y   = +  3or 0 2 x   π  ≤ ≤ ! ' %ars

<sin& t$e sa%e a*is8 setc$ a suita;"e "ine an( $ence state t$e nu%;er o3 so"utions 3or 

t$e euation*

2 cos 1 xπ  

= −   3or 0 2 x   π  ≤ ≤ !

2 %ars

Section B

4! Ta;"e ;e"ow s$ows t$e #a"ues o3 #aria;"es * an( , o;taine( 3ro% an e*)eri%ent

 x 1 2 ' 4 /

 y 6!4 '!4 1!= 0!9 0!4

It is nown t$at * an( , are re"ate( ;, t$e euation*-1 a; y   =  wit$ a an( ; are

constants!

a P"ot a &ra)$ o3 "o&10 , a&ainst 7 * - 1 usin& a sca"e o3 2 c% to 1 unit on t$e

7*-1+a*is an( 2 c% to 0!2 unit on t$e "o&10  ,+a*is! Hence8 (raw t$e "ine o3 ;est 3it

/ %ars

c ro% ,our &ra)$ 8 3in(

T$e #a"ue o3 a an( ;!

  /

%ars

1

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/! Dia&ra% ;e"ow s$ows a trian&"e POQ! >i#en t$at  pOP   = an(   qOQ   = ! Point X  "ies on OP

suc$ t$at 1:2:   = XP OX   an( )oint Y  is a )oint on OQ w$ere 1:':   =YQOY  !T$e strai&$t "ine

QX  an( PY  intersect at )oint C !

7a E*)ress  PY   an( QX    in ter%s o3   p  an( q !

7; E*)ress OC 

7i in ter%s o3 m8   p  an( q  i3 QX  mOQOC    +=

7iiin ter%s o3 n8   p  an( q  i3  PY  nOP OC    +=

7c Hence8 3in( t$e #a"ues o3 m an( n!

Section C

! A ;aer, uses 4 t,)es o3 in&re(ients A8 ?8 C an( D to )ro(uce a certain t,)e o3 cae!

T$e in&re(ients are %i*e( accor(in& to t$eir %asses in t$e ratio o3 20:'0:40:10! T$e

3o""owin& ta;"e s$ows t$e )rices an( )rice in(ices 3or a"" t$e in&re(ients!

In&re(ient A ? C D

Price in 199= !00 * 4!/0 12!/0

Price in 1996 =!/0 9!00 /!6/ ,

Price in(ices in 1996 ;ase

on 199=

12/ 120 @ 110

7a in( t$e #a"ue o3 *8 , an( @!

7; Ca"cu"ate t$e co%)osite in(e* 3or a"" t$e in&re(ients in t$e ,ear 1996 ;ase( on199=!

7c A / & cae was so"( 3or M =/ in t$e ,ear 199=! Ca"cu"ate t$e )rice o3 t$e si%i"ar

/& cae in t$e ,ear 1996!

7( T$e cost o3 t$e in&re(ients increase( ;, 20 B 3ro% t$e ,ear 1996 to t$e ,ear 1999

in( t$e co%)osite in(e* 3or t$e ,ear 1999 ;ase( on t$e ,ear 199=!

=

E

2

Q

Y 

 P 

C 

O X 

A

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  6 c%

  9!/ c%

  //0 

? C =c% D

Dia&ra% a;o#e s$ows a trian&"e A?C an( trian&"e ECD! ?CD is a strai&$t "ine!

Ca"cu"ate

7a T$e "en&t$ o3 ?C

7; An&"e CED

7c T$e "en&t$ o3 AE!

7( T$e area o3 t$e w$o"e (ia&ra%!

ANSES

1! * . +0!1/98 * . +2!96  , .1!22=8 , .+0!4/

2!' 22 4 '

1 4'

14 14

 y x x

b y x

= − +

= − −

'! Nu%;er o3 so"utions . 2

4! a '!'1 ; 0!46

'/!

4

2

'

29 71 9

'

'71 9

4

1 28

2 '

 PY p q

QX q p

b OC m q mp

OC n p nq

m n

= − +

= − +

= − +

= − +

= =

uuur

% %uuur

% %uuur

%uuur

% %

!a * . =!/ , . 1'!=/ @ . 1'0

  ; 124 c 9' ( 146!6

=!a ?C . 6!'=2 ; '9!/0

  c AE . '!=/' ( '6!99

'

1200