Add Mth Answer Spm Johor 09

347211

suLlr 3472t1 Additional Mathematics Kertas 1 September2009 2 Jam JABATANPELAJARANNEGERIJOHOR PERCUBAANSPM 2OO9 PEPERIKSAAN MATHEMATICS ADDITIONAL Kertas 1

JANGANBTJKABUKUJAWAPANSEHINGGADIBERITAHU

MARKING SCHEMEI ------J

6 halamanbercetak l(ertassoalanini mengandungi

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347211tlak CiptaJPNJ 2009

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Submarks

Solution (a) 4 (b) 5

(4 s18)= 1

2

g-'(x) = " =' /

B1

lv

( b)gh(x) = -^ - 6@ equiv a le n t J

qh(x) =z(!-1) + 1 '

'a

(l^

.l

9 l ;-r \J

rJl

^)

\ )

I

1(

g(x) = - TU x

3

11

9(Y) = j5 l

\v

la

) '-l

B2

)

I.

rtl ----1..-*- i= ) \ - 4

)l

\x-r J

i

x- _1 p=12 k=-4

82

3+k=-1 @3k=p@ equiva le n t

B1

J

(a)a=1 (b)b=3 (c) c= -3 6

x<-3 , x>3 \,/ ' t/

*--\ :\--3 \-/

/zi -:+ 3

@equivalent

B1

(x+3)(x-3)"0 @ x=t3

i

'L__

_l

82

'

-L

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fotal marks

Submarks

.x

J

=*-

Totalmarks

I

9 l 8 (3 -) = 2

I

2 7 (3 ',)-g (3 .1 = 2

82

3 *.3 3- 3 *.3 2=2

I

f-

I a i" :, li rog,x =los,zqlu,g,+

I

i

i

oranybase i log,x = 19"sJlog19

I

/"

l--:

I

it

t--T-_lt

I I

ll"

i

l^

I

l) o o = -..-.-

B1

I'

l

l81

iirl

: t-*-f"--, o n=12 il1

i

i

I i

t:*ol_:5rli?_og i ,,' 2(2) i,,-

it

j ;[:r-s)+(i?-l)4j,2oo

i

i1-

I is"200 i--]*tr

i

I tt i

1, .=18:

2

i l, i2

B1

v

it f--T-_1 2 j m=1 6a n d n =1 .7 4 1 1 i ii

J

i

-loezr=-! orlog lo g , n ltos,m=4or

i li

I

tl

A T

log,),= logzat_ "rlog,n

B1

il

L_t_ _-

82

www.tutormuruli.blogspot.com -,-.----_-L

L__

S[b marks T Totalmarks -;-| l =6 a n d s=-2 0 t=6 o r s=--2 0 s + 3 / = -2 ,2 s- 4 t =1 6 (Solvesimultaneous equation) s+l+2t+14

? 1 -'l-l 2'l

.r ol L-ol

82

1lo 2 - _ *'

a

I I li

tl+

a

r

ol

o

4l

I

-r

A (0 ,4 o ) r D (0 ,-1 )

/

2

,l /

B1 either2aor

\ \ .(l

-b

t,

\ !

\ \

(a) k=2 ')*-

a

2 q

=3 ), k +7 =2 )@,: "2 g

(ft+ l )q- ::h = l (2 q -3 [) @ equivalent 2nS = .1.pQ

.--]

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( a)

1 O'-

t. rvl l-

p

lr- p' (01

.e r:;'

srn--=^l ? \l Ll

)

g\ ^( 2sin'l - | \2)

'o:l-

2.044 cmz 1^ 1

4

- _(aX3) :(5)'(0.6435) 2'' 2 l1

1$)'

(0.643s)or

B3

1{+X3)

82

Shaded region = Apon- Apon IPOR = 36.870 or 0.6435racl

.V= 1 .4 4

4

-0.06

B3

-96 6y=--:x0.02 aa -

82

3Z

),, ? = - 4(21)x'5 dx

( -t

or -4(24)x

5

\

| -:.4 I

I

I

I

I II

\2

I

4

)

d'y . d2,, -;* = -;2 ?r.rd -=4 ,= -16

dx'

x''

dx'

B3

,l 4-L-'

2 Q- = q- --1-ro equivatent clx

x'

B1

I

t_

82

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I

I

.;4 J

ZA:i--2(--1)- 3 J

,.] l?r"-11' " l

t -" lr l

Ix-l

L

t Izx-:l

;t

?

I

rl_ x-

I

(a)7

B1

(b ) .r=1 x>0

I*:31:J":!l 2 5 +x

,s 2 B1 I

B1

x t-2 8+x

I a

(a) k-0,679

2

P(Z>k;=A.2486

= 82..031 (b) .r .r=8203' I I(b) il I lo67e=tjlo

L _i__

I

II

I

__i_

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B1

SULIT

3472t2

3472t2 Additional Mathematics Kertas2 September2009 2'h Jam JABATAN PELAJARANNEGERIJOHOR PEPERIKSAANPERCUBAANSPM 2OO9

ADDITIONALMATHEMATICS Kertas 2

JANGAN BUKA KERTASSOALAN IN! SEHINGGADIBERITAHU

RKINGSCHEUIE

Kertassoalanini mengandungi17 halamanbercetak

347212 Hak CiptaJPNJ 2009

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SULlT

3472t2

2 A BAHAGIAN Solution

p = 1 -2 q

Sub marks

roGi marks

1- n

Orq =--'. 2ll

l Pl I

Eliminatep or q "( 1 - 2q)' - 3.(1 *2q) + 2q2- 3 = O

Or p2 - 3p +2"(\P )2-3*-0 ?_ or eouivalent Solvetlre quadratic equationby usingthe factorization@

6q '+2q-5=0 3p '-8p-5=0

quadratic formula@ completinq the sq

-z r J+-"(oX-sj 2(6)

ri i&4:ll3&t 2( 3)

q = 0 .7 6 1 ,-1 .0 9 5 or p = - 0 .5 2 3 ,3 .1 8 9

p: - 0.523, 3.189 (9

I

q = 0.761,- 1.095

if the workingclfsolvingquadraticequationis not shown [

]"*_

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347212 llak CiptaJPNJ 2009

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347212

SULIT

Sub marks

Solution

2

Total marks

i ,,

"/ - axt -zx dx

a) 2y=-y+J fl'ln = - -

I I

2

*1r=2 E] substitute x = 1 in to axo-2x = 2 a ( 1)'- 2 (1 )'=2

a=4

Integratey=

P FI

Jt*+"'- 2x)rtx

4 xo 2 x' {-c v'A .)- -----.-+L

x = 1, y= 3 into Substitute 4xa y = :.i

2x2

+ c, to find the

valueof c. y=x4-x2+3.

N qte: .y: J(*4x'' - Zx)h

must have at least 1 ter"mwith powerincreasedby

A

t.

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A "t

- -Solulion

347212

Sutr marks

Total marks

tr_l U se T n -a +(n_1) d 7 2 1 =4 0 +20"( 40)

1 0 00- *840=160 OR othervalidmethod

b)

l.JseSn= !; tz^ + (n - 1 )dl

+(11x40)l I rrrooo)

l ,l

3472t2 Hak Cipta.JPNJ2009

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rUILI

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{

SULIT

---

- --olution

* L=3 9.5or*F=7or

"fr=d

lP l

347212

Sub marks

I

Use medianformula al' \

47 .5 =.39.5 + (

,^

I r'-Y --1l-* (7 \ 2\ '

1

*a

to "L With "f, and F corresponding

or Zl*Zf*= 1425

72917sm

/+ Ffr, 'Zf -- |t-J --- - "1 ' tGt"

f

4

st- t-

,1425,2 *72917.5 -30--'-30-''

y= x+ 1 ii)

3 y + 7 x=3 3 y=x+1

ST = 2TQ Can be impliedas formula. I Pl

s (3,4) I

Usingthe formuiadistance /

,,n7 tr t

Jtr-:)' +(y-4)' = 2JG-7)'+(r'-8)' \:_l

\

- - - - .-

\-_,

l?7--q --]=Y PPvJ 9r1rqv1-qql_[Lihat sebelah S ULI T

347212 Hak Cipta JPNJ 2009

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347212

9UUr

m

[_i-Solution

Use identity C os2A -S in2A =Cos2A

L HS = RHS Ns mistake allowed

b ) i)

Fr*l

GraphSin l p e ri o C i n0 sxs 1 Amplitude

n

tir*l E]

Drawingof the straightlinefrom the equationinvolving ( X and y, eithergradientOR y interceptof straight l-ine must be correct.

rr

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347212 |'lak Cipta JPNJ 2009

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SULIT

3472t2

7 BAHAGIANB

Sub marks

Total

marks

Solvethe simultaneous x?'+2= * x + 8 x 2 + x -6 x=2,x = -3

-n

lntegratex2+2 Uselim it

F2

I'

u

'1

in to .[\x) - + 2xJ J

Or findthe areatriangle % (6)(6)

lntegrate (x'+2) + areaoftriangie

T +rc 3

(24.667)

c) lntegratez x2

n t tt'- 2),1t,

Uselimit /\ \ Kl \*__r/

in ., r Q) | - 2 yl

/

\

I \

I

i

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SULIT

3472t2

Sub marks

Solution

Total marks

8 a)

M Note: lf tableis notshownawardF,lmarkif all the pointsare plottedcorrecily. b)

v lJlotj- agatnstx x \K] (Correctaxes and uniformscales)

/

-r

l--r r'ir )

6 *pointsplottedcorrectly

\

-r

(xr

Line of bestfit

\

\*-"

c)

//Il

! =zax"oLl]_l

x Or implied. i)

Use*m = t"

ii)

8-2

i

I I

Frcm graplr ): againtx

i

/-)

\5!,

(*'

)

a="tL ( 0.7<---),0.8)

ii)

b=-

2.45

[ -2 . S 0 < ---+ -2 . 3 0 ]

t- = -0 .6 x

= -0 .6

1/

y=-0 .7 2

\

Nl

,/

Note:

ss-1if, Part of the scale is not uniformat the x-axisand/0rthe I -axis.

x OR not using the given scales. OR not using graph paper. 347212 Hak CintaJPN.| 2009

10 [Lihat sebelah

$uur

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_i

3472t2

v x I

7

6

5

4

3

2

:i

r^"r ,t' I

.4 ii

r{

ri ..^ t ;!

1i

,l

7212"Hak" ei$t5J]rNru''20f.rs

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347212

10

SULIT

Total marks 9 a)

b)

< TCO = 45" OR 0.7855rad OR

v 4

Use S = r 0 la find the lengthof arc BE

or ODFC

1s(?-) or 5(i)

KI Use Pytho.Teoremto find CO

16.2132 Perimeterof the shadedregion "23.565+ *16.2132+ *3.9275 + 5

4azasr// 48706//

aazt

Use Y, f g to find the area of sectorTODFC Or sectorOEB

Area of shadedregion ( 176.7375 - 112.5)+ I 8188)

'/roaua//74.06

347212 Hak Cipta JPI'IJ 2009

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SULIT

3472t2

lt

Solution

10 a) i) ii) iii)

Usethetrianglelaw for Of or g..t or .V

or =

1)

I

lo. io

;0

t2

--p+;Q OJ

N1

2

f f i = 6 q 1 p * ;q ) J I

QR =n1-c +1n) Use the trianglelaw to find

oQ=on*nO 1 71 j tP o

Jn,q-ntp*nq=q

Comparethe coefficient ,.-*\ / Kt ) of p andq. \*-/ l^1rn .

n=0

JL J

m= --n

2

2 - m+n;1 _) af

-(-n )+ n = 1 J 11a

rJ

n=---.m=-1A .4-

a

{Both are correctl

347212 Hak CiptaJPNJ 2009

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t2

347212

Solution 11 a) i)

Sub marks

Total marks

2 ----1 lpr I

I

0.2048 Usenq I

/^\

6 0 0."). (:) \ J

Kr )

-\-____,

zooI !!1 | b) i)

UseZ

P(2.*#",

r-\ \Kr/ -\

oaorsl_rur I P( Xtk)=0.123

p '(z> L:45 )=0.123 i0 k -4 5 l0

347212 Hak CiptaJPNJ 2009

= 1.16

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13

3472t2

BAHAGIAN C $olution

Sub

rng$s

Total marks

12 a) i) ii)

,1 ,,

Use al = 0. to findt dt 1

t= l

a I

- |,,)y=tr(J

1

*4 G) -2 a I

_)

6

=.---

J

ii i )

v< 0 ( t- .lx3t+l)<0

G) \--

0
lntegratet = -2) (tt Jlott 1t 2t3-2.t2-2t K1

S u b s t it u t e t =l o rt = 3 FindSr or 53.

Ditancetraveled * 2 + 2 + 3 0 =34 c)

Forshapeofthesraph L{ Ip r

| "'

I

andx-intercept I y- intercept

347212 Hak CiptaJPNJ 2009

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sur=rT

14

347212

Solution 13 a)

(l) User=9- xloo n rvv \ - K{

eo

2'6 0

x I00 = 130

\_,

X

Ll!]_] x = RM2.0o

+ (140)3 + ( 1 1 5 ) 2+n ( 1 2 0 ) n Zt w = ( 1 3 0 ) 1 t- t, I

-

Use,=

\ tw

T,

= 125 r--r \.KIJ t\

1_Nln=2

,,=-13,:ttl 100 t32

I :{tl r!$;) +'l:():!_?g) l0

347212 Hak Cipra JPNJ 2009

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t5

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3472t2

Sub marks

SinQ

\rct )

Sin45"

\\

62.12'il 62' zl xrrI

r__--.1

L NI I < e , o b t u s e


a

_ Qg' _ ____6 __ Sin55.76Sin62.12" Q Q' = 5.6115 f;-

.-fll

30---=---_-7-lc, t

Si n l T .1 2 ,S uI r1 7 ^8 y

-f

RQ'= 2.4977

Area of trianglePRQ' = % (V.5)(2.4977)sin45"

6.6230

347212 Hak Cipta JPNJ 2009

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t6

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347212

Sub marks 15 (a)

2x + 5y < 30

or equivalent

3x + 2y s 24 or equivalent x < 2,!

or equivalent

at leastonestraightlinefronithe Drawcorrectly *inequalities x andy. whichinvoves Drawcorrectlyallthree"straightlines Note: Acceptdottedlines. The correctregionR shaded A =4 Furniture ( f r omx=4 i n th e re g i o n ) c) i)

pointat (5,4) Maximum

@ trr:l

Use200x+ 250y for pointin the *regionR RM2000

ss-1i f

In (a) the symboi" - " is not used at ali or morethanthree are given. Inequalities In (b) does not use the scalegiven or does not use graph paperOr interchangebetweenthe x-axisand the y-axis'

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t7

-t t

Y'1

i

t

$t i

l:

,: !l

i{ il

i

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3472t2