\._€

l.

ADDTTTONAL rvrArr@arr c sKertaslSepternber2 jam

MAJLIS PENGETUA SEKOLAH MALAYSIACAWANGAI\ PULAU PINANG

MODUL LATIHAN BERFOKUS SPM 2OI4

MARKSCHEME

ADDITIONAT MATHEMATIC S

Paper I

Two hours

2

ADDITIONAL MATHEMATICS PAPER 1 2AI4

347211

Question Solution and Mark SchemeSub

MarhsTotalMark

I (a)

(b)

t(3 , g), (2, 4), (- 2, 4), (- 3, g))

.f :x->*2 oR f(x)=i2

1

1 2

2 (a)

(b)

L7

-14

81: f-t (x): +

1

2 3

3 (a)

(b)

-3

Bl: m2+6m+11 -2

S(r):i2 +2

Bl : g(Y): (V -3)'+ 6U- 3) + 11

2

2 4

4 (a)

(b)

p:-9

Bl : (3)2 + pQ)+ 18 = 0

x:6

81 : (r-3Xr-6)-0 or 3a:18 or 3+o= (-9)

2

2 4

-J m1-6, m> 6

B2: (- m)2 - 4QXe) >

Bl :x2 -mx+ 9 = Q

3 3

6 (a)

(b)

(b)

x=3

(3,2)

f(x)= (r- 3)'+2

I

1

I 3

3472n

3 34721r

7 3

2/l rr,s

2

B2:2x -3

Bl:5b"51 or 53

3 3

IR

1gP2 or fi-}JF

83 : R2 =26P oR R2 =26

P

B2 : log2 R2 = lo1z26 +log2 P OR,R2rogz T=

81 , l=og'P. or 3log2log24

4 4

9 24

B2: -1 37 +(n- 1X6) 0

B1 : d-6

3 3

10 (a)

(b)

4

20

B2: (2n + 4t)(n-2q - 0

BI;va)*b-1)4I = 820

I

3 4

11 -2r l-+Ll-4,255

B.2: -71- 2

3

B1 ,-23

3 3

34721I

3472tI

tz (a)

(b)

1v

L*'+42

p:2,q-7(both)

B2:p-2 or q-7

Bt : 5 =+p+41or s --G) +4tz

I

3 4

13 r =14,t -+ 0oth)5

82: r =14 ort7+-

5

orBt : 8 _z(r)+t(a)5

, _20)*3(t)o-

5

3 3

t4 2100, 3300

83 : 210o or 3300

B2: (2sinr + l)(sin )c +2)= 0

Bl : (1 -2sin2x)-5sin x=3

4 4

ls (a)

(b)

t6

Bl 1 /. PoR:0.45

344.576 /344.59/344.6

Bl , te6)2(z.6sz)

2

2 4

16 Ji5 /3.6a6

22 +3281 :

2 2

3472tr

3472n

t7 (a)

(b)

-5 a *4b

2

Bz: 5 -s( -L )=z or 4( L)=t' or\3+n) \3+n) 5

-6

81 : -3(2x+ t-4 Q)

2t (a)

(b)

-4

16

81 :

347211

3472n

Br : p=ry--($Ef

23 (a) 144

Bl:4! x3!

72

Bl:31 x2! x3!

24 (a)

(b)

0.1 4AI

Bl : 0. I3l4

1.08

u72lt

3:47212

ADDITIONAL MATFIEh4ATIC S

Kertas 2SepternberAl .z- lam2"

MAJLIS PENGETUA SEKOLAH MALAYSIACAWANGAN PULAU PINAIIG

MODUL LATIHAN BERFOKUS SPM 2OI4

MARKSCHEME

ADI}ITIONAL MATHEMATIC S

Paper 2

Two hours and thirty minutes

2

sEcTroN A ( 40 MARKS )

347212

Ques Mark SchemeSub

MarhsTotaIMark

1. y=2x-6

or

y=x2-15

or x-3+ ry E

_t1Ttiil'_t_:'_:

*z _ *(b_6)

or

.f, .:r)'-y

or

- 15 Solve the Quadr aticEquation

il;il;;= 15

-(e)*Jt-z)z -4(rX-9)2(r)

OR

l-e-15)=32\-(8)t

2(r)

Cqmnletine the sgugfe

(x- i2 - (- D2 -g= Q

OR

b, + 4)2 - 42 -24= o

x = 4.16, -2.16Qr

y:2,32,- 10.32

OR

y :2.32r- 10.32or

x: 4.16, -2,16

Notg:

OW-l if steps to solve quadratic equation arcnot showq.

(8)2 - 4(1) (-24)

347212

Ques Mark SchemeSub

MarksTotalMark

2 (a)

(b)

k+(9-lXx):52A?Po+

(t z-rX')J =474a

Solve simultaneouslinear equations

x- 50

ork: 120

k- 120or

x- 50

* 120 + (n- 1)*(50) : 48 + (n - I)(74)

n- 4

Note : Accept complete listing of the terms

I

I

5

2 7

3 (a)

(b)

(c)

9:t- ytl!?:-.JfllyC- -r/2, fllAD-2

Y -(-3) =*2(x-1)or

-3 : * 2(I)+ c

Y: hc-5

Solve simultaneous equations?x + 4y:20 and *y:2x - 5

D(4,3)

B (t0, o)

use A -%lf )-( )l

;: i-1ro-iib:i;;:i;i +o *r|

22.5

3

2

3

I

8

347212

347212

347212

I

(a) | Use tTtrx ntz: - 1

I

Irll*r:;,mt:-3 substitute x-- 1 into 4

gg-e-gY ?t:19-l(:-3-) ::n;(

!)3 =_r(- r)t

(b) Integr ate 2- ! \ rrt )c

11y---+-+C2xn x

Substitute (- I ,2) into tyto find c

I 17l)=--+-+-'2xax2

3

3 6

s (a)

(b)

Shape of sine

2cyclesfor0<x32n

Arnplitude - 3

Modulus

EE]EIE)Accept

cosine

graph

v

3

2

I

.r 3xy-5--1T

Sketch the straight line

Number of solutions =

4

3 7

347212

It

347212

Ques Mark SchemeSub

MarksTotaIMark

(a) | Height of the bars proportional

I to the frequenciesI

I ruUel the class bound afies or

I midpoints of the classe s orI the class interval

EE

Method to find mode

26

(b)1" @ F: 14 @

*14

"f*

l

-9 EIu0)

; g

22.83

4

347212

347212

347212

7

sEcrroNB(40MARKS)

347212

Ques Mark SchemeSub

MarksFull

Mark

x-I 1 2 3 4 5 6

logroy 0.44 0.63 0.82 1.03 1.20 L.40

(a) EE(b)

&

(c)

Plot logt $/ against ( x - 1)

6 *points plotted correctly

Line of best fit

log;,oy= logr ok + ( r - 1) logrch EUse *c log1s fr

;Use *m: logoh

h - 1.56 lg- 1.74 10

Note : SS-1 if part of the scale'is not uniform or not usingthe scales given or not using the graph paper

347212

347212

logro

347212

347212

Ques Mark SchemeSub

MarksFull

Mark

(a) (i) &(ii) Use Triangle Law for 0R or OS

fr:4q-6v

OS - 4u +3v

(b) 9l: _qetsl_11ry:_"_ ${ _9_T _

Or -9u-4v-3-

se QT = lQR

I

(c) | (i) &(ii)

I tubstitute u =3iandt-I Y - -21+ liinto 'r ffi

OS = 6r_+ lSi

4 6t+Lsj?_a- - -

"llao 4 10

3472n

t0 347212

Mark Scheme

Solve simultaneous equationsy:(x-3)2andy--x*9.

A(0,9)

Integrate (x - 3)'wrt x

B(5,4)

Ar:t- 6xz ,+ 9x32(* -3)3At\/'

3

,f5

Uselimit t

in Ar

OR

Find the arcaof tapezium Az At

Az= l-. *(e + 4)x *(5)2\

l+-'{:or

r25 /Do1/lzo.B3

Integrate n(x - 3)o wrt x

v:rry

Use3

Iin0

o(ry-sf5

243 n ll48:- n // 48.6n

11 341212

Mark Scheme

Use ^r - r0 to find arc

DC or EC

AC:

AC*arcDC+2

19.45 /l

Use / = Y, fg to findarea of sector CBE

15.5

Use A- Yrfe b findarea of sector DBE

A,: *Of(;-Lz4)Use

I * bxh OR lo|sin Cto22

find Ar:LABC or A3: A ABEor A+: L EBC

Az= 17.5', At - 5 .687 ;A+: Il,82

/- *15.5 +As-At-A+

OR

Arc DC-s (Z\\2)

or arc EC - 5(1.2

+72

5.22 ll 5.23

A- A2-Ar-A+* (*t5.5 -A+)

347212

t2 347212

Mark Scheme

(i) 8p:2

D- 1 ll o.z5L4

(ii) Use

* (0.25), (0.75)n-r

Write P(X>Z):P(X - 3) + P(X - 4) +.. .+ P(X: 8)

OR1 P(X: 0) - P(X - 1) - P(X= 2)

(i) z- 2.5 -1.60.8

0.13029 ll 0,1303

fn -1.6 .'r' -'- =1.175

0.8

0.3015

m=2,54

347212

13

SECTTON C (20 MARKS)

347212

Mark Scheme

Integr ate Izt - 8 dt

v- f 8t + c Substitute t: 0 and

v- 12 in * y tofind c

c: 12

Use a -0tofindr

t -4

Solvev.0tofindr

(t-2)(t-6)-0

t:2, 6

!t' -8r+12 dt

s = t- - 4t2 +r2t3 Substitute t = *2 or t - 4 into *s

OR

use I: or I: in *s

Find total distance travelled

fi"at arll:"dtl

t6

347212

L4 347212

Mark Scheme

use Qt x looQo

)c - 125 y=

I=* I2s(27) + l2s(34) + 137 .5(10) + 1s0(10)

27 +34+10+10+19

127.8

use Qn x loo26A0

= *127 .8

RM 3322.8

118 x *1 27 .B

100

t2o(1e)

150.8

347212

t.

l5 3472/2

Ques Mark SchemeSub

MarksFUII

Mark

1,4 (a) (i) Yr-'-: :rlt 19 lt ! 4! 9- ly -!$- B c

BC 2 = 82 +lz2 - 2(8) (lz) cos 25'

BC: 5.83

(ii) y:: :ir :l:11 y:? ::.1i : : ::osin C sin 25=-12 5.83

60.450

(b) I G)

I 19.550

(ii) ADB = 60,45o

(iii) Z DAfi:59. 1o

use |olez)sin

(*5e.1 + 25)

47 "7510

347212

t6 347212

Mark Scheme

x + y S 8 or equivalent

x

600x + 2A0y < 3 000 or equivalent

EEE

Draw coffectly at least one straight linefrom the *inequalities which involvesxand y

Draw coffectly all the three*straight linesNote : Accept dashed lines

The correct region shaded

(i) I

(ii) Maxirnum point: (4, 3)

Use 3bc + 7yfor point in the *region R

Note:ss-l ifin (a), the symbol 63-" is not used at all orrnore than three inequalities are given

in (b), does not use the scale given ordoes not use graph paper orinterchange between r-anis and y-axis

347212

t7 347212

END OF N{ARK SCHEME

347212